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\section{Introduction}
For a fixed $k$-graph $H_0$ of order $m$,
we say that
a given
$k$-graph
$G$ of order $n$
is {\it $H_0$-tileable}
if $G$ contains, as subhypergraphs, $\lfloor n/m \rfloor$ vertex-disjoint copies of $H_0$.
Now, suppose $G$ has vertex set $V$,
and for an integer $1\leq \ell \leq k$, let $U \in \tbinom{V}{\ell}$ be given. As is customary, let
$$
N(U) = N_G(U) = \left\{W \in \binom{V}{k- \ell}: U \cup W \in E(G)\right\}\, \quad \text{and} \quad
\delta_{\ell}(G) = \min \left\{ |N(U)|: U \in \binom{V}{\ell}\right\}
$$
denote, respectively, the neighborhood of $U$ in $G$, and the {\it $\ell$-degree} of $G$.
Define $t_{\ell}^k(n, H_0)$ to be the smallest integer $d$ so that every $k$-graph $G$ of order $n$
for which $\delta_{\ell}(G) \geq d$
holds is $H_0$-tileable.
In the case of graphs ($k=2$), $t_1^2(n, H_0)$ is known, up to an additive constant,
for every fixed graph $H_0$ (see \cite{KO2}). Furthermore, there
are some graphs $H_0$ for which $t_1^2(n, H_0)$ is known exactly.
The most celebrated such result is the Hajnal-Szemer\'edi theorem~\cite{HSz}, which says that for the $r$-clique $H_0 = K_r$ and for $n$ divisible by $r$,
$$
t_1^2(n, K_r) = \left(1 - \frac{1}{r}\right)n.
$$
A recent result of Wang~\cite{W} shows that for all integers $n$ divisible by 4,
$t_1^2(n, C_{4}) =
\tfrac{n}{2}$.
This result is a special case of the well-known El-Zahar conjecture, and had been independently conjectured by Erd\H{o}s and Faudree.
In the case of hypergraphs ($k\geq 3$),
much less is known about tiling problems.
For only
the $k$-edge $H_0 = K_k^k$ (the tiling of which is a perfect matching)
is
$t_{k-1}^k(n, H_0)$ known for all $k \geq 3$.
This significant result is due to
R\"odl, Ruci\'nski and Szemer\'edi~\cite{RRS}, and asserts that for all sufficiently large integers $n$ divisible by $k$,
$$
t_{k-1}^k(n, K_k^k) =
\frac{n}{2} - k + \varepsilon_{k,n}, \quad \text{where} \quad
\varepsilon_{k,n} \in
\left\{ \frac{3}{2}, \, 2,\, \frac{5}{2}, \, 3\right\}
$$
is determined by explicit divisibility conditions on $n$ and $k$.
We are interested in tilings when $k=3$ and $\ell=2$, where some
interesting
results have recently developed.
(In what follows, we abbreviate $t_2^3(n,H_0)$ to $t(n,H_0)$.)
As usual,
let $K_4^3$ denote the complete 3-graph on 4 vertices. Let $K_4^3 - e$ denote its subhypergraph consisting of 3 edges, and let $K_4^3 - 2e$ denote its subhypergraph consisting of 2 edges.
K\"uhn and Osthus~\cite{KO} proved that
$t(n, K_4^3 - 2e) = (1 +o(1)) n/4$. Recently,
Lo and Markstr\"om~\cite{LM1, LM2}
have shown that
$t(n, K_4^3 -e) = (1 + o(1)) n/2$
and that
$t(n, K_4^3) = (1 + o(1)) 3n/4$.
Keevash and Mycroft~\cite{KM} showed the exact counterpart that,
for sufficiently large integers $n$ divisible by 4,
$t(n, K_4^3) = (3n/4) - \varepsilon_n$,
where $\varepsilon_n = 2$ if $8|n$ and $\varepsilon_n = 1$ otherwise.
We shall prove the following exact result for $K_4^3 - 2e$.
\begin{theorem}\label{main}
For all sufficiently large integers $n$ divisible by 4,
$$
t(n, K_4^3-2e) =
\left\{
\begin{array}{cc}
\frac{n}{4} & \text{when $\tfrac{n}{4}$ is odd,} \\
\frac{n}{4} + 1 & \text{when $\tfrac{n}{4}$ is even.}
\end{array}
\right.
$$
\end{theorem}
\noindent The proof of Theorem~\ref{main}
spans
Sections~\ref{section:extreme}
and~\ref{section:non-extreme}.
We mention that an essential ingredient in our proof is the `absorption technique' (see
Section~\ref{section:non-extreme})
of R\"odl, Ruci\'nski and Szemer\'edi.
In the remainder of this
paper, we shall make the abbreviation $D = K_4^3 - 2e$.
(In the papers~\cite{KO, RRS}, $D = K_4^3 - 2e$ was abbreviated by ${\mathcal C}$ and $C_4^{3,1}$, respectively, since for those authors, $K_4^3-2e$ was viewed as a type of cycle.)
In the remainder of this
introduction, we discuss the main concept used in the proof of Theorem~\ref{main}, that of an `$\varepsilon$-extremal' 3-graph (for $D= K_4^3-2e$).
\subsection{Theorem~\ref{main} and~$\varepsilon$-extremal 3-graphs}
To motivate the concept of an $\varepsilon$-extremal 3-graph (stated in the upcoming Definition~\ref{extremaldef}),
we first observe the following constructions for the lower bounds of Theorem~\ref{main}.
Let $A$ be a set of $\tfrac{n}{4} - 1$ vertices, and let $B$ be a set of $\tfrac{3n}{4} + 1$ additional vertices.
Define $G_0 = \tbinom{A \cup B}{3} \setminus \tbinom{B}{3}$, and note that $\delta_2(G_0) = \tfrac{n}{4} - 1$.
When $\tfrac{n}{4}$ is even, add any Steiner triple system\footnote{A {\it Steiner triple system} (STS) is a
3-graph $H$ where $\delta_2(H) = \Delta_2(H) = 1$.
It is well-known that an STS of order $m$ exists if, and only if, $m \equiv 1, 3$ (mod 6).}
on vertex set $B$ to $G_0$, and call this hypergraph $G_1$, where we note that $\delta_2(G_1) = \tfrac{n}{4}$.
Since $G_i[B]$, $i = 0, 1$, is $D$-free,
every copy of $D$ in $G_i$
contains at least one vertex of $A$, and so
$G_i$ is not
$D$-tileable.
\begin{definition}[$\varepsilon$-extremal]
\label{extremaldef}
\rm
Let $\varepsilon> 0$ be given, and
suppose $G$ is a 3-graph
of order $n$.
We say $G$ is {\it $\varepsilon$-extremal}
if there exists $S \subset V(G)$ of size $|S| \geq (1 - \varepsilon) \tfrac{3n}{4}$ for which $G[S]$ is $D$-free.
\end{definition}
While the lower bound
constructions for Theorem~\ref{main}
motivate the concept of Definition~\ref{extremaldef}, the following fact indicates why we choose the terminology `extremal'.
\begin{fact}
\label{noSTS}
Let $G$ be a 3-graph
on $n$ vertices, where $n$ is divisible by 4, satisfying
\begin{equation}
\label{eqn:codeg}
\delta_2(G) \geq
\left\{
\begin{array}{cc}
\frac{n}{4} & \text{when $\tfrac{n}{4}$ is odd,} \\
\frac{n}{4} + 1 & \text{when $\tfrac{n}{4}$ is even.}
\end{array}
\right.
\end{equation}
Then any $S \subset V(G)$ for which $G[S]$ is $D$-free satisfies $|S| \leq \tfrac{3}{4}n$.
\end{fact}
\begin{proof} Since
$G[S]$ is $D$-free, when $\tfrac{n}{4}$ is even,
we have $\tfrac{n}{4} + 1 \leq \delta_2(G) \leq n - (|S| - 1)$, and the result follows.
When
$\tfrac{n}{4}$ is odd,
suppose
some $S_0 \subset V(G)$ exists of size $\tfrac{3n}{4} + 1$ for which $G[S_0]$ is $D$-free.
Since $G[S_0]$ is not an STS (since $\tfrac{3n}{4} + 1 \not\equiv 1, 3$ (mod 6)), some pair $s, s' \in S_0$ satisfies
$N(s, s') \cap S_0 = \emptyset$, in which case $\frac{n}{4} \leq |N(s, s')| \leq n - |S_0|$, and the result follows.
\end{proof}
Now, the upper bounds in Theorem~\ref{main} follow immediately from the following two statements.
\begin{theorem}[Theorem~\ref{main} -- extremal case]
\label{extreme}
There exists $\varepsilon_0 > 0$ so that,
for all sufficiently large integers $n$ divisible by 4, the following holds.
Whenever $G$ is a 3-graph
of order $n$
satisfying~(\ref{eqn:codeg}) and which
is $\varepsilon_0$-extremal, then $G$ is $D$-tileable.
\end{theorem}
\noindent We prove Theorem~\ref{extreme} in Section~\ref{section:extreme}.
\begin{theorem}[Theorem~\ref{main} -- non-extremal case]
\label{non-extreme}
For every $\varepsilon>0$
and for all sufficiently large integers $n$ divisible by 4, the following holds.
Whenever $G$ is a 3-graph of order $n$ satisfying~(\ref{eqn:codeg})
(see Remark~\ref{remark}),
which is not $\varepsilon$-extremal, then $G$ is $D$-tileable.
\end{theorem}
\noindent We prove Theorem~\ref{non-extreme} in Section~\ref{section:non-extreme}.
\begin{remark}
\label{remark}
\rm
We mention that Theorem~\ref{non-extreme} can be proved, for the same money, under a slightly weaker hypothesis than~(\ref{eqn:codeg}).
In particular,
Theorem~\ref{non-extreme} remains valid if one only assumes
that
$\delta_2(G) \geq (n/4) (1 - \gamma)$, for a constant $\gamma > 0$ sufficiently smaller than $\varepsilon$.
\end{remark}
\section{Proof of Theorem~\ref{extreme}}
\label{section:extreme}
We shall use the following theorem of Pikhurko \cite{P}, stated here in a less general form.
\begin{theorem}[\cite{P}, Theorem 3]\label{4graph}
Let $H$ be a $4$-partite $4$-graph with 4-partition $V(H) = V_1\cup V_2\cup V_3\cup V_4$, where $|V_1| = \dots = |V_4| = m$.
Let $\delta(V_1)=\min\{|N(v_1)|: v_1\in V_1\}$ and
$$
\delta(V_2, V_3, V_4)=\min\{|N(v_2, v_3, v_4)|: v_2\in V_2, \, v_3\in V_3, \, v_4\in V_4\}.
$$
For $\gamma > 0$ and a sufficiently large integer $m$, if
$$
m\delta(V_1)+m^3 \delta(V_2, V_3, V_4)\geq (1+\gamma)m^4,
$$
then $H$ contains a perfect matching.
\end{theorem}
To prove Theorem~\ref{extreme}, it suffices to take $\varepsilon_0 = 10^{-18}$, and we shall take $n$ sufficiently large, whenever needed.
We write $n = 4k$
and $\alpha^3 = \varepsilon_0$.
Let $G$ be a 3-graph of order $n$ satisfying~(\ref{eqn:codeg})
which is $\varepsilon_0$-extremal. We prove that $G$ is $D$-tileable, and will construct a $D$-tiling in stages. In particular, we will build
vertex-disjoint partial $D$-tilings ${\mathcal Q}$, ${\mathcal R}$, ${\mathcal S}$ and ${\mathcal T}$ whose union is a $D$-tiling of $G$.
To build these partial tilings, we need a few initial considerations.
To begin, let $Z \subset V(G)$ be a maximal set for which $G[Z]$ is $D$-free.
Define
\begin{equation}
\label{eqn:X}
X = \left\{x \in V(G) \setminus Z: \left|N(x) \cap \binom{Z}{2}\right| \geq (1 - \alpha) \binom{|Z|}{2} \right\},
\end{equation}
and $Y = V(G) \setminus (X \cup Z)$.
We estimate each of the quantities in $|X| + |Y| + |Z| = 4k = n$:
\begin{equation}
\label{Yupper}
k(1 - 4\alpha^2)
\leq
|X|
\leq
k (1 + 3\varepsilon_0), \quad
0 \leq |Y| \leq 4\alpha^2 k, \quad
3k(1 - \varepsilon_0) \leq
|Z|
\leq
3k,
\end{equation}
i.e., $|Y|$ is small, $|X|$ is around $n/4$ and $|Z|$ is around $3n/4$.
Indeed, the third estimate in~(\ref{Yupper}) follows from our hypothesis and Fact~\ref{noSTS}.
To see the second estimate,
for
$W \subset X \cup Y$,
write
$G[Z, Z, W]$ for the
collection of triples of $G$ consisting of two vertices from $Z$ and one vertex from $W$. Then,
$$
(k - 1) \binom{|Z|}{2} \leq \big|G[Z,Z,X\cup Y]\big|
\leq
(1 - \alpha) \binom{|Z|}{2} |Y| + \binom{|Z|}{2} |X|,
$$
so that
$k-1+\alpha|Y|\leq |X|+|Y|$.
The estimate on $|Z|$ implies that
$|X| + |Y| \leq k + 3\varepsilon_0 k$, and so we have
the second estimate of~(\ref{Yupper}).
Finally, our bounds on $|Y|$ and $|Z|$ render the first estimate in~(\ref{Yupper}).
Let us also check that~\eqref{Yupper} implies that
\begin{equation}\label{zzx}
\forall z_1, z_2\in Z, \ |N(z_1, z_2)\cap X|\geq (1-\alpha)|X|.
\end{equation}
Indeed,
since $|N(z_1, z_2) \cap Z| \leq 1$, we have
$$
|N(z_1, z_2) \cap X| \geq k - 1 - |Y|
\stackrel{\text{(\ref{Yupper})}}{\geq}
(1 - 5 \alpha^2) k
\stackrel{\text{(\ref{Yupper})}}{\geq}
\frac{1 - 5 \alpha^2}{1 + 3\varepsilon_0} |X|
\geq
(1 - \alpha) |X|.
$$
We now introduce the first of our partial $D$-tilings, namely, ${\mathcal Q}$. \\
\noindent {\bf The partial $D$-tiling ${\mathcal Q}$.}
Let ${\mathcal Q}$ be a largest
$D$-tiling in $G$ for which each element $D_0 \in {\mathcal Q}$ has three vertices in $Z$ and one vertex in $Y$. Write $q = |{\mathcal Q}|$,
write $Y_{{\mathcal Q}} \subset Y$ for the set of vertices of $Y$ covered by ${\mathcal Q}$,
and write $Z_{{\mathcal Q}} \subset Z$ for the set of vertices of $Z$ covered by ${\mathcal Q}$. Clearly, $|Y_{{\mathcal Q}}| = q$ and $|Z_{{\mathcal Q}}| = 3q$.
Write $\ell = k - |X|$, where we note
from~(\ref{Yupper}) that
\begin{equation}
\label{eqn:sizeofell}
-3 \varepsilon_0 k
\leq
\ell = k - |X| \leq 4 \alpha^2 k.
\end{equation}
For future reference, we make the following two claims.
\begin{claim}
$q\geq \ell = k - |X|$.
\end{claim}
\begin{proof}
If $\ell\leq 0$, there is nothing to show.
If $\ell=1$, we have $|Y\cup Z|=3k+1$, and thus Fact~\ref{noSTS}
implies that $G[Y\cup Z]$ contains a copy of $D$, which requires $|Y| \geq 1$.
Now, if $q = 0$, then we could move a vertex from $Y$ to $Z$, which contradicts the maximality of $Z$.
Finally, suppose $\ell\geq 2$,
and observe that
the quantity
$|G[Z,Z,Y]|
=
|G[Z,Z,Y_{{\mathcal Q}}]| + |G[Z,Z,Y \setminus Y_{{\mathcal Q}}]|$ satisfies
that
\begin{multline*}
(\ell - 1) \binom{|Z|}{2}
\leq |G[Z,Z,Y]|
\leq
|Y_{{\mathcal Q}}| (1 - \alpha) \binom{|Z|}{2} + \left(\frac{|Z| - |Z_{{\mathcal Q}}|}{2} + |Z_{{\mathcal Q}}| |Z|\right)\big|Y\setminus Y_{{\mathcal Q}} \big| \\
=
q (1 - \alpha) \binom{|Z|}{2} + \left(\frac{|Z| - 3q}{2} + 3q |Z|\right)\big(|Y| - q\big)
\stackrel{\text{(\ref{Yupper})}}{\leq}
q (1 - \alpha) \binom{|Z|}{2} + 16 \alpha^2 q |Z| k.
\end{multline*}
Now, if $q \leq \ell - 1$, then
$$
1 \leq 1 - \alpha + 32\frac{\alpha^2 k}{|Z| - 1}
\stackrel{\text{(\ref{Yupper})}}{\leq}
1 - \alpha + 16 \alpha^2,
$$
a contradiction.
\end{proof}
Note that, on account of the claim above,
\begin{equation}
\label{eqn:sizeofq-ell}
0 \leq q - \ell
\stackrel{\text{(\ref{eqn:sizeofell})}}{\leq}
|Y| + 3\varepsilon_0 k
\leq
|Y| + 4\alpha^2 k
\stackrel{\text{(\ref{Yupper})}}{\leq}
8 \alpha^2 k.
\end{equation}
\begin{claim}\label{yzx}
For all $y\in Y\setminus Y_{{\mathcal Q}}$ and $z\in Z\setminus Z_{{\mathcal Q}}$, $|N(y,z)\cap X|\geq (1-\alpha)|X|$.
\end{claim}
\begin{proof}
Fix $y\in Y\setminus Y_{{\mathcal Q}}$ and $z\in Z\setminus Z_{{\mathcal Q}}$. By the maximality of ${\mathcal Q}$, we have $|N(y,z)\cap Z|\leq |Z_{{\mathcal Q}}| +1 = 3q + 1$.
As such, since $|Y| \geq q$,
we have
$$
|N(y, z) \cap X| \geq k - (3q + 1) - (|Y| - 1)
\geq k - 4|Y|
\stackrel{\text{(\ref{Yupper})}}{\geq}
(1 - 16 \alpha^2 ) k
\stackrel{\text{(\ref{Yupper})}}{\geq}
\frac{1 - 16 \alpha^2}{1 + 3\varepsilon_0} |X|
\geq (1 - \alpha)|X|.
$$
\end{proof}
\noindent {\bf The partial $D$-tiling ${\mathcal R}$.}
We now use~\eqref{zzx} and Claim~\ref{yzx} to build
a collection ${\mathcal R}$ of
$|Y\setminus Y_{{\mathcal Q}}|$
vertex-disjoint copies of $D$, each with $1$ vertex in $Y\setminus Y_{{\mathcal Q}}$, $1$ vertex
in $X$, and two vertices in $Z\setminus Z_{{\mathcal Q}}$.
For sake of argument,
assume $|Y \setminus Y_{{\mathcal Q}}| \geq 1$.
Inductively, assume we have obtained $0 \leq i < |Y\setminus Y_{{\mathcal Q}}|$
vertex-disjoint copies of $D$, each with $1$ vertex in $Y\setminus Y_{{\mathcal Q}}$, $1$ vertex
in $X$, and two vertices in $Z\setminus Z_{{\mathcal Q}}$.
Arbitrarily
select an uncovered $y' \in Y\setminus Y_{{\mathcal Q}}$
and uncovered $z_1', z_2' \in Z \setminus Z_{{\mathcal Q}}$, noting that the latter is possible
since at most $|Z_{{\mathcal Q}}| + 2i \leq 5|Y| \leq |Z| - 2$
(cf.~(\ref{Yupper}))
vertices
in $Z$ are unavailable for selection.
Since $|N(y', z_1') \cap N(z_1', z_2') \cap X| \geq (1 - 2\alpha)|X|$,
we have at least $(1 - 2\alpha) |X| - i \geq (1 - 2\alpha ) |X| - |Y| > 0$ (cf.~(\ref{Yupper})) choices for an uncovered
vertex $x' \in X$, to complete the $(i+1)^{\rm st}$ copy of $D$.
Note that all vertices of $Y$ are covered by ${\mathcal Q}$ or ${\mathcal R}$. Let $Z_{{\mathcal Q},{\mathcal R}} \supset Z_{{\mathcal Q}}$ denote the set of vertices of $Z$ covered by ${\mathcal Q}$ or ${\mathcal R}$,
and let $X_{{\mathcal R}}$ denote the set of vertices of $X$ covered by ${\mathcal R}$ (no vertices of $X$ were covered by ${\mathcal Q}$).
Observe that
$$
|X \setminus X_{{\mathcal R}}| = |X| - (|Y| - |Y_{{\mathcal Q}}|)
= k - |Y| + (q - \ell), \text{ and}
$$
\begin{equation}
\label{eqn:sizeofZ-ZQR}
|Z \setminus Z_{{\mathcal Q}, {\mathcal R}}|= |Z| - |Z_{{\mathcal Q}}|-2(|Y|-|Y_{{\mathcal Q}}|) = 3(k-|Y|)-(q-\ell),
\end{equation}
where we used that $|Z| = 4k - |X| - |Y| = 3k + \ell - |Y|$. \\
\noindent {\bf The partial $D$-tiling ${\mathcal S}$.}
We now obtain a collection ${\mathcal S}$ of $q - \ell$ vertex-disjoint copies of $D$, each with 2 vertices in $X \setminus X_{{\mathcal R}}$ and 2 vertices in $Z \setminus Z_{{\mathcal Q}, {\mathcal R}}$.
Indeed, arbitrarily pick vertices $z_1, z_1', \dots, z_{q-\ell}, z_{q - \ell}' \in Z \setminus Z_{{\mathcal Q}, {\mathcal R}}$, which is possible since
$$
|Z \setminus Z_{{\mathcal Q}, {\mathcal R}}|
- 2 (q - \ell)
\stackrel{(\ref{eqn:sizeofZ-ZQR})}{=}
3 (k - |Y| - (q - \ell))
\stackrel{\text{
(\ref{Yupper}),
(\ref{eqn:sizeofq-ell})}}{\geq}
3 k (1 - 12 \alpha^2) \geq 2.
$$
Inductively, assume we have covered
$0 \leq i < q - \ell$
pairs $z_1, z_1', \dots, z_i, z_i'$
by vertex-disjoint copies $D_1, \dots, D_i$ of $D$, where each $D_j$, $0 \leq j \leq i$, has vertices
$\{z_j, z_j', x_j, x_j'\}$, where $x_j, x_j' \in X \setminus X_{{\mathcal R}}$.
We infer from~\eqref{zzx} that
$$
\big|N(z_1, z_1') \cap \big(X \setminus (X_{{\mathcal R}} \cup \{x_1, x_1',\dots, x_i, x_i'\})\big)\big| \geq (1 - \alpha) |X| - |X_{{\mathcal R}}| - 2i
\geq (1 - \alpha) |X| - |Y| - 2(q - \ell)
\geq 2,
$$
where the last inequality holds on account of~(\ref{Yupper})
and~(\ref{eqn:sizeofq-ell}).
We thus obtain
the $(i+1)^{\rm st}$ copy of $D$.
Let $Z_{{\mathcal Q},{\mathcal R}, {\mathcal S}} \supset Z_{{\mathcal Q}, {\mathcal R}}$ denote the set of vertices of $Z$ covered by ${\mathcal Q}$, ${\mathcal R}$ or ${\mathcal S}$, and let $X_{{\mathcal R}, {\mathcal S}} \supset X_{{\mathcal R}}$ denote the set of vertices of $X$ covered by ${\mathcal R}$ or ${\mathcal S}$.
Set $m:= |X \setminus X_{{\mathcal R},{\mathcal S}}|$ and note that
\begin{equation}
\label{ZQRS}
m =
|X \setminus X_{{\mathcal R}, {\mathcal S}}|
\stackrel{\text{(\ref{eqn:sizeofZ-ZQR})}}{=}
k - |Y| - (q - \ell) \quad \text{and} \quad |Z \setminus Z_{{\mathcal Q}, {\mathcal R}, {\mathcal S}}|
\stackrel{\text{(\ref{eqn:sizeofZ-ZQR})}}{=}
3\big(k- |Y| - (q - \ell)\big) = 3m.
\end{equation}
We conclude the proof of Theorem~\ref{extreme} by building the remaining partial $D$-tiling ${\mathcal T}$. \\
\noindent {\bf The partial $D$-tiling ${\mathcal T}$.}
Arbitrarily partition $Z \setminus Z_{{\mathcal Q}, {\mathcal R}, {\mathcal S}} = Z_1 \cup Z_2 \cup Z_3$ into three sets of size $m$, and for simplicity of notation, write $X_0 = X \setminus X_{{\mathcal R}, {\mathcal S}}$.
Define the following auxiliary 4-partite 4-graph $H$ with 4-partition $V(H) = X_0 \cup Z_1 \cup Z_2 \cup Z_3$, obtained by including each $\{x, z_1, z_2, z_3\} \in H$, $x \in X_0$, $z_i \in Z_i$ for $i = 1, 2, 3$,
if
$\{x, z_1, z_2, z_3\}$ spans a copy of $D$ in $G$.
We claim that $H$ satisfies the hypothesis of
Theorem~\ref{4graph} with $\gamma = 1/2$, and hence contains a perfect matching, which will then define ${\mathcal T}$ and finish our proof
of Theorem~\ref{extreme}.
To bound $\delta_H(Z_1, Z_2, Z_3)$,
fix $z_1 \in Z_1, z_2 \in Z_2, z_3 \in Z_3$.
We infer from~\eqref{zzx} that
\begin{multline*}
|N_H(z_1, z_2, z_3)| \geq \big| N_G (z_1, z_2) \cap N_G (z_1, z_3) \cap X_0 \big| \geq (1 - 2\alpha)|X| - |X_{{\mathcal R}, {\mathcal S}}| \\
\geq
(1 - 2\alpha) |X| - |Y| - 2(q - \ell)
\stackrel{\text{(\ref{Yupper}), (\ref{eqn:sizeofq-ell})}}{\geq}
(1 - 2\alpha)|X| - 20 \alpha^2 k
\stackrel{\text{(\ref{Yupper})}}{\geq}
\big((1 -2\alpha)((1 - 4\alpha^2) - 20 \alpha^2\big)) k \\
\stackrel{\text{(\ref{Yupper})}}{\geq}
\frac{1 - 26 \alpha}{1 + 3\varepsilon_0} |X|
\geq (1 - 27 \alpha) |X| \geq (1 - 27 \alpha) |X_0|
= (1 - 27 \alpha) m.
\end{multline*}
Thus, $\delta_H(Z_1, Z_2, Z_3) \geq (1 - 27 \alpha)m$.
To bound $\delta_H(X_0)$,
fix $x \in X_0$, and
for clarity of notation in what follows, write $N_G(x) = G_x$.
By the definition of $X$, we have that $|G_x[Z]| \geq (1 - \alpha) \tbinom{|Z|}{2}$, and so all but at most $\sqrt{\alpha} |Z|$ vertices
$z \in Z$ satisfy that $\deg_{G_x[Z]}(z) \geq (1 - \sqrt{\alpha}) |Z|$.
For each such $z \in Z$ and $i = 1, 2, 3$,
$$
|N_{G_x}(z) \cap Z_i| \geq (1 - \sqrt{\alpha})|Z| - |Z_{{\mathcal Q}, {\mathcal R}, {\mathcal S}}| - 2m
\stackrel{\text{(\ref{ZQRS})}}{=}
m - \sqrt{\alpha} |Z|
=
\left(1 - \sqrt{\alpha}\frac{|Z|}{m}\right)m.
$$
Since, by~(\ref{Yupper}) and~(\ref{ZQRS}), we have
\begin{equation}
\label{m}
3m
=
|Z| - |Z_{{\mathcal Q}, {\mathcal R}, {\mathcal S}}|
=
|Z| - \big(3q + 2(|Y| - q) + 2(q - \ell)\big)
\geq
|Z| - 5|Y| + 2 \ell
\stackrel{\text{(\ref{Yupper}), (\ref{eqn:sizeofell})}}{\geq}
|Z| - 26 \alpha^2 k
\stackrel{\text{(\ref{Yupper})}}{\geq}
\frac{|Z|}{2},
\end{equation}
we conclude that
$$
|N_{G_x}(z) \cap Z_i| \geq (1 - 6\sqrt{\alpha}) m.
$$
As such,
$$
|N_H(x)|
\geq \sum_{z_1 \in Z_1}
|N_{G_x}(z_1) \cap Z_2|
|N_{G_x}(z_1) \cap Z_3|
\geq
\left(m - \sqrt{\alpha}|Z|\right)
\left(\left(1 - 6\sqrt{\alpha}\right) m\right)^2
\stackrel{\text{(\ref{m})}}{\geq}
\left(1 - 6\sqrt{\alpha}\right)^3 m^3,
$$
and so $\delta_H(X_0) \geq (1 - 234\sqrt{\alpha}) m^3$.
The obtained bounds on $\delta_H(Z_1, Z_2, Z_3)$ and $\delta_H(X_0)$ then implies
$$
m \delta_H(X_0) + m^3 \delta_H(Z_1, Z_3, Z_3) \geq
\left(2 - 234\sqrt{\alpha} -
27 \alpha
\right) m^4
\geq
\left(2 - 261\sqrt{\alpha}
\right) m^4
>
\frac{3}{2} m^4
$$
so that, as claimed,
$H$ satisfies the hypothesis of
Theorem~\ref{4graph} with $\gamma = 1/2$.
\section{Proof of Theorem~\ref{non-extreme}}
\label{section:non-extreme}
Our proof of Theorem~\ref{non-extreme} is based on the following two lemmas, the second of which mirrors an `absorption' lemma of R\"odl, Ruci\'nski and Szemer\'edi~\cite{RRS}.
\begin{lemma}\label{almost-perfect}
For all $\gamma > 0$ and sufficiently large integers $m$ divisible by $4$, the following holds. Let $H$ be a 3-graph of order $m$. If $\delta_2(H) \geq \left(\tfrac{1}{4} - \gamma\right) m$ and $H$ is not $(8\gamma)$-extremal, then $H$ admits a $D$-tiling covering all but $50/\gamma$ vertices.
\end{lemma}
\begin{lemma}
\label{absorbing-lemma}
For all $\alpha >0$
and sufficiently large integers $n$ divisible by 4, the following holds. Let $G$ be a 3-graph of order $n$. If $\delta_2(G) \geq n/4$, then there exists $A \subset V(G)$ of size $|A| \leq \alpha n$ so that,
for every $W \subset V \setminus A$ of size $|W| \leq 50/\alpha$
for which $|A \cup W|$ is divisible by 4,
the hypergraph $G[A \cup W]$ is $D$-tileable.
\end{lemma}
\noindent We defer the proofs of
Lemmas \ref{almost-perfect} and \ref{absorbing-lemma} to Sections~\ref{sec:almost-perfect} and~\ref{sec:absorbing-lemma} respectively
in favor of first proving Theorem~\ref{non-extreme}.
\begin{proof}[Proof of Theorem \ref{non-extreme}]
Let $\varepsilon > 0$ be given, together with a sufficiently large integer $n$ which is divisible by 4.
Let $G$ be a $3$-graph of order $n$ satisfying~(\ref{eqn:codeg}) which is not $\varepsilon$-extremal.
For $\alpha = \varepsilon / 9$, let $A \subset V(G)$ be the set given by Lemma~\ref{absorbing-lemma}.
Set $H = G[V \setminus A]$,
and write $m = n - |A|$.
We claim that $H$ satisfies the hypothesis of Lemma~\ref{almost-perfect} with $\gamma = \alpha$. Indeed,
note that
$$
\delta_2(H) \geq \frac{n}{4} - |A| \geq \frac{n}{4} - \alpha n = \left(\frac{1}{4} - \alpha\right)n \geq
\left(\frac{1}{4} - \alpha\right)m.
$$
Observe, moreover, that $H$ is not $(8\alpha)$-extremal. Indeed, if $S \subset V(H)$ satisfies that $H[S]$ is $D$-free, then $G[S]$ is also $D$-free, and
if
$$
|S| \geq
(1 - 8\alpha) \frac{3m}{4} =
(1 - 8\alpha) \frac{3}{4} (n - |A|) \geq
(1 - 8\alpha)(1 - \alpha) \frac{3n}{4}
\geq
(1 - 9\alpha) \frac{3n}{4} =
(1 - \varepsilon) \frac{3n}{4},
$$
then $G$ would be $\varepsilon$-extremal, a contradiction.
Lemma~\ref{almost-perfect} implies that
$H$ admits a $D$-tiling covering all but $50/\alpha$ vertices. Set $W \subset V(H)$ to be the set of vertices (if any)
uncovered by this $D$-tiling. Since
$|V(H) \setminus W|$ is divisible by 4, it must be that $|A \cup W|$ is divisible by 4, and so Lemma~\ref{absorbing-lemma} guarantees that $G[A \cup W]$ is $D$-tileable.
Thus, $G$ is $D$-tileable.
\end{proof}
\subsection{Proof of Lemma~\ref{almost-perfect}}
\label{sec:almost-perfect}
Let $\gamma > 0$ be given, and let
$m$ be a sufficiently large integer which is divisible by 4.
Let $H$ be a 3-graph of order $m$, which is not $(8\gamma)$-extremal, and for which
$\delta_2(H) \geq\left(\tfrac{1}{4} - \gamma\right)m$.
We prove that $H$ contains a $D$-tiling covering all but $50/\gamma$ vertices.
To that end,
let ${\mathcal M}$ be a maximum $D$-tiling in $H$, but assume, on the contrary, that ${\mathcal M}$ leaves more than $50/\gamma$ vertices uncovered.
We use the following notation and terminology. Let $V_{{\mathcal M}}$ denote the set of vertices of $H$ covered by ${\mathcal M}$, and let $W = V(H) \setminus V_{{\mathcal M}}$.
For a vertex $v \in V_{{\mathcal M}}$, write $H_v[W]$ for $N_H(v) \cap \tbinom{W}{2}$,
and say that
$v \in V_{{\mathcal M}}$ is {\it $W$-big} if
$|H_v[W]|
\geq 10|W|$, and {\it $W$-small} otherwise.
Observe that every element $D_0 \in {\mathcal M}$ contains at most one $W$-big vertex. Indeed, assuming otherwise, let $u, v\in V(D_0)$ both be $W$-big vertices.
Since
$|H_u[W]| \geq 10
|W| > |W| /2$,
the graph $H_u[W]$
contains a path of length 2, with vertices denoted by $w_1, w_2, w_3$. The graph
$H_v[W\setminus \{w_1, w_2, w_3\}]$ then has size
\begin{equation}
\label{eqn:2path}
\left| H_v\left[W\setminus \{w_1, w_2, w_3\}\right] \right|
\geq
|H_v[W]| - 3 |W| \geq 7 |W| > |W| /2,
\end{equation}
and so $H_v[W\setminus \{w_1, w_2, w_3\}]$
contains a path of length 2, with vertices denoted by $w_1', w_2', w_3'$.
Then, $\{u, w_1, w_2, w_3\}$ and $\{v, w_1', w_2', w_3'\}$ span vertex-disjoint copies of $D$, which can replace
$D_0$ in ${\mathcal M}$ to contradict that ${\mathcal M}$ was a maximum $D$-tiling in $H$.
Now, write $B$ for the set of $W$-big vertices $v \in V_{{\mathcal M}}$, and write $|B| = b$.
We now observe that $b \geq \left(\tfrac{1}{4} - 2\gamma\right)m$.
Indeed, write $H[W,W,V_{{\mathcal M}}]$ for the set of triples from $H$ containing exactly two vertices from $W$. From our definitions above,
note that
$$
\left| H[W,W,V_{{\mathcal M}}] \right| \leq b \left( 30 |W| + \binom{|W|}{2} \right) + 40 (|{\mathcal M}| - b) |W| \leq
b \binom{|W|}{2} + 40 |{\mathcal M}| |W| \leq b \binom{|W|}{2} + 10 m |W|.
$$
On the other hand, the maximality of ${\mathcal M}$ implies that $H[W]$ is $D$-free, and so
$$
\left| H[W,W,V_{{\mathcal M}}] \right| \geq
\left(\left(\frac{1}{4} - \gamma\right)m - 1\right) \binom{|W|}{2}.
$$
The inequalities above imply that
$$
b \geq \left(\frac{1}{4} - \gamma\right) m - 1 - \frac{20 m}{|W| - 1} \geq
\left(\frac{1}{4} - \gamma\right) m - 1 - \frac{40 m}{|W|},
$$
and by our assumption that $|W| > 50/\gamma$, we infer that $b \geq \left(\tfrac{1}{4} - 2\gamma\right)m$, as claimed.
Now, write ${\mathcal M}_{B} \subset {\mathcal M}$ for elements of ${\mathcal M}$ which contain a $W$-big vertex, and let $V_{{\mathcal M}_B}$
denote the set of vertices of $H$ covered by ${\mathcal M}_B$.
Then, $S_B = V_{{\mathcal M}_B} \setminus B$ consists of $W$-small vertices and we have
$|S_B| = 3|B| \geq (1 - 8\gamma) 3m/4$.
Since $H$ is not $(8\gamma)$-extremal, $H[S_B]$ contains a copy $D_0$ of $D$, say with vertices $v_1, v_2, v_3, v_4$.
Let $u_1, u_2, u_3, u_4$ denote
the
$W$-big vertices corresponding to $v_1, v_2, v_3, v_4$, respectively, in ${\mathcal M}_B$. Among $u_1, \dots, u_4$, at least two and at most 4 are distinct, and so w.l.o.g.,
let $u_1, \dots, u_j$, for some $j \in \{2, 3, 4\}$, denote the distinct vertices of $u_1, \dots, u_4$.
For $1\leq i \leq j$,
let $D_i \in {\mathcal M}_B$ be the element containing $u_i$.
Similarly to~(\ref{eqn:2path}),
the definition of a $W$-big vertex will guarantee, for each $1\leq i \leq j$, the existence of a 2-path
$P_2(u_i) \subset H_{u_i}[W]$ so that
$P_2(u_1), \dots, P_2(u_j)$ are each pair-wise vertex-disjoint.
Indeed, if we already have the desired 2-paths $P_2(u_1), \dots, P_2(u_{i-1})$, where $2\leq i \leq j \leq 4$, then
$$
\Big|H_{u_i}\Big[W \setminus \big(V(P_2(u_1)) \cup \dots \cup V(P_2(u_{i-1})) \big) \Big] \Big|
\geq
|H_{u_i}[W]| - 3(i-1)|W|
\geq |H_{u_i}[W]| - 9|W|
\geq
|W| > |W|/2,
$$
and so there exists a 2-path $P_2(u_i) \subset H_{u_i}[W]$ which is vertex-disjoint from each of $P_2(u_1), \dots, P_2(u_{i-1})$.
Clearly, for each $1\leq i \leq j$, $\{u_i\} \cup V(P_2(u_i))$ spans a copy of $D$, which we shall denote as
$D^{u_i}$. Then, $D^{u_1}, \dots, D^{u_j}$ are pair-wise vertex-disjoint
copies of $D$, and so,
deleting from ${\mathcal M}$
the elements
$D_1, \dots, D_j$ and adding $D_0, D^{u_1}, \dots, D^{u_j}$ contradicts that ${\mathcal M}$ was a maximum $D$-tiling.
This concludes the proof
of Lemma~\ref{almost-perfect}.
\subsection{Proof of Lemma~\ref{absorbing-lemma} -- Absorption}
\label{sec:absorbing-lemma}
We shall prove the following stronger form of Lemma~\ref{absorbing-lemma}, which allows for a smaller co-degree and larger choices of subset $W$.
\begin{lemma}[Lemma~\ref{absorbing-lemma} - strong form]
\label{lem:abs}
For all $\alpha, \delta > 0$, there exists $\omega > 0$ so that for all sufficiently large integers $n$ divisible by 4, the following holds.
Let $G$ be a 3-graph of order $n$. If $\delta_2 (G) \geq \delta n$, then there exists $A \subset V(G)$ of size $|A| \leq \alpha n$ so that, for every
$W \subset V\setminus A$ of size $|W| \leq \omega n$ for which $|A \cup W|$ is divisible by 4, the hypergraph $G[A \cup W]$ is $D$-tileable.
\end{lemma}
Our proof of Lemma~\ref{lem:abs} will be based on
Proposition~\ref{prop:abs}, for which we need the following definition.
\begin{definition}
\label{def:absorb}
\rm
Suppose $G$ is a 3-graph with vertex set $V$, and let $U \in \tbinom{V}{4}$. We say that a set $S \in \tbinom{V \setminus U}{8}$ {\it absorbs}
$U$ if $G[S]$ is $D$-tileable and $G[S \cup U]$ is $D$-tileable.
\end{definition}
\begin{proposition}
\label{prop:abs}
For all $\delta > 0$, there exists $\sigma > 0$
so that for all sufficiently large integers $n$, the following holds. Suppose $G$ is a 3-graph
with vertex set $V$
of order $|V| = n$ for which $\delta_2(G) \geq \delta n$.
For each $U \in \tbinom{V}{4}$, there are $\sigma n^8$ sets $S \in \tbinom{V}{8}$ which absorb $U$.
\end{proposition}
To prove Proposition~\ref{prop:abs}, we require the following well-known `supersaturation' result of Erd\H{o}s~\cite{erdos}
(stated here only in special case form).
\begin{theorem}[Erd\H{o}s~\cite{erdos}]
\label{thm:erd}
For all $c_1 > 0$ there exists $c_2 > 0$ so that for all sufficiently large integers $n$, the following holds.
If $H$ is a 3-graph of order $n$ and size $|H| \geq c_1 n^3$, then $H$ contains at least $c_2 n^9$ copies of
$K^{3}_{3,3,3}$ (the balanced complete 3-partite 3-graph of order 9).
\end{theorem}
\begin{proof}[Proof of Proposition~\ref{prop:abs}]
Let $\delta > 0$ be given.
Let $c_1 = \delta^3 / 36$, and let $c_2 > 0$ be the constant guaranteed by Theorem~\ref{thm:erd}.
We define $\sigma = c_2$,
and in all that follows, we take $n$ to be a sufficiently large integer.
Let $G$ be a 3-graph with vertex set $V$ of order $|V| = n$ for which $\delta_2(G) \geq \delta n$. Fix $U = \{u_1, u_2, u_3, u_4\}
\subset V$. We prove there are $\sigma n^8$
sets $S \in \tbinom{V}{8}$ which absorb $U$.
To that end, define $V_1 = N(u_1, u_2)$, $V_2 = N(u_3, u_4)$ and
$$
V_3 = \bigcup \big\{N(v_1, v_2): (v_1, v_2) \in V_1 \times V_2 \big\}.
$$
Note that $V_1 \cup V_2 \cup V_3$ is not necessarily a partition,
but it will not be difficult to find
pairwise disjoint subsets $W_i \subset V_i$, $i = 1, 2, 3$, for which
$|G[W_1, W_2, W_3]| \geq c_1 n^3$.
To that end, let $W_1 \subset V_1 \setminus \{u_3, u_4\}$ be any set of size (exactly) $\lceil \delta n / 3 \rceil$
(recall $|V_1| \geq \delta n$).
Let $W_2 \subset V_2 \setminus (W_1 \cup \{u_1, u_2\})$ be any set of size
(exactly) $\lceil \delta n / 3 \rceil$ (recall $|V_2| \geq \delta n$).
Now, set $W_3 = V_3 \setminus (W_1 \cup W_2 \cup \{u_1, u_2, u_3, u_4\})$.
Then,
$$
|G[W_1,W_2,W_3]| = \sum_{(w_1, w_2) \in W_1 \times W_2} |N(w_1,w_2) \cap W_3|
\geq
\left\lceil \frac{\delta n}{3} \right\rceil^2 \left( \delta n -
2\left\lceil \frac{\delta n}{3} \right\rceil
- 4
\right)
\geq
\frac{\delta^3n^3 }{36} = c_1 n^3.
$$
Now, set $H = G[W_1, W_2, W_3]$, which we view as a hypergraph of order $n$. Since $H$ has size
$|H| \geq c_1 n^3$, Theorem~\ref{thm:erd} guarantees that $H$ has at least $c_2 n^9 = \sigma n^9$ copies of $K^{3}_{3,3,3}$.
Note that each such copy has exactly 3 vertices in each of $W_1, W_2, W_3$ and that, for some fixed
$w_3 \in W_3$ (it doesn't matter which), at least $\sigma n^8$ such copies
contain the vertex $w_3$.
Let $\{w_1, w_1', w_1'', w_2, w_2', w_2'', w_3, w_3', w_3''\}$ denote the vertex set of such a copy, where
$w_i, w_i', w_i'' \in W_i$, $i = 1, 2, 3$.
We claim that
$$
S_U =
S_U(w_3) =
\{w_1, w_1', w_1'', w_2, w_2', w_2'', w_3', w_3''\}
$$
absorbs the set $U$
(see Figure~\ref{fig-absorbing}).
Indeed,
$$
S_1:= \Big\{ \{w_1, w_2, w_3'\}, \, \{w_1', w_2, w_3'\} \Big\}\, , \quad
S_2:= \Big\{ \{w_1'', w_2', w_3''\}, \, \{w_1'', w_2'', w_3''\} \Big\}
$$
is a $D$-tiling of $G[S_U]$ and
$$
T_1:=\Big\{ \{u_1, u_2, w_1\}, \, \{u_1, u_2, w_1'\} \Big\}\,, ~
T_2:=\Big\{ \{u_3, u_4, w_2\}, \, \{u_3, u_4, w_2'\} \Big\}\,, ~
T_3:=\Big\{ \{w_1'', w_2'', w_3'\}, \, \{w_1'',w_2'',w_3''\} \Big\}
$$
is a $D$-tiling of $G[S_U \cup U]$.
\end{proof}
\begin{figure}
\begin{center}
\scalebox{1}{\includegraphics{figDriver.eps}}
\caption{Absorbing structure.}\label{fig-absorbing}
\end{center}
\end{figure}
Finally we use Proposition \ref{prop:abs} to prove Lemma \ref{lem:abs}.
\begin{proof}[Proof of Lemma~\ref{lem:abs}]
Let $\alpha, \delta > 0$ be given.
Let $\sigma = \sigma (\delta) > 0$ be the constant guaranteed by Proposition~\ref{prop:abs}.
We define
\begin{equation}
\label{eqn:w}
\omega = \frac{\alpha \sigma^2}{128}.
\end{equation}
In all that follows, we take $n$ to be a sufficiently large integer divisible by 4.
Let $G$ be a given 3-graph with vertex set $V$ of order $|V| = n$ for which $\delta_2 (G) \geq \delta n$.
We prove that $G$ admits a set $A \subset V$ described in the conclusion of Lemma~\ref{lem:abs}. To
produce the desired set $A$, we employ the well-known deletion method in probabilistic combinatorics.
To begin,
set $p = (1/16) \alpha \sigma n^{-7}$, and let ${\mathbb H} = {\mathbb H}^{(8)}(n, p)$ be the binomial random 8-uniform hypergraph with $n$-element vertex set $V$.
We note several basic properties of ${\mathbb H}$ (due to the Chernoff inequality, unless otherwise indicated):
\begin{enumerate}
\item
With probability $1 - \exp \{- n/\log n\}$,
$$
|{\mathbb H}| \leq 2 p \binom{n}{8} \leq \frac{1}{8}\alpha n;
$$
\item
Let ${\mathbb H} \otimes {\mathbb H} = \{(S_1, S_2) \in {\mathbb H} \times {\mathbb H}: S_1 \cap S_2 \neq \emptyset\}$.
Then,
$$
{\mathbb E} \left[ |{\mathbb H} \otimes {\mathbb H}| \right] \leq 8\binom{n}{8} \binom{n}{7} p^2 \leq
\frac{1}{256}
\alpha^2 \sigma^2 n.
$$
As such, by the Markov inequality,
$$
\Pr \left[ |{\mathbb H} \otimes {\mathbb H}| \geq \frac{1}{128} \alpha^2 \sigma^2 n \right] \leq \frac{1}{2};
$$
\item
For $U \in \tbinom{V}{4}$, let
${\mathcal A}(U)$ be the collection of sets $S \in \tbinom{V}{8}$ which absorb $U$.
By Proposition~\ref{prop:abs}, $|{\mathcal A}(U)| \geq \sigma n^8$, and so
with probability $1 - \exp \{- n/\log n\}$, ${\mathbb H}$ satisfies that for every $U \in \tbinom{V}{4}$,
$$
|{\mathcal A}(U) \cap {\mathbb H}| \geq \frac{1}{2}
p |{\mathcal A}(U)|
\geq
\frac{1}{32}
\alpha \sigma^2 n.
$$
\end{enumerate}
Let $H$ be an instance of ${\mathbb H}$ for which properties~$(i)$--$(iii)$ hold (and specifically, where
$|H \otimes H| < \alpha^2 \sigma^2 n/ 128$).
Now,
\begin{enumerate}
\item[$(a)$]
delete any $S \in H$ for which there exists $S' \in H$ for which
$S \cap S' \neq \emptyset$. This deletes at most
$$
2 \times \frac{\alpha^2 \sigma^2 n}{128} =
\frac{\alpha^2 \sigma^2 n}{64}
$$
elements $S \in H$;
\item[$(b)$]
delete any $S \in H$
for which no $U \in \tbinom{V}{4}$ has $S \in {\mathcal A}(U)$.
\end{enumerate}
The resulting hypergraph is then, importantly, a (partial) matching $M$ in $V$. Let $m:=|M|$, $\{S_1,\dots, S_m\}=M$, and $A:=\bigcup_{i=1}^mS_i$ (the set of vertices covered by $M$). We now confirm that $A$ satisfies its claimed properties.
Observe from~$(i)$ that $|A| = 8 |M| \leq \alpha n$, as promised.
Now, let $W \subset V \setminus A$ have size $4t:=|W| \leq \omega n$
(cf.~(\ref{eqn:w})) and then arbitrarily partition $W$ into 4-sets $\{W_1,W_2,\dots, W_{t}\}=:\mathcal{W}$.
Note that by (iii), (a), and \eqref{eqn:w} we have that for all $W_i\in \mathcal{W}$, $$|\mathcal{A}(W_i)\cap M|\geq \frac{1}{32} \alpha \sigma^2 n -\frac{1}{64} \alpha^2 \sigma^2 n \geq \frac{1}{64} \alpha \sigma^2 n\geq \frac{\omega n}{4}\geq t.$$
So for each $W_i\in \mathcal{W}$ we can greedily choose some unique $S_i'\in \mathcal{A}(W_i)\subseteq M$, which guarantees that each of $G[S_1' \cup W_1], \dots, G[S_t' \cup W_t]$ are $D$-tileable. Finally, since $G[S]$ is $D$-tilable for all $S\in M$ (by (b) and Definition \ref{def:absorb}), and since $\{S_1,\dots, S_m, W_1,\dots, W_t\}$ is a partition of $A\cup W$, we infer that $G[A\cup W]$ is $D$-tileable as desired.
\end{proof}
\subsection*{Acknowledgements}
Our thanks to the referees for their thoughtful suggestions.
|
1108.4038
|
\section{Introduction} \label{sec:intro}
In recent years, surprising connections have emerged between error correction of quantum information and topological condensed matter phases \cite{kitaev2003, nayak2008}.
At the same time, ideas from quantum information have proved useful in defining topological phases. Two dimensional phases with topological order, such as those
realized in the context of the Fractional Quantum Hall effect, are gapped phases for which the ground state degeneracy depends on the genus of the space on which
they are defined. Recent work has shown that they can be identified by the entanglement properties of their ground state wavefunction \cite{hamma2005,levin2006,kitaev2006}.
The entanglement entropy of a region with a smooth boundary of length $L$ takes the form $S_A=\alpha_1L-b_0 \gamma$, where $\gamma$
is the topological entanglement entropy, $b_0$ is the number of connected components components
of the boundary of region $A$, and we have dropped the subleading terms. In gapped phases
without topological order, such as band insulators, $\gamma=0$ for a smooth boundary. These predictions have been verified
in the context of a number of specific $D=2$ models \cite{haque, melko, frank, furukawa2007} and in $D=3$ $Z_2$ toric code models \cite{chamon2008}. In this paper, we discuss the general structure of entanglement
entropy for gapped topological and non-topological phases in $D\geq 3$.
One notices that in $D = 2$, the topological entanglement entropy depends only on a topological property of the boundary- in
this case the number of connected components. There are two equivalent ways of extracting
the topological entanglement entropy \cite{kitaev2006, levin2006}. First, via the scaling of
the entropy with
boundary size for smooth boundaries, to extract the constant term. The second, by considering a combination of entanglement entropies of three suitably chosen
regions $A,\,B, \, C$, so that $-\gamma = S_A+S_B+S_C-S_{AB}-S_{BC}-S_{AC}+S_{ABC}$.
Here, we will discuss analogous questions in $D\geq 3$. In particular, consider a gapped $D=3$ phase, and a region $A$ with a smooth boundary. (1) If the entanglement
entropy $S_A$ contains a constant term, does it necessarily reflect topological order? (2) The boundary of $A$ is a closed two dimensional surface that has two topological
invariants associated with it - the number of connected components, and the genus (number of handles) of each component. Does this imply there are two distinct types
of TEEs and correspondingly two varieties of topological order in $D=3$? The answer is {\em no} to both these questions, as we elaborate in this paper. We show that
even a trivial gapped phase, with no topological order, one can have a constant term in the entanglement entropy in $D=3$ (and in any other odd dimension). Hence, this by
itself does not signify topological order. Moreover, this constant is generally genus dependent, ruling out a topological origin for a genus dependent entanglement
entropy. This reduces the number of possible TEE to the same as $D=2$. We discuss generalization of the Kitaev-Preskill scheme \cite{kitaev2006} to extract the TEE in $D=3$; and why some
naive extrapolations fail.
A deeper understanding of TEE is obtained by considering higher dimensions. We show that at least one new topological entanglement entropy appears on going up every two
dimensions. Thus, while $D=2 \,{\rm and}\, 3$ are similar, a new topological constant does appear in $D=4$ (in $D=2n \,{\rm and} \,2n+1$, there are thus $n$ constants). These
are related to the Betti numbers \cite{footnotebetti} of the boundary. We construct topological phases that manifest these new TEEs, and explicitly calculate their
value. These are based on a discrete gauge group $G$. In all dimensions, the TEE
for discrete gauge theories is $-\log|G|$ per connected surface component, where $|G|$ is the number of elements in $G$. These theories capture both abelian (like the $Z_2$ toric code) and non-abelian phases
and the ground state of these theories correspond to condensate of closed loops. One can also consider more general abelian discrete
gauge theories where the fluctuating loops are readily generalized to fluctuating $p$-dimensional surfaces. These manifest explicitly in the topological entanglement entropy,
through the appearance of new topological constants that depend on higher Betti numbers. Furthermore, a previously discussed duality between $p$ and $D-p$ theories in $D$
dimensions \cite{bombin2007, dennis2001, hammawen}, is reflected in the structure of TEE.
To isolate topological contributions it is useful to know the structure of entanglement entropy in trivial gapped phases. Since correlations are local in such phases,
we propose an expansion of entanglement entropy $S_A$ based on adding individual contributions from patches on the surface of region A: $S_A=\sum_i S_i$. The entropy
densities $S_i$ will depend on local properties, such as the local curvature of the surface. One can then expand the entropy density in polynomials of curvature
and its derivatives, similar to the Landau expansion of free energy density \cite{chaikin}. The contribution from higher order terms to $S_A$ are subdominant for large surfaces.
Interestingly, not every term is allowed in this expansion. When we divide space into a region $A$ (inside) and $\bar{A}$ (outside), the entanglement entropy of both
are equal i.e. $S_A = S_{\bar{A}}$. This imposes a Z$_2$ symmetry on the expansion that is {\em unique to ground state entanglement entropy}\cite{corner_note}.
This has important consequences. Consider for example, $D=2$. The entropy density is not allowed to depend linearly on the boundary curvature $\kappa$, which changes
sign on interchanging inside and outside. Thus, the expansion of entropy density for a trivial $D=2$ phase is $S_i=a_0+a_2\kappa^2(r_i)+\dots$, which when integrated around
the boundary leads to $S_A=\alpha_1 L+ \alpha_3/L+\dots$, where $L$ is the length of the boundary of region $A$. The first term is the area law, and the next term is two orders of $L$ down, due to the Z$_2$ symmetry,
which eliminates the constant term in total entropy for smooth boundaries. Thus the existence of a constant term in a gapped $D=2$ state implies a {\em non-trivial} phase i.e. topological order.
In general, this method predicts that for an isotropic, parity invariant state without topological order,
the entanglement entropy in even (odd) spatial dimensions depends {\em only} on odd (even) powers of $L$, the linear scale of the boundary.
We will assume it is possible to take the continuum limit for all the phases that we consider in this paper. This assumption exclude phases such as a layered $Z_2$ topological ordered phases in $D = 3$ whose
topological entropy depends on the local geometry of the region $A$.
The paper is organized as follows: in section \ref{sec:decomp} we discuss the general structure of entanglement entropy for gapped phases and explain the basic assumptions underlying our discourse.
In section \ref{sec:EEtrivial} we introduce the aforementioned curvature expansion for entanglement entropy of trivial gapped phases and study its consequences.
In section \ref{sec:EE3d} and \ref{sec:EE4d} we study topological ordered phases in $D = 3$ and $D > 3$ respectively, through extracting the dependence
of entanglement entropy of a region on the topology of its boundary. We also generalize the constructions for extracting topological entropy \cite{kitaev2006, levin2006, chamon2008}.
\section{Structure of Entanglement Entropy for Gapped Phases} \label{sec:decomp}
In this article, we will assume that the entanglement entropy of a region $A$ can be decomposed into two parts:
\begin{equation}
S_A=S_{A,local}+S_{A,topological}
\label{eq:decomposition}
\end{equation}
We postpone the underpinnings of this assumption to Sec.\ref{sec:EE3d} when we study topologically ordered phases. Here $S_{local}$ is defined by the property that it can be written as a sum
over contributions from patches located along the boundary of region $A$:
\begin{equation}
S_{A,local}=\sum_i S_i
\label{eq:local}
\end{equation}
where $S_i$ depends only on the shape of the patch $i$, and not on the rest of the
surface or how it fits with other patches, see Fig.\ref{patch},
$\emph{at least}$ if the edge of the patch
connects smoothly to all other patches.
\begin{figure}
\centering
\includegraphics[width=.4\textwidth]{Fig3.eps}
\caption{The local part of the
entropy of region $A$ is the sum of contributions
of small patches on the boundary.\label{patch}}
\end{figure}
We assume the other contribution
$S_{topological}$ is topologically invariant, i.e., it does not
change as the boundary is deformed unless the topology of the region changes.
If such a term is present and if it cannot be expressed in a local way,
then the phase has long range entanglement, which is the hallmark of topological order \cite{kitaev2003, kitaev2006, levin2006}.
Let us consider the assumptions under which the decomposition (Eqn.\ref{eq:local}) would be possible for a trivial (i.e. not topologically ordered) gapped phase. The reduced density matrix
corresponding to a region $A$ for the ground state wavefunction may be written as
$$
\rho_A = e^{-H_A}/Z
$$
where $H_A$ is the so called \textit{entanglement Hamiltonian} and $Z = tr(e^{-H_A})$ so that $tr(\rho_A) = 1$. Therefore, we can think of $\rho_A$ as the thermal density matrix at temperature $T = 1$ for the
Hamiltonian $H_A$ and the von Neumann entropy $S_A = -tr(\rho_A\, log{\rho_A})$ as the thermal entropy for this system. Let us define $\tilde{\rho}_A(T) = e^{-H_A/T}/Z(T)$ where $Z(T) = tr(e^{-H_A/T})$. Clearly,
$\rho_A = \tilde{\rho_A}(T = 1)$ and $S_A$ obeys the following equation:
\begin{equation}
S_A \equiv S_{A,local} = \int_1^\infty \frac{dT}{T} \frac{\partial \langle H_A \rangle}{\partial T} \label{integral}
\end{equation}
where $\langle H_A \rangle$ denotes the thermal average of $H_A$ at temperature $T$ with respect to the density matrix $\tilde{\rho}(T)$. We claim that the entanglement entropy
would admit an expansion such as Eqn.\ref{eq:local}
if the following conditions are
satisfied.
\begin{itemize}
\item $H_A$ can be written as a sum of local operators $O$'s i.e. $H_A = \sum_x O(x)$.
\item There is no phase transition for the Hamiltonian $H_A$ for $T \geq 1$.
\end{itemize}
The first condition along with the fact that all correlations of local operators are short-ranged in a gapped phase imply that $H_A$ has non-zero support only near the
boundary of region $A$ (within the distance of correlation length). In other words, the degrees of freedom inside and outside of
region $A$ are coupled only through operators that lie within a distance $\sim \xi$ from the boundary. This implies that $ \langle H_A \rangle = \sum_i h_i $ where $i$
denotes a point at the boundary of region $A$
and the $h_i$'s depend solely on the properties of the boundary in the vicinity of point $i$.
The second condition implies that the integral in Eqn.\ref{integral} does not admit any singularity so that all terms $S_i$ in the Eqn.\ref{eq:local} are finite.
Physically, this means that the actual system of
interest is smoothly connected to its $ T = \infty$ zero correlation length system where Eqn.\ref{eq:local} holds trivially.
\section{Entanglement Entropy of trivial gapped phases} \label{sec:EEtrivial}
In this section we will focus on understanding the leading and subleading dependence of $S_{A,local}$ on $L$ for gapped trivial phases of matter. Let us assume that the boundary of region $A$ is smooth, and further
that the phase is isotropic and parity invariant
(consequences of the violation of these assumptions are discussed
at the end of this section and in Appendix \ref{sec:rotpar}).
Then, as we will see below, in the absence of topological order, only alternate terms in the power series expansion of $S_A(L)$ appear:
\begin{equation}
S_{A,local}(L) = \alpha_1 L^{D-1}+\alpha_3 L^{D-3}+\alpha_5 L^{D-5}+\dots \label{expanL}
\end{equation}
i.e. only those with odd co-dimension exponent can appear. This expansion implies a distinction between even and odd dimensions:
in even dimensions, any constant contribution to the entanglement
entropy must come from $S_{A,topological}$, and thus
indicates topological order. In odd spatial dimensions, a constant term may appear in the local entropy,
making it more difficult, though still possible,
to isolate topological contributions (note that these conclusions
apply only to smooth boundaries in rotationally symmetric
systems; a corner can produce a constant term
for even dimensions as well as odd ones).
Let us consider some instances of Eqn.\ref{expanL} that will
motivate its derivation. First, it is well known that in $D=2$,
a constant term in $S_A(L)$ implies the presence of topological order \cite{kitaev2006, levin2006}.
On the other hand, in $D=3$, a constant term can appear for a non-topologically ordered phase, such as a gapped scalar field.
Consider for a moment \cite{casini2009} a {\em massless} field,
where it is known that the entanglement scales as $S_A \sim L^2 + log(L)$
for a spherical ball of radius $L$. Now, providing a mass $m$ to the scalar
field would cut off the $log(L)$ term and instead lead to a constant contribution proportional to $log(1/m)$. We verified this explicitly using a numerical calculation similar to Ref. \cite{srednicki}.
Interestingly, when the surface of region $A$ is flat,
then there is no constant contribution as we show in Appendix A (a flat boundary is possible when the total system has the topology of a three torus $T^3$; then $A$ may be taken
to have the geometry $T^2 \times l$, where $l$ is a line segment).
This indicates that the presence or absence of a constant term may have something to do with the curvature of the boundary of region $A$. In the next subsection, we make this statement
precise and explain the observations made above.
\subsection{Entropy Density Functional} \label{sec:local}
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{inside-outsidesymmetry.eps}
\caption{Illustration of the Z$_2$ symmetry for the curvature expansion discussed in the text.\label{fig:inout}}
\end{figure}
\begin{table*}
\begin{tabular}{|l|c|c|p{4cm}|}
\hline
& \textbf{Full symmetry} & \textbf{All symmetries broken} & \textbf{Broken parity alone}\\
\hline
$\mathbf{D=2}$ & All even terms $L^0,L^{-2}$ & Constant term & All even terms\\ \hline
$\mathbf{D\geq 3}$ \textbf{odd}& All Odd terms $L^{D-2}, L^{D-4},\dots$ & Nothing forbidden &
Odd terms with positive exponents and also $L^{-1}$ if $D\equiv
3(\mathrm{mod\ 4})$\\ \hline
$\mathbf{D\geq 4}$ \textbf{even} & Even terms $L^{D-2},L^{D-4},\dots$ &Nothing forbidden &
Even terms\\
\hline
\end{tabular}
\caption{\label{table:mahogany}Terms in the entropy forbidden by
symmetries.
The three columns describe systems with rotational and parity
symmetry, no spatial symmetry, and rotational but not parity symmetry. The
entries list the scaling of terms $L^k$ that are forbidden
from appearing in the entropy.}
\end{table*}
Let us consider a region in two dimensions for concreteness.
We postulate that the local entropy $S_{A,local}$ is given by the following integral:
\begin{equation}
S_A = \sum_i S_i = \int d\sigma F(\kappa,\partial\kappa,\dots)
\label{eq:continuous}
\end{equation}
where $F(\kappa,\partial\kappa,\dots)$ is the ``entropy density functional".
In a gapped phase, the entropy $S_i$ of a patch larger than
the correlation length
can depend only on the properties
of the patch, such as its length $\Delta\sigma_i$ and
curvature $\kappa_i$, as well as derivatives of the latter,
and must be proportional to $\Delta\sigma_i$.
Hence $S_i=\Delta\sigma_i F(\kappa_i,\partial^n\kappa_i)$. Taking
the limit where the patches become microscopic
compared to $L$ (but greater than $\xi$)
leads to Eq. (\ref{eq:continuous}).
The entropy density functional always satisfies a $Z_2$ symmetry, which
is the key to understanding the $L$-dependence of the entropy.
The symmetry results from the fact that, if $A$ and $B$
are complementary regions, then $S_A=S_B$.
Therefore, changing `inside' to `outside' keeps the entanglement entropy
invariant. Now, under this transformation, radii of curvature clearly
change sign $\kappa \rightarrow -\kappa$, and this constrains
the entropy density functional (Fig.\ref{fig:inout}). As an illustration of this $Z_2$ symmetry, consider the form of the functional $F$ for a gapped two dimensional system.
On a smooth boundary, one can expand the function $F$ in a Taylor series, retaining the first few terms:
\begin{equation}
F(\kappa_i,\partial^n_\sigma\kappa_i)=a_0+a_1\kappa_i + a_2\kappa_i^2+b_2\partial_\sigma\kappa_i+\dots \label{expan2d}
\end{equation}
The first term gives the boundary law $S_A=a_0 L$. The second term, if it is present, would give a constant contribution $\oint d\sigma\kappa = 2\pi$
for the curve shown, which would be
a non-universal constant contribution.
However, such a term is in fact forbidden by the $Z_2$ symmetry, since
the term is odd in $\kappa$.
The term $\kappa^2$ gives the next contribution to the entropy
that is proportional to $\oint d\sigma\kappa^2$. If the shape of region $A$ is kept fixed, then this contribution scales as $1/L$.
The term $\partial_\sigma\kappa$ is allowed by the $Z_2$ symmetry
(since both the derivative and the radius of curvature change sign,
assuming that the direction of the curve is set by a `right hand rule', whereby the arc length increases along a specific direction), yet it still
vanishes because it is a total derivative. Generalizing these arguments,
one finds that $S_A=\sum_{k=0}^\infty \alpha_{2k+1}L^{1-2k}$.
In general dimensions, we find similar results if we continue
to assume rotational, parity, and translational symmetry. These
assumptions imply that $F$ can depend only on the metric
tensor $g^{\alpha\beta}$ and
on the extrinsic curvature
of the surface. The latter tensor does not appear when considering intrinsic properties of a manifold (as in general relativity). However,
entanglement entropy
does depend on the embedding of the boundary $\partial A$, since it is measuring
the entanglement of the degrees of freedom in the space around the surface.
The extrinsic curvature is a tensor $\kappa_{\alpha\beta}$ with two indices
(see Appendix \ref{sec:primer} for a short primer on the requisite differential
geometry).
Thus each term in $F$ contains some number of factors of $\kappa_{\alpha\beta}$
and its covariant derivatives, with all the indices contracted by factors
of $g^{\gamma\delta}$ (if parity is broken, the antisymmetric volume tensor
$\gamma^{\alpha_1\dots\alpha_{D-1}}$ is allowed as well, Appendix \ref{sec:rotpar}).
The inside/outside symmetry further limits the form
of the terms in $F$: it implies
that each term in $F$ includes an even number of factors $n_{\kappa}$ of
$\kappa$. The total order of all the derivatives $n_D$
must also be even. This follows from rotational symmetry. For rotational symmetry to be respected, one has to
contract all the lower indices with the tensor $g^{\gamma\delta}$.
This leads to an even number of lower
indices, that include the derivatives as well as the curvature indices. Since the curvature tensor is of even rank, the number of derivatives
$n_D$ has to be even. Putting everything together, one finds that the contribution to the entropy density $F$ scales as $L^{-(n_\kappa+n_D)}$ that clearly has an even exponent,
explaining why only alternate terms appear in the entropy, Eq. \ref{expanL}.
When rotational or inversion symmetries are broken spontaneously or by applying a field, additional terms appear in
the entropy, as summarized in
Table \ref{table:mahogany}. We provide the details leading to these results in Appendix \ref{sec:rotpar}.
The local entropy can also contain topology-dependent terms e.g. the term $\int G dA=4\pi \chi$
in three dimensions where $G$ is the Gaussian curvature which is the determinant of the matrix $\kappa_{\alpha \beta}$. Note that this term is compatible with the symmetry
$\kappa\rightarrow -\kappa$, since $G$ is quadratic in $\kappa$. Hence,
as mentioned earlier, the presence of a term in the entropy that is proportional to the Euler characteristic does
not necessarily correspond to topological order.
In general dimensions, $\int \mathrm{det}\kappa dA$ is topological,
but it is only symmetric in odd dimensions, where it
is proportional to the Euler characteristic of the boundary
(in general, it is proportional to the Euler characteristic
of the region itself \cite{gauss-bonnet}).
\section{Topological Entanglement Entropy in $D= 3$} \label{sec:EE3d}
We now turn to the topological part of the entanglement entropy. Our starting point is Eqn. \ref{eq:decomposition} which we rewrite here for convenience:
\begin{equation}
S_A=S_{A,local}+S_{A,topological}
\label{eq:decomposition2}
\end{equation}
This decomposition is what enables the extraction of topological entropy using Kitaev-Preskill \cite{kitaev2006} or Levin-Wen \cite{levin2006} constructions for two-dimensional
topological ordered phases. The assumptions underlying this equation are somewhat tricky. Though Eqn.\ref{eq:decomposition2} holds for toric code models in all dimensions and there is strong numerical evidence
that it also holds for
many interesting two dimensional topological ordered states such as $Z_2$ spin liquids, quantum dimer models and various quantum Hall states \cite{furukawa2007, haque, melko, sierra2010, frank},
Eqn.7 surreptitiously rules out
a layered $Z_2$ topologically ordered state. Such a state would lead to a correction in entanglement entropy $\Delta S_A = -\gamma_{2D} L_z$,
for layering perpendicular to the $z$ direction. Here $\gamma_{2D}$ is the topological entanglement entropy associated with the theory living in each layer. Clearly, $\Delta S_A$ is not
topologically invariant. The assumptions underlying Eqn. \ref{eq:decomposition2}
most likely also do not apply to the self-correcting code state of Ref. \cite{haah2011}; this state (whose
ground state degeneracy depends on the divisibility of system size by powers of $2$,
for example) illustrates why we need to make an assumption of this type. Nevertheless, we will briefly discuss the topological entanglement entropy of layered $Z_2$ state in Appendix \ref{sec:layer}.
\textit{Independent contributions to $S_{topological}$:} The boundary $\partial A$ of a three dimensional region $A$ is a compact manifold that is characterized by Betti numbers $b_0$ and $b_1$ (note that for
compact manifolds $b_2 = b_0$). As we show in Appendix \ref{sec:linear} in three dimensions $S_{A,topological}$ is a linear function of $b_0, b_1$,
say, $S_{topological} = -\gamma_0b_0-\gamma_1b_1$ (we assume that the space in which region $A$ is embedded has the topology of $\mathbb{R}^3$, otherwise more complicated
dependence is possible in principle). This might lead one to suspect that there are two different kinds of topological orders in three dimensions, namely,
those corresponding to a non-zero $\gamma_0$ and $\gamma_1$ respectively. However, $b_0, b_1$ are related to the Euler characteristic $\chi$ through
$2b_0 - b_1 = \chi$. Thus, one may redefine $S_{A,local}'=S_{A,local}+\alpha\chi$ and $S_{A,topological}'=S_{A,topological}-\alpha\chi$ without
changing the entropy \textit{and} $\alpha$ may be adjusted so that the $b_1$ dependence of $S_{topological}$ is canceled out.
Here the term $\alpha\chi$ may be thought of as both local and topological. It is local, because the Euler's formula, $\chi=V-E+F$ gives a local expression
for this term, where $V$, $E$, and $F$ are the number of vertices, edges, and faces into which $\partial A$ is divided (alternatively,
in a continuum theory, $\alpha\chi$ can be incorporated into the entropy density $F$ since $\chi$ is the integral of the Gaussian curvature). It is also
topological, because $\alpha\chi$ is independent of how the surface is divided up into regions. The upshot of this discussion is that there is only one kind of topological entropy in three dimensions.
\textit{$Z_2$ string and $Z_2$ membrane models:} As an alternative way to understand the above result, let us study specific models whose
topological entanglement potentially depends on different Betti numbers. Consider the following model of $Z_2$ gauge theories consisting of spin-$1/2$ degrees of freedom that live
on the links of a three-dimensional cubic lattice:
\begin{equation}
H_{string} = -\sum_{\Box} \,\,\prod_{l\in \Box} \tau_{z,l} - h \sum_l \tau_{x,l} \label{hstr3d}
\end{equation}
where $\Box$ denotes a plaquette of the cubic lattice, the operators $\tau_{x,l}, \tau_{z,l}$ live on the links $l$ of the lattice. The above Hamiltonian is supplemented with the constraint (`Gauss law')
$ \prod_{l \in \,vertex} \tau_{x,l} = 1 $ to impose the absence of $Z_2$ charges in the theory. Because of this constraint the gauge invariant degrees of freedom in this model consist of closed loops
$\mathcal{C}$ on the edges of the lattice. In the deconfined phase of the gauge theory, $ |h| \ll 1$, the loops condense
because they do not cost much energy. The entanglement entropy of this model for a region $A$ depends only on the Betti number $b_0$ of $\partial A$, since each component of the boundary places a separate
constraint on the loops
that intersect the boundary $\partial A$. Let us take Kitaev's `toric code limit' of the above model \cite{kitaev2003} by setting $h=0$.
In this limit, the constraint commutes with the Hamiltonian and can be included as a part of it. Hence
the model may be written as
\begin{eqnarray}
H_{string,h=0} & = & -\sum_{\Box} \,\,\prod_{l\in \Box} \tau_{z,l} - \sum_{vertex}\,\, \prod_{vertex \in \,l} \tau_{x,l} \nonumber \\ \label{hstr3d2}
\end{eqnarray}
Interestingly, the ground state of Eqn.\ref{hstr3d2} may be reinterpreted as a superposition of closed \textit{membranes}. This is seen as follows.
The first term in the Hamiltonian may be regarded as the constraint $\prod_{l\in \Box} \tau_{z,l} = 1$. Now consider the dual lattice each of whose plaquettes
is pierced by a link `$l$' of the original cubic lattice. A surface can be defined by the plaquettes of the dual lattice pierced by $\tau_{z,l}=-1$ bonds. Due to the constraint, this surface is closed.
Thus there is no distinction between condensed loops and condensed membranes in this case\cite{bombin2007,hammawen},
consistent with the fact that in three dimensions there is only one kind of topological entanglement entropy.
\textit{Discrete gauge theories in $D=3$:} Before moving on to the discussion of topological entropy in general dimensions, let us derive the entanglement entropy corresponding to a discrete gauge theory with
general gauge group $G$ for
Kitaev model \cite{kitaev2003} on a cubic lattice \cite{thanklevin}
\begin{equation}
H = - t \sum_{p} \delta(g_1 g_2 g_3 g_4 = e) - V \sum_{s, g} L^{1}_g L^{2}_g L^{3}_g L^{4}_g
\end{equation}
Here `p' stands for a plaquette, `s' for a star (i.e. six links emanating from a vertex) while $g$'s are the elements of group $G$ with size of
group being $|G|$. For non-abelian groups one needs to chose an orientation of the links so that for opposite orientations, the group element on a link
is $g$ and $g^{-1}$. The operators $L^{g}$ live on the links and and their action is described by $L_{g_1} |g_2 \rangle = |g_1 g_2 \rangle$ or
$L_{g_1} |g_2 \rangle = |g_2 g_1^{-1} \rangle$ depending on whether $g_1$ points away from or towards the
vertex at which the action of $L_g$ is being considered. The ground state of $|\Phi\rangle$ of $H$ is given by
\begin{equation}
|\Phi\rangle = \sum_{\{g\},g_1 g_2 g_3 g_4 = e \,\forall \,plaquettes} |\{g\}\rangle
\end{equation}
Let us divide the entire system into region $A$ and $B$ and assume that the boundary is made up of plaquettes of the lattice. The links along the boundary are labeled by the group elements $h_1, h_2, ...,h_n$.
The Schmidt decomposition of $|\Phi\rangle$ reads
\begin{equation}
|\Phi\rangle = \sum_{\{h\}} |\phi\rangle^{\{h\}}_{in} \otimes |\phi\rangle^{\{h\}}_{out}
\end{equation}
where
\begin{equation}
|\phi\rangle^{\{h\}}_{in} = \sum_{\substack{\{g\},g_1 g_2 g_3 g_4 = e \,\forall \,plaquettes \in A,\\ g_i = h_i \,\textrm{for}\, i \in \partial A}} |\{g\}\rangle
\end{equation}
and $|\phi\rangle^{\{h\}}_{out}$ is defined similarly. All the states in the Schmidt decomposition enter with the same weight and are orthogonal, therefore the entanglement entropy is
the logarithm of the number of states. These may be counted by finding all the configurations for $\{h\}$ that satisfy the following constraint: the product of the $\{h\}$'s around
any closed loop on the boundary must equal the identity \cite{hamma2005, levin2006}s. This includes contractible as well as noncontractible loops on the surface, and each independent loop reduces the total
number of configurations by a factor of $|G|$, leading to
\begin{eqnarray}
S & = & log (|G|^{V-1}) \nonumber \\
& = & V log(|G|) - \gamma \label{eq:discretetee}
\end{eqnarray}
where $V$ is the number of vertices on the boundary
and $\gamma = log(|G|)$ is the topological entanglement entropy. This result for topological entanglement entropy is identical to that for discrete gauge theories in $D=2$.
\subsection{Extracting Topological Entanglement Entropy in $D =3$}
\begin{figure}
\includegraphics[width=240pt, height=220pt]{3dgamma2.eps}
\caption{Fig.(a) and (b) show two valid $ABC$ constructions (Eq. \ref{kitpres}) in three dimensions that can be used to extract the topological entanglement entropy. In Fig.(a) the cross-section of a torus has
been divided into three tori $A, B$
and $C$ while in Fig.(b) a torus that has been divided into three cylinders $A, B$ and $C$. The Fig.(c) shows an \textit{invalid} construction as explained in the text. In all three figures,
we define region $D$ to be the rest of the system.}
\label{fig:3dtee}
\end{figure}
In the spirit of Ref. \cite{kitaev2006, levin2006}, we would like to combine the total entanglement entropies of
certain regions in such a way that the local part of the entropy cancels out while topological part survives.
The Kitaev-Preskill construction, which succeeds in
this task in two dimensions can be modified so that it works
for three dimensions as well (Ref.\cite{chamon2008} describes an extension of the Levin-Wen scheme to $D=3$ ).
The construction involves three regions $A,B,C$ embedded
inside region $D$:
\begin{equation}
-\gamma_{topo} = S_A + S_B + S_C - S_{AB} - S_{BC} - S_{CA} + S_{ABC} \label{kitpres}
\end{equation}
In two dimensions, the regions are taken to be three $120^\circ$ segments
of a circle. In three dimensions, a direct generalization of the two-dimensional
construction (dividing a \emph{cylinder} into three sectors as in Fig.\ref{fig:3dtee}c) fails to be topologically invariant
because the changes in the entropy
near the points at the top and bottom of
the cylinder where $A,B,C$ and $D$ all meet do not cancel.
However, the regions such as
the two shown in Fig.\ref{fig:3dtee}a and 3b can be used where such points do not exist.
For example, if one deforms the circle at which regions $A,B,D$ all meet
(in either geometry), then
\begin{eqnarray}
\Delta \gamma_{topo} & = & -[\Delta (S_A - S_{CA}) + \Delta (S_B - S_{BC}) + \Delta (S_{ABC} - S_{AB})] \nonumber \\
& = & 0
\end{eqnarray}
The last equation follows because each of the three terms in the brackets could be thought of as the difference between entropies of two regions that differ by addition of region $C$ (that
is located far from the point where $A, B$ and $D$ meet). Since each
region has a single boundary component, $\gamma_{topo}=S_{topological}$.
In Appendix \ref{sec:valid3d}, we detail the general requirements for a construction that would always yield a topological invariant.
Based on our earlier discussion of curvature expansion for entanglement entropy, we note that in the special case of a completely flat boundary between a region
and the rest of the system, the constant term in the entanglement entropy corresponding to that region can indeed be identified with topological entanglement entropy \cite{footnote_degen}. This can
be realized by taking the total system to be $T^3$ and region $A$ as $T^2 \times l$ where $l$ is a line segment (similar to the calculation of
entanglement entropy for a free scalar in Appendix \ref{sec:scalar}).
\section{Topological Entanglement Entropy in $D > 3$} \label{sec:EE4d}
\textit{Independent terms in $S_{topological}$ in arbitrary dimensions:} Following our discussion of topological entanglement entropy in $D = 3$, in this section we study the independent contributions to
$S_{topological}$ in a general dimension $D>3$. The boundary $\partial A$ of a $D$-dimensional region $A$ is a compact manifold that is characterized by Betti numbers, $b_0,\dots,b_{D-1}$ that
describe various orders of connectivity of the surface (see e.g. \cite{nakahara}).
We will assume a linear relationship,
$S_A=-\sum_{k=0}^{D-1} \gamma_k b_k$. In principle, in higher dimensions the entanglement entropy could depend on more subtle
topological properties of the boundary, but we will focus only
on this form.
Further, as we will see below, this form turns out to be sufficient for Kitaev models that describe
discrete $p$-form gauge theories ($p \ge 1$) in arbitrary dimensions.
To see how many types of topological entropy can exist in higher
dimensions, first note that
for compact manifolds, the Betti numbers have a symmetry,
$b_k=b_{D-1-k}$ and hence the sum may be cut short, at
$k=\lfloor\frac{D-1}{2}\rfloor$.
Furthermore, owing to the relation $\chi=\sum_{k=0}^{D-1} (-1)^k b_k$, in all odd dimensions a
part of the topological entropy may be absorbed into the local entropy,
reducing the number of coefficients by one more.
Hence there are $n$ topologically nontrivial
contributions to the entanglement entropy in $2n$ and $2n+1$
dimensions:
\begin{equation}
S_{A,topological}=\begin{cases}
-\gamma_0 b_0 - \gamma_1 b_1 - \dots - \gamma_{\frac{D}{2}-1}b_{\frac D2-1}, & \text{if }D\text{ is even}\\
-\gamma_0 b_0 -\gamma_1 b_1+\dots - \gamma_{\frac{D-3}{2}}b_{\frac {D-3}{2}}, & \text{if }D\text{ is odd}\end{cases}.
\label{gamma_d}
\end{equation}
Precisely such a hierarchy of states associated
with different Betti numbers has been arrived
at by Ref. \cite{bombin2007} by
constructing a sequence of
Kitaev `toric-code' type models where the ground
state is a superposition of all $p$-dimensional manifolds on a lattice (for $1\leq p\leq D-1$).
This state is dual to the superposition of all $q=D-p$ dimensional manifolds,
so the number of distinct models is $\lfloor\frac{D}{2}\rfloor$, the same
as the number of types of topological entropies.
\textit{$S_{topological}$ for gauge theories in arbitrary dimensions:} Similar to three dimensions, one may study models of discrete gauge theories to understand these results.
For example, on a hypercubic lattice in $D=4$, the string and membrane theories describe very different ground states \cite{dennis2001} and unlike $D = 3$, the membrane
theory is now dual to itself, not to the string phase. Explicitly, in the `toric code limit' \cite{kitaev2003, dennis2001} these two theories are given by
\begin{equation}
H_{string} = -\sum_{\Box} \,\,\prod_{l\in \Box} \tau_{z,l} - \sum_{vertices} \, \prod_{vertex \in \, l} \tau_{x,l} \label{hstr3dkit}
\end{equation}
\begin{equation}
H_{membrane} = -\sum_{l} \,\,\prod_{l \in \Box} \sigma_{z,\Box} - \sum_{cubes} \, \prod_{\Box \in \,cube} \sigma_{x, \Box} \label{hmem3dkit}
\end{equation}
As we show now, the entanglement entropy of the model in Eqn. \ref{hstr3dkit} in four dimensions
depends on the Betti number $b_0$ of $\partial A$ while that corresponding to model in Eqn. \ref{hmem3dkit} depends on the difference
$b_1 - b_0$. For the sake of generality, let us derive the entanglement entropy of a generalized toric model in arbitrary spatial dimensions $D$ whose ground state is given by sum over all closed
$d_g$ dimensional membranes. This ground state describes deconfined phase of a $d_g$-form abelian gauge theory.
These membranes intersect the boundary $\partial A$ of region $A$ in closed membranes of dimension $d_g -1$, \textit{with the restriction that these
intersections are always boundaries of a membrane of dimension $d_g$ contained in $\partial A$}. For example,
consider the entanglement of membrane model in Eqn.\ref{hmem3dkit} in $D = 3$ when the boundary of region $A$ is a torus $T^2$ (note that the form of Hamiltonian for membrane theory is identical
in $D = 3$ and $D = 4$).
When a closed membrane intersects $\partial A = T^2$, one sees that one can only obtain an even number of closed loops along any non-contractible cycle of $T^2$, which would therefore form
the boundary of two dimensional membrane. Returning to the general case,
let us denote the number of independent $n$-dimensional membranes that belong to $\partial A$ by $C_n$ and those that are boundary of a $n+1$-dimensional membrane by $B_n$.
Using the definition of Betti numbers \cite{nakahara} and simple linear algebra, one finds that the entanglement entropy $S_A$
\begin{equation}
S_A \propto \sum_{n=0}^{d_g-1} (-)^{d_g-1 + n} C_n - \sum_{n=0}^{d_g-1} (-)^{d_g-1 + n} b_n
\end{equation}
Since the $C_n$ are expressed in terms of local quantities such as the number of edges, vertices etc. that lie on the boundary without any additional constraint,
we identify the first sum as $S_{local}$ and the second as $S_{topological}$. The proportionality constant depends on the
gauge group and akin to three dimensions equals $log(|G|)$ where $|G|$ is the number of elements in the abelian gauge group (note that the calculation of TEE in $D$ = 3 (Eqn. \ref{eq:discretetee})
applies to abelian as well as non-abelian discrete gauge theories). Therefore
\begin{equation}
S_{topological} = - log(|G|) \sum_{n=0}^{d_g-1} (-)^{d_g-1 + n} b_n
\end{equation}
\emph{Extracting topological entropy in Four Dimensions} We will restrict our discussion of extracting $S_{topological}$ to four dimensions.
For a given region $A$, from Eqn. \ref{gamma_d} one has
$S_A = S_{A,local} - b_0 \gamma_0 -b_1 \gamma_1$ and one would like to have a
construction similar to Levin-Wen \cite{levin2006} and/or Kitaev-Preskill \cite{kitaev2006} that enables one
to extract the topological numbers $\gamma_0$ and $\gamma_1$.
We extract $\gamma_1$ by a generalization
of the construction in \cite{levin2006}. Let region $A$ have the topology of $B^2\times S^2$. Region $B$ is $A$ with a channel cut in it and has topology of $B^4$. Finally,
region C has a second identical channel cut out opposite to the first one and has topology $S^1\times B^3$
Now,
$S_A-2S_B+S_C$ is topologically invariant just as in two
dimensions
and the Betti numbers of the bounding surfaces are:
\begin{eqnarray}
b_0(\partial A) & = & 1, \,\,\,\,\,b_1(\partial A) = 1 \nonumber \\
b_0(\partial B) &= & 1, \,\,\,\,\, b_1(\partial B) = 0 \nonumber \\
b_0(\partial C) &= & 1, \,\,\,\,\, b_1(\partial C) = 1 \nonumber
\end{eqnarray}
Hence $(S_A - S_B) - (S_B - S_C) =- 2 \gamma_1$. Since $\gamma_1 \ne 0$ for the membrane Kitaev model $H_{membrane}$ (Eq.\ref{hmem3dkit}) while it is zero for the string model $H_{string}$ (Eq.\ref{hstr3dkit}) in $d = 4$,
this construction measures membrane correlations.
To isolate $\gamma_0$,
the analogous procedure, but with $A$ being $B^3\times S^1$ suffices.
The combination $(S_A-S_B)-(S_B-S_C)$ gives $\gamma_0+\gamma_1$, and
this may be combined with the previous construction to extract
both $\gamma_0$ and $\gamma_1$. This construction selectively measures string correlations since $\gamma_0 + \gamma_1$ is zero for $H_{membrane}$.
The Kitaev-Preskill construction of dividing a disc into three triangles that meet at the center is readily extended to any even dimension. In $D=4$ consider dividing the ball $B^4$ into five `pentahedra'
that meet at the center. The combination
\begin{equation}
\Delta(\{S\}) = \sum_i S_i - \sum_{i<j} S_{ij} + ... + S_{12345}
\end{equation}
is topologically invariant and gives $-\gamma_0$.
Here $S_{i_1 i_2...i_n}$ denotes the entanglement entropy corresponding to the region $A_{i_1} \cup A_{i_2} ... \cup A_{i_n}$.
\section{Discussion and Conclusion}
In this paper, we discussed the qualitative structure of the entanglement entropy for gapped phases.
We introduced the concept of `entanglement entropy density' whose integral over the boundary of a region $A$ yields the
entanglement entropy of region $A$. For gapped trivial phases the symmetry constraints on the entropy density, including the inside-outside exchange symmetry $S_A = S_{\overline{A}}$, naturally lead to the leading
and subleading dependence of the entanglement entropy on the linear size of a given region.
In the second half of the paper, we studied the topological entanglement entropy $S_{topological}$ of topologically ordered systems in various dimensions. A key result was that in $D=3$ there
is a single category of TEE, as in $D=2$, that depends linearly on the number of connected components of the boundary. This constrains the possible forms of topological order in $D=3$.
We briefly discussed TEE in higher dimensions - using generalized Kitaev toric code like models (i.e. deconfined
phases of $p$-form discrete gauge theories) to realize various topologically ordered phases. In $D=4$ we find two categories of TEE . In general, one new category of TEE appears each time
the dimension is raised by two. This even-odd effect is understood as follows. $S_{topological}$ depends on the Betti numbers of the boundary of region $A$. In odd spatial dimensions,
the Gauss-Bonnet theorem relates Betti numbers to the curvature of the boundary of region $A$. This implies that there is one linear combination of Betti numbers that can be
expressed as an integral of a local property of the boundary (such as curvature), and is thus not an independent topological contribution to the entanglement entropy. We also mentioned how
to extract $S_{topological}$ by a generalization of the $D=2$ Kitaev-Preskill and Levin-Wen constructions.
Potentially, in $D\geq 4$, $S_{topological}$ may depend not only on Betti numbers of the boundary manifold, but on more subtle topological properties such as its homotopy group.
If such phases do exist, then entanglement entropy could shed light on the classification of manifolds. Lattice 3D models realize a richer variety of topological phases than the
isotropic phases considered here. For example, there exist layered $Z_2$ topologically ordered phases, which retain a two dimensional character despite coupling between layers.
Another example is the self correcting quantum memory of Ref.\cite{haah2011}. For these, the separation between the topological and the local part of the entanglement entropy is
not obvious. General statements about entanglement in such topological phases remain for future work.
One might also consider a curvature expansion for the fluctuations of a conserved quantity such as particle number or total spin, inside a region $A$.
Intuitively, these would be a property of the boundary of region $A$ \cite{klich2006, klich2006_2, lehur2010,levitov2009}. Indeed, akin to entanglement entropy, one has $F_A = F_{\overline A}$ where
$F_A = \sqrt{\langle ( \sum_{r \in A} O_r )^2 \rangle - \langle\sum_{r \in A} O_r \rangle^2} $ is the variance of $O$ inside the region $A$. Therefore,
a curvature expansion for $F_A$ would inherit many of the arguments we used to derive the leading and sub-leading behavior of the quantity $F_A$, and can provide a framework to understand known
results \cite{lehur2010,klich2006, klich2006_2, levitov2009, helling2011, swingle2010}.
Finally, it may be possible to learn more about the systematics of the size dependence of entanglement entropy in {\em gapless} phases by a generalization of the curvature expansion
under certain conditions. Many gapless systems such as massless scalar/Dirac fermion also follow an area law and have an expression for entropy with interesting parallels to Eqn.\ref{expanL}.
\textit{Acknowledgements:} We thank Michael Levin for illuminating discussions and Matt Hastings for helpful comments on the manuscript. Support from NSF DMR- 0645691 is acknowledged.
|
2301.12856
|
\section{Introduction}
Sample path regularity of random processes and fields is an important and extensively studied topic in the literature, and can be used as a tool, for example, to study convergence of certain statistical estimators or to study convergence of numerical schemes for stochastic partial differential equations. Indeed, various application areas arise from the fact that sharp modulus of continuity estimates can be translated to supremum tail bounds, providing sharp tail behaviour of the supremum.
The problem is particularly well-studied in the case of Gaussian processes and fields. One of the earliest result in this direction is a sufficient condition due to Fernique \cite{Fernique} who provided a sufficient condition for the sample path continuity involving the increment metric
$d_X(s,t) = \left[\mathbb{E}(X_s-X_t)^2\right]^{1/2}$. Later on, Dudley \cite{Dudley1,Dudley2} provided a necessary condition for the continuity of a Gaussian process $X$ by using a metric entropy. While in general Dudley's condition is not sufficient, in the case of stationary Gaussian processes it turns out to be necessary and sufficient. Finally, general necessary and sufficient conditions for general Gaussian processes $X$ were obtained by Talagrand \cite{Talagrand}, in terms of metric entropies. Finally, while Talagrand's necessary and sufficient condition is rather complicated, a simple necessary and sufficient condition for H\"older continuity of Gaussian processes in terms of incremental increment metric
$d_X(s,t) = \left[\mathbb{E}(X_s-X_t)^2\right]^{1/2}$ was obtained in \cite{lauri2014}, where the authors proved that the celebrated general Kolmogorov-Chentshov criterion for H\"older continuity is both necessary and sufficient condition for Gaussian processes.
While the topic is widely studied in the Gaussian case, the literature on modulus of continuity beyond Gaussianity is more limited. One general approach to obtain modulus of continuity for processes is to use Garsia-Rodemich-Rumsey lemma \cite{Garsia-Rodemich-Rumsey-1970}. A multiparameter version was provided in \cite{hu-le2013} where the authors obtained modulus of continuity estimates and joint H\"older continuity of solutions to certain stochastic partial differential equations driven by Gaussian noise. Finally, we mention a closely related article \cite{Viens-Vizcarra2007}, where the authors studied the case of so-called sub-$n$th Gaussian processes or Wiener processes, where $n$ is an arbitrary integer. Such processes arise, roughly speaking, from processes of form $H_n(X_t)$, where $X$ is a Gaussian process and $H_n$ is the $n$th order Hermite polynomial. More precisely, in \cite{Viens-Vizcarra2007} the authors provided sufficient conditions for continuity and, in the spirit of Talagrand, estimates for the tail of the supremum via metric entropy by using Malliavin calculus which is well-suited to the setting of a Gaussian Wiener space.
In this article we study sample path continuity for general hypercontractive processes and fields. That is, we assume that higher order moments of the increments satisfy
$$
\mathbb{E}|X_t-X_s|^p \leq C(p)\left[\mathbb{E}|X_t-X_s|^2\right]^{p/2},
$$
allowing us to deduce conditions in terms of the simple incremental increment metric $\mathbb{E}|X_t-X_s|^2$. We show how the growth of the constant $C(p)$ translates into certain exponential moment bounds and tail estimates for the supremum. As a corollary, we extend the necessary and sufficient Kolmogorov-Chentshov criterion for the H\"older continuity beyond the Gaussian case studied in \cite{lauri2014}. It is worth to emphasize that, while our results can be used to cover and extend the known results on the Gaussian case \cite{lauri2014} and in the case of sub-$n$th processes \cite{Viens-Vizcarra2007}, our results require only hypercontractivity and can be used in a more general framework where, e.g. Malliavin calculus is not at our disposal. Finally, our results extend naturally to random fields, and can be e.g. used to study joint H\"older continuity, in the spirit of \cite{hu-le2013}.
The rest of the article is organised as follows. In Section \ref{sec:process} we introduce our notation and main results, while all the proofs and auxiliary lemmas are postponed to Section \ref{sec:proofs}.
\section{Necessary and sufficient conditions for continuity of hypercontractive processes and fields}
\label{sec:process}
We consider stochastic processes and fields, respectively, given by $X = (X_t)_{t\in K}$, where $K = [0,1]$ or $K=[0,1]^n$. We make use of the following hypercontractivity assumption:
\begin{assumption}
\label{assu:hyper-basic}
We suppose that for all $p\geq 1$ we have
\begin{equation}
\label{eq:hyper-process}
\mathbb{E}|X_t-X_s|^p \leq C_0^p p^{p\iota}\left[\mathbb{E}|X_t-X_s|^2\right]^{\frac{p}{2}},
\end{equation}
where $C_0>0$ is a generic fixed constant and $\iota\geq 0$ is a given parameter.
\end{assumption}
\begin{remark}
In general hypercontractivity is stated as
$$
\mathbb{E}|X_t-X_s|^p \leq C(p)\left[\mathbb{E}|X_t-X_s|^2\right]^{\frac{p}{2}},
$$
where $C(p)$ is a constant depending solely on $p$. Motivated by our examples below, we restrict ourself to the case $C(p) \leq C^p p^{p\iota}$. One could state our results by using a more general form of $C(p)$ with suitable growth in $p$, but this would result in an additional layer of notational complexity.
\end{remark}
\begin{remark}
It is worth to note that our assumption implies the exponential moment assumption (up to an unimportant constant) of \cite{Viens-Vizcarra2007}, in the case $\iota = n$ is an integer, cf. proof of Theorem \ref{thm:sufficient-process}.
\end{remark}
\begin{example}
If $X$ is Gaussian, then it is well-known that
$$
\mathbb{E}|X_t-X_s|^p = \frac{1}{\sqrt{\pi}}2^{\frac{p}{2}}\Gamma\left(\frac{p+1}{2}\right)\left[\mathbb{E}|X_t-X_s|^2\right]^{\frac{p}{2}},
$$
where $\Gamma$ is the Gamma function. By Stirling's approximation, we have
$$
\Gamma(x+1) \sim \sqrt{2\pi x}e^{-x}x^x
$$
for large $x$, and hence we may choose $\iota = \frac12$ in Assumption \ref{assu:hyper-basic}.
\end{example}
\begin{example}
\label{ex:wiener-chaos}
Let \(\mathcal{H}\) be a separable, real Hilbert space, and \(Z = \{Z(h): h \in \mathcal{H}\} \) an isonormal Gaussian process on \(\mathcal{H}\).
We define the \(nth\) Hermite polynomial as \(H_0(x)=1\) and for \(n\geq 1\) by
\[
H_n(x) = (-1)^n e^{x^2/2} \frac{d}{dx}e^{-x^2/2}.
\]
Denote by \(\mathcal{H}_p\) the linear space generated by the class \(\{ H_p(Z(h)): p \geq 0, \, h \in \mathcal{H}, \Vert h\Vert_\mathcal{h} = 1\}.\) This linear space is called the \(pth\) Wiener chaos of \(Z\). Then it is known (see, e.g. \cite{nourdin-peccati2012}) that
\[
\left[\mathbb{E}\lvert F \rvert^q \right]^{1/q} \leq \left[\mathbb{E}\lvert F \rvert^r\right]^{1/r} \leq \left(\frac{r-1}{q-1}\right)^{p/2} \left[\mathbb{E}\lvert F \rvert^q \right]^{1/q}.
\]
Hence if $X$ is a process living in a fixed Wiener chaos of order $p$ (or in a finite linear combination of them with $p$ as the highest chaos), we have \eqref{eq:hyper-process} with $\iota = \frac{p}{2}$ and recover the processes studied in \cite{Viens-Vizcarra2007}. In particular, we recover the Gaussian case by setting $p=1$.
\end{example}
We begin with the following general result providing the modulus of continuity and certain moment estimates.
\begin{theorem}
\label{thm:sufficient-process}
Suppose that a continuous $X = (X_t)_{t\in [0,1]}$ satisfies Assumption \ref{assu:hyper-basic} and suppose that $\mathbb{E}( X_t-X_s)^2\leq \rho(|t-s|)^2$ for some non-decreasing, non-negative continuous function $\rho$, with $\rho(0)=0$. Then
\begin{align}
\label{eq:continuity-general-process}
|X_t - X_s | \leq 8 \int_0^{|s - t|} \beta^{-\iota}\left(\log\left(\frac{4B}{u^2}\right)\right)^{\iota}d\rho(u)
\end{align}
for any $\beta>0$, where
$$
B = \int_0^1\int_0^1 \exp\left(\beta\left(\frac{|X_t-X_s|}{\rho(|t-s|)}\right)^{\frac{1}{\iota}}\right)dsdt.
$$
Moreover, for $\beta \in \left(0,\frac{e\iota}{C_0^{\frac{1}{\iota}}}\right)$, the random variable $B$ has finite $p$-moments for all $p$ satisfying
$$
1 \leq p < \frac{e\iota}{\beta C_0^{\frac{1}{\iota}}}.
$$
\end{theorem}
\begin{remark}
As in \cite{Viens-Vizcarra2007}, one could first study the existence of a continuous version in terms of metric entropies. That is, by considering the number $N_\epsilon$ of balls required to cover the interval $[0,T]$ in a metric $\left[\mathbb{E}(X_t-X_s)^2\right]^{1/2}$. Then one obtains continuity provided that
$$
\int_0^\infty \left|\log N_\epsilon\right|^{\iota} d\epsilon < \infty.
$$
As we are interested in the modulus of continuity, we assume continuity a priori.
\end{remark}
We obtain immediately the following corollary.
\begin{corollary}
\label{cor:holder-process}
Suppose that $X = (X_t)_{t\in [0,1]}$ satisfies Assumption \ref{assu:hyper-basic} and suppose that $\mathbb{E} (X_t-X_s)^2 \leq |t-s|^{2\alpha}$. Let $\beta \in \left(0,\frac{e\iota}{C_0^{\frac{1}{\iota}}}\right)$. Then there exists a random variable $C(\omega)=C(\beta,\omega)$ satisfying $\mathbb{E} \exp(\beta_0 C(\omega)^{1/\iota})<\infty$ for any $\beta_0$ satisfying
\begin{equation}
\label{eq:beta_0-process}
\beta_0 < \frac{e\iota}{\left(8C_0\cdot 3^{\max(\iota-1,0)}\right)^{\frac{1}{\iota}}}
\end{equation}
and a deterministic constant $C_{d}=C_d(\beta)$ such that
\begin{align}
\label{eq:holder-process}
|X_t - X_s | \leq C(\omega)|t-s|^{\alpha} + C_{d}|t-s|^{\alpha}\left(\log \frac{1}{|t-s|}\right)^{\iota}.
\end{align}
In particular, we have
\begin{equation}
\label{eq:lil}
\limsup_{|t-s|\to 0} \frac{|X_t-X_s|}{|t-s|^\alpha\left(\log \frac{1}{|t-s|}\right)^{\iota}} \leq C_{d}.
\end{equation}
\end{corollary}
\begin{corollary} \label{cor:sufficient-process}
Suppose $X = (X_t)_{t \in [0,1]}$ satisfies Assumption \ref{assu:hyper-basic} and assume that $\mathbb{E} (X_t-X_s)^2 \leq |t-s|^{2\alpha}$. Let $\beta \in \left(0,\frac{e\iota}{C_0^{\frac{1}{\iota}}}\right)$. Then for any interval $I\subset [0,1]$ and any $s\in I$ we have
\begin{align*}
P\left(\sup_{t \in I} \abs{X_t - X_s} \geq u\abs{I}^\alpha + C_de^{-\alpha \iota}\iota^\iota\right) \leq C(\beta_0) e^{-\beta_0 u^{1/\iota} },
\end{align*}
where $$C(\beta_0) = C(\beta_0, \beta) = \E4B^{\beta_0 \beta^{-1}\left(8 \cdot 3^{\max(\iota-1,0})\right)^{1/\iota} } < \infty$$ for any $\beta_0$ satisfying \eqref{eq:beta_0-process}.
\end{corollary}
\begin{remark}
The above result is close in spirit to \cite[Theorem 3.1]{Viens-Vizcarra2007}, and generalises naturally the well-known exponential decay of the supremum of Gaussian processes, in which case we would obtain
$$
P(\sup_{t\in I}|X_t-X_s|\geq u|I|^\alpha +C_1) \leq C_2e^{-c_3u^2}
$$
for constants $C_1,C_2$, and $C_3$.
\end{remark}
As our final main theorem, we obtain the following characterisation of H\"older continuity: under Assumption \ref{assu:hyper-basic}, Kolmogorov continuity criterion is a necessary and sufficient condition for H\"older continuity. This extends the results of \cite{lauri2014} beyond Gaussian processes and covers, in particular, processes living in a finite sum of Wiener chaoses, cf. Example \ref{ex:wiener-chaos}.
\begin{theorem}
\label{thm:holder-process-iff}
Suppose that $X = (X_t)_{t\in [0,1]}$ satisfies Assumption \ref{assu:hyper-basic}. Then $X$ is H\"older continuous of any order $\gamma<\alpha$, i.e. for any $\epsilon>0$
$$
|X_t-X_s| \leq C_{\epsilon}(\omega)|t-s|^{\alpha-\epsilon},
$$
if and only if for any $\epsilon>0$ we have
\begin{equation}
\label{eq:holder-iff}
\mathbb{E}(X_t-X_s)^2 \leq C_\epsilon|t-s|^{2\alpha-\epsilon}.
\end{equation}
Moreover, in this case the H\"older constant $C_\epsilon(\omega)$ of $X$ satisfies
\begin{equation}
\label{eq:exp-moments}
\mathbb{E} \exp\left(\beta C_\epsilon(\omega)^{\frac{1}{\iota}}\right) < \infty
\end{equation}
for small enough $\beta>0$ which depends only on $C_0$, $\alpha$, $\iota$, and $\epsilon$.
\end{theorem}
Our results extend naturally to the case of random fields $X = (X_t)_{t\in [0,1]^n}$. We begin with the following result, which extends Theorem \ref{thm:holder-process-iff} to the case of fields in a natural way.
\begin{proposition}
\label{prop:field-holder}
Suppose that $X=(X_t)_{t\in [0,1]^n}$ satisfies Assumption \ref{assu:hyper-basic}. Then $X$ is H\"older continuous of any order $\gamma<\alpha$, i.e. for any $\epsilon > 0$,
$$
|X_t-X_s| \leq C_\epsilon(\omega)|t-s|^{\alpha - \epsilon},
$$
if and only if for any $\epsilon>0$ we have
$$
\mathbb{E}(X_t-X_s)^2 \leq C_\epsilon|t-s|^{2\alpha-\epsilon}.
$$
Moreover, in this case the H\"older constant of $X$ satisfies
\begin{equation*}
\mathbb{E} \exp\left(\beta C_\epsilon(\omega)^{\frac{1}{\iota}}\right) < \infty
\end{equation*}
for small enough $\beta>0$ which depends only on $C_0$, $\alpha$, $\iota$, and $\epsilon$.
\end{proposition}
If one considers rectangular increments and joint continuity, we first need some notation, taken from \cite{hu-le2013}.
Let \( x = (x_1, ...,x_n)\) and \(y = (y_1,...,y_n)\) be two elements in \(\mathbb{R}^d\). For each integer \(k = 1,2,..,n,\) we define
\[
V_{k,y}x := (x_1,...,x_{k-1},y_k,x_{k+1},...,x_n).
\]
Let \(f\) be a \(\mathbb{R}^m\)-valued map on \(\mathbb{R}^n\). We define the operator \(V_{k,y}\) acting on \(f\) as follows:
\[
V_{k,y}f(x) := f(V_{k,y}).
\]
It is simple to verify that \(V_{k,y}V_{k,y}f(x) = V_{k,y}f(x)\) and that \(V_{k,y}V_{l,y}f(x) = V_{l,y}V_{k,y}f(x)\) for any \(f\).
Next, we define the joint (rectangular) increment of a function \(f\) on an $n$-dimensional rectangle,
\begin{align} \label{joint-rect-increment-defn}
\Box_y^n f(x) = \prod_{k=1}^n (I-V_{k,y})f(x),
\end{align}
where \(I\) denotes the identity operator.
For a random field $X=(X_t)_{t\in [0,1]^n}$, let \(d_X(t,s) := \sqrt{\mathbb{E}\lvert \Box^n_t X(s) \rvert^2 }.\) We assume that the following condition, analogous to Assumption \ref{assu:hyper-basic}, is satisfied.
\begin{assumption}
\label{assu:hyper-field}
We suppose that for all $p\geq 1$ we have
\begin{equation}
\label{eq:hyper-field}
\mathbb{E}|\Box_y^n X(s)|^p \leq C_0^p p^{p\iota}d^p_X(t,s),
\end{equation}
where $C_0>0$ is a generic fixed constant and $\iota\geq 0$ is a given parameter.
\end{assumption}
\begin{theorem}
\label{thm:sufficient-field}
Suppose that $X=(X_t)_{t\in [0,1]^n}$ is continuous and satisfies Assumption \ref{assu:hyper-field} and suppose that $d_X(t,s) \leq \prod_{j=1}^n\rho_j(|t_j-s_j|)$. Then
\begin{align*}
|\Box^n_t X(s)| \leq 8^{n} \int_0^{|s_1 - t_1|}\ldots \int_0^{|s_n - t_n|}\beta^{-\iota}\left(\log\left(\frac{4^nB}{u_1^2\ldots u_n^2}\right)\right)^{\iota}d\rho_1(u_1)\ldots d\rho_n(u_n)
\end{align*}
for any $\beta>0$,
$$
B = \int_{[0,1]^n} \int_{[0,1]^n} \exp\left(\beta\left(\frac{|\Box^n_t X(s)|}{\prod_{j=1}^n\rho_j(|t_j-s_j|)}\right)^{\frac{1}{\iota}}\right)dsdt.
$$
Moreover, for $\beta \in \left(0,\frac{e\iota}{C_0^{\frac{1}{\iota}}}\right)$, the random variable $B$ has finite $p$-moments for all $p$ satisfying
$$
1 \leq p < \frac{e\iota}{\beta C_0^{\frac{1}{\iota}}}.
$$
\end{theorem}
The following corollary is analogous to Corollary \ref{cor:holder-process}, and extends some of the results in \cite{hu-le2013} beyond Gaussianity.
\begin{corollary}
\label{cor:holder-field}
Suppose that $X=(X_t)_{t\in [0,1]^n}$ satisfies \ref{assu:hyper-field} and suppose that $d^2_X(t,s) \leq \prod_{j=1}^n|t_j-s_j|^{2\alpha_j}$. Let $\beta \in \left(0,\frac{e\iota}{C_0^{\frac{1}{\iota}}}\right)$. Then there exists a random variable $C(\omega)=C(\beta,\omega)$ satisfying $\mathbb{E} \exp(\beta_0 C^{1/\iota}(\omega))<\infty$ for any $\beta_0$ satisfying
\begin{equation}
\label{eq:beta_0-field}
\beta_0 < \frac{e\iota}{\left(8^nC_0\cdot 3^{\max(\iota-1,0)}\right)^{\frac{1}{\iota}}}
\end{equation}
and a deterministic constant $C_{d}=C_d(\beta)$, such that
\begin{align*}
|\Box^n_t X(s)| \leq C(\omega)\prod_{j=1}^n|t_j-s_j|^{\alpha_j} + C_{d}\prod_{j=1}^n|t_j-s_j|^{\alpha_j}\left(\log \frac{1}{\prod_{j=1}^n|t_j-s_j|}\right)^{\iota}.
\end{align*}
In particular, we have
\begin{equation*}
\limsup_{\max_j|t_j-s_j|\to 0} \frac{|\Box^n_t X(s)|}{\prod_{j=1}^n|t_j-s_j|^{\alpha_j}\left(\log \frac{1}{\prod_{j=1}^n|t_j-s_j|}\right)^{\iota}} \leq C_{d}.
\end{equation*}
\end{corollary}
\begin{corollary} \label{cor:sufficient-field}
Suppose that $X=(X_t)_{t\in [0,1]^n}$ satisfies \ref{assu:hyper-field} and suppose that $d^2_X(t,s) \leq \prod_{j=1}^n|t_j-s_j|^{2\alpha_j}$. Let $\beta \in \left(0,\frac{e\iota}{C_0^{\frac{1}{\iota}}}\right)$. Then for any intervals $I_j \subset [0,1]$ and any $s \in I = I_1 \times \ldots I_n$ we have
\begin{align*}
P\left(\sup_{t \in I} |\Box^n_t X(s)| \geq u\prod_{j=1}^n \abs{I_j}^{\alpha_j} + \tilde{C}\right) \leq C(\beta_0) e^{-\beta_0 u^{1/\iota} },
\end{align*}
where $$C(\beta_0) = C(\beta_0, \beta) = \E4B^{\beta_0 \beta^{-1}\left(8 \cdot 3^{\max(\iota-1,0})\right)^{1/\iota} } < \infty$$ for any $\beta_0$ satisfying \eqref{eq:beta_0-field} and
$$
\tilde{C} = C_d\max_{0\leq x_j\leq 1}\prod_{j=1}^n |x_j|^{\alpha_j}\left(\log \frac{1}{\prod_{j=1}^n |x_j|}\right)^{\iota}.
$$
\end{corollary}
Similarly, Theorem \ref{thm:holder-process-iff} extends in a natural manner to the multiparameter case. Again, the proof is analogous to the proof of Theorem \ref{thm:holder-process-iff} and is thus left to the reader.
\begin{theorem}
\label{thm:holder-field-iff}
Suppose that $X=(X_t)_{t\in [0,1]^n}$ satisfies Assumption \ref{assu:hyper-field}. Then $X$ is jointly H\"older continuous of any order $\gamma = (\gamma_1,\ldots,\gamma_n)$ with $\gamma_j<\alpha_j$, i.e. for any $\epsilon =(\epsilon_1, ..., \epsilon_n)$ with $\epsilon_i > 0$,
$$
|\Box^n_t X(s)| \leq C(\omega)\prod_{j=1}^n |t_j-s_j|^{\gamma_j-\epsilon_j},
$$
if and only if for any $\epsilon =(\epsilon_1, ..., \epsilon_n)$ with $\epsilon_i > 0$ we have
$$
d^2_X(t,s) \leq C_\epsilon\prod_{j=1}^n|t_j-s_j|^{2\alpha_j-\epsilon_j}.
$$
Moreover, in this case the H\"older constant of $X$ satisfies
\begin{equation*}
\mathbb{E} \exp\left(\beta C_\epsilon(\omega)^{\frac{1}{\iota}}\right) < \infty
\end{equation*}
for small enough $\beta>0$ which depends only on $C_0$, $\alpha$, $\iota$, and $\epsilon$.
\end{theorem}
\section{Proofs}
\label{sec:proofs}
Our results in Section \ref{sec:process} are based on the following Garsia-Rodemich-Rumsey inequality \cite{Garsia-Rodemich-Rumsey-1970} and its multiparameter extension \cite{hu-le2013}.
\begin{proposition}
\label{prop:GRR}
Let $\Psi(u)$ be a non-negative, even function on $(- \infty, \infty)$ and $p(u)$ be a non-negative, even function on $[-1,1]$. Assume both $p(u)$ and $\Psi(u)$ are non-decreasing for $u \geq 0$, and $p$ is continuous. Moreover, assume that $\lim_{x \to \infty} \Psi(x) = \infty$ and $p(0)=0$. Let $f(x)$ be continuous on $[0, 1]$ and suppose that
\begin{align}
\notag \int_0^1 \int_0^1 \Psi\left( \frac{f(x) - f(y) }{p(x - y)} \right)dx dy \leq B < \infty.
\end{align}
Then, for all $s, t \in [0, 1]$,
\begin{align}
\notag |f(s) - f(t) | \leq 8 \int_0^{|s - t|} \Psi^{-1} \left( \frac{4B}{u^2} \right) dp(u).
\end{align}
\end{proposition}
As an immediate consequence we obtain the Sobolev embedding theorem in the one-dimensional case.
\begin{corollary}
\label{cor:sobolev-embedding}
For any continuous function $f$ on $[0,1]$ we have
$$
|f(t)-f(s)| \leq C_{\alpha,p}|t-s|^{\alpha p -1} \left(\int_0^1\int_0^1 \frac{|f(x)-f(y)|^p}{|x-y|^{\alpha p +1}}dxdy\right)^{\frac{1}{p}}.
$$
\end{corollary}
The following multiparameter version of Proposition \ref{prop:GRR} was proved in \cite{hu-le2013}.
\begin{proposition}[Theorem 2.3 \cite{hu-le2013}]
\label{prop:GRR-field}
Let $\Psi(u)$ be a non-negative even function on $(- \infty, \infty)$ and $p_j(u),j=1,\ldots,n$ be non-negative even functions on $[-1,1]$. Assume that all $p_j(u),j=1,\ldots,n$ and $\Psi(u)$ are non-decreasing for $u \geq 0$, and that the functions $p_j$ are continuous. Moreover, assume that $\lim_{x \to \infty} \Psi(x) = \infty$ and $p(0)=0$. Let $f$ be a continuous function on $[0,1]^n$ and suppose that
$$
B:=\int_{[0,1]^n}\int_{[0,1]^n} \Psi\left(\frac{|\Box_y^n f(x)|}{\prod_{j=1}^n \rho_j(|x_k-y_k|)}\right)dxdy < \infty.
$$
Then
$$
|\Box_y^n f(x)| \leq 8^n \int_{0}^{|s_1-t_1|}\ldots \int_0^{|s_n-t_n|} \Psi^{-1}\left(\frac{4^nB}{u_1^2\ldots u_n^2}\right)d\rho_1(u_1)\ldots d\rho_n(u_n).
$$
\end{proposition}
\begin{proof}[Proof of Theorem \ref{thm:sufficient-process}]
The bound \eqref{eq:continuity-general-process} follows directly from Proposition \ref{prop:GRR} with the choice $\Psi(x) = \exp\left(\beta |x|^{\frac{1}{\iota}}\right)$, which has the inverse $\Psi^{-1}(x) = \beta^{-\iota}\left(\log x\right)^{-\iota}$, and hence it suffices to prove the claim on the moments. Using Minkowski's integral inequality, we have, for any $p \geq 1,$
$$
\mathbb{E} B^p \leq \left[\int_0^1\int_0^1 \left(\mathbb{E} \exp\left(\beta p\left(\frac{|X_t-X_s|}{\rho(|t-s|)}\right)^{\frac{1}{\iota}}\right)\right)^{\frac{1}{p}}dsdt\right]^p,
$$
where, by \eqref{eq:hyper-process}, we have
\begin{align*}
\mathbb{E}\exp\left(\beta p \left(\frac{|X_t-X_s|}{\rho(|t-s|)}\right)^{\frac{1}{\iota}}\right) &= \mathbb{E} \sum_{k=0}^\infty \frac{(\beta p)^k}{k!}\left(\frac{|X_t-X_s|}{\rho(|t-s|)}\right)^{\frac{k}{\iota}}
\leq \sum_{k=0}^\infty \frac{(\beta p)^k}{k!}C_0^{\frac{k}{\iota}}\left(\frac{k}{\iota}\right)^{k}\\
=&\sum_{k=0}^\infty\left(\frac{\beta pC_0^{\frac{1}{\iota}}}{e\iota}\right)^k e^k \frac{k^k}{k!}.
\end{align*}
From Stirling's approximation we get $k! = \Gamma(k+1) \sim \sqrt{2\pi k} k^ke^{-k}$ for large $k$, allowing us to deduce that
\begin{align} \label{eq:exp-moments-finite}
\sum_{k=0}^\infty\left(\frac{\beta pC_0^{\frac{1}{\iota}}}{e\iota}\right)^k e^k \frac{k^k}{k!}
\leq c \sum_{k=0}^\infty \left(\frac{\beta pC_0^{\frac{1}{\iota}}}{e\iota}\right)^k
= \frac{c}{1-\frac{\beta pC_0^{\frac{1}{\iota}}}{e\iota}} < \infty,
\end{align}
since by assumption
$
\frac{\beta pC_0^{\frac{1}{\iota}}}{e\iota} < 1.
$
This completes the proof.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:holder-process}]
The continuity of $X$ follows from Assumption \ref{assu:hyper-basic} together with $\mathbb{E}(X_t-X_s)^2 \leq |t-s|^{2\alpha}$ and the classical Kolmogorov continuity criterion. Now from Theorem \ref{thm:sufficient-process} we get
\begin{align*}
|X_t - X_s | &\leq 8\alpha \int_0^{|s - t|} \beta^{-\iota}\left(\log\left(\frac{4B}{u^2}\right)\right)^{\iota} u^{\alpha-1}du \\
&= 8\alpha\beta^{-\iota}|t-s|^\alpha\int_0^{1} \left(\log\left(\frac{4B}{|t-s|^2v^2}\right)\right)^{\iota}v^{\alpha-1}dv\\
&= 8\alpha\beta^{-\iota}|t-s|^\alpha\int_0^{1} \left(\log (4B)+2\log v^{-1} + 2\log |t-s|^{-1}\right)^{\iota}v^{\alpha-1}dv \\
&\leq 8\alpha\cdot 3^{\max(\iota-1,0)}\beta^{-\iota}|t-s|^\alpha \int_0^1 \big[\left(\log\max(4B,1)\right)^{\iota} + \left(2\log v^{-1}\right)^{\iota} \\ &\quad + \left(2\log |t-s|^{-1}\right)^{\iota} \big]v^{\alpha-1}dv,
\end{align*}
where we have used Jensen's inequality to obtain the last inequality. Hence we may set
$$
C(\omega) = 8\cdot 3^{\max(\iota-1,0)}\beta^{-\iota}\left(\log \max(4B(\omega),1)\right)^{\iota}
$$
and
$$
C_d = 8\alpha\cdot 2^\iota\cdot 3^{\max(\iota-1,0)}\beta^{-\iota}\left[\alpha^{-1}+\int_0^1 \left(\log v^{-1}\right)^\iota v^{\alpha-1}dv\right]
$$
to obtain \eqref{eq:holder-process}.
For the claim $\mathbb{E}\exp(\beta_0 C(\omega)^{1/\iota})<\infty$, we retrace the arguments at the end of the proof of Theorem \ref{thm:sufficient-process}, and we obtain on the set $\{B>1/4\}$ that
\begin{equation}\label{eq:exponential-moments-B}
\exp(\beta_0 C(\omega)^{1/\iota})
= 4 B(\omega)^{\beta_0\beta^{-1}\left(8\cdot 3^{\max(\iota-1,0)}\right)^{\frac{1}{\iota}}}
\end{equation}
which has finite moments provided that (see \eqref{eq:exp-moments-finite})
$$
\beta_0\beta^{-1}\left(8\cdot 3^{\max(\iota-1,0)}\right)^{\frac{1}{\iota}} \leq p < \frac{e\iota}{\beta C_0^{1/\iota}}.
$$
This translates into \eqref{eq:beta_0-process}.
Finally, \eqref{eq:lil} follows directly from \eqref{eq:holder-process} completing the proof.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:sufficient-process}]
By Corollary \ref{cor:holder-process} we have, for every $s,t\in I$, that
$$
|X_t-X_s| \leq C(\omega)|I|^\alpha + C_d\sup_{0\leq x\leq |I|}x^\alpha \left(\log 1/x\right)^\iota \leq C(\omega)|I|^\alpha + C_d e^{-\iota}\left(\frac{\iota}{\alpha}\right)^\iota,
$$
where we have used the fact that the function
$$
f(x) = x^\alpha (-\log x)^\iota
$$
attains its maximum in $[0,1]$ at $x^* = e^{-\iota}\left(\frac{\iota}{\alpha} \right)^\iota$. Applying also Chebyshev's inequality, it follows that
\begin{align*}
P \Bigg(\sup_{t \in I} \abs{X_t-X_s} \geq u|I|^\alpha + C_d e^{-\iota}\left(\frac{\iota}{\alpha}\right)^\iota \Bigg)
&\leq
P\left( C(\omega) \geq u \right) \\
&\leq
e^{-\beta_0 u^{1/\iota} } \mathbb{E}{e^{\beta_0 C(\omega)^{1/\iota}}} \\
&=
C(\beta_0) e^{-\beta_0 u^{1/\iota}},
\end{align*}
where from \eqref{eq:exponential-moments-B} we set $C(\beta_0) = C(\beta_0,\beta) = \mathbb{E}[4B^{\beta_0 \beta^{-1}\left(8 \cdot 3^{\max(\iota-1,0)}\right)^{1/\iota} } ]<\infty$ for any $\beta_0$ satisfying \eqref{eq:beta_0-process}. This completes the proof.
\end{proof}
Before the proof of Theorem \ref{thm:holder-process-iff}, we need two additional lemmas. The first one is the well-known Paley-Zygmund inequality.
\begin{lemma}[Paley-Zygmund]
Let \(X\) be a non-negative random variable with finite variance, and \(\theta \in [0,1]\). Then
\begin{align}\label{eq:p-z}
P(X > \theta \, \mathbb{E}X) \geq (1-\theta)^2 \frac{[EX]^2}{EX^2}.
\end{align}
\end{lemma}
\begin{lemma}
\label{lem:lemma1-vastine}
Let \((F_i)_{i \in I}\) be a tight collection of non-negative random variables satisfying $\mathbb{E} F_i^4 \leq C(\mathbb{E} F_i^2)^2$ for all $i\in I$, where $C$ is independent of $i$. Then
$$
\sup_{i\in I} \mathbb{E} F_i^2 < \infty.
$$
\end{lemma}
\begin{proof}
We choose \(X = F_i^2, \, \theta = \frac{1}{2}\) in \eqref{eq:p-z} to obtain
\begin{align*}
P(F_i^2 > \mathbb{E}F_i^2/2) \geq \frac{1}{4} \frac{[\mathbb{E} F_i^2]^2}{\mathbb{E} F_i^4} \geq \frac{1}{4C}.
\end{align*}
By assumption the collection \((F_i)_i\) is tight, and thus for any \(\varepsilon>0\) there exists a constant \(K_\varepsilon>0\) such that for all \(i \in I\) we have \(P(\lvert F_i \rvert > K_\varepsilon) < \varepsilon\). Choosing \(\varepsilon = \frac{1}{4C}.\) leads to
\[
P(\lvert F_i \rvert > K_\varepsilon) < \frac{1}{4C} \leq P(F_i^2 > \mathbb{E}F_i^2/2),
\]
and hence \(\mathbb{E}F_i^2 < 2 K^2_\varepsilon\). Here by the tightness of the collection \((F_i)_{i \in I}\), the constant \(K_\varepsilon\) is uniform in \(i\), and hence it follows that \(\sup_{i \in I} \mathbb{E}F_i^2 < \infty, \) completing the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:holder-process-iff}]
Assuming \eqref{eq:holder-iff} and since $\epsilon>0$ is arbitrary, the H\"older continuity of any order $\gamma<\alpha$ follows directly from the Kolmogorov continuity criterion and Assumption \ref{assu:hyper-basic}. For the other direction, set
$$
F_{s,t} = \frac{|X_t-X_s|}{|t-s|^{\alpha-\epsilon}}.
$$
As $F_{s,t}$ is a tight collection by H\"older continuity and satisfies $\mathbb{E} F^4_{s,t} \leq c \left[\mathbb{E} F^2_{s,t}\right]^2$, it follows from Lemma \ref{lem:lemma1-vastine} that then
$$
\sup_{s,t} \mathbb{E} F_{s,t}^2 < \infty.
$$
That is, $\mathbb{E}(X_t-X_s)^2 \leq C_\epsilon |t-s|^{2H-2\epsilon}$. Hence it remains to prove the existence of moments. By H\"older continuity, we have
$$
|X_t-X_s| \leq C_\epsilon(\omega)|t-s|^{\alpha - \epsilon}.
$$
Note that the constant $C_\varepsilon(\omega)$ depends also on the H\"older index $\alpha$ but this dependence is omitted in the notation for simplicity. On the other hand, using Corollary \ref{cor:sobolev-embedding} we obtain
$$
|X_t - X_s| \leq C_{\alpha,p}|t-s|^{\gamma - 1/p}\left(\int_0^1 \int_0^1 \frac{|X_u-X_v|^p}{|u-v|^{1+\gamma p}}dudv\right)^{\frac{1}{p}}
$$
for any $\gamma$ and $p$ such that $\gamma p > 1$. By choosing $\gamma = \alpha - \frac{\epsilon}{2}$ and $p = \frac{2}{\epsilon}$ allows us to choose
$$
C_\epsilon(\omega) = C_\epsilon\left(\int_0^1 \int_0^1 \frac{|X_u-X_v|^{2/\epsilon}}{|u-v|^{2\alpha / \epsilon}}dudv\right)^{\frac{\epsilon}{2}}.
$$
Moreover, by \eqref{eq:holder-iff} we have, for any $\delta$ that
$$
\mathbb{E} (X_t -X_s)^2 \leq C_\delta|t-s|^{2H-\delta}.
$$
Arguing as in \cite{lauri2014} leads to, for every $q\geq \frac{\epsilon}{2}$ and $\delta < \frac{\epsilon}{2}$,
$$
\mathbb{E} C^q_{\epsilon}(\omega) \leq 2C_0^q q^{q\iota}C_\delta \left(\frac{\epsilon}{2\delta}\right)^{\frac{q\epsilon}{2}}\left(1-\frac{\epsilon}{2\delta}\right)^{\frac{q\epsilon}{2}}.
$$
By expanding the exponential as in the proof of Theorem \ref{thm:sufficient-process} and retracing the argument of said proof, we finally obtain \eqref{eq:exp-moments} for sufficiently small $\beta$ (depending on the chosen $\delta$ and $\epsilon$). This completes the whole proof.
\end{proof}
We are now ready to present proofs for our results in the case of fields. As they follow essentially from the same arguments, we only sketch some essential arguments that are required, while the details are left to the reader.
\begin{proof}[Proof of Proposition \ref{prop:field-holder}]
By \cite[p. 564 Eq. (8.8)]{hitchhiker}, we have
$$
|X_t-X_s| \leq C|t-s|^{\frac{\gamma p - n}{p}} \left(\int_{[0,1]^n}\int_{[0,1]^n} \frac{|X_u-X_v|^p}{|u-v|^{d+\gamma p}}dudv\right)^{\frac{1}{p}}
$$
provided that $\gamma p > n$. The claim follows from this by using the same arguments as in the proof of Theorem \ref{thm:holder-process-iff}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:sufficient-field}]
The proof follows analogously to the proof of Theorem \ref{thm:sufficient-process} by applying the multiparameter Garsia-Rodemich Rumsey inequality, Proposition \ref{prop:GRR-field}. We leave the details to the reader.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:holder-field}]
Following the proof of Corollary \ref{cor:holder-process} we obtain that we may choose
$$
C(\omega) = 8^n\cdot 3^{\max(\iota-1,0)}\beta^{-\iota}\left(\log \max(4^n B(\omega),1)\right)^{\iota}
$$
and
$$
C_{d} = 8^n\prod_{j=1}^n\alpha_j\cdot 2^\iota\cdot 3^{\max(\iota-1,0)}\beta^{-\iota}\left[\prod_{j=1}^n\alpha_j^{-1}+\int_{[0,1]^n} \left(\sum_{j=1}^n\log v_j^{-1}\right)^\iota \prod_{j=1}^nv_j^{\alpha_j-1}dv_1\ldots dv_n\right].
$$
The rest of the proof goes analogously to the proof of Corollary \ref{cor:holder-process}.
\end{proof}
The proofs of Corollary \ref{cor:sufficient-field} and Theorem \ref{thm:holder-field-iff} follow now analogously to the proofs of their one parameter counterparts. For this reason we omit the details.
|
2104.06407
|
\chapter{Preface}
\chapter*{Acknowledgements}
The famous saying ``no man is an island'' is doubly-true in Mathematics,
and indeed I've had the good fortune to know and learn from many interesting people,
concerning the contents of this book. Special thanks goes to Ricardo Diaz, my first collaborator along these topics.
I would like to thank the following people, from the bottom of my heart, for their valuable input and interesting
discussions about some of these topics over the years:
Ian Alevy, Artur~Andr\'e, Christine~Bachoc, Tamar~Bar,
Imre~B\'ar\'any, Alexander~Barvinok, Matthias~Beck, Dori~Bejleri,
Luca~Brandolini, Michel~Brion, Henry~Cohn, Leonardo~Colzani, Amalia~Culiuc, Pierre~Deligne,
Jes\'us A. De Loera, Michel~Faleiros, Lenny~Fukshansky, Nick~Gravin,
Martin~Henk, Didier~Henrion, Roberto Hirata Junior, Jeffrey~Hoffstein, \\
Alex~Iosevich, Michael Joswig,
Marvin~Knopp, Mihalis~Kolountzakis, Matthias~K\"oppe, Greg~Kuperberg,
Jean~Bernard~Lasserre,
Nhat~Le~Quang, Rafael~Zuolo~Coppini~Lima, Sameer~Iyer,
Fabr\'icio~Caluza~Machado,
Romanos~Malikiosis, M\'at\'e Matolci, \\
Tyrrell~McAllister,
Victor Moll, Mel~Nathanson,
James~Pommersheim, Jim~Propp, \\
Thales~Paiva, Jill~Pipher, Jorge Luis Ram\'irez Alfons\'in, Ethan~Reiner, Bruce~Reznick,
Tiago~Royer, Nicolas~Salter,
Gerv\'asio~Santos, Richard~Schwartz,
Dima~Shiryaev, \\
Joseph~Silverman, Richard~Stanley, Christophe~Vignat, Sergei~Tabachnikov, \\
Giancarlo~Travaglini, Kevin~Woods, Ren~Yi,
G\"unter~Ziegler, Chuanming~Zong.
\mainmatter
\chapter{\blue{Introduction} }
\begin{wrapfigure}{R}{0.4\textwidth}
\centering
\includegraphics[width=0.34\textwidth]{Fourier1}
\caption{Joseph Fourier} \label{Fourier}
\end{wrapfigure}
What is a Fourier transform? Why is it so useful? How can we apply Fourier transforms and Fourier series - which were originally used by Fourier to study heat diffusion - in order to better understand topics in
discrete and combinatorial geometry, number theory, and sampling theory?
To begin, there are some useful analogies: imagine that you are drinking a milk-shake (lactose-free), and you want to know the ingredients of your tasty drink. You would need to filter out the shake into some of its most basic components. This decomposition into its basic ingredients may be thought of as a sort of
``Fourier transform of the milk-shake''. Once we understand each of the ingredients, we will also be able to
restructure these ingredients in new ways, to form many other types of tasty goodies. To move the analogy back into mathematical language, the milkshake represents a function, and each of its basic ingredients represents for us the basis of sines and cosines; we may also think of a basic ingredient more compactly as a complex exponential $e^{2\pi i nx}$, for some $n\in \mathbb{Z}$. Composing these basic ingredients together in a new way represents a Fourier series.
Mathematically, one of the most basic kinds of milk-shakes is the indicator function of the unit interval, and to break it down into its basic components, mathematicians, Engineers, Computer scientists, and Physicists have used the
sinc function (since the $1800$'s):
\[
{\rm{sinc}}(z):= \frac{\sin(\pi z)}{\pi z}
\]
with great success, because it happens to be the Fourier transform of the unit interval
$[-\frac{1}{2}, \frac{1}{2}]$:
\[
\int_{-\frac{1}{2}}^\frac{1}{2} e^{-2\pi i z x} dx = {\rm{sinc}}(z),
\]
as we will compute shortly in identity \eqref{sinc function formula}.
Somewhat surprisingly, comparatively little energy has been given to some of its higher dimensional
extensions, namely those extensions that arise naturally as Fourier transforms of polytopes.
One motivation for this book is to better understand how this $1$-dimensional function -- which has proved to be extremely powerful in applications -- extends to higher dimensions. Namely, we will build various mathematical structures that are motivated by the question:
\[
\text{ {\bf
What is the Fourier transform of a polytope}? }
\]
Of course, we will ask ``how can we apply it"? An alternate title for this book might have been:
\centerline{ {\bf We're taking Poisson summation and Fourier transforms of polytopes}}
\centerline{ {\bf for a very long ride....}}
Historically, sinc functions were used by Shannon (as well as Hardy, Kotelnikov, and Whittaker) when he published his seminal work on sampling theory and information theory.
In the first part of this book, we will learn how to use the technology of Fourier transforms of polytopes in order to build the (Ehrhart) theory of integer point enumeration in polytopes, to prove some of Minkowski's theorems in the geometry of numbers, and to understand when a polytope tiles Euclidean space by translations.
In the second portion of this book, we give some applications to active research areas which are sometimes considered more applied, including
the sphere-packing problem, and the sampling of signals in higher dimensions.
There are also current research developments of the material developed here, to
the learning of deep neural networks.
In many applied scientific areas, in particular radio astronomy, computational tomography, and magnetic resonance imaging, a frequent theme is the reconstruction of a function from knowledge of its Fourier transform. Somewhat surprisingly,
in various applications we only require very partial/sparse knowledge of its Fourier transform in order to reconstruct the required function, which may represent an image or a signal.
There is a rapidly increasing amount of research focused in these directions in recent years, and it is therefore time to put some of these new findings in one place, making them accessible to a general scientific reader.
The fact that the sinc function is indeed the Fourier transform of the $1$-dimensional line segment
$[-\frac{1}{2}, \frac{1}{2}]$, which is a $1$-dimensional polytope, \index{polytope}
gives us a first hint that
there is a deeper link between the geometry of a polytope and the analysis of its Fourier transform.
Indeed one reason that sampling and information theory, as initiated by Claude Shannon, \index{Shannon, Claude}
works so well is precisely because the Fourier transform of the unit interval has this nice form, and even more-so because of the existence of the Poisson summation formula.
The approach we take here is to gain insight into how the Fourier transform of a polytope \index{polytope}
can be used to solve various specific problems in discrete geometry, combinatorics, optimization, approximation theory, and the Shannon-Whittaker sampling theory in higher dimensions:
\begin{enumerate}[(a)]
\item Analyze tilings of Euclidean space by translations of a polytope
\item Give wonderful formulas for volumes of polytopes
\item Compute discrete volumes of polytopes, which are combinatorial approximations to the continuous volume
\item Introduce the geometry of numbers, via Poisson summation
\item Optimize sphere packings, and get bounds on their optimal densities
\item Study the Shannon-Whittaker sampling theorem and its higher-dimensional siblings
\item Recover a polytope by the inverse problem of knowing enough of its moments
\end{enumerate}
\medskip
Let's see at least one direction that quickly motivates the study of Fourier transforms. In particular, we often begin with
simple-sounding problems that arise naturally in combinatorial enumeration, discrete and computational geometry, and number theory.
Throughout, an {\bf integer point} \index{integer point} is any vector $v:=(v_1, \dots, v_d)\in \mathbb{R}^d$, all of whose coordinates $v_j$ are integers. In other words, $v$ belongs to the integer lattice $\mathbb{Z}^d$.
A {\bf rational point}
\index{rational point} is a point $m$ whose coordinates are rational numbers, in other words
$m \in \mathbb{Q}^d$.
We define the {\bf Fourier transform} of a function $f(x)$:
\begin{align} \label{Fourier transform} \index{Fourier transform}
\hat f(\xi) := \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle \xi, x \rangle} dx,
\end{align}
defined for all $\xi \in \mathbb{R}^d$ for which the latter integral converges, and where we use the standard inner product
$\langle a, b \rangle:= a_1 b_1 + \cdots + a_d b_d$. We will also use the notation
${\mathcal F}(f)$ for the Fourier transform of $f$, which is useful in some typographical contexts, for example when considering ${\mathcal F}^{-1}(f)$.
We introduce one of the main objects of study in this book, the {\bf Fourier transform of a polytope}
\index{Fourier transform of a polytope}
${\mathcal P}$, defined by:
\begin{align} \label{Fourier transform of P}
\hat 1_{\mathcal P}(\xi)
:= \int_{\mathbb{R}^d} 1_{\mathcal P}(x) e^{-2\pi i \langle \xi, x \rangle} dx = \int_{{\mathcal P}} e^{-2\pi i \langle \xi, x \rangle} dx,
\end{align}
where the function $1_{\mathcal P}(x)$ is the {\bf indicator function} of ${\mathcal P}$, defined by
\[
1_{\mathcal P}(x):=
\begin{cases}
1 & \mbox{if } x\in {\mathcal P} \\
0 & \mbox{if not}.
\end{cases}
\]
Thus, the words ``Fourier transform of a polytope ${\mathcal P}$'' will always mean the Fourier transform of the
indicator function \index{indicator function} of ${\mathcal P}$.
\begin{wrapfigure}{R}{0.4\textwidth}
\centering
\includegraphics[width=0.32\textwidth]{Poisson2}
\caption{Sim\'eon Denis Poisson} \label{PoissonHimself}
\end{wrapfigure}
The {\bf Poisson summation formula}, named after Sim\'eon Denis Poisson,
\index{Poisson summation formula}
tells us that for any ``sufficiently nice"
function $f : \mathbb{R}^d \rightarrow \mathbb{C}$ we have:
\[
\sum_{n \in \mathbb{Z}^d} f(n) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi).
\]
In particular, if we were to naively set $f(n) := 1_{{\mathcal P}}(n)$, the indicator function of a polytope ${\mathcal P}$, then we would get:
\begin{align} \label{PoissonSummation1}
\sum_{n \in \mathbb{Z}^d} 1_{{\mathcal P}}(n) = \sum_{\xi \in \mathbb{Z}^d} \hat 1_{{\mathcal P}}(\xi),
\end{align}
which is technically false in general due to the fact that the indicator function $1_{\mathcal P}$ is a discontinuous function on $\mathbb{R}^d$.
However, this technically false statement is very useful! We make this claim because it helps us build intuition for the more rigorous statements that are true, and which we study in later chapters.
For applications to discrete geometry, we are interested in the number of integer points in a closed convex polytope ${\mathcal P}$, namely $|{\mathcal P} \cap \mathbb{Z}^d|$.
The combinatorial-geometric quantity $|{\mathcal P} \cap \mathbb{Z}^d|$
may be regarded as a {\bf discrete volume}
\index{discrete volume}
for ${\mathcal P}$.
From the definition of the indicator function of a polytope, the left-hand-side of \eqref{PoissonSummation1}
counts the number of integer points in ${\mathcal P}$, namely
we have by definition
\begin{equation}
\sum_{n \in \mathbb{Z}^d} 1_{{\mathcal P}}(n) = |{\mathcal P} \cap \mathbb{Z}^d|.
\end{equation}
On the other hand, the right-hand-side of \eqref{PoissonSummation1} allows us to compute this discrete volume of ${\mathcal P}$ in a new way. This is great, because it opens a wonderful window of computation for us in the following sense:
\begin{align} \label{PoissonSummation2}
|{\mathcal P} \cap \mathbb{Z}^d| = \sum_{\xi \in \mathbb{Z}^d} \hat 1_{{\mathcal P}}(\xi).
\end{align}
We notice that for the $\xi = 0$ term, we have
\begin{align} \label{Fourier transform at 0}
\hat 1_{\mathcal P}(0) := \int_{\mathbb{R}^d} 1_{{\mathcal P}}(x) e^{-2\pi i \langle 0, x \rangle} dx =
\int_{{\mathcal P}} dx = \vol({\mathcal P}),
\end{align}
and therefore the {\bf discrepancy} \index{discrepancy}
between the continuous volume of ${\mathcal P}$ and the discrete volume of ${\mathcal P}$ is
\begin{align} \label{PoissonSummation3}
|{\mathcal P} \cap \mathbb{Z}^d| - \vol({\mathcal P}) = \sum_{\xi \in \mathbb{Z}^d-\{0\}} \hat 1_{\mathcal P}(\xi),
\end{align}
showing us very quickly that indeed $|{\mathcal P} \cap \mathbb{Z}^d|$ is a discrete approximation to the classical Lebesgue volume $\vol({\mathcal P})$, and pointing us to the task of finding ways to evaluate the transform $\hat 1_P(\xi)$.
From the trivial but often very useful identity
\[
\hat 1_{\mathcal P}(0) = \vol({\mathcal P}),
\]
we see another important motivation for this book: the Fourier transform of a polytope is a very {\bf natural extension of volume}.
\index{volume}
Computing the volume of a polytope ${\mathcal P}$ captures a bit of information about ${\mathcal P}$, but we also lose a lot of information.
On the other hand, computing the Fourier transform of a polytope
$\hat 1_{\mathcal P}(\xi)$ uniquely determines ${\mathcal P}$, so we do not lose any information at all. Another way of saying this is that the Fourier transform of a polytope is a {\bf complete invariant}.
\index{complete invariant}
In other words,
it is a fact of life that
\[
\hat 1_{\mathcal P}(\xi) = \hat 1_{\mathcal Q}(\xi) \text{ for all } \xi \in \mathbb{R}^d \ \iff \ {\mathcal P} = \mathcal Q.
\]
Combinatorially, there are brilliant identities (notably the Brion identities) that emerge between
the Fourier and Laplace transforms of a given polytope, and its facets and vertex tangent cones.
In Statistics, the moment generating function of any probability distribution is given by a Fourier transform of the indicator function of the distribution, hence Fourier transforms arise very naturally in Statistical applications.
At this point, a natural glaring question naturally comes up:
\begin{equation}
\text{ How do we {\bf compute} the Fourier transform of a polytope }
\hat 1_P(\xi)?
\end{equation}
And how do we use such computations to help us understand the important
``error'' term
\[
\sum_{\xi \in \mathbb{Z}^d-\{0\}} \hat 1_{\mathcal P}(\xi)
\]
that came up naturally in \eqref{PoissonSummation3} above?
There are many applications of the theory that we will build-up. Often, we find it instructive to
sometimes give an informal proof first, because it
brings the intuitive ideas to the
foreground, allowing the reader to gain an overview of the steps. Then, later on, we revisit the same intuitive proof again, making it rigorous.
The Poisson summation formula \index{Poisson summation formula} is one of our main stars,
and has a relatively easy proof. But it constitutes a very first step for many of our explorations. It may even be said that, from this perspective, the Poisson summation formula is to combinatorial analysis as a microscope is to our vision. It enhances our ability to see mathematical facts, and often in a surprisingly simple way. So it's a question of what we do with these tools - where do we point them?
A word about {\bf prerequisites} for this book: {\bf Linear Algebra} is always very useful!
A couple of calculus courses are required as well, with a touch of real analysis. In particular, familiarity with infinite series is assumed. We give new proofs for
some of the main theorems in this theory,
including Theorem \ref{Siegel for general lattices}, Theorem \ref{brion, continuous form}, Theorem \ref{brion2}, and Theorem \ref{brion, discrete form}. These new Fourier-type proofs help streamline the theory, unifying sporadic results in the literature. This
unifying thread will hopefully help the reader put the various results, from both the past and the present, into context.
We will assume some familiarity with the basic definitions of polytopes and their faces, although at places we will remind the reader of some of these definitions.
There are many excellent texts that introduce the student to the classical language of polytopes, in particular the two classics: G\"unter Ziegler's ``Lectures on Polytopes" \cite{Ziegler}, and Branko Gr\"unbaum's ``Convex Polytopes" \cite{Grunbaum}.
For an easy introduction to the interactions between polytopes and lattice point enumeration, the reader is invited to consult ``Computing the continuous discretely: integer point enumeration in polytopes", by Beck and Robins \cite{BeckRobins}.
The level of the current book is aimed at {\bf advanced undergraduates} and {\bf beginning graduate students} in various fields, and in particular Mathematics, Computer Science, Electrical Engineering, and Physics.
Because of the large number of exercises, with solutions to many of them in the back, this book can also be used effectively for self-study.
Finally, this book is still in draft form, and in particular Chapters 9, 10, 11, and 13 are still under revision.
We proceed by developing an intuitive understanding first, using many examples and analogies, and this intuition then points us to a rigorous path for the details of the ensuing proofs.
\bigskip
Sinai Robins \hfill December 2021
IME, University of S\~ao Paulo
\chapter{
A motivating problem: \\ tiling a rectangle with rectangles}
\label{Chapter.Tiling.A.Rectangle}
\index{Tiling a rectangle}
\begin{quote}
``Ripping up carpet is easy -- {\it tiling} is the issue''.
-- Douglas Wilson
\end{quote}
\begin{wrapfigure}{R}{0.5\textwidth}
\centering
\includegraphics[width=0.45\textwidth]{rectangle}
\caption{A rectangle tiled by nice rectangles} \label{nice rectangle}
\end{wrapfigure}
\section{Intuition}
To warm up, we begin with a simple tiling problem in the plane. A rectangle will be called
{\bf nice} if at least one of its sides is an integer.
We prove a now-classical fact about tiling a rectangle with nice rectangles, namely Theorem \ref{Integer.Side.Rectangle}, and we focus on the {\bf method} of the straightforward proof.
This proof brings to the foreground an important idea: by simply taking a Fourier transform of a body $B$, we immediately get interesting geometric consequences for $B$.
In particular, we will see throughout this book various ways in which the Fourier transform of a geometric body is a natural extension of its volume, sometimes in a continuous way, and sometimes in a discrete way.
So in order to study relationships between volumes of bodies, it is very natural and useful to play with their Fourier transforms.
\section{Nice rectangles}
The tilings that we focus on, in this small chapter, are tilings that are composed of smaller rectangles, all of which have their sides parallel to the axes, and all of which are nice.
There are at least $14$ different known proofs \cite{StanWagon} of Theorem \ref{Integer.Side.Rectangle}.
Here we give the proof that uses very basic Fourier tools, from first principles, motivating the chapters that follow.
The idea for this proof goes back to Nicolaas Govert De Bruijn \cite{DeBruijn.Book}.
\index{De Bruijn, Nicolaas Govert}
\begin{thm}[De Bruijn] \label{Integer.Side.Rectangle}
Suppose we tile a fixed rectangle $\mathcal{R}$ with smaller, nice rectangles. \\
Then $\mathcal{R}$ is a nice rectangle.
\end{thm}
\begin{proof}
Suppose that the rectangle $\mathcal{R}$ is tiled with smaller rectangles $\mathcal{R}_1, \dots, \mathcal{R}_N$, as in
Figure~\ref{nice rectangle}.
Due to our tiling
\index{tiling}
hypothesis, we have
\begin{equation} \label{first identity of rectangles}
1_{\mathcal{R}}(x) = \sum_{k=1}^N 1_{\mathcal{R}_k}(x) + \sum (\pm \text{ indicator functions of lower-dimensional polytopes}),
\end{equation}
where the notation $1_S(x)$ always means we are using indicator functions.
To ease the reader into the computations, we
recall that the Fourier transform of the indicator function of any rectangle $R:=[a, b] \times [c, d]$ is defined by:
\begin{equation}
\hat 1_{\mathcal{R}}(\xi) := \int_{\mathbb{R}^2} 1_{\mathcal{R}}(x) e^{-2\pi i \langle \xi, x \rangle} dx
=\int_a^b \int_c^d e^{-2\pi i (\xi_1 x_1 + \xi_2 x_2)}dx_1 dx_2.
\end{equation}
Now we may formally take the Fourier transform of both sides of \eqref{first identity of rectangles}.
In other words we
simply multiply both sides of \eqref{first identity of rectangles}
by the exponential function $e^{-2\pi i \langle \xi, x \rangle} $
and then integrate both sides over $\mathbb{R}^2$, to get:
\begin{equation} \label{sum.of.little.transforms}
\hat 1_{\mathcal{R}}(\xi) = \sum_{k=1}^N \hat 1_{\mathcal{R}_k}(\xi).
\end{equation}
In \eqref{sum.of.little.transforms}, we have used the fact that a $2$-dimensional integral over a $1$-dimensional line segment always vanishes, due to the fact that
a line segment has measure $0$ relative to the $2$-dimensional measure of the $2$-dimensional transform.
Let's compute one of these integrals, over a generic rectangle $\mathcal{R}_k := [a_1, a_2] \times [b_1, b_2]$:
\begin{align} \label{transform.of.a.rectangle}
\hat 1_{\mathcal{R}_k}(\xi) &:= \int_{\mathbb{R}^2} 1_{\mathcal{R}_k}(x) e^{-2\pi i \langle x, \xi \rangle} dx =
\int_{\mathcal{R}_k} e^{-2\pi i \langle x, \xi \rangle} dx \\
&= \int_{b_1}^{b_2} \int_{a_1}^{a_2} e^{-2\pi i \langle x, \xi \rangle} dx \\
&= \int_{a_1}^{a_2} e^{-2\pi i \xi_1 x_1} dx_1 \int_{b_1}^{b_2} e^{-2\pi i\xi_2 x_2} dx_2 \\
&= \frac{ e^{-2\pi i \xi_1 a_2} - e^{-2\pi i \xi_1 a_1} }{-2\pi i \xi_1}
\cdot \frac{ e^{-2\pi i \xi_2 b_2} - e^{-2\pi i \xi_2 b_1} }{-2\pi i \xi_2}\\ \label{last one}
&= \frac{1}{(-2\pi i)^2} \frac{ e^{-2\pi i (\xi_1 a_1 + \xi_2 b_1)} }{\xi_1 \xi_2}
(e^{-2\pi i \xi_1 (a_2-a_1)} - 1 ) (e^{-2\pi i \xi_2 (b_2-b_1)} -1),
\end{align}
valid for all $(\xi_1, \xi_2) \in \mathbb{R}^2$ except for the union of the two lines $\xi_1 = 0$ and $\xi_2 = 0$.
Considering the latter formula for the Fourier transform of a rectangle, we make the following leap of faith:
{\bf Claim}. \ Suppose that $\mathcal{R}$ is a rectangle whose sides are parallel to the axes. Then
\begin{equation} \label{First.case.of.Fourier.tiling.criterion}
\index{tiling}
\mathcal{R} \text{ is a nice rectangle } \iff \hat 1_{\mathcal{R}}\Big(\icol{1\{\bf 1}} \Big) = 0.
\end{equation}
Proof of the claim. \ We consider the last equality \eqref{last one}.
We see that
\begin{equation} \label{twofactors}
\hat 1_{\mathcal{R}_k}(\xi) =0 \iff (e^{-2\pi i \xi_1 (a_2-a_1)} - 1 ) (e^{-2\pi i \xi_2 (b_2-b_1)} -1)=0,
\end{equation}
which is equivalent to having either $e^{-2\pi i \xi_1 (a_2-a_1)} =1$, or $e^{-2\pi i \xi_2 (b_2-b_1)} =1$.
But we know that due to Euler,
$e^{2\pi i \theta} = 1$ if and only if $\theta \in \mathbb{Z}$ (Exercise \ref{TrivialExponential}), so we have
\begin{equation} \label{the last bit}
\hat 1_{\mathcal{R}}(\xi) = 0 \ \iff \xi_1 (a_2-a_1) \in \mathbb{Z} \ \text{ or } \ \xi_2 (b_2-b_1) \in \mathbb{Z}.
\end{equation}
Now, if $\mathcal{R}$ is a nice rectangle, then one of its sides is an integer, say $a_1 - a_2 \in \mathbb{Z}$ without loss of generality. Therefore
$\xi_1 (a_2-a_1) \in \mathbb{Z}$ for $\xi_1 = 1$, and by \eqref{the last bit}, we see that
$ \hat 1_{\mathcal{R}}\Big(\icol{1\{\bf 1}} \Big) = 0$.
Conversely, if we assume that $ \hat 1_{\mathcal{R}}\Big(\icol{1\{\bf 1}} \Big) = 0$, then by
\eqref{the last bit} either \\
$1\cdot (a_2-a_1) \in \mathbb{Z} \text{ or } 1\cdot (b_2-b_1) \in \mathbb{Z}$, proving the claim.
\medskip
To finish the proof of the theorem, by hypothesis each little rectangle $\mathcal{R}_k$ is a nice rectangle, so by the claim above it satisfies
$ \hat 1_{\mathcal{R}_k}\Big(\icol{1\{\bf 1}} \Big) = 0$.
Returning to \eqref{sum.of.little.transforms}, we see that therefore
$\hat 1_{\mathcal{R}}(\xi) = \sum_{k=1}^N \hat 1_{\mathcal{R}_k}(\xi) = 0$, for $\xi = \icol{1\{\bf 1}}$, and using the claim again (the converse part of it this time), we see that $\mathcal{R}$ must be nice.
\end{proof}
The proof of Theorem \ref{Integer.Side.Rectangle} was simple and elegant, motivating the use of Fourier transforms of polytopes in the ensuing chapters. The claim, namely equation \eqref{First.case.of.Fourier.tiling.criterion}, offers an intriguing springboard for deeper investigations - it tells us that we can convert a geometric statement about tiling into a purely analytic statement about the vanishing of a certain integral transform. Later, when we learn about
Theorem \ref{zero set of the FT of a polytope}, we will see that this small initial success of \eqref{First.case.of.Fourier.tiling.criterion} is part of a larger theory. This is the beginning of a beautiful friendship.......
\bigskip
\section{Conventions, and quick basics}
We mention some conventions that we use throughout the book.
First, we note that whenever we are given a complex-valued function $f:\mathbb{R}^d \rightarrow \mathbb{C}$, we may write $f$ in terms of its
real and imaginary parts: $f(x):= u(x) + i v(x)$. The {\bf integral of such an $f$} is defined by
\begin{equation}
\int_{\mathbb{R}^d} f(x) dx := \int_{\mathbb{R}^d} u(x) dx + i \int_{\mathbb{R}^d} v(x) dx,
\end{equation}
so that all of our Fourier transforms are really reduced to the usual integration of real-valued functions on Euclidean space
(see Exercise \ref{definition of complex integral}). This is good news for the reader, because even though we see complex functions in the integrand, elementary calculus suffices.
\medskip
Let $S\subset \mathbb{R}^d$ be a set.
For our purposes, we may call $S$ a {\bf measurable} set if the integral
$
\int_S dx \text{ exists},
$
and in this case we define
\[
{\rm measure}(S) := \int_S dx.
\]
Equivalently, we may call $S$ measurable if the indicator function $1_S$ is an integrable function, by definition of the integral.
A set $S$ is said to have {\bf measure zero} if
\[
\int_S dx = 0.
\]
In $\mathbb{R}$, for example, we may also define a set $S$ of measure $0$ by saying that, given any $\varepsilon >0$,
there exists a countable collection of open intervals $I_n$ that cover all of $S$, and whose total length satisfies
$\sum_{n=1}^\infty |I_n|< \varepsilon$. But we will assume the reader knows the definition(s) of an integral
(either the Riemann integral or the Lebesgue integral),
circumventing discussions about $\sigma$-algebras of sets, so that the background required of the reader is kept to a minimum.
The point we want to make here is that most things are in fact easier than the reader may have previously thought.
We say that a statement $A(x)$ concerning points $x\in \mathbb{R}^d$
{\bf holds for almost every} $x\in \mathbb{R}^d$ (we also use the words {\bf almost everywhere})
if the set of $x\in \mathbb{R}^d$ for which $A(x)$ is false is a set of measure $0$.
In this connection, we will assume the following fact from real analysis:
\[
\int_{\mathbb{R}^d} f(x) dx = \int_{\mathbb{R}^d} g(x) dx \ \iff \ f = g \text{ almost everywhere},
\]
which means that $f(x) = g(x)$ for all $x\in \mathbb{R}^d$, except perhaps on a set of measure $0$.
We also mention our convention/notation for some definitions. Whenever we want to define a new object called $N$, in terms of some combination of previously known mathematical objects called $K$, we will use the notation
\[
N:= K.
\]
For any set $A\subset \mathbb{R}^d$, we define the {\bf closure} of $A$ as the
the smallest (w.r.t containment) closed set that contains $A$, written as $\closure A$.
We define the {\bf interior} of $A$ as the set of all points $x \in A$ such that
there exists a ball of some positive radius $\varepsilon$, centered at $x$, with $B_\varepsilon(x) \subset A$.
We define the {\bf boundary} of $A$, written as $\partial A$, by
\[
\partial A:= \closure A \setminus \interior A.
\]
An important concept is that of the support of a function $f:\mathbb{R}^d\rightarrow \mathbb{C}$, defined by
\begin{equation} \label{def of support}
\supp(f):= \closure \{ x \in \mathbb{R}^d \bigm | f(x) \not=0 \}.
\end{equation}
With this definition, we have for example:
\[
\supp(1_{[0, 1]}) = \supp(1_{(0, 1)}) = [0, 1].
\]
We will also say that a function $f$ is {\bf compactly supported} if the support of $f$ is a
compact set $C$. In particular this means that $f$ vanishes outside of $C$.
\bigskip
\section*{Notes}
\begin{enumerate}[(a)]
\item This little chapter was motivated by the lovely article written by Stan Wagon \cite{StanWagon}, which gives $14$ different proofs of Theorem \ref{Integer.Side.Rectangle}.
The article \cite{StanWagon} is important because it shows how tools from one field can leak into another field, and thus may lead to important discoveries in the future.
\item In a related direction, we might wonder which polygons, and more generally which polytopes, tile Euclidean space by translations with a lattice. It turns out (Theorem~\ref{zero set of the FT of a polytope}) that this question is equivalent to the statement that the Fourier transform of ${\mathcal P}$ vanishes on a (dual) lattice.
\item In the context of the Hilbert space of functions $L^2([0,1])$, Exercise \ref{orthogonality for exponentials}
is one step towards showing that the set of exponentials $\{ e_n(x) \}_{n \in \mathbb{Z}}$ is a basis for $L^2([0,1])$. Namely, the identity above shows that these basis elements are orthogonal to each other - their inner product
$\langle e_a, e_b \rangle := \int_0^1 e_a(x) \overline{e_b(x)} dx $ vanishes
for integers $a \not= b$. Thus, the identity of Exercise \ref{orthogonality for exponentials}
is often called the orthogonality relations
for exponentials, over $L^2([0,1])$. To show that they {\it span} the space of functions in $L^2([0,1])$ is a bit harder, but see \cite{Travaglini} for details.
\item The question in Exercise \ref{Erdos lattice partition problem} for $\mathbb{Z}$ was originally asked by Paul Erd\"os
\index{Erd\"os, Paul} in $1951$, and has an affirmative answer. This question also has
higher-dimensional analogues:
\begin{quote}
Suppose we give a partition of the integer lattice $\mathbb{Z}^d$
into a finite, disjoint union of translated sublattices.
Is it always true that at least two of these sublattices are translates of each other?
\end{quote}
The answer is known to be false for $d \geq 3$, but is still unsolved for $d=2$ (see \cite{FeldmanProppRobins},\cite{BorodzikNguyenRobins}).
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
If there is a problem you can't solve, then there is an easier problem you can't solve: find it.
-- George Polya
\end{quote}
\medskip
\begin{prob} $\clubsuit$ \label{TrivialExponential}
Show that if $x \in \mathbb{C}$, then $e^{2\pi i x} = 1$ if and only if $x \in \mathbb{Z}$.
\end{prob}
\medskip
\begin{prob} \label{bound of the exponential function}
Show that $|e^z| \leq e^{|z|}$, for all complex numbers $z \in \mathbb{C}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{orthogonality for exponentials}
\index{orthogonality of exponentials in $L^2([0,1])$}
Here we prove the {\bf orthogonality relations for the exponential functions} defined by
$e_n(x) := e^{2\pi i n x}$, for each integer $n$.
Recall that the complex conjugate of any complex number $x + iy$ is defined by
\[
\overline{x+ iy} := x - iy,
\]
so that $\overline{e^{i \theta}} := e^{-i \theta}$ for real $\theta$.
Prove that for all integers $a,b$:
\begin{equation}
\int_0^1 e_a(x) \overline{e_b(x)} dx =
\begin{cases}
1 & \mbox{if } a=b \\
0 & \mbox{if not}.
\end{cases}
\end{equation}
\end{prob}
\medskip
\begin{prob} \label{definition of complex integral}
Here the reader may gain some practice with the definitions of integrals that use complex-valued integrands $f(x) := u(x) + iv(x)$.
We recall for the reader the following definition:
\begin{equation} \label{real.and.imaginary.parts}
\int_{\mathbb{R}^d} f(x) dx := \int_{\mathbb{R}^d} \left( u(x) + i v(x) \right) dx := \int_{\mathbb{R}^d} u(x) dx + i \int_{\mathbb{R}^d} v(x) dx,
\end{equation}
a linear combination of two real-valued integrals.
Recalling that by definition,
\[
\hat 1_{[0,1]}(\xi) := \int_{[0,1] } e^{-2\pi i \xi x} dx,
\]
show directly from definition \ref{real.and.imaginary.parts} and from Euler's identity
$e^{i\theta} = \cos \theta + i \sin \theta$,
that for any nonzero $\xi \in \mathbb{R}$, we have
\begin{equation*}
\int_{[0,1]} e^{-2\pi i \xi x} dx = \frac{e^{-2\pi i \xi } -1}{ -2\pi i \xi}.
\end{equation*}
{\rm Notes. Another way of thinking about this exercise is that it extends the `Fundamental theorem of calculus' to complex-valued functions in a rather easy way. The anti-derivative of the integrand $f(x):= e^{-2\pi i \xi x}$ is $F(x):= \frac{e^{-2\pi i \xi x}}{-2\pi i \xi}$, and we are saying that it is ok to use it in place of the usual anti-derivative in Calculus $1$ - it is consistent with definition \ref{real.and.imaginary.parts}.
In the future, we generally do not have to break up complex integrals into their real and imaginary parts,
because we can make use of the fact that antiderivatives of complex-valued functions are often simple, such as the one in this example.
We also note that this is {\bf not} calculus with a complex variable, because the {\bf domains of our integrands}, as well as the measures we are using throughout this book, in order to integrate, are always defined over real Euclidean space $\mathbb{R}^d$. This means we are still using basic Calculus.
}
\end{prob}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{6thRootsOfUnity}
\end{center}
\caption{The $6$'th roots of unity, with $\zeta:= e^{\frac{2\pi i}{6}}$. Geometrically, Exercise \ref{SumOfRootsOfUnity} tells
us that their center of mass is the origin. } \label{6th roots of unity}
\end{figure}
\medskip
\begin{prob} $\clubsuit$ \label{SumOfRootsOfUnity}
We recall that the $N$'th roots of unity are by definition the set of $N$ complex solutions to $z^N =1$, and are given by the set
$\{e^{2\pi i k/N} \mid k = 0, 1, 2, \dots, N-1 \}$ of points on the unit circle. Prove that the sum of all of the $N$'th roots of unity vanishes. Precisely, fix any positive integer $N\geq 2$,
and show that
\[
\sum_{k = 0}^{N-1} e^{\frac{2\pi i k}{N}} = 0.
\]
\end{prob}
\medskip
\begin{prob} \label{DivisibilityUsingExponentials}
Prove that, given positive integers $M, N$, we have
\[
\frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i kM}{N}} =
\begin{cases}
1 & \mbox{if } N \mid M \\
0 & \mbox{if not}.
\end{cases}
\]
\end{prob}
Notes. This result is sometimes referred to as {\bf ``the harmonic detector"} for detecting when a rational
number $\frac{M}{N}$ is an integer; that is, it assigns a value of $1$ to the sum if $\frac{M}{N} \in \mathbb{Z}$, and it assigns a value
of $0$ to the sum if
$\frac{M}{N} \not\in \mathbb{Z}$.
\medskip
\begin{prob} $\clubsuit$ \label{Orthogonality.for.roots.of.unity} \index{orthogonality, roots of unity}
Here we prove the {\bf orthogonality relations for roots of unity}. Namely, fix any two nonnegative integers $a,b$,
and prove that
\begin{equation} \label{12345}
\frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i ka}{N}} e^{-\frac{2\pi i kb}{N}} =
\begin{cases}
1 & \mbox{if } a \equiv b \mod N \\
0 & \mbox{if not}.
\end{cases}
\end{equation}
\end{prob}
Notes. In a later chapter on Euclidean lattices (Chapter \ref{chapter.lattices}), we will see that the identity
\ref{12345} is a special case of the
more general orthogonality relations for characters on lattices. From this perspective, this exercise is the orthogonality relations on the finite cyclic group $\mathbb{Z}/{N\mathbb{Z}}$. There are more general orthogonality relations
for characters of group representations, which play an important role in Number Theory.
\medskip
\begin{prob} \label{trick-write an integer as a product with roots of unity}
Show that for any positive integer $n$, we have
\[
n = \prod_{k=1}^{n-1} (1-\zeta^k),
\]
where $\zeta:= e^{2\pi i / n}$.
\end{prob}
\medskip
\begin{prob} \label{PrimitiveRootsOfUnity} \index{root of unity, primitive}
An $N$'th root of unity is called a {\bf primitive root of unity} if it is not a $k$'th root of unity for some smaller
positive integer $k < N$.
Show that the primitive $N$'th roots of unity are precisely the numbers $e^{2\pi i k/N}$ for which
$\gcd(k, N) = 1$.
\end{prob}
\medskip
\begin{prob} \label{SumOfPrimitiveRootsOfUnity}
The M\"obius $\mu$-function
\index{M\"obius $\mu$-function}
is defined by:
\[
\mu(n) := \begin{cases}
(-1)^{\text{ number of distinct prime factors of } n} & \mbox{if } n > 1 \\
1 & \mbox{if } n=1.
\end{cases}
\]
Prove that the sum of all of the primitive $N$'th roots of unity is equal to the
M\"obius $\mu$-function, evaluated at $N$:
\begin{equation}
\sum_{1\leq k < N \atop \gcd(k, N) = 1} e^{\frac{2\pi i k}{N}} = \mu(N).
\end{equation}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{extension of exponential}
We follow the Weierstrassian approach to defining the complex exponential $e^{z}$
for all complex $z \in \mathbb{C}$:
\begin{equation}
e^{z} := \sum_{n=0}^\infty \frac{1}{n!} z^n,
\end{equation}
which converges absolutely for all $z\in \mathbb{C}$. We also have the (Weierstrassian) definitions
of $\cos z$ and $\sin z$:
\[
\cos z:= \sum_{n=0}^\infty \frac{1}{(2n)!} (-1)^n z^{2n}, \quad
\sin z:= \sum_{n=1}^\infty \frac{1}{(2n-1)!} (-1)^{n-1} z^{2n-1},
\]
both converging absolutely again for all $z \in \mathbb{C}$. Prove that Euler's formula
has the extension:
\[
e^{iz} = \cos z + i \sin z,
\]
valid for all $z \in \mathbb{C}$.
\end{prob}
Notes. \ Karl Weierstrass developed a rigorous and beautiful theory of real and complex functions, beginning with such a power series approach.
\medskip
\begin{prob}
Here the reader needs to know a little bit about the quotient of two groups (this is one of the few exercises that assumes group theory). We
prove that the group of `real numbers mod $1$' under addition, is
isomorphic to the unit circle, under multiplication of complex numbers. Precisely, we can define
$h: \mathbb{R} \rightarrow S^1$ by $h(x) := e^{2\pi i x}$.
\begin{enumerate}[(a)]
\item We recall the definition of the kernel of a map, namely $ker(h):= \{ x\in \mathbb{R} \mid h(x) = 1\}$. Show that $ker(h) = \mathbb{Z}$.
\item Using the first isomorphism Theorem for groups, show that
$\mathbb{R}/\mathbb{Z}$ is isomorphic to the unit circle $S^1$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
Using gymnastics with roots of unity, we recall here a very classical solution to the problem of finding the roots of a cubic polynomial.
\begin{enumerate}[(a)]
\item Let $\omega:= e^{2\pi i/3}$, and show that we have the polynomial identity:
\[
(x + a + b)( x + \omega a + \omega^2 b) (x + \omega^2 a + \omega b)
= x^3 - 3abx + a^3 + b^3.
\]
\item Using the latter identity, solve the cubic polynomial:
$x^3 - px + q = 0$ by substituting $p = 3ab$ and $q= a^3 + b^3$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}\label{zeros of the sin function}
Thinking of the function $\sin(\pi z)$ as a function of a complex variable $z\in \mathbb C$, show that its zeros are precisely the set of integers $\mathbb{Z}$.
\end{prob}
\medskip
\begin{prob}
Here we give another equivalent condition for a rectangle in Theorem \ref{Integer.Side.Rectangle} to be a nice rectangle, using the same
definitions as before.
Let's call $\xi \in \mathbb{Z}^2$ a {\bf generic} integer point
if $\xi$ is not orthogonal to any of the edges of $\mathcal{R}$. In other words, a generic integer vector satisfies
$\langle \xi, p \rangle \not=0$, for all $p\in \mathcal{R}$, and in particular $p=0$ is not generic, nor is any
point $p$ on the $x$-axis or the $y$-axis.
Then
\begin{equation}
\mathcal{R} \text{ is a nice rectangle } \iff \hat 1_{\mathcal{R}}(\xi) = 0, \text{ for all generic points } \xi \in \mathbb{Z}^2.
\end{equation}
\end{prob}
\medskip
\begin{prob}[Erd\"os, 1951] \label{Erdos lattice partition problem}
\rm{
Erd\"os asked: ``Can the set $\mathbb{Z}_{>0}$ of all positive integers be partitioned (that is, written as a disjoint union) into a finite number of
arithmetic progressions, such that no two of the arithmetic progressions will have the same common difference?''
Suppose that we have a list of disjoint arithmetic progressions, each with its common difference $a_k$:
\[
\{ a_1 n + b_1 \mid n \in \mathbb{Z}\}, \dots, \{ a_N n + b_N \mid n \in \mathbb{Z}\},
\]
where $a_1 \leq a_2 \leq \cdots \leq a_N$, and $N\geq 2$.
Prove that in any such partitioning of the integers, there are at least two arithmetic progressions that have the same maximal $a_N$.
Notes. For example, if we write $\mathbb{Z} = \{ 4 n + 1 \mid n \in \mathbb{Z}\} \cup \{ 2 n \mid n \in \mathbb{Z}\} \cup \{ 4 n + 3 \mid n \in \mathbb{Z}\}$, a disjoint union of
$3$ arithmetic progressions, then we see
that the common difference of $4$ appears twice. Erd\"os noticed that such a phenomenon must always occur.
(See also Exercise \ref{Extension of Erdos to dimension d} for an extension to lattices in $\mathbb{R}^d$).
}
\end{prob}
\chapter{Examples that nourish the theory} \label{Chapter.Examples}
\begin{quote}
``A pint of example is worth a gallon of advice.''
-- Anonymous
\end{quote}
\smallskip
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{Bernoulli}
\end{center}
\caption{The first periodic Bernoulli polynomial $P_1(x)$, sometimes called the sawtooth function,
which turns out to be one of the building blocks of integer point enumeration in polytopes }
\label{FirstBernoulli}
\end{figure}
\section{Intuition}
One way to think about the Fourier transform of a polytope ${\mathcal P} \subset \mathbb{R}^d$ is that it simultaneously captures all of the moments of ${\mathcal P}$, thereby uniquely defining ${\mathcal P}$. Here we begin concretely by computing some Fourier transforms of various polytopes in dimensions $1$ and $2$, as well as the Fourier transforms of some simple families of polytopes in dimension $d$ as well.
The $2$-dimensional computations will get the reader more comfortable with the basics. In later chapters, once we learn a little more theory, we will return to these families of polytopes and compute some of their Fourier transforms in general.
We also see, from small examples, that the Bernoulli polynomials immediately enter into the picture, forming natural building blocks.
In this chapter we compute Fourier transforms without thinking too much about convergence issues, to let the reader run with the ideas.
But commencing with the next chapter, we will be more rigorous when using Poisson summation, and with convergence issues.
\section{Dimension $1$ - the classical sinc function}
\bigskip
We begin by computing the classical $1$-dimensional example of the Fourier transform \index{Fourier transform}
of the symmetrized unit interval ${\mathcal P}:= [-\frac{1}{2}, \frac{1}{2}]$:
\begin{align} \label{ClassicalExample}
\hat 1_{{\mathcal P}}(\xi) & := \int_{\mathbb{R}} 1_{\mathcal P}(x) \ e^{-2\pi i x \xi } dx \\
&= \int_{[-\frac{1}{2}, \frac{1}{2}]} e^{-2\pi i x \xi } dx \\
& = \frac{e^{-2\pi i \left( \frac{1}{2} \right) \xi} - e^{-2\pi i \left( \frac{-1}{2} \xi\right) } }{-2\pi i \xi} \\ \label{sinc}
& = \frac{ \cos (-\pi \xi) + i \sin(-\pi \xi) - (\cos(\pi \xi) + i \sin(\pi \xi)) }{-2\pi i \xi} \\ \label{sinc function formula}
&= \frac{\sin(\pi \xi)}{\pi \xi},
\end{align}
valid for all $\xi \not= 0$. The latter function is also known as the {\bf sinc function}.
\index{Sinc function} We notice that $\xi = 0$ is a removable singularity, so that we may define the continuous sinc-function by
\begin{equation}\label{SincFunction}
{\rm{sinc}}(x):= \begin{cases}
\frac{\sin(\pi x)}{\pi x}, &\mbox{if } x \not= 0 \\
1 & \mbox{if } x= 0,
\end{cases}
\end{equation}
which is in fact infinitely smooth, via Lemma \ref{FT of a polytope is entire} below.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.6in]{sinc1}
\end{center}
\caption{The function ${\rm{sinc}}(x)$, which is Fourier transform of the $1$-dimensional
polytope ${\mathcal P} = [-\frac{1}{2}, \frac{1}{2}]$. }
\label{sinc.pic} \index{sinc function}
\end{figure}
\bigskip
\section{The Fourier transform of ${\mathcal P}$ as a complete invariant}
\label{Fourier inversion}
The main goal of this section is to state Lemma \ref{complete invariance of the FT}, which tells us that all of the information
about a polytope is contained in its Fourier transform. To that end, we
introduce
the inverse Fourier transform, \index{inverse Fourier transform} often called the {\bf Fourier inversion formula}.
We'd like to see the fundamental fact that under certain conditions,
the Fourier transform is invertible. First, we call a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ absolutely integrable if
$\int_{\mathbb{R}^d} | f(x) | dx < \infty$, and we write this as $f \in L^1(\mathbb{R}^d)$.
\begin{thm} \label{thm:Inverse Fourier transform}
Given a function $f$ such that both $f \in L^1(\mathbb{R}^d)$ and $\hat f \in L^1(\mathbb{R}^d)$, we have
\begin{equation}\label{second version of Fourier inversion}
f(x) = \int_{\mathbb{R}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle} d\xi,
\end{equation}
for all $x \in \mathbb{R}^d$.
\hfill $\square$
\end{thm}
(see \cite{Travaglini} for a proof).
Equation \eqref{second version of Fourier inversion} tells us that the inverse Fourier transform ${\mathcal F}^{-1}$ exists, and is almost equal to ${\mathcal F}$ itself. A moment's thought reveals that we may rewrite \eqref{second version of Fourier inversion} in the following useful form:
\begin{equation} \label{InverseFourierTransform2}
({\mathcal F} \circ {\mathcal F})f (x) = f(-x).
\end{equation}
\bigskip
\begin{example}\label{Integral.of.sinc}
\rm{
A famous and historically somewhat tricky integral formula for the sinc function is the following fact:
\begin{equation} \label{area under sinc}
\int_{-\infty}^\infty {\rm{sinc}}(x) dx := \int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} dx = 1,
\end{equation}
also known as `the Dirichlet integral'. \index{Dirichlet integral}
The careful reader might notice that the latter integrand is not absolutely convergent, which means that
$\int_{-\infty}^\infty \Big| \frac{\sin(\pi x)}{\pi x} \Big| dx = \infty$ (Exercise \ref{divergence of |sinc|}). So we have to specify what we really mean
by the identity \eqref{area under sinc}. The rigorous claim is:
\begin{equation*}
\lim_{N\rightarrow \infty}
\int_{-N}^N \frac{\sin(\pi x)}{\pi x} dx =1.
\end{equation*}
Let's see an intuitive derivation of \eqref{area under sinc}, where we will be fast-and-loose for the moment. Using \eqref{ClassicalExample},
we've seen above that the Fourier transform of the indicator function of the interval ${\mathcal P} := [-\frac{1}{2}, \frac{1}{2}]$ is:
\begin{equation}
{\mathcal F}(1_{{\mathcal P}})(\xi) = \frac{\sin(\pi \xi)}{\pi \xi},
\end{equation}
so that
\begin{equation}
{\mathcal F} \left( \frac{\sin(\pi \xi)}{\pi \xi} \right)
= ({\mathcal F} \circ {\mathcal F})(1_{{\mathcal P}})(\xi)
= 1_{ {\mathcal P} }(-\xi).
\end{equation}
Using the definition of the Fourier transform, the latter identity is:
\begin{equation}
\int_{\mathbb{R}} \frac{\sin(\pi x)}{\pi x} e^{-2\pi i \xi x} dx
= 1_{{\mathcal P}}(\xi),
\end{equation}
and now evaluating both sides at $\xi = 0$ gives us \eqref{area under sinc}.
Although this derivation appears very convincing, it would not make it past the rigor police. So why not? It is because we applied the Fourier inversion formula
to a function that was {\bf not} in $L^1(\mathbb{R})$, namely the sinc function. So we owe it to ourselves to pursue a rigorous
approach by showing that
\begin{equation}
\lim_{N\rightarrow \infty} \int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} e^{-2\pi i \langle \xi, x \rangle} d\xi = 1_ {[-\frac{1}{2}, \frac{1}{2}]}(x),
\end{equation}
whose validity would give us a variation on Fourier inversion, for a function that is not in $L^1(\mathbb{R})$, namely $\hat 1_ {[-\frac{1}{2}, \frac{1}{2}]}(\xi)= {\rm{sinc}}(\xi)$.
This is tricky business, but such an endeavor is taken up in Exercise \ref{rigorous inversion formula for sinc}.
}
\hfill $\square$
\end{example}
We can extend Example \ref{Integral.of.sinc} in a natural way to all Fourier pairs of functions, $\{f(x), \hat f(\xi)\}$, provided
that we may apply Fourier inversion, as follows.
Simply let $x=0$ in \eqref{second version of Fourier inversion}, to get:
\begin{equation} \label{IntegralTrick}
f(0)=\int_{\mathbb{R}^d} \hat f(x) dx.
\end{equation}
To summarize, Example \ref{Integral.of.sinc} is simply identity \eqref{IntegralTrick} with
$f(x) := 1_{[-\frac{1}{2}, \frac{1}{2}]}(x)$.
Another nice - and very useful - fact about the Fourier transform
of a polytope is that it is an entire function, meaning that it is differentiable everywhere. This differentiability
is already observable in the sinc function above, with its removable singularity at the origin.
\begin{lem} \label{FT of a polytope is entire}
Let ${\mathcal P} \subset \mathbb{R}^d$ be a $d$-dimensional polytope. Then $\hat 1_{\mathcal P}(\xi)$ is an entire function of $\xi \in \mathbb{C}^d$.
\end{lem}
\begin{proof}
Because ${\mathcal P}$ is compact, we can safely differentiate under the integral sign (this is a special case of Lebesgue's Dominated Convergence Theorem). Namely, for any coordinate variable $\xi_k$, we have:
$\frac{d}{d\xi_k} \int_{{\mathcal P}} e^{-2\pi i \langle \xi, x \rangle} dx= \int_{{\mathcal P}} \frac{d}{d\xi_k}e^{-2\pi i \langle \xi, x \rangle} dx
= 2\pi i \int_{{\mathcal P}} x_k e^{-2\pi i \langle \xi, x \rangle} dx$, and it is clear that all possible derivatives exist in this manner, because the integrand is infinitely smooth.
\end{proof}
We also have the very fortuitous fact that the Fourier transform of any polytope ${\mathcal P} \subset \mathbb{R}^d$ is a complete invariant, in the following sense.
We recall that by definition a polytope is in particular a closed set.
\begin{lem} \label{complete invariance of the FT}
Let ${\mathcal P}\subset \mathbb{R}^d$ be a polytope. Then
$\hat 1_{\mathcal P}(\xi)$ uniquely determines ${\mathcal P}$. Precisely, given any two $d$-dimensional polytopes $P, Q\subset \mathbb{R}^d$,
we have
\[
\hat 1_{\mathcal P}(\xi) = \hat 1_{Q}(\xi) \text{ for all } \xi \in \mathbb{R}^d \ \iff \ {\mathcal P} = Q.
\]
In other words, for any polytope ${\mathcal P}$, its Fourier transform $\hat 1_{\mathcal P}$ uniquely determines the polytope.
\end{lem} \label{FT.complete invariant}
\begin{proof} (outline)
If ${\mathcal P}=Q$, it is clear that $\hat 1_{\mathcal P}(\xi) = \hat 1_{Q}(\xi)$ for all $\xi \in \mathbb{R}^d$. Conversely, suppose that
$\hat 1_{\mathcal P}(\xi) = \hat 1_{Q}(\xi)$ for all $\xi \in \mathbb{R}^d$. Using Fourier inversion (see \cite{Podkorytov}),
we may take the Fourier transform of both sides of the latter equation to get $1_{\mathcal P}(-\xi) = 1_Q(-\xi)$, for all $\xi \in \mathbb{R}^d$.
\end{proof}
The reason that the proof above is only an outline is because we have applied the Fourier inversion formula to
$\hat 1_{\mathcal P}$, which is not absolutely integrable
(see Exercise \ref{the FT of 1_P is not in L^1} below, in Chapter \ref{Stokes' formula and transforms}).
However, there is a nice version of the Fourier inversion formula, due to Podkorytov and Minh, that
holds for such functions and nicely patches up this hole (see \cite{Podkorytov}).
The reason we've put Lemma \ref{complete invariance of the FT}
so early in the text is because it offers an extremely strong motivation for the study of Fourier transforms of polytopes, showing
that they are complete invariants.
A fascinating consequence of Lemma \ref{complete invariance of the FT} is that when we take the Fourier transform of a polytope, then {\bf all of the combinatorial and geometric information} of ${\mathcal P}$ is contained in the formula of its transform......
So we may begin to create a complete dictionary between the geometry and combinatorics of a polytope in the space domain, and its Fourier transform in the frequency domain.
\bigskip
\section{Bernoulli polynomials} \index{Bernoulli polynomial}
We introduce the Bernoulli polynomials, which turn out to be a sort of ``glue'' between
discrete geometry and number theory, as we will see throughout the book.
The {\bf Bernoulli polynomials} \index{Bernoulli polynomial}
are defined via the following generating function:
\begin{equation} \label{generating function for Bernoulli polynomials}
\frac{te^{xt}}{e^t-1} = \sum_{k =0}^\infty B_k(x) \frac{t^k}{k!}.
\end{equation}
It's fruitful to sometimes restrict the Bernoulli polynomials to the unit interval $[0,1]$,
and then periodize them. In other words, using
\[
\{x\} := x - \lfloor x \rfloor,
\]
the fractional part of $x$, \index{fractional part}
we may define the $n$'th {\bf periodic Bernoulli polynomial}: \index{periodic Bernoulli polynomial}
\begin{equation} \label{definition of periodic Bernoulli polys}
P_n(x) := B_n(\{x\}),
\end{equation}
for $n\geq 2$. Since $P_n(x)$ is periodic on $\mathbb{R}$ with period $1$,
it has a Fourier series, and in fact:
\begin{equation}
P_n(x) = -\frac{n!}{(2\pi i)^n} \sum_{k \in \mathbb{Z} - \{0\}} \frac{e^{2\pi i k x}}{k^n},
\end{equation}
valid for $x \in \mathbb{R}$ (Exercise \ref{Bernoulli Polynomials}).
When $n=1$, we have the first Bernoulli polynomial
\[
P_1(x):= x - \lfloor x \rfloor - \frac{1}{2},
\]
which
is very special (see Figure \ref{FirstBernoulli}). For one thing,
it is the only periodic Bernoulli polynomial that is not continuous everywhere,
and we note that its Fourier series does not converge absolutely, although it is quite appealing:
\begin{equation} \label{FirstBernoulliPolynomial}
P_1(x) = -\frac{1}{2\pi i} \sum_{k \in \mathbb{Z} - \{0\}} \frac{e^{2\pi i k x}}{k},
\end{equation}
valid for all $x \notin \mathbb{Z}$. Hence special care must be taken with $P_1(x)$. Exercises \ref{brute force Bernoulli polys} through
\ref{vanishing identity for Beroulli numbers} illustrate some of the important properties of
these polynomials. Exercise \ref{rigorous convergence of P_1(x)} provides a rigorous proof
of the convergence of \eqref{FirstBernoulliPolynomial}.
\begin{example} \rm{
The first few Bernoulli polynomials are:
\begin{align}
B_0(x) &= 1 \\
B_1(x) &= x - \frac{1}{2} \\
B_2(x) &= x^2 - x + \frac{1}{6} \\
B_3(x) &= x^3 - \frac{3}{2} x^2 +\frac{1}{2} x \\
B_4(x) &= x^4 -2x^3 + x^2 - \frac{1}{30} \\
B_5(x) &= x^5 - \frac{5}{2} x^4 + \frac{5}{3} x^3 - \frac{1}{6} x \\
B_6(x) &= x^6 - 3x^5 + \frac{5}{2} x^4 - \frac{1}{2} x^2 + \frac{1}{42} \label{B_6}
\end{align}
The {\bf Bernoulli numbers} are defined to be the constant terms of the Bernoulli polynomials:
\[
B_k := B_k(0).
\]
The first few Bernoulli numbers are: \\
\[
B_0 = 1, \ B_1 = -\frac{1}{2}, \ B_2 = \frac{1}{6}, \ B_3 = 0, \ B_4 = - \frac{1}{30}, \ B_5 = 0, \ B_6 = \frac{1}{42}.
\]
It follows quickly from
the definition \ref{generating function for Bernoulli polynomials}
above that for odd $k \geq 3$, $B_k = 0$ (Exercise \ref{odd Bernoulli numbers}).
From the generating function \ref{generating function for Bernoulli polynomials}
the Bernoulli numbers are defined via
\begin{equation} \label{Def. of Bernoulli numbers}
\frac{t}{e^t-1} = \sum_{k =0}^\infty B_k \frac{t^k}{k!}.
\end{equation}
}
\hfill $\square$
\end{example}
Historically, the first appearance of the Bernoulli polynomials occurred while Jakob Bernoulli tried to
compute sums of powers of integers. In particular, Bernoulli showed that:
\[
\sum_{k=1}^{n-1} k^{d-1} = \frac{ B_d(n) - B_d }{ d },
\]
for all integers $d \geq 1$ and $n \geq 2$ (Exercise \ref{historical origin of Bernoulli poly}).
An interesting identity that allows us to compute the Bernoulli numbers recursively rather quickly is:
\[
\sum_{k=0}^n {n+1 \choose k}B_k = 0,
\]
valid for all $n \geq 1$ (Exercise \ref{vanishing identity for Beroulli numbers}).
Some of the most natural, and beautiful, Fourier series arise naturally from the periodized Bernoulli polynomials. The following intuitive application of the Poisson summation formula already suggests an initial connection between periodized Bernoulli polynomials and Fourier transforms of polytopes - even in dimension $1$.
\begin{example} [Intuitive Poisson summation] \label{intuitive Poisson example}
\index{Poisson summation formula}
\rm{
In this example we allow ourselves to be completely intuitive, and unrigorous at this moment,
but often such arguments are useful in pointing us to their rigorous counterparts.
Consider the $1$-dimensional polytope ${\mathcal P}:= [a,b]$, and restrict attention to the case
of $a, b \not\in \mathbb{Z}$. If we could use the Poisson summation formula
\[
\sum_{n \in \mathbb{Z}^d} f(n) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi),
\]
applied to the function $f(x):= 1_{\mathcal P}(x)$, then we would get:
\begin{align*}
\sum_{n \in \mathbb{Z}} 1_{\mathcal P}(n) &``=\text{''} \sum_{\xi \in \mathbb{Z}} \hat 1_{\mathcal P}(\xi)\\
&``=\text{''} \
\hat 1_{\mathcal P}(0)+\sum_{\xi \in \mathbb{Z} - \{0\}} \frac{e^{-2\pi i \xi b} - e^{-2\pi i \xi a} }{-2\pi i \xi} \\
&``=\text{''} \
(b-a) -\frac{1}{2\pi i} \sum_{\xi \in \mathbb{Z} - \{0\}} \frac{e^{-2\pi i \xi b}}{\xi}
+ \frac{1}{2\pi i} \sum_{\xi \in \mathbb{Z}-\{0\}} \frac{e^{-2\pi i \xi a}}{\xi} \\
&``=\text{''} \
(b-a) + \frac{1}{2\pi i} \sum_{\xi \in \mathbb{Z} - \{0\}} \frac{e^{2\pi i \xi b}}{\xi}
- \frac{1}{2\pi i} \sum_{\xi \in \mathbb{Z}-\{0\}} \frac{e^{2\pi i \xi a}}{\xi} \\
&``=\text{''} \
(b-a) -\left( \{b\}- \frac{1}{2} \right) + \left( \{a\} - \frac{1}{2} \right) \\
&``=\text{''} \
b - \{b\} - ( a - \{a\} ) = \lfloor b \rfloor - \lfloor a \rfloor.
\end{align*}
Since we already know how to evaluate the LHS of Poisson summation above, namely that
$\sum_{n \in \mathbb{Z}} 1_{\mathcal P}(n) = \#\left\{ \mathbb{Z} \cap {\mathcal P} \right\} = \lfloor b \rfloor - \lfloor a \rfloor$,
we have confirmed that Poisson summation has given us here the correct formula, in spite of the lack of rigor here.
Why is the intuitive argument above not rigorous yet? In order to plug a function $f$ into
Poisson summation, and consider convergence at each point of the domain,
$f$ and its Fourier transform $\hat f$ must both satisfy some growth conditions at infinity, at the very least
ensuring proper convergence of both sides of the Poisson summation formula.
We will see such conditions later, in Chapter~\ref{Fourier analysis basics}, Theorem \ref{nice2}. Once we learn how to use Poisson summation, we will return to this example (see
Example \ref{rigorous example of P_1}).
}
\hfill $\square$
\end{example}
We recall that a series $\sum_{n\in \mathbb{Z}} a_n$ is said to {\bf converge absolutely} if $\sum_{n\in \mathbb{Z}} |a_n|$ converges. It's easy to see that the series in \eqref{FirstBernoulliPolynomial} for $P_1(x)$ does not converge absolutely. Such convergent series that do not converge absolutely are called {\bf conditionally convergent}.
To prove rigorously that the conditionally convergent series
\eqref{FirstBernoulliPolynomial} does in fact converge, see
Exercises \ref{Abel summation by parts}, \ref{Dirichlet's convergence test}, \ref{exponential sum bound}, and
\ref{rigorous convergence of P_1(x)}, which include the Abel summation formula, and the Dirichlet convergence test (although extremely useful, we will not use them
very much in the ensuing chapters).
\bigskip
\section{The cube, and its Fourier transform}
Perhaps the easiest way to extend the Fourier transform of the unit interval is to consider the
$d$-dimensional unit cube
\[
\square := \left[-\frac{1}{2}, \frac{1}{2} \right]^d.
\]
What is its Fourier transform? When we compute a Fourier transform of a function $f$, we will
say that $\{ f, \hat f\}$ is a {\bf Fourier pair}. We have seen that
$\left\{ 1_{[-\frac{1}{2}, \frac{1}{2}]}(x), {\rm{sinc}}(\xi) \right\}$ is a Fourier pair in dimension $1$.
\bigskip
\begin{example} \label{Example, unit cube}
\rm{
Due to the fact that the cube is the direct product of line segments, it follows that the ensuing integral can be separated into a product of integrals, and so it is the product of $1$-dimensional transforms:
\begin{align}
\hat 1_{\square}(\xi) &= \int_{\mathbb{R}^d} 1_\square(x) e^{-2\pi i \langle x, \xi \rangle} dx \\
&= \int_{\square} e^{-2\pi i(x_1 \xi_1 + \cdots + x_d \xi_d)} dx \\
&= \prod_{k=1}^d \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i x_k \xi_k} dx_k \\
&= \prod_{k=1}^d \frac{\sin(\pi \xi_k)}{\pi \xi_k},
\end{align}
valid for all $\xi \in \mathbb{R}^d$ such that none of their coordinates vanishes. So here we have the Fourier pair
\[
\left\{ 1_\square(x), \, \prod_{k=1}^d \frac{\sin(\pi \xi_k)}{\pi \xi_k} \right\}.
\]
In general, though,
polytopes are not a direct product of lower-dimensional polytopes, so we will need to develop more tools to compute their Fourier transforms.
}
\hfill $\square$
\end{example}
\bigskip
\section{The simplex, and its Fourier transform}
Another basic building block for polytopes is the {\bf standard simplex},
\index{standard simplex}
defined by
\begin{equation}
\RightTriangle := \left\{
x \in \mathbb{R}^d \bigm | \, x_1 + \cdots + x_d \leq 1, \text{ and all } x_k \geq 0
\right\} .
\end{equation}
\begin{figure}[!h]
\centering
\begin{tikzpicture}[scale=1]
\draw (0,0) node[below left] {$0$};
\draw[loosely dotted] (-1,-1) grid (2,2);
\draw[->] (-1.25,0) -- (2.25,0) node[right] {$x$};
\draw[->] (0,-1.25) -- (0,2.25) node[above] {$y$};
\draw[thick] (0,0) -- (1,0) -- (0,1) -- cycle;
\filldraw[nearly transparent, blue] (0,0) -- (1,0) -- (0,1) -- cycle;
\end{tikzpicture}
\caption{The standard simplex in $\mathbb{R}^2$}
\label{standard simplex in two dimensions} \index{standard simplex}
\end{figure}
\begin{example}\label{standard simplex FT}
\index{standard simplex}
\rm{
Just for fun, let's compute the Fourier transform of $\triangle$
for $d=2$, via brute-force.
We may use the following parametrization (called a hyperplane description)
for this standard triangle:
\[
\RightTriangle = \left\{ (x, y) \bigm | x+y \leq 1, \text{ and } x\geq 0, y\geq0 \right\}.
\]
Hence, we have:
\begin{align*}
&\hat 1_{\, \rt}(\xi_1, \xi_2) := \int_{\, \rt} e^{-2\pi i \big(x \xi_1 + y \xi_2\big)} dx dy \\
&= \int_0^1 \int_{y=0}^{y=1-x} e^{-2\pi i \big(x \xi_1 + y \xi_2\big)} dy dx \\
&= \int_0^1 e^{-2\pi i x \xi_1}
\left[
\frac{ e^{-2\pi i y \xi_2 } }{-2\pi i \xi_2 } \Big|_{y=0}^{y=1-x}
\right]
dx \\
&= \frac{1}{-2\pi i \xi_2 } \int_0^1 e^{-2\pi i x \xi_1}
\left( e^{-2\pi i (1-x) \xi_2 } - 1 \right) dx \\
&= \frac{1}{-2\pi i \xi_2 } \int_0^1
\left(
e^{-2\pi i x (\xi_1 -\xi_2)} e^{-2\pi i \xi_2 } - e^{-2\pi i x \xi_1}
\right) dx \\
&= \frac{1}{(-2\pi i)^2}
\frac{ e^{-2\pi i \xi_2 } }{ \xi_2(\xi_1-\xi_2) }
(e^{-2\pi i (\xi_1 -\xi_2)} -1)
-
\frac{1}{(-2\pi i)^2} \frac{ e^{-2\pi i \xi_1 } -1 }{ \xi_1 \xi_2 } \\
&= \frac{1}{(-2\pi i)^2} \left[
\frac{ e^{-2\pi i \xi_1} - e^{-2\pi i \xi_2 } }{ \xi_2(\xi_1-\xi_2) }
-
\frac{ e^{-2\pi i \xi_1 } -1 }{ \xi_1 \xi_2 }
\right].
\end{align*}
We may simplify further by noticing the rational function identity
\[
\frac{ e^{-2\pi i \xi_1 } }{ \xi_2 ( \xi_1 - \xi_2 ) }
-\frac{ e^{-2\pi i \xi_1 } }{ \xi_1 \xi_2 }
= \frac{ e^{-2\pi i \xi_1 } }{ \xi_1 ( \xi_1 - \xi_2 ) },
\]
giving us the symmetric function of $(\xi_1, \xi_2)$:
\begin{equation}\label{actual FT of the standard simplex}
\hat 1_{\rt}(\xi_1, \xi_2) =
\frac{1}{(-2\pi i)^2} \left[
\frac{ e^{-2\pi i \xi_1 } }{ \xi_1 ( \xi_1 - \xi_2 ) }
+ \frac{ e^{-2\pi i \xi_2 } }{ \xi_2(\xi_2-\xi_1) }
+
\frac{ 1 }{ \xi_1 \xi_2 }
\right].
\end{equation}
}
\hfill $\square$
\end{example}
We need the concept of a {\bf convex set} $X\subset \mathbb{R}^d$, defined by the property that for any two points
$x, y \in X$, the line segment joining them also lies in $X$. In other words, the line segment
$\left\{ \lambda x + (1-\lambda)y \bigm | 0\leq \lambda \leq 1 \right\} \subset X$, $\forall x, y \in X$.
Given any finite set of points $S:= \{ v_1, v_2, \dots, v_N\} \subset \mathbb{R}^d$, we can also form the set of all
{\bf convex linear combinations} of $S$ by defining
\begin{equation}
\conv(S):=
\left\{
\lambda_1 v_1 + \lambda_2 v_2+ \dots +\lambda_{N} v_{N}
\bigm |
\sum_{k=1}^N \lambda_k = 1, \text{ where all } \lambda_k \geq 0
\right\}.
\end{equation}
Given any set $U\subset \mathbb{R}^d$ (which is not restricted to be finite), we define the {\bf convex hull} of $U$
\index{convex hull}
as the set of convex linear combinations, taken over all finite subsets of $U$, and denoted by $\conv(U)$.
We define a {\bf polytope} as the convex hull of any finite set of points in $\mathbb{R}^d$. This definition of a polytope is called its {\bf vertex description}.
\index{vertex description of a polytope}
We define a {\bf $k$-simplex} \index{simplex}
$\Delta$ as the convex hull of a finite set of vectors
$\{ v_1, v_2, \dots, v_{k+1} \}$:
\[
\Delta := \conv\{ v_1, v_2, \dots, v_{k+1} \},
\]
where $0 \leq k \leq d$, and $v_2-v_1, v_3-v_1, \dots, v_{k+1} - v_1$ are linearly independent vectors in $\mathbb{R}^d$.
The points $v_1, v_2, \dots, v_{k+1}$ are called the vertices of $\Delta$, and this
object is one of the basic building-blocks of polytopes, especially when triangulating a polytope.
The simplex $\Delta$ is a $k$-dimensional polytope, sitting in $\mathbb{R}^d$. When $k=d$, the dimension of $\Delta$ equals the dimension of the ambient space $\mathbb{R}^d$ - see Figure \ref{simplex}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{simplex}
\end{center}
\caption{A $3$-simplex and its faces, which are lower-dimensional simplices as well}
\label{simplex}
\end{figure}
We have already computed the Fourier transform of a particular $2$-simplex,
in \eqref{actual FT of the standard simplex}.
How do we define a face of a polytope ${\mathcal P}$ more precisely? Here we need a new notion.
A {\bf hyperplane}
\[
H:=\{ x \in \mathbb{R}^d \mid \langle x, n \rangle = b \}
\]
is called a {\bf supporting hyperplane for} ${\mathcal P}$ if ${\mathcal P}$ lies on one side of $H$, in the precise sense that:
\[
{\mathcal P} \subset \{ x \in \mathbb{R}^d \mid \langle x, n \rangle \leq b \} \ \text{ or }
{\mathcal P} \subset \{ x \in \mathbb{R}^d \mid \langle x, n \rangle \geq b \}.
\]
We now call $F\subseteq {\mathcal P}$ a {\bf face of } ${\mathcal P}$ if $F= H \cap {\mathcal P}$, for some supporting hyperplane $H$ of ${\mathcal P}$.
${\mathcal P}$ is also considered a face of ${\mathcal P}$ itself (use a degenerate hyperplane $H$ with $n:=0:=b$); for logical consistencies,
the empty set is also defined to be
a face of ${\mathcal P}$ (use a hyperplane $H$ far away from ${\mathcal P}$).
With these preliminaries, we're now ready to compute the Fourier transform of any $2$-simplex in
$\mathbb{R}^2$.
In order to handle a general triangle, let $\Delta$ be any triangle in the plane, with vertices
\[
v_1:= \icol{ a_1 \\ b_1}, v_2:=\icol{ a_2 \\ b_2} , v_3:= \icol{ a_3 \\ b_3}.
\]
Can we reduce the computation of $\hat 1_{\Delta}$ to our already known formula
for $\hat 1_{\rt}$, given by \eqref{actual FT of the standard simplex}?
We first notice (after a brief cup of coffee) that we can map any triangle in the plane to the standard triangle, by using a linear transformation followed by a translation:
\begin{equation}\label{M followed by T}
\Delta = M ( \, \RightTriangle ) + v_3,
\end{equation}
where $M$ is the $2\times2$ matrix whose columns are $v_1-v_3$ and $v_2-v_3$.
We are now ready to compute the Fourier transform of a general triangle $\Delta$:
\[
\hat 1_{\Delta}(\xi) = \int_{\Delta} e^{-2\pi i \langle \xi, x \rangle} dx
= \int_{M(\rt)+ v_3} e^{-2\pi i \langle \xi, x \rangle} dx.
\]
Making the substitution $x := My+v_3$, with $y \in \rt$, we have $dx = |\det M| dy$, and so
\begin{align*}
& \int_{M(\rt)+v_3} e^{-2\pi i \langle \xi, x \rangle} dx
= |\det M| \int_{\, \rt} e^{-2\pi i \langle \xi, M y +v_3 \rangle} dy \\
&= |\det M| e^{-2\pi i \langle \xi, v_3 \rangle}
\int_{\, \rt} e^{-2\pi i \langle M^{T} \xi, y \rangle} dy \\
&= |\det M| e^{-2\pi i \langle \xi, v_3 \rangle} \hat 1_{\, \rt}(M^{T} \xi) \\
&= |\det M| e^{-2\pi i \langle \xi, v_3 \rangle}
\hat 1_{\, \rt} \big( \langle v_1-v_3, \xi \rangle, \langle v_2-v_3, \xi \rangle \big) \\
&=
|\det M| e^{-2\pi i \langle \xi, v_3 \rangle}
\frac{1}{(-2\pi i)^2} \left[
\frac{ e^{-2\pi i z_1 } }{ z_1 ( z_1 - z_2 ) }
+ \frac{ e^{-2\pi i z_2 } }{ z_2(z_2-z_1) }
+
\frac{ 1 }{ z_1 z_2 }
\right],
\end{align*}
where we've used our formula \eqref{actual FT of the standard simplex} for the FT of the standard triangle (thereby bootstrapping out way to the general case) with
$z_1:=\langle v_1-v_3, \xi \rangle$, and $z_2:= \langle v_2-v_3, \xi \rangle$. Substituting these values into the latter expression, we finally arrive at the FT of our general triangle $\Delta$:
\begin{align}\label{FT of a general triangle}
\hat 1_{\Delta}(\xi)
= \tfrac{ |\det M| }{(-2\pi i)^2}
\left[
\frac{ e^{-2\pi i \langle v_1, \xi \rangle } }{ \langle v_1-v_3, \xi \rangle
\langle v_1-v_2, \xi \rangle }
+ \frac{ e^{-2\pi i \langle v_2, \xi \rangle } }{ \langle v_2-v_3, \xi \rangle
\langle v_2-v_1, \xi \rangle }
+
\frac{ e^{-2\pi i \langle \xi, v_3 \rangle} }{ \langle v_3-v_1, \xi \rangle
\langle v_3-v_2, \xi \rangle }
\right].
\end{align}
We can notice in equation \eqref{FT of a general triangle} many of the same patterns that had already occurred in Example
\ref{cross-polytope example in R^2}. Namely,
the Fourier transform of a triangle has denominators that are products of linear forms in $\xi$,
and it is a finite linear combination of rational functions multiplied by complex exponentials.
Also, in the particular case of
equation \eqref{FT of a general triangle}, $ \hat 1_\Delta(\xi) $ is a symmetric function of $v_1, v_2, v_3$, as we might have expected.
Using exactly the same ideas that were used in equation \eqref{FT of a general triangle}, it is possible to prove (by induction on the dimension) that the Fourier transform of a general $d$-dimensional simplex $\Delta \subset \mathbb{R}^d$ is:
\begin{equation}\label{FT of a d-dimensional simplex}
\hat 1_{\Delta}(\xi) = (\vol \Delta) d! \sum_{j=1}^N
\frac{e^{-2\pi i \langle v_j, \xi \rangle}}{\prod_{k=1}^d \langle v_j-v_k, \xi \rangle }[k \not= j],
\end{equation}
where the vertex set of ${\mathcal P}$ is $\{ v_1, \dots, v_N\}$ (Exercise \ref{FT of a general simplex, brute-force}), and in fact the same formula persists for all complex $\xi \in \mathbb{C}^d$ such that the products of linear forms in the denominators do not vanish.
However, looking back at the computation leading to \eqref{FT of a general triangle},
and the corresponding computation which would give \eqref{FT of a d-dimensional simplex},
the curious reader might be thinking:
\medskip
\centerline{ ``There must be an easier way!'' }
But never fear - indeed there is.
So even though at this point the computation of $\hat 1_\Delta(\xi)$ may be a bit laborious (but still interesting),
computing the Fourier transform of a general simplex will become quite easy once we will revisit it in a later chapter (see Theorem \ref{brion, continuous form}).
\bigskip
\section{Stretching and translating}
\bigskip
The perspicacious reader may have noticed that in order to arrive at the formula
\eqref{FT of a general triangle} above for the FT of a general triangle, we exploited the fact that the Fourier transform interacted peacefully with the linear transformation $M$, and with the translation by the vector $v$. Is this true in general?
Indeed it is, and we record these thoughts in the following two lemmas, which will become our bread and butter for future computations.
In general, given any invertible linear transformation $M :\mathbb{R}^d \rightarrow \mathbb{R}^d$, and any function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ whose FT (Fourier transform) exists, we have the following useful interaction between Fourier transforms and linear transformations.
\begin{lem}[Stretch] \index{stretch lemma}
\label{FT under linear maps}
\begin{equation}\label{The FT under streching}
(\widehat{f \circ M})(\xi)= \frac{1}{|\det M|} \hat f\left(M^{-T}\xi \right)
\end{equation}
\end{lem}
\begin{proof}
By definition, we have
$
(\widehat{f \circ M})(\xi) :=\int_{\mathbb{R}^d} f(Mx) e^{-2\pi i \langle \xi, x \rangle} dx.
$
We perform the change of variable $y:= Mx$, implying that $dy = |\det M| dx$, so that:
\begin{align*}
(\widehat{f \circ M})(\xi) &=
\frac{1}{|\det M|} \int_{\mathbb{R}^d} f(y) e^{-2\pi i \langle \xi, M^{-1}y \rangle} dy \\
&=\frac{1}{|\det M|} \int_{\mathbb{R}^d} f(y) e^{-2\pi i \langle M^{-T}\xi, y \rangle} dy \\
&= \frac{1}{|\det M|} \hat f\left(M^{-T} \xi \right).
\end{align*}
\end{proof}
What about translations? They are even simpler.
\begin{lem}[Translate] \index{translate lemma}
\label{FT under translations}
For any translation $T(x):= x + v$, where $v\in \mathbb{R}^d$ is a fixed vector, we have
\begin{equation}\label{The FT under translations}
(\widehat{f \circ T})(\xi)= e^{2\pi i \langle \xi, v \rangle} \hat f(\xi).
\end{equation}
\end{lem}
\begin{proof}
Again, by definition we have
$
(\widehat{f \circ T})(\xi) := \int_{\mathbb{R}^d} f( Tx) e^{-2\pi i \langle \xi, x \rangle} dx,
$
so that performing the simple change of variable $y = Tx := x + v$, we have $dy = dx$.
The latter integral becomes
\begin{align*}
(\widehat{f \circ T})(\xi)
&= \int_{\mathbb{R}^d} f(y) e^{-2\pi i \langle \xi, y-v \rangle} dy \\
&= e^{2\pi i \langle \xi, v \rangle}
\int_{\mathbb{R}^d} f(y) e^{-2\pi i \langle \xi, y \rangle} dy :=
e^{2\pi i \langle \xi, v \rangle} \hat f(\xi).
\end{align*}
\end{proof}
In general, any function $\phi:\mathbb{R}^d \rightarrow \mathbb{C}$ of the form
\begin{equation}
\phi(x) = Mx+v,
\end{equation}
where $M$ is a fixed linear transformation and $v\in \mathbb{R}^d$ is a fixed vector, is
called an {\bf affine transformation}. \index{affine transformation} For example, we've already seen
in \eqref{M followed by T} that the right triangle $\RightTriangle$ was mapped to the more general triangle
$\Delta$ by an affine transformation.
So the latter two lemmas allow us to compose Fourier transforms very easily with affine transformations.
\begin{example}
\rm{
The simplest example of the Stretch Lemma \ref{FT under linear maps} is obtained
in $\mathbb{R}$, where the matrix $M = r$, a positive real number. So we have $M^{-T} = \frac{1}{r}$.
Considering $f(rx)$ as a function of $x \in \mathbb{R}$, we have by \eqref{The FT under translations}:
\begin{equation} \label{simple example of stretching}
\widehat{f(rx)} := (\widehat{f \circ M})(\xi) = \tfrac{1}{r} \hat f\left( \tfrac{1}{r} \xi \right).
\end{equation}
As an interesting sub-example, let's take $f(x) := 1_{\left[-\tfrac{c}{2}, \tfrac{c}{2} \right]}(x) $,
for a fixed constant $c>0$. What's the easy way to use the Stretch lemma to compute $\hat f(\xi)$? First, we have to make a slight conversion: $1_{\left[-\tfrac{c}{2}, \tfrac{c}{2} \right]}(x)
= 1_{\left[-\tfrac{1}{2}, \tfrac{1}{2} \right]}(\tfrac{1}{c} x)$. Using the FT of the unit interval, equation \eqref{sinc function formula},
together with \eqref{simple example of stretching}, we have:
\begin{equation}\label{Stretch lemma for the sinc function}
\hat 1_{\left[-\tfrac{c}{2}, \tfrac{c}{2} \right]}(\xi) = c\, \hat 1_{\left[-\tfrac{1}{2}, \tfrac{1}{2} \right]}(c \xi)
=c\, {\rm{sinc}}(c\xi) = \frac{\sin(c\pi \xi)}{\pi \xi}.
\end{equation}
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
Consider any set $B\subset \mathbb{R}^d$, for which $1_B$ is integrable, and let's translate $B$ by a fixed vector $v \in \mathbb{R}^d$, and compute
$\hat 1_{B+v}(\xi)$.
We note that because $1_{B+v}(\xi) = 1_{B}(\xi-v)$, the translate lemma applies, but with a minus sign. That is, we can use
$T(x):= x-v$ and $f:= 1_B$ to get:
\begin{equation} \label{The FT of a translate of B}
\hat 1_{B+v}(\xi) = \widehat{ (1_B \circ T) }(\xi) = e^{-2\pi i \langle \xi, v \rangle} \hat 1_B(\xi).
\end{equation}
}
\hfill $\square$
\end{example}
\bigskip
\section{The parallelepiped, and its Fourier transform} \index{parallelepiped}
Now that we know how to compose the FT with affine transformations (translations and linear transformations), we can easily find the FT of
any parallelepiped in $\mathbb{R}^d$ by using our formula for the Fourier transform of the unit cube
$\square := \left[-\frac{1}{2}, \frac{1}{2} \right]^d$, which we derived in Example \ref{Example, unit cube}:
\begin{align}\label{first FT of a cube}
\hat 1_{\square}(\xi) = \prod_{k=1}^d \frac{\sin(\pi \xi_k)}{\pi \xi_k},
\end{align}
for all $\xi \in \mathbb{R}^d$ such that all the coordinates of $\xi$ do not vanish.
First, we translate the cube $\square$
by the vector $(\frac{1}{2}, \cdots, \frac{1}{2})$, to obtain
\[
C:= \square + \left(\frac{1}{2}, \, \cdots, \frac{1}{2} \right) = [0, 1]^d.
\]
It's straightforward to compute its FT as well (Exercise \ref{transform.of.unit.cube}), by using Lemma \ref{FT under translations}, the `translate' lemma:
\begin{equation}\label{second appearance of FT of the cube}
\hat 1_{C}(\xi) = \frac{1}{(2\pi i)^d} \prod_{k=1}^d \frac{ 1- e^{-2\pi i \xi_k} }{ \xi_k }.
\end{equation}
Next, we define a $d$-dimensional {\bf parallelepiped} ${\mathcal P} \subset \mathbb{R}^d$ as an affine image of the unit cube. In other words,
any parallelepiped has the description
\[
{\mathcal P} = M(C) + v,
\]
for some linear transformation $M$, and some translation vector $v$. Geometrically, the cube is stretched and translated
into a parallelepiped.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{3Dparallelepiped}
\end{center}
\caption{Mapping the unit cube to a parallelepiped}
\label{3Dparallelepiped}
\end{figure}
For the sake of concreteness, will will first set $v:= 0$ and compute the Fourier transform of ${\mathcal P}:= M(C)$, where we now give $M$ as a $d \times d$ invertible matrix whose columns are
$w_1, w_2, \dots, w_d$. Because the cube $C$ may be written as a convex linear combination of the basis vectors $e_j$,
we see that ${\mathcal P}$ may be written as a convex linear combination of $M e_j = w_j$.
In other words, we see that the parallelepiped ${\mathcal P}$ has the equivalent vertex description:
\[
{\mathcal P} = \left\{ \sum_{k=1}^d \lambda_k w_k \bigm | \text{ all } \lambda_k \in [0, 1] \right\}.
\]
To review the basics, let's compute the FT of our parallelepiped ${\mathcal P}$ from first principles:
\begin{align} \label{cube composed with M}
\hat 1_{\mathcal P}(\xi) &:= \int_{{\mathcal P}} e^{-2\pi i \langle \xi, x \rangle} dx = \int_{M(C)} e^{-2\pi i \langle \xi, x \rangle} dx\\
&= |\det M| \int_{C} e^{-2\pi i \langle \xi, My \rangle} dy\\
&= |\det M| \int_{C} e^{-2\pi i \langle M^T \xi, y \rangle} dy := |\det M| \, \hat 1_C\left(M^T \xi \right) \\ \label{for Q}
&= \frac{ |\det M| }{(2\pi i)^d} \prod_{k=1}^d \frac{ 1- e^{-2\pi i \langle w_k, \xi \rangle} }{ \langle w_k, \xi \rangle }.
\end{align}
where in the third equality we used the substitution $x:= My$, with $y\in C$, yielding $dx = |\det M| dy$. In the last equality, we used our known formula \eqref{second appearance of FT of the cube} for the FT of the cube $C$, together with the elementary linear algebra fact that the $k$'th coordinate of $M^T \xi$ is given by
$\langle w_k, \xi \rangle$.
Finally, for a general parallelepiped, we have $Q:= {\mathcal P} + v$, so that by definition
\[
Q = \left\{ v+ \sum_{k=1}^d \lambda_k w_k \bigm | \text{ all } \lambda_k \in [0, 1] \right\}.
\]
Noting that $1_{{\mathcal P}+ v}(\xi) = 1_{{\mathcal P}}(\xi -v)$,
we compute the Fourier transform of $Q$ by using the `translate lemma' (Lemma \ref{FT under translations}),
together with formula \eqref{for Q} for the Fourier transform of ${\mathcal P}$:
\begin{equation}\label{FT of a general parallelepiped}
\hat 1_Q(\xi) = e^{-2\pi i \langle \xi, v \rangle}
\frac{ |\det M| }{(2\pi i)^d}
\prod_{k=1}^d \frac{ 1- e^{-2\pi i \langle w_k, \xi \rangle} }{ \langle w_k, \xi \rangle },
\end{equation}
for all $\xi \in \mathbb{R}^d$, except for those $\xi$ that are orthogonal to one of the $w_k$ (which are edge vectors for $Q$).
\begin{example}
\rm{
A straightforward computation shows that if we let $v:= -\frac{w_1 + \cdots + w_d}{2}$, then
$Q:= \{ v+ \sum_{k=1}^d \lambda_k w_k \mid \text{ all } \lambda_k \in [0, 1] \}$ is symmetric about the origin, in the sense that
$x \in Q \iff -x \in Q$ (Exercise \ref{symmetrized parallelepiped}). In other words, the center of mass of this new $Q$ is now the origin.
Geometrically, we've translated the previous parallelepiped by using half its `body diagonal'.
For such a parallelepiped $Q$, centered at the origin, formula \eqref{FT of a general parallelepiped} above gives
\begin{align}
\hat 1_Q(\xi) &= e^{2\pi i \langle \xi, \frac{w_1 + \cdots + w_d}{2} \rangle}
\frac{ |\det M| }{(2\pi i)^d}
\prod_{k=1}^d \frac{ 1- e^{-2\pi i \langle w_k, \xi \rangle} }{ \langle w_k, \xi \rangle } \\
&=
\frac{ |\det M| }{(2\pi i)^d}
\prod_{k=1}^d \frac{ e^{\pi i \langle w_k, \xi \rangle}- e^{-\pi i \langle w_k, \xi \rangle} }{ \langle w_k, \xi \rangle } \\
&= \frac{ |\det M| }{(2\pi i)^d}
\prod_{k=1}^d \frac{ (2i) \sin( \pi \langle w_k, \xi \rangle) }{ \langle w_k, \xi \rangle } \\
&= |\det M|
\prod_{k=1}^d \frac{ \sin( \pi \langle w_k, \xi \rangle) }{ \pi \langle w_k, \xi \rangle }.
\end{align}
To summarize, for a parallelepiped that is symmetric about the origin, we have the Fourier pair
\[
\left\{ 1_Q(x), \ \ | \det M |
\prod_{k=1}^d \frac{ \sin( \pi \langle w_k, \xi \rangle) }{ \pi \langle w_k, \xi \rangle } \right\}.
\]
We could have also computed the latter FT by beginning with our known
Fourier transform \eqref{first FT of a cube} of the cube $\square$, composing the FT
with the same linear transformation $M$ of \eqref{cube composed with M}, and using the `stretch' lemma, so everything is consistent.
}
\hfill $\square$
\end{example}
\bigskip
\section{The cross-polytope}
\index{cross-polytope}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.8in]{Octahedron}
\end{center}
\caption{The cross-polytope $\Diamond$ in $\mathbb{R}^3$ (courtesy of David Austin)} \label{crosspic}
\end{figure}
Another natural convex body in $\mathbb{R}^2$ is the cross-polytope
\begin{equation} \label{2dim.crosspolytope}
\Diamond_2 := \left\{ \left( x_1, x_2 \right) \in \mathbb{R}^2 \bigm | \, \left| x_1 \right| + \left| x_2 \right| \leq 1 \right\} .
\end{equation}
In dimension $d$, the {\bf cross-polytope} $\Diamond_d$ \label{cross polytope}
can be defined similarly by its {\bf hyperplane description}
\begin{equation} \label{crosspolytopehyperplanes}
\Diamond_d := \left\{ \left( x_1, x_2, \dots, x_d \right) \in \mathbb{R}^d \bigm | \, \left| x_1 \right| + \left| x_2 \right| + \dots + \left| x_d \right| \leq 1 \right\} .
\end{equation}
The cross-polytope is also, by definition, the unit ball in the $L_1$-norm on Euclidean space, and from this perspective a very natural object. In $\mathbb R^3$,
the cross-polytope $\Diamond_3$ is often called an {\bf {octahedron}}. \index{octahedron}
In this section we only work out the $2$-dimensional case of the Fourier transfrom of the crosspolytope,
In Chapter \ref{chapter.Brion}, we will work out the
Fourier transform of any $d$-dimensional cross-polytope, $\hat 1_{\Diamond_d}$,
because we will have more tools at our disposal.
Nevertheless, it's instructive to compute $\hat 1_{\Diamond_2}$ via brute-force for $d=2$ here,
in order to gain some practice.
\begin{example} \label{cross-polytope example in R^2}
\rm{
Using the definition of the Fourier transform, we first compute the FT of the $2$-dimensional
cross polytope:
\begin{align}
\hat 1_{\Diamond_2}(\xi) &:= \int_{\Diamond_2} e^{-2\pi i \langle \xi, x \rangle} dx.
\end{align}
In $\mathbb{R}^2$, we may write $\Diamond_2$ as a union of the following $4$ triangles:
\begin{align*}
\Delta_1&:= \conv ( \icol{0\{\bf 0}}, \icol{1\{\bf 0}}, \icol{0\{\bf 1}} )\\
\Delta_2&:= \conv ( \icol{0\{\bf 0}}, \icol{-1\{\bf 0}}, \icol{0\{\bf 1}} )\\
\Delta_3&:= \conv ( \icol{0\{\bf 0}}, \icol{-1\{\bf 0}}, \icol{0\\-1} )\\
\Delta_4&:= \conv ( \icol{0\{\bf 0}}, \icol{1\{\bf 0}}, \icol{0\\-1} ).
\end{align*}
Since these four triangles only intersect in lower-dimensional subsets of $\mathbb{R}^2$, the $2$-dimensional integral vanishes on such lower dimensional subsets, and we have:
\begin{equation} \label{transform of 2d crosspolytope}
\hat 1_{\Diamond_2}(\xi) = \hat 1_{\Delta_1}(\xi) + \hat 1_{\Delta_2}(\xi)
+ \hat 1_{\Delta_3}(\xi) + \hat 1_{\Delta_4}(\xi).
\end{equation}
Recalling from equation \eqref{actual FT of the standard simplex}
of example \ref{standard simplex FT}
that the Fourier transform of the standard simplex
\index{standard simplex}
$\Delta_1$ is
\begin{equation}\label{simplex transform}
\hat 1_{\Delta_1}(\xi) = \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{1}{\xi_1 \xi_2}
+ \frac{\ e^{-2\pi i \xi_1} }{(-\xi_1 + \xi_2) \xi_1}
+ \frac{ \ e^{-2\pi i \xi_2} }{( \xi_1 - \xi_2) \xi_2}
\right),
\end{equation}
we can compute $\hat 1_{\Delta_2}(\xi)$, by reflecting $\Delta_2$ about the $x_2-axis$ (the Jacobian of this transformation is $1$), and
using the already-computed transform \eqref{simplex transform} of $\Delta_1$:
\begin{align*}
\hat 1_{\Delta_2}(\xi_1, \xi_2) &:= \int_{\Delta_2} e^{-2\pi i (x_1 \xi_1 + x_2 \xi_2)} dx \\
&= \int_{\Delta_1} e^{-2\pi i (-x_1 \xi_1 + x_2 \xi_2)} dx \\
&= \int_{\Delta_1} e^{-2\pi i (x_1 (-\xi_1) + x_2 \xi_2)} dx \\
&= \hat 1_{\Delta_1}(-\xi_1, \xi_2)).
\end{align*}
Similarly, we have $\hat 1_{\Delta_3}(\xi_1, \xi_2) = \hat 1_{\Delta_1}(-\xi_1, -\xi_2)$, and
$\hat 1_{\Delta_4}(\xi_1, \xi_2) = \hat 1_{\Delta_1}(\xi_1, -\xi_2)$.
Hence we may continue the computation from equation \ref{transform of 2d crosspolytope} above, putting all the pieces back together:
\begin{align} \label{Fourier transform of 2d crosspolytope}
\hat 1_{\Diamond_2}(\xi) &= \hat 1_{\Delta_1}(\xi_1, \xi_2) + \hat 1_{\Delta_1}(-\xi_1, \xi_2)
+ \hat 1_{\Delta_1}(-\xi_1, -\xi_2) + \hat 1_{\Delta_1}(\xi_1, -\xi_2) \\
&= \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{1}{\xi_1 \xi_2}
+ \frac{-\ e^{2\pi i \xi_1} }{(-\xi_1 + \xi_2) \xi_1}
+ \frac{ - \ e^{2\pi i \xi_2} }{( \xi_1 - \xi_2) \xi_2}
\right) \\
&+ \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{-1}{\xi_1 \xi_2}
+ \frac{\ e^{-2\pi i \xi_1} }{(\xi_1 + \xi_2) \xi_1}
+ \frac{ \ e^{2\pi i \xi_2} }{( \xi_1 + \xi_2) \xi_2}
\right) \\
&+ \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{1}{\xi_1 \xi_2}
+ \frac{e^{-2\pi i \xi_1} }{(\xi_1 - \xi_2) \xi_1}
+ \frac{e^{-2\pi i \xi_2} }{( -\xi_1 + \xi_2) \xi_2}
\right) \\
&+ \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{-1}{\xi_1 \xi_2}
+ \frac{e^{2\pi i \xi_1} }{(\xi_1 + \xi_2) \xi_1}
+ \frac{e^{-2\pi i \xi_2} }{( \xi_1 + \xi_2) \xi_2}
\right) \\
&= -\frac{1}{2\pi^2} \left(
\frac{\cos(2\pi \xi_1) }{(\xi_1 - \xi_2) \xi_1}
+ \frac{\cos(2\pi \xi_2) }{(-\xi_1 + \xi_2) \xi_2}
+ \frac{\cos(2\pi \xi_1) }{(\xi_1 + \xi_2) \xi_1}
+ \frac{\cos(2\pi \xi_2) }{(\xi_1 + \xi_2) \xi_2}
\right) \\ \label{formula 1 for the FT of the 2-d crosspolytope}
&= -\frac{1}{\pi^2} \left(
\frac{ \cos(2\pi \xi_1) - \cos(2\pi \xi_2) }{ (\xi_1 + \xi_2)( \xi_1 - \xi_2) }
\right).
\end{align}
}
\hfill $\square$
\end{example}
It's time to mention another important relationship between the cross-polytope $\Diamond$
\index{cross-polytope}
and the cube ${\mathcal P}~:= ~[-1, 1]^d$.
To see this relationship, we define, for any polytope ${\mathcal P} \subset \mathbb{R}^d$, its {\bf dual polytope}:
\begin{equation}\label{dual polytope, definition}
{\mathcal P}^* := \left\{ x\in \mathbb{R}^d \bigm | \, \langle x, y\rangle \leq 1, \text{ for all } y \in {\mathcal P} \right\}.
\end{equation}
It is an easy fact (Exercise \ref{duals of each other}) that in $\mathbb{R}^d$, the cross-polytope $\Diamond$ and the cube ${\mathcal P}:= [-1, 1]^d$ are dual to each other, as in the figure below.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.0in]{duals.jpg}
\end{center}
\caption{Left: a page from
Kepler's book, \emph{Harmonices Mundi} ($1619$), showing the author's interest in various dual polytopes, over $400$ years ago. Right: The cube and the cross-polytope as duals of each other. } \label{duals}
\end{figure}
\bigskip
\section{Observations and questions}
Now we can make several observations about all of the formulas that we found so far,
for the Fourier transforms of various polytopes. For the $2$-dimensional cross-polytope, we found that
\begin{equation} \label{again the FT of a 2d crosspolytope}
\hat 1_{\Diamond_2}(\xi) = -\frac{1}{\pi^2} \left(
\frac{ \cos(2\pi \xi_1) - \cos(2\pi \xi_2) }{ (\xi_1 + \xi_2)( \xi_1 - \xi_2) }
\right).
\end{equation}
\begin{enumerate}[(a)]
\item \ It is real-valued for all $\xi \in \mathbb{R}^2$, and this is due to the fact that $\Diamond_2$ is symmetric about the origin (see section \ref{Centrally symmetric polytopes}).
\begin{question}
Is it true that {\emph any} symmetric property of a polytope ${\mathcal P}$ is somehow mirrored by a corresponding symmetric property of its Fourier transform?
\end{question}
Although this question is not well-defined at the moment (it depends on how we define `symmetric property'),
it does sound exciting, and we can morph it into a few well-defined questions later.
\item \ The only apparent singularities of the FT in \eqref{again the FT of a 2d crosspolytope}
(though they are in fact removable singularities) are the two lines
$\xi_1 - \xi_2=0$ and $\xi_1 + \xi_2=0$, and these two
lines are {\it perpendicular} to the facets of $\Diamond_2$,
which is not a coincidence (see Chapter \ref{Stokes' formula and transforms}).
\item \ It is always true that the Fourier transform of a polytope is an entire function, by
Lemma \ref{FT of a polytope is entire}, so that the singularities
in the denominator $ (\xi_1 + \xi_2)( \xi_1 - \xi_2)$ of
\eqref{again the FT of a 2d crosspolytope} must be removable singularities!
\item The denominators of all of the FT's so far are always products of {\bf linear forms} in $\xi$.
\begin{question} \rm{[Rhetorical]}
Is it true that the Fourier transform of any polytope is always a finite sum of rational functions times an exponential, where the denominators of the rational functions are always products of linear forms?
\end{question}
{\bf Answer}: (spoiler alert) Yes! It's too early to prove this here, but we will do so in Theorem \ref{brion2}.
\item \ We may retrieve the volume of $\Diamond_2$ by letting $\xi_1$ and $\xi_2$ tend to zero
(Exercise \ref{retreiving volume of 2d.crosspolytope}), as always. Doing so, we obtain
$\lim_{\xi \rightarrow 0} \hat 1_{\Diamond_2}(\xi) = 2 = \text{Area}(\Diamond_2)$.
\end{enumerate}
\section*{Notes}
\begin{enumerate}[(a)]
\item Another way to compute $1_{\Diamond}(\xi)$ for the $2$-dimensional cross-polytope $\Diamond$ is by starting with the square $[-\frac{1}{2}, \frac{1}{2}]^2$ and applying a rotation of the plane by $\pi/4$, followed by a simple dilation. Because we know that linear transformations interact in a very elegant way with the FT, this method gives an alternate approach
for the Example \ref{cross-polytope example in R^2} in $\mathbb{R}^2$.
However, this method no longer works for the cross-polytope in dimensions $d \geq 3$,
where it is not (yet) known if there is a simple way to go from the FT of the cube to the FT of the cross-polytope.
\index{cross-polytope}
More generally, one may ask:
\begin{question}
is there a nice relationship between the FT of a polytope ${\mathcal P}$ and the FT of its dual?
\end{question}
\item We note that $P_1({x})$ is defined to be
equal to $0$ at the integers, because its Fourier series naturally converges to the mean of
the discontinuity of the function, at each integer.
\item It has been known since the work of Riemann that the Bernoulli numbers occur as special values of the Riemann zeta function (see Exercise \ref{Riemann zeta function, and Bernoulli numbers}). Similarly, the
Hurwitz zeta function, defined for each fixed $x >0$ by
\[
\zeta(s, x) := \sum_{n =0}^\infty \frac{1}{(n+x)^s},
\]
has a meromorphic continuation to all of $\mathbb{C}$, and its special values at the negative integers are the Bernoulli polynomials $B_n(x)$ (up to a multiplicative constant).
\item There are sometimes very unusual (yet useful) formulations for the Fourier transform of certain functions.
Ramanujan \cite{Ramanujan1} discovered the following remarkable formula
for the Fourier transform of the Gamma function:
\begin{equation}
\int_{\mathbb{R}} |\Gamma(a + iy)| e^{-2\pi i \xi y} dy =
\frac{ \sqrt{\pi} \ \Gamma(a) \Gamma( a + \frac{1}{2})}{\cosh(\pi \xi)^{2a}},
\end{equation}
valid for $a>0$. For example with $a:= \frac{1}{2}$, in the language of this chapter we have the Fourier pair $\{ |\Gamma(\frac{1}{2} + iy)|, \frac{\pi}{\cosh(\pi \xi)} \}$.
\bigskip
\end{enumerate}
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
Problems worthy of attack prove their worth by fighting back.
-- Paul Erd\"os
\end{quote}
\medskip
\begin{prob} \label{transform.of.interval.a.to.b} $\clubsuit$
Show that the Fourier transform of the closed interval $[a, b]$ is:
\[
\hat 1_{[a,b]}(\xi) =\frac{ e^{-2\pi i \xi a} - e^{-2\pi i \xi b} }{2\pi i \xi},
\]
for $\xi \not=0$.
\end{prob}
\medskip
\begin{prob} \label{transform.of.unit.cube}
Show that the Fourier transform of the unit cube $C:= [0,1]^d \subset \mathbb{R}^d$ is:
\begin{equation}
\hat 1_{C}(\xi) = \frac{1}{(2\pi i)^d} \prod_{k=1}^d \frac{ 1- e^{-2\pi i \xi_k} }{ \xi_k },
\end{equation}
valid for all $\xi \in \mathbb{R}^d$, except for the union of hyperplanes defined by \\
$H := \left\{ x \in \mathbb{R}^d \bigm | \xi_1 = 0 \text{ or } \xi_2 = 0 \dots \text{ or } \xi_d = 0 \right\}$.
\end{prob}
\medskip
\begin{prob}
Suppose we are given two polynomials $p(x)$ and $q(x)$, of degree $d$. If there are $d+1$ distinct points
$\{z_1, \dots, z_{d+1}\}$ in the complex plane such that $p(z_k) = q(z_k)$ for $k = 1, \dots, d+1$, show that the two polynomials are identical. (Hint: consider $(p-q)(z_k)$)
\end{prob}
\medskip
\begin{prob} \label{brute force Bernoulli polys}
To gain some facility with generating functions, show by a brute-force computation with Taylor series that
the coefficients on the right-hand-side of equation \eqref{generating function for Bernoulli polynomials}, which are
called $B_n(x)$ by definition, must in fact be polynomials in $x$.
In fact, your direct computations will show that for all $n \geq 1$, we have
\[
B_n(x) = \sum_{k=0}^n {n \choose k} B_{n-k} \ x^k,
\]
where $B_j$ is the $j$'th Bernoulli number. \index{Bernoulli number}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Reflection property for B_n(x)}
Show that for all $n \geq 1$, we have
\[
B_n(1-x) = (-1)^n B_n(x).
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{difference of Bernoulli polys}
Show that for all $n \geq 1$, we have
\[
B_n(x+1) - B_n(x) = n x^{n-1}.
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{derivative of Bernoulli polys}
Show that for all $n \geq 1$, we have
\[
\frac{d}{dx} B_n(x) = n B_{n-1}(x).
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{historical origin of Bernoulli poly}
Prove that:
\[
\sum_{k=1}^{n-1} k^{d-1} = \frac{ B_d(n) - B_d }{ d },
\]
for all integers $d \geq 1$ and $n \geq 2$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{Bernoulli Polynomials} \index{Bernoulli polynomial}
Show that the periodic Bernoulli polynomials $P_n(x) := B_n(\{ x \})$, for all $n\geq 2$, have the following Fourier series:
\begin{equation}
P_n(x) = -\frac{n!}{(2\pi i)^n} \sum_{k \not=0} \frac{e^{2\pi i k x}}{k^n},
\end{equation}
valid for all $x \in \mathbb{R}$. For $n \geq 2$, these series are absolutely convergent. We note that from the definition above, $B_n(x) = P_n(x)$ when $x\in (0,1)$.
\end{prob}
\medskip
\begin{prob} \label{Raabe's identity for { } via Fourier series}
Show that the greatest integer function $\floor{x}$ (often called the `floor function') enjoys the
property:
\[
\sum_{k=0}^{N-1} \floor{ x + \frac{k}{N} } = \floor{Nx},
\]
for all $x\in \mathbb{R}$, and all positive integers $N$, and that in the same range we also have
\[
\sum_{k=0}^{N-1} \left\{ x + \frac{k}{N} \right\} = \left\{ Nx \right\}.
\]
\end{prob}
\medskip
\begin{prob}
Show that the Bernoulli polynomials enjoy the following identity, proved by Joseph Ludwig Raabe in 1851:
\[
B_n(Nx) = N^{n-1} \sum_{k=0}^{N-1} B_n\left( x + \frac{k}{N} \right),
\]
for all $x\in \mathbb{R}$, all positive integers $N$, and for each $n \geq 1$.
Notes. Such formulas, in these last two exercises, are also called ``multiplication Theorems'', and they hold
for many other functions, including the Gamma function, the dilogarithm, the Hurwitz zeta function, and many more.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{another definition for Bernoulli polynomials}
Here we give a different method for defining the Bernoulli polynomials, based on the following three properties that they enjoy:
\begin{enumerate}
\item $B_0(x) = 1$.
\item For all $n \geq 1, \frac{d}{dx} B_n(x) = n B_{n-1}(x)$.
\item For all $n \geq 1$, we have $\int_0^1 B_n(x) dx = 0$.
\end{enumerate}
Show that the latter three properties imply the original defining property of the Bernoulli polynomials
\eqref{generating function for Bernoulli polynomials}.
\end{prob}
\medskip
\begin{prob}
Here is a more explicit, useful recursion for computing the Bernoulli polynomials. Show that
\[
\sum_{k=0}^{n-1} {n \choose k} B_k(x) = n x^{n-1},
\]
for all $n \geq 2$.
\end{prob}
\medskip
\begin{prob} \label{B_7}
Use the previous exercise, together with the known list the first $6$ Bernoulli polynomials that appear
in equation \ref{B_6}, to compute $B_7(x)$.
\end{prob}
\medskip
\begin{prob} \label{odd Bernoulli numbers}
Show that for odd $k \geq 3$, we have $B_k = 0$.
\end{prob}
\medskip
\begin{prob} \label{ Bernoulli numbers alternate in sign}
Show that the even Bernoulli numbers alternate in sign. More precisely, show that
\[
(-1)^{n+1} B_{2n} \geq 0,
\]
for each positive integer $n$.
\end{prob}
\medskip
\begin{prob} \label{vanishing identity for Beroulli numbers}
Show that the Bernoulli numbers enjoy the recursive property:
\[
\sum_{k=0}^n {n+1 \choose k}B_k = 0,
\]
for all $n \geq 1$.
\end{prob}
\medskip
\begin{prob} \label{asymptotics for Beroulli numbers}
Show that the Bernoulli numbers enjoy the following asymptotics:
\[
B_{2n} \sim 2 \frac{(2n)!}{(2\pi)^{2n}}
\]
as $n\rightarrow \infty$. Here we are using the usual notation for asymptotic functions, namely that
$f(n) \sim g(n)$ as $n\rightarrow \infty$ if $\lim_{n\rightarrow \infty} \frac{f(n)}{g(n)} \rightarrow 1$.
\end{prob}
\medskip
\begin{prob} \label{Fresnel} $\clubsuit$
Show that the following integrals converge and have the closed forms:
\begin{align}
\int_{-\infty}^\infty \cos(x^2) dx &= \sqrt{\frac{\pi}{2}}, \\
\int_{-\infty}^\infty \sin(x^2) dx &= \sqrt{\frac{\pi}{2}}.
\end{align}
Notes. These integrals are called Fresnel integrals, and they are related to the Cornu spiral, which was created by Marie Alfred Cornu. Marie used the spiral
as a tool for computing diffraction patterns that arise naturally in optics.
\end{prob}
\medskip
\begin{prob}
Prove the following Gamma function identity, using the sinc function:
\[
\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1+x)\Gamma(1-x)},
\]
for all $x \not\in \mathbb{Z}$.
Notes. This identity is often called Euler's reflection formula. $\Gamma(x):= \int_0^\infty e^{-t} t^{x-1} dt$ is by definition
the Gamma function,
where the integral converges for all $x > 0$ (see Section \ref{Volume of the ball, the Gamma function} for more on the $\Gamma$ function).
\end{prob}
\medskip
\begin{prob}
\label{retreiving volume of 2d.crosspolytope}
$\clubsuit$ Using the formula for the Fourier transform of the $2$-dimensional cross-polytope $\Diamond$, derived in the text,
namely
\[
\hat 1_{\Diamond}(\xi) =
-\frac{1}{\pi^2} \left(
\frac{ \cos(2\pi \xi_1) - \cos(2\pi \xi_2) }{ \xi_1^2 - \xi_2^2}
\right),
\]
find the area of $\Diamond$ by letting $\xi \rightarrow 0$ in the latter formula.
\end{prob}
\medskip
\begin{prob}
\label{Elementary bounds for sin(x), sinc(x)}
Some elementary but very useful bounds for trig functions are developed here.
\begin{enumerate}[(a)]
\item Prove that
\[
\frac{2}{\pi} < \frac{\sin x}{x} \leq 1,
\]
where the left inequality holds for $ 0 < x < \frac{\pi}{2}$, and the right inequality holds for $x\in \mathbb{R}$.
\item \label{Elementary trig bounds, part b}
Prove that
\[
\frac{2x}{\pi} \leq |1-e^{ix} | \leq |x|,
\]
where the left inequality holds for $|x| \leq \pi$, and the right inequality holds for $x\in \mathbb{R}$.
\item Prove that
\[
\frac{2x^2}{\pi^2} \leq |1-\cos x | \leq \frac{x^2}{2},
\]
where the left inequality holds for $|x| \leq \pi$, and the right inequality holds for $x\in \mathbb{R}$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
\label{divergence of |sinc|}
$\clubsuit$
Show that
$\int_{-\infty}^\infty \Big| \frac{\sin(\pi x)}{\pi x} \Big| dx = \infty$.
\end{prob}
\medskip
\begin{prob}
There are (at least) two different ways of periodizing a given function $f:\mathbb{R} \rightarrow \mathbb{C}$
with respect to $\mathbb{Z}$.
First, we can define $F_1(x) := f(\{x\})$, so that $F_1$ is periodic on $\mathbb{R}$ with period~$1$.
Second, we may also define $F_2(x) := \sum_{n\in \mathbb{Z}} f(x+n)$, which is also a periodic function on $\mathbb{R}$ with period $1$.
Find an integrable (meaning that $\int_\mathbb{R} f(x)dx$ converges) function $f$ for which $F_1 \not= F_2$, as functions.
Notes. \ In Chapter \ref{Fourier analysis basics}, we will see that the latter function $F_2(x) := \sum_{n\in \mathbb{Z}} f(x+n)$ captures a lot more information about $f$, and often captures all of $f$ as well.
\end{prob}
\medskip
\begin{prob} \label{symmetrized parallelepiped} \index{parallelepiped}
Given linearly independent vectors $w_1, \dots, w_d \in \mathbb{R}^d$, let $v:= -\frac{w_1 + \cdots + w_d}{2}$, and define
$Q:= \{ v+ \sum_{k=1}^d \lambda_k w_k \mid \text{ all } \lambda_k \in [0, 1] \}$, a parallelepiped.
Show that $Q$ is symmetric about the origin, in the sense that
$x \in Q \iff -x \in Q$.
\end{prob}
\medskip
\begin{prob}
\label{duals of each other}
$\clubsuit$ Show that the $d$-dimensional cross-polytope $\Diamond$ and the cube
$\square:= [-1, 1]^d$ are dual to each other.
\end{prob}
\medskip
\begin{prob}
\label{2-dim'l formula for triangle}
Prove the following $2$-dimensional integral formula:
\begin{align}
\int_{\lambda_1, \lambda_2 \geq 0 \atop \lambda_1 + \lambda_2 \leq 1}
e^{A \lambda_1 } e^{B \lambda_2 } d\lambda_1 d\lambda_2
= \frac{ B e^A - A e^B }{AB(A-B)} +\frac{1}{AB},
\end{align}
valid for all $A, B \in \mathbb{C}$ such that $AB(A-B) \not=0$.
\end{prob}
\medskip
\begin{prob}
\label{FT of a general simplex, brute-force}
Using the ideas of Example \ref{FT of a general triangle}, prove (by induction on the dimension) that the Fourier transform of a general $d$-dimensional simplex $\Delta \subset \mathbb{R}^d$ is given by:
\begin{equation}
\hat 1_{\Delta}(\xi) = (\vol \Delta) d!
\sum_{j=1}^N
\frac{e^{-2\pi i \langle v_j, \xi \rangle}}{\prod_{1\leq k \leq d} \langle v_j-v_k, \xi \rangle }
[k\not=j],
\end{equation}
for all $\xi \in \mathbb{R}^d$, where the vertex set of ${\mathcal P}$ is $\{ v_1, \dots, v_N\}$.
\end{prob}
\medskip
\begin{prob}[Abel summation by parts] $\clubsuit$ \label{Abel summation by parts}
Here we prove the straightforward but very useful technique of Niels Abel, called {\bf Abel summation by parts}.
\index{Abel, Niels} \index{Abel summation by parts}
Suppose we are given two sequences $\{a_n\}_{n=1}^\infty$, and $\{b_n\}_{n=1}^\infty$. We define the finite partial sums
$B_n:= \sum_{k=1}^n b_k$. Then we have
\begin{equation} \label{actual Abel summation}
\sum_{k=1}^n a_k b_k = a_n B_n + \sum_{k=1}^{n-1} B_k(a_k - a_{k+1}),
\end{equation}
for all $n\geq 2$.
\end{prob}
Notes. \ Using the forward difference operator, it's easy to recognize identity \eqref{actual Abel summation}
as a discrete version of integration by parts.
\medskip
\begin{prob}[{\bf Dirichlet's convergence test}] $\clubsuit$ \label{Dirichlet's convergence test}
\index{Dirichlet's convergence test}
Suppose we are given a real sequence $\{a_n\}_{n=1}^\infty$, and a complex sequence
$\{b_n\}_{n=1}^\infty$,
such that
\begin{enumerate}[(a)]
\item $\{a_n\}$ is monotonically decreasing to $0$, and
\item $| \sum_{k=1}^n b_k | \leq M$, for some positive constant $M$, and all $n \geq 1$.
\end{enumerate}
Then $\sum_{k=1}^\infty a_k b_k$ converges.
\end{prob}
\medskip
\begin{prob} \label{first Dirichlet kernel}
Prove that for all $x \in \mathbb{R}-\mathbb{Z}$, we have the following important identity, called the ``Dirichlet kernel'',
\index{Dirichlet kernel}
named after Peter Gustav Lejeune Dirichlet:
\begin{equation}
\sum_{k= -n}^n e^{2\pi i k x} = \frac{\sin \left( 2\pi x(n + \frac{1}{2}) \right) }{\sin(\pi x)}.
\end{equation}
\end{prob}
\medskip
\begin{prob} \label{exponential sum bound}
For any fixed $x \in \mathbb{R}-\mathbb{Z}$, show that we have the bound on the following exponential sum:
\begin{equation}
\left | \sum_{k= 1}^n e^{2\pi i k x} \right | \leq \frac{1}{ | \sin(\pi x) | }.
\end{equation}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{rigorous convergence of P_1(x)}
Prove that $\sum_{m = 1}^\infty \frac{e^{2\pi i m a}}{m}$ converges, given any fixed $a \in \mathbb{R} - \mathbb{Z}$.
Notes. \ We see that, although $\sum_{m = 1}^\infty \frac{e^{2\pi i m a}}{m}$ does not converge absolutely,
Abel's summation formula \eqref{actual Abel summation} gives us
\[
\sum_{k = 1}^n \frac{e^{2\pi i k a}}{k} = \frac{1}{n}\sum_{r=1}^n e^{2\pi i r a}
+ \sum_{k=1}^{n-1} \Big( \sum_{r=1}^k e^{2\pi i r a} \Big) \frac{1}{k(k+1)},
\]
and the latter series {\bf does converge absolutely}, as $n \rightarrow + \infty$. So we see that Abel summation transforms one series (that barely converges at all) into
another series that converges more rapidly.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{integral of sinc is 1}
Here we'll prove that
\begin{equation} \label{exercise:the Dirichlet integral}
\int_{-\infty}^\infty \frac{ \sin(\pi t) }{\pi t} dt = 1,
\end{equation}
and this integral is known as ``the Dirichlet integral''.
We notice that due to the evenness of the integrand, after a change of variable
it suffices to prove that
$\int_0^ \infty \frac{ \sin t }{ t} dt =\frac{\pi}{2}$. Comparing this Dirichlet integral with
Exercise \ref{divergence of |sinc|}, we see that there is something very subtle going on here.
\begin{enumerate}[(a)]
\item Define
\begin{equation} \label{Laplace transform of sinc}
F(s):= \int_0^\infty e^{-st} \frac{ \sin t }{ t} dt,
\end{equation}
for each $s>0$. Justify differentiation under the integral sign, and show that
\[
\frac{dF}{ds} = -\int_0^\infty e^{-st} \sin t dt,
\]
\item Show that $\int_0^\infty e^{-st} \sin t dt = \frac{1}{1+s^2}$.
\item Show that $F(s) = C - \tan^{-1} s$, and then show that the constant $C= \frac{\pi}{2}$.
\item Prove that $F$ is a continuous function of $s\in \mathbb{R}_{>0}$, and finally prove that
\[
\lim_{s\rightarrow 0} F(s) = \frac{\pi}{2},
\]
which is the desired result (Here you might want to integrate by parts first, and then use the Dominated convergence theorem).
\end{enumerate}
\end{prob}
Notes. There are many proofs of this famous identity \eqref{exercise:the Dirichlet integral}; here we are only assuming
knowledge of some real analysis. The expression in \eqref{Laplace transform of sinc} is also known as the Laplace transform of the sinc function, and it is a variation of the Fourier transform that we will return to when studying similar transforms of cones in
Section \ref{Fourier Laplace transforms of cones}.
\medskip
\begin{prob} $\clubsuit$
\label{rigorous inversion formula for sinc}
\rm{
Here we give a rigorous proof of the tricky business that for all $x\in \mathbb{R}$, we have
\begin{equation*}
\lim_{N\rightarrow \infty} \int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} e^{-2\pi i \xi x} d\xi = 1_ {[-\frac{1}{2}, \frac{1}{2}]}(x),
\end{equation*}
following an approach of S. Bochner \cite{BochnerBook}.
We begin by noticing that this integral can be easily reduced to a real-valued integral:
\begin{equation*}
\int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} e^{-2\pi i \xi x} d\xi
= \int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} \cos( 2\pi \xi x) d\xi,
\end{equation*}
because for each $x\in \mathbb{R}$, $\int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} \sin( 2\pi \xi x) d\xi = 0$ owing to the oddness of the integrand.
\begin{enumerate}[(a)]
\item \label{tricky middle part with alpha}
Using the result from Exercise \ref{integral of sinc is 1}, prove that
\[
\lim_{N\rightarrow \infty} \int_{-N}^N \frac{\sin(\pi \alpha t)}{\pi t} dt=
\begin{cases}
\ \ 1 & \mbox{if } \alpha >0, \\
\ \ 0 & \mbox{if } \alpha =0, \\
-1 & \mbox{if } \alpha <0.
\end{cases}
\]
\item Finish up by using \ $2\sin t \cos(\alpha t) = \sin(1- \alpha)t + \sin(1+\alpha)t$, thereby showing that the desired
integral
\[
\lim_{N\rightarrow \infty} \int_{-N}^N \frac{\sin(\pi t)}{\pi t} \cos( 2\pi t x) dt
\]
reduces to part \ref{tricky middle part with alpha}.
\end{enumerate}
}
\end{prob}
\chapter{The basics of Fourier analysis}
\label{Fourier analysis basics}
\index{Fourier analysis}
\begin{quote}
``If a function is periodic, then we should try to expand it into its Fourier series, and wonderful things will begin to happen....."
-- Erich Hecke
\end{quote}
\begin{quote}
``. . . Fourier's great mathematical poem.''
[Referring to Fourier's mathematical theory of the conduction of heat]
-- William Thomson Kelvin \index{Kelvin, William Thomson}
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.4in]{cube}
\end{center}
\caption{The unit cube $\square:= [0,1]^3$, in $\mathbb{R}^3$, which tiles the space by translations.
Which other polytopes tile by translations? How can we make mathematical use of such tilings?
In particular, can we give an explicit basis of exponentials for functions defined on $\square$?} \label{crosspic}
\end{figure}
\bigskip
\section{Intuition}
Because we will use tools from Fourier analysis throughout, we introduce them here as an outline of the field,
with the goal of {\it applying} them to the discrete geometry of polytopes, lattices, and their interactions.
We will sometimes introduce a concept by using an intuitive argument, which we call ``fast and loose", but after such
an intuitive argument, we state the precise version of the corresponding theorem.
In this chapter, we will sometimes point to the literature for some of the proofs.
Our goal is to use the necessary tools of Fourier analysis in order to tackle problems in the enumerative combinatorics of polytopes,
in number theory, discrete geometry, and in some other fields.
We emphasize that the Poisson summation formula allows us to {\bf discretize integrals},
in a sense that will be made precise in later chapters.
One pattern that the reader may have already noticed, among all of the examples of Fourier transforms of polytopes computed thus far, is that each of them is a linear combination of a very special kind of rational function of $\xi$, multiplied by a complex exponential that involves a vertex of the polytope:
\begin{equation}\label{structure 1}
\hat 1_{\mathcal P}(\xi) =
\sum_{k=1}^M \frac{1}{\prod_{j=1}^d \left\langle \omega_{j,k}(v_k), \xi \right\rangle} \,
e^{2\pi i \langle v_k, \xi \rangle},
\end{equation}
where the vertices of ${\mathcal P}$ are $v_1, \dots, v_N$, and where $M \geq N$. We observed that in all of our examples thus far, the denominators
are in fact products of linear forms, as in \eqref{structure 1}.
We will be able to see some of the more precise geometric structure for these products of linear forms, which
come from the edges of the polytope, once we learn more about Fourier-Laplace transforms of cones.
It is rather astounding that every single fact about a given polytope ${\mathcal P}$ is somehow
hiding inside these {\bf rational-exponential} functions given by \eqref{structure 1}, due to the fact that the Fourier transform $\hat 1_{\mathcal P}$ is a complete invariant (Lemma \ref{complete invariance of the FT}).
\bigskip
\section{Introducing the Fourier transform on $L^1(\mathbb{R}^d)$}
In the spirit of bringing the reader very quickly up to speed, regarding the applications of Fourier analytic tools, we outline the basics
of the field, and prove some of them. Nowadays, there are many good texts on Fourier analysis, and the reader is encouraged to peruse some of these books (see Note \ref{Fourier books}).
One of the most useful tools for us is the Poisson summation formula. We provide several versions of Poisson summation, each of
which uses a different set of sufficient conditions.
As we will see, the Fourier transform \index{Fourier transform} is a very friendly creature, allowing us to travel back and forth between the ``space domain'' and the ``frequency domain'' to obtain many useful results.
The readers who are already familiar with basics of Fourier analysis may easily skip
this chapter without impeding their understanding of the rest of the book.
Although we enjoy thinking about the warm and cozy Hilbert spaces $L^2(\mathbb{R}^d)$ and $L^2([0, 1]^d)$, there are many subtle convergence issues (and divergence issues) of Fourier series, a whole field onto itself. We won't go there.
However, the very basic convergence issues are still important for us as well, and we want to get the reader up and running.
The function space that immediately come up very naturally is the
the space of {\bf absolutely integrable functions} on $\mathbb{R}^d$:
\[
L^1(\mathbb{R}^d) :=\left\{ f: \mathbb{R}^d \rightarrow \mathbb{C} \bigm | \ \int_{\mathbb{R}^d} |f(x)| dx < \infty \right\}.
\]
Secondly, the space of
{\bf square-integrable functions} on $\mathbb{R}^d$ is defined by:
\[
L^2(\mathbb{R}^d) := \left\{ f: \mathbb{R}^d \rightarrow \mathbb{C} \bigm | \ \int_{\mathbb{R}^d} |f(x)|^2 dx < \infty \right\}.
\]
The usual theory of Fourier transforms progresses by first defining the Fourier transform for functions belonging to $L^1(\mathbb{R}^d)$, which is quite a natural condition, and then later extending the Fourier transform to the $L^2(\mathbb{R}^d)$ space by taking appropriate limits. We initially restrict attention to functions $f \in L^1(\mathbb{R}^d)$.
There are many fascinating facts about all of these functions spaces. For practice, let's ask:
\begin{question} \rm{[Rhetorical]}
Given two functions $f, g \in L^2(\mathbb{R}^d)$, is their product always in $L^1(\mathbb{R}^d)$?'
\end{question}
Well, by the Cauchy-Schwartz inequality for the
Hilbert space $L^2(\mathbb{R}^d)$, we have:
\begin{equation}\label{product of two L^2 functions is L^1}
\int_{\mathbb{R}^d} |f(x) g(x)|dx \leq
\left(\int_{\mathbb{R}^d} |f(x)|^2 dx \right)^{\frac{1}{2}} \left(\int_{\mathbb{R}^d} |g(x)|^2 dx\right)^{\frac{1}{2}} < \infty,
\end{equation}
the latter inequality holding by the assumption $f, g \in L^2(\mathbb{R}^d)$. So the product $f(x) g(x)$ is indeed in $L^1(\mathbb{R}^d)$. This is the first sign that there
are fascinating links between $L^1$ functions and $L^2$ functions.
Moreover, the utility of the Cauchy-Schwarz inequality should never be underestimated.
It's interesting that $L^1(\mathbb{R}^d)$ is not a Hilbert space, as we can easily show by exhibiting a counter-example
to the Cauchy-Schwarz inequality, as follows.
\bigskip
\begin{example} \label{cool CS counterexample}
\rm{
We claim that the Cauchy-Schwarz inequality
is false in $L^1(\mathbb{R})$. If the Cauchy-Schwarz inequality was true here, then \eqref{product of two L^2 functions is L^1} would be valid
for all functions
$f, g \in L^1(\mathbb{R})$. But as a counterexample, let
\[
f(x):= 1_{(0,1)}(x) \frac{1}{\sqrt x}.
\]
It's easy to see that $f \in L^1(\mathbb{R})$:
\[
\int_{\mathbb{R}} 1_{(0,1)}(x) \frac{1}{\sqrt x} dx = \int_0^1 \frac{1}{\sqrt x} dx = \frac{1}{2}.
\]
But $ \int_{\mathbb{R}} f(x) \cdot f(x) dx = \int_0^1 \frac{1}{x} dx$ diverges, so that we do not have a Cauchy-Schwarz inequality in $L^1(\mathbb{R})$, because here both the left-hand-side and the right-hand-side of such an inequality do not even converge.
However, if two functions $f, g$ are bounded on $\mathbb{R}$, and absolutely integrable on $\mathbb{R}$, then we do have a Cauchy-Schwartz inequality
for the pair $f, g$, and we let the reader enjoy its verification.
}
\hfill $\square$
\end{example}
An easy but extremely important inequality is the {\bf triangle inequality for integrals}, as follows.
\begin{thm} \label{triangle inequality for integrals}
For any $f\in L^1(\mathbb{R}^d)$, and any measurable subset $S \subset \mathbb{R}^d$,
we have:
\begin{equation} \index{triangle inequality for integrals}
\Big| \int_S f(x) dx \Big| \leq \int_S | f(x) | dx.
\end{equation}
\end{thm}
\begin{proof}
Letting $z:= \int_S f(x) dx \in \mathbb{C}$, we may write $|z| = \alpha z$, for a (unique) complex $\alpha$ on the unit circle.
We let $u$ be the real part of $\alpha f:= u + iv $, so that $u \leq \sqrt{u^2 + v^2} = |\alpha f| = |f|$.
Altogether, we have:
\begin{equation*}
\Big| \int_S f(x) dx \Big| = \alpha \int_S f(x) dx = \int_S \alpha f(x) dx = \int_S u(x) dx
\leq \int_S |f(x)| dx.
\end{equation*}
In the third equality, we used the fact that $\int_S \alpha f(x) dx$ is real, which follows
from the first two equalities:
$\int_S \alpha f(x) dx = \Big| \int_S f(x) dx \Big|$.
\end{proof}
\medskip
\begin{cor} If $f$ is bounded on a measurable set $S\subset \mathbb{R}^d$ by a constant $M>0$, then:
\begin{equation}
\Big| \int_S f(x) dx \Big| \leq {\rm measure} (S) \cdot M.
\end{equation}
\begin{proof}
\[
\Big| \int_S f(x) dx \Big| \leq \int_S | f(x) | dx \leq \int_S M dx = {\rm measure} (S) \cdot M,
\]
where the first inequality uses the triangle inequality for integrals, namely Theorem \eqref{triangle inequality for integrals}, and the second inequality uses
the boundedness assumption on $f$.
\end{proof}
\end{cor}
\medskip
We've defined the Fourier transform before, and we remind the reader that for any function $f\in L^1(\mathbb{R}^d)$,
the {\bf Fourier transform} of $f$ \index{Fourier transform} is
\begin{equation}
\hat f(\xi) := \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, \xi \rangle} dx.
\end{equation}
Where does this definition really come from? One motivation comes from the inner product for functions
(in $L^2(\mathbb{R}^d)$),
where we project a function $f$ onto each exponential function:
\[
\langle f, e^{2\pi i \langle x, \xi \rangle} \rangle := \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, \xi \rangle} dx.
\]
Another motivation comes from the proof of the Poisson summation formula - eq. \eqref{finally, the FT} below, which
shows a crucial connection between the Fourier transform of $f$ and the Fourier coefficients of the periodized function
$\sum_{n\in \mathbb{Z}^d} f(x+n)$.
One of the first things we might notice is:
\begin{claim}
The Fourier transform is a bounded linear operator.
\end{claim}
The Fourier transform is a linear operator, by the linearity
of the integral: $\widehat{(f+g)} = \hat f + \hat g$, and it is a bounded operator
due to the elementary estimate in \eqref{boundedness of FT} below.
A natural question is: where does the Fourier transform take a function $f\in L^1(\mathbb{R}^d)$?
An immediate partial answer is that for any $f\in L^1(\mathbb{R}^d)$, we have:
\[
\hat f \in B(\mathbb{R}^d),
\]
where $B(\mathbb{R}^d):=
\{f:\mathbb{R}^d \rightarrow \mathbb{C} \bigm | \, \exists M>0 \text{ such that } |f(x)| < M, \text{ for all } x \in \mathbb{R}^d \}$ is the space of bounded functions on $\mathbb{R}^d$. Here the constant $M$ depends only on $f$.
To see this, consider:
\begin{align}
| \hat f(\xi) | &:= \left | \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, \xi \rangle} dx \right |
\leq
\int_{\mathbb{R}^d} \left | f(x) e^{-2\pi i \langle x, \xi \rangle} \right | dx \\ \label{boundedness of FT}
&= \int_{\mathbb{R}^d} \left | f(x) \right | dx := \| f \|_{L^1(\mathbb{R}^d)},
\end{align}
where we used Theorem \ref{triangle inequality for integrals}, the triangle inequality for integrals, together with the fact that
$ \left | e^{-2\pi i \langle x, \xi \rangle} \right | =1$.
\begin{example}
\rm{
Let's bound the Fourier transform of an integrable indicator function $1_S$, for any measurable set $S\subset \mathbb{R}^d$:
\[
| \hat 1_S(\xi) |:= \left | \int_{S} e^{-2\pi i \langle x, \xi \rangle} dx \right |
\leq \int_{S} \left | e^{-2\pi i \langle x, \xi \rangle} \right | dx = \int_{S} dx = {\rm measure} (S).
\]
In particular, for any polytope ${\mathcal P}\subset \mathbb{R}^d$,
\[
|\hat 1_{\mathcal P}(\xi) | \leq \vol {\mathcal P}, \text{ for all } \xi \in \mathbb{R}^d.
\]
We already know that $\hat 1_{\mathcal P}(0) = \vol {\mathcal P}$, so it's natural to ask whether the maximum allowed value of $\vol {\mathcal P}$ can also be achieved by a nonzero $\xi \in \mathbb{R}^d$; or perhaps it may be the case that we always have the strict inequality
$|\hat 1_{\mathcal P}(\xi) | < \vol {\mathcal P}, \text{ for all nonzero } \xi \in \mathbb{R}^d$?
(Exercise \ref{strictly less than the FT at zero}).
}
\hfill $\square$
\end{example}
But a lot more is true for absolutely integrable functions.
\medskip
\begin{lem} \label{uniform continuity}
If $f\in L^1(\mathbb{R}^d)$, then $\hat f$ is uniformly continuous on $\mathbb{R}^d$.
\begin{proof}
We fix any $\xi \in \mathbb{R}^d$, and $h \in \mathbb{R}^d$, and we compute:
\begin{align*}
\hat f(\xi + h) - f(\xi) &:=
\int_{\mathbb{R}^d} f(x) \Big( e^{-2\pi i \langle x, \xi + h \rangle} - e^{-2\pi i \langle x, \xi \rangle} \Big) dx \\
&= \int_{\mathbb{R}^d} f(x)e^{-2\pi i \langle x, \xi \rangle} \Big( e^{-2\pi i \langle x, h \rangle} -1 \Big) dx,
\end{align*}
so by the triangle inequality for integrals, we have
\begin{equation} \label{pre-uniform conv}
| \hat f(\xi + h) - f(\xi)| \leq \int_{\mathbb{R}^d} |f(x)| | e^{-2\pi i \langle x, h \rangle} -1 | dx.
\end{equation}
Letting $g_h(x):= f(x) \Big( e^{-2\pi i \langle x, h \rangle} -1 \Big)$, we see that
\[
|g_h(x)| \leq 2 |f(x)|, \text{ and } \lim_{h \rightarrow 0} |g_h(x)| =0,
\]
using $ |e^{-2\pi i \langle x, h \rangle} -1| \leq 2$. We may now use the dominated convergence theorem, because the functions
$g_h$ are dominated by the absolutely integrable function $2 f$. So we get:
\[
\lim_{h\rightarrow 0} \int_{\mathbb{R}^d} |f(x)| | e^{-2\pi i \langle x, h \rangle} -1 | dx = \int_{\mathbb{R}^d} \lim_{h\rightarrow 0} |f(x)| | e^{-2\pi i \langle x, h \rangle} -1 | dx
= 0.
\]
Because the latter limit is independent of $\xi$, \eqref{pre-uniform conv} tells us that $| \hat f(\xi + h) - f(\xi)| \rightarrow 0$, as $h\rightarrow 0$, uniformly
in $\xi \in \mathbb{R}^d$.
\end{proof}
\end{lem}
It turns out that sometimes we need to measure distance between functions in a manner different than just pointwise convergence. We therefore introduce convergence in the $L^2$~norm. We say that a sequence of functions $f_n:\mathbb{R}^d \rightarrow \mathbb{C}$ converges to a function $f$
{\bf in the $L^2$ norm} if
\begin{equation} \label{convergence in the L^2 norm}
\index{convergence in the $L^2$ norm}
\int_{\mathbb{R}^d} \left| f_n(x) - f(x) \right |^2 dx \rightarrow 0, \text{ as } n \rightarrow \infty,
\end{equation}
for which we also use the notation $\lim_{n\rightarrow \infty} \| f_n - f\|_2 =0$.
It is also very useful to define the $L^p(\mathbb{R}^d)$ spaces, for each $1\leq p < \infty$:
\begin{equation}
L^p(\mathbb{R}^d):= \{ f:\mathbb{R}^d \rightarrow \mathbb{C} \bigm | \int_{\mathbb{R}^d} |f(x)|^p dx < \infty \},
\end{equation}
which naturally extend the $L^1$ and $L^2$ spaces. It is well-known that among all of the $L^p(\mathbb{R}^d)$ spaces, the only one that is a Hilbert space is $L^2(\mathbb{R}^d)$. For the curious reader, the other $L^p(\mathbb{R}^d)$ spaces, for $p \not=2$, also possess some additional structure, namely they are Banach algebras, after identifying two functions that are equal a.e. (see \cite{EinsiedlerWardBook} for details). The development of $L^p$ spaces is very important for Fourier analysis; for the sake of simplicity of exposition, here we will mostly work with $p=1$ and $p=2$.
Similarly to \eqref{convergence in the L^2 norm}, we define {\bf convergence in the $L^p$ norm},
for $1\leq p < \infty$ by
\begin{equation} \label{convergence in the L^p norm}
\int_{\mathbb{R}^d} \left| f_n(x) - f(x) \right |^p dx \rightarrow 0, \text{ as } n \rightarrow \infty,
\end{equation}
for which we also use the notation
\[
\lim_{n\rightarrow \infty} \| f_n - f \|_p = 0.
\]
For a review of some of these various forms of convergence, see the Appendix - Chapter \ref{Appendix A}.
The celebrated {\bf Riemann--Lebesgue lemma}
gives us the basic decay property of the Fourier transform $\hat f(\xi)$ as $|\xi| \rightarrow \infty$.
To prove it, we will use the fact that we can approximate any function $f\in L^1(\mathbb{R}^d)$ with arbitrary precision by using
`step functions' in $\mathbb{R}^d$. More precisely, let a {\bf box in $\mathbb{R}^d$} \index{box}
be defined by ${\mathcal P}:= [a_1, b_1]\times \cdots \times [a_d, b_d]$, and consider
the indicator function $1_{\mathcal P}$ of this box. If we consider the set of all finite sums, taken over all such indicator functions (varying over all boxes), with arbitrary real coefficients, then this set turns out to be dense in $L^1(\mathbb{R}^d)$, in the $L^1$ norm. We record this fact as a lemma.
\begin{lem} \label{box functions dense in L^1}
If $f \in L^1(\mathbb{R}^d)$, then there is a finite sum of indicator functions of boxes that approaches $f$, in the $L^1$ norm.
\hfill $\square$
\end{lem}
\index{Riemann-Lebesgue lemma}
\begin{lem}[Riemann-Lebesgue] \label{Riemann--Lebesgue lemma}
If $f \in L^1(\mathbb{R}^d)$, then:
\[
\lim_{|\xi| \rightarrow \infty} \hat f(\xi) = 0.
\]
\begin{proof}
We first show the result in the case that $f$ is the indicator function of a box. We already know, via
Exercise \ref{transform.of.interval.a.to.b}, that if
${\mathcal P}:= [a_1, b_1]\times \cdots \times [a_d, b_d]$, then
\begin{equation} \label{FT of boxes...}
\hat 1_{\mathcal P}(\xi) = \prod_{k=1}^d
\frac{ e^{-2\pi i \xi_k a_k} - e^{-2\pi i \xi_k b_k} }{2\pi i \xi_k}.
\end{equation}
As $|\xi| \rightarrow \infty$, $\prod_{k=1}^d \xi_k \rightarrow \infty$, while the numerator of \eqref{FT of boxes...} stays bounded, so we've proved
the lemma for indicator functions of boxes. Since $f \in L^1(\mathbb{R}^d)$, we know by Lemma \ref{box functions dense in L^1} that
there exists a sequence of functions $g_n \in L^1(\mathbb{R}^d)$ such that $\|f -g_n\|_1 \rightarrow 0$, as $n\rightarrow \infty$.
Also, by \eqref{FT of boxes...} we know that this sequence already satisfies $\lim_{|\xi| \rightarrow \infty} \hat g_n(\xi) =0$.
Using the elementary inequality \eqref{boundedness of FT},
we get:
\[
\big| \hat f(\xi) - \hat g_n(\xi) \big|
= \big| \widehat{(f - g_n)}(\xi) \big|
\leq \|f -g_n\|_1 \rightarrow 0,
\]
as $n\rightarrow \infty$. Therefore $\lim_{|\xi| \rightarrow \infty} \hat f(\xi) = 0$.
\end{proof}
\end{lem}
With all of the above properties, it is now natural to consider the space of all uniformly continuous functions on $\mathbb{R}^d$ that go to $0$ at infinity:
\begin{equation}
C_0(\mathbb{R}^d):= \{ f: \mathbb{R}^d \rightarrow \mathbb{C} \bigm | f \text{ is uniformly continuous on } \mathbb{R}^d, \text{ and } \lim_{|x| \rightarrow \infty} |f| = 0 \}.
\end{equation}
So although the Fourier transform does not map the space $L^1(\mathbb{R}^d)$ into itself, all of the above results may be summarized as follows.
\begin{lem}
If $f \in L^1(\mathbb{R}^d)$, then $\hat f \in C_0(\mathbb{R}^d)$.
\end{lem}
\begin{proof}
The boundedness of $\hat f$ was given by the inequality $ | \hat f(\xi) | \leq \|f\|_1$ \eqref{boundedness of FT}, the uniform continuity by Lemma \ref{uniform continuity},
and the decay to zero at infinity by Lemma \ref{Riemann--Lebesgue lemma}.
\index{Riemann-Lebesgue lemma}
\end{proof}
Interestingly, there exists an even more precise statement (using convolutions) that involves the $L^2(\mathbb{R}^d)$ space, for the image of the space $L^1(\mathbb{R}^d)$ under the Fourier transform; we state it in \eqref{RudinAmazingConvolutions} below. This
dance beween the $L^1$ and $L^2$ spaces has more to offer.
\begin{lem} \label{both f and its FT in L^1 implies L^2}
If $f \in L^1(\mathbb{R}^d)$ and $\hat f \in L^1(\mathbb{R}^d)$, then both $f, \hat f \in L^2(\mathbb{R}^d)$.
\end{lem}
\begin{proof}
Because $\hat f \in L^1(\mathbb{R}^d)$, we know by the basic inequality \eqref{boundedness of FT} that $f$ must be bounded on $\mathbb{R}^d$: $|f(x)| \leq M$ for some $M>0$. We now compute:
\[
\int_{\mathbb{R}^d} |f(x)|^2 dx \leq \int_{\mathbb{R}^d} M |f(x)| dx= M \int_{\mathbb{R}^d} |f(x)| dx < \infty,
\]
where the last inequality holds because $f \in L^1(\mathbb{R}^d)$ by assumption. So $f \in L^2(\mathbb{R}^d)$.
Finally, because both $f, \hat f \in L^1(\mathbb{R}^d)$, we can invoke Fourier inversion,
namely Theorem \ref{thm:Inverse Fourier transform}, and therefore exactly the same reasoning applies to $\hat f$.
\end{proof}
\section{The torus $\mathbb{R}^d/\mathbb{Z}^d$}
Suppose a function $f:\mathbb{R} \rightarrow \mathbb{C}$ is {\bf periodic on the real line}, with period $1$: $f(x + 1) = f(x)$, for all $x\in \mathbb{R}$. Then we may think
of $f$ as `living' on the unit circle, via the map $x\rightarrow e^{2\pi i x}$ which wraps the real line onto the circle. In this setting, we may also think of the circle as the quotient group $\mathbb{R}/\mathbb{Z}$ (though as we promised, group theory will not be assumed of the reader here).
We may travel along these ideas in the other direction: commencing with any function $g$ whose domain is just $[0, 1)$, we can always extend $g$ by periodicity to the whole real line by
defining $G(x):= \{ x\}$, the fractional part of $x$, for all $x \in \mathbb{R}$. Then $G(x) = g(x)$ for all $x \in \mathbb T$, $G$ is periodic on $\mathbb{R}$,
and therefore we may think of
$g$ as living on the circle $\mathbb T$.
More generally,
we may think of a {\bf periodic function $f:\mathbb{R}^d \rightarrow \mathbb{C}$} as living on the cube $\square := [0,1]^d$,
if we insist that $f$ is periodic in the following sense:
\[
f(x) = f(x + e_k), \text{ for all } x\in \square, \text{ and all } 1\leq k \leq d.
\]
In this case, the $1$-dimensional circle is replaced by the $d$-dimensional torus
\[
{\mathbb T^d}:= \mathbb{R}^d/\mathbb{Z}^d,
\]
which we may also think of as the unit cube $[0, 1]^d$, but with opposite facets `glued together'.
Here we define another infinite-dimensional vector space, namely:
\begin{equation}
L^2({\mathbb T^d}):= \{ f:{\mathbb T^d} \rightarrow \mathbb{C} \bigm | \int_{[0, 1]^d} |f(x)|^2 dx < \infty \}.
\end{equation}
We notice that the domains of the integrals in $L^2({\mathbb T^d})$ are cubes, and hence always compact. So we may therefore expect nicer phenomena to occur in this space.
We also have the space of functions
\begin{equation}
L^1({\mathbb T^d}):= \{ f:{\mathbb T^d} \rightarrow \mathbb{C} \bigm | \int_{[0, 1]^d} |f(x)| dx < \infty \},
\end{equation}
which plays a simpler role than the analogous $L^1(\mathbb{R}^d)$ space we had before. And finally
we also define the useful space of $k$-differentiable functions on the torus:
\begin{equation} \label{k derivatives on the torus}
C^k({\mathbb T^d}):= \{ f:{\mathbb T^d} \rightarrow \mathbb{C} \bigm | f \text{ has $k$ continuous derivatives} \}.
\end{equation}
As a special case, we'll simply denote by $C({\mathbb T^d})$ the space of all continuous functions on the torus.
We emphasize that by definition, all of the latter function spaces consist
of \emph{periodic functions} on the cube $[0, 1]^d$.
Similarly to the inner product on $L^2(\mathbb{R}^d)$, we also have in this new context a natural inner product for the space of square-integrable functions
$f\in L^2({\mathbb T^d})$, defined by:
\begin{equation}
\langle f, g \rangle := \int_{[0,1]^d} f(x) \overline{g(x)} dx,
\end{equation}
making $L^2({\mathbb T^d})$ a Hilbert space.
For each $n\in \mathbb{Z}^d$, we define $e_n: \mathbb{R}^d \rightarrow \mathbb{C}$ by:
\begin{equation} \label{basis for Hilbert space}
e_n(x):= e^{2\pi i \langle n, x\rangle}.
\end{equation}
This countable collection of exponentials turns out to form a complete orthonormal basis for $L^2({\mathbb T^d})$.
The orthogonality is the first step, which we prove next. For the proof that the exponentials span $L^2({\mathbb T^d})$ and are complete,
we refer the reader to \cite{EinsiedlerWardBook}.
\bigskip
\begin{thm} [{\bf Orthogonality relations for the exponentials $e_n(x)$ on the torus}]
\index{orthogonality relations for the exponentials $e_n(x)$}
\label{Orthogonality relations for the exponentials $e_n(x)$}
\begin{equation}
\int_{[0,1]^d} e_n(x) \overline{e_m(x)} dx =
\begin{cases}
1 & \mbox{if } n=m \\
0 & \mbox{if not}.
\end{cases}
\end{equation}
\end{thm}
\begin{proof}
Because of the geometry of the cube, we can proceed in this case by separating the variables. If $n \not= m$, then there is at least one index $k$ for which $n_k \not= m_k$. We compute:
\begin{align*}
\int_{[0,1]^d} e_n(x) \overline{e_m(x)} dx
&= \int_{[0,1]^d} e^{2\pi i \langle n-m, x \rangle} dx \\
&= \int_0^1 e^{2\pi i (n_k -m_k)x_k} dx \int_{[0,1]^{d-1}} \prod_{j\not=k} e^{2\pi i (n_j -m_j)x_j} dx \\
&= \int_0^1 e^{2\pi i (n_k -m_k)x_k} dx \int_{[0,1]^{d-1}} \prod_{j\not=k} e^{2\pi i (n_j -m_j)x_j} dx \\
&= \left( \frac{e^{2\pi i (n_k-m_k)}-1}{2\pi i (n_k-m_k)} \right) \int_{[0,1]^{d-1}} \prod_{j\not=k} e^{2\pi i (n_j -m_j)x_j} dx =0,
\end{align*}
because $n_k -m_k$ is a nonzero integer.
\end{proof}
Because $L^2({\mathbb T^d})$ is also an inner product space, it still enjoys the Cauchy-Schwartz inequality.
Intuitively, the space $L^2({\mathbb T^d})$ should be a cozier little space than $L^1({\mathbb T^d})$. This intuition can be made more
rigorous by the following Lemma, despite the fact that $L^2(\mathbb{R}^d) \not\subset L^1(\mathbb{R}^d)$.
\begin{lem} \label{proper containment of L^2 in L^1 for torus}
We have the proper containment of spaces:
\begin{equation}\label{proper containment of spaces over the torus}
L^2({\mathbb T^d}) \subset L^1({\mathbb T^d}).
\end{equation}
\end{lem}
\begin{proof}
Given $f \in L^2({\mathbb T^d})$, we must show that $f \in L^1({\mathbb T^d})$. Using the Cauchy-Schwartz inequality for
$ L^2({\mathbb T^d})$, with $f$ and the constant function $h(x) \equiv 1$, we have:
\begin{align*}
\int_{\mathbb T^d} |f(x)| dx &= \int_{\mathbb T^d} |f(x) h(x) |dx \\
& \leq
\left( \int_{\mathbb T^d} |f(x)|^2 dx \right)^{\frac{1}{2}}
\left( \int_{\mathbb T^d} |h(x)|^2 dx \right)^{\frac{1}{2}} \\
&= \left( \int_{\mathbb T^d} |f(x)|^2 dx \right)^{\frac{1}{2}},
\end{align*}
so we see that $f$ is absolutely integrable over the torus ${\mathbb T^d}$. To show that the containment in \eqref{proper containment of spaces over the torus} is proper, we can consider the following function on
$[0, 1]$:
\[
f(x):=
\begin{cases}
\frac{1}{\sqrt x} & \text{ if } x \in (0, 1], \\
0 & \text{ if } x=0.
\end{cases}
\]
So $\int_0^1 f(x) dx = x^{\frac{1}{2}} \Big|_0^1=1$, but $\int_0^1 |f(x)|^2 dx =
\int_0^1 \frac{1}{x} dx = \infty$.
\end{proof}
\bigskip
\subsection{Fourier series: fast and loose}
Let's see how we can expand (certain) functions in a Fourier series, as well as find a formula for their series coefficients, in a foot-loose and carefree way - i.e. abandoning all rigor for the moment.
Given that the sequence of exponential functions $\{e_n(x) \}_{n \in \mathbb{Z}^d}$ forms a basis
for the infinite dimensional vector space $V:=L^2({\mathbb T^d})$,
we know from Linear Algebra that any function $f \in V$ may be written in terms of this basis:
\begin{equation}
f(x) = \sum_{n \in \mathbb{Z}^d} a_n e_n(x).
\end{equation}
How do we compute the Fourier coefficients $a_n$? Let's go through the intuitive process here, ignoring convergence issues. Well, again by Linear Algebra, we take the inner product of both sides with a fixed basis element $e_k(x)$:
\begin{align*}
\langle f(x), e_k(x) \rangle &= \langle \sum_{n \in \mathbb{Z}^d} a_n e_n(x), e_k(x) \rangle \\
&= \sum_{n \in \mathbb{Z}^d} a_n \langle e_n(x), e_k(x) \rangle \\
&= \sum_{n \in \mathbb{Z}^d} a_n \, \delta(n,k) \\
&= a_k
\end{align*}
where we've used the orthogonality relations, Theorem \ref{Orthogonality relations for the exponentials $e_n(x)$} above, in the third equality. We also used the standard notation $\delta(n,k) := 0$ if $n\not=k$, and $\delta(n,k):=1$ if $n=k$.
Therefore, it must be the case that
\begin{align*}
a_k &= \langle f(x), e_k(x) \rangle \\
&:= \int_{[0,1]^d} f(x) \overline{ e^{2\pi i \langle k, x \rangle} } dx \\
&= \int_{[0,1]^d} f(x) e^{-2\pi i \langle k, x \rangle} dx,
\end{align*}
also called the
{\bf Fourier coefficients} of $f$.
\bigskip
\subsection{Fourier series: slow and rigorous}
Let's record now the rigorous statements of the intuitive arguments that we constructed in the previous section. We may think of a periodic function on $\mathbb{R}^d$ as a function belonging to $L^2({\mathbb T^d})$.
\begin{thm}[{\bf Fourier series for functions on ${\mathbb T^d}$}]
\label{Fourier series for periodic functions}
\index{Fourier series for periodic functions}
The set of exponentials
\[
\{ e_n(x) \bigm | n \in \mathbb{Z}^d \}
\]
form a {\bf complete orthonormal basis} for $L^2({\mathbb T^d})$.
Moreover, we have the following:
\begin{enumerate}[(a)]
\item Every function $g \in L^2({\mathbb T^d})$ has a {\bf Fourier series}
\begin{equation} \label{The Fourier series}
g(x) = \sum_{n\in \mathbb{Z}^d} c_n e^{2\pi i \langle n, x \rangle},
\end{equation}
where the convergence in \eqref{The Fourier series} takes place in the $L^2$ norm on the torus ${\mathbb T^d}$.
\item The {\bf Fourier coefficients} $c_n$ may be computed via the formula:
\begin{equation} \label{Fourier coefficients}
c_n = \int_{[0,1]^d} g(t) e^{-2\pi i \langle n, t\rangle} dt,
\end{equation}
for all $n\in \mathbb{Z}^d$.
\item \rm{(\bf The Parseval identity})
The function $g\in L^2({\mathbb T^d})$ in \eqref{The Fourier series} satisfies
\begin{equation} \label{True Parseval identity}
\int_{[0, 1]^d} | g(x)|^2 dx = \sum_{n \in \mathbb{Z}^d} |c_n|^2.
\end{equation}
\end{enumerate}
\hfill $\square$
\end{thm}
(For a proof, see \cite{EinsiedlerWardBook}, p. 96)
At the risk of overstating the obvious, we note that the equality in \eqref{True Parseval identity} is simply equality between real numbers.
We also note that the Fourier coefficients above are integrals over the unit cube
$[0,1]^d$, and may also be thought of as $c_n = \langle g, e_n \rangle$, the projection of $g$ onto each basis element. To summarize, we've encountered the following types of transforms so far:
\begin{equation} \label{both integrals}
\int_{[0,1]^d} g(t) e^{-2\pi i \langle n, t\rangle} dt, \text{ and } \int_{\mathbb{R}^d} g(t) e^{-2\pi i \langle n, t\rangle} dt.
\end{equation}
To disambiguate, the first integral in \eqref{both integrals} arises from periodic functions on $\mathbb{R}^d$, and it appears as a Fourier coefficient in
Theorem \ref{Fourier series for periodic functions}.
The second integral is our old friend the Fourier transform. How are the two integrals related to each other? This is exactly the magic of the Poisson summation formula, Theorem \ref{Poisson.Summation}.
In the pretty proof of Poisson summation, we begin with a Fourier series of a periodized version of $f$,
and end up showing that its Fourier coefficients, by a small miracle of nature, turn out to also be Fourier transforms of $f$.
A natural question is:
\begin{question}
Which functions have a pointwise convergent Fourier series?
\end{question}
But this question turns out to be rather difficult, and many lifetimes have been devoted to related questions.
It is a fact of life that the Fourier series of an arbitrary continuous function on $\mathbb{R}^d$ may fail to converge uniformly, or even pointwise.
However, there is some good news. As it turns out, if we impose some smoothness conditions on $f$, then $f$ does have a Fourier series which converges pointwise. The next theorem gives
a useful criterion of this type in dimension $1$.
For the real line, we have the following refined version of Theorem \ref{Fourier series for periodic functions}.
We use the standard notation
$f(x_0^+):= \lim_{\varepsilon \rightarrow 0} f(x_0 + \varepsilon)$, and
$f(x_0^-):= \lim_{\varepsilon \rightarrow 0} f(x_0 - \varepsilon)$, where $\varepsilon$ is always chosen to be positive.
We call $f$ piecewise smooth on $[0, 1]$ if
$f'$ is a piecewise continuous function on $[0, 1]$.
\bigskip
\begin{thm} \label{theorem:Fourier series convergence to the mean}
Let $f:\mathbb{R} \rightarrow \mathbb{C}$ be a periodic function, with domain $[0 , 1]$, and piecewise smooth on $\mathbb{R}$.
Then, for each $t \in \mathbb{R}$, we have
\begin{equation} \label{convergence of Fourier series to the mean}
\lim_{N\rightarrow \infty} \sum_{n= -N}^N c_n e^{2\pi i n t} =
\frac{ f(t^+) + f(t^-) }{2},
\end{equation}
where $c_n:= \int_0^1 f(x) e^{-2\pi i x n} dx$ are the Fourier coefficients of $f$.
\rm{(For a proof of Theorem \ref{theorem:Fourier series convergence to the mean} see \cite{Travaglini})}.
\hfill $\square$
\end{thm}
We will come back to these {\bf partial Fourier sums}, \index{partial Fourier sums}
occurring in Theorem \ref{theorem:Fourier series convergence to the mean},
and defined by
\begin{equation} \label{partial sums}
S_N f(t):= \sum_{n= -N}^N c_n e^{2\pi i n t}.
\end{equation}
There is also a natural and easy extension of Parseval's identity \eqref{True Parseval identity}.
Given any two functions $f, g\in L^2({\mathbb T^d})$, we've seen in \eqref{The Fourier series}
that
\[
f(x) = \sum_{n\in \mathbb{Z}^d} a_n \, e^{2\pi i \langle n, x \rangle}, \text{ and }
g(x) = \sum_{n\in \mathbb{Z}^d} b_n \, e^{2\pi i \langle n, x \rangle},
\]
both converging in the $L^2({\mathbb T^d})$ norm.
\begin{thm}
If $f, g, \in L^2({\mathbb T^d})$, then with the notation above we have
\[
\int_{{\mathbb T^d}} f(x) \overline{g(x)} dx = \sum_{n\in \mathbb{Z}^d} a_n \overline{b_n}.
\]
\hfill $\square$
\end{thm}
\subsection{The first periodic Bernoulli polynomial}
To see a concrete instance of Theorem \ref{Fourier series for periodic functions}, we study the
function $P_1(x)$, which we've briefly encountered before, as the first periodic Bernoulli polynomial. This function turns out to be so important that it deserves its own section here. We recall its definition:
\begin{equation} \label{def of P_1 again}
P_1(x):=
\begin{cases}
\{ x \} - \frac{1}{2} &\mbox{if } x \notin \mathbb{Z}, \\
0 & \mbox{if } x \in \mathbb{Z}.
\end{cases}
\end{equation}
It's easy to see that
$P_1 \in L^1(\mathbb T)$, so it has a Fourier series, by Theorem \ref{Fourier series for periodic functions}, part (a):
\begin{equation}
P_1(x) = \sum_{n \in \mathbb{Z}} c_n e^{2\pi i n x},
\end{equation}
and the equality here means equality in the $L^2(\mathbb T)$ norm.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.8in]{Bernoulli}
\end{center}
\caption{The first periodic Bernoulli polynomial $P_1(x)$ }
\end{figure}
Let's compute the Fourier coefficients of $P_1$, according to
Theorem \ref{Fourier series for periodic functions}, part (b). We will use integration by parts:
\begin{align*}
c_n&= \int_0^1 \left( \{ x \} - \tfrac{1}{2} \right) e^{-2\pi i n x} dx
= \int_0^1 x e^{-2\pi i n x} dx - \tfrac{1}{2} \int_0^1 e^{-2\pi i n x} dx \\
&= x \frac{ e^{-2\pi i n x} }{-2\pi i n } \Big|_0^1 - \int_0^1 \frac{ e^{-2\pi i n x} }{-2\pi i n } dx
= \frac{ 1 }{-2\pi i n } - 0 = \frac{ 1 }{-2\pi i n },
\end{align*}
when $n \not=0$. For $n=0$, we have $c_0 = \int_0^1 (x - \tfrac{1}{2})dx = 0$.
Hence we have the Fourier series
\begin{equation} \label{Fourier series of P_1 in the norm}
P_1(x) =\{ x \} -\frac{1}{2}= - \frac{ 1 }{2\pi i } \sum_{n \in \mathbb{Z} \atop n\not=0} \frac{ 1 }{ n } e^{2\pi i n x},
\end{equation}
where the latter equality means convergence in the $L^2({\mathbb T^d})$ norm.
If we want to get pointwise convergence of this Fourier series, we may apply
Theorem \ref{theorem:Fourier series convergence to the mean}, which allows us to conclude that we have pointwise convergent sums:
\begin{align} \label{example of theorem on pointwise convergence}
\lim_{N\rightarrow \infty} -\frac{1}{2\pi i} \sum_{-N \leq n \leq N \atop n\not=0} \frac{1}{n} e^{2\pi i n x}
&= \frac{ P_1(x^+) + P_1(x^-) }{2} \\
&= \{ x \} -\frac{1}{2},
\end{align}
when $x\notin \mathbb{Z}$. For $x \in \mathbb{Z}$, we can also check that the equality
\eqref{example of theorem on pointwise convergence} holds by observing that
\[
\sum_{-N \leq n \leq N \atop n\not=0} \frac{1}{n} e^{2\pi i n x} =
\sum_{-N \leq n \leq N \atop n\not=0} \frac{1}{n} = 0,
\]
while $\frac{ P_1(x^+) + P_1(x^-) }{2} =\tfrac{1}{2}\left(- \frac{1}{2} + \frac{1}{2}\right) =0 $ as well, which is consistent with the definition
\eqref{def of P_1 again}
of $P_1(x)$ at the integers.
Next, we can give a classical application of the Fourier series \eqref{Fourier series of P_1 in the norm}
using Parseval's identity \eqref{True Parseval identity}:
\[
\int_0^1 |P_1(u)|^2 du = \sum_{n \in \mathbb{Z}} |a_n|^2.
\]
Let's simplify both sides:
\begin{align*}
\sum_{n \in \mathbb{Z}} |a_n|^2 &= \frac{1}{4\pi^2} \sum_{n \in \mathbb{Z}-\{0\}} \frac{1}{n^2} =
\frac{1}{2\pi^2} \sum_{n \geq 1} \frac{1}{n^2},
\end{align*}
while
\begin{align*}
\int_0^1 |P_1(u)|^2 du = \int_0^1 \left( \{ x \} - \frac{1}{2} \right)^2 dx
&=
\int_0^1 \left(x - \frac{1}{2}\right)^2 dx
= \frac{1}{12}.
\end{align*}
Therefore
\[
\sum_{n \geq 1} \frac{1}{n^2} = \frac{\pi^2}{6},
\]
a number-theoretic identity that goes back to Euler.
In a similar manner one can evaluate the Riemann zeta function at all positive even integers,
using the cotangent function
(Exercise \ref{Riemann zeta function, and Bernoulli numbers}).
\begin{comment}
Now that we have the Fourier series for $P_1(x)$, let's see what happens if we integrate both sides
of \eqref{Fourier series of P_1 in the norm}, assuming for the moment that we may integrate the RHS of \eqref{Fourier series of P_1 in the norm} term-by-term:
\begin{align*}
-\frac{ 1 }{(2\pi i)^2 } \sum_{n \in \mathbb{Z} \atop n\not=0} \frac{ e^{2\pi i n t} }{ n^2 } =
\int_0^t P_1(x) dx =
\end{align*}
using the fact that $\frac{d}{dx} B_n(x) = n B_{n-1}(x)$ for all $n \geq 1$
(Exercise \ref{derivative of Bernoulli polys}), and hence the same is true for $P_n(x)$.
\end{comment}
\bigskip
Another natural question arises.
\begin{question}
What sort of functions $f$ are uniquely determined by all of their Fourier coefficients?
\end{question}
To describe a partial answer, we recall the space of all continuous functions on the torus:
\begin{equation}
C({\mathbb T^d}):= \{ f: {\mathbb T^d} \rightarrow \mathbb{C} \bigm | f \text{ is continuous on } {\mathbb T^d} \}.
\end{equation}
\begin{thm}
Let $f \in C({\mathbb T^d})$, and suppose that $\hat f(n) = 0$ for all $n \in \mathbb{Z}^d$. Then $f(x)=0$,
for all $ x \in [0, 1]^d$.
In particular, if $f, g \in C({\mathbb T^d})$ and $\hat f(n) =\hat g(n)$ for all $n \in \mathbb{Z}^d$,
then $f(x) = g(x)$ for all $x \in [0, 1]^d$.
\hfill $\square$
\end{thm}
In other words, a continuous function on the torus is uniquely determined by its Fourier coefficients
(see \cite{EinsiedlerWardBook} for a proof).
\begin{comment}
We might hope that the same is true for a function that is piecewise continuous on the torus ${\mathbb T^d}$.
\begin{align}
\hat F(n) &= \int_0^{1/2} e^{-2\pi i n x} dx - \int_{1/2}^1 e^{-2\pi i n x} dx =
\frac{e^{-2\pi i n x}}{-2\pi i n}\Big|_0^{1/2}- \frac{e^{-2\pi i n x}}{-2\pi i n}\Big|_{1/2}^{0} \\
&= \frac{1}{-2\pi i n}
\Big(
e^{-\pi i n} - 1 - ( 1 - e^{-\pi i n} )
\Big)
=\frac{1 - e^{-\pi i n}}{\pi i n}=\frac{1 - (-1)^n}{\pi i n}.
\end{align}
\end{comment}
\bigskip
\section{As $f$ gets smoother, $\hat f$ decays faster}
\label{As f gets smoother, the FT decays faster}
There is a very basic and important relationship between the level of smoothness of $f$,
and the speed with which $\hat f$ tends to $0$ as $x \rightarrow \infty$. To capture this relation very concretely, let's compute things on the real line,
to see how the FT interacts with the derivative.
\begin{lem} \label{FT interacts with derivative}
Let $f \in L^1(\mathbb{R})$.
\begin{enumerate}[(a)]
\item \label{FT of derivative}
If $f$ is piecewise smooth, and also enjoys $f' \in L^1(\mathbb{R})$, then:
\[
\widehat{ f' }(\xi) = (2\pi i) \xi \hat f(\xi).
\]
\item \label{explicit decay rate}
More generally, let $k\geq 0$, suppose that $f$ has $k$ derivatives, $f^{(k)}$ is piecewise smooth,
and that
we also have $f^{(k+1)} \in L^1(\mathbb{R})$.
Then:
\[
\widehat{ f^{(k+1)} }(\xi) = (2\pi i \xi)^{k+1} \hat f(\xi).
\]
\item \label{derivative of the FT}
Now we suppose that $x f(x) \in L^1(\mathbb{R})$. Then:
\[
\frac{d}{d\xi} {\mathcal F}(f)(\xi) = (-2\pi i) \, {\mathcal F}(x f(x))(\xi).
\]
\end{enumerate}
\end{lem}
\begin{proof}
To prove part \ref{FT of derivative}, we notice that $\lim_{x\rightarrow \infty} f(x) = f(0) + \int_0^\infty f'(x) dx$, using the hypothesis $f' \in L^1(\mathbb{R})$.
Using the hypothesis $f \in L^1(\mathbb{R})$, we know that the Riemann-Lebesgue Lemma \ref{Riemann--Lebesgue lemma} implies that $\lim_{x\rightarrow \infty} f(x)=0$.
Similarly, $\lim_{x\rightarrow -\infty} f(x) =0$. Integration by parts now gives us:
\begin{align*}
\widehat{ f' }(\xi) &= \int_{\mathbb{R}} f'(x) e^{-2\pi i x \xi} dx
= f(x)e^{-2\pi i x \xi}\Big |_{-\infty}^\infty - \int_{\mathbb{R}} f(x) (-2\pi i \xi) e^{-2\pi i x \xi} dx \\
&= 2\pi i \xi \int_{\mathbb{R}} f(x) e^{-2\pi i x \xi} dx := 2\pi i \xi \hat f(\xi).
\end{align*}
Part \ref{explicit decay rate} follows from \ref{FT of derivative} by induction on $k$. To prove part \ref{derivative of the FT}, we have:
\begin{align*}
{\mathcal F}(x f(x))(\xi) &:= \int_{\mathbb{R}} xf(x) e^{-2\pi i x \xi} dx = \frac{1}{-2\pi i} \int_{\mathbb{R}} \frac{d}{d\xi} f(x) e^{-2\pi i x \xi} dx\\
&=-\frac{1}{2\pi i} \frac{d}{d\xi} \int_{\mathbb{R}} f(x) e^{-2\pi i x \xi} dx
=-\frac{1}{2\pi i} \frac{d}{d\xi} \hat f(\xi).
\end{align*}
\end{proof}
It follows from Theorem \ref{FT interacts with derivative}, part \ref{explicit decay rate}, that we have an explicit decay rate for the Fourier coefficients of a periodic function $f$,
assuming that $f$ is sufficiently smooth. To obtain the following Corollary, we can simply use the fact that $f^{(k+1)} \in L^1(\mathbb{R})$ implies that $ \widehat{ f^{(k+1)}}$ is uniformly bounded:
$\Big| \frac{1}{(2\pi )^{k+1}} \widehat{ f^{(k+1)} }(\xi) \Big| < C$, for a positive constant $C$.
\begin{cor} \label{cor: f smoother implies FT of F decays faster}
If $f$ has $k$ continuous derivatives, and we also have $f^{(k+1)} \in L^1(\mathbb{R})$,
then there is a constant $C>0$ such that:
\begin{equation}
| \hat f(\xi) | < C \frac{1}{ | \xi|^{k+1}},
\end{equation}
for all $\xi \not=0$.
\hfill $\square$
\end{cor}
In other words, we now understand
the dictum ``as $f$ gets smoother, $\hat f$ decays faster'' in a precise quantitative manner:
if $f$ has $k$ derivatives, then $\hat f$ decays faster than a polynomial of degree $k$.
\bigskip
\section{How fast do Fourier coefficients decay?}
In a manner completely analogous to the previous Section \ref{As f gets smoother, the FT decays faster},
we can repeat the important idea of integration by parts to see how fast
Fourier coefficients decay, and here we may expect even better results, because we will integrate over the compact unit cube, rather than over the non-compact space $\mathbb{R}^d$. We first work things out in dimension $1$, recalling that
the Fourier coefficients of $f$ are defined by $c_n:= \int_0^1 f(x) e^{-2\pi i n x} dx$, for all $n \in \mathbb{Z}$.
For the sake of the reader, we recall the space of functions $C^k(\mathbb T)$ from
\ref{k derivatives on the torus}, which have $k$ continuous derivatives. We also recall that
$f \in L^1(\mathbb T)$ means $\int_0^1f(x) dx$ is finite, and that $f(x+1) = f(x)$,
for all $x\in [0, 1]$. Finally, we note that the same conclusion of the Riemann-Lebesgue lemma \ref{Riemann--Lebesgue lemma} also holds for
functions $f \in L^1({\mathbb T^d})$, with exactly the same proof that we gave in Lemma \ref{Riemann--Lebesgue lemma}.
\begin{thm} \label{decay of Fourier coefficients}
Let $f\in L^1(\mathbb T)$.
\begin{enumerate}[(a)]
\item \label{decay of Fourier coefficients for C^1}
If $f \in C^1(\mathbb T)$, then its Fourier coefficients satisfy
\begin{equation}
\lim_{|n|\rightarrow \infty} |n c_n| = 0.
\end{equation}
In other words, $|c_n| = o\left( \frac{1}{n} \right)$.
\item \label{decay of Fourier coefficients for C^k}
More generally, fix an integer $k\geq 1$. If $f \in C^k(\mathbb T)$, then its Fourier coefficients satisfy
\begin{equation}
\lim_{|n|\rightarrow \infty} |n^k c_n| = 0.
\end{equation}
In other words, $|c_n| = o\left( \frac{1}{n^k} \right)$.
\end{enumerate}
\end{thm}
\begin{proof}
We compute the Fourier coefficients using integration by parts. For each $n \not=0$, we have:
\begin{align*}
c_n &:= \int_0^1 f(x) e^{-2\pi i n x} dx
= \left[ f(x) \frac{ e^{-2\pi i n x} }{-2\pi i n}\right] \Big|_0^1 + \frac{ 1}{2\pi i n}\int_0^1 f'(x) e^{-2\pi i n x} dx \\
&= \frac{ f(1)-f(0) }{ -2\pi i n } + \frac{ 1}{2\pi i n}\int_0^1 f'(x) e^{-2\pi i n x} dx \\
&= \frac{ 1}{2\pi i n}\int_0^1 f'(x) e^{-2\pi i n x} dx,
\end{align*}
using the periodicity of $f$. Because $f'$ is continuous, the Riemann-Lebesgue lemma on $L^1(\mathbb T)$ gives us
$\lim_{|n|\rightarrow \infty} \int_0^1 f'(x) e^{-2\pi i n x} dx = 0$. So we see that
\[
|n c_n| \rightarrow 0, \ \text{ as } |n| \rightarrow \infty,
\]
completing part \ref{decay of Fourier coefficients for C^1}. Part \ref{decay of Fourier coefficients for C^k} follows easily by induction on $k$, repeating exactly the same integration by parts computation above.
\end{proof}
We note that the same proof works with even weaker hypotheses in part \ref{decay of Fourier coefficients for C^k}. Namely, given an integer $k\geq 1$, all we require is that
$f^{(j)}$ is continuous on $\mathbb T$, for $0\leq j < k$, and $f^{(k)} \in L^1(\mathbb T)$.
Let's see a concrete application of these ideas (see Note \ref{Note:GregKuperberg}).
\begin{thm} \label{Euler-Maclaurin type identity}
Suppose that $f\in C^k(\mathbb T)$, for a fixed integer $k\geq 1$. Then:
\begin{equation}
\int_0^1 f(x) dx = \frac{1}{N} \sum_{m=0}^{N-1} f\left(\tfrac{m}{N}\right) + o\left(\tfrac{1}{N^{k}}\right),
\end{equation}
as $N\rightarrow \infty$.
\end{thm}
\begin{proof}
Because $f$ is periodic on $\mathbb{R}$, we follow ``Hecke's dictum''; namely, we first expand $f$ into its Fourier series, which is guaranteed by
Theorem \ref{Fourier series for periodic functions}:
\[
f(x) = \sum_{n\in \mathbb{Z}} c_n e^{2\pi i n x}.
\]
Since this Fourier series converges absolutely, we may interchange the finite sum with the series:
\begin{align*} \label{first step of finite sum approximation}
\frac{1}{N} \sum_{m=0}^{N-1} f\left(\tfrac{m}{N}\right) &= \frac{1}{N} \sum_{m=0}^{N-1}
\sum_{n\in \mathbb{Z}} c_n e^{2\pi i n \tfrac{m}{N}} \\
&= \sum_{n\in \mathbb{Z}} c_n
\Big( \tfrac{1}{N} \sum_{m=0}^{N-1} e^{2\pi i n \tfrac{m}{N}} \Big) \\
&=\sum_{n\in \mathbb{Z}} c_{Nn},
\end{align*}
using Exercise \ref{DivisibilityUsingExponentials} (the harmonic detector for divisibility).
Next, we recall that the constant term is $c_0= \int_0^1 f(x) dx$, and we separate out this term from the latter series:
\begin{align*}
\frac{1}{N} \sum_{m=0}^{N-1} f\left(\tfrac{m}{N}\right)
= \int_0^1 f(x) dx + \sum_{n\in \mathbb{Z} \atop {n \not=0 } } c_{Nn},
\end{align*}
Now we can use the (little-o) rate of decay of the Fourier coefficients, given by
Theorem \ref{decay of Fourier coefficients}, part \ref{decay of Fourier coefficients for C^k}, to write
$|c_{Nn}| < \frac{C}{(Nn)^k}$ for \emph{all} constants $C>0$. We conclude that
\[
\sum_{n\in \mathbb{Z} \atop {n \not=0 } } |c_{Nn}|
< C \sum_{n\in \mathbb{Z} \atop {n \not=0 }} \frac{1}{ N^{k} |n|^{k}}
< 2 C\, \zeta(k) \frac{1}{ N^{k}},
\]
for all constants $C>0$. So as $N\rightarrow \infty$, the error term
$\sum_{n\in \mathbb{Z} \atop {n \not=0 } } c_{Nn}$ is $o\left( \frac{1}{ N^k} \right)$, as claimed.
\end{proof}
\begin{comment}
\begin{example}
\rm{
Let's study the sum $ \frac{1}{N} \sum_{m=0}^{N-1} f\left(\tfrac{m}{N}\right) $, where
$f(x) := \sqrt{\{x\}+\frac{1}{N}}$, and $N$ is a fixed positive integer, and $\{x\}$ is the fractional part of $x$. Because $f$ is infinitely smooth
on $\mathbb T$, and periodic on $\mathbb{R}$,
the error term in Theorem \ref{Euler-Maclaurin type identity} holds for all positive integers $k$,
giving us super-exponential decay in $N$ (and the latter conclusion holds for any $C^\infty$ function on the circle $\mathbb T$).
By Theorem \ref{Euler-Maclaurin type identity}, we have
\begin{align*}
\frac{1}{N} \sum_{m=0}^{N-1} \sqrt{ \frac{m}{N} +\frac{1}{N} }
&=\int_0^1 \sqrt{x +\frac{1}{N}} \, dx + o\left(\frac{1}{N^{k}}\right)
=\tfrac{2}{3} \left(1+ \frac{1}{N} \right)^{\frac{3}{2}} - \tfrac{2}{3} \left( \frac{1}{N} \right)^{\frac{3}{2}} + o\left(\frac{1}{N^{k}}\right),
\end{align*}
or equivalently
\begin{equation}
1+ \sqrt 2 + \cdots + \sqrt N
=\tfrac{2}{3} \left(N+1 \right)^{\frac{3}{2}} - \tfrac{2}{3} + o\left(\frac{1}{N^{k-\frac{3}{2}}}\right),
\end{equation}
which is valid for all $k \geq 1$.
Although this might seem `too good to be true',
by definition we may equivalently write:
\begin{align*}
\lim_{N \rightarrow \infty} N^{k-\frac{3}{2}}
\left | \sum_{m=1}^{N} \sqrt m - \left(N+1 \right)^{\frac{3}{2}} + \tfrac{2}{3} \right | =0,
\end{align*}
for each $k\geq 1$.
}
\hfill $\square$
\end{example}
\end{comment}
It is worth mentioning that although our proof of Theorem \ref{Euler-Maclaurin type identity}
does not cover the case `$k=0$', this case
is also true because it represents the Riemann sum approximation to the integral.
\section{The Schwartz space} \label{nice functions}
\index{Schwartz space}
We saw in Section \ref{As f gets smoother, the FT decays faster} that a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ in the space domain, that has $k$ derivatives, corresponds to a function $\hat f$ in the Fourier transform domain.
If we `take this idea to the limit', so to speak, What does that last adjective mean?
Following Laurent Schwartz, we can make rigorous sense of the words `rapidly decreasing', as follows.
We recall that our definition of a `nice function' was any function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ for which the Poisson summation formula holds. Here we give our first family of sufficient conditions for a function $f$ to be nice.
A {\bf Schwartz function} $f: \mathbb{R} \rightarrow \mathbb{C}$ is defined as any infinitely smooth function ($f \in C^\infty(\mathbb{R})$)
that satisfies the following
growth condition:
\begin{equation}
|x^a \frac{d}{dx^k} f(x)| \text{ is bounded on } \mathbb{R},
\end{equation}
for all integers $a, k \geq 0$.
In particular, a Schwartz function decreases faster than any polynomial function, as $|x|$ tends to infinity.
\medskip
\begin{example}
\rm{
The Gaussian function $G_t(x) := e^{-t ||x||^2}$ is a Schwartz function, for each fixed
$t>0$. To see this, we first consider $\mathbb{R}^1$, where we note that the $1$-dimensional Gaussian
is a Schwartz function, as follows. We observe that for all positive integers $k$,
$ \frac{d}{dx^k} G_t(x) = H_n(x) G_t(x)$, where $H_n(x)$ is a univariate polynomial in $x$ (which also depends on the parameter $t$, but we think of $t$ as a constant).
Since $\lim_{x\rightarrow \infty} \frac{x^a \, H_n(x)}{ e^{t ||x||^2}} =0$, for all positive integers $a$, we see that $G_t(x)$ is a Schwartz function.
Now we note that the product of Schwartz functions is again a Schwartz function; hence the $d$-dimensional Gaussian, $G_t(x) := e^{-t ||x||^2} = \prod_{k=1}^d e^{-t x_k^2} $, a product of $1$-dimensional Gaussians, is a Schwartz function.
Some might say the Gaussian is the quintessential Schwartz function, partly because it is also an eigenfunction of the Fourier transform, as we'll see below.
}
\hfill $\square$
\end{example}
\medskip
\begin{example} \label{example: the abs value exponential}
\rm{
We define $f(x) := e^{-2\pi t |x|}$ on the real line, for a fixed $t >0$.
To see that $f$ is \emph{not} a Schwarz function, we merely have to observe that $f$ is not differentiable at $x=0$.
To be a Schwartz function, $f$ would have to be infinitely differentiable everywhere on $\mathbb{R}$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{exponential}
\end{center}
\caption{The function $e^{-|x|}$. }
\end{figure}
Interestingly, we can also see that $f$ is not a Schwartz function in another way - by computing its Fourier transform and observing that it is not rapidly decreasing:
\begin{align*}
\hat f(\xi) &:= \int_\mathbb{R} e^{-2\pi t |x| -2\pi i x \xi }dx \\
&= \int_{-\infty}^0 e^{2\pi t x -2\pi i x \xi } dx
+ \int_0^{+\infty} e^{-2\pi t x -2\pi i x \xi } dx \\
&= \int_{-\infty}^0 e^{2\pi x(t - i \xi) } dx
+ \int_0^{+\infty} e^{-2\pi x (t+ i \xi) } dx \\
&= \frac{ e^{2\pi x(t - i \xi)} }{2\pi (t-i\xi)}\Big|_{x=-\infty}^{x=0} +
\frac{ e^{-2\pi x(t + i \xi)} }{-2\pi (t+i\xi)}\Big|_{x=0}^{x=\infty} \\
&= \frac{ 1 }{2\pi (t-i\xi)} +
\frac{ 1 }{2\pi (t+i\xi)} \\
&=
\frac{ t }{\pi (t^2 + \xi^2)},
\end{align*}
valid for all $\xi \in \mathbb{R}$. Because the Fourier transform
\begin{equation} \label{FT of the abs value exponential}
\frac{ t }{\pi (t^2 + \xi^2)}
\end{equation}
is not a rapidly decreasing function, we have another proof that $f$ is not a Schwartz function.
This example is interesting in that $f$ is infinitely differentiable everywhere, except at one point, namely
$x =0$. Yet this local lack of smoothness - at only a single point - is enough to cause a global change in decay for its Fourier transform.
}
\hfill $\square$
\end{example}
It is just as easy to define Schwartz functions on $\mathbb{R}^d$ as well.
For any $k:= (k_1, \dots, k_d) \in \mathbb{Z}_{\geq 0}^d$, we can define the multivariable differential operator
\[
D_k := \frac{\partial}{\partial x_1^{k_1} \cdots \partial x_d^{k_d}}.
\]
\begin{example}
In $\mathbb{R}^1$, this is the usual $k$'th derivative, namely
$D_k f(x) := \frac{d}{dx^k}f(x)$.
In $\mathbb{R}^2$, for example, we have $D_{(1,7)} f(x) := \frac{\partial}{\partial x_1 \partial x_2^7}f(x)$.
\hfill $\square$
\end{example}
The {\bf order} of the differential operator
$D_k$ is by definition $|k|:= k_1+ \cdots + k_d$.
To define spaces of differentiable functions, we call a function $f : \mathbb{R}^d \rightarrow \mathbb{C}$
a $C^m$-function if all partial derivatives
$D_k f$ of order $|k| \leq m$ exists and are continuous. We denote the collection of all such
$C^m$-functions on Euclidean space by $C^m(\mathbb{R}^d)$.
When considering {\bf infinitely-differentiable functions on Euclidean space}, we denote this space by $C^\infty(\mathbb{R}^d)$.
So we see that in $\mathbb{R}^d$, we can define {\bf Schwartz functions} \index{Schwartz function}
similarly to our previous definition: they are infinitely differentiable
functions $f:\mathbb{R}^d \rightarrow \mathbb{C}$ such that for all vectors $a, k \in \mathbb{Z}_{\geq 0}^d$
we have:
\begin{equation}
|x^a D_k f(x)| \text{ is bounded on } \mathbb{R}^d,
\end{equation}
where $x^a:= x_1^{a_1} \cdots x_d^{a_d}$ is the standard multi-index notation.
We also define the {\bf Schwartz space} $S(\mathbb{R}^d)$ to be set of all Schwartz functions $f:\mathbb{R}^d \rightarrow \mathbb{C}$.
\begin{thm} \label{Schwartz goes to Schwartz}
The Fourier transform maps the Schwartz space $S(\mathbb{R}^d)$ one-to-one, onto itself.
(See Exercise \ref{Schwartz space convolution invariance})
\end{thm}
In fact, more is true: the mapping $f \rightarrow \hat f$ from $S(\mathbb{R}^d)$ to itself is an isometry.
The proof of this
fact uses the Parseval relation below.
And now that we know the definition of rapid decay, we see that an obvious consequence of
Corollary \ref{cor: f smoother implies FT of F decays faster}
is the following:
\begin{equation} \label{smooth implies FT is rapidly decreasing}
\text{ If } f \text{ is infinitely smooth, then } \hat f \text{ is rapidly decreasing}.
\end{equation}
In fact, we can combine some of the ideas above to record another useful fact.
\begin{lem} \label{useful Schwartz fact}
Let $\phi:\mathbb{R}^d \rightarrow \mathbb{C}$ be compactly supported and infinitely smooth.
Then
\[
\phi \in {\mathcal S}(\mathbb{R}^d).
\]
\end{lem}
\begin{proof}
Because $\phi$ is compactly supported, we know that $\hat \phi$ is infinitely smooth (differentiation under the integral). Moreover, the assumption that $\phi$ is infinitely smooth implies that
$\hat \phi$ is rapidly descreasing, by \eqref{smooth implies FT is rapidly decreasing}. So now we know that
$\hat \phi$ is both rapidly decreasing and infinitely smooth - i.e. a Schwartz function.
Applying Theorem \ref{Schwartz goes to Schwartz}, we see that its Fourier transform is also a Schwartz function. Namely, using Fourier inversion, we conclude that $\hat{\hat \phi}(-x) = \phi(x) \in S(\mathbb{R}^d)$.
\end{proof}
The functions satisfying the conditions of Lemma \ref{useful Schwartz fact} are also called {\bf bump functions}.
\section{Poisson Summation I} \label{Poisson Summation section}
\index{Poisson summation formula}
We introduce the Poisson summation formula, one of the most useful tools in analytic number theory, and in discrete / combinatorial geometry. This version of Poisson summation holds for Schwartz functions.
There are many different families of sufficient conditions that a function $f$ can satisfy, in order for Poisson summation to be applicable to $f$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.8in]{NiceFunctions}
\end{center}
\caption{Spaces of functions for Poisson summation}
\label{nice functions, containment}
\end{figure}
\bigskip
\begin{thm}[Poisson summation formula, I]
\index{Poisson summation formula} \index{Poisson summation formula}
\label{Poisson.Summation}
Given a {\it Schwartz function} $f~:~\mathbb{R}^d \rightarrow \mathbb{C}$, we have
\begin{equation} \label{Poisson.summation1}
\sum_{n \in \mathbb{Z}^d} f(n+ x) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle},
\end{equation}
valid for all $x \in \mathbb{R}^d$.
In particular, we have:
\begin{equation} \label{Poisson.summation2}
\sum_{n \in \mathbb{Z}^d} f(n) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi).
\end{equation}
Both sides of \eqref{Poisson.summation1} converge absolutely, and are continuous functions on $\mathbb{R}^d$.
\end{thm}
\begin{proof}
If we let $F(x):= \sum_{n \in \mathbb{Z}^d} f(n+ x)$, then we notice that $F$ is periodic on $\mathbb{R}^d$, with the cube $[0,1)^d$ as a fundamental domain. The argument is easy: fix any $m \in \mathbb{Z}^d$. Then
$F(x + m) = \sum_{n \in \mathbb{Z}^d} f(n+ x + m) = \sum_{k \in \mathbb{Z}^d} f(x + k)$, because $\mathbb{Z}^d + m = \mathbb{Z}^d$.
By Theorem \ref{Fourier series for periodic functions}, $F$ has a fourier series, so let's compute it:
\[
F(x) := \sum_{k \in \mathbb{Z}^d} a_k e^{2\pi i \langle k, x \rangle},
\]
where $a_k = \int_{[0,1)^d} F(u) e^{2\pi i \langle k, u \rangle}du$ for each fixed $k\in \mathbb{Z}^d$.
Let's see what happens if we massage $a_k$ a bit:
\begin{align}
a_k &:= \int_{[0,1)^d} F(u) e^{-2\pi i \langle k, u \rangle} du \\
&=\int_{[0,1)^d} \sum_{n \in \mathbb{Z}^d} f(n+ u)
e^{-2\pi i \langle k, u \rangle} du \\
&=\sum_{n \in \mathbb{Z}^d} \int_{[0,1)^d} f(n+ u) \label{outersum}
e^{-2\pi i \langle k, u \rangle} du.
\end{align}
The interchange of summation and integral in the latter step is allowed by Theorem \ref{Application of dominated convergence}, which is an application of the dominated convergence theorem,
because the integrand satisfies
$ | f(n+ u) e^{-2\pi i \langle k, u \rangle} | = | f(n+ u) | \in L^1(\mathbb{R}^d)$. The latter absolute integrability of $f$ is
due to the fact that $f$ is a Schwartz function.
Now we fix an $n\in\mathbb{Z}^d$ in the outer sum of \eqref{outersum}, and make the change of variable in the integral:
$n+u := w$, so that $du = dw$.
A critical step in this proof is the fact that as $u$ varies over the cube ${[0,1)^d}$, $w:= n+u$ varies
over all of $\mathbb{R}^d$ because we have a tiling
\index{tiling}
of Euclidean space by the unit cube: ${[0,1)^d} + \mathbb{Z}^d = \mathbb{R}^d$.
We note that under this change of variable,
$e^{-2\pi i \langle k, u \rangle} = e^{-2\pi i \langle k, w-n \rangle} =
e^{-2\pi i \langle k, w \rangle}$, because $k, n \in \mathbb{Z}^d$ and hence
$e^{2\pi i \langle k, n \rangle} =1$.
Therefore, we finally have:
\begin{align}\label{finally, the FT}
a_k = \sum_{n \in \mathbb{Z}^d} \int_{n+ [0,1)^d} f(w)
e^{-2\pi i \langle k, w \rangle} dw
= \int_{\mathbb{R}^d} f(w)
e^{-2\pi i \langle k, w \rangle} dw := \hat f(k),
\end{align}
so that $F(x) = \sum_{k \in \mathbb{Z}^d} a_k e^{2\pi i \langle k, x \rangle}
= \sum_{k \in \mathbb{Z}^d} \hat f(k) e^{2\pi i \langle k, x \rangle}$.
\end{proof}
We define a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ to be a {\bf nice function}
if both $f, \hat f \in L^1(\mathbb{R}^d)$, and if the Poisson summation formula
\begin{equation} \label{nice functions}
\sum_{n \in \mathbb{Z}^d} f(n+ x) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle}
\end{equation}
holds for $f$ pointwise, for each $x \in \mathbb{R}^d$.
We will give various different sets of sufficient conditions for a
function $f$ to be nice.
Figure \ref{nice functions, containment} suggests a simple containment relation between some of these function spaces,
as we will easily prove.
There are a few things to notice about the classical, and pretty proof of Theorem \ref{Poisson.summation1}. The first is that we began with any square-integrable function $f$ defined on all of $\mathbb{R}^d$,
and forced a periodization of it, which was by definition $F$. This is known as the ``folding'' part of the proof.
Then, at the end of the proof, there was the ``unfolding'' process, where we summed an integral over a lattice, and because the cube tiles $\mathbb{R}^d$, the sum of the integrals transformed into a single integral over $\mathbb{R}^d$.
The second thing we notice is that the integral $\int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, \xi \rangle} dx$,
which is by definition the Fourier transform of $f$,
appears quite naturally due to the tiling of $\mathbb{R}^d$ by the unit cube $[0,1)^d$.
Hopefully there will now be no confusion as to the difference between the integral over the cube, and the integral over $\mathbb{R}^d$, both appearing together in this proof.
\section{Useful convergence lemmas, in preparation for Poisson summation II}
To prepare ourselves for Poisson's original summation formula, which we give in the next section, we will see here Poisson's hypotheses for the growth of $f$ and $\hat f$, together with the immediate convergence consequences they carry.
\begin{lem} \label{Poisson bound implies L^1}
Let $f:\mathbb R^d \rightarrow \mathbb{C}$ be a function that enjoys the bound
\[
|f(x)|\leq\frac{C}{(1+||x||)^{d+\delta}},
\]
for all $x\in \mathbb R^d$, and for constants $C, \delta >0$ that are independent of $x$.
Then $f\in L^1(\mathbb{R}^d)$.
\end{lem}
\begin{proof}
Consider the cube
$Q_n:= [-n,n]^d$ and let $D_n:=Q_{n+1} - Q_n$ denote the set difference; in other words, $D_n$ is the cubical shell between the cube $Q_n$ and the cube $Q_{n+1}$.
We have $\mathbb{R}^d=\bigcup_{n\geq 0} D_n$, and $D_0=Q_1$. Also, we note that on each shell $D_n$, $\frac{1}{\|x\|} \leq \frac{1}{n}$, so that:
\begin{align}
\int_{\mathbb{R}^d}|f(x)|dx
&=\sum_{n\geq0}\int_{D_n}|f(x)|dx\\
&= \int_{D_0}|f(x)|dx +
\sum_{n\geq 1}\int_{D_n}|f(x)|dx \\
&\leq\frac{C}{2^{d+\delta}}+\sum_{n\geq1}\int_{D_n}\frac{C}{(1+n)^{d+\delta}}dx \\
&=\frac{C}{2^{d+\delta}}+\sum_{n\geq1} \frac{C}{(1+n)^{d+\delta}} \int_{D_n} dx \\
&=\frac{C}{2^{d+\delta}}+\sum_{n\geq1}\frac{C}{(1+n)^{d+\delta}}\left((2n+2)^{d}-(2n)^{d}\right)\\
&=\frac{C}{2^{d+\delta}}+2^dC\sum_{n\geq1}\frac{1}{(1+n)^{d+\delta}}\left((n+1)^{d}-n^{d}\right)\\ \label{Big-O}
&=\frac{C}{2^{d+\delta}}+\sum_{n\geq1}\frac{O(n^{d-1})}{(1+n)^{d+\delta}}\\
&=\frac{C}{2^{d+\delta}}+\sum_{n\geq1}O\left(\frac{1}{n^{1+\delta}}\right)\\
&=\frac{C}{2^{d+\delta}}+O\left(\sum_{n\geq1}\frac{1}{n^{1+\delta}}\right)<\infty,
\end{align}
where we've used the fact that the constant in the Big-O of equation \eqref{Big-O} is independent of $n$, so that we can move the series inside.
\end{proof}
For the absolute summability of functions satisfying the same growth condition of the previous lemma, we have the following.
\begin{lem}
\label{Poisson bound implies absolutely summable}
Let $f:\mathbb R^d \rightarrow \mathbb{C}$ be a function that enjoys the bound
\[
|f(x)|\leq\frac{C}{(1+||x||)^{d+\delta}},
\]
for all $x\in \mathbb R^d$, and for constants $C, \delta >0$ that are independent of $x$.
Then the series
\[
\sum_{k\in \mathbb{Z}^d} f(x+k)
\]
converges uniformly and absolutely, for $x\in\mathbb{R}^d$.
\end{lem}
\begin{proof}
We will restrict attention to $x \in [0, 1)^d$, because the function $F(x):= \sum_{k\in \mathbb{Z}^d} f(k+x)$, if convergent, forms
a periodic function of $x \in \mathbb{R}^d$, with the unit cube $[0, 1)^d$ being a period. We also note for all $x \in [0, 1)^d$,
we have the bound $\|x\| \leq \sqrt d$.
We consider the tail of the series, for any given $N>0$:
\begin{align}
| \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} f(k+x) |
& \leq \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \left | f(k+x) \right |
\leq C \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \frac{1}{(1+\|k+x\|)^{d+\delta}} \\
& \leq \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \frac{1}{ \left(1+\frac{\|k+x\|}{1+\sqrt d} \right)^{d+\delta}}
\\ \label{second equality here}
&=\sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \frac{\left(1+\sqrt{d}\right)^{d+\delta}}{\left(1+\sqrt{d}+\|k+x\| \right)^{d+\delta}}
\\ \label{third line here}
&\leq C_{d, \delta} \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \frac{1}{(1+\|k\|)^{d+\delta}}
\\ \label{surface area of sphere approximation}
&= C_{d, \delta} \sum_{n \geq N}\frac{1}{(1+n)^{d+\delta}} O \left( n^{d-1} \right) \\
&=\sum_{n \geq N}O\left(\frac{1}{n^{1+\delta}} \right)\\
&=O\left(\sum_{n \geq N}\frac{1}{n^{1+\delta}} \right) \rightarrow 0, \text{ as } N\rightarrow \infty,
\end{align}
and the last bound is independent of $x$. In passing from \eqref{second equality here} to \eqref{third line here},
we used the estimate
$\|k + x \| \geq \|k \| - \| x\| \geq \|k\|-\sqrt{d}$, and $C_{d, \delta}:= \left(1+\sqrt{d}\right)^{d+\delta}$. The equality in \eqref{surface area of sphere approximation} is due to the fact that the number of integer points $k \in \mathbb{Z}^d$ that lie on a sphere of radius $n$ is $O\left(\text{surface area of } nS^{d-1} \right)
= O \left( n^{d-1} \right) $. We've shown that the series $\sum_{k\in \mathbb{Z}^d} f(x+k)$ converges uniformly on $\mathbb{R}^d$.
\end{proof}
We note that the only reason for having $(1+ \|x\|)^{d+ \delta}$ in the denominators of the bounds, instead of simply $\|x\|^{d+ \delta}$,
is to give simultaneously a bound at the origin, as well as any nonzero $x$.
\bigskip
\section{Poisson summation II, \'a la Poisson}
There are various different families of functions for which the adjective `nice' applies, in \eqref{nice functions},
and one of the simplest to understand is the Schwartz class of functions.
But there is a more general family of nice functions that is extremely useful, given by Poisson himself, as follows.
\begin{thm}[Poisson summation formula, II] \index{Poisson summation formula}
\label{nice2}
Suppose that for some positive constants $\delta$, $C$, and for all $x \in \mathbb{R}^d$, we have the bounds:
\begin{align} \label{growth conditions for Poisson}
|f(x)| < \frac{C}{ (1+\|x\|)^{d+\delta} }
\text{ \, and \, } |\hat{f}(x)| < \frac{C}{ ( 1+\|x\|)^{d+\delta} }.
\end{align}
Then we have the pointwise equality:
\begin{equation} \label{Poisson summation, take 2}
\sum_{n \in \mathbb{Z}^d} f(n+ x) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle},
\end{equation}
for each $x\in \mathbb{R}^d$. In addition, both sides of \eqref{Poisson summation, take 2}
converge absolutely, and are continuous functions on $\mathbb{R}^d$.
\end{thm}
\begin{proof}
Step $1$. \ The growth conditions \eqref{growth conditions for Poisson} allow us to conclude that both $f, \hat f \in L^1(\mathbb{R}^d)$, by Lemma \ref{Poisson bound implies L^1}.
This implies that both $f, \hat f \in L^2(\mathbb{R}^d)$, by the elementary Lemma \ref{both f and its FT in L^1 implies L^2}.
We also know that the Fourier transform of an $L^1$ function must be uniformly continuous on $\mathbb{R}^d$, and so both $f$ and $\hat f$ are uniformly continuous (Lemma \ref{uniform continuity}).
Step $2$. \ The hypothesis regarding the growth conditions \eqref{growth conditions for Poisson} implies that the series defined by
$F(x):= \sum_{n \in \mathbb{Z}^d} f(n+ x)$ converges uniformly on $[0, 1]^d$, as we showed in Lemma \ref{Poisson bound implies absolutely summable}.
It follows that this series must also converge in the $L^2$-norm on $[0, 1]^d$. So $F \in L^2({\mathbb T^d})$, and it must therefore
possess a Fourier series, which converges to it in the $L^2$-norm:
\begin{equation}
F(x) = \sum_{n \in \mathbb{Z}^d} a_n \, e^{2\pi i \langle n, x \rangle}.
\end{equation}
Step $3$. \ Next, we compute the Fourier coefficients $a_k$.
This is almost the same step that already appeared in the proof of Theorem \ref{Poisson.Summation}, but we repeat it for completeness, and also because the interchange of sum and integral below is justified in a different way.
\begin{align}
a_k &:= \int_{[0,1)^d} F(u) e^{-2\pi i \langle k, u \rangle} du \\
&=\int_{[0,1)^d} \sum_{n \in \mathbb{Z}^d} f(n+ u)
e^{-2\pi i \langle k, u \rangle} du \\
&=\sum_{n \in \mathbb{Z}^d} \int_{[0,1)^d} f(n+ u) \label{outersum}
e^{-2\pi i \langle k, u \rangle} du.
\end{align}
The interchange of summation and integral in the latter step is allowed by the uniform convergence of the series
$ \sum_{n \in \mathbb{Z}^d} f(n+ x)$.
We fix an $n\in\mathbb{Z}^d$ in the outer sum of \eqref{outersum}, and make the change of variable in the integral:
$n+u := w$. As $u$ varies over the cube ${[0,1)^d}$, $w:= n+u$ varies
over all of $\mathbb{R}^d$ because the unit cube tiles the whole space:
\[
{[0,1)^d} + \mathbb{Z}^d = \mathbb{R}^d.
\]
We also have
$e^{-2\pi i \langle k, u \rangle} = e^{-2\pi i \langle k, w-n \rangle} =
e^{-2\pi i \langle k, w \rangle}$, because $k, n \in \mathbb{Z}^d$ and hence
$e^{2\pi i \langle k, n \rangle} =1$.
Finally:
$a_k = \sum_{n \in \mathbb{Z}^d} \int_{n+ [0,1)^d} f(w)
e^{-2\pi i \langle k, w \rangle} dw
= \int_{\mathbb{R}^d} f(w)
e^{-2\pi i \langle k, w \rangle} dw := \hat f(k)$.
Step $4$. \ Since each summand $f(n+x)$ is a continuous function of $x$, and since the convergence is uniform, the function $F(x)$ must also be continuous.
Finally, we'd like to pass from the convergence of the Fourier series in the $L^2$-norm, to pointwise and uniform convergence. For this task we can use
Lemma \ref{norm convergence plus absolute convergence implies equality},
assuming that we can show the absolute convergence of the Fourier series
$\sum_{n \in \mathbb{Z}^d} a_n \, e^{2\pi i \langle n, x \rangle} =
\sum_{n \in \mathbb{Z}^d} \hat f(n) e^{2\pi i \langle n, x \rangle}$.
But this absolute convergence follows from the same Lemma \ref{Poisson bound implies absolutely summable}, with $f$ replaced by $\hat f$, because the same growth bounds \eqref{growth conditions for Poisson}
are also assumed for $\hat f$.
To summarize this last step, we know that $F$ is continuous, and the previous remarks allow us to use Lemma \ref{norm convergence plus absolute convergence implies equality} to conclude that the Fourier series converges pointwise and uniformly to $F(x)$.
\end{proof}
We call a function that enjoys property \eqref{growth conditions for Poisson} a {\bf Poisson function},
because Sim\'eon Denis Poisson proved Theorem \ref{nice2}
between $1823$ and $1827$ \cite{Travaglini}.
Poisson's Theorem \ref{nice2} is a {\bf stronger} version of Poisson summation
than Theorem \ref{Poisson.Summation} above.
To justify this latter claim, we need to show that any Schwartz function also satisfies the growth conditions \eqref{growth conditions for Poisson}, but this is clear because Schwartz functions (and their transforms) decay faster than any polynomial, hence faster than the bounds given by \eqref{growth conditions for Poisson}.
We call the space of functions that satisfy the hypotheses of Theorem \ref{nice2}, the {\bf Poisson space} of functions, in honor of the mathematician that discovered this class. As we've just seen, the suggestion of Figure \ref{nice functions, containment} is correct, showing that the Schwartz space is contained in the Poisson space.
\begin{question}
Are there some natural necessary and sufficient conditions for Poisson summation?
\end{question}
This is an important open question.
In other words, we may ask what are the inherent limitations of functions that satisfy Poisson summation? Although there are well
over $20$ different versions of sufficient conditions in the literature on Poisson summation, there are currently no known necessary and sufficient conditions for Poisson summation to hold.
It is natural to wonder if `nice' functions might include all functions $f:\mathbb{R}^d\rightarrow \mathbb{C}$ such that
\[
f\in L^1(\mathbb{R}^d) \text{ and } \hat f\in L^1(\mathbb{R}^d)?
\]
Sadly, the answer is ``no'' in general, and there is an important counterexample,
by Yitzhak Katznelson (\cite{Katznelson}, Ch. VI, p. 143, Exercise 15).
There are many other families of nice functions in the literature, which include hypotheses such as `functions of bounded variation', and `absolutely continuous' functions. We'll not delve into these other families here, but the reader may glance at
Figure \ref{Refined nice functions} for a slightly more refined relationship between nice functions and the $L^1$ and $L^2$ spaces.
To justify the new containments that is suggested by Figure \ref{Refined nice functions}, we recall that a nice function $f$ was defined in \eqref{nice functions} to include the property that both $f, \hat f \in L^1(\mathbb{R}^d)$.
By Lemma \ref{both f and its FT in L^1 implies L^2}, we know that therefore both
$f, \hat f \in L^2(\mathbb{R}^d)$ as well, so the Figure is correct.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.8in]{RefinedPoisson}
\end{center}
\caption{A slightly more detailed Venn diagram for function spaces related to nice functions}
\label{Refined nice functions}
\end{figure}
We will use a slightly more general version of the Poisson summation formula, which holds for any lattice, and which follows rather quickly from the Poisson summation formula above.
We define a (full-rank) lattice ${\mathcal L}:= M(\mathbb{Z}^d) \subset \mathbb{R}^d$, the image of the integer lattice under an invertible linear transformation $M$.
The {\bf dual lattice} \index{dual lattice} of ${\mathcal L}$ is defined by
${\mathcal L}^* := M^{-T}(\mathbb{Z}^d)$, where $M^{-T}$ is the inverse transpose matrix of the real matrix $M$ (see Section \ref{dual lattice} for more on dual lattices).
As we've seen in Lemma \ref{FT under linear maps}, Fourier Transforms behave beautifully under compositions with any linear transformation.
We will use this fact again in the proof of the following extension of Poisson summation, which holds for
all lattices ${\mathcal L}$ and is quite standard. We recall that a Poisson function $f$ by definition
satisfies the growth conditions \eqref{growth conditions for Poisson}.
\begin{thm} [Poisson summation formula, III]
\label{The Poisson Summation Formula, for lattices}
\index{Poisson summation formula for lattices}
Given a full-rank lattice ${\mathcal L}\subset \mathbb{R}^d$, and a Poisson function $f: \mathbb{R}^d \rightarrow \mathbb{C}$, we have
\begin{equation} \label{PoissonSummationForLattices}
\sum_{n \in {\mathcal L}} f(n+x) = \frac{1}{\det {\mathcal L}} \sum_{m \in {\mathcal L}^*}
\hat f(m) e^{2\pi i \langle x, m \rangle},
\end{equation}
valid for all $x \in \mathbb{R}^d$. In particular, we have
\begin{equation} \label{Poisson.summation3} \index{Poisson summation formula}
\sum_{n \in {\mathcal L}} f(n) = \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi).
\end{equation}
Both sides of \eqref{PoissonSummationForLattices} converge absolutely and are continuous functions on $\mathbb{R}^d$.
\end{thm}
\begin{proof}
Any lattice (full-rank) may be written as ${\mathcal L} := M(\mathbb{Z}^d)$, so that $\det {\mathcal L} := |\det M|$.
Using the Poisson summation formula \eqref{Poisson.summation1}, with the change of variable $n = Mk$,
with $k \in \mathbb{Z}^d$, we have:
\begin{align*}
\sum_{n \in {\mathcal L}} f(n) &= \sum_{k \in \mathbb{Z}^d} (f\circ M)(k) \\
&= \sum_{\xi \in \mathbb{Z}^d} \widehat{(f\circ M)}(\xi) \\
&= \frac{1}{|\det M|} \sum_{\xi \in \mathbb{Z}^d} \hat f\left(M^{-T} \xi \right) \\
&=\frac{1}{\det {\mathcal L}} \sum_{m \in {\mathcal L}^*} \hat f(m).
\end{align*}
where in the third equality we used the elementary `Stretch' Lemma \ref{FT under linear maps}, and in the fourth equality we used the definition of the dual lattice ${\mathcal L}^*:= M^{-T} \mathbb{Z}^d$.
\end{proof}
As an afterthought, it turns out that the special case \eqref{Poisson.summation3}
also easily implies the general case, namely \eqref{PoissonSummationForLattices}
(Exercise \ref{going backwards in Poisson summation}).
A traditional application of the Poisson summation formula is the quick derivation of
the functional equation of the theta function.
We first define the Gaussian function by:
\begin{equation} \index{Gaussian}
G_t (x) := t^{-\frac{d}{2}} e^{ -\frac{\pi}{t} || x ||^2 },
\end{equation}
for each fixed $t >0$, and for all $x \in \mathbb{R}^d$, as depicted in Figure \ref{pic of Gaussians}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.4in]{Gaussians}
\end{center}
\caption{The Gaussian family of functions $G_t(x)$ with $t = 1, t=.5, t= .3, $ and $t=.1$ respectively. }
\label{pic of Gaussians}
\end{figure}
Two immediately interesting properties of the Gaussian are:
\begin{equation}
\int_{\mathbb{R}^d} G_{t} (x) dx = 1,
\end{equation}
for each $t>0$, and
\begin{equation}\label{transform of the Gaussian}
\hat G_t( m ) = e^{ -\pi t || m ||^2 },
\end{equation}
properties which are important in Statistics as well
(Exercises \ref{Gaussian1} and \ref{Gaussian2}). Each fixed $\varepsilon$ gives us one Gaussian function and intuitively, as $\varepsilon \rightarrow 0$, this sequence of Gaussians approaches the ``Dirac delta function'' at the origin, which is really known as a ``generalized function'', or ``distribution'' (Note \ref{Dirac delta}).
\begin{example}
\rm{
The classical theta function \index{theta function} (for the integer lattice) is defined by:
\begin{equation} \label{theta function for the integer lattice}
\theta(t) = \sum_{n\in \mathbb{Z}^d} e^{ -\pi t || n ||^2 }.
\end{equation}
This function plays a major role in analytic number theory. One of its first historical applications was carried out
by Riemann himself, who proved its functional equation (eq. \eqref{theta functional equation} below) and then applied a ``Mellin transform'' to it, to prove the functional equation of the Riemann zeta function $\zeta(s):= \sum_{n=1}^\infty \frac{1}{n^s}$.
We claim that the theta function has the functional equation
\begin{equation} \label{theta functional equation}
\theta\left( \frac{1}{t} \right) = t^{\frac{d}{2}} \theta(t),
\end{equation}
for all $t>0$. This will follow immediately from the Poisson summation formula \ref{Poisson.summation2} by using \index{Poisson summation formula}
$f(x):= G_t(x)$. Using our knowledge of its FT, from
\ref{transform of the Gaussian}, we have:
\begin{align*}
\sum_{n\in \mathbb{Z}^d} G_t(n) &= \sum_{\xi \in \mathbb{Z}^d} \hat G_t(\xi) \\
&= \sum_{\xi \in \mathbb{Z}^d} e^{ -\pi t || \xi ||^2 } := \theta(t).
\end{align*}
Since by definition $\sum_{n\in \mathbb{Z}^d} G_t(n) :=
t^{-\frac{d}{2}} \sum_{n\in \mathbb{Z}^d} e^{ -\frac{\pi}{t} ||n||^2 }
:= t^{-\frac{d}{2}} \theta\left(\frac{1}{t}\right)$, \eqref{theta functional equation} is proved.
}
\hfill $\square$
\end{example}
\bigskip
\section{The convolution operation} \label{* is born}
\index{convolution}
For $f,g \in L^1(\mathbb{R}^d)$, their {\bf convolution} is defined by
\begin{equation} \label{def of convolution}
(f * g)(x) = \int_{\mathbb{R}^d} f(x-y) g(y) dy,
\end{equation}
We will also use definition \eqref{def of convolution} to include more general functions $f, g$, for which the latter
integral still converges (see Examples \ref{ex heaviside}, \ref{ex ramp} below).
It is possible to think intuitively of this analogue of multiplication as: ``this is how waves like to multiply", via Lemma \ref{convolution theorem} \ref{convolution under FT}. We have the following basic relations for the convolution operation.
\begin{lem} \label{convolution theorem}
For all $f, g, h \in L^1(\mathbb{R}^d)$, we have:
\begin{enumerate}[(a)]
\item $f*g \in L^1(\mathbb{R}^d)$.
\label{part 1:convolution theorem}
\item $ \widehat{(f * g)}(\xi) = {\hat f}(\xi) {\hat g}(\xi)$.
\label{convolution under FT}
\item $ f*g = g*f, \ \ f*(g*h)= (f*g)*h $, and $ \ f*(g+h) = f*g + f*h$.
\label{part 3:convolution theorem}
\item $\|f*g\|_1 \leq \|f\|_1 \|g\|_1$.
\label{part 4:convolution theorem}
\item More generally, when $f\in L^p(\mathbb{R}^d), \ \ g \in L^1(\mathbb{R}^d)$, with $1\leq p <\infty$, then we have
$f*g \in L^p(\mathbb{R}^d)$ and
\[
\|f*g\|_p \leq \|f\|_p \|g\|_1.
\]
\end{enumerate}
\end{lem}
\begin{proof}
To prove part \ref{convolution under FT}, we use Fubini's Theorem
(Theorem \ref{Fubini} in the Appendix):
\begin{align*}
\widehat{(f * g)}(\xi) &:= \int_{\mathbb{R}^d} e^{-2\pi i \langle x, \xi \rangle} \left(\int_{\mathbb{R}^d}f(x-y) g(y) dy \right) dx \\
&= \int_{\mathbb{R}^d} g(y) e^{-2\pi i \langle y, \xi \rangle} dy \int_{\mathbb{R}^d}f(x-y) e^{-2\pi i \langle x-y, \xi \rangle} dx \\
&= \int_{\mathbb{R}^d} g(y) e^{-2\pi i \langle y, \xi \rangle} dy \int_{\mathbb{R}^d}f(x) e^{-2\pi i \langle x, \xi \rangle} dx \\
&:= {\hat f}(\xi) {\hat g}(\xi),
\end{align*}
where we've used the translation invariance of the measure, in the penultimate equality.
To prove part \ref{part 4:convolution theorem}, we use Fubini's theorem again, and the triangle inequality for integrals:
\begin{align*}
\| f * g\|_1&:= \int_{\mathbb{R}^d} \left | \int_{\mathbb{R}^d}f(x-y) g(y) dy \right | dx \\
&\leq \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \left | f(x-y) g(y) \right | dy dx \\
&=\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \left | f(y) g(y) \right | dy dx \\
&= \int_{\mathbb{R}^d} \left | f(y) \right | dy \int_{\mathbb{R}^d} \left | g(y) \right | dx \\
&:= \|f\|_1 \|g\|_1.
\end{align*}
For the proofs of the remaining parts, we recommend Rudin's book \cite{RudinGreenBook}.
\end{proof}
Lemma \ref{convolution theorem} \ref{convolution under FT}
means that convolution of functions in the space domain corresponds to
the usual multiplication of functions in the frequency domain (and vice-versa).
\begin{example}
\rm{
When ${\mathcal P} := [-\frac{1}{2}, \frac{1}{2}]$, the convolution of $1_{\mathcal P}$ with itself is drawn in Figure \ref{pic of convolution of indicator}. We can already see that this convolution is a continuous function,
hence a little smoother than the discontinuous function $1_{\mathcal P}$.
Using Lemma \ref{convolution theorem} we have
\[
\widehat{ (1_{\mathcal P} * 1_{\mathcal P}) }(\xi) = \hat{1}_{\mathcal P}(\xi) \hat{1}_{\mathcal P}(\xi) =
\left( \frac{\sin(\pi \xi)}{\pi \xi} \right)^2.
\]
We've used equation \ref{ClassicalExample} in the last equality, for the Fourier transform of our interval ${\mathcal P}$ here. Considering the graph in Figure \ref{pic of sinc2}, for the Fourier transform of the convolution $(1_{\mathcal P} * 1_{\mathcal P})$, we see that this positive function is already much more tightly concentrated near the origin, as compared with
$\rm{sinc}(x):= \hat 1_{\mathcal P}(\xi)$. We work out all of the details for this $1$-dimensional function, and generalize it,
in Example \ref{hat function} below.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.4in]{ConvolutionOfIndicator}
\end{center}
\caption{The function $\left( 1_{\mathcal P} * 1_{\mathcal P} \right) (x)$, with ${\mathcal P}:= \left[ -\frac{1}{2}, \frac{1}{2} \right]$ }
\label{pic of convolution of indicator}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.5in]{sinc2}
\end{center}
\caption{The Fourier transform $ \widehat{ \left( 1_{\mathcal P} * 1_{\mathcal P} \right) } (\xi)$,
which is equal to the infinitely smooth, nonnegative function
$\left( \frac{\sin(\pi \xi)}{\pi \xi} \right)^2 := \rm{sinc}^2(\xi)$. }
\label{pic of sinc2}
\end{figure}
}
\hfill $\square$
\end{example}
Another useful bit of intuition about convolutions is that they are a kind of averaging process, and that
the convolution of two functions becomes smoother than either one of them.
For our applications, when we consider the indicator function $1_{\mathcal P}(x)$ for a polytope ${\mathcal P}$,
then this function is not continuous on $\mathbb{R}^d$, so that the Poisson summation formula does not
necessarily hold for it. But if we consider the convolution of $1_{\mathcal P}(x)$ with a Gaussian, for example,
then we arrive at the $C^\infty$ function
\[
(1_{\mathcal P} * G_t)(x),
\]
for which the Poisson summation
does hold. In the sequel, we will use the latter convolved function in tandem with Poisson summation to study ``solid angles".
\medskip
\begin{example} \label{convolution of general bodies}
\rm{
For any bounded measurable sets $K, L \subset \mathbb{R}^d$, we have
\begin{align}
(1_K * 1_L)(y) &:= \int_{\mathbb{R}^d} 1_{K}(x) 1_{L}(y-x) dx \\
&= \int_{\mathbb{R}^d} 1_{K \cap (-L + y)}(x) dx \\
&= \int_{K \cap (-L + y)} dx \\ \label{volume formula for convolution}
& = \vol\left( K \cap (-L + y) \right),
\end{align}
so that the convolution of indicator functions gives volumes, and this simple connection is one of the entry points into convex geometry.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.5in]{HatFunction}
\end{center}
\caption{The hat function $ 1_{\left[ -r, r \right]} * 1_{\left[ -r, r \right]}$ of Example \ref{hat function}, with $r = 3.5$.}
\label{pic of hat function, take 2}
\end{figure}
\medskip
\begin{example} \label{hat function}
\rm{
As a special case of Example \ref{convolution of general bodies}, consider the case
$K= L := \left[ -r, r \right] \subset \mathbb{R}$. So we now know,
by \eqref{volume formula for convolution}, that
\begin{align}
g(x):=\left(1_{\left[ -r, r \right]} * 1_{\left[ -r, r \right]} \right)(x) =
\vol\Big( \left[ -r, r \right] \cap (\left[ -r, r \right] + x) \Big),
\end{align}
making it clear that for $x \leq -2r$ and $x \geq 2r$, we have
$\vol\Big( \left[ -r, r \right] \cap (\left[ -r, r \right] + x) \Big)=0$.
Precisely, when $x \in [-2r, 0]$, we have the function
\[
g(x):= \vol\Big( \left[ -r, r \right] \cap (\left[ -r, r \right] + x) \Big) = | x-2r | = x+ 2r,
\]
Finally, when $x \in [0, 2r]$, we have the function
$g(x):= \vol\Big( \left[ -r, r \right] \cap (\left[ -r, r \right] + x) \Big) = | x-2r | = 2r - x $. To summarize, we have
\[
g(x) =
\begin{cases}
2r-|x| & \text{ if } x \in [-2r, 2r] \\
0 & \text{ if not.}
\end{cases}
\]
Due to its shape, $g$ is sometimes called the {\bf hat function}.
The hat function is extremely useful in many applications. For example, we can use it to build up functions that are compactly supported on $\mathbb{R}$, and yet whose Fourier transform is
\emph{strictly positive} on $\mathbb{R}$ - see Exercise \ref{positive FT over R}.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.4in]{HeavisideFunction}
\end{center}
\caption{The heaviside function $H_0(x)$ }
\label{Heaviside function}
\end{figure}
\bigskip
\begin{example} \label{ex heaviside}
\rm{
The {\bf Heaviside function} is defined by
\begin{equation} \label{def of heaviside}
H_a(x):= \begin{cases}
1 &\mbox{if } x \geq a \\
0 & \mbox{if } x < a,
\end{cases}
\end{equation}
where $a$ is any fixed real number. Although the Heaviside function is clearly not absolutely integrable over $\mathbb{R}$,
we may still use the same definition \eqref{def of convolution} for its convolution with a function $f\in L^1(\mathbb{R})$:
\begin{equation}\label{Heaviside convolution}
(f*H_0)(x):= \int_{\mathbb{R}} f(x-y) H_0(y) dy=\int_{0}^\infty f(x-y) dy=\int_{-\infty}^x f(t) dt,
\end{equation}
a convergent integral.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.4in]{RampFunction}
\end{center}
\caption{The ramp function $r_5(x)$ }
\label{Ramp function}
\end{figure}
\bigskip
\begin{example} \label{ex ramp}
\rm{
The {\bf ramp function} is defined by
\begin{equation} \label{def of ramp}
r_a(x):= \begin{cases}
x &\mbox{if } x \geq a \\
0 & \mbox{if } x < a,
\end{cases}
\end{equation}
where $a$ is any fixed real number. It is evident that we also have
$r_0(x) = \max\{ x, 0\}$. It is also clear that $r'_a(x) = H_a(x)$. The ramp function is ubiquitous in the analysis of machine learning algorithms, where it is called
the ReLu (Rectified Linear Unit) function.
There is an elegant relationship between the ramp function and the Heaviside function:
\begin{equation} \label{claim for two heavisides}
H_0*H_0 = r_0,
\end{equation}
so we see that convolution makes sense here despite the fact that none of these functions are in $L^1(\mathbb{R})$!
To check the latter claim \eqref{claim for two heavisides}, we use \eqref{Heaviside convolution} above:
\begin{align*}
H_0*H_0(x)&:= \int_{-\infty}^x H_0(t) dt =
\begin{cases}
\int_0^x dx &\mbox{if } x \geq 0 \\
0 & \mbox{if } x<0
\end{cases} \\
&=
\begin{cases}
x &\mbox{if } x \geq 0 \\
0 & \mbox{if } x<0
\end{cases}
:= r_0(x).
\end{align*}
There is also a straightforward extension: $H_a*H_b = r_{a+b}$ (Exercise \ref{Heaviside and ramp}).
}
\hfill $\square$
\end{example}
Having seen convolutions, with various examples, we can now return to the question:
\begin{question}
What is the image of the space $L^1(\mathbb{R}^d)$ under the Fourier transform?
\end{question}
It seems that there is no known `complete' answer to this question yet; however, an apparently lesser-known
but elegant result, due to W. Rudin, is the following correspondence.
\begin{thm}[Rudin] \label{RudinAmazingConvolutions}
\begin{equation}
f \in L^1(\mathbb{R}^d) \iff \hat f = g*h, \text{ with } g, h \in L^2(\mathbb{R}^d).
\end{equation}
\hfill $\square$
\end{thm}
In words, Theorem \ref{RudinAmazingConvolutions} tells us that
the image of $L^1(\mathbb{R}^d)$ under the Fourier transform consists precisely of the set of convolutions
$g*h$, where $g, h \in L^2(\mathbb{R}^d)$
(See \cite{RudinGroups}, Theorem 1.6.3, p.~27).
Here is an outline of a proof for the easy direction: suppose that $g, h \in L^2(\mathbb{R}^d)$. Because we want to find
a solution in $f$, to the equation $\hat f = g*h$, it's natural to try $f := \widehat{g*h} = \hat g \cdot \hat h$. Let's try it, by defining
\[
f:= \hat g \cdot \hat h.
\]
Because the Fourier transform acting on $L^2(\mathbb{R}^d)$ is an isometry,
we have $\hat g, \hat h \in L^2(\mathbb{R}^d)$. Also, the product of two $L^2$ functions in an $L^1$ function
(eq. \eqref{product of two L^2 functions is L^1}), so we conclude that
$f:= \hat g \cdot \hat h \in L^1(\mathbb{R}^d)$, as required.
\bigskip
\subsection{How natural is the Fourier transform?}
We close by thinking a bit about another natural question. We've seen that if $f, \hat f \in L^1(\mathbb{R}^d)$, then the map
\[
\Phi_\xi: f \rightarrow \hat f(\xi),
\]
for each fixed $\xi \in \mathbb{R}^d$,
is a homomorphism from $L^1(\mathbb{R}^d)$ to $\mathbb{C}$, due to
Lemma \ref{convolution theorem} \ref{convolution under FT}.
Are there other transforms that act on $L^1(\mathbb{R}^d)$ as a homomorphism? It turns out there are not!
The Fourier transform is the unique homomorphism here, and we record this fact as a lemma, whose proof appears in
\cite{RudinGroups}, Theorem 1.2.2, p.~7).
\begin{lem}
Suppose $\phi: L^1(\mathbb{R}^d) \rightarrow \mathbb{C}$ is a nonzero complex homomorphism.
Then $\phi = \Phi_\xi$, for some $\xi \in \mathbb{R}^d$.
\hfill $\square$
\end{lem}
\begin{comment}
Sometimes it is not necessary for $f$ or $g$ to be in $L^1(\mathbb{R}^d)$ in order for their convolution to make sense, as we've already seen in Examples \ref{ex heaviside} and \ref{ex ramp}. Here is another case where such a construction is sometimes useful.
\begin{example}
\rm{
Suppose we have two functions $\psi, f:\mathbb{R}^d\rightarrow \mathbb{C}^d$ that are both compactly supported,
and $\psi(x) = 1$ for all $x \in \supp(f)$. Here we notice that by construction we have $f(x) = \psi(x) f(x), \forall x \in \mathbb{R}^d$.
Even in the case that we're out of luck, and
$\hat f \notin L^1(\mathbb{R}^d)$, let's ask if we might still have the identity
\begin{equation}
\widehat{(\psi \cdot f)} = (\hat \psi * \hat f) \,?
\end{equation}
We can compute:
\begin{align*}
(\hat \psi * \hat f)(t) &:= \int_{\mathbb{R}^d} \hat \psi(t-x) \hat f(x) dx \\
&= \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \psi(u) e^{-2\pi i \langle u, t-x \rangle} du
\int_{\mathbb{R}^d} f(v) e^{-2\pi i \langle v, x \rangle} dv dx \\
\end{align*}
}
\end{example}
\end{comment}
\bigskip
\subsection{The Dirichlet Kernel}
Using convolutions, we may now also go back to the partial sums of a Fourier series, which we have defined in \eqref{partial sums}
by
\begin{equation}
S_N f(t):= \sum_{n= -N}^N \hat f(n) e^{2\pi i n t}.
\end{equation}
We compute:
\begin{align*}
S_N f(t)
&:=
\sum_{n= -N}^N \hat f(n) e^{2\pi i n t} =
\sum_{n= -N}^N \int_0^1 f(x) e^{-2\pi i x n} dx \, e^{2\pi i n t} \\
&=
\int_0^1 f(x) \sum_{n= -N}^N e^{2\pi i (t-x) n} dx\\
&:= (f*D_N)(t),
\end{align*}
where this convolution is defined on the $1$-Torus (the circle), and where we introduced the important definition
\begin{equation}
D_N(x):= \sum_{n= -N}^N e^{2\pi i x n},
\end{equation}
known as the {\bf Dirichlet kernel}.
But look how naturally another convolution came up! We've just proved the following elementary Lemma.
\begin{lem}
If $f \in L^2(\mathbb T)$, then
\[
S_N f(t) = (f*D_N)(t),
\]
where this convolution is taken over $[0, 1]$.
\hfill $\square$
\end{lem}
It's therefore very natural to study the behavior of the Dirichlet kernel on its own. In Exercise \ref{first Dirichlet kernel}, we showed that
the Dirichlet kernel has the closed form
\[
D_N(x) = \frac{\sin \left( \pi x(2N + 1) \right) }{\sin(\pi x)}.
\]
\index{Dirichlet kernel}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.4in]{DirichletKernel}
\end{center}
\caption{The Dirichlet Kernel $D_{20}(x)$, restricted to the interval $[-1, 1]$ }
\label{pic of Dirichlet Kernel}
\end{figure}
It's clear from the definition of $D_N(x)$ that it is a periodic function of $x$, with period $1$, and if we restrict our attention to the interval $[-1, 1]$, then its graph appears in Figure \ref{pic of Dirichlet Kernel}. It turns out the the $L^1$ norm of the Dirichlet kernel becomes unbounded as $N\rightarrow \infty$, and this phenomenon is responsible for a lot of results about pointwise divergence of Fourier series, a very delicate subject that is replete with technical subtleties.
There are even examples of continuous functions $f$ whose partial Fourier sums
\index{partial Fourier sums}
$S_N f ( x )$ do not converge anywhere (\cite{Travaglini}, Theorem 4.19).
However, the Dirichlet kernel is also useful for proving pointwise
convergence theorems, such as the important
Theorem \ref{theorem:Fourier series convergence to the mean}.
\bigskip
\section{Plancherel} \index{Plancherel Theorem}
One of the main results in Fourier analysis is the {\bf Plancherel Theorem},
which tells us that the Fourier transform, acting on $L^2(\mathbb{R}^d)$, is an isometry.
In other words, we'll show that the Fourier transform preserves norms of functions: $\| \hat f \|_2 = \| f \|_2$.
\begin{thm}[Plancherel]
\label{thm:Plancherel}
Let $f, g \in L^2(\mathbb{R}^d)$. Then we have:
\begin{enumerate}[(a)]
\item
\begin{equation}
\int_{\mathbb{R}^d} |\hat f(\xi)|^2 d\xi = \int_{\mathbb{R}^d} |f(x)|^2 dx.
\end{equation}
\item More generally, we have:
\begin{equation} \label{Plancherel identity}
\int_{\mathbb{R}^d} f(x) \overline{g(x)} dx = \int_{\mathbb{R}^d} \hat f(x) \overline{\hat g(x)} dx.
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}
We prove a slightly weaker claim, assuming that we have the additional hypothesis
$f, g \in L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$ as well, so that we may use Lemma \ref{convolution theorem}.
In this way we may get to used to some of the ideas involved without all of the machinery that is required in the case of the
strict $L^2(\mathbb{R}^d)$ assumption
(for a proof under the more general hypothesis of the functions belonging to $L^2(\mathbb{R}^d)$, see \cite{Folland}, for example).
We let $g(x) := \overline{f(-x)}$, so that
\begin{align*}
\hat g(\xi) &= \int_{\mathbb{R}^d} \overline{f(-x)} e^{-2\pi i \langle x, \xi \rangle} dx \\
&= \overline{ \int_{\mathbb{R}^d} f(-x) e^{2\pi i \langle x, \xi \rangle} dx } \\
&= \overline{ \hat f(\xi)}.
\end{align*}
We define $h := f*g$, and by Lemma \ref{convolution theorem} we have $\hat h(\xi) = \hat f(\xi)\hat g(\xi)$,
so that $\hat h(\xi) = \|\hat f(\xi)\|^2$.
Now, $h(0) := \int_{\mathbb{R}^d} f(0-x) g(x)dx = \int_{\mathbb{R}^d} f(-x) \overline{f(-x)} dx =
\int_{\mathbb{R}^d} |f(x)|^2$.
On the other hand, $h(0) = \int_{\mathbb{R}^d} \hat h(\xi) d\xi = \int_{\mathbb{R}^d} |\hat f(\xi)|^2 d\xi $. We therefore have
\[
\int_{\mathbb{R}^d} |\hat f(\xi)|^2 d\xi = \int_{\mathbb{R}^d} |f(x)|^2 dx.
\]
The proof of part (b) is quite similar, and we do not want to deprive the reader of the pleasure (Exercise \ref{Plancherel extended}).
\end{proof}
\medskip
\begin{example}
\rm{
Let's fix any compact set $Q\subset \mathbb{R}^d$.
Because $1_Q \in L^2(\mathbb{R}^d)$, we also have $\hat 1_Q \in L^2(\mathbb{R}^d)$, by Plancherel. In other words, Theorem \ref{thm:Plancherel} gives us
\begin{equation} \label{consequence of Plancherel for indicator}
\int_{\mathbb{R}^d} |\hat 1_Q (\xi)|^2 d\xi = \int_{\mathbb{R}^d} |1_Q (x)|^2 dx = \int_Q dx = \vol Q < \infty.
\end{equation}
Let's ask a question:
\begin{question} \rm{[Rhetorical]}
Does the function $g:= 1_Q * 1_{-Q}$ belong to $L^1(\mathbb{R}^d)$?
\end{question}
Well, we also know that trivially $1_Q, 1_{-Q} \in L^1(\mathbb{R}^d)$, which implies that
$\hat g := {\mathcal F}\left( 1_Q * 1_{-Q} \right) = \hat 1_Q \hat 1_{-Q}$, via Lemma \ref{convolution theorem} \ref{convolution under FT}.
The Cauchy-Schwarz inequality gives us:
\[
\int_{\mathbb{R}^d} | \hat g(\xi) | \, d \xi
= \int_{\mathbb{R}^d} | \hat 1_Q(\xi) | |\hat 1_{-Q}(\xi) | \, d \xi
\leq \left( \int_{\mathbb{R}^d} | \hat 1_Q (\xi) |^2 \,d \xi \right)^{1/2}
\left( \int_{\mathbb{R}^d} | \hat 1_{-Q} (\xi) |^2 \,d \xi \right)^{1/2}
< \infty,
\]
the last inequality owing itself to \eqref{consequence of Plancherel for indicator}.
}
\hfill $\square$
\end{example}
\medskip
\begin{example}
\rm{
As we recall, the sinc function, defined by
\begin{equation*}
{\rm{sinc}}(x):= \begin{cases}
\frac{\sin(\pi x)}{\pi x}, &\mbox{if } x \not= 0 \\
1 & \mbox{if } x= 0,
\end{cases}
\end{equation*}
plays an important role (in many fields).
Here we'll glimpse another aspect of its importance, as an application of Plancherel's identity
\eqref{Plancherel identity} above. Let's show that
\begin{equation}
\int_\mathbb{R} {\rm{sinc}}(x-n) {\rm{sinc}}(x-m) dx =
\begin{cases}
1 & \mbox{if } n=m \\
0 & \mbox{if } n\not=m.
\end{cases}
\end{equation}
Although ${\rm{sinc}}(x) \notin L^1(\mathbb{R})$, it is true that ${\rm{sinc}}(x) \in L^2(\mathbb{R})$.
Using Plancherel, we have
\begin{align*}
\int_\mathbb{R} {\rm{sinc}}(x-n) {\rm{sinc}}(x-m) dx & = \int_\mathbb{R} {\mathcal F}({\rm{sinc}}(x-n))(\xi) \overline{ {\mathcal F}( {\rm{sinc}}(x-m) )(\xi) } d\xi \\
&= \int_\mathbb{R} 1_{\mathcal P}(\xi) e^{2\pi i \xi n} 1_{\mathcal P}(\xi) \overline{ e^{2\pi i \xi m} }d\xi\\
&= \int_{\mathcal P} e^{2\pi i \xi (n-m)} d\xi \\
&= \delta(n, m),
\end{align*}
where ${\mathcal P}:= [-\frac{1}{2}, \frac{1}{2}]$, and where we've used
the orthogonality of the exponentials over ${\mathcal P}:= [-\frac{1}{2}, \frac{1}{2}]$
(Exercise \ref{orthogonality for exponentials}).
So we see that the collection of functions
\[
\left\{ {\rm{sinc}}(x - n) \bigm | n \in \mathbb{Z} \right\}
\]
forms an orthonormal collection of functions in the Hilbert space $L^2([-\tfrac{1}{2}, \tfrac{1}{2}] )$, relative to its norm. It turns out that when one studies
Shannon's sampling theorem, these translated sinc functions are in fact a complete orthonormal basis for the Hilbert subspace
of $L^2(\mathbb{R})$ that consists of `bandlimited functions' (see Theorem \ref{Shannon}).
}
\hfill $\square$
\end{example}
\section{Approximate identities}
It is a sad fact of life that there is no identity in $L^1(\mathbb{R}^d)$ for the convolution product - in other words, there is no function
$h \in L^1(\mathbb{R}^d)$ such that
\begin{equation}\label{if there was an identity}
f*h = f
\end{equation}
for all $f \in L^1(\mathbb{R}^d)$.
Why is that? Suppose there was such a function $h\in L^1(\mathbb{R}^d)$. Then taking the Fourier transform of both sides of \eqref{if there was an identity}, we would also
have
\begin{equation} \label{taking FT's...}
\hat f \ \hat h= \widehat{f*h} = \hat f,
\end{equation}
for all $f\in L^1(\mathbb{R}^d)$. Picking an $f$ whose transform is nowhere zero, we can divide both sides of \eqref{taking FT's...}
by $\hat f$, to conclude that $\hat h \equiv 1$, the constant function. But by the Riemann-Lebesgue Lemma \ref{Riemann--Lebesgue lemma}, we know that $\hat h$ must go to $0$ as $|x| \rightarrow \infty$, which is a contradiction.
\index{Riemann-Lebesgue lemma}
Nevertheless, it is still interesting to think about what would happen if we were able to apply the
inverse Fourier transform to $\hat h$, formally applying the Fourier transform to the
equation $\hat h = 1$ to get:
\begin{equation}
h(x) = \int_{\mathbb{R}^d} e^{2\pi i \langle x, \xi \rangle} dx,
\end{equation}
an extremely interesting integral that unfortunately diverges. In note \ref{Dirac delta}, we mention briefly that such observations became
critically important for the development of generalized functions that do play the role of the identity for convolutions, and much more.
Although there is no identity element for convolutions, it turns out that using sequences of functions we can get close! Here is how we may do it, and as a consequence we will be able to rigorously apply the Poisson summation formula to a wider class of functions, including smoothed versions of the indicator function of a polytope.
Fix a function $\phi \in L^1(\mathbb{R}^d)$, such that $\int_{\mathbb{R}^d} \phi(x) dx = 1$.
Beginning with any such function $\phi$, we construct an {\bf approximate identity} by defining the sequence of functions
\begin{equation}\label{approximate identity}
\phi_n(x):= n^d \phi(n x),
\end{equation}
for each $n = 1, 2, 3, \dots$.
It's easy to check that we also have $\int_{\mathbb{R}^d}\phi_n(x) dx = 1$, for all $n\geq~1$ (Exercise \ref{total mass 1}).
So scaling $\phi$ by these $n$'s has the effect of
squeezing $\phi$ so that it is becomes concentrated near the origin, while maintaining a total mass of $1$.
Then intuitively a sequence of
such $\phi_n$ functions approach the ``Dirac delta-function" at the origin (which is a distribution, not a function).
There are many families of functions that give an approximate identity.
In practice, we will seldom have to specify exactly which sequence $\phi_n$ we pick, because we will merely use the existence of such a sequence to facilitate the use of Poisson summation.
Returning now to the motivation of this section, we can recover the next-best-thing to an identity for the convolution product, as follows.
\begin{thm}\label{approximate identity convolution}
Suppose we are given a function $f \in L^1(\mathbb{R}^d)$, such that $p \in \mathbb{R}^d$ is a point of continuity for $f$.
Fix an approximate identity $\phi_n(x)$, and assume $f*\phi$ exists. Then we have:
\begin{equation}\label{basic smoothing}
\lim_{n \rightarrow \infty} \left(f * \phi_n \right)(p) = f(p).
\end{equation}
\end{thm}
\begin{proof}
We begin by massaging the convolution product:
\begin{align*}
(\phi_n*f)(p) &:= \int_{\mathbb{R}^d} \phi_n(x) f(p-x) dx \\
&= \int_{\mathbb{R}^d} \phi_n(x) \Big(f(p-x) - f(p) + f(p) \Big) dx \\
&= \int_{\mathbb{R}^d} \phi_n(x) \Big(f(p-x) - f(p) \Big) dx +
f(p) \int_{\mathbb{R}^d} \phi_n(x) dx \\
&= f(p) + \int_{\mathbb{R}^d} \phi_n(x) \Big(f(p-x) - f(p) \Big) dx,
\end{align*}
using the assumption that $\int_{\mathbb{R}^d} \phi_n(x) dx=1$.
Using the definition of
$\phi_n(x):= n^{d} \phi(n x)$, and making a change of variable
$u= n x$ in the latter integral, we have:
\begin{align*}
(\phi_n*f)(p) &:= f(p) + \int_{\mathbb{R}^d} \phi(u) \Big( f\left(p- \frac{1}{n} u\right) - f(p) \Big) du.
\end{align*}
In the second part of the proof, we will show that as $n \rightarrow \infty$, the latter integral tends to zero.
We will do this in two steps, first bounding the tails of the integral in a neighborhood of infinity, and then bounding the integral in a neighborhood of the origin.
Step $1$. \ Given any $\varepsilon >0$, we note that the latter integral converges, so the `tails are arbitrarily small'. In other words, there exists an $r > 0$ such that
\[
\left| \int_{\| u \| > r} \phi(u) \left(f\left(p- \frac{1}{n} u\right) - f(p) \right) du \right| < \varepsilon.
\]
Step $2$. \ Now we want to bound $\int_{\| u \| < r} \phi(u) \left( f\left(p-\frac{1}{n} u\right) - f(p) \right) du$.
We will use the fact that $ \int_{\mathbb{R}^d} | \phi(u) | du = M$, a constant. Also, by continuity of $f$ at $p$,
we can pick an $n$ sufficiently large, such that:
\[
\left| f\left(p-\frac{1}{n} u\right) - f(p) \right| < \frac{\varepsilon}{M},
\]
when $\| \frac{1}{n} u \| < r$. Putting all of this together, and using the triangle inequality for integrals,
we have the bound
\begin{align*}
\Big| \int_{\| u \| < r} \phi(u) \left(f\left(p-\frac{1}{n} u\right) - f(p) \right) du \Big|
&\leq \int_{\| u \| < r} | \phi(u) | \left| f\left(p-\frac{1}{n} u\right) - f(p) \right| du < \varepsilon.
\end{align*}
Therefore, as $n \rightarrow \infty $, we have $(\phi_n*f)(p) \longrightarrow f(p)$.
\end{proof}
We note that a point of discontinuity of $f$, Theorem \ref{approximate identity convolution}
may be false even in dimension $1$, as the next example shows.
\begin{example}
Let $f(x):= 1_{[0,1]}(x)$, which is discontinuous at $x=0$ and $x=1$. We claim that
for $p=1$, for example, we have
\[
\lim_{n \rightarrow \infty} (f * \phi_n)(p) = \frac{1}{2} f(p),
\]
so that the result of Theorem \ref{approximate identity convolution}
does not hold at this particular $p$, because $p$ lies on the boundary
of the $1$-dimensional polytope $[0,1]$. When $p \in \interior([0,1])$, however, Theorem
\ref{approximate identity convolution} does hold.
\hfill $\square$
\end{example}
\section{A practical Poisson summation formula}
In practice, we want to apply Poisson summation to indicator functions $1_{\mathcal P}$ of polytopes and general convex bodies.
With this in mind, it's useful for us to have our own, home-cooked version of Poisson summation that is made for this culinary purpose.
Throughout this section, we fix any compactly supported, nonnegative function $\varphi \in L^2(\mathbb{R}^d)$, with
$\int_{\mathbb{R}^d} \varphi(x) dx = 1$,
and we
set $\varphi_\varepsilon(x) := \frac{1}{\varepsilon^d} \varphi \left( \frac{x}{\varepsilon} \right)$, for each $\varepsilon > 0$.
\bigskip
\begin{thm}[Poisson summation formula IV] \label{PracticalPoisson}
\index{Poisson summation formula}
Let $f(x) \in L^2(\mathbb{R}^{d})$ be a compactly supported function, and suppose that for each $x\in \mathbb{R}^d$,
we have:
\begin{equation} \label{hypothesis, practical Poisson}
f(x) =\lim_{\varepsilon\rightarrow0^{+}}
\left(
\varphi_{\varepsilon}\ast f
\right)(x).
\end{equation}
Then the following hold:
\begin{enumerate}[(a)]
\item
For each $\varepsilon>0$, we have absolute convergence:
$
\sum_{m\in\mathbb{Z}^d} \left| \widehat{\varphi}\left( \varepsilon
m\right) \widehat{f}\left( m\right) \right| <+\infty.
$
\item
For all sufficiently small $\varepsilon >0$, and for each fixed $x\in \mathbb{R}^d$, we have the
pointwise equality:
\begin{equation} \label{practical Poisson summation, first version}
\sum_{n\in\mathbb{Z}^{d}}
\left(
\varphi_{\varepsilon}\ast f
\right)\left( n+x\right)
=
\sum_{m\in\mathbb{Z}^d} \widehat{\varphi}\left(
\varepsilon m\right) \widehat{f}\left( m\right)
e^{2\pi i \langle m, x \rangle}.
\end{equation}
\item
\begin{equation} \label{practical Poisson summation, second version}
\sum_{n\in\mathbb{Z}^{d}}f\left( n+x\right)
=
\lim_{\varepsilon \rightarrow 0}
\sum_{m\in\mathbb{Z}^d} \widehat{\varphi}\left(
\varepsilon m\right) \widehat{f}\left( m\right)
e^{2\pi i \langle m, x \rangle}.
\end{equation}
\end{enumerate}
\end{thm}
Because both $f$ and $\varphi_\varepsilon$ are compactly supported,
the left-hand-sides of equations
\eqref{practical Poisson summation, first version} and
\eqref{practical Poisson summation, second version}
are finite sums.
\hfill $\square$
For a detailed proof of Theorem \ref{PracticalPoisson}, see \cite{BrandoliniColzaniTravagliniRobins1}.
An interesting aspect of this version of Poisson summation is that it can sometimes even apply to functions $f$
that are only piecewise continuous on $\mathbb{R}^d$, as long as \eqref{hypothesis, practical Poisson} holds. Our prime example is of course
\[
f(x):= 1_{\mathcal P}(x),
\]
the indicator function of a polytope ${\mathcal P}$, and more generally $1_Q$ for a compact set $Q$ with reasonable behavior, such as a convex body.
In Chapter \ref{Chapter.Minkowski}, we will use this version of Poisson summation, Theorem \ref{PracticalPoisson},
to prove Theorem \ref{zero set of the FT of a polytope}.
An interesting tool that gets used in the proof of Theorem \ref{PracticalPoisson} is a
{\bf Plancherel-Polya type inequality}, as follows.
\begin{lem}
Suppose that $ f \in L^1(\mathbb{R}^d), \hat f \in L^1(\mathbb{R}^d)$, and $f$ is compactly supported. Then there exists a constant $c > 0$, depending on the support of $f$, such that
\begin{equation}\label{Plancherel-Polya inequality}
\sum_{n \in \mathbb{Z}^d}
| \hat f(n) | \leq c \int_{\mathbb{R}^d} | \hat f(\xi) | d \xi.
\end{equation}
\end{lem}
\begin{proof}
We define a new function $\psi$, which is infinitely smooth, and compactly supported, with $\psi(x) = 1$
for all $x$ in the support of $f$. So we have $f(x) = \psi(x) f(x), \forall x \in \mathbb{R}^d$, and
therefore $\hat f(\xi) =(\hat\psi * \hat f)(\xi)$ (using $\hat f \in L^1(\mathbb{R}^d)$). Because $\psi$ is smooth, we know that $\hat \psi$ is rapidly decreasing (by Corollary \ref{cor: f smoother implies FT of F decays faster}),
and we have
\begin{align}
\sum_{n \in \mathbb{Z}^d} | \hat f(n) | &=
\sum_{n \in \mathbb{Z}^d} \left | \int_{\mathbb{R}^d} \hat \psi( n - \xi) \hat f(\xi) d\xi \right | \\
&\leq \sum_{n \in \mathbb{Z}^d} \int_{\mathbb{R}^d} \left | \hat \psi( n - \xi) \hat f(\xi) \right | d\xi \\
&= \int_{\mathbb{R}^d} \sum_{n \in \mathbb{Z}^d} \left | \hat \psi( n - \xi) \right | \left | \hat f(\xi) \right | d\xi \\
& \leq \sup_{\xi \in \mathbb{R}^d} \left( \sum_{n \in \mathbb{Z}^d} \left | \hat \psi( n - \xi) \right | \right)
\int_{\mathbb{R}^d} | \hat f(\xi) | d\xi \\
&\leq c \int_{\mathbb{R}^d} | \hat f(\xi) | d\xi.
\end{align}
The constant $c$ depends on $\psi$, and hence on the support of $f$. To justify the last step, we note that
$g(\xi):= \sum_{n \in \mathbb{Z}^d} | \hat \psi( n - \xi) |$
is a periodic function of $\xi$, with the unit cube $[0, 1]^d$ being a fundamental domain, so it suffices to show that $g$ is bounded on the unit cube.
But due to the rapid decay of $\hat \psi$, we may apply the Weierstrass $M$-test to conclude that
the series $g$ is a uniformly convergent sum of continuous functions; hence $g$ is itself a continuous function on a compact set (the cube), and in fact achieves its maximum there.
\end{proof}
The reader may consult \cite{SchmeisserSickel2000}, for example, for more information about related Plancherel-Polya type inequalities.
In general, there are many functions $f \in L^1(\mathbb{R})$ such that $\sum_{n \in \mathbb{Z}} | \hat f(n) | $ diverges, yet
$ \int_{\mathbb{R}} | \hat f(\xi) | d \xi$ converges, so that \eqref{Plancherel-Polya inequality} is false for these functions
(Exercise \ref{exercise:Plancherel-Polya type inequalities}).
\bigskip
\section{Uncertainty principles}
\label{section:uncertainy principles}
Perhaps the most basic type of \emph{uncertainy principle} is the fact that if a function $f$ is compactly supported,
then its Fourier transform $\hat f$ cannot be compactly supported - Theorem \ref{basic uncertainty principle} below.
Similar impossible constraints, placed simultaneously on both $f$ and $\hat f$,
have become known as {\bf uncertainty principles}. Perhaps the most famous of these, originating in quantum mechanics, is Heisenberg's
discovery, as follows.
\begin{thm}[Heisenberg uncertainty principle]
\index{uncertainty principle, Heisenberg}
Let $f\in L^2(\mathbb{R}^d)$, with the normalization assumption that $\int_{\mathbb{R}^d} |f(x)|^2 dx=1$. Then:
\begin{equation}
\int_{\mathbb{R}^d} \|x\|^2 |f(x)|^2 dx \int_{\mathbb{R}^d} \|x\|^2 |\hat f(x)|^2 dx \geq \frac{1}{16 \pi^2},
\end{equation}
with equality holding if and only if $f$ is equal to a Gaussian.
\rm{(For a proof see \cite{OsgoodBook}, or \cite{DymMcKean}. )}
\hfill $\square$
\end{thm}
\bigskip
\begin{thm}[Hardy uncertainty principle]
\label{Hardy uncertainty principle} \index{uncertainty principle, Hardy}
Let $f\in L^1(\mathbb{R}^d)$ be a function that enjoys the property that
\begin{equation*}
|f(x)| \leq A e^{-\pi c x^2 } \text{ and } \ |\hat f(\xi) | \leq B e^{-\pi \xi^2/c},
\end{equation*}
for all $x, \xi\in \mathbb{R}^d$, and for some constants $A, B, c >0$.
Then $f(x)$ is a scalar multiple of the Gaussian $e^{-\pi c x^2}$.
\rm{(For a proof see \cite{Hardy.uncertainty})}
\hfill $\square$
\end{thm}
\bigskip
\begin{thm} \label{basic uncertainty principle}
Let $f\in L^1(\mathbb{R}^d)$ be a function that is supported on a compact set in $\mathbb{R}^d$.
Then $\hat f$ is not supported on any compact set in $\mathbb{R}^d$.
\rm{(For a proof see \cite{EpsteinBook})}
\hfill $\square$
\end{thm}
\section*{Notes}
\begin{enumerate}[(a)]
\item \label{Fourier books}
There are some wonderful introductory books that develop Fourier analysis from first principles, such as the books by
Stein and Shakarchi \cite{SteinShakarchi} and Giancarlo Travaglini \cite{Travaglini}.
The reader is also encouraged to read more advanced but fundamental introductions to Fourier analysis, in particular the books by Mark Pinsky \cite{MarkPinsky}, Edward Charles Titchmarsh \cite{Titchmarsh},
Einsiedler and Ward \cite{EinsiedlerWardBook}, Dym and McKean \cite{DymMcKean},
and of course the classic: Stein and Weiss \cite{SteinWeiss}. In addition, the book \cite{Terras} by Audrey Terras is a good introduction to Fourier analysis
on finite groups, with applications.
A more informal introduction to Fourier analysis, focusing on various applications, is given by Brad Osgood \cite{OsgoodBook}.
\item There are some ``elementary'' techniques that we will use, from the calculus of a complex variable, but which require essentially no previous knowledge in this field. In particular, suppose we have two analytic functions
$f:\mathbb{C} \rightarrow \mathbb{C}$ and $g:\mathbb{C} \rightarrow \mathbb{C}$, such that $f(z_k) = g(z_k)$ for a convergent sequence of complex numbers $z_k \rightarrow L$, where $L$ is any fixed complex number. Then $f(z) = g(z)$ for all
$z \in \mathbb{C}$.
The same conclusion is true even if the hypothesis is relaxed to the assumption that
both $f$ and $g$ are meromorphic functions, as long as the sequence and its limit stay away from the poles of $f$ and $g$.
\item \label{Dirac delta}
The ``Dirac delta function" is part of the theory of ``generalized functions'' and may be intuitively defined by the full sequence
of Gaussians $G_t (x) := t^{-\frac{d}{2}} e^{ -\frac{\pi}{t} || x ||^2 }$, taken over all $t>0$. The observation that there is no identity for the convolution product
on $\mathbb{R}^d$ is a clear motivation for a theory of generalized functions, beginning with the Dirac delta function.
Another intuitive way of ``defining'' the Dirac delta function is:
\[
\delta_0(x) :=
\begin{cases}
\infty & \mbox{if } x=0 \\
0 & \mbox{if not},
\end{cases}
\]
even though this is not a function. But in the sense of distributions (i.e. generalized functions), we have
$\lim_{\rightarrow 0} G_t(x) = \delta_0(x)$.
More rigorously, the $\delta$-function belongs to a theory of distributions that was developed by Laurent Schwartz
\index{Schwartz, Laurent}
in the 1950's and by S.L. Sobolev in 1936, where we can think of generalized functions as linear functionals on the space of all bump functions on $\mathbb{R}^d$ (see the book by Lighthill \cite{Lighthill} for a nice introduction to generalized functions).
Such generalized functions were originally used by the Physicist Paul Dirac in 1920, before the rigorous mathematical theory was even created for it, in order to better understand quantum mechanics. \index{Dirac, Paul}
\item \label{Note:GregKuperberg} I'd like to thank Greg Kuperberg for very helpeful comments, and in particular for introducing
me to Theorem \ref{Euler-Maclaurin type identity}, for which we still cannot find a published reference.
\item
It is sometimes interesting to derive analogues between norms in $\mathbb{R}^d$ and norms in an infinite dimensional function space.
Among the many norm relations in $\mathbb{R}^d$, we mention one elementary but interesting relation:
\[
\| x \|_1 \leq \sqrt{n} \ \| x \|_2,
\]
for all vectors $ x \in \mathbb{R}^d$, where $\|x\|_1:= |x_1|+\cdots + |x_d|$, and $\|x\|_2:= \sqrt{x_1^2+\cdots + x_d^2}$.
(see Exercise \ref{elementary norm relations} for more practice with related norm relations).
At this point the curious reader might wonder ``are there any other inner products on $\mathbb{R}^d$, besides the usual inner product
$\langle x, y \rangle:= \sum_{k=1}^d x_k y_k$?" A classification of all inner products that exist on $\mathbb{R}^d$ is given in Exercise \ref{All norms on Euclidean space}.
\item
Of great practical importance, and historical significance, a {\bf bump function} is defined as any infinitely smooth function
on $\mathbb{R}^d$, which is compactly supported. In other words, a bump function enjoys the following properties:
\begin{itemize}
\item $\phi$ has compact support on $\mathbb{R}^d$.
\item $\phi \in C^\infty(\mathbb{R}^d)$.
\end{itemize}
Bump functions are also called {\bf test functions}, and if we consider the set of all bump functions
on $\mathbb{R}^d$, under addition, we get a vector space $V$, whose dual vector space is called the space of {\bf distributions on $\mathbb{R}^d$}.
\item The cotangent function, appearing in some of the exercises below, is the unique {\it meromorphic function} that has a simple pole at every integer, with residue 1 (up to multiplication by an entire function with the same residues). The cotangent function also forms an entry point for Eisenstein series in number theory, through the corresponding partial fraction expansion of its derivatives.
\item A deeper exploration into projections and sections of the unit cube in $\mathbb{R}^d$ can be found in
``The cube - a window to convex and discrete geometry'', by Chuangming~Zong \cite{Zong.book}.
In \cite{KoldobskyBook}, Alexander Koldobsky gives a thorough introduction to sections of convex bodies, intersection bodies, and the Busemann-Petty problem.
\item There are numerous other identities throughout mathematics that are equivalent to special cases of Poisson summation, such as the Euler-MacLaurin summation formula, the Abel-Plana
formula, and the Approximate sampling formula of signal analysis
(see \cite{Butzer.etal} for a nice treatment of such equivalences for functions of $1$ real variable, and functions of $1$ complex variable).
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
``In theory, there is no difference between theory and practice; but in practice, there is! '' \ \ \ \ -- Walter J. Savitch \\
\end{quote}
\medskip
\begin{prob} $\clubsuit$ \label{elementary norm relations}
On $\mathbb{R}^d$ the $L^2$-norm is defined by
$\|x\|_2:= \sqrt{ x_1^2 + \cdots + x_d^2}$, the $L^1$-norm is defined by
$\|x\|_1:= |x_1| + \cdots + |x_d|$, and the $L^\infty$-norm is defined by
$\|x\|_\infty:= \max\{ |x_1|, \dots, |x_d| \}$.
Prove the following four norm relations:
\[
\|x\|_\infty \leq \| x \|_2 \leq \| x \|_1 \leq \sqrt{d} \, \| x \|_2 \leq d \, \|x\|_\infty,
\]
for all $x \in \mathbb{R}^d$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{CS inequality for integrals}
Show that the Cauchy-Schwarz inequality holds in the Hilbert space $L^2({\mathbb T^d})$:
\begin{equation}
\int_{{\mathbb T^d}} f(x)\overline{g(x)} dx \leq
\left(\int_{{\mathbb T^d}} |f(x)|^2 dx \right)^{\frac{1}{2}} \left(\int_{{\mathbb T^d}} |g(x)|^2 dx\right)^{\frac{1}{2}},
\end{equation}
for all $f, g \in L^2({\mathbb T^d})$, with equality if and only if $f(x) = C g(x)$ for some constant $C$.
\end{prob}
\medskip
\begin{prob} \label{exercise:hyperbolic cosine and sine}
We know that the functions $u(t) := \cos t = \frac{e^{it} + e^{-it}}{2}$ and $v(t) := \sin t =
\frac{e^{it} - e^{-it}}{2i}$ are natural, partly because they parametrize the unit circle: $u^2 + v^2 = 1$. Here we see that there are other similarly natural functions, parametrizing the hyperbola.
\begin{enumerate}[(a)]
\item Show that the following functions parametrize the hyperbola $u^2 - v^2 = 1$:
\[
u(t) := \frac{e^t + e^{-t}}{2}, \ \ \ v(t) := \frac{e^t - e^{-t}}{2}.
\]
(This is the reason that the function $\cosh t:= \frac{e^t + e^{-t}}{2}$ is called the hyperbolic cosine, and the function
$\sinh t := \frac{e^t - e^{-t}}{2}$ is called the hyperbolic sine)
\item The hyperbolic cotangent is defined as $\coth t := \frac{ \cosh t }{ \sinh t} =
\frac{ e^t + e^{-t}}{e^t - e^{-t}}$. Using Bernoulli numbers, show that $t \coth t$ has the Taylor series:
\[
t \coth t = \sum_{n=0}^\infty \frac{2^{2n}}{(2n)!} B_{2n} t^{2n}.
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{compute FT for exponential of abs value}
Prove that:
\[
\frac{t}{\pi} \sum_{n \in \mathbb{Z}} \frac{1}{n^2 + t^2} = \sum_{m \in \mathbb{Z}} e^{-2\pi t |m|}.
\]
Hint. \ Think of Poisson summation, applied to the function $f(x):= e^{-2\pi t |x|}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Riemann zeta function, and Bernoulli numbers}
Here we evaluate the Riemann zeta function at the positive even integers.
\begin{enumerate}[(a)]
\item Show that
\[
\sum_{n\in \mathbb{Z}} e^{-2 \pi t |n|} =
\frac{ 1 + e^{-2\pi t} }{ 1-e^{-2\pi t} } := \coth(\pi t),
\]
for all $t>0$.
\bigskip
\item Show that the cotangent function has the following well-known partial fraction expansion:
\[
\pi \cot(\pi x) = \frac{1}{x} + 2x \sum_{n=1}^\infty \frac{1}{ x^2 - n^2},
\]
valid for any $x \in \mathbb{R} - \mathbb{Z}$.
\item Let $0 < t < 1$. Show that
\[
\frac{t}{\pi} \sum_{n\in \mathbb{Z}} \frac{1}{n^2 + t^2} = \frac{1}{\pi t} +
\frac{2}{\pi} \sum_{m=1}^\infty (-1)^{m+1} \zeta(2m) \ t^{2m-1},
\]
where $\zeta(s):= \sum_{n=1}^\infty \frac{1}{n^s}$ is the Riemann zeta function, initially defined by the latter series, which is valid for all $s \in \mathbb{C}$ with $Re(s) >1$.
\item Here we show that we may quickly evaluate the Riemann zeta function at all even integers, as follows. We recall the definition of the Bernoulli numbers, namely:
\[
\frac{z}{e^z - 1} = 1 - \frac{z}{2} + \sum_{m \geq 1} \frac{B_{2m}}{2m!} z^{2m}.
\]
Prove that for all $m \geq 1$,
\[
\zeta(2m) = \frac{(-1)^{m+1} }{2} \frac{ (2\pi)^{2m}}{ (2m)!} B_{2m}.
\]
Thus, for example, using the first $3$ Bernoulli numbers, we have: $\zeta(2) = \frac{\pi^2}{6}$, $\zeta(4) = \frac{\pi^4}{90}$, and $\zeta(6) = \frac{\pi^6}{945}$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{strictly less than the FT at zero}
Given a $d$-dimensional polytope ${\mathcal P} \subset \mathbb{R}^d$, prove the strict inequality
\[
|\hat 1_{\mathcal P}(\xi) | < \vol {\mathcal P}, \text{ for all nonzero } \xi \in \mathbb{R}^d.
\]
\end{prob}
\medskip
\begin{prob} \label{Chebyshev polys}
For each $n\geq 1$, let $T_n(x) = \cos(nx)$.
For example, $T_2(x) = \cos(2x) =2 \cos^2(x) - 1$, so $T_2(x) = 2u^2 -1$, a polynomial in $u:= \cos x$.
\begin{enumerate}[(a)]
\item Show that for all $n\geq 1$, $T_n(x)$ is a polynomial in $\cos x$.
\item Can you write $x^n + \frac{1}{x^n}$ as a polynomial in the variable $x + \frac{1}{x}$?
Would your answer be related to the polynomial $T_n(x)$? What's the relationship in general?
For example, $x^2 + \frac{1}{x^2} = \Big( x + \frac{1}{x}\Big)^2 - 2$.
\end{enumerate}
\end{prob}
Notes. The polynomials $T_n(x)$ are very important in applied fields such as approximation theory, and optimization,
because they have many useful extremal properties.
They are called Chebyshev polynomials. \index{Chebyshev polynomials}
\medskip
\begin{prob} \label{sec - its own Fourier transform}
The hyperbolic secant is defined by
\[
{\rm sech}(\pi x) := \frac{2}{e^{\pi x} + e^{-\pi x}}, \text{ for } x \in \mathbb{R}.
\]
\begin{enumerate}[(a)]
\item \label{eigenfunction of FT}
Show that ${\rm sech}(\pi x)$ is its own Fourier transform:
\[
{\mathcal F}({\rm sech})(\xi) = {\rm sech}(\xi),
\]
for all $\xi \in \mathbb{R}$.
\item \label{bounded above by Gaussian}
Show that ${\rm sech}(\pi x)$ can never be bounded above by any Gaussian, in the precise sense that the following claim is impossible: there exists a constant $c>0$ such that for all $x \in \mathbb{R}$ we have:
\[
{\rm sech}(\pi x) \leq e^{-cx^2}.
\]
\end{enumerate}
Notes. For part \ref{eigenfunction of FT}, the reader may need some background in complex analysis for this exercise. For part \ref{bounded above by Gaussian}, it may be helpful to look at Hardy's uncertainty principle, Theorem \ref{Hardy uncertainty principle}. We can also conclude from Hardy's uncertainty principle that any eigenfunction $f$ of the Fourier transform cannot be bounded above by a Gaussian, aside from the case that $f$ is itself a Gaussian.
\end{prob}
\medskip
\begin{prob}
Using the previous exercise, conclude that
\[
\int_{\mathbb{R}} \frac{1}{e^{\pi x} + e^{-\pi x}} dx = \frac{1}{2}.
\]
\end{prob}
\medskip
The following exercises give more practice in computing/handling general Fourier transforms and their important properties. Throughout, we assume that the Fourier transform of $f$ exists, where
$f:\mathbb{R} \rightarrow \mathbb{C}$ is any integrable function.
\medskip
\begin{prob} $\clubsuit$
Prove that:
\[
\int_0^1 P_1(ax) P_1(bx) dx = \frac{ 1 }{12 \ \rm{gcd}^2(a, b)}.
\]
for all positive integers $a, b$. Here $P_1(x) := x - \{x\} - \frac{1}{2}$ is the first periodic Bernoulli polynomial.
Notes. This integral is called a {\bf Franel integral}, and there is a substantial literature about related integrals.
In 1924, J\'er\^ome Franel related this integral to the Riemann hypothesis, and to Farey fractions.
\end{prob}
\medskip
\begin{prob}
Using Theorem \ref{Euler-Maclaurin type identity}, obtain the little-o asymptotics (with $N\rightarrow \infty$)
for the finite sums
\[
\sum_{m=0}^{N-1} \sin \left(\tfrac{m \pi }{N}\right),
\]
which is simple enough that it also offers an independent verification.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Schwartz space convolution invariance}
Let $f: \mathbb{R} \rightarrow \mathbb{C}$ belong to the Schwarz class of functions on $\mathbb{R}$, denoted by $S(\mathbb{R})$.
Show that $\hat f \in S(\mathbb{R})$ as well.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{All norms on Euclidean space}
Here we answer the very natural question ``What are the other inner products on $\mathbb{R}^d$, besides the usual inner product $\langle x, y \rangle:= \sum_{k=1}^d x_k y_k$ ?"
The fact is that all inner products are related to each other via positive definite matrices, as follows. We recall from Linear Algebra that a symmetric
matrix is called positive definite if all of its eigenvalues are positive.
Prove that the following two conditions are equivalent:
\begin{enumerate}
\item $ \langle x, y \rangle$ is an inner product on $\mathbb{R}^d$.
\item $ \langle x, y \rangle := x^T M y$, for some positive definite matrix $M$.
\end{enumerate}
\end{prob}
\bigskip
\begin{prob}
For any positive real numbers $a < b < c < d$, define
\[
f(x) := 1_{[a, b]}(x) + 1_{[c, d]}(x).
\]
Can you find $a,b,c,d$ such that $\hat f(\xi)$ is nonzero for all $\xi \in \mathbb{R}$?
\end{prob}
\medskip
\begin{prob}
\end{prob}
\medskip
\begin{prob} $\clubsuit$
Show that for $f, \hat f \in L^1(\mathbb{R}^d)$, the only eigenvalues of the linear operator
\[
f \rightarrow \hat f
\]
are $\{ 1, -1, i, -i \}$, and show that each of these eigenvalues is achieved by some function $f \in L^1(\mathbb{R}^d)$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{going backwards in Poisson summation}
Show that the special case of Poisson summation, \ref{Poisson.summation3}, implies the general case, Theorem \ref{The Poisson Summation Formula, for lattices}. \index{Poisson summation formula}
\end{prob}
\medskip
\begin{prob} \label{Gaussian1} $\clubsuit$
We define the Gaussian, for each fixed $\varepsilon >0$, and for all $x \in \mathbb{R}^d$, by
\begin{equation} \index{Gaussian}
G_{\varepsilon} (x) := \frac{1}{\varepsilon^{\frac{d}{2}}} e^{ -\frac{\pi}{\varepsilon} || x ||^2 }.
\end{equation}
Show that:
\[
\int_{\mathbb{R}^d} G_{\varepsilon} (x) dx = 1.
\]
\end{prob}
\medskip
\begin{prob} \label{Gaussian2} $\clubsuit$
Show that, for all $m \in \mathbb{R}^d$, the Fourier transform of the Gaussian
$G_{\varepsilon}(x)$ is:
\[
\hat G_\varepsilon( m ) = e^{ -\pi \varepsilon || m ||^2 }.
\]
\end{prob}
\medskip
\begin{prob}
Prove that if $f, g \in L^1(\mathbb{R}^d)$ are bounded functions, then $f*g$ is continuous on $\mathbb{R}^d$.
\end{prob}
Notes. In particular, this exercise shows that if $A, B \subset \mathbb{R}^d$ are convex bodies, then \\
$(1_A*1_B)(x)= \vol\left( A \cap (-B + x) \right)$ is a continuous function of $x \in \mathbb{R}^d$.
\medskip
\begin{prob} \label{Plancherel extended} $\clubsuit$
For all $f, g \in S(\mathbb{R}^d)$ (the Schwartz space), show that $\langle f, g\rangle = \langle \hat f, \hat g \rangle$.
\end{prob}
\medskip
\begin{prob} \label{total mass 1} $\clubsuit$
Given any approximate identity sequence $\phi_\varepsilon$, as defined in \eqref{approximate identity},
show that for each~$\varepsilon~>~0$,
\[
\int_{\mathbb{R}^d}\phi_\varepsilon(x) dx = 1.
\]
\end{prob}
\medskip
\begin{prob}
Show that the ramp function, defined in \eqref{def of ramp}, also has the representation:
\begin{equation}
r_0(x) = \frac{ x + |x| }{2},
\end{equation}
for all $x\in \mathbb{R}$.
Notes. \ Some books, particularly in approximation theory, use the notation $r_0(x) := x_+$.
\end{prob}
\medskip
\begin{prob}
\label{positive FT over R} $\clubsuit$
\rm{
Here we show that there exist compactly supported functions $f:\mathbb{R}\rightarrow \mathbb{C}$
whose Fourier transform is {\bf strictly positive} on all of $\mathbb{R}$.
Fix any two incommensurable real numbers $r, s$ (meaning that $\frac{r}{s} \notin \mathbb{Q}$),
and define
\[
f:= 1_{[-r, r]}*1_{[-r, r]} + 1_{[-s, s]}*1_{[-s, s]},
\]
which is a sum of two hat functions, as depicted in Figure \ref{A sum of two hat functions}.
Prove that for all $\xi \in \mathbb{R}$, we have $ \hat f(\xi) >0$.
}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{SumOfTwoHatFunctions}
\end{center}
\caption{The function $f$ of Exercise \ref{positive FT over R}, a sum of two hat functions,
with $s = \sqrt{\frac{2}{3}}$, and $r= 1.9$}
\label{A sum of two hat functions}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.5in]{StrictlyPositive}
\end{center}
\caption{The \emph{strictly positive} Fourier transform $\hat f(\xi)$ of Exercise \ref{positive FT over R}, with the two incommensurable numbers $s = \sqrt{\frac{2}{3}}$, and $r= 1.9$}
\label{strictly positive FT on R}
\end{figure}
Notes. This construction can be extended to higher dimensions, once we know more about the Fourier transforms of balls in $\mathbb{R}^d$ - see Exercise \ref{positive FT over R^d}.
\end{prob}
\medskip
\begin{prob}
\label{Heaviside and ramp}
$\clubsuit$
Show that for all $a, b \in \mathbb{R}$, we have:
\[
H_a*H_b = r_{a+b},
\]
where $H_a$ is the heaviside function of \eqref{def of heaviside}, and $r_a$ is the ramp function of \eqref{def of ramp}.
\end{prob}
\medskip
\begin{prob}
\label{exercise:Plancherel-Polya type inequalities} $\clubsuit$
Here we show that the absolute convergence of a series, and the absolute convergence of the corresponding integral, are independent of each other.
\begin{enumerate} [(a)]
\item
Find a function $f :\mathbb{R} \rightarrow \mathbb{C}$ such that $\sum_{n \in \mathbb{Z}} | \hat f(n) | $ diverges, yet
$ \int_{\mathbb{R}} | \hat f(\xi) | d \xi$ converges.
\item On the other hand, find a function $f :\mathbb{R} \rightarrow \mathbb{C}$ such that $ \int_{\mathbb{R}} | \hat f(\xi) | d \xi$ diverges, yet
$\sum_{n \in \mathbb{Z}} | \hat f(n) | $ converges.
\end{enumerate}
\end{prob}
Notes. This exercise shows that there the Plancherel-Polya inequality holds only for a special class of functions.
\medskip
\begin{prob} \label{tricky application of Poisson summation}
Here is a tiny variation on Poisson summation. If $g:\mathbb{R}^d\rightarrow \mathbb{C}$ is infinitely smooth, and compactly supported, prove that
\[
\sum_{n \in \mathbb{Z}^d} \hat g(n) = \sum_{n \in \mathbb{Z}^d} g(n),
\]
and the right-hand-side is a finite sum.
\end{prob}
\chapter{The geometry of numbers - \\
Minkowski's first theorem, and Siegel's extension} \label{Chapter.Minkowski}
\begin{wrapfigure}{R}{0.51\textwidth}
\centering
\includegraphics[width=0.24\textwidth]{Minkowski}
\caption{Hermann Minkowski}
\end{wrapfigure}
\label{Geometry of numbers}
\index{Siegel's formula} \index{Minkowski}
\begin{quote}
``Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.''
-- Hermann Minkowski \index{Minkowski, Hermann}
\end{quote}
\section{Intuition}
To see a wonderful and fun application of Poisson summation, we give a relatively easy proof of Minkowski's first theorem, in the Geometry of Numbers.
Minkowski's theorem gives the existence of an integer point inside symmetric
bodies in $\mathbb{R}^d$, once we know their volume is sufficiently large.
In fact we first prove a more powerful identity which
is a classical result of Carl Ludwig Siegel (Theorem \ref{Siegel}), yielding an identity between Fourier transforms of convex bodies and their volume. Our proof of this identity of Siegel uses Poisson summation, applied to the convolution of an indicator function with itself. \index{Poisson summation formula}
The geometry of numbers is an incredibly beautiful field, and too vast to encompass in just one chapter (see note \ref{new books, geometry of numbers}). We hope this chapter, a small bite of a giant fruit, gives the reader motivation to pursue the interactions between convex bodies and lattices even further.
\bigskip
\section{Minkowski's convex body Theorem}
\begin{wrapfigure}{L}{0.55\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{convexbody}
\caption{A convex, symmetric body in $\mathbb{R}^2$, with area bigger than $4$, containing two nonzero integer points.} \label{convex body}
\end{wrapfigure}
Minkowski initiated the field that we call today `the geometry of numbers', around 1890. To begin, we define a {\bf body} ${\mathcal P}$ in $\mathbb R^d$ as a compact set. In other words, ${\mathcal P}$ is a bounded, closed set. Most of the time, it is useful to work with convex bodies that enjoy the following symmetry.
We call a body ${\mathcal P}$ {\bf centrally symmetric}, also called {\bf symmetric about the origin}, if for all ${\bf x} \in \mathbb{R}^d$ we have
\begin{equation} \label{definition of symmetric body}
{\bf x} \in {\mathcal P} \iff -{\bf x} \in {\mathcal P}.
\end{equation}
\bigskip
A body ${\mathcal P}$ is called {\bf symmetric} if some translation of ${\mathcal P}$ is symmetric about the origin.
For example, the ball
$\{ x\in \mathbb{R}^d \mid \| x\| \leq 1\}$ is centrally symmetric, and the translated ball
$
\{ x\in \mathbb{R}^d \mid \| x- w\| \leq 1\}
$
is symmetric, but not centrally symmetric. An initial, motivating question in the geometry of numbers is:
\begin{question}\label{Rhetorical question, centrally symmetric}
{\rm[Rhetorical]} How large does a convex body ${\mathcal P}$ have to be in order to contain a nonzero integer point?
\end{question}
However, if we are not careful, then Figure \ref{nonexample}, for example, shows that ${\mathcal P}$ can be as large as we like, and yet never contain an integer point. So without further hypotheses, there are no positive answers to Question \ref{Rhetorical question, centrally symmetric}. Therefore, it is natural to assume that our body ${\mathcal P}$ is positioned in a `nice' way relative to the integer lattice, and centrally symmetry is a natural assumption in this respect.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{BodyCounterexample}
\end{center}
\caption{A convex symmetric body in $\mathbb{R}^2$, which is not centered at the origin, may be constructed with arbitrarily large volume and simultaneously with
no integer points.} \label{nonexample}
\end{figure}
\bigskip
\begin{thm}[Minkowski's convex body Theorem for $\mathbb{Z}^d$]
\label{Minkowski's convex body theorem, for Z^d}
Let $B$ be a $d$-dimensional convex body in $\mathbb R^d$, symmetric about the origin.
\begin{equation}
\text{ If } \vol B > 2^d, \text{ then } B
\text{ must contain a nonzero integer point in its interior}.
\end{equation}
\hfill $\square$
\end{thm}
Sometimes this classical and very useful result of Minkowski is stated in its contrapositive form:
Let $B \subset \mathbb{R}^d$ be any convex body, symmetric about the origin.
\begin{equation}
\text{ If the only integer point
in the interior of } B \text{ is the origin, then } \vol B \leq 2^d.
\end{equation}
It is natural, and straightforward, to extend this result to any lattice ${\mathcal L}:= M(\mathbb{Z}^d)$, by simply applying the linear transformation $M$ to both the integer lattice, and to the convex body $B$. The conclusion is the following, which is the version that we will prove as a consequence of Siegel's Theorem \ref{Siegel}.
\begin{thm}[Minkowski's convex body Theorem for a lattice ${\mathcal L}$]
\label{Minkowski convex body Theorem for L}
Let $B$ be a $d$-dimensional convex body in $\mathbb R^d$, symmetric about the origin, and let
${\mathcal L}$ be a (full rank) lattice in $\mathbb{R}^d$.
\begin{equation} \label{Minkowski 2}
\text{ If } \vol B > 2^d (\det {\mathcal L}), \text{ then } B
\text{ must contain a nonzero point of } {\mathcal L} \text{ in its interior}.
\end{equation}
\end{thm}
\begin{proof}
The proof appears below - see \eqref{ACTUAL proof of Minkowski's convex body theorem for lattices}.
\end{proof}
These very important initial results of Minkowski \cite{Minkowski} have found applications in algebraic number theory, diophantine analysis, combinatorial optimization, and other fields.
In the next section we show that Minkowski's result \eqref{Minkowski 2} follows as a special case of Siegel's formula.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.8in]{Rhombicdodecahedron}
\end{center}
\caption{The Rhombic dodecahedron, a $3$-dimensional symmetric polytope that tiles $\mathbb{R}^3$ by translations, and is another extremal body for Minkowski's convex body Theorem. }
\label{The Rhombic dodecahedron}
\end{figure}
\section{Siegel's extension of Minkowski: \\
a Fourier transform identity for convex bodies}
An important construction in the geometry of number is the {\bf Minkowski sum} of convex bodies. \index{Minkowski sum}
Given two convex bodies $K, L \subset \mathbb{R}^d$, their Minkowski sum is defined by
\[
K + L := \{ x + y \mid x \in K, y \in L\}.
\]
Another related construction, appearing in some of the results below, is
\[
K- L := \{ x-y \mid x \in K, y \in L\},
\]
the Minkowski difference of $K$ and $L$.
A very useful special case is the gadget known as the
{\bf Minkowski symmetrized body} of $K$, \index{symmetrized body}
defined by
\begin{equation}
\frac{1}{2} K - \frac{1}{2} K,
\end{equation}
and often also called the {\bf difference body} of $\frac{1}{2}K$.
Given any set $K\subset \mathbb{R}^d$, the difference body $K-K$ is centrally symmetric. To see this, suppose $x\in K-K$, so we may write
$x= y-z$, with $y, z \in K$. Then $-x = z-y \in K-K$.
In addition, we have the fortuitous and easy fact that a convex set $K\subset \mathbb{R}^d$ is centrally symmetric if and only if we have the equality
\begin{equation} \label{centrally symmetric set}
\frac{1}{2} K - \frac{1}{2} K=K.
\end{equation}
(Exercise \ref{c.s. C equals its symmetrized body}). Now suppose we are given two convex bodies $K, L\subset \mathbb{R}^d$. Then the resulting bodies $K+L$, $K-L$ turn out to also be convex (Exercise \ref{convexity of K-K}).
Another important geometric notion is the dilation of a convex body by a positive real number~$t$:
\[
tB := \{ t x \mid x\in B\},
\]
The most basic version of Siegel's theorem is the following identity, which assumes that a convex body $K$ is symmetric about the origin.
\bigskip
\begin{thm} [Siegel] \label{Siegel} \index{Siegel's formula}
Let $B$ be any $d$-dimensional convex body in $\mathbb R^d$, symmetric about the origin, and suppose that the only integer point in the interior of $B$ is the origin. Then
\begin{align} \label{Siegel, version 1}
2^d
&= \vol B + \frac{4^d}{ \vol B } \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} B}(\xi) \right|^2.
\end{align}
\hfill $\square$
\end{thm}
We now prove the following extension of Siegel's Theorem \eqref{Siegel}, namely \eqref{Siegel.formula} below, which applies to bodies that are not necessarily convex, nor necessarily symmetric about the origin.
Our proof of Theorem \ref{Siegel for general lattices} below
consists of yet another application of Poisson summation. \index{Poisson summation formula}
It turns out that if $K$ is any convex body, then $f:= 1_{\frac{1}{2} K}*1_{-\frac{1}{2} K}$ is a nice function (Exercise \ref{convolution of indicators is a nice function}),
in the sense that Poisson summation \eqref{nice functions} holds for $f$.
So Theorem \ref{Siegel} is a consequence of the following extension to bodies that are
not necessarily convex or symmetric.
\medskip
\begin{thm}[Siegel's formula, for a general body $K$, and a lattice ${\mathcal L}$]
\label{Siegel for general lattices}
\index{Siegel's formula}
Let $K\subset \mathbb{R}^d$ be a body (compact set) for which
the convolution $1_{\frac{1}{2} K}*1_{-\frac{1}{2} K}$ is a nice function.
If the only integer point in the interior of the difference body
$ \frac{1}{2}K - \frac{1}{2}K$ is the origin, then
\begin{equation}\label{Siegel.formula}
2^d
= \vol K + \frac{4^d}{ \vol K } \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{equation}
More generally, if we replace the lattice $\mathbb{Z}^d$ by any full-rank lattice ${\mathcal L}$, and assume that the only lattice point of ${\mathcal L}$ in the interior of $ \frac{1}{2}K - \frac{1}{2}K$ is the origin, then we have:
\begin{equation}\label{Siegel formula 2}
2^d \det {\mathcal L}
= \vol K + \frac{4^d}{ \vol K } \sum_{\xi \in {\mathcal L}^* - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{equation}
\end{thm}
\begin{proof}
We start with the function
\begin{equation}
f(x):= \left( 1_{\frac{1}{2} K}*1_{-\frac{1}{2} K} \right) (x),
\end{equation}
which is continuous on $\mathbb R^d$, and we plug $f$ into Poisson summation
\eqref{Poisson.summation2}:
\index{Poisson summation formula}
\begin{align}
\sum_{n \in \mathbb{Z}^d} f(n) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi).
\end{align}
We first compute the left-hand-side of Poisson summation, using the definition of $f$:
\begin{align}
\sum_{n \in \mathbb{Z}^d} f(n) &= \sum_{n \in \mathbb{Z}^d} \int_{\mathbb{R}^d} 1_{\frac{1}{2} K}(y) 1_{-\frac{1}{2} K}(n - y) dy \\
&= \sum_{n \in \mathbb{Z}^d} \int_{\mathbb{R}^d} 1_{\frac{1}{2} \rm{int} K}(y) 1_{-\frac{1}{2} \rm{int} K}(n - y) dy,
\end{align}
where the last step follows from the fact that the integral does not distinguish between a convex set or its closure.
Now we follow the definition of containment: $y \in \frac{1}{2} K$ and $n - y \in -\frac{1}{2} K$ imply that the integer point $n \in \frac{1}{2} K -\frac{1}{2} K$. But by hypothesis $ \frac{1}{2} K -\frac{1}{2} K$ contains the origin as its {\em only} interior integer point, so the left-hand-side of the
Poisson summation formula contains only one term, namely the $n=0$ term:
\begin{align}
\sum_{n \in \mathbb{Z}^d} f(n) &= \sum_{n \in \mathbb{Z}^d} \int_{\mathbb{R}^d} 1_{\frac{1}{2} K}(y) 1_{-\frac{1}{2} K}(n - y) dy \\
&= \int_{\mathbb{R}^d} 1_{\frac{1}{2} K}(y) 1_{-\frac{1}{2} K}(- y) dy \\
&= \int_{\mathbb{R}^d} 1_{\frac{1}{2} K}(y) dy \\
&= \vol \left( {\frac{1}{2} K} \right) = \frac{\vol K}{2^d}.
\end{align}
\noindent
On the other hand, the right-hand-side of Poisson summation gives us:
\begin{align}
\sum_{\xi \in \mathbb{Z}^d} \hat f(\xi)
&= \sum_{\xi \in \mathbb{Z}^d} {\hat 1}_{\frac{1}{2} K}(\xi) {\hat 1}_{-\frac{1}{2} K}(\xi) \\
&= \sum_{\xi \in \mathbb{Z}^d} \int_{\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx
\int_{-\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx \\
&= \sum_{\xi \in \mathbb{Z}^d} \int_{\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx
\int_{\frac{1}{2} K} e^{2\pi i \langle -\xi, x \rangle} dx \\
&= \sum_{\xi \in \mathbb{Z}^d} \int_{\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx
\ \overline{
\int_{\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx
} \\ \label{pulling out the zero term}
&= \sum_{\xi \in \mathbb{Z}^d} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2 \\
&= \left| \hat 1_{\frac{1}{2} K}(0) \right|^2 + \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2 \\
&= \frac{\vol^2 K}{4^d} + \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2,
\end{align}
where we have pulled out the $\xi=0$ term from the series \eqref{pulling out the zero term}.
So we've arrived at
\begin{align*}
\frac{\vol K}{2^d}
&= \frac{\vol^2 K}{4^d} + \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2,
\end{align*}
yielding the required identity:
\begin{align*}
2^d &= \vol K + \frac{4^d}{\vol K}\sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{align*}
Finally, to prove the stated extension to all lattices ${\mathcal L}$, we use the slightly more general
form of Poisson summation, Theorem \ref{The Poisson Summation Formula, for lattices},
valid for any lattice ${\mathcal L}$:
\begin{align}
\sum_{n \in {\mathcal L}} f(n) = \frac{1}{\det {\mathcal L}}\sum_{\xi \in {\mathcal L}^*} \hat f(\xi).
\end{align}
All the steps of the proof above are identical, except for the factor of $\frac{1}{\det {\mathcal L}}$, so that we arrive at
the required identity of Siegel for arbitrary lattices:
\begin{align}\label{Siegel, take 2}
\frac{\vol K}{2^d}
&= \frac{\vol^2 K}{4^d \det {\mathcal L}} + \frac{1}{\det {\mathcal L}}\sum_{\xi \in {\mathcal L}^* - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{align}
\end{proof}
The proof of Minkowski's convex body Theorem for lattices, namely Theorem
\ref{Minkowski convex body Theorem for L} above, now follows immediately.
\begin{proof}[Proof of Theorem \ref{Minkowski convex body Theorem for L}]
\rm{[Minkowski's convex body Theorem for a lattice} ${\mathcal L}$]
Applying Siegel's Theorem \ref{Siegel for general lattices} to the centrally symmetric body $B:=K$,
we see that the lattice sum on the right-hand-side of identity \eqref{Siegel.formula}
contains only non-negative terms. It follows that we immediately get the analogue of Minkowski's result for a given cenetrally symmetric body $B$ and a lattice ${\mathcal L}$, in its contrapositive form:
\begin{align} \label{ACTUAL proof of Minkowski's convex body theorem for lattices}
&\text{If the only lattice point of ${\mathcal L}$ in the interior of $B$ is the origin, then } 2^d \det {\mathcal L} \geq \vol B.
\end{align}
\end{proof}
In fact, we can easily extend Minkowski's Theorem \ref{Minkowski convex body Theorem for L}, using the same ideas of the latter proof, by using Siegel's Theorem \ref{Siegel for general lattices}
so that it applies to non-symmetric bodies as well (but there's a small `catch' - see Exercise \ref{Extending Minkowski to nonconvex bodies}).
\bigskip
\section{Tiling and multi-tiling Euclidean space by translations of polytopes}
\index{tiling}
First, we give a `spectral' equivalence for being able to tile Euclidean space by a single polytope, using only translations by a lattice. It will turn out that the case of equality in Minkowski's convex body Theorem is characterized precisely by the polytopes that tile $\mathbb{R}^d$ by translations. These bodies are called extremal bodies.
More generally, we would like to also consider the notion of multi-tiling, as follows. We say that a polytope
${\mathcal P}$ {\bf $k$-tiles $\mathbb{R}^d$ by using a set of translations ${\mathcal L}$} if
\begin{equation}
\sum_{n \in {\mathcal L}} 1_{{\mathcal P} + n}(x) = k,
\end{equation}
for all $x \in \mathbb{R}^d$, except those points $x$ that lie on the boundary of ${\mathcal P}$ or its translates under ${\mathcal L}$ (and of course these exceptions form a set of measure $0$ in $\mathbb{R}^d$). In other words,
${\mathcal P}$ is a $k$-tiling body if almost every $x \in \mathbb{R}^d$ is covered by exactly $k$ translates of ${\mathcal P}$.
Other synonyms for $k$-tilings in the literature are {\bf multi-tilings} of
$\mathbb{R}^d$, or {\bf tiling at level $k$}.
When ${\mathcal L}$ is a lattice, we will say that such a $k$-tiling is {\bf periodic}.
A common research theme is to search for tilings which are not necessarily periodic, but this is a difficult problem in general.
The classical notion of tiling, such that there are no overlaps between the interiors of any two tiles, corresponds here to the case $k=1$. We have the following dictionary between multi-tiling or Euclidean space by translations of a convex body ${\mathcal P}$, and a property of the zero set of the Fourier transform of ${\mathcal P}$, due to Kolountzakis \cite{Kolountzakis1}.
\bigskip
\begin{thm} \label{zero set of the FT of a polytope}
Suppose that ${\mathcal P}\subset \mathbb{R}^d$ is a compact set. The following two properties are equivalent:
\begin{enumerate}[(a)]
\item ${\mathcal P}$ $k$-tiles $\mathbb{R}^d$ by translations with a lattice ${\mathcal L}$.
\item
$\hat 1_{\mathcal P}(\xi) = 0$ for all nonzero $\xi \in {\mathcal L}^*$, the dual lattice.
\end{enumerate}
Either of these conditions also implies that $k = \frac{\vol {\mathcal P}}{ \det {\mathcal L}}$, an integer.
\end{thm}
\begin{proof}
We begin with the definition of multi-tiling, so that by assumption
\begin{equation} \label{by definition of k-tiling}
\sum_{n \in {\mathcal L}} 1_{{\mathcal P} + n}(x) = k,
\end{equation}
for all $x \in \mathbb{R}^d$ except those points $x$ that lie on the boundary of ${\mathcal P}$ or its translates under ${\mathcal L}$ (and of course these exceptions form a set of measure $0$ in $\mathbb{R}^d$). A trivial but useful
observation is that
\[
1_{{\mathcal P} + n}(x) =1 \iff 1_{{\mathcal P}}(x-n) =1,
\]
so we can rewrite the defining identity \eqref{by definition of k-tiling} as $\sum_{n \in {\mathcal L}} 1_{{\mathcal P}}(x-n) = k$. Now we notice that the left-hand-side is a periodic function of $x$, namely
\[
F(x) := \sum_{n \in {\mathcal L}} 1_{{\mathcal P}}(x-n)
\]
is periodic in $x$ with ${\mathcal L}$ as its set of periods. This is easy to see: if we let $l \in {\mathcal L}$, then $F(x + l) = \sum_{n \in {\mathcal L}} 1_{{\mathcal P}}(x+ l -n) = \sum_{m \in {\mathcal L}} 1_{{\mathcal P}}(x+ m) = F(x)$, because the lattice ${\mathcal L}$ is invariant under a translation by any vector that belongs to it.
The following `intuitive proof' would in fact be rigorous if we were allowed to use `generalized functions',
but since we do not use them in this book, we label this part of the proof as `intuitive', and we then give a rigorous proof, using functions rather than generalized functions.
[{\bf Intuitive proof}] \ By Theorem \ref{Fourier series for periodic functions}, we may expand $F$ into its Fourier series, because it is a periodic function on $\mathbb{R}^d$. Now by Poisson summation, namely
Theorem \ref{The Poisson Summation Formula, for lattices}, we know that its Fourier coefficients are the following:
\begin{equation}\label{Poisson version of k-tiling}
\sum_{m \in {\mathcal L}} 1_{{\mathcal P}}(x+m) = \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*} \hat 1_{{\mathcal P}}(\xi) e^{2\pi i \langle \xi, x \rangle},
\end{equation}
If we now make the assumption that $\hat 1_{{\mathcal P}}(\xi)=0$ for all nonzero $\xi \in {\mathcal L}^*$, then by \eqref{Poisson version of k-tiling} this assumption is equivalent to
\[
\sum_{m \in {\mathcal L}} 1_{{\mathcal P}}(x+m) = \frac{\hat 1_{\mathcal P}(0)}{\det {\mathcal L}} = \frac{\vol {\mathcal P}}{\det {\mathcal L}}.
\]
This relation means that we have a $k$-tiling, where $k:= \frac{\vol {\mathcal P}}{\det {\mathcal L}}$.
Now we replace the intuitive portion of the proof with a rigorous proof.
[{\bf Rigorous proof}] \ In order to apply Poisson summation, \index{Poisson summation formula}
it is technically necessary to replace $1_P(x)$ by
a smoothed version of it, in \eqref{Poisson version of k-tiling}. Because this process is so common and useful in applications, this proof is instructive.
We pick an approximate identity $\phi_n$, which is also compactly supported and continuous. Applying the Poisson summation formula of Theorem \ref{PracticalPoisson} to the smoothed function $1_P*\phi_n$, we get:
\begin{align}\label{rigorous limit for multi-tiling}
\sum_{m \in {\mathcal L}} \left( 1_{\mathcal P}*\phi_n \right)(x+m) &= \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*} \hat 1_{{\mathcal P}}(\xi) \hat \phi_n(\xi) e^{2\pi i \langle \xi, x \rangle}.
\end{align}
Using the fact that the convolution of two compactly supported functions is itself compactly supported, we see that $1_{\mathcal P}*\phi_n $ is again compactly supported. Thus the sum on the LHS of \eqref{rigorous limit for multi-tiling} is a finite sum.
Performing a separate computation, we take the limit as $n\rightarrow \infty$ inside this finite sum, and using
Theorem \ref{approximate identity convolution} (due to the continuity of $1_{\mathcal P}*\phi_n$), we obtain
\[
\lim_{n\rightarrow \infty} \sum_{m \in {\mathcal L}} \left( 1_{\mathcal P}*\phi_n \right)(x+m)
= \sum_{m \in {\mathcal L}} \lim_{n\rightarrow \infty} \left( 1_{\mathcal P}*\phi_n \right)(x+m)
=\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m),
\]
and moreover by \eqref{practical Poisson summation, first version}, we have
\begin{align} \label{smoothed-out RHS}
\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) = \frac{1}{\det {\mathcal L}}
\sum_{\xi \in {\mathcal L}^*} \hat 1_{{\mathcal P}}(\xi) \hat \phi_n(\xi)
\, e^{2\pi i \langle \xi, x \rangle}.
\end{align}
for all sufficiently large values of $n$. Separating the term $\xi=0$ on the RHS of this Poisson summation formula, we have:
\begin{align}
\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) &= \frac{\hat 1_{\mathcal P}(0)}{\det {\mathcal L}}
+ \sum_{\xi \in {\mathcal L}^*-\{0\}} \hat 1_{{\mathcal P}}(\xi) \hat \phi_n(\xi)
\, e^{2\pi i \langle \xi, x \rangle}\\ \label{equivalent condition for tiling}
&=
\frac{\vol {\mathcal P}}{\det {\mathcal L}}
+ \sum_{\xi \in {\mathcal L}^*-\{0\}} \hat 1_{{\mathcal P}}(\xi) \hat \phi_n(\xi)
\, e^{2\pi i \langle \xi, x \rangle}.
\end{align}
Now, $\hat 1_{{\mathcal P}}(\xi) =0$ for all $\xi \in {\mathcal L}^*-\{0\}$ in
\eqref{equivalent condition for tiling} will hold
\begin{align*}
&\iff
\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) = \frac{\vol {\mathcal P}}{\det {\mathcal L}},
\end{align*}
an equivalent condition which we may write as $\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) = k$, where necessarily
$k:= \frac{\vol {\mathcal P}}{\det {\mathcal L}}$. The condition $\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) = k$ means that ${\mathcal P}$ $k$-tiles
$\mathbb{R}^d$ by translations with the lattice ${\mathcal L}$, and also implies that $k$ must be an integer.
\end{proof}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.7in]{ExtremalBody}
\end{center}
\caption{An extremal body in $\mathbb{R}^2$, relative to the integer lattice, which is a hexagon. It has
area $4$, and no integer points in its interior. We also get a $2$-parameter family of such extremal bodies, parametrized by the point $p\in \mathbb{R}^2$ in the figure. It is clear from the picture
that this family of extremal bodies consists of either symmetric hexagons, or symmetric quadrilaterals.}
\label{Extremal body in R^2}
\end{figure}
In 1905, Minkowski gave necessary conditions for a polytope ${\mathcal P}$ to tile $\mathbb{R}^d$ by translations. Later, Venkov and independently McMullen found sufficient conditions as well, culminating in the following fundamental result.
\begin{thm}[Minkowski-Venkov-McMullen] \label{Minkowski-Venkov-McMullen}
A polytope ${\mathcal P}$ tiles $\mathbb{R}^d$ by translations if and only if the following $3$ conditions hold:
\begin{enumerate}
\item ${\mathcal P}$ is a symmetric polytope.
\item The facets of ${\mathcal P}$ are symmetric polytopes.
\item Fix any face $F\subset {\mathcal P}$ of codimension $2$, and project ${\mathcal P}$ onto the $2$-dimensional plane that is orthogonal to the $(d-2)$-dimensional affine span of $F$. Then this projection is either a parallelogram, or a centrally symmetric hexagon.
\end{enumerate}
\end{thm}
\bigskip
\section{Extremal bodies} \label{extremal bodies}
\index{extremal body}
An {\bf extremal body} is a convex, symmetric body $K$ for which we have equality in Minkowski's convex body Theorem:
\[
\vol K = 2^d(\det {\mathcal L}).
\]
If we just look at equation \ref{Siegel.formula} a bit more closely, we quickly get a nice corollary that
arises by combining Theorem \ref{zero set of the FT of a polytope} and Siegel's Theorem \ref{Siegel}.
Namely, equality occurs in Minkowski's convex body theorem if and only if
$K$ tiles \index{tiling} $\mathbb{R}^d$ by translations.
Let's prove this.
\begin{thm}[Extremal bodies] \label{thm:extremal bodies}
Let $K$ be any convex, centrally symmetric subset of $\mathbb{R}^d$, and fix a full-rank lattice ${\mathcal L} \subset \mathbb{R}^d$.
Suppose that the only point of ${\mathcal L}$ in the interior of $K$ is the origin.
Then:
\begin{quote}
$2^d \det {\mathcal L}= \vol K \ \iff \ \frac{1}{2}K$ tiles $\mathbb{R}^d$ by translations with the lattice ${\mathcal L}$.
\end{quote}
\end{thm}
\begin{proof}
By Siegel's formula \eqref{Siegel formula 2}, we have
\begin{equation}
2^d \det {\mathcal L}
= \vol K + \frac{4^d}{ \vol K } \sum_{\xi \in {\mathcal L}^* - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{equation}
Therefore, the assumption $2^d \det {\mathcal L} = \vol K$ holds $\iff$
\begin{equation}
0 = \frac{4^d}{ \vol K } \sum_{\xi \in {\mathcal L}^* - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2,
\end{equation}
$\iff$ all of the non-negative summands $ \hat 1_{\frac{1}{2} K}(\xi) =0$, for all nonzero $\xi \in {\mathcal L}^*$. Now we would like to
use Theorem \ref{zero set of the FT of a polytope} to show the required tiling equivalence, namely that
$\frac{1}{2} K$ tiles $\mathbb{R}^d$ by translations with the lattice ${\mathcal L}$. We have already verified condition (a) of Theorem \ref{zero set of the FT of a polytope}, applied to the body $\frac{1}{2} K$, namely that
$\hat 1_{\frac{1}{2} K}(\xi) =0$, for all nonzero $\xi \in {\mathcal L}^*$.
To verify condition (b) of Theorem
\ref{zero set of the FT of a polytope}, we notice that because $\vol\left( \frac{1}{2} K \right)= \frac{1}{2^d}\vol K$, it follows that
$2^d \det {\mathcal L}= \vol K$ is equivalent to
$1 = \frac{ \vol \left( \frac{1}{2} K \right)}{ \det {\mathcal L}}$, so that we may apply Theorem
\ref{zero set of the FT of a polytope} with ${\mathcal P}:= \frac{1}{2} K$, and with the multiplicity $k:=1$.
\end{proof}
\medskip
There is an extension of theorem \ref{Minkowski-Venkov-McMullen}, the Minkowski-Venkov-McMullen result, to multi-tilings.
\begin{thm}\cite{GravinShiryaevRobins} \label{k-tiling theorem, GravinShiryaevRobins}
If a polytope ${\mathcal P}$ multi-tiles $\mathbb{R}^d$ by translations with a discrete set of vectors, then
\begin{enumerate}
\item ${\mathcal P}$ is a symmetric polytope.
\item The facets of ${\mathcal P}$ are symmetric polytopes.
\end{enumerate}
\end{thm}
In the case that ${\mathcal P}\subset \mathbb{R}^d$ is a rational polytope, meaning that all the vertices of ${\mathcal P}$ have rational coordinates, the latter two necessary conditions for multi-tiling become sufficient conditions as well \cite{GravinShiryaevRobins}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{TruncatedOctahedron}
\end{center}
\caption{The truncated Octahedron, one of the $3$-dimensional polytopes that tiles $\mathbb{R}^3$ by translations. }
\label{TruncatedOctahedron}
\end{figure}
\section{More about centrally symmetric polytopes} \label{Centrally symmetric polytopes}
\index{centrally symmetric polytope}
It's both fun and instructive to begin by seeing how very simple Fourier methods can give us deeper insight into the geometry of symmetric polytopes. The reader may glance at the definitions above, in
\eqref{definition of symmetric body}.
\bigskip
\begin{example}
\rm{
Consider the cross-polytope $\Diamond \subset \mathbb{R}^3$,
\index{cross-polytope}
defined in Chapter \ref{Chapter.Examples}. This is a centrally symmetric polytope, but each of its facets is {\em not} a symmetric polytope, because its facets are triangles.
}
\hfill $\square$
\end{example}
If \emph{all} of the $k$-dimensional faces of a polytope ${\mathcal P}$ are symmetric, for each $1\leq k \leq d$,
then ${\mathcal P}$ is called a {\bf zonotope}.
\index{zonotope}
Zonotopes form an extremely important class of polytopes, and have various equivalent formulations.
\begin{lem}
A polytope ${\mathcal P} \subset \mathbb{R}^d$ is a zonotope $\iff$ ${\mathcal P}$ has one of the following properties.
\begin{enumerate}[(a)]
\item ${\mathcal P}$ is a projection of some $n$-dimensional cube.
\item ${\mathcal P}$ is the Minkowski sum of a finite number of line segments.
\end{enumerate}
\end{lem}
A projection here means any affine transformation of ${\mathcal P}$, where the rank of the associated matrix may be less than $d$.
Zonotopes
have been very useful in the study of tilings (\cite{Ziegler}, \cite{BeckRobins}).
\index{tiling}
For instance, in dimension $3$, the only polytopes that tile $\mathbb{R}^3$ by translations with a lattice are zonotopes, and there is a list of $5$ of them (up to an isomorphism of their face posets), called the {\bf Fedorov solids}, and drawn in Figure \ref{Fedorov solids} (also see our Note \ref{Fedorov Note} below). \index{Fedorov solids}
By definition, any zonotope is a symmetric polytope, but the converse is not true; for example,
the cross-polytope
\index{cross-polytope}
is symmetric, but it has triangular faces, which are not symmetric, so the crosspolytope is not a zonotope.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{rhombic}
\end{center}
\caption{A $3$-dimensional zonotope, called the rhombic dodecahedron, showing in bold its $4$ line segments whose Minkowski sum generate the object. }
\label{first 3d zonotope pic}
\end{figure}
\index{Minkowski sum}
\bigskip
\begin{example}
\rm{
Consider the following $3$ line segments in $\mathbb{R}^2$:
$\conv\{ \icol{0\{\bf 0}}, \icol{1 \{\bf 0}} \}, \conv\{ \icol{0\{\bf 0}}, \icol{2 \{\bf 1}} \}$, and
$\conv\{ \icol{0\{\bf 0}}, \icol{1 \\ 3} \}$.
The Minkowski sum of these three line segments, by definition a zonotope in $\mathbb{R}^2$, is
the symmetric hexagon whose vertices are $\icol{0\{\bf 0}}, \icol{1 \{\bf 0}}, \icol{2 \{\bf 1}}, \icol{3 \\3}, \icol{3 \{\bf 1}}, \icol{4 \\3}$.
Notice that once we graph it, in Figure \ref{a zonotope}, the graph is hinting to us
that this body is a projection of a $3$-dimensional cube, and indeed this turns
out to be always true for Minkowski sums of line segments.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.4in]{zonotope1}
\end{center}
\caption{The Minkowski sum of $3$ line segments in the plane, forming a $2$-dimensional zonotope. }
\label{a zonotope}
\end{figure}
\index{Minkowski sum}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.1in]{Fedorov}
\end{center}
\caption{The Fedorov solids, the only $3$-dimensional polytopes that tile $\mathbb{R}^3$ by translations. All $5$ of them are zonotopes, and they are also extreme bodies for Minkowski's convex body theorem.
The top three, from left to right, are: the Truncated octahedron, the Rhombic dodecahedron, and the Hexarhombic dodecahedron. The bottom two are the cube and the hexagonal prism. }
\label{Fedorov solids} \index{Fedorov solids}
\end{figure}
\begin{example} \label{truncated octahedron}
\rm{
A particular embedding of the truncated octahedron ${\mathcal P}$, drawn in Figure \ref{TruncatedOctahedron},
is given by the convex hull of the set of $24$ vertices defined by all permutations of $(0, \pm 1, \pm 2)$.
We note that this set of vertices can also be thought of as the orbit of just the one point $(0, 1, 2)\in \mathbb{R}^3$ under the hyperoctahedral group (see
\cite{BillChen} for more on the hyperoctahedral group). It turns out that this truncated octahedron ${\mathcal P}$ tiles $\mathbb{R}^3$ by translations with a lattice
(Exercise \ref{tiling using the truncated octrahedron}).
}
\hfill $\square$
\end{example}
As the following Lemma shows, it is easy to detect/prove whether or not $S$ is centrally symmetric by just observing whether or not its Fourier transform is real-valued.
To make the proof go through more easily, we will assume that $\hat 1_S$ is absolutely integrable, so that the usual inverse Fourier transform applies, and we call such a set admissible. But the curious reader might consider extensions to more general sets.
\bigskip
\begin{lem}\label{symmetric iff FT is real}
An admissible set $S \subset \mathbb{R}^d$ is symmetric about the origin $\iff$
\[
\hat 1_{S}(\xi) \in \mathbb{R},
\]
\text{ for all } $\xi \in \mathbb{R}^d$.
\end{lem}
\begin{proof}
Suppose that the set $S$ is centrally symmetric. Then we have
\begin{align}
\overline{ \hat 1_S(\xi)} := \overline{ \int_{S} e^{2\pi i \langle \xi, x \rangle} dx}
&= \int_{S} e^{-2\pi i \langle \xi, x \rangle} dx \\
&= \int_{-S} e^{2\pi i \langle \xi, x \rangle} dx \\
&= \int_{S} e^{2\pi i \langle \xi, x \rangle} dx := \hat 1_S(\xi), \\
\end{align}
showing that the complex conjugate of $\hat 1_S$ is itself, hence that it is real-valued.
Conversely, suppose that $\hat 1_{S}(\xi) \in \mathbb{R}$, for all $\xi \in \mathbb{R}^d$. We use the fact that the Fourier transform $\hat 1_S$ is invertible, so that by Theorem \ref{thm:Inverse Fourier transform} we have:
\begin{equation} \label{Fourier inversion of indicator}
({\mathcal F} \circ {\mathcal F})(1_S)(x) = 1_S(-x),
\end{equation}
for all $x \in \mathbb{R}^d$.
To show that $S$ is centrally symmetric, we need to show that $1_{-S}(x) = 1_{S}(x)$, for all $x \in \mathbb{R}^d$. Further, by \ref{Fourier inversion of indicator}, it now suffices to show that
$\hat 1_{-S}(\xi) = \hat 1_{S}(\xi)$, for all $\xi \in \mathbb{R}^d$.
We therefore compute:
\begin{align}
\hat 1_{-S}(\xi) := \int_{-S} e^{2\pi i \langle \xi, x \rangle} dx
&= \int_{S} e^{-2\pi i \langle \xi, x \rangle} dx \\
&= \overline{ \int_{S} e^{2\pi i \langle \xi, y \rangle} dy } \\
&:= \overline{ \hat 1_S(\xi) } \\
&= \hat 1_S(\xi),
\end{align}
for all $\xi \in \mathbb{R}^d$, where we have used the assumption that $ \hat 1_S(\xi)$ is real-valued in the last equality.
\end{proof}
\bigskip
\begin{example} \rm{
The interval ${\mathcal P}:= [-\frac{1}{2}, \frac{1}{2}]$ is a symmetric polytope, and indeed we can see that
its Fourier transform
$\hat 1_{\mathcal P}(\xi)$ is real-valued, namely we have $\hat 1_{\mathcal P}(\xi) = {\rm{sinc}}(\xi)$, as we saw in equation
\eqref{SincFunction}.}
\hfill $\square$
\end{example}
\bigskip
\begin{example} \rm{
The cross-polytope $\Diamond_2$ is a symmetric polytope, and as we verified in dimension $2$,
equation \eqref {Fourier transform of 2d crosspolytope},
its Fourier transform
$1_{\Diamond_2}(\xi)$ is real-valued. }
\hfill $\square$
\end{example}
Alexandrov \cite{Alexandrov}, and independently Shephard \cite{ShephardSymmetricPolytopes}, proved the following remarkable fact.
\begin{thm}[Alexandrov and Shephard] \label{cs1} \index{Alexandrov, A. D. } \index{Shephard}
\label{Alexandrov-Shepard thm}
Let $P$ be any real, $d$-dimensional polytope, with $d \geq 3$. If all of the facets of $P$
are centrally symmetric, then $P$ is centrally symmetric.
\hfill $\square$
\end{thm}
\begin{example}
\rm{
The converse to the latter result is clearly false, as demonstrated by the cross-polytope in dimension $d > 2$: it is centrally symmetric, but its facets are not symmetric because they are simplices and we know that
no simplex (of dimension $\geq 2$) is symmetric
(Exercise \ref{no simplex is symmetric}).
}
\hfill $\square$
\end{example}
\begin{wrapfigure}{R}{0.49\textwidth}
\centering
\includegraphics[width=0.20\textwidth]{zonotope2}
\caption{A $3$-dimensional zonotope that does not tile $\mathbb{R}^3$ by translations. }
\label{complex zonotope}
\end{wrapfigure}
Suppose we consider $3$-dimensional polytopes ${\mathcal P}$, and ask which ones enjoy the property that all of their $2$-dimensional faces are symmetric? Because $1$-dimensional faces are always symmetric, and because Theorem \ref{Alexandrov-Shepard thm} tells us that ${\mathcal P}$ itself must also be symmetric, the answer is that ${\mathcal P}$ must be a zonotope - in other words all of its faces are symmetric.
Moving up to $4$-dimensional polytopes, our curiosity might take the next step:
which $4$-dimensional polytopes enjoy the property that all of their $3$-dimensional faces are symmetric? Must they also be zonotopes? The $24$-cell is a good counterexample, because it has triangular $2$-dimensional faces, and hence is not a zonotope. On the other hand, the $24$-cell tiles $\mathbb{R}^4$ by translations with a lattice (it is the Voronoi cell of the D$4$ lattice), and therefore by Theorem \ref{Minkowski-Venkov-McMullen} its $3$-dimensional faces must be symmetric.
What if we ask which $4$-dimensional polytopes enjoy the property that all of their $2$-dimensional faces are symmetric?
Peter McMullen \cite{McMullen4}
discovered the wonderful conclusion that all of their faces must be symmetric - in other words they must be zonotopes - and that much more is true.
\begin{thm}[McMullen] \label{McMullen extension to Alexandrov}
Let $P$ be any real, $d$-dimensional polytope, with $d \geq 4$.
Fix any positive integer $k$ with
$2 \leq k \leq d-2$.
If the $k$-dimensional faces of ${\mathcal P}$ are symmetric, then ${\mathcal P}$ is a zonotope.
\hfill $\square$
\end{thm}
\bigskip
One might wonder what happens if we `discretize the volume' of a symmetric body $K$, by counting integer points, and then ask for an analogue of Minkowski Theorem \ref{Minkowski's convex body theorem, for Z^d}.
In fact,
Minkowski already had a result about this too
(and he had so many beautiful ideas that it's hard to put them all in one place!). We give Minkowski's own elegant and short proof.
\begin{thm}[Minkowski, 1910] \label{Minkowski's 3^d theorem}
Let $K\subset \mathbb{R}^d$ be any $d$-dimensional, convex, centrally symmetric set.
If the only integer point in the interior of $K$ is the origin, then
\begin{equation} \label{Minkowski, $3^d$}
\left | K \cap \mathbb{Z}^d \right | \leq 3^d.
\end{equation}
\end{thm}
\begin{proof}
We define the map $\phi: \mathbb{Z}^d \rightarrow \left( \mathbb{Z}/3\mathbb{Z} \right)^d$, by reducing each coordinate modulo $3$.
Now we claim that when restricted to the set $K\cap \mathbb{Z}^d$, our map $\phi$ is $1-1$. The statement of the theorem follows directly from this claim. So let $x,y \in K\cap \mathbb{Z}^d$, and suppose $\phi(x) = \phi(y)$. Then, by definition of the map $\phi$, we have
\begin{equation} \label{z point}
n:= \frac{1}{3}(x-y) \in \mathbb{Z}^d,
\end{equation}
Now we define $C$ to be the {\bf interior} of the convex hull of $x, -y$, and $0$. Because $K$ is symmetric, and $x, y\in K$, we know that $-y \in K$ as well, so that $C \subset \text{int}(K)$. Now using the convexity of $C$, we also see that $n \in C$, because $n$ is a non-trivial convex linear combination of $0, x, -y$.
Therefore $n \in \text{int}(K)$ as well. Altogether, $n \in \text{int}(K) \cap \mathbb{Z}^d = \{0\}$, which forces $n =0$.
Hence $x-y=0$.
\end{proof}
Theorem \ref{Minkowski's 3^d theorem} is often called { \bf Minkowski's $3^d$ theorem}. \index{Minkowski's $3^d$ theorem}
An immediate and natural question is: which bodies account for the `equality case'? One direction is easy to see: if $K$ is the integer cube $[-1, 1]^d$, then it is clear that $K$ is symmetric about the origin, and the only integer point in its interior is the origin.
In addition, $\vol K = 2^d$, and $K$ contains precisely $3^d$ integer points. It is a bit surprising, perhaps, that only in 2012 was it proved that this integer cube is the only case of equality in Minkowski's $3^d$ theorem \cite{DraismaMcAllisterNill}.
In a different direction, it turns out that the volume of the difference body
\index{symmetrized body}
$\frac{1}{2} K - \frac{1}{2} K$, which appeared quite naturally in some of the proofs above, can be related in a rather precise manner to the volume of $K$ itself.
The consequence is the following inequality, known as the {\bf Rogers-Shephard inequality}
\cite{RogersShephard},
\begin{equation} \label{Rogers-Shephard inequality}
\vol K \leq \vol \left( \frac{1}{2} K - \frac{1}{2} K \right) \leq {2d \choose d} \vol K,
\end{equation}
where equality on the left holds $\iff$ $K$ is a symmetric body, and equality on the right
holds $\iff$ $K$ is a simplex (see Cassels \cite{CasselsBook}). There is also an extension of the Rogers-Shephard inequality to two distinct convex bodies $K, L\subset \mathbb{R}^d$:
\begin{equation}
\vol \left( K - L \right) \vol \left( K \cap L \right) \leq {2d \choose d} \vol K \vol L.
\end{equation}
(\cite{RogersShephard} and \cite{Gutierrez.Jimenez.Villa}).
A quick way of proving \eqref{Rogers-Shephard inequality} is by using the ubiquitous
{\bf Brunn-Minkowski inequality} (\cite{Schneider.book}, section $7.1$)
\index{Brunn-Minkowski inequality}
which tells us the following. Two sets $A, B\subset \mathbb{R}^d$ are called homothetic if $A = \lambda B + v$, for some
fixed $v\in \mathbb{R}^d$, and some $\lambda >0$ (or either $A$ or $B$ consist of just one point).
\begin{thm}
If $K$ and $L$ are convex subsets of $\mathbb{R}^d$, then
\begin{equation}
\vol(K+L)^\frac{1}{d} \geq \vol(K)^\frac{1}{d} + \vol(L)^\frac{1}{d},
\end{equation}
with equality if and only if $K$ and $L$ lie in parallel hyperplanes or are homothetic to each other.
\hfill $\square$
\end{thm}
\bigskip
\section*{Notes}
\begin{enumerate}[(a)]
\item Siegel's original proof of Theorem \ref{Siegel} used Parseval's identity, but the spirit of the two proofs is similar.
\item In Exercise \ref{equivalent statements for unimodular triangles} below, we see three equivalent conditions for a $2$-simplex to be unimodular. In higher dimensions, a $d$-simplex will not satisfy all
three conditions, and hence this exercise shows one important `breaking point' between $2$-dimensional and $3$-dimensional discrete geometry.
\item \label{new books, geometry of numbers}
There are a growing number of interesting books on the geometry of numbers. One encyclopedic text
that contains many other connections to the geometry of numbers is Peter Gruber's book \cite{GruberBook}.
Two other excellent and classic introductions are Siegel's book \cite{SiegelBook}, and Cassels' book \cite{CasselsBook}. An expository introduction to some of the elements of the Geometry of numbers, at a level that is even appropriate for high school students, is given by Olds, Lax, and Davidoff \cite{OldsBook}.
For upcoming books, the reader may also consult Martin Henk's lecture notes `Introduction to geometry of numbers' \cite{Henk3}, and
the book by Lenny Fukshansky and Stephan Ramon Garcia, `Geometry of Numbers' \cite{FukshanskyBook}.
\item \label{Brunn-Minkowski} The Brunn-Minkowski inequality is fundamental to many branches of mathematics, including the geometry of numbers. A wonderful and encyclopedic treatment of the
Brunn-Minkowski inequality, with its many interconnections, appears in \cite{Schneider.book}.
\item \label{Fedorov Note}
The Fedorov solids are depicted, and explained via the modern ideas of Conway and Sloan, in an excellent
expository article by David Austin \cite{DavidAustin}.
For a view into the life and work of Evgraf Stepanovich Fedorov, as well as
a fascinating account of how Fedorov himself thought about the $5$ parallelohedra,
the reader may
consult the article by Marjorie Senechal and R. V. Galiulin \cite{SenechalGaliulin}. The authors of
\cite{SenechalGaliulin} also discuss the original book of Fedorov, called
\emph{An Introduction to the Theory of Figures}, published in 1885, which is now considered a pinnacle of modern crystallography.
Fedorov later became one of the great crystallographers of his time.
In $\mathbb{R}^4$, it is known that there are $52$ different combinatorial types of $4$-dimensional parallelohedra. In $\mathbb{R}^5$, the complete classification of all the combinatorial types of
$5$-dimensional paralellohedra was completed in 2016 \cite{Dutour et al.parallelohedra}, where the authors found $110, 244$ of them.
\item The field of multi-tiling is still growing. One of the first important papers in this field was by Mihalis Koloutzakis \cite{Kolountzakis1}, who related the multi-tiling problem to a famous technique known as the idempotent theorem, and thereby proved
that if we have a multi-tiling in $\mathbb{R}^2$ with any discrete set of translations, then we also have a multi-tiling with a finite union of lattices.
A recent advance is an equivalence between multi-tiling and
certain Hadwiger-type invariants, given by Nir Lev and Bochen Liu \cite{LevLiu}. Here the authors show as well that
for a generalized polytope ${\mathcal P} \subset \mathbb{R}^d$ (not necessarily convex or connected), if ${\mathcal P}$ is spectral, then ${\mathcal P}$
is equidecomposable by translations to a cube of equal volume.
Another natural question in multi-tiling, which is still open, is the following:
\begin{question} \label{multi-tiling - what is the discrete set of translations}
Suppose that ${\mathcal P}$ multi-tiles with a discrete set of translations $D$. Do we really need the set $D$ of translates of ${\mathcal P}$ to be a very complicated discrete set,
or is it true that just a finite union of lattices suffices?
Even better, perhaps one lattice always suffices?
\end{question}
In this direction, Liu proved recently that if we assume that ${\mathcal P}$ multi-tiles with a finite union of lattice, then ${\mathcal P}$ also multi-tiles with a single lattice \cite{Liu}. This is big step in the direction of answering Question
\ref{multi-tiling - what is the discrete set of translations} in general. An earlier, and smaller step, was taken in \cite{GravinKolountzakisRobinsShiryaev}, where the authors answered part of Question \ref{multi-tiling - what is the discrete set of translations} in $\mathbb{R}^3$, reducing the search from an arbitrary discrete set of translations, to translations by a finite union of lattices. Taken together, the latter two steps imply that in $\mathbb{R}^3$ (and in $\mathbb{R}^2$), any multi-tiling with a discrete set of translations also occurs with just a one lattice.
In a different direction, the work of Gennadiy Averkov \cite{Averkov} analyzes the equality cases for an extension of Minkowski's theorem, relating those extremal bodies to multi-tilers.
In \cite{YangZong}, Qi Yang and Chuanming Zong show that
the smallest $k$ for which we can obtain a nontrivial $k$-tiling in $\mathbb{R}^2$ is $k=5$, and the authors characterize
those $5$-tiling bodies, showing in particular that if a convex polygon is a $5$-tiler, then it must be either an octagon, or a decagon.
\begin{question}\label{smallest k}
In $\mathbb{R}^d$, what is the smallest integer $k$ such that there exists a $d$-dimensional polytope
${\mathcal P}$ that $k$-tiles $\mathbb{R}^d$ by translations?
\end{question}
\item \label{Fuglede conjecture}
We say that a body ${\mathcal P}$ (any compact subset of $\mathbb{R}^d$)
is `spectral' if the function space $L^2({\mathcal P})$ possesses an orthonormal, complete basis of exponentials.
There is a fascinating and vast literature about such spectral bodies, relating them to tiling,
\index{tiling}
and multi-tiling problems.
One of the most interesting and natural questions in this direction is the following conjecture, by Bent Fuglede \cite{Fuglede74}.
The Fuglede conjecture asks whether the following is true.
\begin{question}\label{Fuglede}
${\mathcal P}$ tiles $\mathbb{R}^d$ by translations $\iff$ ${\mathcal P}$ is spectral?
\end{question}
Terry Tao disproved the Fuglede conjecture for some nonconvex bodies, but in the case that ${\mathcal P}$ is convex one might hope that more is true. Indeed, in 2003
Alex Iosevich, Nets Katz, and Terry Tao \cite{IosevichKatzTao} proved that
the Fuglede conjecture is true for all convex domains in $\mathbb{R}^2$.
In 2021, this conjecture was proved for all convex domains (which must necessarily be polytopes by
an additional simple argument), in the work of Nir Lev and M\'at\'e Matolcsi \cite{LevMatolcsi}.
In a related direction, Sigrid Grepstad and Nir Lev \cite{GrepstadLev} showed that for any bounded, measurable subset $S\subset \mathbb{R}^d$, if $S$ multi-tiles by translations with a discrete set, then
$S$ has a Riesz basis of exponentials.
\item We have seen
that the zero set of the Fourier transform of a polytope is very important, in that Theorem \ref{zero set of the FT of a polytope}
gave us a necessary and sufficient condition for multi-tiling. But the zero set of the FT also gives more information, and an interesting application of the information content in the zero set is the Pompeiu problem. \index{Pompeiu problem}
The Pompeiu problem is an ancient problem (defined in 1929 by Pompeiu) that asks the following:
which bodies ${\mathcal P} \in \mathbb{R}^d$ are uniquely characterized by the collection of their integrals over ${\mathcal P}$, and over all rigid motions of ${\mathcal P}$?
An equivalent formulation is the following.
\begin{question} \label{Pompeiu conjecture}
Given a body ${\mathcal P}$ with nonempty interior, does there exist a nonzero continuous function $f$ that allows for the
the vanishing of all of the integrals
\begin{equation}\label{Pompeiu question}
\int_{M({\mathcal P})} f(x) dx = 0,
\end{equation}
taken over all rigid motions $M$, including translations?
\end{question}
A body ${\mathcal P} \subset \mathbb{R}^d$, for which the answer to the question above is affirmative, is said to have the Pompeiu property.
Even for convex bodies ${\mathcal P}$, it is still an open problem in general dimension whether
${\mathcal P}$ has the Pompeiu property.
It is known, by the work of Brown, Schreiber, and Taylor \cite{BrownSchreiberTaylor} that ${\mathcal P}$ has the Pompeiu property $\iff$
the collection of Fourier transforms $\hat 1_{\sigma({\mathcal P})}(z)$, taken over all rigid motions
$\sigma$ of $\mathbb{R}^d$,
have a common zero $z$. It was also known that all polytopes have the Pompeiu property.
Recently, in \cite{FabricioSinai1}, Fabricio Machado and SR showed that the zero set of the FT does not contain (almost all) circles, and as a consequence we get a simple new proof that all polytopes have the `Pompeiu property'.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
``Every problem has a creative solution''.
-- Folklore
\end{quote}
\begin{quote}
``Every problem has a solution that is simple, neat, and wrong''.
-- Mark Twain
\end{quote}
\medskip
\medskip
\begin{prob}
Suppose that in $\mathbb{R}^2$, we are given a symmetric, convex body $K$ of area $4$, which contains only the origin.
Prove that $B$ must tile $\mathbb{R}^2$ by translations.
\end{prob}
\medskip
\begin{prob} \label{convexity of K-K} $\clubsuit$
Given convex $d$-dimensional bodies $K, L \subset \mathbb{R}^d$, prove that $K+L$ is convex, and that $K-L$ is convex.
\end{prob}
\medskip
\begin{prob} \label{c.s. C equals its symmetrized body}
$\clubsuit$
Suppose initially that $C \subset \mathbb{R}^d$ is any set.
\begin{enumerate}[(a)]
\item Show that
\begin{equation}
\frac{1}{2}C - \frac{1}{2}C = C
\ \implies \text{ $C$ is centrally symmetric}.
\end{equation}
\item
Now suppose that $C$ is convex. Show that
\begin{equation} \label{equivalence for a convex set}
C \text{ is centrally symmetric } \iff \frac{1}{2} C - \frac{1}{2} C = C.
\end{equation}
\item Find an example of a centrally symmetric set $C$ that is not convex, and satisfies
\[
\frac{1}{2} C - \frac{1}{2} C \not= C.
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob}\label{support of convolution} $\clubsuit$
Recalling the definition of the support of a function $f$ from \eqref{def of support}, show that:
\begin{enumerate}[(a)]
\item
Suppose that we are given two closed, convex bodies $A, B \subset \mathbb{R}^d$. Show that
\[
\supp ( 1_A * 1_B) = A + B,
\]
where the addition is the Minkowski addition of sets.
\index{Minkowski sum}
\item
More generally, if two functions $f, g:\mathbb{R}^d \rightarrow \mathbb{C}$ are compactly supported,
show that
\[
\supp(f*g)\subseteq \closure\left( \supp(f) + \supp(g) \right),
\]
the closure of the Minkowski sum of their individual supports.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{equivalent statements for unimodular triangles}
\rm{
Suppose we have a triangle $\Delta$ whose vertices $v_1, v_2, v_3$ are integer points. Prove that the following
properties are equivalent:
\begin{enumerate}[(a)]
\item $\Delta$ has no other integer points inside or on its boundary (besides its vertices).
\item $Area(\Delta) = \frac{1}{2}$.
\item $\Delta$ is a unimodular triangle, which in this case means that $v_3 - v_1$ and $v_2- v_1$ form a basis for $\mathbb{Z}^2$.
\end{enumerate}
(Hint: You might begin by ``doubling'' the triangle to form a parallelogram.)
}
\end{prob}
\medskip
\begin{prob}
Show that in $\mathbb{R}^d$, an integer simplex $\Delta$ is unimodular if and only if $\vol(\Delta) = \frac{1}{d!}$.
\end{prob}
\medskip
\begin{prob} \label{non-unimodular but empty simplex}
In $\mathbb{R}^3$, find an integer simplex $\Delta$ that has no other integer points inside or on its boundary (other than its vertices of course), but
such that $\Delta$ is not a unimodular simplex.
\end{prob}
\medskip
\begin{prob} \label{FT of a polytope is not Schwartz}
Prove that for any polytope ${\mathcal P}$, $\hat 1_{{\mathcal P}}$ is not a Schwartz function.
\end{prob}
\medskip
\begin{prob} \label{convolution of indicators is a nice function}
$\clubsuit$
(hard-ish) Show that if $K$ is any convex body, then $1_K*1_{-K}$ is a nice function, in the sense of \eqref{nice functions}.
In other words, show that the Poisson summation formula holds for the function $f(x):= \left( 1_K*1_{-K} \right)(x)$.
Hint. Use the Parseval identity, valid for functions $f \in L^2(\mathbb{R}^d)$. For this particular exercise, feel free to use the results of all of the later sections (though in general we refrain from such a `look ahead').
\end{prob}
\medskip
\begin{prob} \label{Cantor set}
We first define the following sets recursively:
\[
C_0 := [0, 1], \ C_1 := [0, \tfrac{1}{3}] \cup [\tfrac{2}{3}, 1], \dots ,
C_n:= \tfrac{1}{3} C_{n-1} \cup \left\{ \tfrac{1}{3} C_{n-1} + \tfrac{2}{3} \right\},
\]
and now the {\bf Cantor set} is defined by their infinite intersection:
\[
\mathcal C:= \cap_{n=0}^\infty C_n.
\]
It is a standard fact (which you may assume here) that the Cantor set $\mathcal C$ is compact, uncountable, and has measure $0$. Despite these facts, show that its difference body satisfies the somewhat surprising identity:
\[
\mathcal C-\mathcal C = [-1, 1].
\]
\end{prob}
\medskip
\begin{prob} \label{tiling using the truncated octrahedron}
Show that the truncated octahedron, defined in Example \ref{truncated octahedron}, tiles $\mathbb{R}^3$ by using only
translations with a lattice. Which lattice can you use for this tiling?
\end{prob}
\medskip
\begin{prob} \label{an application of Cauchy-Schwartz 1}
Define $f(x):= a \sin x + b \cos x$, for constants $a,b\in \mathbb{R}$.
Show that the maximum value of $f$ is $ \sqrt{ a^ 2 + b ^2 } $, and occurs
when $\tan x = \frac{ a}{b}$.
\end{prob}
\medskip
\begin{prob}
\rm{
Find an example of a symmetric polygon ${\mathcal P} \subset \mathbb{R}^2$ that multi-tiles (nontrivially) with multiplicity $k = 5$.
Notes. A trivial multi-tiling for ${\mathcal P}$ is by definition a multi-tiling that uses ${\mathcal P}$, with some multiplicity $k>1$, but such that
there also exists a $1$-tiling (classical) using the same ${\mathcal P}$.
}
\end{prob}
\medskip
\begin{prob}
Let $K\subset \mathbb{R}^d$ be centrally symmetric. Show that
\begin{equation*}
\frac{1}{2}K \cap \left( \frac{1}{2}K + n \right) \not= \phi \iff n \in K.
\end{equation*}
\end{prob}
\medskip
\begin{prob} \label{Extending Minkowski to nonconvex bodies}
$\clubsuit$
\rm{
Here we use Siegel's theorem \ref{Siegel for general lattices} to give the following extension of
Minkowski's classical Theorem \ref{Minkowski convex body Theorem for L}, but
for bodies $K$ that are not necessarily symmetric, nor necessarily convex.
Namely, let $K$ be any bounded, measurable subset of $\mathbb{R}^d$, with positive $d$-dimensional volume.
Let $B:= \frac{1}{2}K - \frac{1}{2}K$ be the symmetrized body of $K$ (hence $B$ is a centrally symmetric set containing the origin).
Let ${\mathcal L}$ be a (full rank) lattice in $\mathbb{R}^d$. Prove the following statement:
\begin{equation*}
\text{ If } \vol K > 2^d (\det {\mathcal L}), \text{ then } B
\text{ must contain a nonzero point of } {\mathcal L} \text{ in its interior}.
\end{equation*}
Notes. We note that the positive conclusion of the existence of a nonzero integer point holds only for the symmetrized body $B$,
with no guarantees for any integer points in $K$.
}
\end{prob}
\chapter{An introduction to Euclidean lattices} \label{chapter.lattices}
\index{lattice} \index{integer lattice}
\begin{quote}
``Lattices quantify the idea of periodic structures.''
-- Anonymous
\end{quote}
\begin{quote}
``Less is more........more or less.''
-- Ludwig Mies van der Rohe
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.6in]{parallelepiped1}
\end{center}
\caption{A fundamental parallelepiped (half-open), for a lattice ${\mathcal L}$, generated by the vectors $v_1$ and $v_2$.} \label{parallelepiped1}
\end{figure}
\bigskip
\section{Intuition}
We introduce Euclidean lattices here, which may be thought of intuitively as regularly-spaced points in
$\mathbb{R}^d$, with some hidden number-theoretic structure. Another intuitive way to think of lattices is that they are one of the most natural ways to {\bf discretize Euclidean space}.
A lattice in $\mathbb{R}^d$ is also the most natural extension of an infinite set of equally-spaced points on the real line. In the real-world, lattices come up very naturally when we study crystals, for example.
It is perhaps not surprising that number theory comes in through
study of the integer lattice $\mathbb{Z}^d$, as it is the $d$-dimensional extension of the integers $\mathbb{Z}$. Moreover, whenever we
study almost any periodic behavior, lattices naturally come up, essentially from the
definition of {\bf periodicity} in Euclidean space. \index{periodicity} And of course, where there are lattices, there are Fourier series, as we also saw in Chapter \ref{Fourier analysis basics}.
\section{Introduction to lattices}
\begin{defi}
A {\bf lattice} \index{lattice} is defined by the integer linear span of a fixed set of linearly independent vectors $\{ v_1, \dots, v_m \} \subset \mathbb{R}^d$:
\begin{equation}\label{def.lattice}
{\mathcal L} := \left\{ n_1 v_1 + \cdots + n_m v_m \in \mathbb{R}^d \bigm | \text{ all } n_j \in \mathbb{Z} \right\}.
\end{equation}
\end{defi}
The most common lattice is the {\bf integer lattice} \index{integer lattice}
\[
\mathbb{Z}^d:= \left\{ (x_1, \dots, x_d) \in \mathbb{R}^d \bigm | \text{ all } x_j \in \mathbb{Z} \right\}.
\]
However, we often encounter different types of lattices, occurring very naturally in practice, and it is natural to ask how they are related to each other.
The first thing we might notice is that, by Definition \ref{def.lattice}, a lattice may also be written as follows:
\begin{equation}
{\mathcal L} := \left\{
\begin{pmatrix} | & | & ... & | \\
v_1 & v_2 & ...& v_m \\
| & | & ... & | \\
\end{pmatrix}
\begin{pmatrix}
n_1 \\
\vdots \\
n_m \\
\end{pmatrix}
\ \biggm | \
\begin{pmatrix}
n_1 \\
\vdots \\
n_m \\
\end{pmatrix}
\in \mathbb{Z}^m
\right\}
:= M(\mathbb{Z}^m),
\end{equation}
where by definition, $M$ is the $d \times m$ matrix whose columns are the vectors $v_1, \dots, v_m$. This set of basis vectors
$\{ v_1, \dots, v_m\}$ is called a {\bf basis} \index{lattice basis}
for the lattice ${\mathcal L}$, and $m$ is called the {\bf rank} of the lattice ${\mathcal L}$.
In this context, we also use the notation ${\rm rank}({\mathcal L}) = m$.
We will call $M$ a {\bf basis matrix} \index{basis matrix}
for the lattice ${\mathcal L}$.
But there are always infinitely many other bases for ${\mathcal L}$ as well, and Lemma \ref{changing basis matrices} below shows how they are related to each other.
Most of the time, we will be interested in {\bf full-rank} \index{full rank lattice}
lattices, which means that $m=d$; however, sometimes we will also be interested in lattices that have lower rank, and it is important to understand them.
The {\bf determinant} of a full-rank lattice ${\mathcal L} := M(\mathbb{Z}^d)$ is defined by
\[
\det {\mathcal L} := |\det M|.
\]
It is easy to prove that this definition is independent of the choice of basis matrix $M$, which is the content of Lemma
\ref{changing basis matrices} below.
\begin{example}
\rm{
In $\mathbb{R}^1$, we have the integer lattice $\mathbb{Z}$, but we also have lattices of the form $r\mathbb{Z}$, for any real number $r$.
It's easy to show that any lattice in $\mathbb{R}^1$ is of this latter type (Exercise \ref{lattices in R^1}).
For example, if $r = \sqrt 2$, then all integer multiples of $\sqrt 2$ form a $1$-dimensional lattice. }
\hfill $\square$
\end{example}
\begin{example}
\rm{
In $\mathbb{R}^2$, consider the lattice ${\mathcal L}$ generated by the two integer vectors
$v_1:=\icol{-1\\3}$
and $v_2:= \icol{-4\\ 1}$, drawn in Figure \ref{parallelepiped1}.
A different choice of basis for the same lattice ${\mathcal L}$ is $\{ \icol{-3\\-2}, \icol{-8\\ -9} \}$, drawn in
Figure \ref{parallelepiped2}. We note that $\det {\mathcal L} = 11$, and indeed the areas of both half-open parallelepipeds equals $11$. }
\hfill $\square$
\end{example}
\medskip
A {\bf fundamental parallelepiped} \index{fundamental parallelepiped}
for a lattice ${\mathcal L}$ with basis $\{ v_1, \dots, v_m \}$ is:
\begin{equation} \label{def:half-open parallelepiped}
D:= \left\{ \lambda_1 v_1 + \cdots + \lambda_m v_m \bigm | \text{ all } 0 \leq \lambda_k < 1 \right\},
\end{equation}
also known as a {\bf half-open parallelepiped}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{parallelepiped2}
\end{center}
\caption{A second fundamental parallelepiped for the same lattice ${\mathcal L}$ as in
Figure \ref{parallelepiped1}} \label{parallelepiped2}
\end{figure}
We have the pleasant property
that D tiles $\mathbb{R}^d$ by translations with vectors
from ${\mathcal L}$, and with no overlaps. Let's make this intuition more precise, in the following lemma.
We'll use the standard notation that for any real $\alpha$, $\lfloor \alpha \rfloor$ is the greatest integer not exceeding $\alpha$, and $\{ \alpha\}$ is the fractional part of $\alpha$.
\begin{lem}
Suppose we are given a full rank lattice ${\mathcal L}\subset \mathbb{R}^d$, and a fundamental parallelepiped $D$ for ${\mathcal L}$,
as in Definition \eqref{def:half-open parallelepiped}.
Then any $x\in \mathbb{R}^d$ may be written uniquely as
\[
x = n + y
\]
where $n \in {\mathcal L}$, and $y \in D$.
\end{lem}
\begin{proof}
We know that $D$ is formed by a basis for the lattice ${\mathcal L}$, and we can label the basis elements by
$v_1, \dots, v_d$. These $d$ vectors also form a basis for $\mathbb{R}^d$,
so in particular any $x\in \mathbb{R}^d$ may be written as
\[
x = \sum_{j=1}^d \alpha_j v_j.
\]
Writing each $\alpha_j:= \lfloor \alpha_j \rfloor + \{ \alpha_j \}$, we have
\[
x = \sum_{j=1}^d \lfloor \alpha_j \rfloor v_j + \sum_{j=1}^d \{ \alpha_j \} v_j := n + y,
\]
where we've defined $n := \sum_{j=1}^d \lfloor \alpha_j \rfloor v_j$, and
$y:= \sum_{j=1}^d \{ \alpha_j \} v_j$. Since $\lfloor \alpha_j \rfloor \in \mathbb{Z}$, we see that
$n \in {\mathcal L}$. Since $0\leq \{ \alpha_j \} < 1$, we see that $y \in D$.
To prove uniqueness, suppose we are given $x:= n_1 + y_1 = n_2 + y_2$, where $n_1, n_2 \in {\mathcal L}$ and $y_1, y_2 \in D$. So by definition
$y_1 = \sum_{j=1}^d \{ \alpha_{j, 1} \} v_j$ and $y_2=\sum_{j=1}^d \{ \alpha_{j, 2} \} v_j$.
Then $y_1 - y_2 = n_2 - n_1 \in {\mathcal L}$, which means that $ \alpha_{j, 1} - \alpha_{j, 2} \in \mathbb{Z}$. But $0 \leq \alpha_{j, 1}<1$ and $0 \leq \alpha_{j, 2}<1$ implies that $ \alpha_{j, 1} - \alpha_{j, 2}=0$. Therefore $y_1 = y_2$, and so $n_1 = n_2$.
\end{proof}
How do we define the determinant of a general lattice ${\mathcal L}\subset \mathbb{R}^d$ of rank $r$?
We can start by observing how the squared lengths of vectors in ${\mathcal L}$ behave w.r.t. a given basis of ${\mathcal L}$:
\begin{equation}
\| x \|^2 = \left\langle \sum_{j=1}^r c_j v_j, \, \sum_{k=1}^r c_k v_k \right\rangle =
\sum_{1\leq j, k \leq r} c_j c_k \langle v_j, \, v_k \rangle := c^T M^T M c,
\end{equation}
where $M^TM$ is an $r\times r$ matrix whose columns are basis vectors of ${\mathcal L}$.
With this as motivation, we define:
\begin{equation}\label{def. of sublattice determinant}
\det {\mathcal L} := \sqrt{ M^T M},
\end{equation}
called the {\bf determinant of the lattice} ${\mathcal L}$. \index{determinant of a general lattice}
This definition coincides, as it turns out,
with the Lebesgue measure of any fundamental parallelepiped of ${\mathcal L}$ (Exercise \ref{equivalence between determinants of a sublattice}).
\bigskip
\section{Sublattices}
Given two lattices ${\mathcal L}\subset \mathbb{R}^d$, and ${\mathcal M} \subset \mathbb{R}^d$, such that
\[
{\mathcal L} \subseteq {\mathcal M},
\]
we say that {\bf ${\mathcal L}$ is a sublattice of ${\mathcal M}$}. \index{sublattice}
For example,
Figure \ref{sublattice, rank 1} shows a rank $1$ sublattice of the integer lattice $\mathbb{Z}^2$, together with its determinant.
On the other hand, sublattices that have the same rank are very interesting, and quite useful in applications. Given a sublattice ${\mathcal L}$ of ${\mathcal M}$, both of the same rank, a crucial idea is to think of all of the translates
of ${\mathcal L}$ by an element of the coarser lattice ${\mathcal M}$, which we call:
\begin{equation}
{\mathcal M} / {\mathcal L} := \left\{ {\mathcal L} + m \bigm | m \in {\mathcal M} \right\}.
\end{equation}
Each such translate ${\mathcal L} + m$ is called a {\bf coset} \index{coset}
of ${\mathcal L}$ in ${\mathcal M}$, and the collection of all of these cosets, namely ${\mathcal M} / {\mathcal L}$, is called the
{\bf quotient lattice} (or quotient group) \index{quotient lattice}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.2in]{sublattice1}
\end{center}
\caption{
A sublattice ${\mathcal L} \subset \mathbb{Z}^2$ of rank $1$, which has just one basis vector.
Here ${\mathcal L}$ has a $1$-dimensional fundamental parallelepiped, showing that
$\det {\mathcal L} = \sqrt{v^T v} = \sqrt 5$, consistent with Definition \ref{def. of sublattice determinant}.
}
\label{sublattice, rank 1}
\end{figure}
\begin{thm} \label{sublattice index}
Let ${\mathcal L} \subseteq {\mathcal M}$ be any two lattices of the same rank. Then
\begin{enumerate}
\item $\frac{ \det {\mathcal L} }{ \det {\mathcal M} }$ is an integer.
\item The positive integer $\frac{ \det {\mathcal L} }{ \det {\mathcal M} }$ is equal to the number of
cosets of ${\mathcal L}$ in ${\mathcal M}$. In other words, $\left| {\mathcal M} / {\mathcal L} \right| = \frac{ \det {\mathcal L} }{ \det {\mathcal M} }$.
\end{enumerate}
\end{thm}
For a proof of Theorem \ref{sublattice index}, see \cite{FukshanskyBook}.
\begin{example}
{\rm
Let ${\mathcal M}:= \mathbb{Z}^d$, and ${\mathcal L}:= 2\mathbb{Z}^d$, the sublattice consisting of vectors all of whose coordinates are even integers. So ${\mathcal L} \subset {\mathcal M}$, and the quotient lattice ${\mathcal M}/ {\mathcal L}$ consists of the sets
$\left\{ 2\mathbb{Z}^d + n \bigm | n \in \mathbb{Z}^d \right\}$. It is (almost) apparent that the number of elements of the latter set is exactly $2^d$, so in our new notation we have $\left| \mathbb{Z}^d / 2\mathbb{Z}^d \right| = 2^d$.
We may also think of this quotient lattice $ \mathbb{Z}^d / 2\mathbb{Z}^d $ as the discrete unit cube, namely
$\left\{ 0, 1 \right\}^d$, a common object in theoretical computer science.
\hfill $\square$
}
\end{example}
\section{Discrete subgroups - \\ an alternate definition of a lattice}
The goal here is to give another useful way to define a lattice. The reader does not need any background in group theory, because the ideas here are self-contained, given some background in basic linear algebra.
\smallskip
\begin{defi} \label{discrete subgroup}
\begin{enumerate}[(a)]
We define a {\bf discrete subgroup} \index{discrete subgroup}
of $\mathbb{R}^d$ as a set $S \subset \mathbb{R}^d$, together with the operation of vector addition between all of its elements, which enjoys the following two properties.
\item {\bf [The subgroup property]} If $x, y \in S$, then $ x-y \in S$. \\
\label{discrete subgroup.first part}
\item {\bf [The discrete property]} There exists a positive real number $\delta >0$,
such that \\
the distance between any two distinct points of $S$
is at least $\delta$. \\ \label{discrete subgroup.second part}
\end{enumerate}
\end{defi}
In particular, it follows from Definition \ref{discrete subgroup} \ref{discrete subgroup.first part}
that the zero vector must be in $S$, because for any $x \in S$, it must be the case that $x - x \in S$.
The distance function that we alluded to in Definition \ref{discrete subgroup} \ref{discrete subgroup.second part}
is the usual Euclidean distance function, which we denote here by
\[
\|x-y\|_2:= \sqrt{ \sum_{k=1}^d (x_k - y_k)^2}.
\]
\begin{example}
\rm{
The lattice $\mathbb{Z}^d$ is a discrete subgroup of $\mathbb{R}^d$. In dimension $1$, the lattice
$r\mathbb{Z}$ is a discrete subgroup of $\mathbb{R}$, for any fixed $r>0$. Can we think of discrete subgroups that are not lattices? The answer is given by Lemma \ref{discrete subgroup equivalence} below.
}
\hfill $\square$
\end{example}
The magic here is the following very useful way of going back and forth between this new notion of a discrete subgroup of $\mathbb{R}^d$, and our Definition \ref{def.lattice} of a lattice. The idea of using this alternate Definition \ref{discrete subgroup}, as opposed to our previous Definition \ref{def.lattice} of a lattice, is that it gives us a {\bf basis-free} \index{basis-free} way of proving and discovering facts about lattices.
\bigskip
\begin{lem}\label{discrete subgroup equivalence}
${\mathcal L} \subset \mathbb{R}^d$ is a lattice $\iff$ ${\mathcal L}$ is a discrete subgroup of $\mathbb{R}^d$.
\hfill $\square$
\end{lem}
(For a proof see \cite{GruberBook}).
\bigskip
\begin{example}
\rm{
Given any two lattices ${\mathcal L}_1, {\mathcal L}_2 \subset \mathbb{R}^d$, let's show that $S := {\mathcal L}_1 \cap {\mathcal L}_2$ is also a lattice.
First, any lattice contains the zero vector, and it may be the case that their intersection consists of only the zero vector. For any vectors $x, y \in S$, we also have $x,y \in {\mathcal L}_1$, and $x, y \in {\mathcal L}_2$, hence
by the subgroup property of ${\mathcal L}_1 $ and of ${\mathcal L}_2$, we know that both $x-y \in {\mathcal L}_1$, and $x-y \in {\mathcal L}_2$. In other words, $x-y \in {\mathcal L}_1 \cap {\mathcal L}_2:= S$. To see why the discrete property of
Definition \ref{discrete subgroup} holds here, we just notice that since $x-y \in {\mathcal L}_1$, we already know that
$| x-y | > \delta_1$, for some $\delta_1>0$; similarly, because $x-y \in {\mathcal L}_2$, we know that
$| x-y | > \delta_2$ for some $\delta_2>0$. So we let $\delta:= \min(\delta_1, \delta_2\}$, and we have shown that $S$ is a discrete subgroup of $\mathbb{R}^d$. By Lemma \ref{discrete subgroup equivalence}, we see that $S$ is a lattice.
If we had used Definition \ref{def.lattice} of a lattice to show that $S$ is indeed a lattice, it would require us to work with bases, and this proof would be longer and less transparent.
}
\hfill $\square$
\end{example}
\bigskip
\begin{example} \label{A_d example}
\rm{
Consider the following discrete set of points in $\mathbb{R}^d$:
\[
A_{d-1}:= \left\{ x \in \mathbb{Z}^d \bigm | \sum_{k=1}^d x_k =0 \right\},
\]
for any $d\geq 2$, as depicted in Figure \ref{A_d}. Is $A_d$ a lattice? Using the definition
\ref{def.lattice} of a lattice, it is not obvious that $A_d$ is a lattice, because we would have to exhibit a basis, but
it turns out that the following set of vectors may be shown to be a basis: $ \left\{e_2 - e_1, e_3 - e_1, \cdots e_d - e_1 \right\}$, and hence $A_d$ is a sublattice
of $\mathbb{Z}^d$, of rank $d-1$ (Exercise \ref{basis for A_d}).
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.8in]{A_d}
\end{center}
\caption{The lattice $A_1$, and
the lattice $A_2$, with basis $\left\{ v_1, v_2 \right\}$} \label{A_d}
\end{figure}
Just for fun, we will use Lemma \ref{discrete subgroup equivalence} to show that
$A_d$ is indeed a lattice. To verify the subgroup property of
Definition \ref{discrete subgroup} \ref{discrete subgroup.first part}
suppose that $x, y \in A_d$. Then by definition we have
$\sum_{k=1}^d x_k =0$ and $ \sum_{k=1}^d y_k =0$. So $\sum_{k=1}^d (x_k - y_k)=0$, implying that $x-y \in A_d$.
To verify the discrete property of
Definition \ref{discrete subgroup} \ref{discrete subgroup.second part}
suppose we are given two distinct points $x, y \in A_d$. We can first compute their ``cab metric'' distance function, in other
words the $L^1$-norm defined by
\[
\| x-y \|_1:= |x_1 - y_1| + \cdots + |x_d - y_d|,
\]
By assumption, there is at least one coordinate where $x$ and $y$ differ, say the $k$'th coordinate. Then
$ \| x-y \|_1 := |x_1 - y_1| + \cdots + |x_d - y_d| \geq 1$, because all of the coordinates are integers, and $x_k \not= y_k$ by assumption. Since the $L^1$-norm and the $L^2$-norm
are only off
by $\sqrt d$ (by Exercise \ref{elementary norm relations}), we have:
\[
\sqrt{d} \| x-y \|_2 \geq \| x-y \|_1 \geq 1,
\]
so the property \ref{discrete subgroup} \ref{discrete subgroup.second part}
is satisfied with $\delta := \frac{1}{\sqrt{d}}$, and we've shown that $A_d$ is a lattice.
\hfill $\square$
}
\end{example}
We note that the lattices $A_d$ defined in Example \ref{A_d example} are very important in many fields of Mathematics, including Lie algebras (root lattices), Combinatorial geometry, and Number theory.
\section{Lattices defined by congruences}
In this section we develop some of the theory in a concrete manner. A classic example of a lattice defined by an auxiliary algebraic construction is the following.
Suppose we are given a constant integer vector $(c_1, \dots, c_d) \in \mathbb{Z}^d$, where we further assume
that $\gcd(c_1, \dots, c_d) = 1$. Let
\begin{equation}\label{Lattice from congruences}
C := \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d \equiv 0 \mod N \right\},
\end{equation}
where $N$ is a fixed positive integer.
Is $C$ a lattice?
Indeed, we can see that $C$ is a lattice by first checking
Definition \ref{discrete subgroup} \ref{discrete subgroup.first part}. For any $x, y \in C$, we have
$c_1 x_1 + \cdots + c_d x_d \equiv 0 \mod N$ and $c_1 y_1 + \cdots + c_d y_d \equiv 0 \mod N$.
Subtracting these two congruences gives us
$c_1 (x_1-y_1) + \cdots + c_d (x_d-y_d) \equiv 0 \mod N$, so that $x-y \in C$. The verification
of Definition \ref{discrete subgroup} \ref{discrete subgroup.second part} if left to the reader, and its logic is similar to Example \ref{A_d example}.
There is even a simple formula for the volume of a fundamental parallelepiped for $C$:
\begin{equation}
\det C = N,
\end{equation}
as we prove below, in \eqref{proof of det C = N}.
Perhaps we can solve an easier problem first. Consider the {\bf discrete hyperplane} defined by:
\[
H:= \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =0 \right\},
\]
Is $H$ a lattice? We claim that $H$ itself is indeed a sublattice of $\mathbb{Z}^d$, and has rank $d-1$.
Since this verification is quite similar to the arguments above, we leave this as Exercise \ref{hyperplane lattice}.
The fundamental parallelepiped (which is $(d-1)$-dimensional) of $H$ also has a wonderful formula, as follows. First, we recall
a general fact (from Calculus/analytic geometry) about hyperplanes,
namely that the distance $\delta$ between any two parallel hyperplanes \\
$c_1 x_1 + \cdots + c_d x_d = k_1$ and
$c_1 x_1 + \cdots + c_d x_d = k_2$ is given by
\begin{equation} \label{distance between two hyperplanes}
\delta = \frac{ |k_1 - k_2|}{\sqrt{ c_1^2 + \cdots + c_d^2}}.
\end{equation}
(see Exercise \ref{distance between hyperplanes})
\medskip
\begin{lem} \label{wonderful hyperplane determinant formula}
For any latttice defined by a discrete hyperplane \\
$H:= \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =0 \right\}$, with
$\gcd(c_1, \dots, c_d) = 1$, we have:
\begin{equation} \label{tricky volume of sublattice}
\det H = \sqrt{ c_1^2 + \cdots + c_d^2}.
\end{equation}
\end{lem}
\begin{proof}
We first fix a basis $\{v_1, \dots, v_{d-1}\}$ for the $(d-1)$-dimensional sublattice defined by
$H:= \left\{ x \in \mathbb{Z}^d \mid c_1 x_1 + \cdots + c_d x_d =0 \right\}$. We adjoin to this basis one new vector, namely any integer vector $w$ that translates $H$ to its `hyperplane companion'
$H + w$, which we define by
\[
H+w:= ~\left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =1 \right\}.
\]
It's easy - and fun - to see that there are no integer points strictly between these two hyperplanes $H$ and $H+w$ (Exercise \ref{tiling the integer lattice with hyperplanes}), and so
the parallelepiped ${\mathcal P}$ formed by the edge vectors $v_1, \dots, v_{d-1}, w$ is a fundamental domain for $\mathbb{Z}^d$, hence has volume $1$.
On the other hand, we may also calculate the volume of ${\mathcal P}$ by multiplying the volume of its base times its height, using \eqref{distance between two hyperplanes}:
\begin{align}
1= \vol {\mathcal P} &= (\text{volume of the base of } {\mathcal P})(\text{height of } {\mathcal P}) \\
&= (\det H)\cdot \delta \\
&= (\det H) \frac{1}{\sqrt{c_1^2 + \cdots + c_d^2}},
\end{align}
and so $\det H = \sqrt{ c_1^2 + \cdots + c_d^2}$.
\end{proof}
It follows directly from the definition \ref{Lattice from congruences} of $C$ that we may write the lattice $C$ as a countable, disjoint union of translates of $H$:
\begin{equation}
C := \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d = kN, \text{ where } k = 1, 2, 3, \dots \right\}.
\end{equation}
To be concrete, let's work out some examples.
\begin{example}
\rm{
Using Lemma \ref{wonderful hyperplane determinant formula}, we can easily compute the determinant of the $A_d$ lattice from Example \ref{A_d example}:
\[
\det A_d = \sqrt{1 + 1 + \cdots + 1} = \sqrt d.
\]
}
\end{example}
\medskip
\begin{example}\label{Congruence lattice 2d}
\rm{
As in Figure \ref{lattice 2d},
consider the set of all integer points $(m, n) \in \mathbb{R}^2$ that satisfy
\[
2m + 3n \equiv 0 \mod 4.
\]
In this case the related hyperplane is the line $2x+3y = 0$, and the solutions to the latter congruence may be thought of as a union of discrete lines:
\[
C = \left\{ {x\choose y} \in \mathbb{Z}^2 \bigm | 2x+3y = 4k, \text{ and } k \in \mathbb{Z} \right\}.
\]
}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.7in]{LatticeCongruence}
\end{center}
\caption{The lattice of Example \ref{Congruence lattice 2d}} \label{lattice 2d}
\end{figure}
\rm{
In other words, our lattice $C$, a special case of \eqref{Lattice from congruences}, can in this case be visualized
in Figure \ref{lattice 2d} as a disjoint union of discrete lines. If we denote the distance between any two of these
adjacent discrete lines by $\delta$, then using \eqref{distance between two hyperplanes} we have:
\begin{equation}
\delta = \frac{4}{\sqrt{3^2 + 2^2}}.
\end{equation}
Finally, the determinant of our lattice $C$ here is the area of the
shaded parallelepiped:
\begin{equation}
\det C = \delta \sqrt{3^2 + 2^2} = 4.
\end{equation}
}
\hfill $\square$
\end{example}
Eager to prove the volume relation $ \det C = N$,
we can use the ideas of Example \ref{Congruence lattice 2d} as a springboard for this generalization.
Indeed, Example \ref{Congruence lattice 2d} and the proof of Lemma
\ref{wonderful hyperplane determinant formula} both
suggest that we should compute the volume of a fundamental parallelepiped ${\mathcal P}$, for the lattice $C$ (as opposed to the lattice $\mathbb{Z}^d$),
by using a fundamental domain for its base, and then by multiplying its volume by the height of ${\mathcal P}$.
\begin{lem}\label{lemma:lattice defined by congruence}
Given a constant integer vector $(c_1, \dots, c_d) \in \mathbb{Z}^d$, with $\gcd(c_1, \dots, c_d) = 1$,
let
\begin{equation}
C := \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d \equiv 0 \mod N \right\},
\end{equation}
where $N$ is a fixed positive integer. Then $C$ is a lattice, and
\[
\det C = N.
\]
\end{lem}
\begin{proof}
We fix a basis $\{v_1, \dots, v_{d-1}\}$ for the $(d-1)$-dimensional sublattice defined by
$H:= \left\{ x \in \mathbb{Z}^d \mid c_1 x_1 + \cdots + c_d x_d =0 \right\}$, and we adjoin to this basis one new vector, namely any integer vector $w$ that translates $H$ to its nearest discrete hyperplane companion
\[
H+w:= \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =N \right\}.
\]
Together, the set of vectors
$ \{ v_1, \dots, v_{d-1}, w \} $
form the edge vectors of a fundamental parallelepiped ${\mathcal P}$ for the lattice $C$, whose hight $\delta$ is the distance between these two parallel
hyperplanes $H$ and $H+w$. Using \eqref{distance between two hyperplanes}, we can
may calculate the volume of ${\mathcal P}$ (which is by definition equal to $\det C$) by multiplying the volume of its `base' times its `height':
\begin{align}
\det C &= (\text{volume of the base of } {\mathcal P})(\text{height of } {\mathcal P}) = (\det H)\delta \\ \label{proof of det C = N}
&= (\det H) \frac{N}{\sqrt{c_1^2 + \cdots + c_d^2}} = N,
\end{align}
using the fact that $\det H = \sqrt{c_1^2 + \cdots + c_d^2}$ from
Lemma \ref{wonderful hyperplane determinant formula}.
\end{proof}
\bigskip
\section{The Gram matrix}
There is another very natural matrix that we may use to study lattices, which we can motivate as follows.
Suppose we are given any basis for a lattice ${\mathcal L}\subset \mathbb{R}^d$, say $\beta:= \{ v_1, \dots, v_r \}$,
where $1 \leq r \leq d$. By definition ${\mathcal L} = M(\mathbb{Z}^d)$, and ${\rm rank}({\mathcal L}) = r$,
where the columns of $M$ are defined by the basis vectors from $\beta$, and so $M$ is a $d\times r$ matrix.
We can therefore represent any $x\in \mathbb{R}^d$ uniquely in terms of the basis $\beta$ like this:
\begin{equation} \label{writing a vector in terms of a basis}
x = c_1 v_1 + \cdots + c_r v_r,
\end{equation}
and the squared length of $x$ is:
\begin{equation} \label{Gram matrix positive semidefinite}
\| x \|^2 = \left\langle \sum_{j=1}^r c_j v_j, \, \sum_{k=1}^r c_k v_k \right\rangle =
\sum_{1\leq j, k \leq r} c_j c_k \langle v_j, \, v_k \rangle := c^T M^T M c,
\end{equation}
where $c:= (c_1, \dots, c_r)^T$ is the coefficient vector defined by
\eqref{writing a vector in terms of a basis}.
It's therefore very natural to focus on the matrix $M^T M$, whose entries are the inner products
$\langle v_j, v_k \rangle$ of all the basis vectors of the lattice ${\mathcal L}$, so
we define
\[
G:= M^TM,
\]
a {\bf Gram matrix} for ${\mathcal L}$. It's clear from the computation above in
\eqref{Gram matrix positive semidefinite} that $G$ is positive definite. Although
$G$ does depend on which basis of ${\mathcal L}$ we choose, it is an elementary fact that $\det G$ is independent of the basis of ${\mathcal L}$.
Because we are always feeling the urge to learn more Linear Algebra, we would like to see
why any real symmetric matrix $B$ is the Gram matrix of some set of vectors. To see this, we apply the
Spectral Theorem: $B = P D P^T$, for some orthogonal matrix $P$ and a diagonal matrix $D$ with nonnegative diagonal elements. So we can write $B = (P \sqrt D) (P \sqrt D)^T := M^T M$, where we defined the matrix $M:= (P \sqrt D)^T$, so that the columns of $M$ are the vectors whose corresponding dot products form the symmetric matrix $B$, and now $B$ is a Gram matrix.
To review some more linear algebra, suppose we are given a real symmetric matrix $A$.
We recall that such a matrix is called {\bf positive definite} if in addition we have the positivity condition
\[
x^T A x > 0,
\]
for all $x \in \mathbb{R}^d$. Equivalently, all of the eigenvalues of $A$ are positive. The reason is easy:
$Ax = \lambda x$ for a non-zero vector $x \in \mathbb{R}^d$
implies that
\[
x^T A x := \langle x, Ax \rangle = \langle x, \lambda x \rangle = \lambda \| x \|^2,
\]
so that $x^T A x >0 $ if and only if $\lambda > 0$. In the sequel, if we only require a symmetric matrix
$A$ that enjoys the property $x^T A x \geq 0$ for all
$x\in \mathbb{R}^d$, then we call such a matrix {\bf positive semidefinite}.
Also, for a full-rank lattice, we see that $B:= M^T M$ will be positive definite if and only if $M$ is invertible, so that the columns of $M$ are a basis. Since a positive definite matrix is symmetric by definition, we've proved:
\begin{lem} Suppose we are given a real symmetric matrix $B$. Then:
\begin{enumerate}[(a)]
\item $B$ is positive definite if and only if it is the Gram matrix of a basis.
\item $B$ is positive semidefinite if and only if it is the Gram matrix of some set of vectors.
\end{enumerate}
\hfill $\square$
\end{lem}
What about reconstructing a lattice, knowing only one of its Gram matrices? This is almost possible to accomplish, up to an orthogonal transformation, as follows.
\begin{lem}\label{reconstructing a lattice basis from its Gram matrix}
Suppose that $G$ is an invertible matrix, whose spectral decomposition is
\[
G = P D P^T.
\]
Then
\begin{equation}
G = X^T X \quad \iff \quad X = Q (\sqrt{D} P^T),
\end{equation}
for some orthogonal matrix $Q$.
\end{lem}
\begin{proof}
The assumption $G = X^T X$ guarantees that $G$ is symmetric and has positive eigenvalues, so by the Spectral Theorem we have:
\[
G = P D P^T,
\]
where $D$ is a diagonal matrix consisting of the positive eigenvalues of $G$, and $P$ is an orthogonal matrix consisting of eigenvectors of $G$. Setting $X^T X = P D P^T$, we must have
\begin{equation} \label{technical orthogonal identity}
I = X^{-T} PDP^T X^{-1} = (X^{-T} P \sqrt D) (X^{-T} P \sqrt D)^T,
\end{equation}
where we define $\sqrt D$ to be the diagonal matrix whose diagonal elements are the positive square roots of the eigenvalues of $G$. From \ref{technical orthogonal identity}, it follows that
$X^{-T} P \sqrt D$ is an orthogonal matrix, let's call it $Q^{-T}$. Finally,
$X^{-T} P \sqrt D = Q^{-T}$ implies that $X = Q \sqrt D P^T$.
\end{proof}
So Lemma \ref{reconstructing a lattice basis from its Gram matrix} allows us to reconstruct a lattice ${\mathcal L}$, up to an orthogonal transformation, by only knowing one of its Gram matrices.
To better understand lattices, we need the {\bf unimodular group}, which we write as
${\rm SL}_d(\mathbb{Z})$, \index{unimodular matrix} under matrix multiplication:
\begin{equation}\label{Definition of the unimodular group}
{\rm SL}_d(\mathbb{Z}) :=
\left\{ M \bigm | M \text{ is a }
d \times d \text{ integer matrix, with} \ |\det M| = 1
\right\}.
\end{equation}
\index{unimodular group}
\noindent
The elements of $\rm{SL_d}(\mathbb{Z})$ are called {\bf unimodular matrices}. \index{unimodular matrix}
\begin{example}
\rm{
Some typical elements of $\rm{SL_2}(\mathbb{Z})$ are
\[
S = \big(\begin{smallmatrix}
\ 0 & 1 \\
-1 & 0
\end{smallmatrix}
\big),
T:= \big(\begin{smallmatrix}
1 & 1 \\
1 & 0
\end{smallmatrix}
\big),
\text{ and \ }
-I := \big(\begin{smallmatrix}
-1 & \ 0 \\
\ 0 & -1
\end{smallmatrix}
\big),
\]
so we include matrices with determinant equal to $-1$ as well as $1$.}
\hfill $\square$
\end{example}
Any lattice ${\mathcal L}$ has infinitely many fundamental parallelepipeds and (Exercise \ref{fundamental domains})
it is a nice fact that they are all images of one another by the unimodular group.
Now, suppose a lattice ${\mathcal L}$ is defined by two different basis matrices: ${\mathcal L} = M_1(\mathbb{Z}^d)$ and
${\mathcal L} = M_2(\mathbb{Z}^d)$. Is there a nice relationship between $M_1$ and $M_2$?
\begin{lem}\label{changing basis matrices}
If a full-rank lattice ${\mathcal L} \subset \mathbb{R}^d$ is defined by two different basis matrices $M_1$, and $M_2$,
then
\[
M_1 = M_2 U,
\]
where $U \in \rm{SL_d}(\mathbb{Z})$, a unimodular matrix.
In particular, $\det {\mathcal L}$ is independent of the
choice of basis matrix $M$.
\end{lem}
\begin{proof}
By hypothesis, we know that the columns of $M_1$, say $v_1, \dots, v_d$, form a basis of ${\mathcal L}$, and that the columns of $M_2$, say $w_1, \dots, w_d$, also form a basis of ${\mathcal L}$. So we can begin by writing each fixed basis vector $v_j$ in terms of all the basis vectors $w_k$:
\[
v_j = \sum_{k=1}^d c_{j,k} w_k,
\]
for each $j = 1, \dots, d$, and for some $c_{j,k} \in \mathbb{Z}$. We may collect all $d$ of these identities into matrix form:
\[
M_1 = M_2 C,
\]
where $C$ is the integer matrix whose entries are defined by the integer coefficients $c_{j,k}$ above.
Conversely, we may also write each basis vector $w_j$ in terms of the basis vectors $v_k$:
$w_j = \sum_{k=1}^d d_{j,k} v_k$, for some $d_{j,k}\in\mathbb{Z}$, getting another matrix identity:
\[
M_2 = M_1 D.
\]
Altogether we have
\[
M_1 = M_2 C = (M_1 D) C,
\]
and since $M_1^{-1}$ exists by assumption, we get $DC= I$, the identity matrix. Taking determinants, we see that
\[
| \det D | | \det C | = 1,
\]
and since both $C$ and $D$ are integer matrices, they must belong to $\rm{SL_d}(\mathbb{Z})$, by definition.
Finally, because, because a unimodular matrix $U$ has $|\det U|=1$, we see that any two basis $M_1, M_2$ matrices satisfy
$|\det M_1 | = |\det M_2 |$.
\end{proof}
Using similar techniques, it is not hard to show the following fact
(Exercise \ref{Theorem: Automorphisms of lattices}).
\begin{thm} \label{Automorphisms of lattices}
Fix a full-rank lattice ${\mathcal L} \subset \mathbb{R}^d$. The group of one-to-one, onto, linear transformations from ${\mathcal L}$ to itself
is equal to
the unimodular group $SL_d(\mathbb{Z})$. \index{unimodular group}
\end{thm}
\bigskip
\section{Dual lattices} \index{dual lattice} \label{dual lattice}
Every lattice ${\mathcal L}:= M(\mathbb{Z}^d)$ has a {\bf dual lattice}, which we have already encountered in the Poisson summation formula
for arbitrary lattices. The dual lattice of a full-rank lattice ${\mathcal L}$ was defined by:
\begin{equation}\label{first definition of dual lattice}
{\mathcal L} ^*= M^{-T}(\mathbb{Z}^d).
\end{equation}
But there is another way to define the dual lattice of a lattice ${\mathcal L} \subset \mathbb{R}^d$ (of any rank), which is coordinate-free:
\begin{equation}\label{second definition of dual lattice}
{\mathcal L}^* := \left\{ x \in \mathbb{R}^d \mid \langle x, n \rangle \in \mathbb{Z}, \text{ for all } n \in {\mathcal L} \right\}.
\end{equation}
\begin{lem}
The two definitions above, \eqref{first definition of dual lattice} and \eqref{second definition of dual lattice}, are equivalent.
\begin{proof}
We let $A := {\mathcal L}^*:= M^{-T}(\mathbb{Z}^d)$, and
$B:= \left\{ x \in \mathbb{R}^d \bigm | \langle x, n \rangle \in \mathbb{Z}, \text{ for all } n \in {\mathcal L} \right\}$.
We first fix any $x \in A$. To show $x \in B$, we fix any
$n\in {\mathcal L}$, and we now have to verify that $\langle x, n \rangle \in \mathbb{Z}$.
By assumption, $x = M^{-T}m$ for some $m \in \mathbb{Z}^d$, and $n= M k$, for some $k\in\mathbb{Z}^d$. Therefore
\[
\langle x, n \rangle =\langle M^{-T}m, n \rangle = \langle m, M^{-1}n \rangle
= \langle m, k \rangle \in \mathbb{Z},
\]
because both $m,k \in \mathbb{Z}^d$. So we have $A \subset B$.
For the other direction, suppose that $y\in B$, so by definition
\begin{equation*}\label{def. of B}
\langle y, n \rangle \in \mathbb{Z}, \text{ for all } n \in {\mathcal L}.
\end{equation*}
We need to show that $y = M^{-T} k$ for some $k\in \mathbb{Z}^d$, which is equivalent to $M^T y \in \mathbb{Z}^d$.
Noticing that the $k$'th element of $M^T y$ is
$\langle n, y \rangle$ with $n$ belonging to a basis of ${\mathcal L}$, we are done, by \eqref{def. of B}.
Therefore $A=B$.
\end{proof}
\end{lem}
\bigskip
\begin{example}
\rm{
Let ${\mathcal L} := r \mathbb{Z}^d$, the integer lattice dilated by a positive real number $r$. It's dual lattice is ${\mathcal L}^* = \frac{1}{r} {\mathcal L}$, because a basis for ${\mathcal L}$ is $M := rI$, implying that a basis matrix for ${\mathcal L}^*$ is $M^{-T} = \frac{1}{r} I$.
We also notice that $\det {\mathcal L} = r^d$, while $\det {\mathcal L}^* = \frac{1}{r^d}$.
}
\hfill $\square$
\end{example}
A fundamental relation between a full-rank lattice and its dual follows immediately from Definition
\ref{first definition of dual lattice}: $\det({\mathcal L}^*) := \det(M^{-T}) = \frac{1}{\det M}= \frac{1}{\det {\mathcal L}}$,
which we record as:
\begin{equation} \label{FundamentalDualRelation}
(\det {\mathcal L} )(\det {\mathcal L}^*) = 1.
\end{equation}
If we consider any integer sublattice of $\mathbb{Z}^d$, say ${\mathcal L} \subset \mathbb{Z}^d$, together with its dual lattice ${\mathcal L}^*$ in the same space, some interesting relations unfold between them. Let's consider an example.
\bigskip
\begin{example} \label{dual lattice example}
\rm{
In $\mathbb{R}^2$, let
${\mathcal L} := \left\{m \icol{1\{\bf 1}} + n \icol{1\\ 4} \bigm | m,n \in \mathbb{Z} \right\}$,
a lattice with $\det {\mathcal L} = 3$ that is depicted in Figure \ref{DualLattice} by the larger green balls.
Its dual lattice is
\[
{\mathcal L}^*:= \left\{
\frac{1}{3} \left( a\icol{\ 4\\-1} +b \icol{ \ \, -1\\ \ \ \ 1} \right) \bigm | a, b \in \mathbb{Z}
\right\},
\]
whose determinant equals $\frac{1}{3}$, and is depicted in Figure
\ref{DualLattice} by the smaller orange balls. So ${\mathcal L}$ is a coarser lattice than ${\mathcal L}^*$.
}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.3in]{DualLattice}
\end{center}
\caption{The lattice of Example \ref{dual lattice example}, together with its dual lattice}
\label{DualLattice}
\end{figure}
We can verify that the relation \eqref{FundamentalDualRelation} works for this example: $\det {\mathcal L}^* = \frac{1}{3} = \frac{1}{\det {\mathcal L}}$. We also notice that ${\mathcal L}$ is a sublattice of ${\mathcal L}^*$.
We may notice here that
${\mathcal L}^*/{\mathcal L}$ forms a finite group of order $9 = (\det {\mathcal L})^2$, which is equal to the number of cosets of
the coarser lattice ${\mathcal L}$ in the finer lattice ${\mathcal L}^*$.
\hfill $\square$
\end{example}
The dual lattice also appears as the kernel of a certain map,
as follows. Suppose that for each point $n \in {\mathcal L}$, we define a function called a {\bf character} of ${\mathcal L}$:
\begin{equation}
\chi_n(y) := e^{2\pi i \langle n, y \rangle},
\end{equation}
whose domain is the whole space $\mathbb{R}^d$. To see a connection with the dual lattice ${\mathcal L}^*$, we may consider the simultaneous kernel of all of these functions taken together:
\[
\rm{Ker}:= \{ x \in \mathbb{R}^d \mid \chi_n(x)=1, \text{ for all } n \in {\mathcal L}\}.
\]
It's clear that $\rm{Ker} = {\mathcal L}^*$,
because $e^{2\pi i z} = 1$ if and only if $z \in \mathbb{Z}$.
Now let's consider the collection of all of these characters:
\begin{equation}
G_{\mathcal L} := \{ \chi_n \mid n \in {\mathcal L} \}.
\end{equation}
If we multiply these character together by defining $\chi_n \chi_m:= \chi_{n+m}$, then
$G_{\mathcal L}$ forms a group, called the {\bf group of characters} of ${\mathcal L}$. To see that this multiplication makes sense, we can compute:
\[
(\chi_n \chi_m) (x) =
e^{2\pi i \langle n, x \rangle} e^{2\pi i \langle m, x \rangle}
=e^{2\pi i \langle n+m, x \rangle} =
\chi_{n+m}(x).
\]
Even more is true: $G_{\mathcal L}$ is isomorphic, as a group, to the lattice ${\mathcal L}$ (Exercise \ref{character group}) via
the map $n \rightarrow \chi_n$.
Intuitively, one of the great benefits of group characters is that by using the magic of just two-dimensional complex numbers, we can study high-dimensional lattices.
\begin{example}
For the integer lattice $\mathbb{Z}^d$, its group of characters is composed of the following functions, by definition:
\[
\chi_n(x):= e^{2\pi i \langle n, x \rangle},
\]
for each $n \in \mathbb{Z}^d$.
\hfill $\square$
\end{example}
\bigskip
\section{The successive minima of a lattice: \\
more geometry of numbers}
To warm up, we recall a very classical inequality of Hadamard, giving a bound on determinants.
Intuitively, Hadamard's inequality tells us that if we keep all the lengths of the sides of a parallelepiped constant, and consider all possible parallelepipeds ${\mathcal P}$
with these fixed side lengths, then the volume of ${\mathcal P}$ is maximized exactly when ${\mathcal P}$ is rectangular.
\begin{thm}[Hadamard's inequality] \label{Hadamard inequality} \index{Hadamard's inequality}
Given a non-singular matrix $M$, over the reals, whose column vectors are
$v_1, \dots, v_d$, we have:
\[
|\det M| \leq \| v_1 \| \|v_2\| \cdots \|v_d\|,
\]
\end{thm}
with equality if and only if all of the $v_k$'s are pairwise orthogonal.
\begin{proof}
We use the following matrix decomposition from Linear Algebra: $M = QR$, where $Q$ is an orthogonal
matrix, $R:= [r_{i,j}]$ is an upper-triangular matrix, and $r_{kk} > 0$ (this decomposition is a well-known consequence of the Gram-Schmidt process applied to the columns of M). So now we know that $|\det Q| = 1$, and $\det R = \prod_{k=1}^d r_{kk}$, and it follows that
\[
|\det M| = |\det Q \det R| = \det R = \prod_{k=1}^d r_{kk}.
\]
Let's label the columns of $Q$ by $Q_k$, and the columns of $R$ by $R_k$. We now consider the matrix
$M^T M = R^T Q^T Q R = R^T R$. Comparing the diagonal elements on both sides of $M^T M = R^T R$,
we see that $\| Q_k\|^2 = \| R_K \|^2$. But we also have $\| R_K \|^2 \geq r_{kk}^2$, so that
$\|Q_k \| \geq r_{kk}$.
Altogether we have
\begin{equation} \label{product formula}
|\det M| = \prod_{k=1}^d r_{kk} \leq \prod_{k=1}^d \|Q_k \|.
\end{equation}
The case of equality occurs if and only if $\| R_K \|^2 = r_{kk}^2$ for all $1\leq k\leq d$, and this case of equality would mean
that $R$ is a diagonal matrix. Thus, we have equality in inequality \eqref{product formula} if and only if
$M^T M = R^T R$ is a diagonal matrix, which means that the columns of $M$ are mutually orthogonal.
\end{proof}
A very important characteristic of a lattice ${\mathcal L}$ is the {\bf length of its shortest nonzero vector}:
\index{shortest nonzero vector in a lattice}
\[
\lambda_1({\mathcal L}):=\min \left\{ \| v \| \biggm | v \in {\mathcal L}-\{0\} \right\}.
\]
Every lattice has at least two shortest nonzero vectors, because if $v \in {\mathcal L}$, then $-v \in {\mathcal L}$. Thus, when we use the words `its shortest vector', we always mean that we are free to make a choice between any of its vectors that have the same shortest, nonzero length. A natural question, which has many applications, is ``how short is the shortest nonzero vector in ${\mathcal L}$, as we somehow
vary over all normalized lattices ${\mathcal L}$?"
\begin{example}
\rm{
We define the following lattice in $\mathbb{R}^2$:
\[
{\mathcal L} := \left\{ m \icol{102 \\ 11 } + n \icol{200\\16} \bigm | m,n\in\mathbb{Z} \right\}.
\]
What is the shortest nonzero vector in this lattice ${\mathcal L}$? Without using any fancy
Theorems, we might still try simple subtraction, sort of
mimicking the Euclidean algorithm. So for example, we might try $\icol{200\\16} - 2 \icol{102 \\ 11 } = \icol{-4 \\ -6 }$, which is pretty short. So we seem to have gotten lucky - we found a relatively short vector. But here
comes the impending question:
\begin{question}
How do we know whether or not this is really the shortest nonzero vector in our lattice ${\mathcal L}$?
Can we find an even shorter vector in ${\mathcal L}$?
\end{question}
This is not easy to answer in general, and we need to learn a bit more theory even to approach it in $\mathbb{R}^2$.
Moreover, in dimensions $d\geq 3$, the corresponding problem of finding a shortest nonzero vector in any given lattice is terribly difficult - it is considered to be one of the most difficult problems in computational number theory.
}
\hfill $\square$
\end{example}
To capture the notion of the second-smallest vector in a lattice, and third-smallest vector, etc, we begin by
imagining balls of increasing radii, centered at the origin, and we can (at least theoretically) keep track of how
they intersect ${\mathcal L}$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.6in]{SuccesiveMinima}
\end{center}
\caption{The two successive minima for this lattice ${\mathcal L}$ are $\lambda_1({\mathcal L}) = \sqrt 2$, and $\lambda_2({\mathcal L}) = \sqrt 5$.} \label{Successive Minima}
\end{figure}
Let $B^d(r)$ be the closed ball of radius $r$, centered at the origin.
For each fixed $j$, with $1 \leq j \leq d$, let $r$ be the smallest positive real number such that
$B^d(r)$ contains at least $j$ linearly independent lattice points of ${\mathcal L}$.
This value of $r$ is called $\lambda_j({\mathcal L})$,
the {\bf $j$'th successive minima} of the lattice.
Another way of saying the same thing is:
\begin{equation}
\lambda_j({\mathcal L}) := \min \left\{ r >0 \bigm |
\dim \big( \text{span }({\mathcal L} \cap {B^d(r)}) \big) \geq j \right\}.
\end{equation}
By definition, we have $| \lambda_1({\mathcal L}) | \leq | \lambda_2({\mathcal L}) | \leq \cdots \leq | \lambda_d({\mathcal L}) |$.
\medskip
\begin{example}
\rm{
For ${\mathcal L}:= \mathbb{Z}^d$, the shortest nonzero vector has length $\lambda_1(\mathbb{Z}^d) = 1$, and the successive minima for $\mathbb{Z}^d$ all have the same value. One choice for their corresponding vectors is
$v_1:= {\bf e_1}, \dots, v_d:= {\bf e_d}$, the standard basis vectors.
}
\hfill $\square$
\end{example}
\medskip
\begin{example}
\rm{
In $\mathbb{R}^2$, there is a very special
lattice, sometimes called the {\bf hexagonal lattice} (also known as the {\bf Eisenstein lattice}):
\[
{\mathcal L} := \left\{ m { \frac{\sqrt{3}}{2} \choose \frac{1}{2} } + n {1 \choose 0} \mid m,n\in\mathbb{Z} \right\}.
\]
This lattice has $\det {\mathcal L} = \frac{\sqrt{3}}{2}$ and is generated by the $6$'th roots of unity (Exercise \ref{Eisenstein lattice}). Given the basis above, we see that here we have $\lambda_1({\mathcal L}) = \lambda_2({\mathcal L}) =1$.
It also turns out to be an
extremal lattice in the sense that it (more precisely a dilate of it) is the lattice that achieves Hermite's constant $\gamma_2$, below, over all lattices in $\mathbb{R}^2$.
(Exercise \ref{minimal lattice in R^2}).
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.0in]{EisensteinLattice}
\end{center}
\caption{The hexagonal lattice, also known as the Eisenstein lattice} \label{Eisenstein Lattice}
\end{figure}
\medskip
\begin{example} \label{a curve in the space of lattice}
\rm{
Let's define the following family of $2$-dimensional lattices. For each $t > 0$, we let
\[
M:= \begin{pmatrix}
e^t & 0 \\
& e^{-t} \end{pmatrix}, \text{ and we let } {\mathcal L}_t:= M(\mathbb{Z}^d),
\]
so that we get a parametrized family of lattices. While all of the lattices in this family have $\det {\mathcal L}=1$, their shortest nonzero vectors approach $0$ as $t\rightarrow \infty$, since $\lambda_1({\mathcal L}_t) = e^{-t}$.
So we see that it does not necessarily make sense to talk about the shortest nonzero vector among a collection of lattices, but it will make sense to consider a ``max-min problem'' of this type
(Hermite's constant \eqref{Hermite's constant} below).
}
\hfill $\square$
\end{example}
For each dimension $d$, we define {\bf Hermite's constant} as follows:
\begin{equation}\label{Hermite's constant}
\gamma_d := \max \left\{ \lambda_1({\mathcal L})^2 \bigm | {\mathcal L} \text{ is a full-rank lattice in $\mathbb{R}^d$, with } \det {\mathcal L} =1 \right\}.
\end{equation}
In words, Hermite's constant is retrieved by varying over all normalized lattices in $\mathbb{R}^d$, which have determinant $1$, picking out the smallest squared norm of any nonzero vector in each lattice, and then
taking the maximum of these smallest norms. In a later chapter, on sphere packings, we will see an interesting interpretation of Hermite's constant in terms of the densest lattice packing of spheres.
We next give a simple bound, in Theorem \ref{First successive minima bound} below, for the shortest nonzero
vector in a lattice and hence for Hermite's constant. But first we need to give
a simple lower bound for the volume of the unit ball, in Lemma \ref{volume bound for the ball}.
Curiously, Hermite's constant $\gamma_d$ is only known
precisely for $1 \leq d \leq 8$, and $d=24$, as of this writing.
\bigskip
\begin{lem}\label{volume bound for the ball}
\[
\vol B^d(r) \geq \left( \frac{2r}{\sqrt d} \right)^d.
\]
\end{lem}
\begin{proof}
The cube $C:= \left\{ x\in \mathbb{R}^d \bigm |
\text{ all } |x_k| \leq \frac{r}{\sqrt d} \right\}$ is contained in the ball $B^d(r)$:
if $x \in C$ then $\sum_{k=1}^d x_k^2 \leq d \left( \frac{r}{\sqrt d} \right)^2 = r^2$. So the volume of the ball $B^d(r)$ is greater than the volume of the cube, which is equal to $\left( \frac{2r}{\sqrt d} \right)^d$.
\end{proof}
\bigskip
The following result of Minkowski gives a bound for the shortest nonzero vector in a lattice.
\begin{thm}[Minkowski] \label{First successive minima bound}
Suppose that ${\mathcal L} \subset \mathbb{R}^d$ is a full-rank lattice. Then the shortest nonzero vector $v \in {\mathcal L}$ satisfies
\begin{equation}\label{shortest vector in a lattice bound}
\|v\| \leq \sqrt{d} (\det {\mathcal L})^{\frac{1}{d}}.
\end{equation}
Equivalently, we will often write
\[
\lambda_1({\mathcal L}) \leq \sqrt{d} (\det {\mathcal L})^{\frac{1}{d}}.
\]
\end{thm}
\begin{proof}
The idea is to apply Minkowski's convex body Theorem \ref{Minkowski convex body Theorem for L}
to a ball of sufficiently large radius.
Let $r:= \lambda_1({\mathcal L})$ be the length of the shortest nonzero vector in ${\mathcal L}$, and consider the
ball $B^d(r)$ of radius $r$. By definition, $B^d(r)$ does not contain any lattice points of ${\mathcal L}$ in its interior.
So by Minkowski's convex body Theorem, and Lemma \ref{volume bound for the ball},
\[
\left( \frac{2 \lambda_1({\mathcal L})}{\sqrt d} \right)^d
\leq
\vol B^d(r)
\leq
2^d \det {\mathcal L}.
\]
It follows that $\lambda_1({\mathcal L}) \leq \sqrt{d} \left( \det {\mathcal L} \right)^{\frac{1}{d}}$, proving the claim. \\
\end{proof}
Despite the bound \eqref{shortest vector in a lattice bound} on the shortest nonzero vector in a lattice, there are currently no known efficient algorithms to find such a vector for an arbitrary lattice, and it is thought to be one of the most difficult problems we face today. In practice, researchers often use the LLL algorithm
to find a `relatively short' vector in a given lattice, and the same algorithm even finds a relatively short basis for ${\mathcal L}$.
We already have enough knowledge to relate the length of a shortest nonzero vector of a lattice ${\mathcal L}$ to the length of a
shortest nonzero vector of its dual lattice ${\mathcal L}^*$, as follows.
\begin{cor}
Let ${\mathcal L}\subset \mathbb{R}^d$ be a full-rank lattice, and let ${\mathcal L}^*$ be its dual lattice. Then
\[
\lambda_1({\mathcal L}) \lambda_1({\mathcal L}^*) \leq d.
\]
\end{cor}
\begin{proof}
By Minkowski's bound, namely Theorem \ref{First successive minima bound}, applied to both ${\mathcal L}$ and ${\mathcal L}^*$, we have:
\[
\lambda_1({\mathcal L}) \lambda_1({\mathcal L}^*) \leq \sqrt{d} (\det {\mathcal L})^{\frac{1}{d}} \sqrt{d} (\det {\mathcal L}^*)^{\frac{1}{d}} = d,
\]
using the relation $(\det {\mathcal L})( \det {\mathcal L}^*) = 1$.
\end{proof}
Such relations are called {\bf transference theorems}, as they can transfer the complexity of computing a lattice parameter to the complexity of computing a (perhaps different) parameter in the dual lattice.
While we may not know explicitly all of the short vectors in a given lattice, it is often still useful to construct
an ellipsoid that is based on the successive minima of a lattice, as we do below. In the spirit of reviewing basic concepts from Linear Algebra, an
{\bf ellipsoid boundary} \index{ellipsoid}
centered at the origin is defined by the $(d-1)$-dimensional body
\begin{equation} \label{ellipsoid}
\left\{ x\in \mathbb{R}^d \bigm | \sum_{j=1}^d \frac{{\langle x, b_j\rangle}^2}{c_j^2} =1 \right\},
\end{equation}
for some fixed orthonormal basis
$\{ b_1, \dots, b_d \}$ of $\mathbb{R}^d$. Here the vectors $b_j$ are called the {\bf principal axes} of the ellipsoid, and
the $c_j$'s are the lengths along the principal axes of the ellipsoid.
A more geometric way of
defining an ellipsoid (which turns out to be equivalent to our definition above) is attained
by applying a linear transformation $M$ to the unit sphere
$S^{d-1} \subset \mathbb{R}^d$ (Exercise \ref{Ellipsoid problem}). For the next couple of lemmas, we follow the approach taken by Regev.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{Ellipsoid}
\end{center}
\caption{An ellipsoid in $\mathbb{R}^3$.} \label{Ellipsoid}
\end{figure}
\bigskip
\begin{lem}\label{Empty Ellipsoid}
\rm{
Corresponding to the successive minima of a full-rank lattice ${\mathcal L}$, we have
$d$ linearly independent vectors
$v_1, \dots, v_d$, so that by definition $\| v_k \| := \lambda_k({\mathcal L})$. We apply the
Gram-Schmidt algorithm to this set of vectors $\{v_1, \dots, v_d\}$, obtaining a corresponding orthonormal basis
$\{b_1, \dots, b_d\}$ for $\mathbb{R}^d$.
Now, we define the following open {\bf ellipsoid} by:
\begin{equation}\label{open ellipsoid}
E:=\left\{ x\in \mathbb{R}^d \bigm | \sum_{k=1}^d \frac{{\langle x, b_k \rangle}^2}{{\lambda_k}^2}
< 1
\right\},
\end{equation}
whose axes are the $b_k$'s, and whose radii are the $\lambda_k:= \lambda_k({\mathcal L})$.
We claim that $E$ does not contain any lattice points of ${\mathcal L}$.
}
\end{lem}
\begin{proof}
We fix any vector $v \in {\mathcal L}$.
Let $1\leq k \leq d$ be the maximal index such that $\lambda_k({\mathcal L}) \leq \|v\|$.
We may write $v = \sum_{j=1}^d \langle v, b_j \rangle b_j$, so that
$\|v\|^2 = \sum_{j=1}^d {\langle v, b_j \rangle}^2$.
Now $v$ must lie in $\text{span}\{v_1, \dots v_k\} = \text{span}\{b_1, \dots b_k\}$, for some $1\leq k\leq d$.
Hence we may write $v = \sum_{j=1}^d \langle v, b_j \rangle b_j = \sum_{j=1}^k \langle v, b_j \rangle b_j $,
so that
$\|v\|^2 = \sum_{j=1}^k | \langle v, b_j \rangle |^2$.
We now check if $v$ is contained in $E$:
\[
\sum_{j=1}^d \frac{{\langle v, b_j \rangle}^2}{{\lambda_j}^2} =
\sum_{j=1}^k \frac{{\langle v, b_j \rangle}^2}{{\lambda_j}^2} \geq
\frac{1}{ {\lambda_k}^2 } \sum_{j=1}^k {\langle v, b_j \rangle}^2
= \frac{ \|v\|^2 }{ {\lambda_k}^2 } \geq 1,
\]
so that $v\not\in E$.
\end{proof}
More generally, we have the following refinement of Theorem \ref{First successive minima bound}, which
gives us a bound on the first $d$ shortest (nonzero) vectors in a lattice.
\begin{thm}[Minkowski's second theorem] \label{successive minima bound}
The successive minima of a full-rank lattice ${\mathcal L}$ enjoy the property:
\[
\Big( \| \lambda_1({\mathcal L}) \| \cdots \|\lambda_d({\mathcal L}) \| \Big)^{\frac{1}{d}}
\leq \sqrt{d} \Big(\det {\mathcal L}\Big)^{\frac{1}{d}}.
\]
\end{thm}
\begin{proof} Using Lemma \ref{Empty Ellipsoid}, the ellipsoid $E$ contains no lattice points belonging to
${\mathcal L}$, so that by Minkowski's convex body Theorem, we have $\vol E \leq 2^d \det {\mathcal L}$.
We also know that
\[
\vol E = \left( \prod_{j=1}^d \lambda_j \right) \vol B_1 \geq
\left( \prod_{j=1}^d \lambda_j \right) \left(\frac{2}{\sqrt{d}} \right)^d.
\]
Altogether, we have
\[
2^d \det {\mathcal L} \geq \vol E \geq \left( \prod_{j=1}^d \lambda_j \right) \left(\frac{2}{\sqrt{d}} \right)^d,
\]
arriving at the desired inequality.
\end{proof}
We notice that
$ \Big(\| \lambda_1({\mathcal L}) \| \cdots \|\lambda_d({\mathcal L}) \| \Big)^{\frac{1}{d}}
\geq \| \lambda_1({\mathcal L}) \|$, because $\| \lambda_1({\mathcal L}) \| \leq \| \lambda_k({\mathcal L}) \| $
for all indices $1 < k \leq d$. We therefore see that Theorem \ref{successive minima bound} is indeed a refinement
of Theorem \ref{First successive minima bound}.
\medskip
\begin{example}
\rm{
The $E_8$ lattice is defined by
\begin{equation} \label{E_8}
E_8 := \left\{ (x_1, x_2, \cdots x_8) \in \mathbb{Z}^8 \cup \Big(\mathbb{Z} + \frac{1}{2} \Big)^8 \ \bigm | \ \sum_{k=1}^8 x_k
\equiv 0 \mod 2
\right\}.
\end{equation}
It turns out that the $E_8$ lattice gives the optimal solution to the sphere packing problem, as well as the optimal solution for the kissing number problem in $\mathbb{R}^8$.
}
\hfill $\square$
\end{example}
\begin{comment}
\section{The packing radius, and the covering radius}
Suppose we are given a convex body $K \subset \mathbb{R}^d$, containing the origin, and a full-rank lattice ${\mathcal L}\subset \mathbb{R}^d$.
We define
the {\bf packing radius of the lattice ${\mathcal L}$}, relative to $K$, written as $\mu(K, {\mathcal L})$,
as the largest $r>0$ such that
$(r K + l_1) \cap (r K + l_2) \not= \phi$, for all $l_1, l_2 \in {\mathcal L}$. In other words, the packing radius is the largest dilate of $K$ such that after translating $K$ by all elements of the lattice ${\mathcal L}$, we do not have any overlapping bodies.
More compactly, we may also give the following description for the covering radius:
\begin{align}
\mu(K, \mathcal L)
&:= \min \{ r \geq 0 \mid r K + \mathcal L = \mathbb{R}^d \} \\
&= \min \{ r \geq 0 \mid r K + \mathcal L \text{ is a covering} \}
\end{align}
In a of dual manner, the packing radius of $K \subset \mathbb{R}^d$ with respect to a lattice ${\mathcal L}$ is defined by
In a somewhat dual fashion, we also define the {\bf covering radius of the lattice ${\mathcal L}$}, relative to $K$, as the
largest $r>0$ such that every point
When $K:= B(r)$, the open ball of radius $r$, it is traditional to omit $K$ in the notation, and
we simply write the packing radius in this case as $\mu(K, {\mathcal L}):= \mu({\mathcal L})$. In words, when $K$ is an open ball,
the packing radius is the largest $r > 0$ such that the collection of
open balls $B(v, r)$ centered at all lattice points of ${\mathcal L}$ do not intersect.
It is immediate to see that the packing radius of a lattice equals precisely half the minimum distance.
The covering radius $\mu({\mathcal L})$ is defined by the smallest $\mu >0$ s.t. the collection of open balls
centered at all lattice points of ${\mathcal L}$ cover $\mathbb{R}^d$.
The packing radius $\mu(L)$ equals the inradius of $V(L)$, the covering radius............. equals
the circumradius of $V(L)$, and the quantizer constant is $G(L) = (\det {\mathcal L})^{1+ 2/n} \int_{V(L)} \|x\|^2 dx$.
\end{comment}
\section{Hermite normal form}
We call a lattice ${\mathcal L}$ an {\bf integral lattice} if ${\mathcal L} \subset \mathbb{Z}^d$.
Further, we may recall that any lattice ${\mathcal L} \subset \mathbb{R}^d$ has infinitely many bases, so it may seem impossible at first to associate a single matrix with a given lattice. However, there is an elegant way to do this, as follows.
\begin{example}\label{first ex. of HNF}
{\rm
Suppose we are given a lattice ${\mathcal L}$ as the integral span of the vectors
\[
v_1:= \icol{3\{\bf 1}}, v_2:= \icol{-2\\ \ 2},
\]
which clearly has determinant $8$.
Then any integer linear combinations of $v_1$ and $v_2$ is still in ${\mathcal L}$. In particular, mimicking Gaussian elimination, we place $v_1$ and $v_2$ as rows of a matrix, and row-reduce over the integers:
\begin{align*}
\begin{pmatrix}
\ \ 3 & \ 1 \\
-2 & \ 2
\end{pmatrix}
\rightarrow
\begin{pmatrix}
3 & \ 1 \\
1 & \ 3
\end{pmatrix}
\rightarrow
\begin{pmatrix}
0 & \ -8 \\
1 & \ \ \ 3
\end{pmatrix}
\rightarrow
\begin{pmatrix}
1 & \ \ 3 \\
0 & -8
\end{pmatrix}
\rightarrow
\begin{pmatrix}
1 & 3 \\
0 & 8
\end{pmatrix},
\end{align*}
where at each step we performed row operations (over $\mathbb{Z}$) that did not change the lattice. Hence we have a reduced basis for ${\mathcal L}$, consisting of $\icol{1\\3}$ and $\icol{0\\8}$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.2in]{HNF}
\end{center}
\caption{
The lattice ${\mathcal L}$ of Example \ref{first ex. of HNF}, depicted by the bold green points, and showing the
original basis $\{ v_1, v_2\}$ of ${\mathcal L}$, and the Hermite-reduced basis of ${\mathcal L}$
}
\label{HNF.pic}
\end{figure}
We notice that the resulting matrix is upper-triangular, with positive integers on the diagonal, nonnegative integers elsewhere, and in each column the diagonal element is the largest element in that column.
There is another way to interpret the matrix reductions above, by using unimodular matrices, as follows. The first reduction step can be accomplished by the multiplication on the left by a unimodular matrix:
\[
\begin{pmatrix}
1 & \ 0 \\
1 & \ 1
\end{pmatrix}
\begin{pmatrix}
\ 3 & \ 1 \\
-2 & \ 2
\end{pmatrix}
=
\begin{pmatrix}
3 & \ 1 \\
1 & \ 3
\end{pmatrix}
\]
Similarly, each step in the reduction process can be interpreted by multiplying on the left by some new unimodular matrix, so that at the end of the process we have a product of unimodular matrices times
our original matrix $\begin{pmatrix}
\ 3 & \ 1 \\
-2 & \ 2
\end{pmatrix}$. Because a product of unimodular matrices is yet another unimodular matrix, we can see that we arrived at a reduction of the form:
\[
U
\begin{pmatrix}
\ 3 & \ 1 \\
-2 & \ 2
\end{pmatrix}
=
\begin{pmatrix}
1 & \ 3 \\
0 & 8
\end{pmatrix},
\]
where $U$ is a unimodular matrix.
\hfill $\square$
}
\end{example}
The point of Example \ref{first ex. of HNF} is that a similar matrix reduction persists for all integer lattices, culminating in the following result, which just hinges on the fact that $\mathbb{Z}$ has a division algorithm.
\begin{thm} \label{theorem.HNF}
Given an invertible integer $d\times d$ matrix $M$, there exists a unimodular matrix $U$ with
$UM = H$, such that $H$ satisfies the following conditions:
\begin{enumerate}
\item $[H]_{i, j} = 0$ if $i>j$.
\item $[H]_{i, i} > 0$, for each $1\leq i \leq d$.
\item $0 \leq [H]_{i, j} < [H]_{i, i} $, for each $i >j $. \label{third property}
\end{enumerate}
Property \ref{third property} tells us that each diagonal element $[H]_{i, i}$ in the $i$'th column of $H$ is the largest element in the $i$'th column.
Moreover, the matrix $H$ is the unique integer matrix that satisfies the above conditions.
\hfill $\square$
\end{thm}
The matrix $H$ in Theorem \ref{theorem.HNF} is called the {\bf Hermite normal form} of $M$.
To associate a unique matrix to a given integral full-rank lattice ${\mathcal L} \subset \mathbb{R}^d$, we first choose
any basis of ${\mathcal L}$, and we then construct a $d\times d$ integer matrix $M$ whose rows are the basis vectors that we chose. We then apply Theorem \ref{theorem.HNF} to $M$, arriving at an integer matrix $H$ whose rows are another basis of ${\mathcal L}$, called the {\bf Hermite-reduced basis}.
\begin{cor}
There is a one-to-one correspondence between full-rank integral lattices in $\mathbb{R}^d$ and
integer $d \times d$ matrices in their Hermite Normal Form.
\hfill $\square$
\end{cor}
\section{The Voronoi cell of a lattice}
The {\bf Voronoi cell} of a lattice ${\mathcal L}$, at the origin, is defined by
\begin{equation}
\text{Vor}_0({\mathcal L}) := \left\{ x \in \mathbb{R}^d \bigm | \|x\| \leq \|x - v\|, \ \text{ for all } v \in {\mathcal L} \right\}.
\end{equation}
In other words, the Voronoi cell $\text{Vor}_0({\mathcal L})$ of a lattice ${\mathcal L}$ is the set of all point in space that are closer to the origin than to any other lattice point in ${\mathcal L}$. Because the origin wins the battle of minimizing this particular distance function, it is also possible to construct the Voronoi cell by using half-spaces. Namely, for each $v\in {\mathcal L}$, we define the half-space
\[
H_v:= \left\{ x \in \mathbb{R}^d \bigm | \langle x, v \rangle \leq \tfrac{1}{2} \|v\| \right\},
\]
and we observe that the Voronoi cell may also be given by
\[
{\rm Vor}_0({\mathcal L}) = \bigcap_{v\in {\mathcal L}- \{0\}} H_v,
\]
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.2in]{VoronoiConstruction}
\end{center}
\caption{Top left: a sublattice ${\mathcal L}$ of $\mathbb{Z}^2$, of index $3$. Top right: $v \in {\mathcal L}$ is one of the $6$ relevant vectors, with its corresponding half-plane $H_v$, helping to define the Voronoi cell at the origin.
Bottom: The Voronoi cell $\text{Vor}_0({\mathcal L})$, \index{Voronoi cell} a symmetric hexagon of area $3$, with its $6$ relevant (heavy blue) lattice points of ${\mathcal L}$.
}
\label{VoronoiConstruction}
\end{figure}
as drawn in Figure \ref{VoronoiConstruction}.
It is easy to observe that the Voronoi cell of a lattice is symmetric about the origin, convex, and compact (Exercise \ref{facts about Vor cell}). So we may expect that Minkowski's theorems apply to $\text{Vor}_0({\mathcal L})$, as we see in the proof of Lemma \ref{basic Voronoi lemma} below.
It's also useful to define an analogous Voronoi cell located at each lattice point $m \in {\mathcal L}$:
\begin{equation}
\text{Vor}_m({\mathcal L}) := \left\{ x \in \mathbb{R}^d \bigm | \|x - m \| \leq \|x - v\|, \ \text{ for all } v \in {\mathcal L} \right\}.
\end{equation}
A moment's thought (but this is good practice - Exercise \ref{translating the Voronoi cell around})
reveals that a translation of the Voronoi cell at the origin is exactly the Voronoi cell at another lattice point of ${\mathcal L}$, namely:
\begin{equation} \label{translated Voronoi cells}
\text{Vor}_0({\mathcal L}) + m = \text{Vor}_m({\mathcal L}).
\end{equation}
\begin{lem} \label{basic Voronoi lemma}
Given a full-rank lattice ${\mathcal L}\subset \mathbb{R}^d$, whose Voronoi cell at the origin is $K$, we have:
\begin{enumerate}[(a)]
\item $K$ tiles $\mathbb{R}^d$ by translations with ${\mathcal L}$. \label{part 1 of Voronoi}
\item
$
\vol (K) = \det {\mathcal L}. \label{part 2 of Voronoi}
$
\end{enumerate}
\end{lem}
\begin{proof}
Part \ref{part 1 of Voronoi} follows from the observation that
any $x \in \mathbb{R}^d$, there exists a lattice point $m \in {\mathcal L}$ that is at least as close to $x$ as it is to any other lattice point of ${\mathcal L}$.
In other words, $ \|x - m\| \leq \|x - v\|, \forall v\in {\mathcal L}$, and so
$x \in \text{Vor}_m({\mathcal L})$.
From \eqref{translated Voronoi cells} we see that $x$ is covered by the translate $\text{Vor}_0({\mathcal L}) + m$. It's also clear that as $n$ varies over ${\mathcal L}$, all of the interiors of the translates $\text{Vor}_0({\mathcal L}) + n$ are disjoint, so that
$K:= \text{Vor}_0({\mathcal L})$ tiles $\mathbb{R}^d$ by translations with ${\mathcal L}$.
To prove part \ref{part 2 of Voronoi}, we let $B:= 2K$.
By Theorem \ref{thm:extremal bodies} (regarding extremal bodies), we know that
$\frac{1}{2}B = K$ tiles $\mathbb{R}^d$ with the lattice ${\mathcal L}$ if and only if $ \vol(B) = 2^d \det {\mathcal L}$. Since
\ref{part 1 of Voronoi} tells us that
$K=\frac{1}{2}B$ tiles with the lattice ${\mathcal L}$,
we see that
$\vol K = \vol\Big(\frac{1}{2}B\Big) = \frac{1}{2^d} \vol B = \det {\mathcal L}$.
\end{proof}
The proof above shows that the Voronoi cell of ${\mathcal L}$ is also an extremal body for ${\mathcal L}$, according to Theorem \ref{thm:extremal bodies}.
\medskip
\begin{example}\label{D_n lattices}
\rm{
The $D_n$ lattice is defined by
\[
D_n:= \left\{ x\in \mathbb{Z}^n \bigm | \sum_{k=1}^n x_k \equiv 0 \mod 2 \right\}.
\]
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{D2andD3}
\end{center}
\caption{Left: the $D_2$ lattice. Right: the $12$ shortest nonzero vectors of the $D_3$ lattice, inscribed in the cube $[-1, 1]^3$.}
\label{The $D_2$ and $D_3$ lattices}
\end{figure}
In $\mathbb{R}^4$, the $D_4$ lattice turns out to be a fascinating object of study.
The Voronoi cell $\text{Vor}_0(D_4)$ is called the {\bf $24$-cell}, \index{$24$-cell}
and is depicted in Figure \ref{24-cell}. It is a $4$-dimensional polytope with some wonderful properties - for example, it is one of the few polytopes that is self-dual. It is also an example of a polytope ${\mathcal P}$ in the lowest possible dimension $d$ (namely $d=4$) such that ${\mathcal P}$ tiles $\mathbb{R}^d$ by translations, and yet ${\mathcal P}$ is not a zonotope.
By Lemma \ref{lemma:lattice defined by congruence},
we see that $\det D_n = 2$. }
The lattice $D_n$ is often called the ``checkerboard'' lattice, because $\det D_n = 2$ means there are exactly two cosets
in $\mathbb{Z}^d / D_n$. Finally, the dual lattice $D_n^*$ is equal to the lattice
\[
\\mathbb{Z}^d \cup \left( \mathbb{Z}^d + \left( \tfrac{1}{2}, \cdots, \tfrac{1}{2} \right)^T \right),
\]
which we leave for the pleasure of the reader (Exercise \ref{dual of D_n}).
\hfill $\square$
\end{example}
A fascinating open problem is the Voronoi conjecture, named after
the Ukrainian mathematician Georgy Voronoi, who formulated it in 1908.
Two polytopes ${\mathcal P}, Q$ are called {\bf affinely equivalent} \index{affinely equivalent} if
${\mathcal P} = M(Q) + v$, where $M \in GL_d(\mathbb{R})$, and $v\in \mathbb{R}^d$.
\medskip
\begin{conjecture}[Voronoi] \index{Voronoi conjecture}
A polytope ${\mathcal P}$ tiles $\mathbb{R}^d$ by translations if and only if ${\mathcal P}$ is the Voronoi cell of some lattice ${\mathcal L}$, or ${\mathcal P}$ is affinely equivalent
to such a Voronoi cell.
\end{conjecture}
The main difficulty in the Voronoi conjecture is the apriori search among all of the (infinitely many) possible affinely equivalent images of such a Voronoi cell.
\medskip
\begin{example}
\rm{
For the lattice $A_n \subset \mathbb{R}^{n+1}$ defined in Example \ref{A_d example}, its Voronoi cell turns out to have beautiful and important properties: $A_2 \subset \mathbb{R}^3$ is a hexagon, $A_3 \subset \mathbb{R}^4$ is a truncated octahedron (one of the Fedorov solids), and so on (see Conway and Sloane \cite{ConwaySloan.book}).
}
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{24-cell}
\end{center}
\caption{The Voronoi cell of the $D_4$ lattice in $\mathbb{R}^4$, known as the $24$-cell.} \label{24-cell}
\end{figure}
\bigskip
\section{Quadratic forms and lattices} \label{quadratic forms}
The study of lattices is in a strong sense equivalent to the study of positive definite quadratic forms,
over integer point
inputs, for the following
simple reason. Any positive definite quadratic form $f:\mathbb{R}^d \rightarrow \mathbb{R}$
is defined by $f(x):= x^T A x$, where
$A$ is a positive definite matrix, so the image of the integer lattice under $f$ is
\[
\{ x^T A x \mid x \in \mathbb{Z}^d\}.
\]
On the other hand, any full-rank lattice in $\mathbb{R}^d$ is given by
${\mathcal L} := M(\mathbb{Z}^d)$, for some real non-singular matrix $M$. By definition, this implies that the square of the norm of any vector
in ${\mathcal L}$ has the following shape: $\| v\|^2 = v^T v = x^T M^T M x$, for some $x \in \mathbb{Z}^d$. We notice that
$M^T M$ in the last identity is positive definite.
We may summarize this discussion as follows. Given any lattice ${\mathcal L}:= M(\mathbb{Z}^d)$, we have
\[
\left\{ \| v\|^2 \bigm | v \in {\mathcal L} \} = \{ x^T A x \bigm | x \in \mathbb{Z}^d \right\},
\]
where $A:= M^T M$ is positive definite.
So the distribution of the (squared) norms of all vectors in a given lattice is equivalent to the image of $\mathbb{Z}^d$
under a positive
definite quadratic form.
Interestingly, despite this equivalence, for an arbitrary given lattice ${\mathcal L}$ it is not known in general whether the knowledge of
the norms of all vectors in ${\mathcal L}$ uniquely determines the lattice ${\mathcal L}$. In very small dimensions it is true,
but for dimensions $\geq 4$ there are some counterexamples due to Alexander Schiemann (\cite{Schiemann1}, \cite{Schiemann2}).
The above equivalence between lattices in $\mathbb{R}^d$ and quadratic forms is straightforward but often useful, because it allows both algebraic and analytic methods to come to bear on important problems involving lattices.
Gauss initiated the systematic study of finding the minimum value of positive definite, binary quadratic forms $f(x, y) := a x^2 + 2b xy + cy^2$, over
all integer inputs $(x, y) \in \mathbb{Z}^2$. Gauss' theory is also known as a reduction theory for positive definite binary quadratic forms, and is now a popular topic that can be found in many standard Number Theory books. By the discussion of this short section,
it is clear that minimizing positive definite quadratic forms is essentially equivalent
to finding a vector of smallest nonzero length in a lattice.
So far we worked with one lattice at a time, but it turns out to be fruitful to work with infinite sets of lattices simultaneously.
\begin{thm}[Mahler] \label{Mahler}
Fix $\rho >0, C>0$. Then any infinite sequence of lattices
${\mathcal L} \subset \mathbb{R}^d$ such that
\[
\min \left\{ \|x\| \bigm | x \in {\mathcal L}-\{0\} \right\} \geq \rho, \text{ and } \det {\mathcal L} \leq C,
\]
has an infinite convergent subsequence of lattices.
\end{thm}
In other words, Mahler realized that among all lattices of volume $1$,
if a sequence of lattices diverges, then it must be true that the lengths of the shortest nonzero vectors
of these lattices tend to zero.
To complete the story, we should define what it means for a sequence of lattices $\{ {\mathcal L}_n \}_{n=1}^\infty$ to converge to a fixed lattice $L$. One way to define this convergence is to say that there exists a
sequence of bases $\beta_n$ of the lattices ${\mathcal L}_n$ that converge to a basis $\beta$ of $L$, in the sense that the $j$'th basis vector of $\beta_n$ converges to the $j$'th basis vector of $\beta$.
\bigskip
\section*{Notes}
\begin{enumerate}[(a)]
\item Kurt Mahler \index{Mahler, Kurt} was one of the main contributors to the development of the Geometry of Numbers.
His Theorem \ref{Mahler} is often called Mahler's compactness theorem (also known as Mahler's selection theorem).
\item There is a well-known meme in Mathematics: ``Can one hear the shape of a drum?", which is the title of Mark Kac's famous paper regarding the desire to discern the shape of a drum from its `frequencies'.
An analogous question for lattices, studied by John Conway, is
``which properties of quadratic forms are determined by their representation numbers?''.
For further reading, there is the lovely little book by Conway called ``The sensual quadratic form'', which draws connections between quadratic forms and
many different fields of Mathematics \cite{Conway.Book.SensualForm}.
Of course, no library is complete without the important and biblical ``Sphere Packings, Lattices and Groups", by John H. Conway and Neil Sloane \cite{ConwaySloan.book}.
\item The idea of periodicity, as embodied by any lattice in $\mathbb{R}^d$, also occurs on other manifolds, besides Euclidean space. If we consider a closed geodesic on a manifold, then it's intuitively clear that
as we flow along that geodesic, we have a periodic orbit along that geodesic.
One important family of manifolds where this type of periodicity occurs naturally is the family of Hyperbolic manifolds. Following the philosophy that `if we have periodicity, then we have Fourier-like series',
we discover that there is also an hyperbolic analogue of the Poisson summation formula, known as the Selberg trace formula, and this type of number theory has proved extremely fruitful.
\item A strong bound for Hermite's constant in dimension $d$ was given by Blichfeldt \cite{Blichfeldt1}: \index{Blichfeldt}
\[
\gamma_d \leq \left( \frac{2}{\pi} \right) \Gamma \left( 2 + \frac{ d}{2} \right)^{\frac{2}{d}}.
\]
\item Related to the Hermite Normal Form is another extremely important reduction, called the Smith Normal Form, which we will cover in depth in a future version of this book (see \cite{MorrisNewman}).
\item The family of diagonal matrices in Example \ref{a curve in the space of lattice} is very important in the study of homogeneous dynamics, because it acts by multiplication on the left,
on the space of all lattices that have $\det {\mathcal L} = 1$. This fascinating action is sometimes
called the ``modular flow'', and was studied intensively by Etienne Ghys.
A beautiful result in this direction is that the periodic orbits of the modular flow are in bijection with the conjugacy classes of hyperbolic elements in the modular group $SL_2(\mathbb{Z})$, and furthermore that
these periodic orbits
produce incredible knots in the complement of the trefoil knot.
\item
It is clear that because lattices offer a very natural way to discretize $\mathbb{R}^d$, they continue to be of paramount importance to modern research. In particular, the theory of modular forms, with linear (Hecke) operators that are defined using lattices and their fixed finite index sublattices, is crucial for modern number theory. Euclidean lattices are also the bread-and-butter of crystallographers.
\end{enumerate}
\bigskip \bigskip \bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
My dear Watson, once you eliminate the impossible, then whatever remains - no matter how improbable - must be the truth.
-- Arthur Conan Doyle (in his book Sherlock Holmes)
\end{quote}
\medskip
\begin{prob} \label{Dual of the integer lattice} $\clubsuit$
We say that a lattice ${\mathcal L}$ is {\bf self dual} if ${\mathcal L}^* = {\mathcal L}$.
\begin{enumerate}[(a)]
\item Prove that the integer lattice is self dual: $(\mathbb{Z}^d)^* = \mathbb{Z}^d$.
\item Prove that for any lattice ${\mathcal L} \subset \mathbb{R}^d$, we have $({\mathcal L}^*)^* = {\mathcal L}$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{distance between hyperplanes}
Show that the distance $\delta$ between any two parallel hyperplanes, described by
$c_1 x_1 + \cdots + c_d x_d = k_1$ and
$c_1 x_1 + \cdots + c_d x_d = k_2$, is equal to:
\begin{equation*}
\delta = \frac{ |k_1 - k_2|}{\sqrt{ c_1^2 + \cdots + c_d^2}}.
\end{equation*}
\end{prob}
\medskip
\begin{prob}
Suppose we are given a full-rank sublattice of the integer lattice: ${\mathcal L} \subset \mathbb{Z}^d$.
Prove that there is point of ${\mathcal L}$ on the $x$-axis.
\end{prob}
\medskip
\begin{prob} \label{lattices in R^1} $\clubsuit$
Let ${\mathcal L}$ be a lattice in $\mathbb{R}^1$. Show that ${\mathcal L} = r\mathbb{Z}$ for some real number $r$.
\end{prob}
\medskip
\begin{prob} \label{dual of E^8}
Show that the $8$-dimensional lattice $E_8$, defined in \eqref{E_8}, is self-dual:
$(E_8)^* = E_8$.
\end{prob}
\medskip
\begin{prob} \label{matrix form for det of dual lattice}
Suppose we are given a rank $k$ lattice ${\mathcal L}\subset \mathbb{R}^d$, with $1\leq k \leq d$.
If $M$ is a basis matrix for ${\mathcal L}$, then prove that the matrix $ M(M^TM)^{-1}$ gives a basis for the dual lattice ${\mathcal L}^*$.
\end{prob}
\medskip
\begin{prob}
Show that for any two lattices $L, M\subset \mathbb{R}^d$, we have $L\subseteq M \iff M^* \subseteq L^*$.
\end{prob}
\medskip
\begin{prob} \label{dual of D_n}
Prove that we have the following description for the dual lattice of $D_n$:
\[
D_n^* = \mathbb{Z}^d \cup \left( \mathbb{Z}^d + \left( \tfrac{1}{2}, \cdots, \tfrac{1}{2} \right)^T \right).
\]
\end{prob}
\medskip
\begin{prob} \label{Eisenstein lattice}
The {\bf hexagonal lattice} is the $2$-dimensional lattice defined by
\[
{\mathcal L} := \{ m + n \omega \mid m,n \in \mathbb{Z}\}, \text{ where } \omega:= e^{2\pi i/3}.
\]
Prove that $\det {\mathcal L} = \frac{\sqrt 3}{2}$, and give a description of the dual lattice to the hexagonal lattice.
\end{prob}
\medskip
\begin{prob} [hard] \label{minimal lattice in R^2}
Show that the hexagonal lattice attains the minimal value for Hermite's constant
in $\mathbb{R}^2$, namely $\gamma_2^2 = \frac{2}{\sqrt{3}}$.
\end{prob}
\medskip
\begin{prob} \label{special basis in R^2}
Let ${\mathcal L} \subset \mathbb{R}^2$ be any rank $2$ lattice. Show that there exists a basis $\beta:= \{ v, w \}$ of ${\mathcal L}$ such that
the angle $\theta_\beta$ between $v$ and $w$ satisfies
\[
\frac{\pi}{3} \leq \theta_\beta \leq \frac{\pi}{2}.
\]
\end{prob}
\medskip
\begin{prob} \label{Hadamard's inequality, exercise}
Suppose that $M$ is a $d\times d$ matrix, all of whose $d^2$ elements are bounded by $B$.
Show that $|\det M| \leq B^d d^{\frac{d}{2}} $.
\end{prob}
(Hint: consider Hadamard's inequality \ref{Hadamard inequality})
Notes. It follows from this exercise that if all of the elements of $M$ are $\pm 1$, then
$|\det M| \leq d^{\frac{d}{2}} $. Such matrices are important in combinatorics and are called Hadamard matrices.
It is known that if $d > 2$, then Hadamard matrices can only possibly exist when $4 \mid d$. But for each $d = 4m$, it is not known whether a
$d\times d$ Hadamard matrix exists, except for very small cases.
\medskip
\begin{prob} $\clubsuit$ \label{basis for A_d}
Show that the following set of vectors is a basis for $A_d$:
\[
\left\{e_2 - e_1, \ e_3 - e_1, \cdots , \ e_d - e_1 \right\},
\]
where the $e_j$ are the standard basis vectors. Hence $A_d$ is a rank-$(d-1)$
sublattice of $\mathbb{Z}^d$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{character group}
Recall that $G_{\mathcal L}$ is the group of characters of the lattice ${\mathcal L}$, under the usual multiplication of complex numbers, and that the lattice ${\mathcal L}$ is a group under the usual operation of vector addition.
Show that they are isomoprhic as groups: $G_{\mathcal L} \simeq {\mathcal L}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Orthogonality.For.Characters.Of.A.sublattice}
\index{orthogonality relations for lattices}
Here we prove the {\bf orthogonality relations for characters of a lattice ${\mathcal L}$}. We will do it for any sublattice ${\mathcal L} \subset \mathbb{Z}^d$. Let $D$ be a fundamental parallelepiped for ${\mathcal L}$.
Using the notation in Exercise \ref{character group}, prove that for any two characters $\chi_a, \chi_b \in G_{\mathcal L}$,
we have:
\begin{equation}
\frac{1}{\det {\mathcal L}} \sum_{n \in D \cap \mathbb{Z}^d } \chi_a(n) \overline{\chi_b(n)} =
\begin{cases}
1 & \mbox{if } \chi_a = \chi_b \\
0 & \mbox{if not}.
\end{cases}
\end{equation}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Theorem: Automorphisms of lattices}
Prove Theorem \ref{Automorphisms of lattices}.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{fundamental domains}
Prove that any two fundamental parallelepipeds (as defined in the text) of ${\mathcal L}$, say $D_1$ and $D_2$, must be related to each other by an element of the unimodular group:
\[
D_1 = M(D_2),
\]
for some $M \in SL_d(\mathbb{Z})$.
\end{prob}
\medskip
\begin{prob} \label{number of integer sublattices of index n, R^2}
Let $f(n)$ be the number of distinct integer sublattices of index $n$ in $\mathbb{Z}^2$.
We recall from elementary number theory the function $\sigma(n) := \sum_{d | n} d$, the sum of the divisors of $n$.
Show that
\[
f(n) = \sigma(n).
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{equivalence between determinants of a sublattice}
Given a sublattice ${\mathcal L}\subset \mathbb{R}^d$ of rank $r$, show that our definition of its determinant,
namely $\det {\mathcal L} := \sqrt{M^T M}$, conincides with the Lebesgue measure of any of its fundamental parallelepipeds.
(Here $M$ is a $d\times r$ matrix whose columns are basis vectors of ${\mathcal L}$)
\end{prob}
\medskip
\begin{prob}
Show that a set of vectors $v_1, \dots, v_m \in \mathbb{R}^d$, where $1\leq m \leq d$, are linearly independent
$\iff$ their Gram matrix is nonsingular.
\end{prob}
\medskip
\begin{prob}
Prove that for any given lattice ${\mathcal L} \subset \mathbb{R}^2$, any two(nonzero) shortest linearly independent
vectors for ${\mathcal L}$ generate the lattice ${\mathcal L}$.
Note. \ As a reminder, the first two shortest nonzero vectors of ${\mathcal L}$ may have equal length.
We note that in dimensions $d \geq 5$, such a claim is false in general, as problem \ref{counterexample in RË5} below shows.
\end{prob}
\medskip
\begin{prob} \label{counterexample in RË5}
Find a lattice ${\mathcal L} \subset \mathbb{R}^5$ such that any set of five shortest nonzero vectors of ${\mathcal L}$ do not generate ${\mathcal L}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{hyperplane lattice}
Consider the {\bf discrete hyperplane} defined by:
\[
H:= \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =0 \right\},
\]
Show that $H$ is a sublattice of $\mathbb{Z}^d$, and has rank $d-1$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{tiling the integer lattice with hyperplanes}
\index{discrete hyperplane}
Suppose we are given a discrete hyperplane $H$, as in Exercise \ref{hyperplane lattice}.
\begin{enumerate}[(a)]
\item Prove there exists a vector $w\in \mathbb{R}^d$ such that
\[
\{ H+ kw \bigm | k \in \mathbb{Z} \} = \mathbb{Z}^d.
\]
\item Prove that there are no integer points strictly between $H$ and $H + w$.
\end{enumerate}
\end{prob}
Notes.
You may assume Bezout's identity. \index{Bezout's identity}
Namely, if $\gcd(c_1, \dots, c_d)=1$
then there exists
an integer vector $(m_1, \dots, m_d)$ such that $c_1 m_1 + \cdots + c_d m_d = 1$.
This exercise shows that we can tile the integer lattice with discrete
translates of a discrete hyperplane.
\medskip
\begin{prob}\label{Ellipsoid problem}
Here we give the details for \eqref{ellipsoid}, the definition of an ellipsoid in $\mathbb{R}^d$. Starting over again,
we fix an orthonormal basis $\{ b_1, \dots, b_d\}$ for $\mathbb{R}^d$, and
we define the following matrix:
\[
M := \begin{pmatrix} | & | & ... & | \\
c_1 b_1 & c_2 b_2 & ...& c_d b_d \\
| & | & ... & | \\
\end{pmatrix},
\]
where the $c_k$'s are positive scalars.
We now apply the linear transformation $M$ to the unit sphere $S^{d-1}:= \{ x \in \mathbb{R}^d \mid \| x \|^2 = 1\}$ in
$\mathbb{R}^d$, and we recall what this means. Now we define the $\text{Ellipsoid}_M:=
M(S^{d-1})$, a $(d-1)$-dimensional object. In the spirit of review, we recall the definition
$M(S^{d-1}) := \{ u \in \mathbb{R}^d \mid u = Mx, x \in S^{d-1} \}$.
\begin{enumerate}[(a)]
\item Show that
\begin{equation} \label{equation of ellipsoid}
\text{Ellipsoid}_M =
\left\{ x\in \mathbb{R}^d \bigm | \sum_{j=1}^d \frac{{\langle x, b_j\rangle}^2}{c_j^2} =1 \right\}.
\end{equation}
\item We recall that the unit ball in $\mathbb{R}^d$ is defined by
$B := \left\{ x \in \mathbb{R}^d \bigm | \| x \|^2 \leq 1\right\}$.
Show that for the open ellipsoid body $E$ (a $d$-dimensional object), as defined in \eqref{open ellipsoid}, we have the
$d$-dimensional volume formula:
\[
\vol(E) = \vol B \prod_{j=1}^d c_j.
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
We will use the equation \eqref{equation of ellipsoid} definition of an ellipsoid, from above.
We can extend the previous exercise in the following way. Let $A$ be {\bf any} $d \times d$ real matrix,
and look at the action of $A$ on the unit sphere
$S^{d-1} \subset \mathbb{R}^d$.
Suppose that $rank(A) = r$. Show:
(a) If $r = d$, then $A(S^{d-1})$ is a $d$-dimensional ellipsoid, defined by an equation of the form
\eqref{equation of ellipsoid}.
(b) If $r < d$, then $A(S^{d-1})$ is an $r$-dimensional ellipsoid.
\end{prob}
\medskip
\begin{prob} \label{square root of a matrix}
Suppose that $A$ is a positive definite, real matrix.
Solve for (i.e. characterize) all matrices $X$ that are the `square roots' of $A$:
\[
A = X^2.
\]
\end{prob}
\medskip
\begin{prob}
Suppose that a certain $2$-dimensional lattice ${\mathcal L}$ has a Gram matrix
\[
G := \begin{pmatrix}
\ 2 & -1 \\
-1 & \ 2 \end{pmatrix} .
\]
Reconstruct ${\mathcal L}$ (i.e. find a basis for ${\mathcal L}$), up to an orthogonal transformation.
\end{prob}
\medskip
\begin{prob}
Find a $2$ by $2$ matrix $M$ that enjoys one of the properties of a positive semidefinite
matrix, namely that $x^T M x \geq 0$, for all $x\in \mathbb{R}^2$, but such that $M$ is not symmetric.
\end{prob}
\medskip
\begin{prob} \label{exercise:2by2 positive definite matrix}
Show that any real $2$ by $2$ matrix $A$ is positive definite if and only if both
$\rm{trace}(A) >0$ and
$\det A > 0$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{facts about Vor cell}
Show that $\rm{Vor}_0({\mathcal L})$ is symmetric about the origin, convex, and compact.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{translating the Voronoi cell around}
Given a full rank lattice ${\mathcal L}\subset \mathbb{R}^d$, and any $m \in {\mathcal L}$, show that
\[
{\rm Vor}_0({\mathcal L}) + m = {\rm Vor}_m({\mathcal L}).
\]
\end{prob}
\medskip
\begin{prob} \label{Extension of Erdos to dimension d}
\rm{
(hard) Erd\"os' question, given in Exercise \ref{Erdos lattice partition problem}, possesses a natural extension to dimension $d$.
\begin{question}\label{tiling the lattice with translated sublattices}
Suppose that the integer lattice $\mathbb{Z}^d$ is partitioned into a disjoint union of a finite number of translates of integer
sublattices, say:
\[
\mathbb{Z}^d = \{ {\mathcal L}_1 +v_1 \} \cup \{ {\mathcal L}_2 + v_2 \} \cup \dots \cup \{ {\mathcal L}_N +v_N\}.
\]
Is it true that there are at least two integer sublattices, say ${\mathcal L}_j, {\mathcal L}_k$,
that enjoy the property that ${\mathcal L}_k = {\mathcal L}_j + w$, for some integer vector $w$?
\end{question}
Here we prove that in $\mathbb{R}^3$, Question \ref{tiling the lattice with translated sublattices}
has a negative answer. In particular, find a partition of $\mathbb{Z}^3$ into $4$ integer sublattices, such that no two of them are integer translates of one another. Using an easy extension to $d >3$, also show that the answer to the question above is `no', if $d \geq 3$.
Notes. Question \ref{tiling the lattice with translated sublattices} remains unsolved in dimension $d=2$ \cite{FeldmanProppRobins}.
}
\end{prob}
\chapter{The Fourier transform of a polytope via its vertex description: \\ The Brion theorems} \label{chapter.Brion}
\begin{quote}
``See in {\bf nature} the cylinder, the sphere, the cone.''
-- Paul C\'ezanne
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{Dodecahedron}
\end{center}
\caption{The Dodecahedron in $\mathbb{R}^3$, an example of a simple polytope.
In Exercise \ref{FT of a Dodecahedron}, we compute its Fourier-Laplace
transform by using Theorem \ref{brion, continuous form} below.} \label{Dodecahedron}
\end{figure}
\bigskip
\section{Intuition}
Here we introduce the basic tools for computing precise expressions for the Fourier transform of a polytope. To compute transforms here, we assume that we
are given the vertices of a polytope ${\mathcal P}$ , together with the local geometric information at each vertex of ${\mathcal P}$, namely its neighboring vertices
in ${\mathcal P} \subset \mathbb{R}^d$. It turns out that computing the Fourier-Laplace transform of the tangent cone at each vertex of ${\mathcal P}$ completely characterizes the Fourier transform of ${\mathcal P}$.
One of the basic results here, called the discrete version of Brion's Theorem (\ref{brion, discrete form}),
may be viewed as an extension of the finite geometric sum in dimension $1$, to sums in integer cones, in dimension $d$.
Some basic families of polytopes are introduced, including simple polytopes and their duals, which are simplicial polytopes. These families of polytopes play an important role in the development of Fourier analysis on polytopes.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{C60}
\end{center}
\caption{The C60 Carbon molecule, also known as a buckeyball, is another example of a simple polytope. The nickname ``buckeyball' came from Buckminster Fuller, who
used this molecule as a model for many other tensegrity structures.
\index{simple polytope}
(the graphic is used with permission from
Nanografi, at https://phys.org/news/2015-07-scientists-advance-tunable-carbon-capture-materials.html) } \label{C60}
\end{figure}
\bigskip
\section{Cones, simple polytopes, and simplicial polytopes}
One of the most important concepts in combinatorial geometry is the definition of a {\bf cone ${\mathcal K} \subset \mathbb{R}^d$, with an apex $v$}, defined by;
\begin{equation} \label{def of a cone}
{\mathcal K}:= \left\{ v+ \sum_{k=1}^N \lambda_k w_k \mid \lambda _k \geq 0 \right \}.
\end{equation}
The {\bf edge vectors} of ${\mathcal K}$ are those vectors among the $w_1, \dots, w_N$ (not necessarily all of them) which belong to the boundary
$\partial {\mathcal K}$ of ${\mathcal K}$.
A fun exercise is to show that the following two conditions are equivalent:
\begin{enumerate}[(a)]
\item A cone ${\mathcal K}$ has an apex at the origin.
\item ${\mathcal K}$ is a cone that enjoys the property $\lambda {\mathcal K} = {\mathcal K}$, for all $\lambda >0$.
\end{enumerate}
(Exercise \ref{cone equivalence}).
We note that according to definition \eqref{def of a cone}, an apex need not be unique - in Figure \ref{Cones, pointed and unpointed}, the cone on the left has a unique apex, while the cone on the right has infinitely many apices.
If the vectors $w_1, \dots, w_N$ span a $k$-dimensional subspace of $\mathbb{R}^d$, we say that the
cone ${\mathcal K}$ has {{\bf dimension $k$}.
When a $k$-dimensional cone ${\mathcal K} \subset \mathbb{R}^d$ has exactly $k$ linearly independent edge vectors $w_1, \dots w_k \in \mathbb{R}^d$,
we call such a cone a {\bf simplicial cone}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.6in]{Cones1}
\end{center}
\caption{The cone on the left is pointed, and has edges $w_1, w_2$. The cone on the right, with edges $w_1, w_2$, is also a half-space and it is not pointed.
\index{cones} } \label{Cones, pointed and unpointed}
\end{figure}
A {\bf pointed cone}
\index{cone, pointed}
is a cone ${\mathcal K}\subset \mathbb{R}^d$ with apex $v$, such that its edge vectors $w_1, \dots w_N$ are linearly independent. The following $4$ conditions give equivalent characterizations of a pointed cone~${\mathcal K}$:
\begin{enumerate}[(a)]
\item There exists a hyperplane $H$ such that $H\cap {\mathcal K} = v$.
\item The translated cone $C:= {\mathcal K}-v$, with apex at the origin, enjoys $C \cap (-C) = \{0\}$.
\item ${\mathcal K}$ has a unique apex.
\item ${\mathcal K}$ does not contain an entire line. \label{non-pointed cone contains a line}
\end{enumerate}
(Exercise \ref{pointed cone equivalence}). Part \ref{non-pointed cone contains a line} is equivalent to the statement that for a non-pointed cone ${\mathcal K}$,
there exists a vector $u\in \mathbb{R}^d$ such that ${\mathcal K} + u = {\mathcal K}$. We note that every cone has an apex, it's just that the apex may not be unique, for example when ${\mathcal K}$ is a half-space. All cones are unbounded regions, by definition, so some care will have to be taken when integrating over them. On the other hand, they are `almost linear', because for a cone with apex at the origin, we have
\[
x, y \in {\mathcal K} \ \implies \ x+y \in {\mathcal K}.
\]
This closure property, which does not exist for polytopes, makes cones extremely helpful in the analysis of polytopes (for example, Section \ref{section.Brianchon-Gram}).
An $n$-dimensional polytope ${\mathcal P}\subset \mathbb{R}^d$ is called a {\bf simplicial polytope} if every facet of ${\mathcal P}$ is a simplex. Equivalently:
\begin{enumerate}[(a)]
\item Each facet of ${\mathcal P}$ has exactly $n$ vertices.
\item Each $k$-dimensional face of ${\mathcal P}$ has exactly $k+1$ vertices, for $0\leq k \leq n-1$.
\end{enumerate}
It is a fun exercise to show that any simplicial cone is always a pointed cone (Exercise \ref{simplicial implies pointed}), but the converse is clearly false.
By contrast with the notion of a simplicial polytope, we have the following `dual' family of polytopes.
An $n$-dimensional polytope ${\mathcal P}\subset \mathbb{R}^d$ is called a {\bf simple} polytope if every vertex is contained in exactly $n$ edges of ${\mathcal P}$.
Equivalently:
\begin{enumerate}[(a)]
\item Each vertex of ${\mathcal P}$ is contained in exactly $n$ of its facets.
\item Each $k$-dimensional face of ${\mathcal P}$ is contained in exactly $d-k$ facets, for all $k \geq 0$.
\end{enumerate}
\medskip
\begin{example}
\rm{
Any $d$-dimensional simplex $\Delta$ is a simple polytope. In fact, any $k$-dimensional
face of the simplex $\Delta$ is also a simplex, and hence a simple polytope of lower dimension.
The $3$-dimensional dodecahedron, in Figure \ref{Dodecahedron}, is also a simple polytope. Its edge graph, which
is always a planar graph for a convex polytope, in this case consists of $20$ vertices, $30$ edges, and $12$ faces.
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
A $d$-dimensional simplex also happens to be a simplicial polytope. The $3$-dimensional icosahedron is a simplicial polytope.
}
\hfill $\square$
\end{example}
It is a nice exercise to show that the only polytopes which are both simple and simplicial are either
simplices, or $2$-dimensional polygons (Exercise \ref{simplicial AND simple}).
\begin{example}
\rm{
The $d$-dimensional cube $[0,1]^d$ is a simple polytope. Its dual polytope, which is the cross-polytope
$\Diamond$ (see \eqref{cross polytope}), is a simplicial polytope.
}
\hfill $\square$
\end{example}
One might ask: are the facets of a simple polytope necessarily simplicial polytopes?
Again, an example helps here.
\begin{example}
\rm{
The $120$-cell is a $4$-dimensional polytope whose $3$-dimensional boundary is composed of $120$ dodecahedra \cite{SchleimerSegerman}.
The $120$-cell is a simple polytope, but because all of its facets are dodecahedra, it does not have any simplicial facets.
}
\hfill $\square$
\end{example}
As becomes apparent after comparing the notion of a simple polytope with that of a simplicial polytope, these two types of polytopes are indeed dual to each other, in the sense of duality that we've already encountered in definition \eqref{dual polytope, definition}
\begin{lem}
${\mathcal P}\subset \mathbb{R}^d$ is a simple polytope $\iff$ ${\mathcal P}^*$ is a simplicial polytope.
\end{lem}
(see Gr\"unbaum \cite{Grunbaum} for a thorough study of this duality).
This duality between simple and simplicial polytopes suggests a stronger connection between our geometric structures thus far, and the combinatorics inherent in the partially ordered set of faces of ${\mathcal P}$. Indeed, Gr\"unbaum put it elegantly:
\begin{quote}
``In my opinion, the most satisfying way to approach the definition of polyhedra
is to distinguish between the combinatorial structure of a polyhedron,
and the geometric realizations of this combinatorial structure.'' \cite{Grunbaum2}
\end{quote}
\bigskip
\section{Tangent cones, and the Fourier transform of a simple polytope}
An important step for us is to work with the Fourier-Laplace transform of a cone, and then build some
theorems that allow us to simplify many geometric computations, by using the frequency domain on the Fourier transform side.
We may define the {\bf tangent cone} \index{tangent cone} of each face ${\mathcal F} \subset {\mathcal P}$ as follows:
\begin{equation}\label{tangentcone}
{\mathcal K}_{{\mathcal F}} = \left\{ q+ \lambda(p-q) \mid q\in {\mathcal F}, p\in {\mathcal P}, \lambda \in \mathbb{R}_{\geq 0} \right\}.
\end{equation}
We note that in general ${\mathcal K}_{{\mathcal F}}$ does not necessarily contain the origin.
The tangent cone is also known as the {\bf cone of feasible directions}. Intuitively, we can imagine standing at the point
$q\in {\mathcal F}$, and looking in the direction of
all points that belong to $P$. Then we take the union of all of these directions.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{TangentCones1}
\end{center}
\caption{The triangle ${\mathcal P}$ has three vertex tangent cones: ${\mathcal K}_{v_1}, {\mathcal K}_{v_2}, {\mathcal K}_{v_3}$. The picture is meant to signify that
these cones are, of course, unbounded.
\index{tangent cones} }
\label{Tangent Cones1}
\end{figure}
In the case that the face $F$ is a vertex of ${\mathcal P}$, we call this tangent cone a {\bf vertex tangent cone}.
The vertex tangent cone ${\mathcal K}_v$, which is a cone with apex $v$, may also be
generated by the edge vectors $v_k - v$, where $ [v_k, v]$ is an edge of ${\mathcal P}$:
\begin{equation} \label{main definition of tangent cone}
{\mathcal K}_v = \{ v+ \sum_{k=1}^N \lambda_k (v_k-v) \mid \text{ all } \lambda_k \geq 0, \text{ and the }
v_k \text{ are the neighboring vertices of } v\},
\end{equation}
a construction we will often use in practice.
The tangent cone of an edge of a $3$-dimensional convex polytope is an infinite wedge containing the whole line passing through that edge, while the tangent cone of a vertex (for a convex polytope) never contains a whole line (Exercise \ref{Exercise.tangent cone of a vertex}).
For non-convex polytopes, there are many competing definition for the vertices, and not all of them agree.
One definition for the vertices of non-convex polytopes appears in \cite{BaranyAkopyanRobins}, using Fourier transforms of cones. But in this chapter we focus mainly on convex polytopes.
\begin{example}
\rm{
For the unit cube $\square := [0,1]^d$, the tangent cone at the vertex $v=0$ is
\[
{\mathcal K}_0 = \left\{ \lambda_1 {\bf e_1}+ \lambda_2 {\bf e_2} +
\lambda_3 {\bf e_3} + \cdots + \lambda_d {\bf e_d} \mid \lambda_k \geq 0
\right\},
\]
which also happens to be the {\bf positive orthant} $\mathbb{R}^d_{\geq 0}$.
On the other hand, the tangent cone of $\square$ at the vertex $v =(1,0, \dots, 0)$ is:
\[
{\mathcal K}_v = v + \left\{ \lambda_1 (-{\bf e_1})+ \lambda_2 {\bf e_2} +
\lambda_3 {\bf e_3} + \cdots + \lambda_d {\bf e_d} \mid \lambda_k \geq 0\right\} ,
\]
where ${\bf e_j}$ is the standard unit vector along the $j$'th axis.
}
\hfill $\square$
\end{example}
\begin{example}
\rm{ To relate some of these definitions, consider a $d$-dimensional simplex $\Delta\subset \mathbb{R}^d$.
Located at each of its vertices $v \in \Delta$, we have a tangent cone $K_v$, as in \eqref{main definition of tangent cone}, and here $K_v$ is a simplicial cone. The simplex $\Delta$ is both a simple polytope and a simplicial polytope. }
\hfill $\square$
\end{example}
\bigskip
\section{The Brianchon-Gram identity} \label{section.Brianchon-Gram}
\bigskip
The following combinatorial identity, called the Brianchon-Gram identity, may be thought of as a geometric inclusion-exclusion principle. This identity is quite general, holding true for any convex polytope, simple or not. For a proof of the following result see, for example, \cite{BarvinokEhrhartbook} or \cite{BeckRobins}.
\begin{thm}[Brianchon-Gram identity]\label{Brianchon}
\index{Brianchon-Gram identity}
Let ${\mathcal P}$ be any convex polytope. Then
\begin{equation}\label{BG}
1_{\mathcal P} = \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{dim {\mathcal F}} 1_{{\mathcal K}_F},
\end{equation}
where the sum takes place over all faces of ${\mathcal P}$, including ${\mathcal P}$ itself.
\hfill $\square$
\end{thm}
\medskip
It turns out that the Brianchon-Gram relations \eqref{BG} can be shown to be equivalent (in the sense that one easily implies the other)
to the {\bf Euler-Poincare relation} \index{Euler-Poincare relation} (Exercise \ref{Euler equivalent to Brianchon-Gram})
for the face-numbers \index{face-numbers}
of a convex polytope, which says that
\begin{equation}\label{Euler}
f_0 - f_1 + f_2 - \cdots + (-1)^{d-1} f_{d-1} + (-1)^{d} f_{d}= 1.
\end{equation}
Here $f_k$ is the number of faces of ${\mathcal P}$ of dimension $k$.
\medskip
\begin{example}
\rm{
If we let ${\mathcal P}$ be a $2$-dimensional polygon (including its interior of course) with $V$ vertices, then if must also have $V$ edges, and exactly $1$ face, so that \eqref{Euler} tells us that $V - V + 1 = 1$, which is
not very enlightening, but true.
}
\hfill $\square$
\end{example}
\medskip
\begin{example}
\rm{
If we let ${\mathcal P}$ be a $3$-dimensional polytope with $V$ vertices, $E$ edge, and $F$ facets, then \eqref{Euler} tells us that $f_0 - f_1 + f_2 - f_3 = 1$, which means that $V - E + F - 1 = 1$. So we've retrieved Euler's well known formula
\[
V-E+F=2
\] for the Euler characteristic of $3$-dimensional polytopes.
}
\hfill $\square$
\end{example}
\medskip
\begin{example}
\rm{
To gain some facility with the Euler characteristic, we consider if it is possible to construct a polytope in $\mathbb{R}^3$ all of whose facets are hexagons (which are not necessarily regular). We claim that this is impossible:
{\bf Claim}. \ There can be no convex polytope ${\mathcal P}\subset \mathbb{R}^3$ with only hexagonal facets.
\begin{proof}
By assumption, all the faces of ${\mathcal P}$ are hexagons (not necessarily regular), and of course each edge bounds exactly two facets. To relate the facets to the edges, consider that
each facet contains $6$ edges, giving us $6F=2E$. Combining this latter identity with Euler's formula, we obtain
$V-E+F= V-2F$.
Now we relate the facets to the vertices. Each vertex meets at least three facets, and each hexagonal facet contains exactly six vertices. From the perspective of the facets towards the vertices, we get
$6F \geq 3V$, so that $V \leq 2F$. Putting things together, we arrive at
\[
2= V-E+F= V-2F \leq 0,
\]
and this contradiction finishes the proof.
\end{proof}
}
\end{example}
\section{Brion's formula for the Fourier transform
\\ of a simple polytope}
Brion \index{Brion} proved the following extremely useful result, Theorem \ref{brion, continuous form},
concerning the Fourier-Laplace transform
of a {\it simple polytope} ${\mathcal P}$. To describe the result, we consider each vertex $v$ of ${\mathcal P}$, and we fix the $d$
edge vectors $w_1(v), \dots, w_d(v)$ that emanate from $v$. We recall that the nonnegative real span of the edge vectors $w_k(v)$ generate the vertex
tangent cone ${\mathcal K}_v$, and that these
edge vectors are not necessarily required to be unit vectors. Placing these edge vectors
as columns of a matrix $M_v$, we define
\[
\det {\mathcal K}_v := | \det M_v |,
\]
the absolute value of the determinant of the ensuing matrix.
\medskip
\begin{thm}[{\bf Brion's theorem - the continuous form, 1988}]
\label{brion, continuous form}
\index{Brion's theorem - the continuous form}
Let ${\mathcal P} \subset \mathbb{R}^d$ be a simple, $d$-dimensional real polytope.
Then
\begin{equation}\label{transform formula for a simple polytope}
\int_{\mathcal P} e^{-2\pi i \langle u, \xi \rangle} \, du =
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ e^{-2\pi i \langle v, \xi \rangle} \det {\mathcal K}_v}
{ \prod_{ k=1 }^d \langle w_k(v), \xi \rangle }
\end{equation}
for all $\xi \in \mathbb{R}^d$ such that the denominators on the right-hand side do not vanish.
\hfill $\square$
\end{thm}
Brion's Theorem \ref{brion, continuous form} is one of the cornerstones of Fourier transforms of polytopes.
We note that the determinant $\det {\mathcal K}_v$ clearly depends on our choice of edge vectors $w_1, \dots, w_d$ for the cone ${\mathcal K}_v$, but it is straightforward (and interesting)
that the quotient $ \frac{ \det {\mathcal K}_v}{ \prod_{ k=1 }^d \langle w_k(v), \xi \rangle }$ does not depend on the choice of edge vectors (Exercise \ref{independent of edge vectors}).
This new proof of Brion's theorem uses some of the Fourier techniques that we've developed so far.
Because we promised a friendly approach, we first give a short outline of the relatively simple ideas of the proof:
{\bf Step $1$}. \ We begin with the Brianchon-Gram identity (a standard first step) involving the indicator functions of all of
the tangent cones of ${\mathcal P}$.
\medskip
{\bf Step $2$}. \ We now multiply both sides of the Brianchon-Gram identity \eqref{BG} with the function
$e^{2\pi i \langle x, \xi \rangle - \varepsilon \| x \|^2}$, where we fix an $\varepsilon >0$, and then we will
integrate over all $x \in \mathbb R^d$. Using these integrals, due to the damped Gaussians for each fixed $\varepsilon >0$, we are able to keep the {\it same domain of convergence} for all of our ensuing functions.
\medskip
{\bf Step $3$}. \ Now we let $\varepsilon \rightarrow 0$ and prove that the limit of each integral gives us something meaningful.
Using integration by parts, we prove that for any vertex tangent cone ${\mathcal K}$
the corresponding integral
$\int_{{\mathcal K}} e^{-2\pi i \langle x, \xi \rangle - \varepsilon \| x \|^2} dx$
converges, as $\varepsilon \rightarrow 0$, to the desired exponential-rational function.
In an analogous but easier manner, we will also prove that the corresponding integral over a
non-pointed cone (which includes all faces of positive dimension) converges to zero, completing the proof.
\medskip \noindent
In many of the traditional proofs of Theorem \ref{brion, continuous form}, the relevant Fourier-Laplace integrals over the vertex tangent cones have disjoint domains of convergence, lending the feeling that something magical is going on with the disjoint domains of convergence. Getting around this problem by defining functions that have the same domain of convergence (throughout the proof) was exactly the motivation for this proof.
We favor a slightly longer but clearer expositional proof over a shorter, more obscure proof. The reader familiar with some physics might notice that this proof idea resembles simulated annealing with a Gaussian.
We also note that throughout the proof we will work over $\xi \in \mathbb{R}^d$, and we don't require any analytic continuation.
Onto the rigorous details of the proof. First, a technical but crucial Lemma.
\begin{lem}\label{IntegByParts}
Let ${\mathcal K}_v$ be a $d$-dim'l simplicial pointed cone, with apex $v$, and edge vectors $w_1, \dots, w_d \in \mathbb{R}^d$.
Then
\begin{equation}\label{LimitDim.d}
\lim_{\varepsilon \rightarrow 0}
\int_{{\mathcal K}_v} e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
=
\ \left( \frac{1}{2\pi i} \right)^d \frac{ e^{-2\pi i \langle v, \xi \rangle} \det {\mathcal K}_v }
{ \prod_{ k=1 }^d \langle w_k(v) , \xi \rangle },
\end{equation}
for all $\xi \in \mathbb{R}^d$ such that $ \prod_{ k=1 }^d \langle w_k(v) , \xi \rangle \not=0$.
\end{lem}
\begin{proof}
We begin by noticing that we may prove the conclusion in the case that $v=0$, the origin, and for simplicity write ${\mathcal K}_v := {\mathcal K}$ in this case.
First we make a change of variables, mapping the simplicial cone ${\mathcal K}$ to the nonnegative orthant $\mathbb{R}^d_{\geq 0}$ by the matrix $M^{-1}$, where $M$ is the $d$ by $d$ matrix whose columns are precisely the vectors $w_k$. Thus, in the integral of \eqref{LimitDim.d}, we let $x:= My$,
with $y \in \mathbb{R}_{\geq 0}^d$, so that $dx = \left| \det M \right| dy$. Recalling that by definition $\det {\mathcal K} =| \det M |$, we have
\begin{equation}
\int_{{\mathcal K}} e^{-2\pi i \langle x, \xi \rangle - \varepsilon ||x||^2} dx
=
\left| \det {\mathcal K} \right|
\int_{\mathbb{R}_{\geq 0}^d} e^{-2\pi i \langle My, \xi \rangle - \varepsilon ||M y||^2} dy.
\end{equation}
It is sufficient to therefore show the following limiting identity:
\begin{equation}\label{SimplerLimit}
\lim_{\varepsilon \rightarrow 0}
\int_{\mathbb{R}_{\geq 0}^d} e^{-2\pi i \langle My, \xi \rangle - \varepsilon ||My||^2} dy
=
\ \left( \frac{1}{2\pi i} \right)^d \frac{ 1}{ \prod_{ k=1 }^d \langle w_k(v) , \xi \rangle }.
\end{equation}
To see things very clearly, we first prove the $d=1$ case. Here we must show that
\begin{equation}\label{LimitDim.1}
\lim_{\varepsilon \rightarrow 0}
\int_{0}^\infty e^{-2\pi i x \xi - \varepsilon x^2} dx
=
\frac{1}{2\pi i \xi},
\end{equation}
for all $\xi \in \mathbb{R}-\{0\}$, and we see that even this $1$-dimensional case is interesting. We proceed with integration by parts by letting
$dv:= e^{-2\pi i x \xi}dx$ and $u:= e^{ - \varepsilon x^2}$, to get
\begin{align}
\int_{0}^\infty e^{-2\pi i x \xi - \varepsilon x^2} dx
&= e^{ - \varepsilon x^2} \frac{e^{-2\pi i x \xi}}{-2\pi i \xi} \Big |_{x=0}^{x=+\infty} -
\int_0^\infty \frac{e^{-2\pi i x \xi}}{-2\pi i \xi} (-2\varepsilon x) e^{ - \varepsilon x^2} dx \\
&= \frac{1}{2\pi i \xi} - \frac{\varepsilon}{\pi i \xi}
\int_{0}^\infty x e^{-2\pi i x \xi - \varepsilon x^2} dx \\ \label{last nasty integral}
&= \frac{1}{2\pi i \xi} - \frac{1 }{\pi i \xi}
\int_{0}^{\infty} e^{-2\pi i \frac{u}{\sqrt{\varepsilon}} \xi } u e^{-u^2} du
\end{align}
where we've used the substitution $u:= \sqrt{\varepsilon} x$ in the last equality \eqref{last nasty integral}.
We now notice that
\[
\lim_{\varepsilon \rightarrow 0} \int_{0}^{\infty} e^{-2\pi i \frac{u}{\sqrt{\varepsilon}} \xi } u e^{-u^2} du
=\lim_{\epsilon \rightarrow 0} \hat g\Big(\frac{\xi}{\sqrt\epsilon}\Big),
\]
where $g(u):=u e^{-u^2}1_{[0, +\infty]}(u)$ is an absolutely integrable function.
Luckily, we know by the Riemann--Lebesgue lemma \ref{Riemann--Lebesgue lemma}
\index{Riemann-Lebesgue lemma}
that
\[
\lim_{w\rightarrow \infty} \hat g(w) =0,
\]
and so we arrive at the desired limit \eqref{LimitDim.1}.
We now proceed with the general case, which just uses the $1$-dimensional idea above several times. To prove \eqref{SimplerLimit}, we first fix the variables $y_2, \dots, y_d$ and perform integration by parts on $y_1$ first. Thus, we let
\begin{align}
dv_1 &:= e^{-2\pi i \langle My, \xi \rangle} dy_1=
e^{-2\pi i \langle y, M^t \xi \rangle} dy_1= e^{-2\pi i \Big( y_1 \langle w_1, \xi \rangle + \cdots +
y_d \langle w_d, \xi \rangle \Big)}dy_1,
\end{align}
thought of as a function of only $y_1$.
Carrying out the integration in the variable $y_1$, we have
$v_1 = e^{-2\pi i \langle y, M^t \xi \rangle} / \left(- 2\pi i \langle w_1, \xi \rangle \right)$.
We let $u_1:= e^{- \varepsilon ||My||^2} $, also thought of as a function of $y_1$ alone.
We have $du_1 = -\varepsilon L(y) e^{- \varepsilon ||My||^2} dy_1$, where $L(y)$ is a real polynomial in $y$, whose coefficients come from the entries of $M$.
Integrating by parts in the variable $y_1$ now gives us
\begin{align}\label{SimplerLimitProof}
& \int_{\mathbb{R}_{\geq 0}^d} e^{-2\pi i \langle My, \xi \rangle - \varepsilon ||My||^2} dy
=
\int_{\mathbb{R}_{\geq 0}^{d-1}} dy_2 \cdots dy_d \left[ u_1 v_1 \Big |_0^\infty - \int_0^\infty v_1 du_1 \right] \\
& = \int_{\mathbb{R}_{\geq 0}^{d-1}} dy_2 \cdots dy_d \left[
\frac{ e^{-2\pi i \langle y, M^t \xi \rangle - \varepsilon ||M y||^2} }{ -2\pi i \langle w_1, \xi \rangle }
\Big |_{y_1=0}^{y_1 = \infty} + \frac{\varepsilon}{-2\pi i \langle w_1, \xi \rangle}
\int_0^\infty L(y) e^{-2\pi i \langle y, M^t \xi \rangle - \varepsilon ||My||^2} dy_1
\right] \\
& = \int_{\mathbb{R}_{\geq 0}^{d-1}}
\frac{ e^{2\pi i \langle t, M^t \xi \rangle - \varepsilon ||M t||^2} }{ 2\pi i \langle w_1, \xi \rangle }dt
- \frac{\varepsilon}{2\pi i \langle w_1, \xi \rangle}
\int_{\mathbb{R}_{\geq 0}^{d}} L(y) e^{-2\pi i \langle y, M^t \xi \rangle - \varepsilon || M y ||^2} dy \\
& = \frac{1}{ 2\pi i \langle w_1, \xi \rangle } \int_{\mathbb{R}_{\geq 0}^{d-1}}
e^{-2\pi i \langle t, M^t \xi \rangle - \varepsilon || M t ||^2} dt
- \frac{\varepsilon}{2\pi i \langle w_1, \xi \rangle}
\int_{\mathbb{R}_{\geq 0}^{d}} L(y) e^{-2\pi i \langle y, M^t \xi \rangle - \varepsilon || M y ||^2} dy,
\end{align}
where we've used $t:= (y_2, \dots, y_d)$ in the $3$'rd equality. We repeat exactly the same process of integration by parts as in \eqref{last nasty integral}, one variable at a time. We observe that after $d$ iterations we get a sum of $d$ terms, where the first term does not contain any $\varepsilon$ factors, while
all the other terms do contain $\varepsilon$ factors in the exponents. Therefore, when we complete the $d$-many integration by parts iteratively, and
finally let $\varepsilon$ tend to zero, only the leading term remains, namely
$\left( \frac{-1}{2\pi i} \right)^d \frac{ 1}{ \prod_{ k=1 }^d \langle w_k, \xi \rangle } $. We've shown that \eqref{SimplerLimit} is true.
\end{proof}
\bigskip
\begin{proof}(of Theorem \ref{brion, continuous form})
We begin with the Brianchon Gram identity:
\begin{equation}\label{BG2}
1_{\mathcal P} = \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{dim {\mathcal F}} 1_{K_F}.
\end{equation}
We fix any $\xi \in \mathbb{R}^d$, and any $\varepsilon > 0$. Multiplying both sides of \eqref{BG2} by
$e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2}$, and integrate over all $x \in \mathbb{R}^d$, we have:
\begin{equation}
\int_{\mathbb{R}^d} 1_{\mathcal P}(x) e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
= \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{dim {\mathcal F}} \int_{\mathbb{R}^d} 1_{K_F}(x)
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx.
\end{equation}
\noindent
Equivalently,
\begin{equation}\label{IntegratingBrianchon}
\int_{{\mathcal P}} e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
= \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{dim {\mathcal F}}
\int_{{\mathcal K}_F}
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx.
\end{equation}
For each fixed $\varepsilon > 0$, all integrands in \eqref{IntegratingBrianchon}
are Schwartz functions, and so all of the integrals in the latter identity now converge absolutely (and rapidly).
We identify two types of tangent cones that may occur on the right-hand side of \eqref{IntegratingBrianchon}, for each face ${\mathcal F} \subseteq {\mathcal P}$.
{\bf Case $1$}. When ${\mathcal F} = v$, a vertex, we have the vertex tangent cone ${\mathcal K}_v$: these are the tangent cones that exist for each vertex of ${\mathcal P}$.
It is a standard fact that all of these vertex tangent cones are pointed cones. By hypothesis, all of our vertex tangent cones are simplicial cones, so letting $\varepsilon \rightarrow 0$ and calling on
Lemma \ref{IntegByParts}, we obtain the required limit for
$\int_{{\mathcal K}_v} e^{2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx$.
\smallskip
{\bf Case $2$}. When ${\mathcal F}$ is not a vertex, we have the tangent cone ${\mathcal K}_{\mathcal F}$, and it is a standard fact
that in this case ${\mathcal K}_{\mathcal F}$ always contains a line.
Another standard fact in the land of polytopes is that each tangent cone in this case
may be written as ${\mathcal K}_{{\mathcal F}} = \mathbb{R}^k \oplus {\mathcal K}_p$, the direct sum of a copy of Euclidean space with a pointed cone ${\mathcal K}_p$ for any point $p \in {\mathcal F}$. (as a side-note, it is also true that $\dim {\mathcal F} = k + \dim( {\mathcal K}_p ))$.
We would like to show that for all faces ${\mathcal F}$ that are not vertices of ${\mathcal P}$, the associated integrals tend to $0$:
\[
\int_{{\mathcal K}_F} e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx \rightarrow 0,
\]
as $\varepsilon \rightarrow 0$. Indeed,
\begin{align}
\int_{{\mathcal K}_F}
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx & =
\int_{ \mathbb{R}^k \oplus {\mathcal K}_p } e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx \\
& = \int_{ \mathbb{R}^k } e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
\int_{ {\mathcal K}_p } e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx. \label{product}
\end{align}
The integral $ \int_{ \mathbb{R}^k } e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx$
is precisely the usual Fourier transform of a Gaussian, which is known to be the Gaussian
$G_{\varepsilon}(x):= \varepsilon^{-k/2} e^{-\frac{\pi}{\varepsilon} \|x\|^2}$ by Exercise \ref{Gaussian2}.
It is apparent that for any fixed nonzero value of $x\in \mathbb{R}^k$, we have
$\lim_{\varepsilon \rightarrow 0}G_{\varepsilon}(x)=0$.
Finally, by Lemma \ref{IntegByParts} again, the limit
$\lim_{\varepsilon \rightarrow 0} \int_{ {\mathcal K}_p }
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx$ is finite, because ${\mathcal K}_p$ is another pointed cone.
Therefore the product of the integrals in
\eqref{product} tends to zero, completing the proof.
\end{proof}
Brion's theorem is particularly useful whenever we are given a polytope in terms of its local data at the vertices - including the edge vectors for each vertex tangent cone. We can then easily write down the Fourier transform of a simple polytope, by Theorem \ref{brion, continuous form}.
What happens, though, for non-simple polytopes?
There is the following natural extension of Brion's Theorem \ref{brion, continuous form} to all real polytopes.
\medskip
\begin{thm}[{\bf Fourier-Laplace transform of any real polytope}]
\label{brion2}
Let ${\mathcal P} \subset \mathbb{R}^d$ be any $d$-dimensional polytope. Then:
\begin{equation}
\int_{\mathcal P} e^{-2\pi i \langle u, \xi \rangle} \, du =
\sum_{v \in V}
\frac{e^{-2\pi i \langle v, \xi \rangle} }{(2\pi i)^d}
\sum_{j=1}^{M(v)} \frac{\det {\mathcal K}_j(v) }{\prod_{k=1}^d \langle w_{j, k}(v), \xi \rangle},
\end{equation}
for all $\xi \in \mathbb{R}^d$ such that all of the denominators
$ \prod_{k=1}^d \langle w_{j, k}(v), \xi \rangle \not=0$.
\end{thm}
\begin{proof}
The proof here is identical in almost every aspect to the proof of Theorem \ref{brion, continuous form}, except for {\bf Case} $1$ of its proof, above. By constrast with the proof above of {\bf Case} $1$, here our vertex tangent cones ${\mathcal K}_v$ need not be simplicial. However, we may triangulate each vertex tangent cone ${\mathcal K}_v$ into simplicial cones ${\mathcal K}_{1, v}$, \dots ${\mathcal K}_{M(v), v}$, so that we have the disjoint union
${\mathcal K}_v = {\mathcal K}_{1, v}\cup \dots \cup {\mathcal K}_{M(v), v}$. Therefore
\begin{align*}
\lim_{\varepsilon \rightarrow 0}
\int_{{\mathcal K}_v}
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
&= \lim_{\varepsilon \rightarrow 0}
\sum_{j=1}^{M(v)}
\int_{ {\mathcal K}_{j,v} }
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx \\
&= \sum_{j=1}^{M(v)}
\lim_{\varepsilon \rightarrow 0}
\int_{ {\mathcal K}_{j,v} }
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx \\
&= \ \left( \frac{-1}{2\pi i} \right)^d
\sum_{j=1}^{M(v)}
\frac{ e^{-2\pi i \langle v, \xi \rangle} \det {\mathcal K}_{j, v} }
{ \prod_{ k=1 }^d \langle w_{j, k}(v) , \xi \rangle },
\end{align*}
where we've used Lemma \ref{IntegByParts} in the last equality, owing to the fact that all of the cones
${\mathcal K}_{j, v}$ are simplicial.
The calculation above is valid
for each $\xi \in \mathbb{R}^d$ such that $ \prod_{ k=1 }^d \langle w_{j, k}(v) , \xi \rangle \not=0$ for all vertices $v$ and all $j = 1, \dots, M(v)$.
\end{proof}
\bigskip
\section{Fourier-Laplace transforms of cones}
\label{Fourier Laplace transforms of cones}
What about the Fourier transform of a cone? Well, if we naively try to use the same integrand over a cone, the integral will diverge.
But there is a way to fix this divergence by replacing the real vector
$\xi \in \mathbb{R}^d$ by a complex vector $z \in \mathbb{C}^d$.
Let's consider what would happen if we formally replace the variable $\xi \in \mathbb{R}^d$ by a
complex vector $z := x+iy \in \mathbb{C}^d$, to obtain the transform:
\[
1_{\mathcal P}(z):= \int_{\mathcal P} e^{-2\pi i \langle u, z\rangle} \, du.
\]
Our inner product $\langle u, z \rangle := u_1 z_1 + \cdots + u_d z_d$ is always
the usual inner product on $\mathbb{R}^d$, defined
without using the Hermitian inner product here. In other words, we simply use the usual inner product on $\mathbb{R}^d$, and then formally substitute complex numbers $z_k$ into it.
This means, by definition, that
\begin{align}
\int_{\mathcal P} e^{-2\pi i \langle u, z\rangle} \, du &= \int_{\mathcal P} e^{-2\pi i \langle u, x+iy \rangle} \\
&= \int_{\mathcal P} e^{-2\pi i \langle u, x\rangle} e^{2\pi \langle u, y\rangle} \, du,
\end{align}
so that we have an extra useful real factor of $e^{2\pi \langle u, y\rangle}$ that makes the integral
converge quite rapidly over unbounded domains, provided that
$ \langle u, y \rangle < 0$.
If we set $y=0$, then it's clear that we retrieve the usual Fourier transform of ${\mathcal P}$, while if we set $x=0$, we get a new integral, which we call the {\bf Laplace transform} of ${\mathcal P}$. Finally, the {\bf Fourier-Laplace transform} \index{Fourier-Laplace transform}
of ${\mathcal P}$ is defined by:
\[
\hat 1_{\mathcal P}(z) := \int_{\mathcal P} e^{-2\pi i \langle u, z\rangle} \, du
\]
valid for any $z \in \mathbb{C}^d$ for which the integral converges.
One clear reason for the use and flexibility of the full Fourier-Laplace transform is the fact that for a cone ${\mathcal K}$,
its usual Fourier transform diverges. But if we allow a complex variable $z\in \mathbb{C}^d$, then the integral does converge on a restricted domain. Namely, the Fourier-Laplace transform of
a cone ${\mathcal K}$ is defined by:
\[
\hat 1_{\mathcal K}(z) := \int_{\mathcal K} e^{-2\pi i \langle u, z\rangle} \, du,
\]
for a certain set of $z\in \mathbb{C}^d$, but we can easily understand its precise domain of convergence.
For an arbitrary cone ${\mathcal K} \subset \mathbb{R}^d$, we define its {\bf polar cone}
\index{polar cone} by:
\[
{\mathcal K}^o := \{ y \in \mathbb{R}^d \mid \langle y, u \rangle < 0 \text{ for all } u\in {\mathcal K} \},
\]
which is an open cone. As one might expect, there is the following duality.
If ${\mathcal K}_1 \subset {\mathcal K}_2$, then ${\mathcal K}_2^o \subset {\mathcal K}_1^o$ (Exercise \ref{duality of polar cone}).
\bigskip
\begin{example}
\rm{
Given the $1$-dimensional cone ${\mathcal K}_0 := \mathbb{R}_{\geq 0}$, we compute its Fourier-Laplace transform:
\begin{align*}
\int_{{\mathcal K}_0} e^{-2\pi i u z} \, du = \int_0^\infty e^{-2\pi i u z} \, du = &= \frac{1}{-2\pi i z} e^{-2\pi i u (x+iy)}\Big|_{u=0}^{u=\infty} \\
&= \frac{1}{-2\pi i z} e^{-2\pi i ux} e^{2\pi uy}\Big|_{u=0}^{u=\infty} \\
&= \frac{1}{-2\pi i z} (0-1)
= \frac{1}{2\pi i} \frac{1 }{ z },
\end{align*}
valid for all $z:= x + iy\in \mathbb{C}$ such that $y < 0$. We note that for such a fixed complex
$z$, $| e^{-2\pi i u z} | = e^{2\pi u y }$ is a rapidly decreasing function of $u\in \mathbb{R}_{>0}$,
because $y<0$.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{cone1}
\end{center}
\caption{A simplicial, pointed cone in $\mathbb{R}^3$, with apex $v$ and edge vectors $w_1, w_2, w_3$} \label{Simplicial cone}
\end{figure}
Now let's work out the Fourier-Laplace transform of a $d$-dimensional cone whose apex is the origin.
\begin{lem} \label{F-L transform of a simplicial cone, apex at o}
Let ${\mathcal K} \subset \mathbb{R}^d$ be a simplicial, $d$-dimensional cone, with apex at the origin.
If the edges of ${\mathcal K}$ are labelled $w_1, \dots, w_d$, then
\[
\hat 1_K(z) := \int_{\mathcal K} e^{-2\pi i \langle u, z\rangle} \, du = \frac{1}{(2\pi i)^d} \frac{\det {\mathcal K} }{\prod_{k=1}^d \langle w_k, z \rangle}.
\]
Furthermore, the {\bf domain of convergence} for the latter integral is naturally associated with the polar cone, \index{polar cone} and it is given by:
\[
\{ z:= x + iy \in \mathbb{C}^d \mid \ y \in {\mathcal K}^o \}.
\]
\end{lem}
\begin{proof}
We first compute the Fourier-Laplace transform of the positive orthant ${\mathcal K}_0 := \mathbb{R}_{\geq 0}^d$, with a complex vector $z = x + i y \in \mathbb{C}^d$:
\begin{align} \label{transform of a cone}
\hat 1_{{\mathcal K}_0}(z) &:= \int_{{\mathcal K}_0} e^{-2\pi i \langle z, u \rangle} du \\
&= \int_{\mathbb{R}_{\geq 0}} e^{-2\pi i z_1 u_1} du_1 \cdots \int_{\mathbb{R}_{\geq 0}} e^{-2\pi i z_d u_d} d u_d \\
&= \prod_{k=1}^d \frac{ 0- 1}{-2\pi i z_k}
= \left( \frac{1}{2\pi i}\right)^d \frac{1}{ z_1 z_2 \cdots z_d}. \label{trick2}
\end{align}
Next, the positive orthant ${\mathcal K}_0$ may be mapped to the cone ${\mathcal K}$ by a linear transformation. Namely, we may use the matrix $M$ whose columns are defined to be the edges of ${\mathcal K}$, so that by definition ${\mathcal K} = M({\mathcal K}_0)$. Using this mapping, we have:
\begin{align*}
\hat 1_{{\mathcal K}}(z) &:= \int_{{\mathcal K}} e^{-2\pi i \langle z, u \rangle} du \\
&= |\det M| \int_{{\mathcal K}_0} e^{-2\pi i \langle z, M t \rangle} dt \\
&= |\det M| \int_{{\mathcal K}_0} e^{-2\pi i \langle M^T z, t \rangle} dt \\
&= \left( \frac{1}{2\pi i}\right)^d \frac{|\det M| }{\prod_{k=1}^d \langle w_k, z \rangle}.
\end{align*}
where in the second equality we've made the substitution $u = Mt$, with $t\in {\mathcal K}_0, u \in {\mathcal K}$, and
$du = |\det M| dt$. In the final equality, we used equation \eqref{trick2} above, noting that the $k$'th
element of the vector $M^Tz$ is $\langle w_k, z \rangle$, and we note that by definition $|\det M| = \det {\mathcal K}$.
For the domain of convergence of the integral, we observe that
\[
e^{-2\pi i \langle u, z\rangle} = e^{-2\pi i \langle u, x+iy\rangle} =
e^{-2\pi i \langle u, x\rangle} e^{2\pi \langle u, y\rangle},
\]
and because $ \left| e^{-2\pi i \langle u, x\rangle} \right| = 1$, the integral
$\int_{\mathcal K} e^{-2\pi i \langle u, z\rangle} du$
converges $\iff \langle u, y \rangle < 0$ for all $u\in {\mathcal K}$. But by definition of the polar cone, this means that $y \in {\mathcal K}^o$.
\end{proof}
\bigskip
\begin{example}
\rm{
Given the $2$-dimensional cone
${\mathcal K} := \{
\lambda_1 \big(\begin{smallmatrix}
1 \\
5 \\
\end{smallmatrix}
\big) +
\lambda_2 \big(\begin{smallmatrix}
-3 \\
\ 2 \\
\end{smallmatrix}
\big) \mid \lambda_1, \lambda_2 \in \mathbb{R}_{\geq 0} \}$, we compute its Fourier-Laplace transform, and find its domain of convergence. By Lemma \ref{F-L transform of a simplicial cone, apex at o},
\begin{align*}
\hat 1_{{\mathcal K}}(z):= \int_{\mathcal K} e^{-2\pi i \langle u, z \rangle} \, du &=
\frac{1}{(2\pi i)^2} \frac{17}{(z_1 + 5z_2)(-3z_1 + 2z_2)},
\end{align*}
valid for all $z = \icol{z_1\{\bf z}_2} := x + iy$ such that $ y \in {\mathcal K}^o$. Here the polar cone is
given here by \\
${\mathcal K}^o = \interior\{
\lambda_1 \big(\begin{smallmatrix}
\ 5 \\
-1 \\
\end{smallmatrix}
\big) +
\lambda_1 \big(\begin{smallmatrix}
-2 \\
-3 \\
\end{smallmatrix}
\big) \mid \lambda_1, \lambda_2 \in \mathbb{R}_{\geq 0} \}$.
}
\hfill $\square$
\end{example}
To compute the Fourier-Laplace transform of a simplicial cone ${\mathcal K}$ whose apex is $v \in \mathbb{R}^d$, we may first compute the transform of the translated
cone ${\mathcal K}_0:= {\mathcal K} - v$, whose apex is at the origin, using the previous lemma.
We can then use the fact that the Fourier transform behaves in a simple way under translations, namely
\[
\hat 1_{K + v}(z) = e^{2\pi i \langle z, v \rangle} \hat 1_K(z),
\]
to obtain the following result (Exercise \ref{translating a cone}).
\bigskip
\begin{cor} \label{transform of a translated cone}
Let ${\mathcal K}_v \subset \mathbb{R}^d$ be a simplicial $d$-dimensional cone, whose apex is $v \in \mathbb{R}^d$. Then
\begin{equation} \label{IMPORTANT cone transform}
{\hat 1}_{{\mathcal K}_v}(z) := \int_{{\mathcal K}_v} e^{-2\pi i \langle u, z\rangle} \, du = \frac{1}{(2\pi i)^d}
\frac{ e^{-2\pi i \langle v, z \rangle} \det {\mathcal K}_v }{\prod_{k=1}^d \langle w_k, z \rangle},
\end{equation}
a rational-exponential function.
More generally, for any $d$-dimensional cone ${\mathcal K}_v\subset \mathbb{R}^d$ with apex $v$, we can
always triangulate ${\mathcal K}_v$ into $M(v)$ simplicial subcones ${\mathcal K}_j(v)$ \cite{DRS}, and apply the previous result to each simplicial
subcone, obtaining:
\begin{equation} \label{general cone transform}
{\hat 1}_{{\mathcal K}_v}(z) := \int_{{\mathcal K}_v} e^{-2\pi i \langle u, z\rangle} \, du =
\frac{e^{-2\pi i \langle v, z \rangle} }{(2\pi i)^d}
\sum_{j=1}^{M(v)} \frac{\det {\mathcal K}_j(v) }{\prod_{k=1}^d \langle w_{j, k}(v), z \rangle},
\end{equation}
a rational-exponential function.
\hfill $\square$
\end{cor}
For a non-simple polytope, the question of computing efficiently the Fourier-Laplace transforms of all of its tangent cones becomes unwieldy, as far as we know (this problem is related to the $P \not= NP$ problem). In fact, even computing the volume of a polytope is already known to be NP-hard in general, and the volume is just the Fourier transform evaluated at one point: $\vol {\mathcal P} = 1_{\mathcal P}(0)$.
\medskip
\begin{example}
\rm{
Let's work out a $2$-dim'l example of Brion's Theorem \ref{brion, continuous form}, using Fourier-Laplace transforms of tangent cones.
We will find the rational-exponential function for the Fourier-Laplace transform of the triangle $\Delta$, whose vertices are defined by
$v_1:= \icol{0\{\bf 0}}$, $v_2:= \icol{a\{\bf 0}}$, and $v_3:= \icol{0\{\bf b}}$, with $a>0, b>0$.
First, the tangent cone at the vertex $v_1:= \icol{0\{\bf 0}} $ is simply the nonnegative orthant in this case, with edge vectors $w_1 = \icol{1\{\bf 0}}$ and $w_2 = \icol{0\{\bf 1}}$. Its determinant, given these two edge vectors, is equal to $1$. Its Fourier-Laplace transform is
\begin{equation}
\int_{{\mathcal K}_{v_1}} e^{-2\pi i \langle x, z\rangle} \, dx =
\ \frac{1}{(2\pi i)^2} \, \frac{1}{z_1 z_2},
\end{equation}
and note that here we must have both $\Im(z_1)>0$ and $\Im( z_2)>0$ in order to make the integral converge. Here we use the standard notation $\Im(z)$ is the imaginary part of $z$.
The second tangent cone at vertex $v_2$ has edges $w_1 = \icol{-a\\ \ b}$ and
$w_2 = \icol{ \ 0\\ -b}$ (recall that we don't have to normalize the edge vectors at all). Its determinant has absolute value equal to $ab$, and its
Fourier-Laplace transform is
\begin{equation}
\int_{{\mathcal K}_{v_2}} e^{-2\pi i \langle x, z\rangle} \, dx =
\left( \frac{1}{2\pi i} \right)^2 \frac{(ab) e^{-2\pi i a z_1} }{(-a z_1 + b z_2)(-a z_1)},
\end{equation}
and here the integral converges only for those $z$ for which $\Im( -az_1 + bz_2) >0$ and $\Im( -a z_1 ) >0$.
Finally, the third tangent cone at vertex $v_3$ has edges $w_1 = \icol{ \ a\\ -b}$ and
$w_2 =\icol{\ 0\\ -b}$. Its determinant has absolute value equal to $ab$,
and its Fourier-Laplace transform is
\begin{equation}
\int_{{\mathcal K}_{v_3}} e^{-2\pi i \langle x, z\rangle} \, dx =
\left( \frac{1}{2\pi i} \right)^2 \frac{(ab) e^{-2\pi i b z_2} }{(a z_1 - b z_2)(-b z_2)}.
\end{equation}
and here the integral converges only for those $z$ for which $\Im( az_1 - bz_2) >0$ and $\Im( -b z_2 ) >0$.
We can again see quite explicitly the disjoint domains of convergence in this example, so that there is not even one value of $z \in \mathbb{C}^2$ for which all three Fourier-Laplace transforms of all the tangent cones converge simultaneously. Despite this apparent shortcoming, Brion's identity \eqref{brion, continuous form} still tells us that we may somehow still add these local contributions of the integrals at the vertices combine to give us a formula for the Fourier-Laplace transform of the triangle:
\begin{equation}
\hat 1_{\Delta}(z) := \int_{\Delta} e^{-2\pi i \langle x, z\rangle} dx =
\left( \frac{1}{2\pi i} \right)^2
\left(
\frac{1}{z_1 z_2}
+ \frac{-b\ e^{-2\pi i a z_1} }{(-a z_1 + b z_2) z_1}
+ \frac{ -a \ e^{-2\pi i b z_2} }{(a z_1 - b z_2) z_2}
\right),
\end{equation}
which is \emph{now} magically valid for all generic $(z_1, z_2) \in \mathbb{C}^2$; in other words, it is now valid for all
$(z_1, z_2) \in \mathbb{C}^2$ except those values which make the denominators vanish.
}
\hfill $\square$
\end{example}
\begin{example}\label{FT of a symmetric hexagon}
\rm{
What is the Fourier transform of a hexagon?
Suppose we have a hexagon $H$ that is symmetric about the origin; then we know that its Fourier transform is real-valued, by Lemma \ref{symmetric iff FT is real}. In this case it makes sense to form a $3$-dimensional graph of the points
$(x, y, \hat 1_H(x,y))$, as in Figure \ref{HexagonPic}.
To be concrete, let's define a (parametrized) hexagon $H$ with the following vertices:
\[
v_1 = \Big(\frac{2c}{\sqrt{3}}, 0\Big),\ \
v_2 = \Big(\frac{c}{\sqrt{3}}, c\Big),\ \
v_3 = \Big(\frac{-c}{\sqrt{3}}, c\Big),\ \
v_4 = -v_1,\ \ v_5 = -v_2,\ \ v_6 = -v_3,
\]
for each fixed parameter $c>0$.
Just for fun, our hexagon is scaled so that it has an inscribed circle of radius $c$, which may be useful in future applications.
To use Brion's theorem, we compute the Fourier Transforms of the $6$ vertex tangent cones of $H$.
For $v_1$, the two rays defining $K_{v_1}$ are $w_1 := v_2 - v_1 = (-\frac{c}{\sqrt{3}}, c)$ and $w_2 := v_6-v_1 = (-\frac{c}{\sqrt{3}}, -c)$, so the Fourier Transform of $K_{v_1}$ is:
\[
\hat 1_{K_{v_1}}(z)
= \frac{e^{-2\pi i \frac{2c}{\sqrt{3}}z_1}}{(-2\pi i)^2} \frac{\frac{2c^2}{\sqrt{3}} } {(-\frac{c}{\sqrt{3}}z_1 + c z_2)(-\frac{c}{\sqrt{3}}z_1 - c z_2)}
= \frac{2\sqrt 3 }{(2\pi )^2} \frac{ e^{ -\frac{4\pi i c }{\sqrt{3}} z_1}}
{(-z_1 + \sqrt 3 z_2)(z_1 + \sqrt 3 z_2)}.
\]
For $v_2$, the two rays are $w_1 := v_3 - v_2 = (-\frac{2c}{\sqrt{3}}, 0)$ and $w_2 := v_1 - v_2 = (\frac{c}{\sqrt{3}}, -c)$, giving us:
\[
\hat 1_{K_{v_2}}(z)
= \frac{e^{-2\pi i (\frac{c}{\sqrt{3}}z_1+cz_2)}}{(-2\pi i)^2} \frac{\frac{2c^2}{\sqrt{3}} } {\frac{-2c}{\sqrt{3}}z_1(\frac{c}{\sqrt{3}}z_1 - c z_2)}
= \frac{\sqrt 3}{(2\pi )^2} \frac{ e^{-2\pi c i(\frac{1}{\sqrt{3}} z_1+ z_2)}}
{z_1( z_1 - \sqrt 3 z_2)}.
\]
For $v_3$, the two rays are $w_1 := v_4 - v_3 = (-\frac{c}{\sqrt{3}}, -c)$ and $w_2 := v_2 - v_3 = (\frac{2c}{\sqrt{3}}, 0)$, giving us:
\[
\hat 1_{K_{v_3}}(z)
= \frac{e^{-2\pi i (-\frac{c}{\sqrt{3}}z_1+ cz_2)}}{(-2\pi i)^2} \frac{\frac{c}{\sqrt{3}} } {(-\frac{c}{\sqrt{3}}z_1 - c z_2) \frac{2c}{\sqrt{3}}z_1}
= \frac{\sqrt 3}{(2\pi )^2} \frac{ e^{-2\pi c i(-\frac{1}{\sqrt{3}}z_1+ z_2)}}
{z_1(z_1 +\sqrt 3z_2)}.
\]
By the inherent symmetry of our hexagon $H$, the computations for the other tangent cones are just $\hat 1_{K_{-v}}(z) = 1_{K_{v}}(-z)$, so we have:
\begin{equation}\label{eq:sinc-hexagonal}
\begin{aligned}
&\hat 1_H(z_1, z_2) := \int_{H}e^{-2\pi i \langle \xi, z \rangle} d\xi \\
&= \hat 1_{K_{v_1}}(z) + \hat 1_{K_{v_1}}(-z)
+ \hat 1_{K_{v_2}}(z) + \hat 1_{K_{v_2}}(-z)
+ \hat 1_{K_{v_3}}(z) + \hat 1_{K_{v_3}}(-z) \\
&=
\frac{\sqrt{3}}{2\pi^2}\left(
\frac{2 \cos(\frac{4\pi c}{\sqrt{3}} z_1)}{(-z_1 + \sqrt 3 z_2)(z_1 + \sqrt 3 z_2)}
+
\frac{\cos\big(\frac{2\pi c}{\sqrt{3}} z_1 + 2\pi c z_2 \big)}{z_1(z_1 - \sqrt 3 z_2)}
+
\frac{\cos\big(\frac{2\pi c}{\sqrt{3}} z_1 - 2\pi c z_2 \big)}{z_1(z_1 + \sqrt 3 z_2)} \right).
\end{aligned}
\end{equation}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{HexagonPic}
\end{center}
\caption{A graph of the Fourier transform $\hat 1_H(x,y)$ of the symmetric hexagon $H$
in Example \ref{FT of a symmetric hexagon} }
\label{HexagonPic}
\end{figure}
}
\hfill $\square$
\end{example}
\section{An application of transforms to the volume of a simple polytope, and for its moments} \index{moments}
\index{simple polytope}
\index{volume}
The following somewhat surprising formula for the volume of a simple polytope gives us a very rapid algorithm
for computing volumes of simple polytopes. We note that it is an NP-hard problem \cite{Barany}
to compute volumes of general polytopes, without fixing the dimension.
Nevertheless, there are various other families of polytopes whose volumes possess tractable algorithms.
\bigskip
\begin{thm}[Lawrence \cite{LawrenceVolume}]
\label{volume of a simple polytope}
Suppose ${\mathcal P} \subset \mathbb{R}^d$ is a simple, $d$-dimensional polytope.
For a vertex tangent cone ${\mathcal K}_v$ of ${\mathcal P}$, fix a set of edges of the cone, say
$w_1(v), w_2(v), \dots, w_d(v) \in \mathbb{R}^d$.
Then \index{volume of a simple polytope}
\begin{equation} \label{formula for the volume of simple polytope}
\vol {\mathcal P} = \frac{ (-1)^d }{ d! } \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ {\langle v, z \rangle}^{d} \det {\mathcal K}_v }
{ \prod_{ k=1 }^d \langle w_k(v), z \rangle}
\end{equation}
for all $z\in \mathbb{C}^d$ such that the denominators on the right-hand side do not vanish.
More generally, for any integer $k \ge 0$, we have the {\bf moment formulas}:
\index{moment formulas}
\begin{equation}
\int_{\mathcal P} {\langle x, z \rangle}^{k} dx = \frac{ (-1)^d k! }{ (k +d)! } \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ {\langle v, z \rangle}^{k+d}
\det {\mathcal K}_v }{ \prod_{ m=1 }^d \langle w_m(v), z \rangle} \, .
\end{equation}
\end{thm}
\bigskip
\begin{proof}
We begin with Brion's identity \eqref{transform formula for a simple polytope}, and we substitute $z := t z_0$ for a fixed
complex vector $z_0\in \mathbb{C}^d$, and any positive real value of $t$:
\begin{equation*}\label{transform formula for a simple polytope}
\int_{\mathcal P} e^{-2\pi i \langle u, z_0\rangle t} \, du =
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ e^{-2\pi i \langle v, z_0\rangle t} \det {\mathcal K}_v}
{ t^d \prod_{ m=1 }^d \langle w_m(v), z_0 \rangle }.
\end{equation*}
Now we expand both sides in their Taylor series about $t=0$. The left-hand-side becomes:
\begin{align*}
\int_{\mathcal P} \sum_{k=0}^\infty \frac{1}{k!} (-2\pi i \langle u, z_0\rangle t)^k \, du
&=
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ \sum_{j=0}^\infty \frac{1}{j!} (-2\pi i \langle v, z_0\rangle t)^j \det {\mathcal K}_v}
{ t^d \prod_{ m=1 }^d \langle w_m(v), z_0 \rangle }
\end{align*}
Integrating term-by-term on the left-hand-side, we get:
\begin{align*}
\sum_{k=0}^\infty \frac{t^k}{k!} (-2\pi i )^k \int_{\mathcal P} \langle u, z_0\rangle^k \, du
&=
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ \det {\mathcal K}_v}{\prod_{ m=1 }^d \langle w_m(v), z_0 \rangle}
\sum_{j=0}^\infty \frac{t^{j-d}}{j!} (-2\pi i )^j {\langle v, z_0\rangle}^j.
\end{align*}
Comparing the coefficients of $t^k$ on both sides, we have:
\begin{equation*}
\frac{(-2\pi i )^k}{k!} \int_{\mathcal P} \langle u, z_0\rangle^k \, du =
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ \det {\mathcal K}_v}{\prod_{ m=1 }^d \langle w_m(v), z_0 \rangle}
\frac{1}{(k+d)!} (-2\pi i )^{k+d} {\langle v, z_0 \rangle}^{k+d},
\end{equation*}
and simplifying, we arrive at the moment formulas:
\begin{equation*}
\int_{\mathcal P} \langle u, z_0\rangle^k \, du = (-1)^d \frac{k!}{(k+d)!}
\sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ {\langle v, z_0 \rangle}^{k+d} \det {\mathcal K}_v }{\prod_{ m=1 }^d \langle w_m(v), z_0 \rangle }.
\end{equation*}
In particular, when $k=0$, we get the volume formula \eqref{formula for the volume of simple polytope}.
\end{proof}
Now that we know about the Fourier-Laplace transforms of cones, we can reinterpret
Theorem \ref{brion2} by meromorphically continuing the real vector $\xi$ as follows.
Suppose we are given any $d$-dimensional polytope
${\mathcal P} \subset \mathbb{R}^d$. Using the notation of Theorem \ref{brion2},
we define
\begin{equation} \label{meromorphic continuation identity}
F_{K_v}(z) :=
\frac{e^{-2\pi i \langle v, z \rangle} }{(2\pi i)^d}
\sum_{j=1}^{M(v)} \frac{\det {\mathcal K}_j(v) }{\prod_{k=1}^d \langle w_{j, k}(v), z \rangle},
\end{equation}
for all $z \in \mathbb{C}^d$ such that all of the denominators
$ \prod_{k=1}^d \langle w_{j, k}(v), z \rangle \not=0$.
Because the function on the right-hand-side of \eqref{meromorphic continuation identity}
is a meromorphic function of $z$, we see that
$ F_{K_v}(z)$ is the meromorphic continuation of the Fourier-Laplace transform of the
vertex tangent cone $K_v$, and we know by Corollary
\ref{general cone transform} that
\begin{equation} \label{rewriting the continuous theorem of Brion}
F_{K_v}(z) = {\hat 1}_{{\mathcal K}_v}(z),
\end{equation}
on the restricted domain
\[
\{ z:= x + iy \in \mathbb{C}^d \mid \ y \in {\mathcal K}_v^o \}.
\]
With this notation we may rewrite Theorem \ref{brion2} as follows:
\begin{equation}
\int_{\mathcal P} e^{-2\pi i \langle u, z\rangle} \, du =
\sum_{v\in V} F_{K_v}(z),
\end{equation}
valid for almost all $z \in \mathbb{C}^d$.
\bigskip
\section*{Notes} \label{Notes.chapter.Brion}
\begin{enumerate}[(a)]
\item There is a large literature devoted to triangulations of cones, polytopes, and general point-sets,
and the reader is invited to consult the excellent and encyclopedic book on triangulations, by
Jes\'us de Loera, J\"org Rambau, and Francisco Santos \cite{DRS}.
\item The notion of a {\bf random polytope} has a large literature as well, and although we do not go into this topic here, one classic survey paper is by Imre B\'ar\'any \cite{Barany}.
\item The attempt to extend Ehrhart theory to non-rational polytopes, whose vertices have some irrational coordinates, is ongoing.
The pioneering papers of Burton Randol \cite{Randol1} \cite{Randol2} extended integer point counting to algebraic polytopes, meaning that their vertices are allowed to have coordinates that are algebraic numbers. Recently, a growing number of papers are considering all real dilates of a rational polytope, which is still rather close to the Ehrhart theory of rational polytopes.
In this direction, it is natural to ask how much more of the geometry of a given polytope ${\mathcal P}$ can be captured by
counting integer points in all of its positive real dilates. Suppose we translate a
$d$-dimensional integer polytope ${\mathcal P} \subset \mathbb{R}^d$ by an integer vector $n \in \mathbb{Z}^d$.
The standard Ehrhart theory gives us an invariance principle, namely the equality of the Ehrhart polynomials for ${\mathcal P}$ and ${\mathcal P} + n$:
\[
L_{{\mathcal P}+n}(t) = L_{\mathcal P}(t),
\]
for all \emph{integer} dilates $t>0$.
However, when we allow $t$ to be a positive real number, then it is in general {\bf false} that
\[
L_{{\mathcal P}+n}(t) = L_{\mathcal P}(t) \text{ for all } t > 0.
\]
In fact, these two Ehrhart functions are so different in general, that by the very recent breakthrough of Tiago Royer \cite{Tiago1},
it's even possible to uniquely reconstruct the polytope ${\mathcal P}$ if we know all the counting quasi-polynomials
$L_{{\mathcal P}+n}(t)$, for all integer translates $n \in \mathbb{Z}^d$. In other words, the work of \cite{Tiago1} shows that for two
rational polytopes ${\mathcal P}, Q \subset \mathbb{R}^d$, the equality $L_{{\mathcal P}+n}(t) = L_{Q+n}(t)$ holds for all integer translates $n \in \mathbb{Z}^d \iff {\mathcal P} = Q$.
It is rather astounding that just by counting integer points in sufficiently many translates of ${\mathcal P}$, we may completely reconstruct the whole polytope ${\mathcal P}$ uniquely. Royer further demonstrated \cite{Tiago2} that such an idea also works if we replace a polytope by any symmetric convex body. It is now natural to try to prove the following extended question.
\begin{question}
\rm{Suppose we are given polytopes ${\mathcal P}, Q \subset \mathbb{R}^d$. Can we always find a finite subset
$S\subset \mathbb{Z}^d$ (which may depend on ${\mathcal P}$ and Q)
such that
\[
L_{{\mathcal P}+n}(t) = L_{Q+n}(t) \text{ for all } n \in S, \text{ and all } t >0 \ \iff \ {\mathcal P} = Q?
\]
}
\end{question}
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
``It is better to solve one problem five different ways, than to solve five problems one way.''
-- George P\'olya
\end{quote}
\medskip
\begin{prob} $\clubsuit$ \label{independent of edge vectors}
\rm{
Although $\det {\mathcal K}_v$ depends on the choice of the length of each edge of ${\mathcal K}_v$, show that the ratio
$ \frac{ |\det {\mathcal K}_v| }{\prod_{k=1}^d \langle w_k(v), z \rangle}$
remains invariant if we replace each edge $w_k(v)$ of a simplicial cone by a constant
positive multiple of it, say $\alpha_k w_k(v)$ with $\alpha_k>0$.
(Here $z$ is any generic complex vector, meaning that $\langle w_k(v), z \rangle \not=0$).
}
\end{prob}
\medskip
\begin{prob}
Consider the regular hexagon ${\mathcal P} \subset \mathbb{R}^2$, whose vertices are the $6$'th roots of unity.
\begin{enumerate}[(a)]
\item Compute the area of ${\mathcal P}$ using Theorem \ref{volume of a simple polytope}.
\item Compute all of the moments of ${\mathcal P}$, as in Theorem \ref{volume of a simple polytope}.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
Compute the Fourier transform of the triangle $\Delta$ whose vertices are given by
\[
(1, 0), (0, 1), (-c, -c),
\]
where $c>0$.
\end{prob}
\medskip
\begin{prob} \label{translating a cone}
$\clubsuit$
Prove Corollary \ref{transform of a translated cone} for a simplicial cone ${\mathcal K}_v$, whose apex is $v$, by
translating a cone whose vertex is at the origin, to get:
\[
{\hat 1}_{{\mathcal K}_v}(z) := \int_{{\mathcal K}_v} e^{-2\pi i \langle u, z\rangle} \, du = \frac{1}{(2\pi i)^d}
\frac{ e^{-2\pi i \langle v, z \rangle} \det {\mathcal K}_v }{\prod_{k=1}^d \langle w_k, z \rangle}.
\]
\end{prob}
\medskip
\begin{prob}
Using some of the idea in Lemma \ref{LimitDim.d}, prove the following:
\begin{enumerate}[(a)]
\item For all nonzero $\alpha \in \mathbb{R}$,
\[
\lim_{\varepsilon \rightarrow 0} \int_0^\infty \cos(\alpha x) \, e^{-\varepsilon |x|^2} dx = 0.
\]
\item For all nonzero $\alpha \in \mathbb{R}$,
\[
\lim_{\varepsilon \rightarrow 0} \int_0^\infty \sin(\alpha x) \, e^{-\varepsilon |x|^2} dx = \frac{1}{\alpha}.
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{Pyramid over a square}
Consider the following $3$-dimensional polytope ${\mathcal P}$, whose vertices are as follows:
\[
\{ (0, 0, 0), \ (1, 0, 0), \ (0, 1, 0), \ (1, 1, 0), \ (0, 0, 1) \}.
\]
``a pyramid over a square". Compute its Fourier-Laplace transform $\hat 1_{\mathcal P}(z)$.
\end{prob}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.1in]{pyramid}
\end{center}
\caption{The pyramid over a square, in Exercise \ref{Pyramid over a square} }
\label{pyramid over a square}
\end{figure}
\medskip
\begin{prob}
We recall that the $3$-dimensional cross-polytope (also called an octahedron)
was defined by $\Diamond:=\left\{ \left( x_1, x_2, x_3 \right) \in \mathbb{R}^d \mid
\, \left| x_1 \right| + \left| x_2 \right| + \left| x_3 \right| \leq 1 \right\}$.
Compute the Fourier-Laplace transform of $\Diamond$ by using
Theorem \ref{brion2}.
\index{cross-polytope}
(Here not all of the tangent cones are simplicial cones, but we may triangulate each vertex tangent cones into simplicial cones).
\end{prob}
\medskip
\begin{prob}[hard-ish] \label{FT of a Dodecahedron}
Here we will find the Fourier transform of a dodecahedron ${\mathcal P}$, centered at the origin.
Suppose we fix the following $20$ vertices of ${\mathcal P}$:
\[
\{ (\pm 1,\ \pm 1, \ \pm 1), \
(0, \ \pm \phi, \ \pm \frac{1}{\phi}), \ (\pm \frac{1}{\phi}, \ 0, \ \pm \phi), \
( \pm \phi, \ \pm \frac{1}{\phi}, \ 0)
\},
\]
where $\phi:= \frac{1+\sqrt{5}}{2}$. It turns out that ${\mathcal P}$ is a simple polytope. Compute its Fourier-Laplace transform using Theorem \ref{brion, continuous form}.
Notes. All of the vertices of ${\mathcal P}$ given here can easily be seen to lie on a sphere $S$ of radius $\sqrt{3}$, and
this is a regular embedding of the dodecahedron. It is also true (though a more difficult fact) that these $20$ points maximize the volume of any polytope whose $20$ vertices lie on the surface of this sphere $S$.
\end{prob}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.3in]{Dodecahedron_climbing_wall}
\end{center}
\caption{A climbing wall in Sweden, made up of Dodecahedrons, showing one of their real-life applications} \label{Dodecahedron}
\end{figure}
\medskip
\begin{prob}
Define the $3$-dimensional polytope
${\mathcal P} := \rm{ conv }\{ (0,0,0), (1,0,0), (0,1,0), (0,0, 1), (a,b,c) \}$, where we fix real the positive real numbers $a,b,c$. Compute $\hat 1_{\mathcal P}(z)$, by computing the Fourier-Laplace transforms of its tangent cones.
(Note. Here, not all of the tangent cones are simplicial cones).
\end{prob}
\medskip
\begin{prob} \label{Pyramid over a cube}
This exercise extends Exercise \ref{Pyramid over a square} to $\mathbb{R}^d$, as follows.
Consider the $d$-dimensional polytope ${\mathcal P}$, called a ``pyramid over a cube", defined by the convex
hull of the unit cube $[0,1]^{d-1} \subset \mathbb{R}^{d-1}$, with the point $(0, 0, \dots, 0, 1) \in \mathbb{R}^{d}$.
Compute its Fourier-Laplace transform $\hat 1_{\mathcal P}(z)$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{cone equivalence}
Show the following two conditions are equivalent:
\begin{enumerate}[(a)]
\item A cone ${\mathcal K}$ has an apex at the origin.
\item ${\mathcal K}$ is a cone that enjoys the property $\lambda {\mathcal K} = {\mathcal K}$, for all $\lambda >0$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{simplicial implies pointed}
Suppose we are given a $d$-dimensional simplicial cone ${\mathcal K} \subset \mathbb{R}^d$ (so be definition ${\mathcal K}$ has exactly $d$ edges).
Show that ${\mathcal K}$ must be pointed.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Exercise.tangent cone of a vertex}
Show that for any polytope ${\mathcal P}\subset \mathbb{R}^d$, a vertex tangent cone ${\mathcal K}_v$ never contains a whole line.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{pointed cone equivalence}
\index{cone, pointed}
Show that if ${\mathcal K}$ is a cone with an apex $v$ (not necessarily a unique apex), the
following conditions are equivalent:
\begin{enumerate}[(a)]
\item ${\mathcal K}$ is a pointed cone.
\item There exists a hyperplane $H$ such that $H\cap {\mathcal K} = v$.
\item The translated cone $C:= {\mathcal K}-v$, with apex at the origin, enjoys $C \cap (-C) = \{0\}$.
\item ${\mathcal K}$ has a unique apex.
\item ${\mathcal K}$ does not contain an entire line.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{simplicial AND simple}
Show that the only polytopes that are both simple and simplicial are either
simplices, or $2$-dimensional polygons.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{duality of polar cone}
Show that if we have reverse inclusions for polar cones. Namely, if we have
cones ${\mathcal K}_1 \subset {\mathcal K}_2$, then ${\mathcal K}_2^o \subset {\mathcal K}_1^o$.
\end{prob}
\medskip
\begin{prob} \label{polar cones and Minkowski sums}
\index{Minkowski sum}
Show that if we take the Minkowski sum $K_1 + K_2$ of two cones ${\mathcal K}_1, {\mathcal K}_2 \subset \mathbb{R}^d$, then
polarity interacts with Minkowski sums in the following pleasant way:
\[
\left({\mathcal K}_1 +{\mathcal K}_2\right)^o = {\mathcal K}_1^o \cap {\mathcal K}_2^o.
\]
\end{prob}
\medskip
\begin{prob} \label{polytope from pentagons}
Suppose we try to construct a polytope ${\mathcal P} \subset \mathbb{R}^3$ all of whose facets are pentagons (not necessarily regular).
Show that
$
F\geq 12,
$
where $F$ is the number of facets of ${\mathcal P}$.
\end{prob}
\medskip
\begin{prob} \label{Euler equivalent to Brianchon-Gram} $\clubsuit$
\rm{
\begin{enumerate}[(a)]
\item Show that the Brianchon-Gram relations \eqref{BG} imply
the Euler-Poincare relation for the face-numbers of a convex polytope ${\mathcal P}$:
\begin{equation}
f_0 - f_1 + f_2 - \cdots + (-1)^{d-1} f_{d-1} + (-1)^{d} f_{d}= 1,
\end{equation}
where $f_k$ is the number of faces of ${\mathcal P}$ of dimension $k$.
\item \label{b} (hard) \ Conversely, given a $d$-dimensional polytope ${\mathcal P}\subset \mathbb{R}^d$, show that the Euler-Poincare relation above implies the Brianchon-Gram relations:
\[
1_{\mathcal P}(x) = \sum_{{\mathcal F} \subset {\mathcal P}} (-1)^{dim {\mathcal F}} 1_{{\mathcal K}_F}(x),
\]
for all $x\in \mathbb{R}^d$.
\end{enumerate}
Notes. Interestingly, even though the above two conditions are equivalent, condition \ref{b} is often more useful in practice, because we have a free variable $x$, over which we may sum or integrate.
}
\end{prob}
\chapter{The discrete Brion theorem: Poisson summation strikes again}
\label{chapter:Discrete Brion}
\index{Poisson summation} \index{discrete Brion theorem}
\begin{quote}
``Everything you've learned in school as `obvious' becomes less and less obvious as you begin to study the universe.
For example, there are no solids in the universe.
There's not even a suggestion of a solid.
There are no absolute continuums.
There are no surfaces.
There are no straight lines.''
-- Buckminster Fuller
\end{quote}
\bigskip
\section{Intuition}
As we saw in Theorem \ref{brion, continuous form}, there exists a wonderful way to decompose the Fourier transform of a polytope in terms of the Fourier-Laplace transforms of its vertex tangent cones.
We can now ask:
\begin{question}{\rm [Rhetorical]
\label{question:discrete Brion}
Is there a natural way to {\bf discretize} this continuous identity for the FT of a polytope?}
\end{question}
Another basic question we could ask is:
\begin{question} {\rm [Rhetorical]
How does the finite geometric sum in dimension $1$ extend to dimension $d$? }
\end{question}
As we'll see, these two questions are intertwined, and one answers the other.
One useful way to make sense of Question \ref{question:discrete Brion} is to replace integrals with sums over the integer lattice:
\begin{equation}
\int_{\mathcal P} e^{-2\pi i \langle u, z \rangle} \, du \longrightarrow
\sum_{n\in \mathbb{Z}^d} e^{2\pi i \langle z, n \rangle}.
\end{equation}
Such a descretization will lead us to a discrete version of Brion's Theorem, namely Theorem \ref{brion, discrete form} below.
\bigskip
\section{Discretizing the Fourier-Laplace transform of a cone}
We may also replace the integer lattice by any lattice ${\mathcal L}$, and the ensuing function is very similar.
But since this is only a cosmetic change of variable, we can simplify life and work with the integer lattice.
To this discrete end, we define the {\bf integer point transform}
\index{integer point transform} of a rational polytope ${\mathcal P}$ by
\[
\sigma_{\mathcal P}(z) := \sum_{n \in {\mathcal P} \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle},
\]
a discretization of the Fourier transform of ${\mathcal P}$.
We may also think of the discretized sum $ \sum_{n\in \mathbb{Z}^d} e^{2\pi i \langle z, n \rangle}$
more combinatorially by making the change of variable $q_1:= e^{2\pi i z_1}, \dots, q_d:= e^{2\pi i z_d}$, so that
we have $q_1^{n_1} q_2^{n_2} \cdots q_d^{n_d} = e^{2\pi i n_1 z_1 + \cdots + 2\pi i n_d z_d} := e^{2\pi i \langle n, z \rangle}$.
with this notation in mind, we define the {\bf multinomial notation} for a monomial in several variables:
\[
q^n:= q_1^{n_1} q_2^{n_2} \cdots q_d^{n_d}.
\]
We will therefore sometimes use the equivalent definition
\[
\sigma_{\mathcal P}(q):= \sum_{n \in {\mathcal P} \cap \mathbb{Z}^d} q^n.
\]
We similarly define the {\bf integer point transform of a rational cone} ${\mathcal K}_v$ by the series
\begin{equation}\label{def of integer point transform of a cone}
\sigma_{{\mathcal K}_v}(z) := \sum_{n \in {\mathcal K}_v \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle}.
\end{equation}
But even in dimension $1$ things can get interesting, so let's see an example.
\begin{example}[Finite geometric sums]
\rm{
Consider the $1$-dimensional polytope ${\mathcal P} := [a,b]$, where $a, b\in \mathbb{Z}$. The problem is to compute the finite
geometric series:
\begin{align*}
\sum_{n \in {\mathcal P} \cap \mathbb{Z}} e^{ 2\pi i n z} &= \sum_{a \leq n \leq b} q^n,
\end{align*}
where we've set $q:= e^{2\pi i z}$. Of course, we already know that it possesses a
`closed form' of the type:
\begin{align} \label{verifying1}
\sum_{a \leq n \leq b} q^n &= \frac{q^{b+1} -q^{a} }{q - 1} \\
&= \frac{q^{b+1}}{q-1} - \frac{ q^{a} }{q - 1}, \label{verifying2}
\end{align}
because we already recognize this formula for a {\bf finite geometric sum}.
On the other hand, anticipating the discrete form of Brion's theorem below, we first compute the discrete
sum corresponding to the vertex tangent cone at the vertex $a$, namely $\sum_{a \leq n} q^n$:
\begin{equation}
\label{cone identity1}
q^a+ q^{a+1} + \cdots = \frac{q^a}{1-q}.
\end{equation}
Now we compute the the sum corresponding to the vertex tangent cone at vertex $b$, namely $\sum_{n \leq b} q^n$:
\begin{equation} \label{cone identity2}
q^b+ q^{b-1} + \cdots = \frac{q^b}{1-q^{-1}} =
\frac{q^{b+1}}{q-1} .
\end{equation}
Summing these two contributions, one from each vertex tangent cone, we get:
\begin{align*}
\frac{q^a}{1-q} + \frac{q^{b+1}}{q-1} = \sum_{a \leq n \leq b} q^n,
\end{align*}
by the finite geometric sum identity, thereby
verifying Theorem \ref{brion, discrete form} for this example.
This example shows that Brion's Theorem \ref{brion, discrete form} (the discrete version) may be thought of as a $d$-dimensional extension of the finite geometric sum.
But something is still very wrong here - namely, identity \eqref{cone identity1} converges for $|q|< 1$,
while identity \eqref{cone identity2} converges only for $|q|>1$, so there is not even one value of $q$ for which the required identity \eqref{verifying2} is true. So how can we make sense of these completely {\bf disjoint domains of convergence} ?!
}
\hfill $\square$
\end{example}
\bigskip
To resolve these conundrums, the very useful result of
Michel Brion \cite{Brion} comes to the rescue.
Our proof of Theorem \ref{brion, discrete form} discretizes the continuous form of Brion's Theorem \ref{brion, continuous form},
using the Poisson summation formula, to arrive at a discrete form of Brion's Theorem.
First, we need a slightly technical but easy Lemma.
\begin{lem}\label{technical lemma1}
Let ${\mathcal K}_v$ be a rational cone, with apex at $v$. We pick any compactly supported and smooth
approximate identity $\phi_\varepsilon$. Then:
\begin{align} \label{lem:technical claim for limiting series}
\lim_{\varepsilon \rightarrow 0}
\sum_{n \in \mathbb{Z}^d }
\left( 1_{\interior {\mathcal K}_v}(x) e^{2\pi i \langle x, z\rangle}*\phi_\varepsilon \right)(n)
= \sum_{n \in \mathbb{Z}^d \cap \interior {\mathcal K}_v } e^{2\pi i \langle n, z\rangle} := \sigma_{ \interior {\mathcal K}_v}(z).
\end{align}
\end{lem}
\begin{proof}
We first note that by our assumptions on $\phi_\varepsilon$, it lies in the Schwartz space $S(\mathbb{R}^d)$, by
Lemma \ref{useful Schwartz fact}.
So $\phi_\varepsilon$ is rapidly decreasing. Using the Weierstrass $M$-test,
we see that the series
$\sum_{n \in \mathbb{Z}^d }
\left( 1_{\interior {\mathcal K}_v}(x) e^{2\pi i \langle x, z\rangle}*\phi_\varepsilon \right)(n)$
converges uniformly in $\epsilon$, and because the summands are continuous functions of $\epsilon$, so is the whole series. So we may take the limit as $\epsilon \rightarrow 0$ inside the series.
Finally, using Lemma \ref{approximate identity convolution}, and the continuity of the function
$1_{\interior {\mathcal K}_v}(x) e^{2\pi i \langle x, z\rangle}*\phi_\varepsilon$ at all $x \in \mathbb{R}^d$,
we have
$\lim_{\varepsilon \rightarrow 0}
\left( 1_{\interior {\mathcal K}_v}(x) e^{2\pi i \langle x, z\rangle}*\phi_\varepsilon \right)(n)
= 1_{\interior {\mathcal K}_v}(n) e^{2\pi i \langle n, z\rangle}$, from which
\eqref{lem:technical claim for limiting series} follows.
\end{proof}
It turns out that the continuous form of Brion's theorem, namely Theorem \ref{brion, continuous form},
can be used to prove the discrete form of Brion's theorem, namely Theorem \ref{brion, discrete form} below.
\bigskip
\begin{thm}[{\bf Brion's theorem - the discrete form, 1988}] \label{brion, discrete form}
\index{Brion's theorem - the discrete form}
Let ${\mathcal P} \subset \mathbb{R}^d$ be a rational, $d$-dimensional polytope, and let $N$ be the number of vertices of ${\mathcal P}$.
For each vertex $v$ of ${\mathcal P}$, we consider the open vertex tangent cone $\interior {\mathcal K}_v$ of $ \interior {\mathcal P}$, the interior of ${\mathcal P}$.
Then
\begin{equation} \label{Discrete formula, Brion's theorem}
\sigma_{ \interior {\mathcal P}}(z) = \sigma_{ \interior {\mathcal K}_{v_1}}(z) + \cdots + \sigma_{ \interior {\mathcal K}_{v_N}}(z).
\end{equation}
for all $z \in \mathbb{C}^d - S$, where $S$ is the hyperplane arrangement defined by the (removable) singularities of all of the transforms $\hat 1_{{\mathcal K}_{v_j}}(z)$.
\end{thm}
\begin{proof} We will use the continuous version of Brion, namely Theorem \ref{brion, continuous form}, together with
the Poisson summation formula, to deduce the discrete version here. In a sense, the Poisson summation formula allows us to discretize the integrals. \index{Poisson summation formula}
Step $1$. [{\bf Intuition - fast and loose}] \ To begin, in order to motivate the rigorous proof that follows, we will use Poisson summation on a function $1_{{\mathcal P}}(n) e^{2\pi i \langle n, z\rangle}$
that ``doesn't have the right"
to be used in Poisson summation, because $\hat 1_{{\mathcal P}} \notin L^1(\mathbb{R}^d)$ . But this first step brings the intuition to the foreground.
Then, in Step $2$, we will literally ``smooth'' out the lack of rigor in Step 1, by smoothing $1_{\mathcal P}$ with an approximate identity.
\begin{align*}
\sum_{n \in {\mathcal P} \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle} &:= \sum_{n \in \mathbb{Z}^d} 1_{{\mathcal P}}(n) e^{2\pi i \langle n, z\rangle} \\
&= \sum_{\xi \in \mathbb{Z}^d} \hat 1_{\mathcal P}(z+ \xi) \\
&= \sum_{\xi \in \mathbb{Z}^d} \left( \hat 1_{K_{v_1}} (z+ \xi) + \cdots + \hat 1_{K_{v_1}} (z+ \xi) \right) \\
&= \sum_{\xi \in \mathbb{Z}^d} \hat 1_{K_{v_1}} (z+ \xi) + \cdots + \sum_{\xi \in \mathbb{Z}^d} \hat 1_{K_{v_N}} (z+ \xi) \\
&= \sum_{n \in \mathbb{Z}^d} 1_{K_{v_1}} (n) e^{2\pi i \langle n, z\rangle} + \cdots +
\sum_{n\in \mathbb{Z}^d} 1_{K_{v_N}} (n) e^{2\pi i \langle n, z\rangle} \\
&:= \sum_{n \in \mathbb{Z}^d \cap K_{v_1} } e^{2\pi i \langle n, z\rangle} + \cdots +
\sum_{n \in \mathbb{Z}^d \cap K_{v_N} } e^{2\pi i \langle n, z\rangle},
\end{align*}
where we have used the Poisson summation formula in the second and fifth equalities.
The third equality uses Brion's Theorem \ref{brion, continuous form} for the Fourier transform of ${\mathcal P}$.
Step $2$ [{\bf Rigorous proof}]. \ To make Step $1$ rigorous, we pick any compactly supported approximate identity
$\phi_\varepsilon$, and form a smoothed
version of the function in step $1$. Namely we let
\[
f_\varepsilon(x):= (1_{{\mathcal P}}(x) e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon(x),
\]
so that now we are allowed to apply Poisson summation \index{Poisson summation formula}
to $f_\varepsilon$, because our choice of a smooth and compactly supported $\phi_\varepsilon$ implies that $f_\varepsilon$ is a Schwartz function.
Recalling Theorem \ref{approximate identity convolution}, we know that at a point $x\in \mathbb{R}^d$ of continuity of
$1_{{\mathcal P}}(x) e^{2\pi i \langle x, z\rangle}$, we have
\[
\lim_{\varepsilon \rightarrow 0} f_\varepsilon(x) = 1_{{\mathcal P}}(x) e^{2\pi i \langle x, z\rangle}.
\]
To proceed further,
it is therefore natural to consider points $x \in \interior {\mathcal P}$, the interior of ${\mathcal P}$, because $1_{\mathcal P}$ is continuous there, while it is not continuous on the boundary of ${\mathcal P}$. To recap, we have so far the equalities
\begin{align*}
\sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle} &:=
\sum_{n \in \mathbb{Z}^d} 1_{ \interior {\mathcal P}}(x) e^{2\pi i \langle x, z\rangle} \\
&= \sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} \lim_{\varepsilon \rightarrow 0} f_\varepsilon(n) \\
&= \lim_{\varepsilon \rightarrow 0} \sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} f_\varepsilon(n),
\end{align*}
where we've used the fact that $f_\varepsilon$ is compactly supported, because it is the convolution of two compactly supported functions. So the exchange above, of the sum with the limit, is trivial because the sum is finite.
With this in mind, the Poisson summation formula, applied to the Schwarz function $f_\varepsilon$,
gives us:
\begin{align*}
&\sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle} =
\lim_{\varepsilon \rightarrow 0} \sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} f_\varepsilon(n)
= \lim_{\varepsilon \rightarrow 0} \sum_{n \in \mathbb{Z}^d} \left(1_{ \interior {\mathcal P}} \
e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon\right) (n) \\
&=\lim_{\varepsilon \rightarrow 0} \sum_{n \in \mathbb{Z}^d} {\mathcal F}{ \big( (1_{ \interior {\mathcal P}} \
e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon \big) }(\xi) \\
&= \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in \mathbb{Z}^d} \hat 1_{ \interior {\mathcal P}}(z+ \xi) \hat \phi_\varepsilon(\xi) \\
&= \lim_{\varepsilon \rightarrow 0}
\sum_{\xi \in \mathbb{Z}^d} \left( \hat 1_{ \interior {\mathcal K}_{v_1}} (z+ \xi) + \cdots +
\hat 1_{ \interior {\mathcal K}_{v_1}} (z+ \xi) \right)
\hat \phi_\varepsilon(\xi) \\
&= \lim_{\varepsilon \rightarrow 0}
\sum_{\xi \in \mathbb{Z}^d} {\mathcal F}{ \big( (1_{ \interior {\mathcal K}_{v_1}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon \big) }(\xi)
+ \cdots +
\lim_{\varepsilon \rightarrow 0} \sum_{\xi \in \mathbb{Z}^d}
{\mathcal F}{ \big( (1_{ \interior {\mathcal K}_{v_N}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon \big) }(\xi) \\
&= \lim_{\varepsilon \rightarrow 0}
\sum_{n \in \mathbb{Z}^d} (1_{ \interior {\mathcal K}_{v_1}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon (n)
+ \cdots +
\lim_{\varepsilon \rightarrow 0}
\sum_{\xi \in \mathbb{Z}^d} (1_{ \interior {\mathcal K}_{v_N}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon (n) \\
&= \sigma_{ \interior {\mathcal K}_{v_1}}(z) + \cdots + \sigma_{ \interior {\mathcal K}_{v_N}}(z),
\end{align*}
We've applied Theorem \ref{approximate identity convolution} to $f(n) := 1_{\interior {\mathcal K}_v}(n)$,
for each $n \in \interior {\mathcal K}_v$, because $f$ is continuous at all such points.
The conclusion of Theorem \ref{approximate identity convolution} is that
\[
\lim_{\varepsilon \rightarrow 0}
\Big( (1_{ \interior {\mathcal K}_{v_1}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon \Big)(n) =
1_{ \interior {\mathcal K}_{v_1}}(n) \ e^{2\pi i \langle n, z\rangle},
\]
and by Lemma \ref{technical lemma1} the last equality, in the long string of equalities above, is justified.
\end{proof}
\bigskip
\begin{example} \label{example:standard triangle integer point transform}
\rm{
We compute the integer point transform of the {\bf standard triangle} in the plane, using Brion's Theorem \ref{brion, discrete form}. Namely,
for the standard triangle
\[
\Delta:= \conv( \icol{0\{\bf 0}}, \icol{1\{\bf 0}}, \icol{0\{\bf 1}}),
\]
as depicted in Figure \ref{standard triangle}, we find $\sigma_{\Delta}(z)$.
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.7in]{triangle}
\end{center}
\caption{The standard triangle, with its vertex tangent cones} \label{standard triangle}
\end{figure}
By definition, the integer point transform of its vertex tangent cone ${\mathcal K}_{v_1}$ is
\begin{align*}
\sigma_{{\mathcal K}_{v_1}}(z) &:= \sum_{n \in {\mathcal K}_{v_1} \cap \mathbb{Z}^d} e^{\langle n, z\rangle}
=
\sum_{n_1 \geq 0, n_2 \geq 0}
e^{ \langle n_1 \icol{1\{\bf 0}} + n_2\icol{0\{\bf 1}} ,
z\rangle} \\
& = \sum_{n_1 \geq 0} e^{ n_1 z_1} \sum_{n_2 \geq 0} e^{ n_2 z_2} \\
& = \frac{1}{(1- e^{z_1} )(1- e^{z_2})}.
\end{align*}
For the vertex tangent cone ${\mathcal K}_{v_2}$, we have
\begin{align*}
\sigma_{{\mathcal K}_{v_2}}(z) &:= \sum_{n \in {\mathcal K}_{v_2} \cap \mathbb{Z}^d} e^{\langle n, z\rangle}
=
\sum_{n_1 \geq 0, n_2 \geq 0}
e^{\langle \icol{1\{\bf 0}} + n_1 \icol{-1\\ \ 0} + n_2\icol{-1\\ \ 1} ,
z\rangle} \\
& = e^{z_1} \sum_{n_1 \geq 0} e^{ n_1 (-z_1)}
\sum_{n_2 \geq 0} e^{ n_2 (-z_1+z_2)} \\
& = \frac{e^{z_1}}{(1- e^{-z_1} )(1- e^{-z_1+z_2})}.
\end{align*}
Finally, for the vertex tangent cone ${\mathcal K}_{v_3}$, we have
\begin{align*}
\sigma_{{\mathcal K}_{v_3}}(z) &:= \sum_{n_1 \geq 0, n_2 \geq 0}
e^{\langle \icol{0\{\bf 1}} + n_1 \icol{ \ 0\\-1} + n_2\icol{\ 1\\-1}, z \rangle} \\
& = e^{z_2} \sum_{n_1 \geq 0} e^{ n_1 (-z_2)}
\sum_{n_2 \geq 0} e^{ n_2 (z_1-z_2)} \\
& = \frac{e^{z_2}}{(1- e^{-z_2} )(1- e^{z_1-z_2})}.
\end{align*}
Altogether, using \ref{simplified discrete Brion identity} we have
\begin{align}
\sigma_{{\mathcal P}}(z) &= \sigma_{{\mathcal K}_{v_1}}(z) + \sigma_{{\mathcal K}_{v_2}}(z) +\sigma_{{\mathcal K}_{v_3}}(z) \\
&= \frac{1}{(1- e^{z_1} )(1- e^{z_2})} + \frac{e^{z_1}}{(1- e^{-z_1} )(1- e^{-z_1+z_2})}
+\frac{e^{z_2}}{(1- e^{-z_2} )(1- e^{z_1-z_2})}. \label{last line}
\end{align}
}
\hfill $\square$
\end{example}
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.3in]{Example2}
\end{center}
\caption{A triangle with vertices $v_1, v_2, v_3$, and its vertex tangent cones} \label{triangle}
\end{figure}
\bigskip
\section{Examples, examples, examples}
\bigskip
\begin{example}\label{triangle1}
Here we will compute the integer point transform of the triangle $\Delta$ defined by the convex hull of the points $\icol{0\{\bf 0}}, \icol{3\{\bf 1}}, \icol{3\\6}$, as shown in Figure \ref{triangle}.
We first compute the integer point transforms of all of its tangent cones. For the vertex $v_1$, we already computed the integer point transform of its tangent cone in the previous example.
For the vertex $v_2$, we notice that its vertex tangent cone is a unimodular cone, because
$| \det
\big(\begin{smallmatrix}
0 & -1 \\
-1 & -2
\end{smallmatrix}
\big) | = 1$. Its integer point transform is:
\begin{align*}
\sigma_{{\mathcal K}_{v_2}}(z) &:= \sum_{n \in {\mathcal K}_{v_2} \cap \mathbb{Z}^d} e^{\langle n, \ z\rangle}
=
\sum_{n_1 \geq 0, n_2 \geq 0}
e^{ \langle
\icol{3\\6} + n_1 \icol{0\\-1} + n_2\icol{-1\\-2} ,
z\rangle} \\
& = e^{ 3z_1 + 6z_2)} \sum_{n_1 \geq 0, n_2 \geq 0}
e^{ n_1 (-z_2)} e^{ n_2(-z_1 -2z_2)} \\
& = \frac{ e^{3z_1 + 6z_2} }
{
(1- e^{-z_2} )(1- e^{ -z_1 -2z_2} )
}.
\end{align*}
Equivalently, using the notation from Example \ref{integer point transform of cone} above,
\[
\sigma_{{\mathcal K}_{v_2}}(z) := \sum_{n \in {\mathcal K}_{v_2} \cap \mathbb{Z}^d} q^n
=\frac{ q_1^3 q_2^6}
{
(1- q_2^{-1} ) (1- q_1^{-1} q_2^{-2}).
}
\]
For vertex $v_3$, the computation is similar to vertex tangent cone ${\mathcal K}_{v_1}$, and
we have:
\begin{align*}
\sigma_{{\mathcal K}_{v_3}}(z) &:= \sum_{n \in {\mathcal K}_{v_3} \cap \mathbb{Z}^d} e^{\langle n, \ z\rangle}
= \sum_{n_1 \geq 0, n_2 \geq 0} e^{\langle
\icol{3\{\bf 1}} + n_1 \icol{-3\\-1} + n_2\icol{0\{\bf 1}}, \
z\rangle} \\
&= e^{3z_1 + z_2} \sum_{n_1 \geq 0, n_2 \geq 0} e^{(-3z_1-z_2) n_1}
e^{2\pi i (z_2) n_2} \\
&= e^{3z_1 + z_2} \frac{ 1+ e^{-z_1} + e^{-2z_1} }
{
(1- e^{ 3z_1 + z_2} )(1- e^{ z_2} )
} \\
&= \frac{ e^{3z_1 + z_2} + e^{2z_1 + z_2} + e^{z_1 + z_2} }
{
(1- e^{3z_1 + z_2} )(1- e^{z_2} )
} \\
&= \frac{ q_1^3 q_2 + q_1^2 q_2 + q_1q_2 }
{
(1- q_1^{-3} q_2^{-1} )(1- q_2)
}.
\end{align*}
Finally, putting all of the three vertex tangent cone contributions together,
Theorem \ref{brion, discrete form} gives us:
\begin{align*}
\sigma_{\Delta}(z) &= \sigma_{{\mathcal K}_{v_1}}(z) + \sigma_{{\mathcal K}_{v_2}}(z) + \sigma_{{\mathcal K}_{v_3}}(z) \\
&= \frac{ 1+ q_1 q_2 + {q_1}^2 q_2 + {q_1}^2 {q_2}^2 + q_1^3 q_2^2
}
{ (1- q_1^3 q_2 ) (1- q_1 q_2^2)
}
+
\frac{ q_1^3 q_2^6}
{
(1- q_2^{-1} ) (1- q_1^{-1} q_2^{-2})
}
+
\frac{ q_1^3 q_2 + q_1^2 q_2 + q_1q_2 }
{
(1- q_1^{-3} q_2^{-1} )(1- q_2)
}.
\end{align*}
\hfill $\square$
\end{example}
\bigskip
\begin{example}\label{integer point transform of cone}
\rm{
We work out the integer point transform $\sigma_{\mathcal K}(z)$ of the cone
\[
{\mathcal K} := \{
\lambda_1 \big(\begin{smallmatrix}
3 \\
1 \\
\end{smallmatrix}
\big) +
\lambda_2 \big(\begin{smallmatrix}
1 \\
2 \\
\end{smallmatrix}
\big) \mid \lambda_1, \lambda_2 \in \mathbb{R}_{\geq 0} \},
\]
Drawn in the figures below. We note that here $\det {\mathcal K} = 5$, and that there are indeed $5$ integer points in
$D$, its half-open fundamental parallelepiped.
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{slide2}
\end{center}
\caption{The $5$ integer points in a fundamental parallelepiped $D$ of the cone ${\mathcal K}$.} \label{cone2}
\end{figure}
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{slide3}
\end{center}
\caption{The point ${1\choose 1}$ in $D$, with its images in ${\mathcal K}$ under translations by the edge vectors of ${\mathcal K}$.}
\label{cone3}
\end{figure}
We may `divide and conquer' the integer point transform $\sigma_{\mathcal K}(z)$,
by breaking it up into $5$ infinite series,
one for each integer point in $D$, as follows:
\[
\sigma_{\mathcal K}(z) := \sum_{n \in {\mathcal K}\cap \mathbb{Z}^d} e^{\langle n, z\rangle}
:= \sum_{\icol{0\{\bf 0}}} + \sum_{\icol{1\{\bf 1}}} + \sum_{\icol{2\{\bf 1}}} + \sum_{\icol{2\\2}} + \sum_{\icol{3\\2}},
\]
where
\begin{align*}
\sum_{\icol{1\{\bf 1}}} &:= \sum_{n_1 \geq 0, n_2 \geq 0}
e^{\langle \big(\begin{smallmatrix}
1 \\
1 \\
\end{smallmatrix} \big) +
n_1 \big(\begin{smallmatrix}
3 \\
1 \\
\end{smallmatrix}
\big) +
n_2 \big(\begin{smallmatrix}
1 \\
2 \\
\end{smallmatrix}
\big) , z \rangle} \\
&=
e^{ \langle \big(\begin{smallmatrix}
1 \\
1 \\
\end{smallmatrix} \big) , z \rangle}
\sum_{n_1 \geq 0, n_2 \geq 0}
e^{ \langle
n_1 \big(\begin{smallmatrix}
3 \\
1 \\
\end{smallmatrix}
\big) +
n_2 \big(\begin{smallmatrix}
1 \\
2 \\
\end{smallmatrix}
\big) , z\rangle} \\
&=
e^{ \langle \big(\begin{smallmatrix}
1 \\
1 \\
\end{smallmatrix} \big) , z \rangle}
\sum_{n_1 \geq 0}
e^{n_1 \langle
\big(\begin{smallmatrix}
3 \\
1 \\
\end{smallmatrix}
\big) , z\rangle}
\sum_{n_2 \geq 0}
e^{n_2 \langle
\big(\begin{smallmatrix}
1 \\
2 \\
\end{smallmatrix}
\big) , z\rangle} \\
&=
\frac{e^{ z_1 + z_2}
}
{ (1-e^{3z_1 + z_2})(1-e^{z_1 + 2z_2})
},
\end{align*}
and similarly we have
\[
\sum_{\icol{2\{\bf 1}}}=
\frac{e^{2z_1 + z_2}
}
{ (1-e^{ 3z_1 + z_2})(1-e^{z_1 + 2z_2})
},
\]
\[
\sum_{\icol{2\\2}} =
\frac{e^{ 2z_1 + 2z_2}
}
{ (1-e^{ 3z_1 + z_2})(1-e^{z_1 + 2z_2})
},
\]
\[
\sum_{\icol{3\\2}} =
\frac{e^{ 3z_1 + 2z_2}
}
{ (1-e^{ 3z_1 + z_2})(1-e^{z_1 + 2z_2})
},
\]
and finally
\[
\sum_{\icol{0\{\bf 0}}} =
\frac{1}
{ (1-e^{3z_1 + z_2})(1-e^{z_1 + 2z_2})
}.
\]
To summarize, we have the following expression:
\[
\sum_{n \in {\mathcal K}\cap \mathbb{Z}^d} e^{ \langle n, z\rangle}
=\frac{ 1+ e^{z_1 + z_2} + e^{ 2z_1 + z_2)} + e^{2z_1 + 2z_2} + e^{3z_1 + 2z_2 }
}
{ (1-e^{3z_1 + z_2})(1-e^{z_1 + 2z_2})
}.
\]
Equivalently, using our multinomial notation $q_j:= e^{ z_j}$, we have
\[
\sum_{n \in {\mathcal K}\cap \mathbb{Z}^d} q^n
=\frac{ 1+ q_1 q_2 + {q_1}^2 q_2 + {q_1}^2 {q_2}^2 + q_1^3 q_2^2
}
{ (1- q_1^3 q_2 ) (1- q_1 q_2^2)
}.
\]
}
\hfill $\square$
\end{example}
\section{Integer point transforms of cones in general: \\ rational functions}
The Examples \ref{triangle1} and \ref{integer point transform of cone} above suggest a general pattern, namely that integer point transforms are always rational functions, and that their numerators are polynomials that encode the integer points inside a fundamental parallelepiped $\Pi$
that sits at the vertex of each vertex tangent cone. The proof of this general fact will be fairly easy - we only need to put several geometric series together, as in Figure \ref{cone3}. Now that we've seen some examples, we can prove things in general.
First, given any $d$-dimensional simplicial rational cone ${\mathcal K}\subset \mathbb{R}^d$, with integer edge vectors
$w_1, \dots, w_d \in \mathbb{Z}^d$, and apex $v\in \mathbb{R}^d$, we define the {\bf fundamental parallelepiped} of ${\mathcal K}$ by:
\begin{equation}
\Pi := \{ \lambda_1 w_1 + \cdots + \lambda_d w_d \mid
\text{ all } 0 \leq \lambda_j < 1 \},
\end{equation}
a half-open, integer parallelepiped.
In the same way that we've encoded integer points in polytopes using
$\sigma_{\mathcal P}(z)$, we can encode the integer points in $\Pi$ by defining
\[
\sigma_\Pi(z) := \sum_{n \in \mathbb{Z}^d \cap \Pi} e^{\langle z, n \rangle}.
\]
For a rational simplicial cone $K_v$, it turns out that its integer point transform
\[
\sigma_{K_v}(z):= \sum_{n \in {\mathcal K}_v \cap \mathbb{Z}^d} e^{ \langle n, z \rangle}
\]
has a pretty structure theorem - it is a rational function of the variables $e^{z_1}, \dots, e^{z_d}$, as follows.
\begin{thm} \label{closed form for integer point transform of a cone}
Given a $d$-dimensional simplicial cone ${\mathcal K}_v \subset \mathbb{R}^d$, with apex $v \in \mathbb{R}^d$, and with
$d$ linearly independent {\bf integer} edge vectors
$w_1(v), w_2(v), \dots, w_d(v) \in \mathbb{Z}^d$. Then:
\begin{equation}
\sigma_{K_v}(z)
=
\frac{ \sigma_{ \Pi + v}(z) }
{ \prod_{k=1}^d \left( 1 - e^{ \langle w_k , z \rangle} \right) }.
\end{equation}
\end{thm}
\begin{proof}
We claim that we can parametrize all of the integer points in the cone ${\mathcal K}_v$ precisely by
\begin{equation}\label{claim:the cone integer points}
{\mathcal K}_v \cap \mathbb{Z}^d = \{ p + m_1 w_1 + \cdots + m_d w_d \mid p \in (\Pi+ v)\cap \mathbb{Z}^d, \text{ and all } m_j \in \mathbb{Z}_{\geq 0}\}.
\end{equation}
To prove \eqref{claim:the cone integer points}, we begin by writing each $m\in {\mathcal K}_v \cap \mathbb{Z}^d$, by definition of the cone ${\mathcal K}_v$,
as follows:
\[
m = v+ \lambda_1 w_1 + \cdots + \lambda_d w_d,
\]
with the $\lambda_k \geq 0$. This representation of $m$ is unique,
because $w_1, \dots, w_d$ is a basis for $\mathbb{R}^d$. Now we use the fact that each
$\lambda_k = \lfloor \lambda_k \rfloor + \{ \lambda_k \}$, where $\{x\}$ is the fractional part of $x$:
\begin{align*}
m &= v+ \Big( \{ \lambda_1 \} w_1 + \cdots + \{ \lambda_d \} w_d \Big) +
\lfloor \lambda_1 \rfloor w_1 + \cdots + \lfloor \lambda_d \rfloor w_d \\
&:=p + \lfloor \lambda_1 \rfloor w_1 + \cdots + \lfloor \lambda_d \rfloor w_d,
\end{align*}
where we've defined $p:= v+ \Big( \{ \lambda_1 \} w_1 + \cdots + \{ \lambda_d \} w_d \Big)$. We now notice that
$p \in v+ \Pi$, and in fact $p \in \mathbb{Z}^d$, because $m, w_1, \dots, w_d \in \mathbb{Z}^d$.
Since $\Pi$ tiles the cone ${\mathcal K}_v$ precisely by the translation vectors $w_1, \dots, w_d$, we see
that the set of all integer points in ${\mathcal K}_v$ is precisely
the disjoint union of the sets
\begin{equation} \label{typical integer points in the cone}
\{ p + k_1 w_1 + \dots + k_d w_d \mid \, k_1, \dots, k_d \in \mathbb{Z}_{\geq 0} \}
\end{equation}
(which we may think of as `multidimensional arithmetic progressions') , as $p$ varies over the integer points of $\Pi$ .
Finally, we expand each denominator in the following rational function as a geometric series to get:
\[
\frac{ \sigma_{ \Pi + v}(q) }
{ \prod_{j=1}^d \left( 1 - q^{ w_j} \right) } =
\left(
\sum_{p \in (\Pi + v) \cap \mathbb{Z}^d} q^p
\right)
\left(
\sum_{k_1 \geq 0} q^{k_1 w_1}
\right)
\cdots
\left(
\sum_{k_d \geq 0} q^{k_d w_d}
\right).
\]
Multiplying out all of these geometric series together, we see that the exponents look like
the points in \eqref{typical integer points in the cone}.
\end{proof}
\bigskip
\section*{Notes} \label{Notes.chapter.Brion}
\begin{enumerate}[(a)]
\item In the development of the current book, we saw that the discrete version of Brion's theorem
(Theorem \ref{brion, discrete form}) followed from the continuous version of Brion's theorem
(Theorem \ref{brion, continuous form}). The tool we used in order to discretize Theorem \ref{brion, continuous form} was the Poisson summation formula.
In our previous book \cite{BeckRobins}, the ideas developed in exactly the opposite direction: in that context we first proved the discrete Brion theorem, and then derived the continuous version from it.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\medskip
\begin{prob} \label{unimodular cone, integer point transform}
Suppose that ${\mathcal P}\subset \mathbb{R}^d$ is a unimodular polytope, with vertex set $V$. Using Theorem \ref{brion, discrete form}, show that its integer point transform is:
\begin{equation}
\sigma_{\mathcal P}(z) = \sum_{v \in V}
\frac{e^{\langle v, z \rangle} }{\prod_{k=1}^d \left( 1 - e^{\langle w_k, z\rangle} \right) }.
\end{equation}
\end{prob}
\medskip
\begin{prob}
Fix a positive integer $m > 1$, and let ${\mathcal P}$ be the $2$-dimensional triangle
whose vertices are given by
$(0, 0), (0, 1)$, and $(m, 0)$. First compute the integer point transforms
$\sigma_{K_v}(z)$ for its three vertex tangent cones, and then compute the integer point transform $\sigma_{\mathcal P}(z)$.
\end{prob}
\medskip
\begin{prob} \label{bound for integer point transform}
Let ${\mathcal P} \subset \mathbb{R}^d$ be the $d$-dimensional polytope. We recall that for $x \in \mathbb{R}^d$, we have the definition
$\sigma_{\mathcal P}(2\pi i x) := \sum_{n\in {\mathcal P} \cap \mathbb{Z}^d } e^{2\pi i \langle x, n \rangle}$.
Prove that
\[
\left | \sigma_{\mathcal P}(2\pi i x) \right | \leq \left |\mathbb{Z}^d \cap {\mathcal P} \right |,
\]
for all $x \in \mathbb{R}^d$.
\end{prob}
\medskip
\begin{prob}
Let ${\mathcal P}$ be the $3$-dimensional simplex whose vertices are given by
$(0, 0, 0), (1, 1, 0), (1, 0, 1)$, and $(0, 1, 1)$. Compute its integer point transforms
$\sigma_{K_v}(z)$ for all of its four vertex tangent cones, and then compute its integer point transform $\sigma_{\mathcal P}(z)$.
\end{prob}
\medskip
\begin{prob}
Suppose we are given a $2$-dimensional simplicial integer cone ${\mathcal K} \subset \mathbb{R}^2$, together with its polar cone ${\mathcal K}^o$.
Is there a simple relationship between
the integer point transforms $\sigma_{{\mathcal K}}(z)$ and $\sigma_{{\mathcal K}^o}(z)$?
\end{prob}
Notes. It's worth thinking about the relationship between the edge vectors of the
fundamental parallelepipeds for ${\mathcal K}$ and ${\mathcal K}^o$.
\chapter{Counting integer points in polytopes - the Ehrhart theory} \label{Ehrhart theory}
\index{Ehrhart theory}
\begin{quote}
``How wonderful that we have met with a paradox. Now we have some hope of making progress. ''
-- Niels Bohr \index{Niels Bohr}
\end{quote}
\begin{wrapfigure}{R}{0.58\textwidth}
\centering
\includegraphics[width=0.62\textwidth]{DiscreteArea}
\end{wrapfigure}
\section{Intuition}
A basic question in discrete geometry is ``how do we discretize volume?"
One method of discretizing the volume of ${\mathcal P}$ is to count the number of integer points in ${\mathcal P}$.
Even in $\mathbb{R}^2$, this question may be highly non-trivial, depending on the arithmetic properties of the vertices of ${\mathcal P}$. Ehrhart first considered integer dilations of a fixed, integer polytope ${\mathcal P}$, and
defined:
\begin{equation} \label{combinatorial discrete volume}
L_{{\mathcal P}}(t):= | \mathbb{Z}^d \cap t{\mathcal P} |,
\end{equation}
where $t{\mathcal P}$ is the $t$'th dilate of ${\mathcal P}$, and $t$ is a positive integer. Ehrhart showed
that $L_{{\mathcal P}}(t)$ is a polynomial in the positive integer parameter $t$, known as the
{\bf Ehrhart polynomial} of ${\mathcal P}$.
Viewed from the lens of Fourier analysis, Ehrhart polynomials may be computed by `averaging'
the Fourier transform of a polytope over the full integer lattice:
\begin{equation}
L_{{\mathcal P}}(t):= | \mathbb{Z}^d \cap t{\mathcal P} | = \sum_{n\in \mathbb{Z}^d} 1_{t{\mathcal P}}(n) = \sum_{\xi \in \mathbb{Z}^d} \hat 1_{t{\mathcal P}}(\xi),
\end{equation}
where the latter identity uses Poisson summation, but because we may not use indicator functions directly in
Poisson summation, \index{Poisson summation formula}
some care is required and the process of smoothing may be applied to $1_{\mathcal P}$.
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{Ehrhart1}
\end{center}
\caption{Here the polytope ${\mathcal P}$ is the unit square, and we also have its $5$'th dilate $5{\mathcal P}$.} \label{Ehrhart1}
\end{figure}
We can also compare this combinatorial method to discretize volume, namely equation \eqref{combinatorial discrete volume},
and the discrete volumes of the previous chapter which used solid angles.
More generally, given a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$, we may sum the values of $f$
at all integer points and observe how close this sum gets to the integral of $f$ over ${\mathcal P}$. This approach is known as Euler-Maclaurin summation \index{Euler-Maclaurin summation}
over polytopes, and is a current and exciting topic of a growing literature (see Note \ref{EM summation note} below).
\bigskip
\section{Computing integer points in polytopes via the discrete Brion Theorem}
\begin{example}
\rm{
Probably the simplest example in $\mathbb{R}^2$ is the unit square ${\mathcal P}:= [0, 1]^2$.
As Figure \ref{Ehrhart1} suggests,
the $t$-dilate $t{\mathcal P}$ here contains $(t+1)^2 = t^2 + 2t + 1$ points of the integer lattice $\mathbb{Z}^2$. Here it was easy to conclude that
$L_{\mathcal P}(t)$ was a polynomial function of $t \in \mathbb{Z}_{>0}$, but by a small miracle of nature
a similar phenomenon occurs for \emph{all integer polytopes} in $\mathbb{R}^d$.
\hfill $\square$
}
\end{example}
If all of the vertices of ${\mathcal P}$ have integer coordinates, we call ${\mathcal P}$ an {\bf integer polytope}.
On the other hand, if all of the vertices of a polytope ${\mathcal P}$ have rational coordinates, we call ${\mathcal P}$ a {\bf rational polytope} \index{rational polytope}.
Let ${\mathcal P} \subset \mathbb{R}^d$ be a rational, $d$-dimensional polytope, and let $N$ be the number of its vertices.
For each vertex $v$ of ${\mathcal P}$, we consider the vertex tangent cone ${\mathcal K}_v$ of $ {\mathcal P}$. Once we dilate ${\mathcal P}$ by $t$, each
vertex $v$ of ${\mathcal P}$ gets dilated to become $tv$, and so each of the vertex tangent cones ${\mathcal K}_v$ of ${\mathcal P}$ simply get shifted to the
corresponding vertex tangent cone ${\mathcal K}_{tv}$ of $t{\mathcal P}$. Using the discrete Brion theorem (Theorem \ref{brion, discrete form}), we have
\begin{equation} \label{simplified discrete Brion identity}
\sum_{n \in t{\mathcal P} \cap \mathbb{Z}^d} e^{ \langle n, z\rangle} =
\sum_{n \in {\mathcal K}_{tv_1} \cap \mathbb{Z}^d} e^{ \langle n, z\rangle} + \cdots +
\sum_{n \in {\mathcal K}_{tv_N} \cap \mathbb{Z}^d} e^{\langle n, z\rangle},
\end{equation}
for all $z \in \mathbb{C}^d - S$, where $S$ is the hyperplane arrangement defined by the (removable) singularities of all of the transforms $\hat 1_{{\mathcal K}_{v_j}}(z)$.
To simplify notation, we have absorbed the constant $-2\pi i$ into the complex vector $z$ by replacing
$z$ by $-\frac{1}{2\pi i } z$.
We recall that we rewrote \eqref{simplified discrete Brion identity} by using the notation:
\begin{equation}\label{discrete Brion, with rational functions}
\sigma_{\mathcal P}(z) = \sigma_{{\mathcal K}_{v_1}}(z) + \cdots + \sigma_{{\mathcal K}_{v_N}}(z).
\end{equation}
And now we notice that when $z = 0$, the left-hand-side gives us precisely
\[
\sum_{n \in t{\mathcal P} \cap \mathbb{Z}^d} 1 := | \mathbb{Z}^d \cap t{\mathcal P} |,
\]
which is good news - it is the Ehrhart polynomial $L_{{\mathcal P}}(t)$, by definition. The bad news is that $z=0$ is a singularity of the right-hand-side of
\eqref{discrete Brion, with rational functions}. But then again, there is still more good news - we already saw in the previous chapter
that it is a removable singularity. So we may let $z\rightarrow 0$, and discover what happens.
\bigskip
\begin{example} \label{Ehrhart poly for the standard triangle}
\rm{
We can find a formula for the Ehrhart polynomial $L_{{\mathcal P}}(t) := | \mathbb{Z}^2 \cap t{\mathcal P} |$ of the standard triangle, continuing Example \ref{example:standard triangle integer point transform}. It turns out
that the method we use in this example is universal - it can always be used to find the Ehrhart polynomial of any rational polytope. We will formalize this statement in the ensuing section.
In this example we are lucky in that we may use brute-force to compute it, since the number of integer points in the $t$-dilate of ${\mathcal P}$ may be computed along the diagonals:
\[
L_{{\mathcal P}}(t) = 1 + 2 + 3 + \cdots + (t+1) = \frac{(t+1)( t+2)}{2} =
\frac{1}{2} t^2 + \frac{3}{2}t + 1.
\]
Now we can confirm this lucky answer with our brand new machine, as follows.
Using \eqref{simplified discrete Brion identity},
and the formulation \eqref{last line} from
Example \ref{example:standard triangle integer point transform}.,
we have the integer point transform for the dilates of ${\mathcal P}$:
\begin{align}
& \sum_{n \in t\Delta \cap \mathbb{Z}^d} e^{ \langle n, z\rangle} =
\sum_{n \in {\mathcal K}_{tv_1} \cap \mathbb{Z}^d} e^{ \langle n, z\rangle}
+ \sum_{n \in {\mathcal K}_{tv_1} \cap \mathbb{Z}^d} e^{ \langle n, z\rangle}
+ \sum_{n \in {\mathcal K}_{tv_3} \cap \mathbb{Z}^d} e^{\langle n, z\rangle} \\
\label{integer point transform for standard triangle}
&= \frac{1}{(1- e^{z_1} )(1- e^{z_2})} + \frac{e^{t z_1}}{(e^{-z_1}-1 ) ( e^{-z_1+z_2} -1)}
+\frac{e^{t z_2}}{(e^{-z_2}-1)(e^{z_1-z_2}-1)} \\ \label{symmetric rationals}
& := F_1(z) + F_2(z) + F_3(z),
\end{align}
where we have defined $F_1, F_2, F_3$ by the last equality. We can let $z\rightarrow 0$ along almost any direction, but it turns out that we can simplify our computations by taking advantage of the symmetry of this polytope, so we will pick $z = \icol{ \ x \\ -x} $, which will simplify our computations
(see Note \ref{Michel Faleiros}). Here is our plan:
\begin{enumerate}[(a)]
\item We pick $z:= \icol{ \ x\\ -x} $.
\item We expand all three meromorphic functions $F_1, F_2, F_ 3$
in terms of their Laurent series in $x$, giving us Bernoulli numbers.
\item Finally, we let $x \rightarrow 0$, to retrieve the constant term (which will be
a polynomial function of $t$) of the resulting Laurent series.
\end{enumerate}
To expand $F_1(z), F_2(z), F_3(z)$ in their Laurent series, we recall the definition
\ref{Def. of Bernoulli numbers}
of the Bernoulli numbers in terms of their generating function, namely
$ \frac{t}{e^t-1} = \sum_{k =0}^\infty B_k \frac{t^k}{k!}$:
\begin{align*}
F_1(x, -x) &= \frac{-1}{x^2} \sum_{m \geq 0} B_m \frac{x^m}{m!}
\ \sum_{n \geq 0} B_n \frac{(-x)^n}{n!} \\
&= \frac{-1}{x^2} \left( 1 - \frac{x}{2} + \frac{x^2}{12} + O(x^3)\right)
\left( 1 + \frac{x}{2} + \frac{x^2}{12} + O(x^3)\right) \\
&= \frac{-1}{x^2} - \frac{1}{3} + O(x)
\end{align*}
Similarly, we have
\begin{align*}
F_2(x, -x) &= \frac{1 + t x + \frac{t^2}{2!} x^2 + O(x^3) }{2x^2}
\left(1 + \frac{x}{2} + \frac{x^2}{12} + O(x^3) \right)
\left(1 + \frac{(2x)}{2} + \frac{(2x)^2}{12} + O(x^3) \right) \\
&= \frac{1}{2x^2} + \frac{3}{4x} + \frac{2}{3} + \frac{t}{2x} + \frac{3t}{4} + \frac{t^2}{4} + O(x)
\end{align*}
Now, by symmetry we see that $F_3(x,-x) = F_2(-x, x)$, so that by \eqref{symmetric rationals}
and the latter expansions, we finally have:
\[
\sum_{n \in t\Delta \cap \mathbb{Z}^d} e^{ \langle n, \icol{ \ x\\ -x} \rangle} =
F_1(x, -x) + F_2(x, -x) + F_2(-x, x) = 1 + \frac{3}{2} t + \frac{1}{2} t^2 + O(x).
\]
Letting $z:= \icol{ \ x \\ -x} \rightarrow 0$ in the latter
computation, we retrieve the (Ehrhart) polynomial:
\[
\sum_{n \in t\Delta \cap \mathbb{Z}^d} 1 = L_\Delta(t) = 1 + \frac{3}{2} t + \frac{1}{2} t^2,
\]
as desired.
}
\hfill $\square$
\end{example}
\bigskip
\section{The Ehrhart polynomial of an integer polytope, and the Ehrhart quasi-polynomial of a rational polytope}
\begin{wrapfigure}{R}{0.39\textwidth}
\centering
\includegraphics[width=0.30\textwidth]{Ehrhart}
\caption{Eugene Ehrhart} \label{Ehrhart}
\end{wrapfigure}
Eugene Ehrhart initiated a systematic study of the integer point enumerator
\[
L_{\mathcal P}(t):= \left| t{\mathcal P} \cap \mathbb{Z}^d \right|,
\]
for an integer polytope ${\mathcal P}$, which Ehrhart proved was always a polynomial function of the positive integer dilation parameter $t$.
Ehrhart also proved
that for a rational polytope ${\mathcal P} \subset \mathbb{R}^d$, the integer point enumerator
$L_{\mathcal P}(t)$ is a {\bf quasi-polynomial} in the positive integer parameter $t$, which means by definition that
\begin{equation}
L_{\mathcal P}(t) = c_d t^d + c_{d-1}(t) t^{d-1} + \cdots + c_1(t) t + c_0(t),
\end{equation}
where each $c_j(t)$ is a periodic function of $t \in \mathbb{Z}_{>0}$.
The study of Ehrhart polynomials and Ehrhart quasi-polynomials has enjoyed a renaissance in recent years (\cite{BarvinokEhrhartbook}, \cite{BeckRobins}), and has some suprising connections to many branches of science, and even to voting theory, for example.
\bigskip
\begin{thm}[Ehrhart] \label{Ehrhart's main result}
For an integer polytope \\
${\mathcal P} \subset \mathbb{R}^d$, its discrete volume $L_{\mathcal P}(t)$
is a polynomial functions of $t$, for all positive integer values of the dilation parameter $t$. Moreover, we have
\begin{equation}
L_{\mathcal P}(t) = (\vol {\mathcal P}) t^d + c_{d-1} t^{d-1} + \cdots + c_1 t + 1.
\end{equation}
\end{thm}
\hfill $\square$
Ehrhart's Theorem \ref{Ehrhart's main result} has an extension to rational polytopes, as follows.
We will derive the more general Theorem \ref{Ehrhart's rational polytope theorem} of Ehrhart, by using
the discrete Brion Theorem \ref{brion, discrete form}.
\bigskip
\begin{thm}[Ehrhart] \label{Ehrhart's rational polytope theorem}
For a rational polytope \\
${\mathcal P} \subset \mathbb{R}^d$, its discrete volume $L_{\mathcal P}(t)$
is a quasi-polynomial function of $t$, for all positive integer values of the dilation parameter $t$.
In particular, we have
\begin{equation}
L_{\mathcal P}(t) = (\vol {\mathcal P}) t^d + c_{d-1}(t) t^{d-1} + \cdots + c_1(t) t + c_0(t),
\end{equation}
where each {\bf quasi-coefficient} $c_k(t)$ is a periodic function of $t\in \mathbb{Z}_{>0}$.
\end{thm}
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.3in]{Ehrhart2}
\end{center}
\caption{Left: A triangle ${\mathcal P}$, and its dilate $2{\mathcal P}$. \, Right: the vertex tangent cones ${\mathcal K}_v$ and ${\mathcal K}_{2v}$ have
the same edge vectors $w_1, w_2$. } \label{Ehrhart2}
\end{figure}
\begin{proof}
To begin, suppose that $p$ the least common denominator of the coordinates of all the rational vertices of ${\mathcal P}$.
We need to show that, for each fixed $0\leq r < p$, the integer point enumerator $L_{\mathcal P}(r+ pk)$ is a polynomial in the parameter $k\in \mathbb{Z}_{>0}$.
By definition of a quasi-polynomial, this will prove that $L_{\mathcal P}(t)$ is a quasi-polynomial in $t \in \mathbb{Z}_{>0}$. In other words, we restrict attention to each fixed arithmetic progression of dilations in $t:= r + pk$. Now, from the discrete Brion Theorem \ref{brion, discrete form}, we know that
\begin{equation}\label{Brion, for this proof}
\sigma_{ {\mathcal P}}(z) = \sigma_{ {\mathcal K}_{v_1}}(z) + \cdots + \sigma_{ {\mathcal K}_{v_N}}(z),
\end{equation}
and we also know the elementary relation
\[
\sigma_{ t{\mathcal P}}(0) := \sum_{n \in \mathbb{Z}^d \cap t{\mathcal P}} 1 = L_{\mathcal P}(t).
\]
So we'd like to let $z \rightarrow 0$ on both sides of Brion's discrete identity \eqref{Brion, for this proof}:
\begin{equation}\label{first step of Ehrhart in terms of integer point transforms}
L_{\mathcal P}(t) = \lim_{z\rightarrow 0}
\Big(
\sigma_{ t{\mathcal K}_{v_1}}(z) + \cdots + \sigma_{ t {\mathcal K}_{v_N}}(z)
\Big).
\end{equation}
The bad news is that
the right-hand-side of \eqref{Brion, for this proof}
introduces local singularities in the denominators of each
rational-exponential function
\[
\sigma_{ {\mathcal K}_{v_j}}(z) = \sum_{n \in {\mathcal K}_v \cap \mathbb{Z}^d} e^{ \langle n, z\rangle}.
\]
But there is good news too! These singularities must be removable singularities. The reason is easy -
$ \sigma_{{\mathcal P}}(z)$ is a finite sum of exponentials (by compactness of ${\mathcal P}$), and is therefore an analytic function of $z$, so any singularities on the right-hand side of \eqref{Brion, for this proof} must be removable singularities.
To proceed further, we'll begin by writing each vertex tangent cone ${\mathcal K}_v$ in terms of its vertex $v$, and edge vectors $w_j$:
\[
{\mathcal K}_v := \left\{ v + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\}.
\]
Now we consider the dilates of ${\mathcal K}_v$ a bit more carefully, and we will use the fact that the edge
vectors $w_1, \dots, w_{M_v}$ of any dilate of a vertex tangent cone ${\mathcal K}_{tv}$ remain invariant, as in Figure \ref{Ehrhart2}:
\[
{\mathcal K}_{tv} := \left\{t v + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\}.
\]
{\bf Case $1$.} Suppose $r \not=0$. Then:
\begin{align*}
t{\mathcal K}_v &:= (r+pk){\mathcal K}_v = \left\{ (r+pk)v + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\} \\
&= k(pv) + \left\{ rv + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\} \\
&= k(pv) + r {\mathcal K}_v.
\end{align*}
The salient feature of this computation is that $pv$ is an integer vector, by definition of $p$. This implies that
\begin{align*}
\sigma_{ t {\mathcal K}_{v}}(z) &:=\sigma_{ (r+pk) {\mathcal K}_{v}}(z) :=
\sum_{n \in \Big( k(pv) + r {\mathcal K}_v \Big) \cap \mathbb{Z}^d} e^{ \langle n, z\rangle} \\
&= \sum_{m \in r {\mathcal K}_v \cap \mathbb{Z}^d} e^{ \langle k(pv) + m, z\rangle} \\
&= e^{ \langle k(pv), z\rangle} \sum_{m \in r {\mathcal K}_v \cap \mathbb{Z}^d} e^{ \langle m, z\rangle} \\
&:= e^{ \langle k(pv), z\rangle} \sigma_{rK_v}(z) \\
\end{align*}
Summarizing, \eqref{first step of Ehrhart in terms of integer point transforms} gives us:
\[
L_{\mathcal P}(t)= \lim_{z\rightarrow 0}
\sum_{v \in V} e^{ \langle k(pv), z\rangle} \sigma_{rK_v}(z),
\]
and giving a common denominator to all of the rational functions (of $e^{z_j}$) $\sigma_{rK_v}(z)$,
we may apply L'Hospital's rule a finite number of times. Because the integer variable $k$ only appears in the exponents
$e^{ \langle k(pv), z\rangle}$, we see that each time we apply L'Hospital, an extra factor of $k$ comes down, giving us a polynomial function of $k$.
{\bf Case $2$.} Suppose $r =0$. Here the situation is slightly easier: $t = pk$, so
\begin{align*}
t{\mathcal K}_v &:= pk {\mathcal K}_v = \left\{ pk v + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\} \\
&= k(pv) + \left\{ \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\} \\
&= k(pv) + ({\mathcal K}_v - v),
\end{align*}
which is an integer cone because $pv$ is an integer vector, and ${\mathcal K}_v - v$ is an integer cone with apex at the origin.
Similarly to the computation above, we have
\begin{align*}
\sigma_{ t {\mathcal K}_{v}}(z)
&= \sum_{n \in \Big( kpv + \left({\mathcal K}_v-v \right) \Big) \cap \mathbb{Z}^d} e^{ \langle n, z\rangle}
= \sum_{m \in \left({\mathcal K}_v-v \right) \cap \mathbb{Z}^d} e^{ \langle kpv + m, z\rangle} \\
&= e^{ \langle kpv, z\rangle} \sum_{m \in \left({\mathcal K}_v-v \right)\cap \mathbb{Z}^d} e^{ \langle m, z\rangle} \\
&:= e^{ \langle kpv, z\rangle} \sigma_{ \left({\mathcal K}_v-v \right)}(z),
\end{align*}
and the remaining steps are identital to Case $1$.
\end{proof}
We note that for an integer polytope ${\mathcal P}$, the same proof gives us Theorem \ref{Ehrhart's main result}, namely
that $L_{\mathcal P}(t)$ is a polynomial for positive integer dilations $t$;
here we just need Case $2$, with $t:=k$, so that $r=0$ and $p=1$.
We emphasize again that one of the important steps in the latter computation was the fact that in both cases of the proof above, $k(pv)$ was an integer vector, allowing us to rewrite the integer point transform of the cone in a simpler way.
As a first application of Theorem \ref{Ehrhart's main result}, we show that the discrete volume of a (half-open) parallelepiped has a particularly elegant and useful form.
\begin{lem}
Let $D$ be any half-open integer parallelepiped in $\mathbb{R}^d$, defined by
\[
D:= \left\{
\lambda_1 w_1 + \cdots + \lambda_d w_d \mid 0 \leq \lambda_1, \dots, \lambda_d < 1
\right\},
\]
where $w_1, \cdots w_d \in \mathbb{Z}^d$ are linearly independent. Then:
\[
\#\{ \mathbb{Z}^d \cap D\} = \vol D,
\]
and for each positive integer $t$, we also have
\[
\#\{ \mathbb{Z}^d \cap tD\} = \left(\vol D\right) t^d.
\]
\end{lem}
\begin{proof} We can tile $tD$ by using $t^d$ translates of $D$, because $D$ is half-open. Therefore
\[
\#\{ \mathbb{Z}^d \cap tD\} = \#\{ \mathbb{Z}^d \cap D\} t^d,
\]
and by definition $\#\{ \mathbb{Z}^d \cap tD\} = L_D(t)$. On the other hand, we also know by Ehrhart's Theorem \ref{Ehrhart's main result} that $L_D(t)$ is a polynomial for integer values of $t$, whose leading coefficient is $\vol D$. Since $L_D(t)=\#\{ \mathbb{Z}^d \cap D\} t^d$ for all positive integer values of $t$, we conclude that
\[
\#\{ \mathbb{Z}^d \cap D\}= \vol D.
\]
\end{proof}
\bigskip
\section{Unimodular polytopes}
A $d$-dimensional integer simplex $\Delta$ is called a {\bf unimodular simplex} if $\Delta$ is the modular image of the
standard simplex $\Delta_{\rm standard}$, \index{standard simplex}
the convex hull of the points $\{ 0, {\bf e_1}, \dots, {\bf e_d} \} \subset \mathbb{R}^d$, where
${\bf e_k}:= (0, \dots, 0, 1, 0, \dots, 0)$ is the standard unit vector pointing in the direction of the positive axis $x_k$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{upolygon}
\end{center}
\caption{A unimodular polygon - each vertex tangent cone is a unimodular cone. It is clear from the construction in the Figure that we can form arbitrarily large unimodular polygons.}
\label{unimodular polygon}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.3in]{ucone}
\end{center}
\caption{A unimodular cone at $v$, appearing as one of the vertex tangent cones in
Figure \ref{unimodular polygon}. We notice that its half-open fundamental parallelepiped, with vertex
at $v$, does not contain any integer points other than $v$.} \label{unimodular cone}
\end{figure}
\begin{example}
\rm{
Let $\Delta := {\rm conv}\left( \icol{0\{\bf 0}\{\bf 0}}, \icol{1\{\bf 0}\{\bf 0}}, \icol{1\{\bf 1}\{\bf 0}}, \icol{1\{\bf 1} \\ 1}
\right)$, their convex hull. Then $\Delta$ is a unimodular simplex, because the unimodular matrix
$\left(
\begin{smallmatrix}
1 & 1 & 1\\
0 & 1 & 1 \\
0 & 0 & 1
\end{smallmatrix}
\right)$ maps the standard simplex $\Delta_{\rm standard}$ to $\Delta$.
}
\hfill $\square$
\end{example}
It is not difficult to show that the tangent cone of a unimodular simplex possesses edge vectors that
form a lattice basis for $\mathbb{Z}^d$. Thus, it is natural to define a {\bf unimodular cone} ${\mathcal K} \subset \mathbb{R}^d$ as a simplicial cone, possessing the additional property that its $d$ edge vectors form a lattice basis for $\mathbb{Z}^d$.
\begin{example}
\rm{
We consider the polygon ${\mathcal P}$ in Figure \ref{unimodular polygon}. An easy verification shows that
each of its vertex tangent cones is unimodular. For example, focusing on the vertex $v$, we
see from Figure \ref{unimodular cone}, that its vertex tangent cone is
${\mathcal K}_v:= v + \{ \lambda_1 \icol{ \ 1\\-2} + \lambda_2 \icol{-1\\ \ 1} \mid
\lambda_1, \lambda_2 \geq 0 \}$.
${\mathcal K}_v$ is a unimodular cone, because the matrix formed by the its two edges
$\icol{ \ 1\\ -2}$ and $\icol{-1\\ \ 1}$ is a unimodular matrix.
}
\hfill $\square$
\end{example}
More generally, a simple, integer polytope is called a
{\bf unimodular polytope} if each of its vertex tangent cones is a unimodular cone.
Unimodular polytopes are the first testing ground for many conjectures in discrete geometry and number theoery. Indeed, we will see later that the number of integer points in a unimodular polytope, namely
$|\mathbb{Z}^d \cap {\mathcal P}|$, admits a simple and computable formula, if we are given the local tangent cone information at each vertex. By contrast, it is in general thought to be quite difficult to compute the number of integer points $|\mathbb{Z}^d \cap {\mathcal P}|$, even for (general) simple polytopes, a problem that belongs to the NP-hard class of problems (if the dimension $d$ is not fixed).
\begin{lem}
Suppose we have two integer polytopes ${\mathcal P}, \mathcal Q \subset \mathbb{R}^d$, which are unimodular images of each other:
\[
{\mathcal P} = U \mathcal Q,
\]
for some unimodular matrix $U$.
Then $L_{\mathcal P}(t) = L_{\mathcal Q}(t)$, for all $t \in \mathbb{Z}_{\geq 0}$.
\end{lem}
\bigskip
\begin{lem}\label{NEW: unimodular simplex equivalence}
Suppose that $\Delta \subset \mathbb{R}^d$ is a $d$-dimensional integer simplex. Then $\Delta$ is a unimodular simplex $\iff$ $(d-1)\Delta$ does not contain any interior integer points.
\end{lem}
\section{More examples of rational polytopes and quasi-polynomials}
The following properties for the floor function, the ceiling function, and the fractional part function
are often useful.
It's convenient to include the following indicator function, for the full set of integers, as well:
\[
1_{\mathbb{Z}}(x) :=
\begin{cases}
1 & \text{if } x \in \mathbb{Z} \\
0 & \text{if } x \notin \mathbb{Z} \\
\end{cases},
\]
the indicator function for $\mathbb{Z}$. For all $x\in \mathbb{R}$, we have:
\begin{enumerate}[(a)]
\item $\left\lceil x \right\rceil = - \floor{-x}$ \label{fractional part property a}
\item $1_{\mathbb{Z}}(x)= \floor{x} - \left\lceil x \right\rceil +1$
\item $ \{ x \} + \{-x\} = 1- 1_{\mathbb{Z}}(x)$
\item Let $m \in \mathbb{Z}_{>0}, n \in \mathbb{Z}$. Then $\floor{ \frac{n-1}{m} } + 1 = \left\lceil \frac{n}{m} \right\rceil$.
\end{enumerate}
(Exercise \ref{properties of floor, ceiling, fractional part})
\bigskip
\begin{example}
\rm{
Let's find the integer point enumerator $L_{\mathcal P}(t) := | \mathbb{Z} \cap t{\mathcal P} |$ of the rational line segment
$ {\mathcal P} := [\frac{1}{3}, \ 1 ]$. Proceeding by brute-force, for $t \in \mathbb{Z}_{>0}$ we have
\begin{align}
L_{\mathcal P}(t) &= \left| \left[\frac{t}{3}, \ t \right] \cap \mathbb{Z} \ \right| \label{answer comparison}
=\floor{t} - \left\lceil \frac{t}{3} \right\rceil + 1 \\
&= t + \floor{ -\frac{t}{3} } +1 \\
&= t + -\frac{t}{3} - \left\{-\frac{t}{3} \right\} +1\\
&= \frac{2}{3} t - \left\{ -\frac{t}{3} \right\} +1,
\end{align}
a periodic function on $\mathbb{Z}$ with period $3$. Here we used property \ref{fractional part property a} in the third equality.
In fact, here we may let $t$ be any positive real number, and we still obtain the same answer, in this $1$-dimensional case.
Now we will compare this to a new computation, but this time from the perspective of the vertex tangent cones.
For the cone ${\mathcal K}_{tv_1} := [\frac{t}{3}, + \infty)$, we can parametrize the integer points in this cone by
${\mathcal K}_{tv_1} \cap \mathbb{Z} =
\{ \left\lceil \frac{t}{3} \right\rceil , \left\lceil \frac{t}{3} \right\rceil +1, \dots \}$, so that
\begin{align*}
\sigma_{{\mathcal K}_{tv_1}}(z) = e^{ \left\lceil \frac{t}{3} \right\rceil z} \sum_{ n \geq 0 } e^{n z} =
e^{ \left\lceil \frac{t}{3} \right\rceil z} \frac{1}{1-e^{z}}.
\end{align*}
For the cone ${\mathcal K}_{tv_2} := (-\infty , t]$, we can parametrize the integer points in this cone by
${\mathcal K}_{tv_2} \cap \mathbb{Z} = \{ t, t-1, \dots \}$, so that
\begin{align*}
\sigma_{{\mathcal K}_{tv_2}}(z) = e^{ t \cdot z} \sum_{ n \leq 0 } e^{n z} =
e^{tz} \frac{1}{1-e^{-z}}.
\end{align*}
So by the discrete Brion Theorem (which is here essentially a finite geometric sum), we get:
\begin{align*}
\sum_{n \in [\frac{t}{3}, t] } e^{nz} &= e^{ \left\lceil \frac{t}{3} \right\rceil z}
\frac{1}{1-e^{z}} + e^{tz} \frac{1}{1-e^{-z}} \\
& =-\left(1 + \left\lceil \frac{t}{3} \right\rceil z + \left\lceil \frac{t}{3} \right\rceil^2 \frac{z^2}{2!} + \cdots \right)
\left(\frac{1}{z} -\frac{1}{2} + \frac{1}{12} z + \cdots \right) \\
&+\left( 1 + (t+1)z + (t+1)^2 \frac{z^2}{2!} + \cdots \right)
\left(\frac{1}{z} -\frac{1}{2} + \frac{1}{12} z + \cdots \right) \\
&= \frac{1}{2} - \left\lceil \frac{t}{3} \right\rceil + (t+1) -\frac{1}{2} +O(z) \longrightarrow \
t - \left\lceil \frac{t}{3} \right\rceil +1,
\end{align*}
as $z\rightarrow 0$, recovering the same answer \ref{answer comparison} above.
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
Let's find the integer point enumerator $L_{\mathcal P}(t) := | \mathbb{Z}^2 \cap t{\mathcal P} |$ of the rational triangle
\[
{\mathcal P}:= \conv\left( \icol{0\{\bf 0}}, \icol{\frac{1}{2} \{\bf 0}}, \icol{0\\ \frac{1}{2} } \right).
\]
First we will proceed by brute-force (which does not always work well), and then we will use the machinery of \eqref{simplified discrete Brion identity}.
For the brute-force method, we need to consider separately the even integer dilates and the odd integer dilates. Letting $t=2n$ be a positive even integer, it's clear geometrically
that
\begin{align*}
L_{\mathcal P}(t) &:= | \mathbb{Z}^2 \cap 2n{\mathcal P} | = 1+ 2 + \cdots + n \\
&= \frac{n(n+1)}{2} = \frac{ \frac{t}{2} ( \frac{t}{2} +1)}{2} \\
&= \frac{1}{8} t^2 + \frac{1}{4} t.
\end{align*}
On the other hand, if $t= 2n-1$, then we
notice that we never have an integer point on the diagonal face of ${\mathcal P}$, so that in this case
we get:
\[
L_{\mathcal P}(t) := | \mathbb{Z}^2 \cap (2n-1){\mathcal P} | = 1+ 2 + \cdots + n = \frac{ \frac{t+1}{2} ( \frac{t+1}{2} +1)}{2}
= \frac{1}{8}t^2 + \frac{1}{2} t + \frac{3}{8}.
\]
Alternatively, we may also rederive the same answer by using the Brion identity
\eqref{simplified discrete Brion identity}. We can proceed as in
Example \ref{Ehrhart poly for the standard triangle}. The only difference now is that the vertex tangent cones have rational apices. So although we may still use the same edge vectors to parametrize the integer points in
${\mathcal K}_{tv_3} \cap \mathbb{Z}^d$, we now have a new problem:
the rational vertex $v_3 = \icol{0\\ \frac{1}{2}}$.
But in any case, we get:
${\mathcal K}_{tv_3} \cap \mathbb{Z}^d= \left\{
\icol{0 \\ \frac{t}{2} } + n_1 \icol{ \ 0 \\ -1} + n_2\icol{ \ 1\\-1} \mid n_1, n_2 \in \mathbb{Z}_{\geq 0} \right\}$.
}
We invite the reader to complete this alternate derivation of the Ehrhart quasi-polynomial $L_{\mathcal P}(t)$ in this case.
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.7in]{rationaltriangle1}
\end{center}
\caption{A rational triangle, which happens to be a {\bf rational dilate} of the standard simplex.} \label{rational triangle 1}
\end{figure}
\section{Ehrhart reciprocity} \index{Ehrhart reciprocity}
There is a wonderful, and somewhat mysterious, relation between the Ehrhart polynomial of the (closed) polytope ${\mathcal P}$, and the Ehrhart polynomial
of its interior, called $\interior {\mathcal P}$. We recall our convention that all polytopes are, by definition, closed polytopes.
We first compute $L_{\mathcal P}(t)$, for positive integers $t$, and once we have this polynomial in $t$, we formally replace $t$ by $-t$. So by definition, we form $L_{\mathcal P}(-t)$ algebraically, and then embark on a search for its new combinatorial meaning.
\bigskip \bigskip
\begin{thm}[Ehrhart reciprocity]
Given a $d$-dimensional rational polytope ${\mathcal P}\subset \mathbb{R}^d$, let
$
L_{\interior {\mathcal P}}(t) := | \mathbb{Z}^d \cap \interior {\mathcal P} |,
$
the integer point enumerator of its interior. Then
\begin{equation}
L_{ {\mathcal P}}(-t)= (-1)^d L_{\interior {\mathcal P}}(t),
\end{equation}
for all $t\in \mathbb{Z}$.
\end{thm}
Offhand, it seems like `a kind of magic', and indeed Ehrhart reciprocity is one of the most elegant geometric inclusion-exclusion principles we have. Some examples are in order.
\begin{example}
\rm{
For the unit cube $\square:= [0, 1]^d$, we can easily compute from first principles
$L_\square(t) = (t+1)^d= \sum_{k=0}^d {d\choose k} t^k$.
For the open cube $\interior \square$, we can also easily compute
\begin{align*}
L_{\interior \square}(t) &= (t-1)^d = \sum_{k=0}^d {d\choose k} t^k(-1)^{d-k} \\
&= (-1)^d \sum_{k=0}^d {d\choose k} (-t)^k \\
&= (-1)^d L_\square(-t),
\end{align*}
using our known polynomial $L_\square(t) = (t+1)^d$.
}
\hfill $\square$
\end{example}
\begin{example}
\rm{
For the standard simplex
\index{standard simplex}
$\Delta$, we consider its $t$-dilate, given by
\[
t\Delta := \{ (x_1, \dots, x_d) \in \mathbb{R}^d \mid \sum_{k=1}^d x_i \leq t, \text{ and all } x_k \geq 0\}.
\]
We can easily compute its Ehrhart polynomial, by using combinatorics. We need to find the number of nonnegative
integer solutions to
\[
x_1 + \cdots + x_d \leq t,
\]
which is equal to $L_\Delta(t)$, for a fixed positive integer $t$. We can introduce a `slack variable'
\index{slack variable}
$z$, to transform the
latter inequality to an equality: $x_1 + \cdots + x_d + z = t$, where $ 0 \leq z \leq t$. By a very classical and pretty argument, (involving placing $t$ balls into urns that are separated by $d$ walls) this number is equal to
${t+d \choose d}$ (Exercise \ref{Ehrhart poly for closure of standard simplex}).
So we found that
\begin{equation}\label{Ehrhart poly for closed standard simplex}
L_\Delta = {t+d \choose d} = \frac{ (t+d) (t+d-1) \cdots (t+1)}{d!},
\end{equation}
a degree $d$ polynomial, valid for all positive integers $t$.
What about the interior of $\Delta$? Here we need to find the number of {\bf positive} integer solutions to
$x_1 + \cdots + x_d < t$, for each positive integer $t$.
It turns out that by a very similar argument as above (Exercise \ref{Ehrhart poly for interior of standard simplex}), the number of positive integer solutions is ${t-1 \choose d} = L_{\interior \Delta}(t)$.
So is it really true that
\[
(-1)^d {d-t \choose d} = {t-1 \choose d} \ ?
\]
Let's compute, substituting $-t$ for $t$ in \eqref{Ehrhart poly for closed standard simplex} to get:
\begin{align*}
L_\Delta(-t) = {-t+d \choose d} &= \frac{ (-t+d) (-t+d-1) \cdots (-t+1)}{d!} \\
&= (-1)^d\frac{ (t-d) (t-d+1) \cdots (t-1)}{d!} \\
&= (-1)^d {t-1 \choose d} = (-1)^d L_{\interior \Delta}(t),
\end{align*}
confirming that Ehrhart reciprocity works here as well.
}
\hfill $\square$
\end{example}
\bigskip
\section{The M\"obius inversion formula for the face poset}
\bigskip
Given a polytope ${\mathcal P}\subset \mathbb{R}^d$,
the collection of all faces $F$ of ${\mathcal P}$ - including the empty set and ${\mathcal P}$ itself - is ordered by inclusion.
This ordering forms a partially ordered set, and is called the {\bf face poset}. \index{face poset}
There is a particularly useful inversion formula on this face poset.
\begin{thm}[M\"obius inversion formula for the face poset]
\label{Mobius inversion}
\index{M\"obius inversion formula}
Given any function $g: {\mathcal P}\rightarrow \mathbb{C}$, we may define a sum over the face poset of ${\mathcal P}$:
\begin{equation}
h({\mathcal P}) := \sum_{F\subseteq {\mathcal P}} g(F).
\end{equation}
We then have the following inversion formula:
\begin{equation} \label{Mobius inversion formula}
g({\mathcal P}) := \sum_{F\subseteq {\mathcal P}} (-1)^{\dim F} h(F).
\end{equation}
\end{thm}
\hfill $\square$
To prove (again) that for positive integer values of $t$, the angle polynomial $A_{\mathcal P}(t)$ is indeed a polynomial in $t$, we
may use the following useful little relation between solid angle sums and integer point sums.
We recall that for any polytope ${\mathcal F}$, the integer point enumerator for the relative interior of ${\mathcal F}$ was defined by $L_{\interior {\mathcal F}}(t):= | \mathbb{Z}^d \cap \interior {\mathcal F} |$.
For each face ${\mathcal F} \subseteq {\mathcal P}$, we define the $d$-dimensional {\bf solid angle of the face} ${\mathcal F}$ by picking any point $x$ inside the relative interior of ${\mathcal F}$ and denoting
\[
\omega_{\mathcal P}({\mathcal F}) := \omega_{\mathcal P}(x).
\]
\begin{thm}\label{lemma: relation between angle sum and integer sum}
Let ${\mathcal P}$ be a $d$-dimensional polytope in $\mathbb{R}^d$. Then we have
\begin{equation}
A_{\mathcal P}(t) = \sum_{{\mathcal F} \subseteq {\mathcal P}} \omega_{\mathcal P}({\mathcal F}) L_{\interior {\mathcal F}}(t).
\end{equation}
\end{thm}
\begin{proof}
The polytope ${\mathcal P}$ is the disjoint union of its relatively open faces ${\mathcal F} \subseteq {\mathcal P}$, and similarly
the dilated polytope $t{\mathcal P}$ is the disjoint union of its relatively open faces $t{\mathcal F} \subseteq t{\mathcal P}$. We therefore have:
\[
A_{\mathcal P}(t)=\sum_{n \in \mathbb{Z}^d} \omega_{t{\mathcal P}}(n) = \sum_{{\mathcal F} \subseteq{\mathcal P}} \sum_{n \in \mathbb{Z}^d}
\omega_{t{\mathcal P}}(n) 1_{\interior(t{\mathcal F})}(n).
\]
But by definition each $\omega_{t{\mathcal P}}(n)$ is constant on the relatively open face $\interior(t{\mathcal F})$ of $t{\mathcal P}$, and we denoted this constant by $\omega_{\mathcal P}({\mathcal F})$. Altogether, we have:
\[
A_{\mathcal P}(t)= \sum_{{\mathcal F} \subseteq{\mathcal P}} \omega_{\mathcal P}({\mathcal F}) \sum_{n \in \mathbb{Z}^d} 1_{\interior(t{\mathcal F})}(n)
:= \sum_{{\mathcal F} \subseteq {\mathcal P}} \omega_{\mathcal P}({\mathcal F}) L_{\interior {\mathcal F}}(t).
\]
\end{proof}
\begin{thm}
Given an integer polytope ${\mathcal P} \subset \mathbb{R}^d$, the discrete volume $A_{\mathcal P}(t)$ is a polynomial in $t$, for integer values of the dilation parameter $t$.
\end{thm}
\begin{proof}
\ \ By Ehrhart's Theorem \ref{Ehrhart's main result}, we know that for each face
${\mathcal F}~\subseteq~{\mathcal P}$,
$L_{\interior {\mathcal F}}(t)$ is a polynomial function of $t$, for positive integers $t$. By
Theorem \ref{lemma: relation between angle sum and integer sum}, we see that $A_{\mathcal P}(t)$ is a finite linear combination of polynomials, with constant coefficients, and is therefore a polynomial in $t$.
\end{proof}
We may apply Theorem \ref{Mobius inversion} to invert the
relationship in Theorem
\ref{lemma: relation between angle sum and integer sum}
between solid angle sums and local Ehrhart polynomials, to get the following consequence of the M\"obius inversion formula.
\begin{cor}\label{lemma: Mobius mu-function for angle sum and integer sum}
Let ${\mathcal P} \subset \mathbb{R}^d$ be a $d$-dimensional polytope. Then we have
\begin{equation}
L_{\interior {\mathcal P}}(t) = \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{\dim F} A_F(t).
\end{equation}
\end{cor}
\begin{proof}
We begin with the identity of
Theorem \ref{lemma: relation between angle sum and integer sum}:
\begin{equation}
A_{\mathcal P}(t) = \sum_{{\mathcal F} \subseteq {\mathcal P}} \omega_{\mathcal P}({\mathcal F}) \, L_{\interior {\mathcal F}}(t),
\end{equation}
and we use the M\"obius inversion formula \eqref{Mobius inversion formula}
to get:
\begin{equation}
\omega_{\mathcal P}({\mathcal P}) L_{\interior {\mathcal P}}(t) = \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{\dim F} A_F(t).
\end{equation}
But $ \omega_{\mathcal P}({\mathcal P})=1$, by definition, and so we are done.
\end{proof}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.7in]{Triangle2}
\end{center}
\caption{An integer triangle, for which we compute $A_F(1)$ for each face $F\subset {\mathcal P}$ in Example \ref{Inverting a solid angle sum using Mobius inversion}, and use M\"obius inversion to find
$L_{int {\mathcal P}}(1)$}
\label{Triangle2}
\end{figure}
\begin{example} \label{Inverting a solid angle sum using Mobius inversion}
\rm{
Let's work out a special case of Corollary \ref{lemma: Mobius mu-function for angle sum and integer sum},
in $\mathbb{R}^2$, for the triangle ${\mathcal P}$ appearing in Figure \ref{Triangle2}, with $t=1$.
${\mathcal P}$ has vertices $v_1:= \icol{-1\\ \ 3}, v_2:=\icol{ \ 2\\ -1}, v_3:= \icol{4\{\bf 1}}$, and edges $E_1, E_2, E_3$.
We have to compute $A_F(1)$ for each face $F \subset {\mathcal P}$.
At the vertices, we have $A_{v_1}(1) = \theta_1$, $A_{v_2}(1) = \theta_2$, and $A_{v_3}(1) = \theta_3$.
For the edges of ${\mathcal P}$, we have:
\[
A_{E_1}(1) = \theta_{v_2} + \tfrac{1}{2} + \theta_{v_3},
\]
\[
A_{E_2}(1) = \theta_{v_3} + \theta_{v_1},
\]
\[
A_{E_3}(1) = \theta_{v_1} + \theta_{v_2}.
\]
Finally, for ${\mathcal P}$ itself, we have
\[
A_{\mathcal P}(1) = 6 + \tfrac{1}{2} + \theta_{v_1} + \theta_{v_2} + \theta_{v_3}
= 7.
\]
Putting everything together, we have:
\begin{align*}
\sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{\dim F} A_F(1) &=
\Big( A_{v_1}(1) + A_{v_2}(1) + A_{v_3}(1) \Big)
- \Big( A_{E_1}(1) + A_{E_2}(1) + A_{E_3}(1) \Big)
+ A_{{\mathcal P}}(1) \\
&= \Big( \theta_{v_1} + \theta_{v_2} + \theta_{v_3} \Big)
- \Big(
\theta_{v_2} + \tfrac{1}{2} + \theta_{v_3} + \theta_{v_3} + \theta_{v_1} + \theta_{v_1} + \theta_{v_2}
\Big) +7 \\
&= \frac{1}{2} - \frac{3}{2} + 7 = 6 = L_{\interior {\mathcal P}}(1),
\end{align*}
the number of interior integer points in ${\mathcal P}$.
}
\hfill $\square$
\end{example}
Finally, we mention a fascinating open problem by Ehrhart.
\index{Ehrhart conjecture}
\begin{question}[Ehrhart, 1964] \label{Ehrhart conjecture}
Let $B \subset \mathbb{R}^d$ be a d-dimensional convex body with the origin as its
barycenter. If the origin is the only interior integer point in $B$, then
\[
\vol B \leq \frac{ (d+1)d}{d!},
\]
and futhermore the equality holds if and only if $B$ is unimodularly equivalent to $(d + 1)\Delta$,
where $\Delta$ is the $d$-dimensional standard simplex.
\index{standard simplex}
\end{question}
Ehrhart proved the upper bound for all $d$-dimensional simplices, and also for all convex bodies in dimension $2$. But Question \ref{Ehrhart conjecture} remains open in general
(see \cite{Nill.and.Paffenholz} for more details).
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.Ehrhart}
\begin{enumerate}[(a)]
\item Ehrhart theory has a fascinating history, commencing with the fundamental work of Ehrhart \cite{Ehrhart1}, \cite{Ehrhart2},
\cite{Ehrhartbook}, in the 1960's. Danilov \cite{Danilov} made a strong contribution to the field, but after that
the field of Ehrhart theory lay more-or-less dormant, until it was rekindled by Jamie Pommersheim in 1993 \cite{Pommersheim}, giving it strong connections to Toric varieties. Using the Todd operators to discretize certain volume deformations of polytopes, Khovanskii and Pukhlikov discovered a wonderful result that helped develop the theory further (see Theorem 12.6 of \cite{BeckRobins}).
In 1993, Alexander Barvinok \cite{Barvinok.algorithm} \index{Barvinok, Alexander}
gave the first polynomial-time algorithm for counting integer points in polytopes in fixed dimension.
In recent years, Ehrhart theory has enjoyed an enthusiastic renaissance (for example, the books \cite{BarvinokEhrhartbook}, \cite{BeckRobins}, \cite{FultonBook}). For more relations with combinatorics, the reader may enjoy reading
Chapter $4$ of the classic book ``Enumerative Combinatorics'', \cite{StanleyBook} by Richard Stanley.
\item Regarding the computational complexity of counting integer points in polytopes, Alexander Barvinok settled the problem
in \cite{Barvinok.algorithm} by showing that for a fixed dimension $d$, there is a polynomial-time algorithm, as a function of the
`bit capacity' of any given
rational polytope ${\mathcal P} \subset \mathbb{R}^d$, for counting the number of integer points in ${\mathcal P}$.
\item It is also true that for integer polytopes which are not necessarily convex (for example simplicial complexes),
the integer point enumerator makes sense as well. In this more general context,
the constant term of the corresponding integer point enumerator equals the (reduced) Euler characteristic of the simplicial complex.
\item For more information about the rapidly expanding field of Euler-MacLaurin summation over polytopes, a brief (and by no means complete) list of paper in this direction consists of the work by Berligne and Vergne \cite{BerlineVergne}, Baldoni, Berline, and Vergne \cite{BaldoniBerlineVergne}, Garoufalidis and Pommersheim \cite{GaroufalidisPommersheim}, Brandolini, Colzani, Travaglini, and Robins \cite{BrandoliniColzaniTravagliniRobins2},
Karshon, Sternberg, and Weitsman (\cite{KarshonSternbergWeitsman1}, \cite{KarshonSternbergWeitsman2}), and very recently Fischer and Pommersheim \cite{FischerPommersheim}.
\item There are some fascinating relations between an integer polytope ${\mathcal P}$ and its dual polytope ${\mathcal P}^*$. In particular, let ${\mathcal P} \subset \mathbb{R}^2$ be an integer polygon (convex) whose only interior integer point is the origin.
Such polygons are called reflexive polygons, and up to unimodular transformations there are only a finite number of them in each dimension. If we let $B({\mathcal P})$ be the number of integer points on the boundary of ${\mathcal P}$, then Bjorn Poonen and Fernando Villegas proved \cite{PoonenVillegas} that
\[
B({\mathcal P}) + B({\mathcal P}^*) = 12.
\]
One way to see why we get the number ``12'' is to consider Bernoulli numbers and Dedekind sums, but in \cite{PoonenVillegas} the authors give 4 different proofs, including Toric varieties and modular forms.
\item \label{Michel Faleiros} The trick used in Example \ref{Ehrhart poly for the standard triangle}
of picking the particular vector $z := (x, -x)$, which turns out to simplify the computations a lot, is due to Michel Faleiros.
\item \label{EM summation note}
In a future version of this book, we will also delve into Dedekind sums, which arise very naturally when considering the Fourier series of certain rational-exponential functions. To define a general version of these sums, let
${\mathcal L}$ be a $d$-dimensional lattice in $\mathbb{R}^d$, let $w_1, \dots, w_d$ be linearly independent vectors from
${\mathcal L}^*$, and let $W$ be a matrix with the $w_j$'s as columns. For any $d$-tuple $e = (e_1, \dots, e_d)$ of positive integers $e_j$,
define $|e| := \sum_{j = 1}^k e_j$. Then, for all $x \in \mathbb{R}^d$, a lattice Dedekind sum is defined by
\begin{equation}
L_{\mathcal L}(W, e; x) :=
\lim_{\varepsilon \rightarrow 0}
\frac{1}{(2\pi i)^{|e|}}
\sum_{\substack{\xi \in {\mathcal L} \\ \langle w_j, \xi \rangle \neq 0, \forall j}}
\frac{e^{-2\pi i \langle x, \xi \rangle }}{\prod_{j = 1}^k \langle w_j, \xi \rangle^{e_j}} e^{-\pi \varepsilon \|\xi\|^2}.
\end{equation}
Gunnells and Sczech \cite{GunnellsSczech} have an interesting reduction theorem for these sums,
giving a polynomial-time complexity algorithm for them, for fixed dimension $d$.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\medskip
\begin{prob} \label{exercise:warm up dual lattice}
In $\mathbb{R}$, consider the $1$-dimensional polytope ${\mathcal P}:= [a,b]$, for any
$a,b \in \mathbb{Z}$.
\begin{enumerate}[(a)]
\item Show that the Ehrhart polynomial of ${\mathcal P}$ is $L_{\mathcal P}(t) = (b-a)t + 1$.
\item Find the Ehrhart quasi-polynomial $L_{\mathcal P}(t)$
for the rational segment $\mathcal Q:= [\frac{1}{3}, \frac{1}{2}]$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
Fix positive integers $a, b$.
Working in $\mathbb{R}^2$, show that the closed line segment ${\mathcal P} \subset \mathbb{R}^2$, whose vertices are the origin and $(a, b)$,
contains exactly $\gcd(a, b) + 1$ integer points of $\mathbb{Z}^2$. Conclude that we have the lower-dimensional
Ehrhart polynomial $L_{\mathcal P}(t) = \gcd(a, b) t + 1$.
\end{prob}
\medskip
\begin{prob} \label{2-d cross polytope Ehrhart}
We recall that the $d$-dimensional cross-polytope was defined by
\[
\Diamond:=\left\{ \left( x_1, x_2, \dots, x_d \right) \in \mathbb{R}^d \mid
\, \left| x_1 \right| + \left| x_2 \right| + \cdots + \left| x_d \right| \leq 1 \right\}.
\]
For $d=2$, find the Ehrhart polynomial $L_\Diamond(t)$.
\end{prob}
\medskip
\begin{prob}
Extending Exercise \ref{2-d cross polytope Ehrhart}, show that the Ehrhart polynomial of $\Diamond$ in $\mathbb{R}^d$ is
\[
L_{\Diamond}(t) = \sum_{k=0}^d \binom{d}{k} \binom{t-k+d}{d},
\]
for all $t \in \mathbb{Z}_{>0}$.
\end{prob}
\medskip
\begin{prob}
Let $d=2$, and consider the cross-polytope $\Diamond \subset \mathbb{R}^2$. Find the Ehrhart quasi-polynomial $L_{\mathcal P}(t)$ for the rational polygon ${\mathcal P} := \frac{1}{2} \Diamond$.
\end{prob}
\medskip
\begin{prob}
Suppose $\Delta$ is the standard simplex in $\mathbb{R}^d$. Show that the first $d$ dilations of $\Delta$ do not contain
any integer points in their interior:
\[
t(\interior \Delta) \cap \mathbb{Z}^d = \phi,
\]
for $t = 1, 2, \dots, d$.
In other words, show that $L_{\interior {\mathcal P}}(1) = L_{\interior {\mathcal P}}(2) = \cdots =
L_{\interior {\mathcal P}}(d) = 0$. Conclude that the same statement is true for any unimodular simplex.
\end{prob}
\medskip
\begin{prob} \label{Bernoulli polynomial as an Ehrhart polynomial}
Here we show that the Bernoulli polynomial $B_d(t)$, is essentially equal to the Ehrhart polynomial
$L_{\mathcal P}(t)$ for the ``Pyramid over a cube" (as defined in Exercise
\ref{Pyramid over a square}). We recall the definition: let $C:=[0,1]^{d-1}$ be the $d-1$-dimensional cube,
considered as a subset of $\mathbb{R}^d$, and let ${\bf e_d}$ be the unit vector pointing in the $x_d$-direction.
Now we define ${\mathcal P}:= \conv\{ C, {\bf e_d} \}$, a pyramid over the unit cube.
Show that its Ehrhart polynomial is
\[
L_{{\mathcal P}}(t) = \frac{1}{d}(B_d(t+2) - B_d),
\]
for $t \in \mathbb{Z}_{>0}$.
\end{prob}
\medskip
\begin{prob} \label{Pick's formula, generalization to d dimensions}
For any integer $d$-dimensional (convex) polytope ${\mathcal P} \subset \mathbb{R}^d$, show that
\begin{equation} \label{Volume in terms of forward differences}
\vol {\mathcal P} = \frac{(-1)^d}{d!} \left( 1 + \sum_{k=1}^d {d\choose k} (-1)^k L_{\mathcal P}(k) \right),
\end{equation}
which can be thought of as a generalization of Pick's formula to $\mathbb{R}^d$.
\index{Pick's formula, generalization}
Note. \ Using iterations of the forward difference operator
\[
\Delta f(n):= f(n+1) - f(n),
\]
the latter identity
may be thought of a {\bf discrete analogue} of the $d$'th derivative of the Ehrhart polynomial. This idea in fact gives another method of proving \eqref{Volume in terms of forward differences}.
\end{prob}
\medskip
\begin{prob} \label{Pick's formula from general Ehrhart exercise}
Show that Pick's formula is the special case of
Exercise \ref{Pick's formula, generalization to d dimensions} when the dimension $d=2$.
That is, given an integer polygon ${\mathcal P} \subset \mathbb{R}^2$, we have
\[
\rm{Area } {\mathcal P} = I + \frac{1}{2} B -1,
\]
where $I$ is the number of interior integer points in ${\mathcal P}$, and B is the number of boundary integer points of ${\mathcal P}$.
\end{prob}
\medskip
\begin{prob} \label{convolution of the indicator function with a Gaussian}
Fix $\epsilon >0$. Show that the convolution of the indicator function $1_{\mathcal P}$
with the heat kernel $G_{\varepsilon}$, as in equation \eqref{Gaussian smoothing},
is a Schwartz function (of $x \in \mathbb{R}^d$).
\end{prob}
\medskip
\begin{prob} \label{unimodular triangle}
Show that any unimodular triangle has area equal to $\frac{1}{2}$.
\end{prob}
\medskip
\begin{prob} \label{unimodular triangle, Ehrhart poly}
Show that the Ehrhart polynomial of the standard simplex
\index{standard simplex}
$\Delta \subset \mathbb{R}^d$ is
\[
L_{\Delta}(t) = \binom{t+d}{d}.
\]
\end{prob}
\medskip
\begin{prob}
Consulting Figure \ref{unimodular polygon}:
\begin{enumerate}[(a)]
\item Find the integer point transform of the unimodular polygon in the Figure.
\item Find the Ehrhart polynomial $L_{{\mathcal P}}(t)$ of the integer polygon ${\mathcal P}$ from part (a).
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{two definitions for a solid angle}
Show that \eqref{def. of solid angle} is equivalent to the following definition, using balls instead of spheres.
Recall that the unit ball in $\mathbb{R}^d$ is define by $B^d:= \{ x\in \mathbb{R}^d \mid \| x\| \leq 1 \}$, and similarly
the ball of radius $\varepsilon$, centered at $x\in \mathbb{R}^d$, is denoted by $B^d(x, \varepsilon)$. Show that
for all sufficiently small $\varepsilon$, we have
\begin{equation*}
\frac{\vol(S^{d-1}(x,\varepsilon) \cap {\mathcal P})}{\vol(S^{d-1}(x,\varepsilon))} =
\frac{\vol(B^{d}(x,\varepsilon) \cap {\mathcal P})}{\vol(B^{d}(x,\varepsilon))}.
\end{equation*}
\end{prob}
\medskip
\begin{prob} \label{properties of floor, ceiling, fractional part}
Here we gain some practice with `floors', `ceilings', and `fractional parts'.
First, we recall that by definition, the fractional part of any real number $x$ is $\{x\} := x - \floor{x}$.
Next, we recall the indicator function of $\mathbb{Z}$, defined by:
$
1_{\mathbb{Z}}(x) :=
\begin{cases}
1 & \text{if } x \in \mathbb{Z} \\
0 & \text{if } x \notin \mathbb{Z} \\
\end{cases}.
$
Show that:
\begin{enumerate}[(a)]
\item $\left\lceil x \right\rceil = - \floor{-x}$
\label{ex:part 1 of fractional parts}
\item $1_{\mathbb{Z}}(x)= \floor{x} - \left\lceil x \right\rceil +1$
\label{ex:part 2 of fractional parts}
\item $ \{ x \} + \{-x\} = 1- 1_{\mathbb{Z}}(x)$
\label{ex:part 3 of fractional parts}
\item Let $m \in \mathbb{Z}_{>0}, n \in \mathbb{Z}$. Then $\floor{ \frac{n-1}{m} } + 1 = \left\lceil \frac{n}{m} \right\rceil$.
\label{ex:part 4 of fractional parts}
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Ehrhart poly for closure of standard simplex}
\index{standard simplex}
Show that the number of nonnegative integer solutions $x_1, \dots, x_d, z \in \mathbb{Z}_{\geq 0}$
to
\[
x_1 + \cdots + x_d + z = t,
\]
with $ 0 \leq z \leq t$, equals ${t+d \choose d}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Ehrhart poly for interior of standard simplex}
Show that for each positive integer $t$, the number of {\bf positive} integer solutions to
$x_1 + \cdots + x_d < t$ is equal to ${t-1 \choose d}$.
\end{prob}
\medskip
\begin{prob}
We define the rational triangle whose vertices are $(0, 0), (1, \frac{N-1}{N}), (N, 0)$, where $N \geq 2$ is a fixed integer.
Prove that the Ehrhart quasi-polynomial is in this case
\[
L_{\mathcal P}(t) = \frac{p-1}{2} t^2 + \frac{p+1}{2} t + 1,
\]
for all $t \in \mathbb{Z}_>0$.
Notes. So we see here a phenomenon known as `period collapse', where we expect a quasi-polynomial behavior, with some nontrivial period, but in fact we observe a strict polynomial.
\end{prob}
\medskip
\begin{prob}
Here we show that the Ehrhart polymomial $L_{\mathcal P}(t)$ remains invariant under the full unimodular group $SL_d(\mathbb{Z})$.
In particular, recalling definition \ref{Definition of the unimodular group}, of a unimodular matrix, show that:
\begin{enumerate}[(a)]
\item Every element of $SL_d(\mathbb{Z})$ acts on the integer lattice $\mathbb{Z}^d$ bijectively.
\item \label{invariance of Ehrhart under the unimodular group}
Let ${\mathcal P}$ be an integral polytope, and let $Q := A({\mathcal P})$, where $A \in SL_d(\mathbb{Z})$. Thus, by definition ${\mathcal P}$ and $Q$
are unimodular images of each other. Prove that
\[
L_{{\mathcal P}}(t) = L_Q(t),
\]
for all $t \in \mathbb{Z}_{>0}$.
\item Is the converse of part \ref{invariance of Ehrhart under the unimodular group} true? In other words, given integer polytopes ${\mathcal P}, Q$, suppose that $L_{{\mathcal P}}(t) = L_Q(t)$, for all positive integers $t$. Does it necessarily follow that
$Q := A({\mathcal P})$, for some unimodular matrix $A \in SL_d(\mathbb{Z})$?
\end{enumerate}
\end{prob}
\chapter{The Fourier transform of a polytope via its hyperplane description: \\
the divergence Theorem}
\label{Stokes' formula and transforms}
\index{Stoke's formula} \index{face poset}
\begin{quote}
``Like a zen koan, Stokes' Theorem tells us that in the end, what happens on the outside
is purely a function of the change within.''
--Keenan Crane
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{VectorField}
\end{center}
\caption{A real vector field in $\mathbb{R}^2$} \label{vector field}
\end{figure}
\section{Intuition}
The divergence theorem is a multi-dimensional version of ``integration by parts'', a very useful tool in
$1$-dimensional calculus. When we apply the divergence theorem, described below, to a polytope, we obtain
a kind of combinatorial version of the divergence theorem, allowing us to transfer some of the complexity of computing the Fourier transform of a polytope to the complexity of computing corresponding Fourier transforms of its facets. This kind of game can be iterated, yielding interesting geometric identities and results for polytopes, as well as for discrete volumes of polytopes.
In the process, we also obtain another useful way to compute the Fourier transform of a polytope in its own right.
\section{The divergence theorem, and a combinatorial \\ divergence theorem for polytopes}
\index{combinatorial divergence theorem}
To warm up, we recall the divergence theorem, with some initial examples.
A {\bf vector field} on Euclidean space is a
function $F:\mathbb{R}^d \rightarrow \mathbb{C}^d$ that assigns to each point in $\mathbb{R}^d$ another vector in $\mathbb{C}^d$, which we will denote by
\[
F(x) := (F_1(x), F_2(x), \dots, F_d(x)) \in \mathbb{C}^d.
\]
If $F$ is a continuous (respectively, smooth) function, we say that $F$ is a
{\bf continuous vector field} (respectively, {\bf smooth vector field}).
If all of the coordinate functions $F_j$ are real-valued functions, we say that we have a {\bf real vector field}.
We define the {\bf divergence} of $F$ at each $x := (x_1, \dots, x_d) \in \mathbb{R}^d$ by
\[
\rm{div} F(x) := \frac{\partial F_1}{{\partial} x_1} + \cdots + \frac{\partial F_d}{\partial x_d},
\]
assuming that $F$ is a smooth (or at least once-differentiable) vector field. This divergence of $F$ is a measure of the local change (sink versus source) of the vector field at each point $x\in \mathbb{R}^d$.
Given a surface $S \subset \mathbb{R}^d$, and an outward pointing unit normal vector ${\bf n}$, defined at each point $x\in S$, we also define the {\bf flux} of the vector field $F$ across the surface $S$ by
\[
\int_S F\cdot {\bf n} \ dS,
\]
where $dS$ denotes the Lebesgue measure of the surface $S$, and where the dot product
$F\cdot {\bf n}$ is the usual inner product $\langle F, {\bf n} \rangle := \sum_{k=1}^d F_k n_k$. We will apply the divergence theorem (which is technically a special case of Stokes' Theorem)
to a polytope ${\mathcal P}\subset \mathbb{R}^d$, and its $(d-1)$-dimensional bounding surface $\partial {\mathcal P}$. Intuitively,
the divergence theorem tells us that the total divergence of a vector field $F$ inside a manifold is equal to the total
flux of $F$ across its boundary.
\begin{thm}[The Divergence Theorem]
Let $M \subset \mathbb{R}^d$ be a piecewise smooth manifold, and let $F$ be a smooth vector field.
Then
\begin{equation}\label{Divergence Theorem} \index{divergence Theorem}
\int_M \rm{div}F(x) dx = \int_S F\cdot {\bf n} \ dS.
\end{equation}
\end{thm}
\begin{example} \label{Pyramid formula via the divergence theorem}
\rm{
Let ${\mathcal P} \subset \mathbb{R}^d$ be a $d$-dimensional polytope, containing the origin, with defining facets $G_1, \dots, G_N$.
Define the real vector field
\[
F(x):= x,
\]
for all $x\in \mathbb{R}^d$. First, we can easily compute
here the divergence of $F$, which turns out to be constant:
\begin{align*}
\rm{div } F(x) &= \frac{\partial F_1}{{\partial} x_1} + \cdots + \frac{\partial F_d}{\partial x_d} =
\frac{\partial x_1}{{\partial} x_1} + \cdots + \frac{\partial x_d}{\partial x_d} = d.
\end{align*}
If we fix any facet $G$ of ${\mathcal P}$ then, due to the piecewise linear structure of the polytope,
every point $x \in G$ has the same constant outward pointing
normal vector to $F$, which we call ${\bf n}_G$.
Computing first the left-hand-side of the divergence theorem, we see that
\begin{equation}
\int_P \rm{div } F(x) dx = d \int_P dx = (\vol {\mathcal P})d.
\end{equation}
Computing now the right-hand-side of the divergence theorem, we get
\begin{align*}
\int_S F\cdot {\bf n} \ dS = \int_{\partial {\mathcal P}} \langle x, {\bf n} \rangle \ dS = \sum_{k=1}^N \int_{G_k} \langle x, {\bf n}_G \rangle \ dS.
\end{align*}
Now it's easy to see that the inner product $ \langle x, n_G \rangle$ is constant on each facet $G \subset {\mathcal P}$, namely
it is the distance from the origin to $G$ (Exercise \ref{distance to a facet}), denoted by $\rm{dist}(G)$.
So we now have
\begin{align*}
\int_{\partial {\mathcal P}} F \cdot n \ dS &= \sum_{k=1}^N \int_{G_k} \langle x, {\bf n}_{G_k} \rangle dS \\
&= \sum_{k=1}^N \rm{dist}(G_k) \int_{G_k} dS = \sum_{k=1}^N \rm{dist}(G_k) \vol G_k,
\end{align*}
so that altogether we the following conclusion from the divergence theorem:
\begin{equation}\label{pyramid formula} \index{pyramid formula}
\vol {\mathcal P} = \frac{1}{d} \sum_{k=1}^N \rm{dist}(G_k) \vol G_k.
\end{equation}
known as ``the pyramid formula'' for a polytope, a classical result in Geometry, which also has a very easy
geometrical proof (Exercise \ref{Pyramid formula}).
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
Let ${\mathcal P} \subset \mathbb{R}^d$ be a $d$-dimensional polytope with defining facets $G_1, \dots, G_N$, and outward pointing unit vectors $n_{G_1}, \dots, n_{G_N}$. We fix any constant vector $\lambda \in \mathbb{C}^d$, and we consider the {\bf constant vector field}
\[
F(x):= \lambda,
\]
defined for all $x\in \mathbb{R}^d$. Here the divergence of
$F$ is $\rm{div} F(x) = 0$, because $F$ is constant, and so the left-hand-side of Theorem
\ref{Divergence Theorem} gives us
\begin{align*}
\int_P \rm{div } F(x) dx = 0.
\end{align*}
Altogether, the divergence theorem gives us:
\begin{align*}
0 = \int_{\partial {\mathcal P}} F \cdot {\bf n} \ dS &= \sum_{k=1}^N \int_{G_k} \langle \lambda, {\bf n}_{G_k} \rangle dS \\
&= \sum_{k=1}^N \langle \lambda, {\bf n}_{G_k} \rangle \int_{G_k} dS \\
&= \langle \lambda, \sum_{k=1}^N \vol G_k {\bf n}_{G_k} \rangle,
\end{align*}
and because this holds for any constant vector $\lambda$, we can conclude that
\begin{equation}\label{Minkowski relation}
\sum_{k=1}^N \vol G_k {\bf n}_{G_k}= 0.
\end{equation}
Identity \eqref{Minkowski relation} is widely known as the {\bf Minkowski relation} for polytopes. There is a marvelous converse to the latter relation, given by Minkowski as well,
for any convex polytope.
[See Theorem \ref{Minkowski's problem for polytopes}]
}
\hfill $\square$
\end{example}
\bigskip
Now we fix $\xi \in \mathbb{R}^d$, and we want to see how to apply the divergence theorem to the vector-field
\begin{equation}\label{First vector field}
F(x) := e^{- 2\pi i \langle x, \xi \rangle} \xi.
\end{equation}
Taking the divergence of the vector field $F(x)$, we have:
\begin{align*}
\rm{div } F(x) &= \frac{\partial \left( e^{- 2\pi i \langle x, \xi \rangle} \xi_1 \right) }{{\partial} x_1} + \cdots +
\frac{\partial ( e^{- 2\pi i \langle x, \xi \rangle} \xi_d )}{\partial x_d} \\
&= (-2\pi i \xi_1^2) e^{- 2\pi i \langle x, \xi \rangle} + \cdots + (-2\pi i \xi_d^2) e^{- 2\pi i \langle x, \xi \rangle} \\
&= -2\pi i \| \xi \|^2 e^{- 2\pi i \langle x, \xi \rangle}.
\end{align*}
So by the divergence theorem we have
\begin{align} \label{initial divergence}
\int_{x\in P} - 2\pi i ||\xi||^2 e^{- 2\pi i \langle x, \xi \rangle} dx
= \int_{x\in P} \text{div} F(x) dx = \int_{\partial P} e^{- 2\pi i \langle x, \xi \rangle}
\langle \xi, {\bf n} \rangle \ dS,
\end{align}
where ${\bf n}$ is the outward-pointing unit normal vector at each point $x\in \partial {\mathcal P}$. When ${\mathcal P}$ is a polytope, these arguments quickly give the following conclusion.
\begin{thm} \label{FT of a polytope, first iteration of divergence}
Given any $d$-dimensional polytope ${\mathcal P}\subset \mathbb{R}^d$, with outward pointing normal vector $n_G$ to each facet $G$ of ${\mathcal P}$, its Fourier transform has the form
\begin{equation} \label{the first step of divergence}
\hat 1_{\mathcal P}(\xi) =\frac{1}{-2\pi i } \sum_{G\subset \partial P} \frac{ \langle \xi, {\bf n}_G \rangle}{ ||\xi ||^2} \hat 1_G(\xi),
\end{equation}
for all nonzero $\xi \in \mathbb{C}^d$. Here the integral that defines each $\hat 1_G$ is taken with respect to Lebesgue measure that matches the dimension of the facet $G \subset \partial P$.
\end{thm}
\begin{proof}
\begin{align*}
\hat 1_{\mathcal P}(\xi) &:= \int_{x\in P} e^{- 2\pi i \langle x, \xi \rangle} dx \\
&= \frac{ 1 }{-2\pi i \|\xi\|^2} \int_{\partial P} \langle \xi, {\bf n} \rangle e^{- 2\pi i \langle x, \xi \rangle} dS
\quad (\text{using} \, \eqref{initial divergence}) \\
&= \frac{ 1}{-2\pi i \|\xi\|^2}
\int_{G_1} \langle \xi, {\bf n}_{G_1} \rangle e^{- 2\pi i \langle x, \xi \rangle} dS
+ \cdots +
\frac{ 1}{-2\pi i \|\xi\|^2}
\int_{G_N} \langle \xi, {\bf n}_{G_N} \rangle e^{- 2\pi i \langle x, \xi \rangle} dS \\
&= \frac{ \langle \xi, {\bf n}_{G_1} \rangle }{-2\pi i \|\xi\|^2} \hat 1_{G_1}(\xi)
+ \cdots +
\frac{ \langle \xi, {\bf n}_{G_N} \rangle }{-2\pi i \|\xi\|^2} \hat 1_{G_N}(\xi),
\end{align*}
where in the third equality we used the fact that the boundary $\partial {\mathcal P}$ of a polytope is a finite union of $(d-1)$-dimensional polytopes (its facets), and hence $\int_{\partial P} = \int_{G_1} + \cdots + \int_{G_N}$, a sum of integrals over the $N$ facets of ${\mathcal P}$.
\end{proof}
This result allows us to reduce the Fourier transform of ${\mathcal P}$ to a finite sum of Fourier transforms of the facets of ${\mathcal P}$. This process can clearly be iterated, until we arrive at the vertices of ${\mathcal P}$. But we will need a few book-keeping devices first.
To simplify the notation that will follow, we can also the {\bf Iverson bracket} notation, defined as follows.
Suppose we have any
boolean property $P(n)$, where $n\in \mathbb{Z}^d$; that is, $P(n)$ is either true or false.
Then the Iverson bracket $[ P ]$ is defined by:
\begin{equation}\label{Iverson bracket}
[P] =
\begin{cases}
1 & \mbox{if P is true } \\
0 & \mbox{if P is false }
\end{cases}
\end{equation}
Now we may rewrite the identity of Theorem \ref{FT of a polytope, first iteration of divergence}
as follows:
\begin{equation}
\hat 1_{\mathcal P}(\xi) =\vol {\mathcal P} \ [\xi = 0] +
\frac{1}{-2\pi i } \sum_{G\subset \partial P} \frac{ \langle \xi, {\bf n}_G \rangle}{ ||\xi ||^2} \hat 1_G(\xi)\
[\xi \not= 0].
\end{equation}
Later, after Theorem \ref{Combinatorial divergence theorem} below, we will return to the Iverson bracket, and be able to use it efficiently.
To proceed further, we need to define the {\bf affine span} \index{affine span} of a face $F$ of ${\mathcal P}$:
\begin{equation} \label{affine span of F}
\rm{aff}(F) := \left\{ \sum_{j=1}^k \lambda_j v_j \mid k>0, v_j \in F,
\lambda_j \in \mathbb{R}, \text{ and } \sum_{j=1}^k \lambda_j = 1 \right\}.
\end{equation}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.3in]{AffineSpan}
\end{center}
\caption{The affine span of a face $F$, its linear span , and the projection of $\xi$ onto $F$. Here
we note that the distance from the origin to $F$ is $\sqrt{20}$.} \label{affine span}
\end{figure}
In other words, we may think of the affine span of a face $F$ of ${\mathcal P}$ as follows. We first translate $F$ by any element $x_0 \in F$. So this translate, call if $F_0:= F - x_0$, contains the origin. Then we take all real linear combinations of points of $F_0$, obtaining a vector subspace of $\mathbb{R}^d$, which we call the {\bf linear span } of $F$. Another way to describe the linear span of a face $F$ of ${\mathcal P}$ is:
\[
{\rm lin}(F):= \left\{ x - y \mid x, y \in F \right\}.
\]
\index{linear span}
Finally, we may translate this subspace $\rm{lin}(F)$ back using the same translation vector $x_0$, to obtain $\rm{aff}(F):= \rm{lin}(F) + x_0$ (see Figure \ref{affine span}).
\begin{example}
\rm{
The affine span of two distinct points in $\mathbb{R}^d$ is the unique line in $\mathbb{R}^d$ passing through them. The affine span of three points in $\mathbb{R}^d$ is the unique $2$-dimensional plane passing through them.
The affine span of a $k$-dimensional polytope $F \subset \mathbb{R}^d$ is a translate of a $k$-dimensional vector subspace of $\mathbb{R}^d$.
Finally, the affine span of a whole $d$-dimensional polytope ${\mathcal P} \subset \mathbb{R}^d$ is all of $\mathbb{R}^d$.
}
\hfill $\square$
\end{example}
In formalizing \eqref{the first step of divergence} further, we will require the notion of the projection of any point
$\xi \in \mathbb{R}^d$ onto the linear span of any face $F\subseteq {\mathcal P}$, which we abbreviate by $\rm{Proj}_F \xi$:
\begin{equation}
\rm{Proj}_F \xi := \rm{Proj}_{\rm{lin}(F)}(\xi).
\end{equation}
(see Figure \ref{affine span})
We will also need the following elementary fact. Let $F$ be any $k$-dimensional polytope in
$\mathbb{R}^d$, and fix the outward-pointing unit normal to $F$, calling it ${\bf n}_F$.
It is straightforward to show that if we take any point $x_F \in F$, then $\langle x_F, {\bf n}_F \rangle$ is the distance from the origin to $F$. Therefore, if $\rm{Proj}_F \xi = 0$, then a straightforward computation shows that
$\langle \xi, x_F \rangle = \| \xi \| \rm{dist}(F)$ (Exercise \ref{distance to a facet}).
We can now extend \eqref{the first step of divergence} to lower-dimensional polytopes, as follows.
\begin{thm}[Combinatorial Divergence Theorem] \label{Combinatorial divergence theorem}
Let $F$ be a polytope in $\mathbb{R}^d$, where $1 \leq \dim F \leq d$. For each facet $G \subseteq F$, we let
${\bf n}(G, F)$ be the unit normal vector to $G$, with respect to $\rm{lin}(F)$.
Then for each $\xi \in \mathbb{R}^d$, we have:
\begin{enumerate}[(a)]
\item If $\rm{Proj}_F \xi = 0$, then
\begin{equation}
\hat 1_F(\xi) = (\vol F) e^{-2\pi i \| \xi \| \rm{dist}(F) }.
\end{equation}
\item If $\rm{Proj}_F \xi \not= 0$, then
\begin{equation}
\hat 1_F(\xi) = \frac{1}{-2\pi i } \sum_{G\subset \partial F}
\frac{ \langle \rm{Proj}_F \xi, {\bf n}(G, F) \rangle}{ ||\rm{Proj}_F \xi ||^2} \hat 1_G(\xi).
\end{equation}
\end{enumerate}
\end{thm}
\hfill $\square$
\bigskip
We notice that, as before, we are getting rational-exponential functions for the Fourier transform of a polytope. But Theorem \ref{Combinatorial divergence theorem} gives us the extra freedom to begin with a lower-dimensional polytope $F$, and then find its Fourier transform in terms of its facets.
We are now set up to iterate this process, defined by Theorem \ref{Combinatorial divergence theorem},
reapplying it to each facet $G \subset \partial {\mathcal P}$. Let's use the Iverson bracket,
defined in \eqref{Iverson bracket},
and apply the combinatorial divergence Theorem \ref{Combinatorial divergence theorem} to ${\mathcal P}$ twice:
\begin{align*}
\hat 1_{\mathcal P}(\xi) &=\vol {\mathcal P} \ [\xi = 0] +
\frac{1}{-2\pi i } \sum_{F_1 \subset \partial P} \frac{ \langle \xi, {\bf n}_{F_1} \rangle}{ ||\xi ||^2} \ [\xi \not= 0] \ \hat 1_{F_1}(\xi) \\
&=\vol {\mathcal P} \ [\xi = 0] +
\frac{1}{-2\pi i } \sum_{{F_1}\subset \partial P} \frac{ \langle \xi, {\bf n}_{F_1} \rangle}{ ||\xi ||^2} [\xi \not= 0] \\
& \cdot \Big( (\vol {F_1}) e^{-2\pi i \langle \xi, x \rangle} \ [\rm{Proj}_{F_1} \xi = 0 ] +
\frac{1}{-2\pi i } \sum_{F_2 \subset \partial {F_1}}
\frac{ \langle \rm{Proj}_{F_2} \xi, {\bf n}(F_2, F_1) \rangle}{ ||\rm{Proj}_{F_2} \xi ||^2} \hat 1_{F_2}(\xi) [\rm{Proj}_{F_1} \xi \not= 0] \Big)
\\
&= \vol {\mathcal P} \ [\xi = 0] +
\frac{1}{-2\pi i } \sum_{F_1 \subset \partial P}
\frac{ \langle \xi, {\bf n}_{F_1} \rangle (\vol F_1) e^{-2\pi i \langle \xi, x \rangle} }{ ||\xi ||^2}
\ [\xi \not= 0][\rm{Proj}_{F_1} \xi = 0 ] \\
& + \frac{1}{(-2\pi i )^2} \sum_{F_1 \subset \partial P} \sum_{F_2 \subset \partial F_1}
\frac{ \langle \xi, {\bf n}_{F_1} \rangle}{ ||\xi ||^2}
\frac{ \langle \rm{Proj}_{F_2} \xi, {\bf n}(F_2, F_1) \rangle}{ ||\rm{Proj}_{F_2} \xi ||^2}
\hat 1_{F_2}(\xi) \ [\xi \not= 0] [\rm{Proj}_{F_1} \xi \not= 0]
\end{align*}
It is an easy fact that the product of two Iverson brackets is the Iverson bracket of their intersection:
$[ P ] [ Q ] = [ P \text{ and } Q ]$ (Exercise \ref{Exercise Iverson bracket}). Hence, if we define
\[
F^\perp := \{ x \in \mathbb{R}^d \mid \langle x, y \rangle = 0 \text{ for all } y \in \rm{lin}F \},
\]
Then we see that ${\mathcal P}^\perp = \{ 0 \}$, and we can rewrite the latter identity as
\begin{align*}
\hat 1_{\mathcal P}(\xi) &= \vol {\mathcal P} \ [\xi \in {\mathcal P}^\perp ] +
\frac{1}{-2\pi i } \sum_{F_1 \subset \partial P}
\frac{ \langle \xi, {\bf n}_{F_1} \rangle (\vol F_1) e^{-2\pi i \langle \xi, x \rangle} }{ ||\xi ||^2}
\ [ \xi \in F_1^\perp - {\mathcal P}^\perp] \\
& + \frac{1}{(-2\pi i )^2} \sum_{F_1 \subset \partial P} \sum_{F_2 \subset \partial F_1}
\frac{ \langle \xi, {\bf n}_{F_1} \rangle}{ ||\xi ||^2}
\frac{ \langle \rm{Proj}_{F_2} \xi, {\bf n}(F_2, F_1) \rangle}{ ||\rm{Proj}_{F_2} \xi ||^2}
\hat 1_{F_2}(\xi) \ [ \xi \not\in F_1^\perp].
\end{align*}
In order to keep track of the iteration process, we will introduce another book-keeping device.
The {\bf face poset} of a polytope ${\mathcal P}$ \index{face poset} is defined to be the partially ordered set (poset) of all faces of ${\mathcal P}$, ordered by inclusion, including ${\mathcal P}$ and the empty set.
\bigskip
\begin{example}
\rm{
Consider a $2$-dimensional polytope ${\mathcal P}$ that is a triangle. We have the following picture for the face poset ${\frak F}_P$ of ${\mathcal P}$, as in Figure \ref{FacePoset1}. It turns out that if we consider a $d$-simplex ${\mathcal P}$, then its face poset ${\frak F}_P$ has the structure of a ``Boolean poset'', which is isomorphic to the edge graph of a $(d+1)$-dimensional cube.
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{FacePoset1}
\caption{The face poset of a triangle} \label{FacePoset1}
\end{figure}
}
\end{example}
We only have to consider rooted chains in the face poset ${\frak F}_P$, which means chains whose root is $P$. The only appearance of non-rooted chains are in the following definition.
If $G$ is a facet of $F$, we attach the following weight to any (local) chain $(F,G)$, of length $1$, in the face poset of $P$:
\begin{equation}\label{weight}
W_{(F,G)}(\xi):=\frac{-1}{2 \pi i} \frac{\langle \proj_{F} (\xi), {\bf n}(G, F) \rangle }{\| \proj_{F} (\xi) \|^2}.
\end{equation}
Note that these weights are functions of $\xi$ rather than constants. Moreover, they are all homogeneous of degree $-1$. Let $\mathbf{T}$ be any rooted chain in ${\frak F}_P$, given by
\[
T:= (P \to F_1 \to F_2, \dots, \to F_{k-1} \to F_k),
\]
so that by definition $\dim(F_j) = d-j$.
We define the {\bf admissible set} $S(\mathbf{T})$ of the rooted chain
$\mathbf{T}$ to be the set of all vectors $\xi\in \mathbb{R}^d$ that are orthogonal to the linear span of $F_k$ but not orthogonal to the linear span of $F_{k-1}$. In other words,
\begin{align*}
S(\mathbf{T}) &:= \{ \xi \in \mathbb{R}^d \mid \xi \perp \rm{lin}(F_k), \text{ but }
\xi \not\perp \rm{lin}(F_{k-1}) \} \\
& = \{ \xi \in \mathbb{R}^d \mid \xi \in F_k^\perp - F_{k-1}^\perp \}.
\end{align*}
Finally, we define the following weights associated to any such rooted chain $\mathbf{T}$:
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{graphG}
\caption{A symbolic depiction of the face poset ${\frak F}_P$, where $P$ is a $3$-dimensional tetrahedron. Here the points and arrows are drawn suggestively, as a directed graph. We can see all the rooted chains, beginning from a symbolic vertex in the center, marked with the color purple. The rooted chains that terminate with the yellow vertices have length $1$, those that terminate with the green vertices have length $2$, and those that terminate with the blue vertices have length $3$. }
\end{figure}
\begin{enumerate}[(a)]
\item The rational weight $\mathcal{R}_{\mathbf{T}}(\xi) = \mathcal{R}_{(P \to ... \to F_{k-1} \to F_k)}(\xi)$ is defined to be the product of weights associated to all the rooted chains $\mathbf{T}$
of length $1$, times the Hausdorff volume of $F_k$ (the last node of the chain $\mathbf{T}$). It is clear from this definition that $\mathcal{R}_{\mathbf{T}}(\xi)$ is a homogenous rational function of $\xi$.
\bigskip
\item The exponential weight
$\mathcal{E}_{\mathbf{T}}(\xi) = \mathcal{E}_{(P \to ... \to F_{k-1} \to F_k)}(\xi)$
is defined to be the evaluation of $e^{-2\pi i\langle\xi,x\rangle}$ at any point $x$ on the face $F_k$:
\begin{equation} \label{exponential.weight}
\mathcal{E}_{\mathbf{T}}(\xi) := e^{-2\pi i\langle\xi,x_0\rangle},
\end{equation}
for any $x_0 \in F_k$. We note that the inner product $\langle\xi,x_0 \rangle$ does not depend on the position of $x_0 \in F_k$.
\bigskip
\item The total weight of a rooted chain $T$ is defined to be the rational-exponential function
\begin{equation}
W_{\mathbf{T}}(\xi) = W_{(P \to ... \to F_{k-1} \to F_k)}(\xi):= \mathcal{R}_{\mathbf{T}}(\xi) \mathcal{E}_{\mathbf{T}}(\xi) \mathbf{1}_{S(\mathbf{T})}(\xi),
\end{equation}
\noindent
where $\mathbf{1}_{S(\mathbf{T})}(\xi)$ is the indicator function of the admissible set $S(\mathbf{T})$ of $\mathbf{T}$.
\end{enumerate}
\bigskip
\noindent
By repeated applications of the combinatorial divergence
Theorem \ref{Combinatorial divergence theorem},
we arrive at a description of the Fourier transform of $P$ as the sum of weights of all the rooted chains of
the face poset ${\frak F}_P$, as follows.
\begin{thm} \label{ingredient1}
\begin{align} \label{explicit Fourier transform of a polytope}
\hat 1_{P}(\xi) = \sum_{\mathbf{T}} W_{\mathbf{T}}(\xi) = \sum_{\mathbf{T}} \mathcal{R}_{\mathbf{T}}(\xi) \mathcal{E}_{\mathbf{T}}(\xi) \mathbf{1}_{S(\mathbf{T})}(\xi),
\end{align}
valid for any fixed $\xi \in \mathbb{R}^d$.
\end{thm}
For a detailed proof of Theorem \ref{ingredient1}, see \cite{RicardoNhatSinai}.
Using this explicit description of the Fourier transform of a polytope, we will see an application of it in the following
section, for the coefficients of Macdonald's angle quasi-polynomial. In the process, equation
\eqref{explicit Fourier transform of a polytope}, which gives an explicit description of the Fourier transform of a polytope, using the facets of ${\mathcal P}$ as well as lower-dimensional faces of ${\mathcal P}$, will become even more explicit with some examples.
\section{Generic frequencies versus special frequencies}
Given a polytope ${\mathcal P} \subset \mathbb{R}^d$, we call a vector $\xi \in \mathbb{R}^d$ a {\bf generic frequency} (relative to ${\mathcal P}$) \index{generic frequencies}
if $\xi$ is not orthogonal to any face of ${\mathcal P}$. All other $\xi \in \mathbb{R}^d$ are orthogonal to some face $F$ of ${\mathcal P}$, and are called {\bf special frequencies}. Let's define the following hyperplane arrangement, given by the finite collection of hyperplanes orthogonal to any edge of ${\mathcal P}$:
\[
\mathcal H := \{ x \in \mathbb{R}^d \mid \langle x, F_1 \rangle = 0, \text{ for any $1$-dimensional edge $F_1$ of ${\mathcal P}$ } \}.
\]
Then it is clear that the special frequencies are exactly those vectors that lie in the hyperplane arrangement
$\mathcal H$. So we see from Theorem \ref{ingredient1} that for a generic frequency $\xi$, we have
\begin{align}
\hat 1_{P}(\xi) = \sum_{\mathbf{T}: P \to ... \to F_{1} \to F_0} \mathcal{R}_{\mathbf{T}}(\xi) e^{-2\pi i \langle \xi, F_0 \rangle},
\end{align}
where the $F_0$ faces are the vertices of ${\mathcal P}$. In other words, for generic frequencies, all of our rooted chains in the face poset of ${\mathcal P}$ go all the way to the vertices. The special frequencies, however, are more complex. But we can collect the special frequencies in `packets', giving
us the following result.
\bigskip
\begin{thm} [Coefficients for Macdonald's angle quasi-polynomial] \cite{RicardoNhatSinai}
\label{thm:main}
Let $P$ be a $d$-dimensional rational polytope in $\mathbb{R}^d$, and let $t$ be a positive real number.
Then we have the quasi-polynomial
\[ A_P(t) =\sum_{i = 0}^d a_i(t)t^i, \]
where, for $0 \leq i \leq d$,
\begin{equation}\label{complicatedcoeff}
a_i(t) := \lim_{\varepsilon\to 0^+} \sum_{\xi\in\mathbb{Z}^d \cap S(\mathbf{T})}
\sum_{l(\mathbf{T}) = d-i} \mathcal{R}_{\mathbf{T}}(\xi) \mathcal{E}_{\mathbf{T}}(t\xi) \ e^{-\pi\varepsilon\|\xi\|^2},
\end{equation}
\end{thm}
\noindent
where $l(\mathbf{T})$ is the length of the rooted chain $\mathbf{T}$ in the face poset of $P$,
$\mathcal{R}_{\mathbf{T}}(\xi)$ is the rational function of $\xi$ defined above, $\mathcal{E}_{\mathbf{T}}(t\xi) $ is the complex exponential defined in \eqref{exponential.weight} above, and
$\mathbb{Z}^d \cap S(\mathbf{T})$ is the set of all integer points that are orthogonal to the last node in the chain $T$,
but not to any of its previous nodes.
See \cite{RicardoNhatSinai} for the detailed proof of Theorem \ref{thm:main}.
We call the coefficients $a_i(t)$ the {\bf quasi-coefficients} of the solid angle sum $A_P(t)$. \index{quasi-coefficients}
As a consequence of Theorem \ref{thm:main}, it turns out that there is a closed form for the codimension-$1$ quasi-coefficient, which extends previous special cases of this coefficient.
We recall our first periodic Bernoulli polynomial, from \eqref{definition of periodic Bernoulli polys}:
\begin{equation}
P_1 (x):=
\begin{cases}
x - \lfloor x \rfloor- \frac{1}{2} & \mbox{if } x \notin \mathbb{Z} \\
0 & \mbox{if } x \in \mathbb{Z},
\end{cases}
\end{equation}
where $\lfloor x \rfloor$ is the integer part of $x$.
\begin{thm}\cite{RicardoNhatSinai}
\label{codim1coeff}
Let $P$ be any real polytope. Then the codimension-1 quasi-coefficient of the solid angle sum $A_P(t)$ has the following closed form:
\begin{equation}
a_{d-1} (t) =
-\sum_{\substack{F \textup{ a facet of } P \\ with \ v_F \neq 0}} \frac{\vol F}{\|v_F\|}
P_1 (\langle v_F, x_F \rangle t),
\end{equation}
where $v_F$ is the primitive integer vector which is an outward-pointing normal vector to $F$, $x_F$ is
any point lying in the affine span of $F$, and $t$ is any positive real number.
\end{thm}
\hfill $\square$
\bigskip
We note that, rather surprisingly,
the latter formula shows in particular that for any real polytope ${\mathcal P}$, the quasi-coefficient $a_{d-1}(t)$ is always a periodic function of $t > 0$, with a period of $1$.
Although it is not necessarily true that for any real polytope the rest of the quasi-coefficients $a_k(t)$
are periodic functions of $t$, it is true that in the case of rational polytopes, the quasi-coefficients
are periodic functions of all real dilations $t$, as we show below.
We recall that zonotopes are projections of cubes or, equivalently, polytopes whose faces (of all dimensions) are symmetric. We also recall the result of Alexandrov and Shephard (Theorem \ref{Alexandrov-Shepard thm}) from chapter \ref{Geometry of numbers}: If all the facets of ${\mathcal P}$ are symmetric, then ${\mathcal P}$ must be symmetric as well. The following result appeared in \cite{BarvinokPommersheim}, and here we give a different proof, using the methods of this chapter.
\begin{thm} \label{cs.facets} \index{Barvinok, Alexander}
Suppose $P$ is a $d$-dimensional integer polytope in $\mathbb{R}^d$
all of whose facets are centrally symmetric. Then
\[
A_{\mathcal P}(t) = (\vol {\mathcal P}) t^d,
\]
for all positive integers $t$.
\end{thm}
\begin{proof}
We recall the formula for the solid angle polynomial $A_{\mathcal P}(t)$.
\begin{equation}\label{eq:APtsum}
A_{\mathcal P}(t) = \lim_{\varepsilon \to 0^+} \sum_{\xi \in \mathbb{Z}^d} \hat{1}_{t{\mathcal P}}(\xi)e^{-\pi\varepsilon\|\xi\|^2}.
\end{equation}
The Fourier transform of the indicator function of a polytope may be written as follows, after one application of the combinatorial divergence formula:
\begin{align}
\hat 1_{t{\mathcal P}}(\xi) = t^d \vol {\mathcal P} \, [\xi =0]
+ \left( \frac{-1}{2 \pi i} \right) t^{d-1}
\sum_{{\substack{F \subseteq {\mathcal P} \\ \dim F = d-1}}}
\frac{\langle \xi, {\bf n}_F \rangle }{\| \xi \|^2} \hat 1_F (t \xi) [\xi \not= 0],
\end{align}
where we sum over all facets $F$ of ${\mathcal P}$. Plugging this into~\eqref{eq:APtsum} we get
\begin{align} \label{imaginary}
A_{\mathcal P}(t) - t^d \vol {\mathcal P}
= \left( \frac{-1}{2 \pi i} \right) t^{d-1}
\lim_{\varepsilon \to 0^+} \sum_{\xi \in \mathbb{Z}^d \setminus\{0\}}
\frac{e^{-\pi\varepsilon\|\xi\|^2}}{\| \xi \|^2} \sum_{{\substack{F \subseteq {\mathcal P} \\ \dim F = d-1}}}
\langle \xi, {\bf n}_F \rangle \hat 1_F(t \xi),
\end{align}
so that it is sufficient to show that the latter sum over the facets vanishes.
The assumption that all facets of ${\mathcal P}$ are centrally symmetric implies that ${\mathcal P}$ itself is also centrally symmetric, by Theorem~\ref{cs1}.
We may therefore combine the facets of ${\mathcal P}$ in pairs of opposite facets $F$ and $F'$. We know that $F' = F + c$, where $c$ is an integer vector, using the fact that the facets are centrally symmetric.
Therefore, since ${\bf n}_F' = - {\bf n}_F $, we have
\begin{align*}
\langle \xi, {\bf n}_F \rangle &\hat 1_{F}(t \xi)
+ \langle \xi, -{\bf n}_F \rangle \hat 1_{F+ c}(t \xi)\\
&= \langle \xi, {\bf n}_F \rangle \hat 1_{F}(t \xi)
- \langle \xi, {\bf n}_F \rangle \hat 1_{F}(t \xi)
e^{-2\pi i\langle t\xi, c \rangle} \\
&= \langle \xi, {\bf n}_F \rangle \hat 1_{F}(t \xi)
\left( 1 - e^{-2\pi i\langle t\xi, c \rangle} \right) = 0,
\end{align*}
because $\langle t\xi, c \rangle \in \mathbb{Z}$ when both $\xi \in \mathbb{Z}^d$ and $t \in \mathbb{Z}$. We conclude that the entire right-hand side of \eqref{imaginary} vanishes, and we are done.
\end{proof}
Fourier analysis can also be used to give yet more general classes of polytopes that satisfy the formula
$A_P(t) = (\vol {\mathcal P}) t^d$, for positive integer values of $t$ (See also \cite{FabricioSinai2}, \cite{DeligneTabachnikovRobins}).
\bigskip
There is a wonderful result of Minkowski that gives a converse to the relation
\eqref{Minkowski relation}, as follows.
\begin{thm}[The Minkowski problem for polytopes] \label{Minkowski's problem for polytopes}
\index{Minkowski problem for polytopes}
Suppose that $u_1, \dots, u_k\in \mathbb{R}^d$ are unit vectors that do not lie in a hyperplane.
Suppose further that we are given positive numbers $\alpha_1, \alpha_2, \dots, \alpha_k >0$ that
satisfy the relation
\[
\alpha_1 u_1 + \cdots + \alpha_k u_k =0.
\]
Then there exists a polytope ${\mathcal P}\subset R^d$, with facet normals $u_1, \dots, u_k\in \mathbb{R}^d$, and
facet areas $\alpha_1, \alpha_2, \dots, \alpha_k$.
Moreover, this polytope ${\mathcal P}$ is unique, up to translations.
\hfill $\square$
\end{thm}
There is a large body of work, since the time of Minkowski, that is devoted to extensions of
Minkowski's Theorem \ref{Minkowski's problem for polytopes}, to other convex bodies, as well as to other manifolds.
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.Divergence}
\begin{enumerate}[(a)]
\item We could also define another useful vector field, for our combinatorial divergence theorem, besides
our vector field in equation \eqref{First vector field}. Namely, if we define
$F(x):= e^{2\pi i \langle x, \xi \rangle} \lambda$,
for a fixed $\lambda\in\mathbb{C}^d$, then we would get the analogous combinatorial divergence formula as shown below in
(Exercise \ref{alternate combinatorial divergence Theorem}),
and such vector fields
have been used, for example, by Alexander Barvinok \cite{Barvinok1} in an effective way.
\index{Barvinok, Alexander}
To the best of our knowledge,
the first researcher to use iterations of Stokes' formula to obtain lattice point asymptotics was Burton Randol \cite{Randol3}, \cite{Randol4}.
\item The Minkowski problem for polytopes can also be related directly to generalized isoperimetric inequalities for mixed volumes, as well as the Brunn-Minkowski inequality for polytopes,
as done by Daniel Klain in \cite{Klain1}.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\medskip
\begin{prob} $\clubsuit$ \label{Pyramid formula}
We define the distance from the origin to $F$, denoted by $\rm{dist}(F)$, as the length of the shortest vector of translation between
$\rm{aff}(F)$ and $\rm{lin}(F)$ (resp. the affine span of $F$ and the linear span of $F$, defined in
\eqref{affine span of F}). Figure \ref{affine span} shows what can happen in such a scenario.
\begin{enumerate}[(a)]
\item Suppose that we consider a facet $F$ of a given polytope ${\mathcal P} \subset \mathbb{R}^d$, and we
let ${\bf n}_F$ be the unit normal vector to $F$. Show that the function
\[
x_F \rightarrow \langle x_F, {\bf n}_F \rangle
\]
is constant for $x_F \in F$, and is in fact equal to the distance from the origin to $F$. In other words, show that
\[
\langle x, {\bf n}_F \rangle = \rm{dist}(F).
\]
\item Show that if $\rm{Proj}_F \xi = 0$, then $\langle \xi, x_F \rangle = \| \xi \| \rm{dist} F$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
Here we prove the elementary geometric formula for a pyramid over a polytope. Namely, suppose we are
given a
$(d-1)$-dimensional polytope ${\mathcal P}$, lying in the vector space defined by the first $d-1$ coordinates.
We define a pyramid over ${\mathcal P}$, of height $h > 0$, as the $d$-dimensional polytope defined by
\[
\rm{Pyr}({\mathcal P}) := \conv\{ {\mathcal P}, \ h \cdot e_{d} \},
\]
where $e_d := (0, 0, \dots, 0, 1) \in \mathbb{R}^d$. Show that
\[
\vol \rm{Pyr}({\mathcal P}) = \frac{h}{d} \vol {\mathcal P}.
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{distance to a facet}
Prove the Pyramid formula, \eqref{pyramid formula} in
Example \ref{Pyramid formula via the divergence theorem},
for a $d$-dimensional polytope ${\mathcal P}$ which contains the origin, but now using just elementary geometry:
\begin{equation}
\vol {\mathcal P} = \frac{1}{d} \sum_{k=1}^N \rm{dist}(G_k) \vol G_k,
\end{equation}
where the $G_k$'s are the facets of ${\mathcal P}$, and $\rm{dist}(G_k)$ is the distance from the origin to $G_k$.
\end{prob}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.2in]{ExerciseDivergence}
\end{center}
\caption{The meaning of Minkowski's relation in dimension $2$ - see Exercise
\ref{Geometric interpretation of Minkowski's relation for d=2} }
\label{Divergence Exercise}
\end{figure}
\medskip
\begin{prob} $\clubsuit$ \label{alternate combinatorial divergence Theorem}
Show that if we replace the vector field in equation \eqref{First vector field} by the
alternative vector field $F(x):= e^{-2\pi i \langle x, \xi \rangle} \lambda$, with a constant nonzero vector $\lambda \in \mathbb{C}^d$, then we get:
\begin{equation} \label{our alternate formula for the transform}
\hat 1_{\mathcal P}(\xi) =\frac{1}{-2\pi i } \sum_{G\subset \partial P}
\frac{ \langle \lambda, {\bf n}_G \rangle}{ \langle \lambda, \xi \rangle}
\hat 1_G(\xi),
\end{equation}
valid for all nonzero $\xi \in \mathbb{R}^d$. Note that one advantage of this formulation
of the Fourier transform
of ${\mathcal P}$ is that each summand in the right-hand-side of
\eqref{our alternate formula for the transform}
is free of singularities, assuming the vector
$\lambda$ has a nonzero imaginary part.
\end{prob}
\medskip
\begin{prob} \label{equivalent identity to the alternate vector field}
Show that the identity \eqref{our alternate formula for the transform} of Exercise \ref{alternate combinatorial divergence Theorem} is equivalent to the vector identity:
\[
\xi \hat 1_{\mathcal P}(\xi) = \frac{1}{-2\pi i } \sum_{G\subset \partial P} {\bf n}_G \hat 1_G(\xi),
\]
valid for all $\xi \in \mathbb{R}^d$.
\end{prob}
\medskip
\begin{prob}
Show that the result of Exercise \ref{equivalent identity to the alternate vector field}
quickly gives us
the Minkowski relation \eqref{Minkowski relation}:
\[
\sum_{ \text{facets } G \text{ of } P} \vol G {\bf n}_{G}= 0.
\]
\end{prob}
\medskip
\begin{prob}
Continuing Exercise \ref{alternate combinatorial divergence Theorem}, show that by iterating this particular version of the Fourier transform of a polytope ${\mathcal P}$, $k$ times, we get:
\begin{equation}
\hat 1_{\mathcal P}(\xi) =\frac{1}{(-2\pi i )^k} \sum_{G_k \subset G_{k-1} \subset \cdots G_1 \subset \partial P}
\prod_{j=1}^k
\frac{
\langle \lambda, {\bf n}_{G_{j}, G_{j-1}} \rangle}{ \langle \lambda, \rm{Proj}_{G_{j-1}} \xi \rangle }
\hat 1_{G_k}(\xi),
\end{equation}
valid for all nonzero $\xi \in \mathbb{R}^d$, and where we sum over all chains $G_k \subset G_{k-1} \subset \cdots G_1$ of length $k$ in the face poset of ${\mathcal P}$, with \rm{codim}$(G_j) = j$.
\end{prob}
\medskip
\begin{prob} \label{Geometric interpretation of Minkowski's relation for d=2}
Show that in the case of polygons in $\mathbb{R}^2$, the Minkowski relation \eqref{Minkowski relation}
has the meaning that
the sum of the pink vectors in Figure \ref{Divergence Exercise} sum to zero. In other words, the geometric interpretation
of the Minkowski relation in dimension $2$ is that the sum of the boundary (pink) vectors
wind around the boundary and close up perfectly.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{no simplex is symmetric}
Let's consider a simplex $\Delta \subset \mathbb{R}^d$ whose dimension satisfies $2 \leq \dim \Delta \leq d$.
Show that $\Delta$ is not a symmetric body.
\end{prob}
\medskip
\begin{prob}
Let $F \subset \mathbb{R}^d$ be a centrally symmetric, integer polytope of dimension $k$. Show that the distance from the origin to $F$ is always a half-integer or an integer. In other words, show that
\[
\rm{dist}(F) \in \frac{1}{2} \mathbb{Z}.
\]
(See Exercise \ref{Pyramid formula} above for the definition of distance of $F$ to the origin)
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Exercise Iverson bracket}
To get more practice with the Iverson bracket, defined in \eqref{Iverson bracket}, show that for all logical statements $P$, we have:
\begin{enumerate}[(a)]
\item $ [ P \rm{ \ and \ } Q ] = [ P ] [ Q ]$.
\item $[ P \rm{ \ or \ } Q ] = [P] + [Q] - [P][Q]$.
\item $[\neg P] = 1 - [P]$, where $\neg P$ means the logical negation of $P$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{the FT of 1_P is not in L^1}
Show that for any polytope ${\mathcal P}\subset \mathbb{R}^d$, its Fourier transform $\hat 1_{\mathcal P}(\xi)$ is not in $L^1(\mathbb{R}^d)$.
\end{prob}
\chapter{The angle polynomial of a polytope}
\label{Angle polynomial}
\index{angle polynomial} \index{solid angle}
\begin{figure}[!h]
\centering
\begin{tikzpicture}[scale=.75]
\draw (0,0) node[below left] {$0$};
\draw[loosely dotted] (-1,-1) grid (7,5);
\draw[->] (-1.25,0) -- (7.25,0) node[right] {$x$};
\draw[->] (0,-1.25) -- (0,5.25) node[above] {$y$};
\draw[fill = green] (3,4) circle (.1cm);
\draw[fill = green] (4,3) circle (.1cm);
\draw[fill = green] (3,3) circle (.1cm);
\draw[fill = green] (3,2) circle (.25cm);
\draw[fill = green] (2,2) circle (.1cm);
\draw[fill = green] (2,3) circle (.25cm);
\draw[fill = green] (4,4) circle (.25cm);
\draw[fill = green] (5,3) circle (.5cm);
\draw[fill = green] (3,5) circle (.5cm);
\draw[fill = green] (1,1) circle (.5cm);
\draw[thick] (1,1) -- (5,3) -- (3,5) -- cycle;
\filldraw[nearly transparent, blue] (1,1) -- (5,3) -- (3,5) -- cycle;
\end{tikzpicture}
\caption{A discrete volume of the triange ${\mathcal P}$, called the angle polynomial of ${\mathcal P}$.
Here we sum local angle weights, relative to ${\mathcal P}$, at all integer points.}
\label{triangle solid angle sum}
\end{figure}
\section{Intuition}
There are infinitely many ways to discretize the classical notion of volume, and here we offer a second path, using `local solid angles'. Given a rational polytope ${\mathcal P}$, we will place small spheres at all integer points in $\mathbb{Z}^d$, and compute the proportion of the local intersection of each small sphere with ${\mathcal P}$. This discrete, finite sum, gives us a new method of discretizing the volume of a polytope, and it turns out to be a more symmetric way of doing so. To go forward, we first discuss how to extend the usual notion of `angle' to higher dimensions, and then use Poisson summation again to pursue the fine detail of this new discrete volume.
\bigskip
\section{What is an angle in higher dimensions?} \label{Chapter.solid.angles}
The question of how an angle in two dimensions extends to higher dimensions is a basic one
in discrete geometry. A natural way to extend the notion of an angle is to consider
a cone ${\mathcal K} \subset \mathbb{R}^d$, place a sphere centered at the apex of ${\mathcal K}$, and then compute the proportion
of the sphere that intersects ${\mathcal K}$. This intuition is captured more rigorously by the following integral:
\begin{equation} \label{integral def. of solid angle}
\omega_{\mathcal K} = \int_{\mathcal K} e^{-\pi \| x \|^2} dx.
\end{equation}
called the {\bf solid angle of the cone} ${\mathcal K}$. \index{solid angle}
The literature has other synonyms for solid angles, arising in different fields,
including the {\bf volumetric moduli} \cite{GourionSeeger},
\index{volumetric moduli}
and the {\bf volume of a spherical polytope}
\index{volume of a spherical polytope}
\cite{BeckRobins}, \cite{DesarioRobins}, \cite{RicardoNhatSinai}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.3in]{solidangle}
\end{center}
\caption{A solid angle in $\mathbb{R}^3$ - note the equivalence with the area of the geodesic triangle on the sphere.}
\end{figure}
We can easily show that the latter definition of a solid angle
is equivalent to the volume of a spherical polytope,
\index{volume of a spherical polytope}
using polar coordinates in $\mathbb{R}^d$, as follows. We denote the
unit sphere by
$S^{d-1}:= \{ x\in \mathbb{R}^d \mid \| x\| = 1 \}$. Then using the fact that the Gaussians give a probability distribution, namely $\int_{\mathbb{R}^d} e^{-\pi ||x||^2} dx = 1$ (which we know by Exercise \ref{Gaussian1}), we have
\begin{align}
\omega_{\mathcal K} &= \frac{\int_{\mathcal K} e^{-\pi\|x\|^2}dx}{\int_{\mathbb{R}^d} e^{-\pi\|x\|^2}dx} \label{second equality}
\ = \ \frac{\int_0^{\infty} e^{-\pi r^2} r^{d-1} dr \int_{S^{d-1} \cap {\mathcal K}} d\theta}{\int_0^{\infty}
e^{-\pi r^2} r^{d-1} dr \int_{S^{d-1}} d\theta} \\
&= \ \frac{\int_{S^{d-1} \cap {\mathcal K}} d\theta}{\int_{S^{d-1}} d\theta} \\ \label{normalized spherical volume}
&= \ \frac{\vol_{d-1} \left({\mathcal K} \cap S^{d-1}\right)}{\vol_{d-1} \left( S^{d-1} \right)},
\end{align}
where $\vol_{d-1}$ denotes the volume measure on the surface of the $(d-1)$-dimensional sphere $S^{d-1}$.
We may think of \eqref{normalized spherical volume} as the {\bf normalized volume} of a spherical polytope defined by the intersection of the cone ${\mathcal K}$
with the unit sphere. Thus for any cone ${\mathcal K}\subset \mathbb{R}^d$, we have
\[
0 \leq \omega_{\mathcal K} \leq 1.
\]
We used polar coordinates in the second equality \eqref{second equality} above:
$x= (r, \theta)$, with $r \geq 0, \ \theta \in {\mathcal S}^{d-1}$.
The Jacobian in the change of variables is $dx = r^{d-1} dr d\theta$.
We note that when ${\mathcal K}= \mathbb{R}^d$, so that the cone is \emph{all} of Euclidean space,
the integral \eqref{integral def. of solid angle}
becomes
\[
\int_{\mathbb{R}^d} e^{-\pi ||x||^2} dx = 1,
\]
by Exercise \ref{Gaussian1}. This computation confirms that we do indeed have the proper normalization
with $\omega_{\mathcal K} = 1$ if and only if ${\mathcal K} = \mathbb{R}^d$.
\begin{example}
\rm{
If ${\mathcal K}\subset \mathbb{R}^d$ is a half-space, then
$\omega_{\mathcal K} = \frac{1}{2}$. If ${\mathcal K}:= \mathbb{R}_{\geq 0}^d$, the positive orthant, then
\begin{align*}
\omega_{\mathcal K} &= \int_{\mathbb{R}^d_{\ge 0}} e^{-\pi ||x||^2} dx =
\left( \int_{\mathbb{R}_{\ge 0}} e^{-\pi u^2} du \right)^d = \frac{1}{2^d}.
\end{align*}
}
\hfill $\square$
\end{example}
\section{Local solid angles for a polytope, and Gaussian smoothing}
Here we want to define solid angles relative to a fixed polytope.
So given any polytope ${\mathcal P} \subset \mathbb{R}^d$, we fix any point $x \in \mathbb{R}^d$ and define a local solid angle relative to ${\mathcal P}$ as follows.
The normalized {\bf solid angle} \index{solid angle} fraction that a
$d$-dimensional polytope ${\mathcal P}$ subtends at any point $x \in \mathbb{R}^d$ is defined by
\begin{equation}\label{def. of solid angle}
\omega_{\mathcal P}(x)=\lim_{\varepsilon \to 0} \frac{\vol(S^{d-1}(x,\varepsilon) \cap {\mathcal P})}{\vol\left(S^{d-1}(x,\varepsilon)\right)}.
\end{equation}
Here, $\omega_{{\mathcal P}}(x)$ measures the fraction of a small $(d-1)$-dimensional sphere
$S^{d-1}(x,\varepsilon)$
centered at $x$, that intersects the polytope ${\mathcal P}$.
We will use the standard notation for the interior of a convex body, namely $\interior({\mathcal P})$, and for the boundary of a convex body, namely $\partial {\mathcal P}$. As a side-note, we mention that balls and spheres can be used interchangeably in this definition, meaning that the fractional weight given by \eqref{def. of solid angle}
is the same using either method (see Exercise \ref{two definitions for a solid angle}).
It follows from the definition of a solid angle that $0 \leq \omega_{\mathcal P}(x) \leq 1$, for all $x \in \mathbb{R}^d$, and that
\[
\omega_{\mathcal P}(x) =
\begin{cases}
1 & \text{if } x \in \interior({\mathcal P}) \\
0 & \text{if } x \notin {\mathcal P}.
\end{cases}
\]
But when $x \in \partial {\mathcal P}$, we have $\omega_{\mathcal P}(x) >0$. For example, if $x$ lies on a
codimension-two face of ${\mathcal P}$, then $\omega_{\mathcal P}(x)$ is the fractional dihedral angle
subtended by ${\mathcal P}$ at $x$.
Returning to discrete volumes, Ehrhart and Macdonald analyzed a different discrete volume for
any polytope ${\mathcal P}$. Namely, for each positive integer $t$, define the finite sum
\begin{equation} \label{anglesum1}
A_{\mathcal P}(t) := \sum_{n\in \mathbb{Z}^d} \omega_{t{\mathcal P}}(n),
\end{equation}
where $t{\mathcal P}$ is the $t$'th dilation of the polytope ${\mathcal P}$.
for ${\mathcal P}$.
In other words, $A_{\mathcal P}(1)$ is
a new discrete volume for ${\mathcal P}$, obtained by placing at each integer point $n\in \mathbb{Z}^d$ the weight
$\omega_{t{\mathcal P}}(x)$, and summing all of the weights.
\begin{example}
\rm{
In Figure \ref{triangle solid angle sum}, the solid angle sum of the polygon ${\mathcal P}$ is
\[
A_{\mathcal P}(1) = \theta_1 + \theta_2 + \theta_3 + 3\left( \tfrac{1}{2} \right)+ 4 = 6.
\]
Here the $\theta_j$'s are the three angles at the vertices of
${\mathcal P}$.
}
\hfill $\square$
\end{example}
Using purely combinatorial methods, Macdonald showed that for any integer polytope ${\mathcal P}$,
and for {\bf positive integer values} of $t$,
\begin{equation} \label{solidanglesum2}
A_{\mathcal P}(t) = (\vol {\mathcal P}) t^d + a_{d-2} t^{d-2} + a_{d-4} t^{d-4} + \cdots +
\begin{cases}
a_1 t & \text{if } d \text{ is odd},\\
a_2 t^2 & \text{if } d \text{ is even}.
\end{cases}
\end{equation}
We will call $A_{\mathcal P}(t)$ the {\bf angle-polynomial} of ${\mathcal P}$, \index{angle polynomial}
for integer polytopes ${\mathcal P}$ and positive
integer dilations $t$. However, when these restrictions are lifted, the sum still captures crucial
geometric information of ${\mathcal P}$, and we will simply call it the (solid) angle-sum of ${\mathcal P}$.
We define the {\bf heat kernel}, \index{heat kernel}
for each fixed positive $\varepsilon$, by
\begin{equation}
G_{\varepsilon}(x) := \varepsilon^{-\frac{d}{2}} e^{-\frac{\pi}{\varepsilon} \| x \|^2},
\end{equation}
for all $x \in \mathbb{R}^d$. By Exercises \ref{Gaussian1} and \ref{Gaussian2}, we know that
$\int_{\mathbb{R}^d} G_{\varepsilon}(x)dx = 1$ for each fixed $\varepsilon$, and that
\begin{equation} \label{Fourier transform of the Gaussian, formal}
\hat G_{\varepsilon}(\xi) = e^{-\varepsilon \pi \| \xi \|^2}.
\end{equation}
The convolution of the indicator function $1_{\mathcal P}$ by the heat kernel $G_{\varepsilon}$ will be called the
{\bf Gaussian smoothing} \index{Gaussian smoothing}
of $1_{\mathcal P}$:
\begin{align} \label{Gaussian smoothing}
(1_{\mathcal P} * G_{\varepsilon})(x) &:= \int_{\mathbb{R}^d} 1_{\mathcal P}(y) G_{\varepsilon} (x-y) dy
= \int_{{\mathcal P}} G_{\varepsilon} (y-x) dy \\
&= \varepsilon^{-\frac{d}{2}} \int_{{\mathcal P}} e^{-\frac{\pi }{\varepsilon} \| y-x \|^2} dy,
\end{align}
a $C^{\infty}$ function of $x\in \mathbb{R}^d$, and in fact a Schwartz function
(Exercise \ref{convolution of the indicator function with a Gaussian}). The following Lemma provides a first
crucial link between the discrete geometry of a local solid angle and the convolution of $1_{\mathcal P}$ with a
Gaussian-based approximate identity.
\bigskip
\begin{lem} \label{basic connection for solid angle}
Let ${\mathcal P}$ be a full-dimensional polytope in $\mathbb{R}^d$. Then for each point $x\in\mathbb{R}^d$, we have
\begin{equation}
\lim_{\varepsilon \rightarrow 0} (1_{\mathcal P} * G_{\varepsilon})(x) = \omega_P(x).
\end{equation}
\end{lem}
\begin{proof}
We have
\begin{align*}
(1_{\mathcal P} * G_{\varepsilon})(x) &= \int_{{\mathcal P}} G_{\varepsilon} (y-x) dy \\
&= \int_{u \in P- x} G_{\varepsilon} (u) du = \int_{\frac{1}{\sqrt{\varepsilon}}(P- x)} G_1(v) dv.
\end{align*}
In the calculation above, we make use of the evenness of $G_{\varepsilon}$ in the second equality. The substitutions $u = y-x$ and $v = u/\sqrt{\varepsilon}$ are also used. Following those substitutions, we change the domain of integration from $P$ to the translation $P- x$,
and to the dilation of $P-x$ by the factor $\frac{1}{\sqrt{\varepsilon}}$.
Now, when $\varepsilon$ approaches $0$,
$\frac{1}{\sqrt{\varepsilon}}(P- x)$ tends to a cone $K$ at the origin, subtended by $P- x$.
The cone $K$ is in fact a translation of the tangent cone \index{tangent cone}
of $P$ at $x$. Thus, we arrive at
\[
\lim_{\varepsilon \to 0} (1_{\mathcal P} * G_{\varepsilon}) (x) = \int_{K} G_1(v)dv = \omega_K(0) = \omega_P(x).
\]
\end{proof}
Putting things together, the definition \ref{anglesum1} and
Lemma \ref{basic connection for solid angle}
above tell us that
\begin{equation}
A_{\mathcal P}(t) = \sum_{n\in\mathbb{Z}^d} \omega_{tP}(x) =
\sum_{n\in\mathbb{Z}^d} \lim_{\varepsilon \rightarrow 0} (1_{t{\mathcal P}} * G_{\varepsilon})(n).
\end{equation}
We would like to interchange a limit with an infinite sum over a lattice, so that we may use Poisson summation, and although this is subtle in general, it's possible to carry out here, because the summands are rapidly decreasing.
\begin{lem} \label{interchanging limit and sum, solid angle sum}
Let ${\mathcal P}$ be a full-dimensional polytope in $\mathbb{R}^d$.
Then
\begin{equation}
A_{\mathcal P}(t) = \lim_{\varepsilon \rightarrow 0} \sum_{n\in\mathbb{Z}^d} (1_{t{\mathcal P}} * G_{\varepsilon})(n).
\end{equation}
\hfill $\square$
\end{lem}
(For a proof of Lemma \ref{interchanging limit and sum, solid angle sum} see \cite{RicardoNhatSinai}).
Next, we apply the Poisson summation formula to the Schwartz function \\
$f(x) := (1_{\mathcal P} * G_{\varepsilon})(x)$:
\begin{align} \label{Gaussian smoothing for Angle sum}
A_P(t) &= \lim_{\varepsilon \rightarrow 0} \sum_{n\in\mathbb{Z}^d} (1_{t{\mathcal P}} * G_{\varepsilon})(n) \\
&= \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}^d} \hat 1_{t{\mathcal P}}(\xi) \hat G_{\varepsilon}(\xi) \\
&= \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}^d} \hat 1_{t{\mathcal P}}(\xi) \ e^{-\varepsilon \pi \| \xi \|^2} \\
&= t^d \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}^d} \hat 1_{{\mathcal P}}( t\xi)
\ e^{-\varepsilon \pi \| \xi \|^2} \\
&= t^d \ \hat 1_{{\mathcal P}}( 0 ) + \lim_{\varepsilon \rightarrow 0} t^d \sum_{\xi \in\mathbb{Z}^d-\{0\}} \hat 1_{{\mathcal P}}( t\xi)
\ e^{-\varepsilon \pi \| \xi \|^2} \\
&= t^d (\vol {\mathcal P}) + \lim_{\varepsilon \rightarrow 0}t^d \sum_{\xi \in\mathbb{Z}^d-\{0\}} \hat 1_{{\mathcal P}}( t\xi)
\ e^{-\varepsilon \pi \| \xi \|^2},
\label{last line of Poisson summation}
\end{align}
where we used the fact that Fourier transforms interact nicely with dilations of the domain:
\[
\hat 1_{t{\mathcal P}}(\xi) = \int_{t{\mathcal P}} e^{-2\pi i \langle \xi, x \rangle} dx =
t^d \int_{{\mathcal P}} e^{-2\pi i \langle \xi, ty \rangle} dy = t^d \int_{{\mathcal P}} e^{-2\pi i \langle t \xi, y \rangle} dy=
t^d \hat 1_{{\mathcal P}}(t\xi).
\]
We also used the simple change of variable $x =t y$, with $y \in {\mathcal P}$, implying that $dx = t^d dy$,
as well as the Fourier transform formula for the heat kernel \eqref{Fourier transform of the Gaussian, formal}.
Altogether, we now have:
\begin{equation} \label{phase 2 of angle polynomial}
A_{\mathcal P}(t) =
t^d (\vol {\mathcal P}) + t^d \lim_{\varepsilon \rightarrow 0}
\sum_{n\in\mathbb{Z}^d-\{0\}} ( \hat 1_{{\mathcal P}}(t\xi) * G_{\varepsilon})(n),
\end{equation}
suggesting a polynomial-like behavior for the angle polynomial $A_{\mathcal P}(t)$.
\medskip
The next step will be to use our knowledge of the Fourier transform of the polytope ${\mathcal P}$, in
the right-hand-side of \eqref{phase 2 of angle polynomial}, for which even
a $1$-dimensional example is interesting.
\medskip
\begin{example} \label{rigorous example of P_1}
\rm{
Let's compute the angle polynomial of the $1$-dimensional polytope ${\mathcal P}:= [a,b]$, with
$a,b \in \mathbb{R}$. We will use our knowledge of the $1$-dimensional Fourier transform of an interval,
from Exercise \ref{transform.of.interval.a.to.b}, to compute:
\begin{align}
A_P(t) &= (b-a) t + \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}} \hat 1_{{\mathcal P}}( t\xi) \ e^{-\varepsilon \pi \xi^2} \\
&= (b-a) t + \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\left( \frac{e^{-2\pi i t\xi b} - e^{-2\pi i t\xi a} }{-2\pi i \xi} \right)
e^{-\varepsilon \pi \xi^2} \\ \label{strange limit1}
&= (b-a) t + \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\frac{e^{-2\pi i tb\xi -\varepsilon \pi \xi^2}}{-2\pi i \xi}
- \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\frac{e^{-2\pi i ta\xi -\varepsilon \pi \xi^2}}{-2\pi i \xi} \\
\end{align}
Throughout this example, all series converge absolutely (and quite rapidly) due to the existence of the Gaussian damping factor~$e^{-\varepsilon \pi \xi^2}$.
Let's see what happens when we specialize the vertices $a$ or $b$ - perhaps we can solve for these
new limits?
case 1. \ $a, b \in \mathbb{Z}$. This is the case of an integer polytope, which in this case is an interval in~$\mathbb{R}^1$. Because we are restricting attention to integer dilates $t$, and since $a, b, \xi \in \mathbb{Z}$, we have $e^{-2\pi i t\xi b} = e^{-2\pi i t\xi a} =1$.
Therefore
\[
A_P(t) = (b-a) t + \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\left( \frac{e^{-2\pi i t\xi b} - e^{-2\pi i t\xi a} }{-2\pi i \xi} \right)
e^{-\varepsilon \pi \xi^2} =(b-a) t + 0.
\]
We arrive at
\[
A_P(t) = (b-a)t,
\]
so that the solid angle sum $A_P(1)$ is exactly the length of the interval we considered. We may compare this discrete volume with the other discrete volume, namely the Ehrhart polynomial of this interval: $L_{[a, b]}(t) = (b-a)t + 1$.
\medskip
case 2. \ $a= 0, b \notin \mathbb{Z}$. Here one of the two series in \eqref{strange limit1} is:
\[
\sum_{\xi \in\mathbb{Z}-\{0\}}
\frac{e^{-2\pi i ta\xi -\varepsilon \pi \xi^2}}{-2\pi i \xi} =
\sum_{\xi \in\mathbb{Z}-\{0\}}
\frac{e^{ -\varepsilon \pi \xi^2}}{-2\pi i \xi} = 0,
\]
because the summand is an odd function of $\xi$. But we already know by direct computation
that in this case
$A_{[0, b]}(t) = \frac{1}{2} + \lfloor bt \rfloor$, we can solve for the other limit:
\[
\frac{1}{2} + \lfloor bt \rfloor= b t +
\lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\left( \frac{e^{-2\pi i t\xi b} }{-2\pi i \xi} \right)
e^{-\varepsilon \pi \xi^2}
\]
So this simple example has given us a nice theoretical result. We record this rigorous proof above as Lemma \ref{rigorous approach for P_1(x)} below, after relabelling $bt := x\in \mathbb{R}$.
}
\hfill $\square$
\end{example}
\medskip
\begin{lem} \label{rigorous approach for P_1(x)}
For any $x\in \mathbb{R}$, we have
\[
\frac{1}{2\pi i} \lim_{\varepsilon \rightarrow0} \sum_{\xi \in \mathbb{Z} - \{0\}}
\frac{e^{-2\pi i x\xi -\varepsilon \pi \xi^2}}{\xi} = x - \lfloor x \rfloor - \frac{1}{2}.
\]
\end{lem}
\bigskip
\begin{thm}
Let ${\mathcal P}$ be an integer polygon. Then the angle polynomial of ${\mathcal P}$ is:
\[
A_{\mathcal P}(t) = (\rm{area}{\mathcal P})t^2,
\]
for all positive integer dilations $t$.
\end{thm}
It turns out that this result, for $A_{\mathcal P}(1)$, is easily
equivalent to the well-known Pick's formula \index{Pick's formula}
for an integer polygon.
\begin{thm}[Pick's formula, 1899]
Let ${\mathcal P}$ be an integer polygon. Then
\[
\rm{Area} {\mathcal P} = I + \frac{1}{2} B -1,
\]
where $I$ is the number of interior integer points in ${\mathcal P}$, and B is the number of boundary integer points in ${\mathcal P}$.
\end{thm}
\begin{figure}[!h]
\centering
\begin{tikzpicture}[scale=0.45]
\draw (0,0) node[below left] {$0$};
\draw[loosely dotted] (-1,-1) grid (7,5);
\draw[->] (-1.25,0) -- (7.25,0) node[right] {$x$};
\draw[->] (0,-1.25) -- (0,5.25) node[above] {$y$};
\draw[fill = green] (3,4) circle (.1cm);
\draw[fill = green] (4,3) circle (.1cm);
\draw[fill = green] (3,3) circle (.1cm);
\draw[fill = green] (3,2) circle (.25cm);
\draw[fill = green] (2,2) circle (.1cm);
\draw[fill = green] (2,3) circle (.25cm);
\draw[fill = green] (4,4) circle (.25cm);
\draw[fill = green] (5,3) circle (.5cm);
\draw[fill = green] (3,5) circle (.5cm);
\draw[fill = green] (1,1) circle (.5cm);
\draw[thick] (1,1) -- (5,3) -- (3,5) -- cycle;
\filldraw[nearly transparent, blue] (1,1) -- (5,3) -- (3,5) -- cycle;
\draw (3,-2) node {$P_1$};
\draw (9,2) node[scale = 2] {$\cup$};
\draw (0+12,0) node[below left] {$0$};
\draw[loosely dotted] (-1+12,-1) grid (7+12,5);
\draw[->] (-1.25+12,0) -- (7.25+12,0) node[right] {$x$};
\draw[->] (0+12,-1.25) -- (0+12,5.25) node[above] {$y$};
\draw[fill = green] (5+12,2) circle (.1cm);
\draw[fill = green] (4+12,2) circle (.1cm);
\draw[fill = green] (3+12,2) circle (.25cm);
\draw[fill = green] (5+12,3) circle (.5cm);
\draw[fill = green] (6+12,2) circle (.5cm);
\draw[fill = green] (1+12,1) circle (.5cm);
\draw[thick] (1+12,1) -- (6+12,2) -- (5+12,3) -- cycle;
\filldraw[semitransparent, blue] (1+12,1) -- (5+12,3) -- (6+12,2) -- cycle;
\draw (3+12,-2) node {$P_2$};
\draw (9+12,2) node[scale = 2] {$=$};
\draw (0+24,0) node[below left] {$0$};
\draw[loosely dotted] (-1+24,-1) grid (7+24,5);
\draw[->] (-1.25+24,0) -- (7.25+24,0) node[right] {$x$};
\draw[->] (0+24,-1.25) -- (0+24,5.25) node[above] {$y$};
\draw[fill = green] (3+24,4) circle (.1cm);
\draw[fill = green] (4+24,3) circle (.1cm);
\draw[fill = green] (3+24,3) circle (.1cm);
\draw[fill = green] (5+24,2) circle (.1cm);
\draw[fill = green] (4+24,2) circle (.1cm);
\draw[fill = green] (3+24,2) circle (.25cm);
\draw[fill = green] (2+24,2) circle (.1cm);
\draw[fill = green] (2+24,3) circle (.25cm);
\draw[fill = green] (4+24,4) circle (.25cm);
\draw[fill = green] (5+24,3) circle (.5cm);
\draw[fill = green] (6+24,2) circle (.5cm);
\draw[fill = green] (3+24,5) circle (.5cm);
\draw[fill = green] (1+24,1) circle (.5cm);
\draw[thick] (1+24,1) -- (6+24,2) -- (3+24,5) -- cycle;
\draw[thick] (1+24,1) -- (5+24,3);
\filldraw[nearly transparent, blue] (1+24,1) -- (5+24,3) -- (3+24,5) -- cycle;
\filldraw[semitransparent, blue] (1+24,1) -- (5+24,3) -- (6+24,2) -- cycle;
\draw (3+24,-2) node {$P_1 \cup P_2$};
\end{tikzpicture}
\caption{Additive property of the angle polynomial}
\end{figure}
There is also a way to characterize the polytopes that $k$-tile $\mathbb{R}^d$ by translations, using
solid angle sums. Gravin, Robins, and Shiryaev~\cite[Theorem 6.1]{GravinShiryaevRobins} gave the following characterization.
\begin{thm}
A polytope $P$ $k$-tiles $\mathbb{R}^d$ by integer translations if and only if
\[
\sum_{\lambda \in \mathbb{Z}^d} \omega_{P + v}(\lambda) = k,
\]
for every $v \in \mathbb{R}^d$.
\end{thm}
\bigskip
\section{The Gram relations for solid angles}
\index{Gram relations}
How does our elementary school identity, giving us the sum of the angles of a triangle, extend to higher dimensions? We describe the extension here, mainly due to Gram. First, for each face
$F$ of a polytope ${\mathcal P} \subset \mathbb{R}^d$, we define the {\bf solid angle of} $F$,
\index{solid angle of a face}
as follows. Fix any
$x_0 \in \interior F$, and let
\[
\omega_F := \omega_P(x_0).
\]
We notice that this definition is independent of $x_0$, as long as we restrict $x_0$ to the relative interior of $F$.
\begin{example}
If ${\mathcal P}$ is the $d$-dimensional cube $[0, 1]^d$, then each of its facets $F$ has
$\omega_F = \frac{1}{2}$. However, it is a fact that for the cube, a face of dimension $k$ has a solid angle of
$\frac{1}{2^{d-k}}$ (Exercise \ref{solid angle of a face of the cube}). In particular
a vertex $v$ of this cube, having dimension $0$, has solid angle $\omega_v = \frac{1}{2^d}$.
\hfill $\square$
\end{example}
\begin{thm}[Gram relations] \label{Gram relations} \index{Gram relations}
Given any $d$-dimensional polytope $P\subset \mathbb{R}^d$, we have
\[
\sum_{F\subset {\mathcal P}} (-1)^{\dim F} \omega_F = 0.
\]
\end{thm}
\hfill $\square$
(For a proof of Lemma \ref{Gram relations} see \cite{BeckRobins}).
\begin{example}
\rm{
Let's see what the Gram relations tell us in the case of a triangle $\Delta$.
For each edge $E$ of $\Delta$, placing a small sphere at a point in the interior of $E$ means half of it is inside $\Delta$ and half of it is outside of $\Delta$, so that $\omega_E = \frac{1}{2}$. Next, each vertex of $\Delta$ has a solid angle equal to the usual (normalized) angle $\theta(v)$ at that vertex.
Finally $\Delta$ itself has a solid angle of $1$, because picking a point $p$ in the interior of $\Delta$, and placing a small sphere centered at $p$, the whole sphere will be contained in $\Delta$. Putting it all together, the Gram relations read:
\begin{align*}
0 &= \sum_{F\subset \Delta} (-1)^{\dim F} \omega_F \\
&= (-1)^0 (\theta(v_1) + \theta(v_2) +\theta(v_3) )
+ (-1)^1 \left(\frac{1}{2} + \frac{1}{2} +\frac{1}{2}\right)
+ (-1)^2 \cdot 1 \\
&= \theta(v_1) + \theta(v_2) +\theta(v_3) -\frac{1}{2},
\end{align*}
which looks familiar! We've retrieved our elementary-school knowledge, namely that the three
angles of a triangle sum to $\pi$ radians. So the Gram relations really are an extension of this fact.
}
\hfill $\square$
\end{example}
What about $\mathbb{R}^3$?
\begin{example}
\rm{
Let's see what hidden secrets lie behind the Gram relations for the standard simplex
$\Delta \subset \mathbb{R}^3$.
\index{standard simplex}
At the origin $v_0 = 0$, the tangent cone is the positive orthant,
so that $\omega(v_0) = \frac{1}{8}$. The other $3$ vertices all ``look alike'', in the sense that
their tangent cones are all isometric, and hence have the same solid angle $\omega_v$.
What about the edges?
In general, it's a fact that the solid angle of an edge equals the dihedral angle between the planes of
its two bounding facets (Exercise \ref{dihedral angle=solid angle}). There are two types of edges here, as in the figure.
For an edge $E$ which lies on the boundary of the skew facet, we have the dihedral angle
$\cos \phi = \left\langle \frac{1}{\sqrt 3}\icol{ 1\{\bf 1}\{\bf 1}}, \icol{ 0\{\bf 1}\{\bf 0}} \right\rangle = \frac{1}{\sqrt 3}$, so that
$\omega_E = \phi = \cos^{-1}\frac{1}{\sqrt 3}$. It's straightforward that for the other type of edge,
each of those $3$ edges has a solid angle of $\frac{1}{4}$. Putting it all together, we see that
\begin{align*}
0 &= \sum_{F\subset \Delta} (-1)^{\dim F} \omega_F \\
&= (-1)^0 \left( \frac{1}{8} + 3\omega_v \right)
+ (-1)^1 \left( 3 \frac{1}{4}+ 3 \cos^{-1}\frac{1}{\sqrt 3} \right)
+ (-1)^2 \frac{1}{2} \cdot 4
+ (-1)^3 \cdot 1.
\end{align*}
Solving for $\omega_v$, we get $\omega_v = \cos^{-1}\frac{1}{\sqrt 3} -\frac{1}{8}$.
So we were able to compute the solid angle of at a vertex of $\Delta$ in $\mathbb{R}^3$, using the
Gram relations, together with a bit of symmetry.
}
\hfill $\square$
\end{example}
Related to the topics above is the fact that the angle polynomial possesses the following fascinating
functional equation
(For a proof of Theorem \ref{Angle polynomial functional equation}, and an extension of it, see \cite{DesarioRobins}).
\begin{thm}[Functional equation for the angle polynomial] \label{Angle polynomial functional equation}
\index{angle polynomial: functional equation}
Given a $d$-dimensional \\
rational polytope ${\mathcal P}\subset \mathbb{R}^d$, we have
\[
A_{{\mathcal P}}(-t) = A_{{\mathcal P}}(t),
\]
for all $t \in \mathbb{Z}$.
\hfill $\square$
\end{thm}
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.angle.polynomial}
\begin{enumerate}[(a)]
\item Let's compare and contrast the two notions of discrete volumes that we have encountered so far.
For a given rational polytope ${\mathcal P}$, we notice that the Ehrhart quasi-polynomial $L_{\mathcal P}(t)$ is invariant when
we map ${\mathcal P}$ to any of its unimodular images. That is, any rational polytope in the whole orbit of the unimodular group
$\rm{SL}_d(\mathbb{Z})({\mathcal P})$ has the same discrete volume $L_{\mathcal P}(t)$. This is false for the second discrete volume $A_{\mathcal P}(t)$ - it is not invariant under the modular group
(Exercise \ref{counterexamples for angle polynomials}). But $A_{\mathcal P}(t)$ is invariant under the
large finite group of the isometries of $\mathbb{R}^d$ that preserve the integer lattice (known as the hyperoctahedral group).
So we see that $A_{\mathcal P}(t)$ is more sensitive to the particular embedding of ${\mathcal P}$ in space, because it is dependent upon a metric. It is reasonable to expect that it can distinguish between ``more'' rational polytopes, but such a question remains to be formalized.
The angle polynomial also has the advantage of being a much more symmetric polynomial, with half as many coefficients that occur
in the Ehrhart polynomial of integer polytopes.
However, $L_{\mathcal P}(t)$ has its advantages as well - to compute a \emph{local summand} for $A_{\mathcal P}(t):= \sum_{n\in\mathbb{Z}^d} \omega_{tP}(x) $ requires finding the volume of a local spherical polytope, while to compute a \emph{local summand} for $L_{\mathcal P}(t):= \sum_{n\in\mathbb{Z}^d} 1$ is quite easy: it is equal to $1$.
But as we have seen, computing the full global sum for $A_{\mathcal P}(t)$ turns out to have its simplifications.
\item The interesting undergraduate dissertation of Nhat Le Quang \cite{Nhat}, from $2010$, gives a thorough analysis of solid angle sums in $\mathbb{R}^2$, for rational polygons.
\item The recent work of Gerv\'asio \cite{GervasioSantos} gives an online implementation for the calculation of solid angles in any dimension, with open source code.
\item In \cite{RicardoNhatSinai}, there is an explicit description for some of the coefficients of the solid angle polynomial
$A_{\mathcal P}(t)$ of a $d$-dimensional polytope, for all positive real dilations $t>0$. Indeed, the approach in \cite{RicardoNhatSinai} uses the Fourier analytic landscape.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\medskip
\begin{prob}
Let $
{\mathcal K} = \{ \lambda_1
\left( \begin{smallmatrix}
1 \\ 0 \\ 0
\end{smallmatrix}
\right) +
\lambda_2
\left( \begin{smallmatrix}
1 \\ 1 \\ 0
\end{smallmatrix}
\right)
+
\lambda_3
\left( \begin{smallmatrix}
1 \\ 1 \\ 1
\end{smallmatrix}
\right) \mid \lambda_1, \lambda_2, \lambda_3 \geq 0 \},
$
a simplicial cone. Show that the solid angle of ${\mathcal K}$ is $\omega_{\mathcal K} = \frac{1}{48}$.
\end{prob}
\medskip
\begin{prob}
We recall the $2$-dimensional cross-polytope
$
\Diamond:=\left\{ \left( x_1, x_2 \right) \in \mathbb{R}^2 \mid
\, \left| x_1 \right| + \left| x_2 \right| \leq 1 \right\}.
$
Find, from first principles, the angle quasi-polynomial for the rational polygon
${\mathcal P}:= \frac{1}{3}\Diamond$,
for all integer dilations of ${\mathcal P}$.
\end{prob}
\medskip
\begin{prob}
We recall that the $3$-dimensional cross-polytope was defined by
\[
\Diamond:=\left\{ \left( x_1, x_2, x_3 \right) \in \mathbb{R}^3 \mid
\, \left| x_1 \right| + \left| x_2 \right| + \left| x_3 \right| \leq 1 \right\}.
\]
Compute the angle polynomial of $A_{\Diamond}(t)$.
\end{prob}
\medskip
\begin{prob}
We recall that the $d$-dimensional cross-polytope \index{cross-polytope}
was defined by
\[
\Diamond:=\left\{ \left( x_1, x_2, \dots, x_d \right) \in \mathbb{R}^d \mid
\, \left| x_1 \right| + \left| x_2 \right| + \cdots + \left| x_d \right| \leq 1 \right\}.
\]
Compute the angle polynomial of $A_{\Diamond}(t)$.
\end{prob}
\medskip
\begin{prob}
Let ${\mathcal P}$ be an integer zonotope. Prove that the angle polynomial of ${\mathcal P}$ is
\[
A_{{\mathcal P}}(t) = (\vol {\mathcal P})t^d,
\]
valid for all positive integers $t$.
\end{prob}
\medskip
\begin{prob}
Let ${\mathcal P}$ be a rational interval $[\frac{a}{c}, \frac{b}{d}]$. Compute the angle quasi-polynomial
$A_{{\mathcal P}}(t)$ here.
\end{prob}
\medskip
\begin{prob}
Define the rational triangle $\Delta$ whose vertices are $(0, 0), (1, \frac{N-1}{N}), (N, 0)$, where $N \geq 2$ is a fixed integer.
Find the angle quasi-polynomial $A_\Delta(t)$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{counterexamples for angle polynomials}
For each dimension $d$, find an example of an integer polytope ${\mathcal P} \subset \mathbb{R}^d$ and a unimodular matrix
$U \in \rm{SL}_d(\mathbb{Z})$, such that the angle quasi-polynomials
$A_{{\mathcal P}}(t)$ and $A_{U({\mathcal P})}(t)$ are not equal to each other for all $t \in \mathbb{Z}_{>0}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{solid angle of a face of the cube}
For the cube $\square:= [0, 1]^d$, show that any face $F\subset \square$ that has dimension $k$ has the solid angle
$\omega_F = \frac{1}{2^{d-k}}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{dihedral angle=solid angle}
Show that the solid angle $\omega_E$ of an edge E ($1$-dimensional face) of a polytope
equals the dihedral angle between the hyperplanes defined by
its two bounding facets. (Hint: use the unit normal vectors for both facets)
\end{prob}
\medskip
\begin{prob}
Using the Gram relations, namely Theorem \ref{Gram relations}, compute the solid angle at any vertex of the following regular tetrahedron:
\[
T:= \conv\Big\{ \icol{1\{\bf 0}\{\bf 0}} \icol{0\{\bf 1}\{\bf 0}}, \icol{0\{\bf 0}\{\bf 1}}, \icol{1\{\bf 1}\{\bf 1}} \Big\}.
\]
\end{prob}
\chapter{Sphere packings} \label{Sphere packings}
\index{sphere packings}
\begin{quote}
The problem of packing, as densely as possible, an unlimited number of equal nonoverlapping circles in a plane
was solved millions of years ago by the bees, who found that the best arrangement consists of circles inscribed
in the hexagons of the regular tessellation. \ -- \ H. S. M. Coxeter \index{Coxeter}
\end{quote}
\begin{quote}
There is geometry in the humming of the strings. There is music in the spacing of the spheres.
\ -- \ Pythagoras \index{Pythagoras}
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.6in]{HexagonalPacking}
\end{center}
\caption{A lattice sphere packing, using the hexagonal lattice, which gives the densest packing in 2 dimensions.} \label{periodic packing, Eisenstein}
\end{figure}
\section{Intuition}
The sphere packing problem traces its roots back to Kepler, and it asks for a packing
of solid spheres in Euclidean space that achieves the maximum possible density. In all of the known cases,
such optimal configurations - for the centers of the spheres - form a lattice. It's natural, therefore, that Fourier analysis comes into the picture. We prove here a result of Cohn and Elkies, from $2003$, which is a beautiful application of Poisson summation, and gives upper bounds for the maximum densities of sphere packings
in $\mathbb{R}^d$.
At this point it may be wise to define carefully all of the terms - what is a packing? what is density?
Who was Kepler?
\begin{wrapfigure}{R}{0.4\textwidth}
\centering
\includegraphics[width=0.27\textwidth]{JohannesKepler}
\caption{Johannes Kepler} \label{Kepler.pic}
\end{wrapfigure}
\section{Definitions}
A {\bf sphere packing} in $\mathbb{R}^d$ is any arrangement
of spheres of fixed radius $r>0$ such that no two interiors overlap, so we do not preclude the possibility that the
spheres may touch one another at some points on their boundary.
A {\bf lattice packing} is a sphere packing with the property that the centers of the spheres form a lattice
${\mathcal L} \subset \mathbb{R}^d$, as in Figure \ref{Sphere Packing 1}.
Relaxing this restriction - in order to allow more general packings - we define
a {\bf periodic packing} by a sphere packing with a lattice ${\mathcal L}$, together with a finite collection of its translates, say ${\mathcal L} + v_1, \dots, {\mathcal L} + v_N$, such that the differences $v_i-v_j \notin L$. This means that the centers of the spheres may be placed at any points belonging to the disjoint
union of ${\mathcal L}$, together with its $N$ translates, as in Figure \ref{periodic packing}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.5in]{SpherePacking1}
\end{center}
\caption{A lattice packing, with small packing density.} \label{Sphere Packing 1}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{PeriodicPacking}
\end{center}
\caption{A periodic packing with two translates of the same lattice. This packing is not a lattice packing.} \label{periodic packing}
\end{figure}
The density of any sphere packing is intuitively the proportion
of Euclidean space covered by the spheres, in an asymptotic sense, but rather than go into these technical
asymptotic details,
we will simply define a density function for lattice packings and for general periodic packings, as follows.
Given a lattice packing, with the lattice ${\mathcal L} \subset \mathbb{R}^d$, and with spheres of radius $r$,
we define its {\bf lattice packing density} by
\begin{equation}
\Delta({\mathcal L}) := \frac{ \vol B^d(r) }{\det {\mathcal L}},
\end{equation}
where $B^d(r) $ is a ball of radius $r$. This lattice packing corresponds to placing a sphere of
radius $r$ at each lattice point of ${\mathcal L}$, guaranteeing that the spheres do not overlap.
\begin{wrapfigure}{R}{0.36\textwidth}
\centering
\includegraphics[width=0.22\textwidth]{IntegerLatticePacking}
\caption{The densest sphere packing for the lattice $\mathbb{Z}^2$, with a packing density of
$\frac{\pi}{4} \approx .785$, which means that about $78.5 \%$ of the plane is covered by this configuration of balls.}
\label{densest integer lattice packing}
\end{wrapfigure}
\begin{example}
\rm{
Consider the integer lattice ${\mathcal L}:= \mathbb{Z}^2$. It is clear that we can place a sphere of radius $r = \frac{1}{2}$ at each integer point, so that we have packing, and it is also clear that any larger radius for our spheres will not work with this lattice (see Figure \ref{densest integer lattice packing}). So this particular packing gives us a sphere packing density of
\[
\frac{ \vol B^2(r) }{\det {\mathcal L}}:= \frac{ \frac{\pi}{4} }{ \det \mathbb{Z}^2 } = \frac{\pi}{4}.
\]
}
\hfill $\square$
\end{example}
More generally, given a period packing with a lattice ${\mathcal L}$ and a set of translates $v_1, \dots, v_N$,
we define its {\bf periodic packing density} by
\begin{equation} \label{periodic packing density}
\Delta_{periodic}({\mathcal L}) := \frac{ N \vol B^d(r) }{\det {\mathcal L}},
\end{equation}
corresponding to placing a sphere of radius $r$ at each point of ${\mathcal L}$, and also at each point
of its translates ${\mathcal L} + v_1, \dots, {\mathcal L} + v_N$. It's not hard to prove that the latter definition
\ref{periodic packing density} matches our intuition that any fixed fundamental parallelepiped of ${\mathcal L}$
intersects this configuration of spheres in a set whose measure is exactly
$N \vol B^d(r) $ (Exercise \ref{volume of periodic packing}).
Henceforth, we use the words `packing density' to mean `periodic packing density', and we always restrict attention to periodic packings - see the Notes for technical remarks involving any sphere arrangement, and why periodic packings are sufficient.
We define the {\bf sphere packing problem} as follows:
\begin{question}
What is the maximum possible packing density, in any periodic packing of spheres?
\end{question}
In other words, the problem asks us to find the maximum density
$\Delta_{periodic}{{\mathcal L}}$, among all lattices ${\mathcal L}$, allowing also any finite collection of translates of ${\mathcal L}$. The sphere packing
problem also asks us to find, if possible, the lattice ${\mathcal L}$ that achieves this optimal density.
Many other questions naturally arise:
\begin{question} \label{lattice or a few lattices?}
Is the densest sphere packing always achieved by using just one lattice, in each dimension $d$?
\end{question}
In other words, are there dimensions $d$ for which we in fact need to use some translates of a lattice?
\begin{question} \label{unique lattice?}
If the answer to Question \ref{lattice or a few lattices?} is affirmative, then is such an optimal lattice unique in each dimension?
\end{question}
The only dimensions $d$ for which we know the answers to Question \ref{lattice or a few lattices?} and
Question \ref{unique lattice?} are $d=1, 2, 3, 8, 24$, and in these known cases the answer is affirmative.
The sphere packing problem is a very important
problem in Geometry, Number theory, Coding theory, and information theory.
\bigskip
\section{The volume of the ball, and of the sphere}
\label{Volume of the ball, the Gamma function}
To warm up, we compute volumes of $d$-dimensional balls and spheres. For these very classical computations, we need the Gamma function:
\begin{equation}
\Gamma(x):= \int_0^\infty e^{-t} t^{x-1} dt,
\end{equation}
valid for all $x>0$. The Gamma function $\Gamma(x)$ interpolates smoothly between the integer values of the factorial function $n!$, in the following sense.
\begin{lem} \label{Gamma properties}
Fix $x>0$. Then
\begin{enumerate}[(a)]
\item $\Gamma(x+1) = x \Gamma(x)$.
\item $\Gamma(n+1) = n!$, for all nonnegative integers $n$.
\item $\Gamma\Big( \frac{1}{2} \Big) = \sqrt \pi$.
\item $\Gamma$ extends to an infinitely smooth function on the complex plane, except at $0$ and at the
negative integers, where it has simple poles.
\end{enumerate}
\end{lem}
The verifications of parts (a), (b), and (c) are good exercises (Exercise \ref{prove Gamma properties}), and we don't want to deprive the reader of that pleasure. Part (d) requires some knowledge of complex analysis, but we include the statement here for general knowledge.
What is the volume of the unit ball $B:= \left\{ x \in \mathbb{R}^d \mid \|x\| \leq 1 \right\}$? And what about the volume of the unit sphere $S^{d-1} := \left\{ x\in \mathbb{R}^d \mid \| x \| \right\}= 1 \|$?
\begin{lem} \label{lem:volume of ball and sphere}
For the unit ball $B$, and unit sphere $S^{d-1}$, we have:
\begin{equation} \label{volume of ball and sphere}
\vol B = \frac{ \pi^{\frac{d}{2}} }{\Gamma\left(\frac{d}{2} +1\right)}, \text{ and }
\vol \left(S^{d-1}\right) = \frac{2 \pi^{\frac{d}{2}} }{ \Gamma\left(\frac{d}{2}\right)}.
\end{equation}
\end{lem}
\begin{proof}
We let $\kappa_{d-1}:= \vol(S^{d-1})$ denote the surface area of the unit sphere
$S^{d-1} \subset \mathbb{R}^d$. We use polar coordinates in $\mathbb{R}^d$, meaning that we may write each $x \in \mathbb{R}^d$ in the form $x = (r, \theta)$, where $r>0$ and $\theta \in S^{d-1}$. Thus $\|x\| = r$, and we also have the calculus fact that $dx = r^{d-1} dr d\theta$.
Returning to our Gaussians $e^{-\pi \|x\|^2}$, we recompute their integrals using polar coordinates in $\mathbb{R}^d$:
\begin{align*}
1=\int_{\mathbb{R}^d} e^{-\pi \|x\|^2} dx &= \int_{S^{d-1} }
\int_0^{\infty} e^{-\pi r^2} r^{d-1} dr \, d\theta \\
&=\kappa_{d-1} \int_0^{\infty} e^{-\pi r^2} r^{d-1} dr \\
&= \kappa_{d-1} \frac{1}{ 2\pi^{\frac{d}{2}} }
\int_0^{\infty} e^{- t} t^{\frac{d}{2} -1} dt \\
\end{align*}
where we've used $t:= \pi r^2$, implying that
$r^{d-1} dr = r^{d-2} r dr =
\Big(\frac{t}{\pi} \Big)^{\frac{d-2}{2}} \frac{dt}{2\pi} $. Recognizing the latter integral as
$\Gamma\left(\frac{d}{2}\right)$, we find that
$1= \frac{ \kappa_{d-1} } { 2\pi^{\frac{d}{2}} } \Gamma\left(\frac{d}{2}\right)$, as desired.
For the volume of the unit ball $B$, we have:
\[
\vol B = \int_0^1 \kappa_{d-1} r^{d-1} dr = \frac{\kappa_{d-1}}{d}
= \frac{\pi^{\frac{d}{2}}}{\frac{d}{2} \Gamma\left(\frac{d}{2}\right) }
= \frac{\pi^{\frac{d}{2}}}{ \Gamma\left(\frac{d}{2} +1\right) }.
\]
\end{proof}
It is easy, but worth mentioning (Exercise \ref{explicit volume for ball and sphere}), that we may also
rewrite the formulas \eqref{volume of ball and sphere} in terms of ratios of factorials, by using the recursive properties of the $\Gamma$ function.
While we are at it, let's dilate the unit ball by $r>0$, and recall our definition of the ball of radius $r$:
\[
B^d(r):= \left\{ x \in \mathbb{R}^d \mid \|x\| \leq r \right\}.
\]
We know that for any $d$-dimensional body $K$, we have $\vol(rK) = r^d \vol K$, so we also get the volumes of the ball of radius $r$, and the sphere of radius $r$:
\begin{equation} \label{dilated volumes of balls and spheres}
\vol B^d(r) = \frac{ \pi^{\frac{d}{2}} }{\Gamma\left(\frac{d}{2} +1\right)} r^d , \text{ and }
\vol \left(r S^{d-1}\right) = \frac{2 \pi^{\frac{d}{2}} }{ \Gamma\left(\frac{d}{2}\right)} r^{d-1}.
\end{equation}
Intuitively, the derivative of the volume is the surface area, and now we can confirm this intuition:
\[
\frac{d}{dr} \vol B^d(r) = \frac{ d \pi^{\frac{d}{2}} }{\Gamma\left(\frac{d}{2} +1\right)} r^{d-1}
=\frac{ 2\frac{d}{2} \pi^{\frac{d}{2}} }{\frac{d}{2} \Gamma\left(\frac{d}{2} \right)} r^{d-1}
=\frac{ 2 \pi^{\frac{d}{2}} }{\Gamma\left(\frac{d}{2}\right)} r^{d-1} =\vol \left(r S^{d-1}\right).
\]
\bigskip
\section{The Fourier transform of the ball}
Whenever considering packing or tiling by a convex body $B$, we have repeatedly seen that taking the Fourier transform of the body, namely
$\hat 1_B$, is very natural, especially from the perspective of Poisson summation.
It's also very natural to consider the FT of a ball in $\mathbb{R}^d$.
To compute the Fourier transform of $1_{B(r)}$, a very classical computation, we first define
the \textbf{Bessel function} \index{Bessel function}
$J_p$ of order $p$ (\cite{EpsteinBook}, page 147), which comes up naturally here:
\begin{equation}\label{Bessel definition}
J_p(x) :=\left(\frac{x}{2} \right)^p \frac{1}{\Gamma\left(p + \frac12\right)\sqrt{\pi}}\int_0^\pi e^{ix\cos\varphi} \sin^{2p}(\varphi) \, d\varphi,
\end{equation}
valid for $p>-\frac{1}{2}$, and all $x \in \mathbb{R}$. We call a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ {\bf radial} \index{radial function}
if it is invariant under all rotations of $\mathbb{R}^d$. In other words, we have the definition
\[
f \text{ is radial } \iff f \circ M = f,
\]
for all $M \in SO_d(\mathbb{R})$, the orthogonal group. Another way of describing a radial function is to say that the function $f$ is constant on each sphere that is centered at the origin, so that a radial function only depends on the norm of its input: $f(x) = f(\|x\|)$, for all $x\in \mathbb{R}^d$.
A very useful fact in various applications of
Fourier analysis (in particular medical imaging) is that the Fourier transform of a radial function is again a radial function (Exercise \ref{radial function transform}).
\bigskip
\begin{lem} \label{FT of the ball}
The Fourier transform of $B^d(r)$, the ball of radius $r$ in $\mathbb{R}^d$ centered at the origin, is
\[\hat{1}_{B^d(r)}(\xi) := \int_{B^d(r)} e^{-2\pi i \langle \xi, x \rangle } dx = \left(\frac{r}{\| \xi\|}\right)^{d/2}J_{d/2}\big(2\pi r\|\xi\|\big).\]
\end{lem}
\begin{proof}
Taking advantage of the inherent rotational symmetry of the ball, and also using the fact that the Fourier transform
of a radial function is again radial (Exercise \ref{radial function transform}), we have:
\[
\hat{1}_{B^d(r)}(\xi) = \hat{1}_{B^d(r)}(0, \dots, 0, \|\xi\|),
\]
for all $\xi \in \mathbb{R}^d$. With $r=1$ for the moment, we therefore have:
\[
\hat{1}_{B}(\xi) =
\int_{\| x\| \leq 1} e^{-2\pi i x_d\|\xi\|}\, dx_1\, \dotsc\, dx_d,
\]
Now we note that for each fixed $x_d$, the function being integrated is constant and the integration domain for the
variables $x_1, \dots, x_{d-1}$ is a $(d-1)$-dimensional ball of radius $(1-x_d^2)^{1/2}$.
By equation \eqref{volume of ball and sphere}, the
volume of this ball is $(1-x_d^2)^{\frac{d-1}{2}} \frac{ \pi^{\frac{d-1}{2}} }{ \Gamma\left(\frac{d+1}{2} \right) }$, we have
\[
\hat{1}_{B}(\xi)
= \frac{ \pi^{\frac{d-1}{2}} }{ \Gamma(\frac{d+1}{2}) }
\int_{-1}^1
e^{-2\pi i x_d\|\xi\|}
(1-x_d^2)^{\frac{d-1}{2}} \,dx_d
= \frac{ \pi^{ \frac{d}{2} }}{\sqrt{\pi}\Gamma\left(\frac{d+1}{2}\right)}
\int_0^\pi
\, e^{2\pi i\|\xi\|\cos\varphi}
\sin^d\varphi \,d\varphi.
\]
Using the definition~\eqref{Bessel definition} of the $J$-Bessel function, we get
\[\hat{1}_{B}(\xi) = \| \xi\|^{-\frac{d}{2} }J_{\frac{d}{2} }\big(2\pi\|\xi\|\big),\]
and consequently
\[\hat{1}_{B^d(r)}(\xi) = \left(\frac{r}{\| \xi\|}\right)^{\frac{d}{2} }J_{\frac{d}{2} }\big(2\pi r\|\xi\|\big).
\qedhere\]
\end{proof}
\bigskip
\begin{example}\label{ex:integral using Bessel functions}
\rm{
Using the $J$-Bessel functions, let's work out the following explicit evaluation of the following interesting
integrals, for all $p>0$:
\begin{equation} \label{identity of J-Bessel example}
\int_0^\pi \sin^{2p}(\varphi) \, d\varphi =
\sqrt \pi \frac{ \Gamma\left(p + \frac12\right) }{ \Gamma\left(p + 1 \right)}.
\end{equation}
Whenever we raise a negative real number to an arbitrary real exponent, some care has to be taken to avoid `branch problems' with the definition of exponentiation. Here we can easily avoid such problems by defining $\sin^{2p}(x) := \left( \sin^2(x)\right)^p$, so that we are always exponentiating a nonnegative real number, and everything is copacetic. We will use the following equivalent formulation for the $J_p$ Bessel function in terms of a hypergeometric series:
\begin{equation} \label{Bessel infinite series}
J_p(x) = \frac{x^p}{2^p} \sum_{k=0}^\infty (-1)^k \frac{x^{2k} }{2^{2k} k! \, \Gamma(p+k+1) }.
\end{equation}
(\cite{EpsteinBook}, p. 684).
Using the definition of the Bessel function \eqref{Bessel definition}, we can rewrite it slightly:
\begin{equation}\label{fancy integral 1}
\frac{J_p(x)}{x^p} 2^p \sqrt{\pi} \, \Gamma\left(p + \frac12\right)
=
\int_0^\pi e^{ix\cos\varphi} \sin^{2p}(\varphi) \, d\varphi.
\end{equation}
Taking the limit as $x\rightarrow 0$, we can safely move this limit inside the integral in \eqref{fancy integral 1}
because we are integrating a differentiable function over a compact interval:
\begin{align*}
\int_0^\pi \sin^{2p}(\varphi) \, d\varphi
&= \lim_{x\rightarrow 0}
\frac{J_p(x)}{x^p} 2^p \sqrt{\pi} \, \Gamma\left(p + \frac12\right).
\end{align*}
So if we knew the asymptotic limit $ \lim_{x\rightarrow 0} \frac{J_p(x)}{x^p}$, we'd be in business.
From \eqref{Bessel infinite series}, we may divide both sides by $x^p$, and then take the limit as $x\rightarrow 0$ to obtain the constant term of the remaining series, giving us
\[
\lim_{x\rightarrow 0} \frac{J_p(x)}{x^p} = \frac{1}{2^p \Gamma(p+1)}.
\]
Altogether, we have
\begin{align*}
\int_0^\pi \sin^{2p}(\varphi) \, d\varphi
&= \lim_{x\rightarrow 0}
\frac{J_p(x)}{x^p} 2^p \sqrt{\pi} \, \Gamma\left(p + \frac12\right) \\
&= \frac{1}{2^p \Gamma(p+1)} 2^p \sqrt{\pi} \, \Gamma\left(p + \frac12\right) \\
&= \sqrt \pi \frac{ \Gamma\left(p + \frac12\right) }{ \Gamma\left(p + 1 \right)},
\end{align*}
valid for all $p>0$.
}
In the special case that $p$ is a positive integer, the latter identity can of course be written in terms of a ratio of factorials (Exercise \ref{the integral of the example in terms of factorials}).
\hfill $\square$
\end{example}
\bigskip
\bigskip
\section{Upper bounds for sphere packings via Poisson summation}
Here we give an exposition of the ground-breaking result of Henry Cohn and Noam Elkies on the sphere packing problem.
This result sets up the machinery for finding certain {\bf magical functions} $f$, as defined
in Theorem \ref{Cohn-Elkies} below,
that allow us to give precise upper bounds on
$\Delta_{periodic}{{\mathcal L}}$. The main tool is Poisson summation again, for arbitrary lattices. We recall that we defined a function $f$ to be \emph{nice} if $f$ satisfies the Poisson summation formula
\[
\sum_{n \in {\mathcal L}} f(n+v) = \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) e^{2\pi i \langle v, \xi \rangle},
\]
pointwise for all $v\in \mathbb{R}^d$.
\begin{thm}[Cohn-Elkies] \label{Cohn-Elkies}
Let $f:\mathbb{R}^d \rightarrow \mathbb{R}$ be a nice function, not identically zero, which enjoys the following
three conditions:
\begin{enumerate}
\item $f(x) \leq 0$, for all $\|x\| \geq r$. \label{condition 1}
\item $\hat f(\xi) \geq 0$, for all $\xi \in \mathbb{R}^d$. \label{condition 2}
\item $f(0) >0$, and $\hat f(0) > 0$. \label{condition 3}
\end{enumerate}
Then the periodic packing density of any $d$-dimensional sphere packing has the upper bound
\[
\Delta_{periodic}({\mathcal L}) \leq \frac{ f(0) }{ \hat f(0) } \vol B^d(r) .
\]
\end{thm}
\begin{proof}
Suppose we have a periodic packing with spheres of radius $r$, a lattice ${\mathcal L}$, and translation vectors
$v_1, \dots, v_N$, so that by definition the packing density is
$\Delta_{periodic}({\mathcal L}) := \frac{ N \vol B^d(r) }{\det {\mathcal L}}$.
By Poisson summation, \index{Poisson summation formula}
we have
\begin{equation}
\sum_{n \in {\mathcal L}} f(n+v) = \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) e^{2\pi i \langle v, \xi \rangle},
\end{equation}
converging absolutely for all $v \in \mathbb{R}^d$. Now we form the following finite sum and rearrange
the right-hand-side of Poisson summation:
\begin{align}\label{fancy Poisson}
\sum_{1\leq i \leq j \leq N}
\sum_{n \in {\mathcal L}} f(n+v_i - v_j) &= \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) \sum_{1\leq i \leq j \leq N} e^{2\pi i \langle v_i - v_j, \xi \rangle} \\
&= \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) \Big| \sum_{1\leq k \leq N} e^{2\pi i \langle v_k, \xi \rangle} \Big|^2.
\label{RHS of Poisson}
\end{align}
Now, every summand on the right-hand-side of \eqref{RHS of Poisson} is nonnegative, because by the second
assumption of the Theorem, we have
$\hat f (\xi) \geq 0$, so that the whole series can be bounded from below by its constant term, which for $\xi = 0$
gives us the bound $ \frac{ \hat f(0) N^2}{\det {\mathcal L}}$.
On the other hand, let's ask what the positive contributions are, from the left-hand-side of \eqref{fancy Poisson}.
Considering the vectors $n+v_i - v_j$ on the left-hand-side of \eqref{fancy Poisson},
suppose we have $\| n+v_i - v_j \| \geq r$. Then the first hypothesis of the Theorem guarantees
that $f(n+v_i - v_j ) \leq 0 $. So we may restrict attention to those vectors that satisfy $\| n+v_i - v_j \| < r$. Here
the vector $n+v_i - v_j$ is contained in the sphere of radius $r$, centered at the origin, but this means (by the packing assumption) that it must be the zero vector: $n + v_i - v_j = 0$. By assumption, the difference between any two translations $v_i-v_j$ is never a nonzero element of ${\mathcal L}$, so we have $i=j$, and now $v_i = v_j \implies n=0$. We conclude that the only positive contribution from the left-hand-side of \eqref{fancy Poisson} is the $n=0$ term, and so the
left-hand-side of \eqref{fancy Poisson} has an upper bound of $N f(0) >0$.
Altogether, Poisson summation gave us the bound:
\[
N f(0)
\geq
| \sum_{1\leq i \leq j \leq N} \sum_{n \in {\mathcal L}} f(n+v_i - v_j) |
=
\frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) \Big| \sum_{1\leq k \leq N} e^{2\pi i \langle v_k, \xi \rangle} \Big|^2
\geq
\frac{ \hat f(0) N^2}{\det {\mathcal L}}.
\]
Simplifying, we have
\[
\frac{ f(0) }{ \hat f(0) } \geq \frac{N}{\det {\mathcal L}} :=
\frac{ \Delta_{periodic}({\mathcal L})}{ \vol B^d(r) }.
\]
\end{proof}
\bigskip
\begin{example} [The trivial bound]
\rm{
Let ${\mathcal L}$ be a full-rank lattice in $\mathbb{R}^d$, whose shortest nonzero vector has length $r>0$.
We define the function
\[
f(x):= 1_{K}(x) * 1_{K}(x),
\]
where $K$ is the ball
of radius $r$, centered at the origin.
We claim that $f$ satisfies all of the conditions of
Theorem \ref{Cohn-Elkies}. Indeed, by the convolution Theorem,
\[
\hat f(\xi) = \widehat{\left(1_{K} * 1_{K}\right)}(\xi)
= \Big( \hat 1_{K}(\xi) \Big)^2 \geq 0,
\]
for all $\xi \in \mathbb{R}^d$, verifying condition \ref{condition 2}. Condition \ref{condition 1} is also easy to verify, because the support of $f$ is equal to the Minkowski sum (by Exercise \ref{support of convolution})
\index{Minkowski sum}
$K + K = 2K$, a sphere of radius $2r$.
It follows that $f$ is identically zero outside a sphere of radius $2r$. For condition \ref{condition 3}, by the definition of convolution we have $f(0) =
\int_{\mathbb{R}^d} 1_{K}(0-x) 1_{K}(x)dx = \int_{\mathbb{R}^d} 1_{K}(x)dx =
\vol K >0$. Finally, $\hat f(0) = \Big( \hat 1_{K}(0) \Big)^2 =
\vol^2( K) > 0$.
By the Cohn-Elkies Theorem \ref{Cohn-Elkies}, we know that the packing density of such a lattice is therefore bounded above by
\[
\frac{ f(0) }{ \hat f(0)} \vol B^d(r)
= \frac{\vol K }{ \vol^2( K)} \vol K = 1,
\]
the trivial bound. So we don't get anything interesting, but all this tells us is that our particular choice of function $f$ above was a poor choice, as far as density bounds are concerned. We need to be more clever in picking our magical $f$.
}
\hfill $\square$
\end{example}
Although it is far from trivial to find magical functions $f$ that satisfy the hypothesis of the
Cohn-Elkies Theorem, and simultaneously give a strong upper bound,
there has been huge success recently in finding exactly such functions - in dimensions $8$ and $24$. These recent magical functions gave the densest sphere packings in these dimensions, knocking off the whole sphere packing problem in dimensions $8$ and $24$.
It also turns out that if we have a magical function $f$ that enjoys all three
hypotheses of the Cohn-Elkies
Theorem \ref{Cohn-Elkies}, then $f\circ \sigma$ also satisfies the same hypotheses,
for any $\sigma \in SO_d(\mathbb{R})$ (Exercise \ref{invariance of magical functions under orthogonal transformations}).
We may therefore take certain radial functions as candidates for magical functions.
This exciting story continues today, and we mention some of the recent spectacular applications of the
Cohn-Elkies Theorem, initiated recently by
Maryna Viazovska for $\mathbb{R}^8$, and then extended by a large joint effort from
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska,
for $\mathbb{R}^{24}$ \cite{Cohn.etal}.
Here is a synopsis of some of their results.
\begin{thm}
The lattice $E_8$ is the densest periodic packing in $\mathbb{R}^8$. The Leech lattice is the densest
periodic packing in $\mathbb{R}^{24}$. In addition, these lattices are unique,
in the sense that there do not exist any
other periodic packings that achieve the same density.
\end{thm}
At the moment, the provably densest packings are known only in dimensions
$1, 2, 3, 8$, and $24$. Each dimension seems to require slightly different methods, and sometimes wildly different methods,
such as $\mathbb{R}^3$.
For $\mathbb{R}^3$, the sphere packing problem was solved by Hales,
and before Hales' proof, it was an open problem since the time of Kepler.
Somewhat surprisingly, the sphere packing problem is still open in all other dimensions.
In $\mathbb{R}^4$, it is very tempting and natural to think of the lattice $D_4$
as a possible candidate for the densest lattice sphere packing in $\mathbb{R}^4$, but this is still unknown.
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.SpherePackings}
\begin{enumerate}[(a)]
\item Each dimension $d$ appears to have a separate theory for sphere packings. This intuition is sometimes tricky to conceptualize, but there are facts that help us do so. For example, it is a fact that the Gram matrix (see \ref{Gram matrix positive semidefinite}) of a lattice ${\mathcal L} \subset \mathbb{R}^d$ consists entirely of integers, with even diagonal elements $\iff \ d$ is divisible by $8$.
For this reason, it turns out that the theta series of a lattice possesses certain functional equations (making it a modular form) if and only if $8 \mid d$, which in turn allows us to build some very nice related `magical' functions $f$ that are sought-after in Theorem \ref{Cohn-Elkies}, at least for $d=8$ and $d=24$ so far.
In dimension $2$, it is an open problem to find such magical functions, even though we have an independent proof that the hexagonal lattice is the optimal sphere packing lattice.
\item Johannes Kepler (1571 --1630) \index{Kepler, Johannes}
was a German astronomer and mathematician.
Kepler's laws of planetary motion motivated Sir Isaac Newton to develop further the theory of gravitational attraction and planetary motion. Kepler conjectured that the densest packing of sphere is given by the
``face-centered cubic'' packing. It was Gauss (1831) \index{Gauss}
who first proved that, if we assume the packing to be a lattice packing, then Kepler's conjecture is true.
In $1998$ Thomas Hales (using an approach initiated by L. Fejes T\'oth (1953)), gave an unconditional proof of the Kepler conjecture.
\item It is also possible, of course, to pack other convex bodies. One such variation is to pack regular tetrahedra in $\mathbb{R}^3$.
The interesting article by Jeffrey Lagarias and Chuanming Zong \cite{LagariasZong} gives a nice account of this story.
\item Regarding lower bounds for the optimal density of sphere packings, Keith Ball \cite{KeithBall.1} discovered the following lower bound in all dimensions:
\[
\Delta_{periodic}({\mathcal L}) \geq \frac{(n-1)}{2^{n-1}} \zeta(n),
\]
where $\zeta(s)$ is the Riemann zeta function.
Akshay Venkatesh \cite{Venkatesh} has given an improvement over the known lower bounds by a multiplicative constant. For all sufficiently large dimensions, this improvement is by a factor of at least $10, 000$.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
``It is better to do the right problem the wrong way, than the wrong problem the right way.''
-- Richard Hamming \index{Hamming, Richard}
\end{quote}
\medskip
\begin{prob} \label{explicit volume for ball and sphere}
Using Lemma \ref{lem:volume of ball and sphere},
show that for the unit ball $B$ and unit sphere $S^{d-1}$ in $\mathbb{R}^d$, we have:
\begin{enumerate}[(a)]
\item
\[
\vol S^{d-1} =
\begin{cases}
\frac{ \left(2 \pi\right)^{\frac{d}{2}} }{ 2\cdot 4 \cdot 6 \cdots (d-2) }, & \text{if } d \text{ is even}, \\
\frac{ 2 \left(2 \pi\right)^{\frac{d-1}{2}} }{ 1\cdot 3 \cdot 5 \cdots (d-2) }, & \text{if } d \text{ is odd}.
\end{cases}
\]
\item
\[
\vol B =
\begin{cases}
\frac{ \left(2 \pi\right)^{\frac{d}{2}} }{ 2\cdot 4 \cdot 6 \cdots d }, & \text{if } d \text{ is even}, \\
\frac{ 2 \left(2 \pi\right)^{\frac{d-1}{2}} }{ 1\cdot 3 \cdot 5 \cdots d }, & \text{if } d \text{ is odd}.
\end{cases}
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{volume of periodic packing}
Given a periodic lattice packing, by $N$ translates of a lattice ${\mathcal L} \subset \mathbb{R}^d$, show that any fixed fundamental
parallelepiped of ${\mathcal L}$
intersects the union of all the spheres in a set of measure $N \vol B^d(r) $, where $r:= \frac{1}{2}\lambda_1({\mathcal L})$.
Thus, we may compute the density of a periodic sphere packing by just considering the
portions of the spheres that lie in one fundamental parallelepiped.
\end{prob}
\medskip
\begin{prob} Here we show that the integer lattice is a very poor choice for sphere packing.
\begin{enumerate}[(a)]
\item Compute the packing density of the integer lattice $\mathbb{Z}^d$.
\item Compute the packing density of the lattices $D_3$ and $D_4$.
\item Compute the packing density of the lattices $D_n$, for $n\geq 5$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{radial function transform}
If $f\in L^1(\mathbb{R}^d)$ is a radial function, prove that its Fourier transform $\hat f$
is also a radial function.
\end{prob}
\medskip
\begin{prob}
Suppose we pack equilateral triangles in the plane, by using only translations of a fixed equilateral triangle.
What is the maximum packing density of such a packing? Do you think it may be the worst possible density among translational packings of any convex body in $\mathbb{R}^2$?
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{prove Gamma properties}
Prove the elementary properties of the $\Gamma$ function, in Lemma \ref{Gamma properties}.
\end{prob}
\medskip
\begin{prob} \label{invariance of magical functions under orthogonal transformations}
Show that if we have a magical function $f$ that enjoys all $3$ hypotheses of Theorem~\ref{Cohn-Elkies},
then $f\circ \sigma$ also satisfies the same hypotheses, for any orthogonal transformation $\sigma \in SO_d(\mathbb{R})$.
\end{prob}
\medskip
\begin{prob}
We define a rigid motion of a compact set $K$ to be any orthogonal transformation of $K$, composed with any translation of $K$.
\begin{enumerate}[(a)]
\item When $d=1$, find a (nontrivial) continuous function $f:\mathbb{R} \rightarrow \mathbb{C}$ such that:
\[
\int_c^{c+R} f(x) dx = 0,
\]
for all constants $c, R>0$.
\item \label{ex:part b, for Pompeiu}
More generally, in any dimension $d$, find a (nontrivial) continuous function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ that allows the following integrals (taken over any ball of radius $r$) to vanish:
\[
\int_{B^d(r)+c} f(x) dx = 0,
\]
for all constants $c, R>0$.
\end{enumerate}
Notes. For part \ref{ex:part b, for Pompeiu},
it's advisable to think about the Fourier transform of the ball.
It is conjectured that for any bounded set $K$ with nonempty interior, the balls in this example are the only examples of objects that allow such nonzero continuous functions $f$ to exist. This is known as the {\bf Pompeiu problem} - see also Question \ref{Pompeiu conjecture}.
\end{prob}
\begin{prob} \label{the integral of the example in terms of factorials}
Show that when $p$ is a positive integer, the identity
\eqref{identity of J-Bessel example} of Example \ref{ex:integral using Bessel functions} simplifies to a ratio of factorials.
\end{prob}
\medskip
\begin{prob}
\rm{ [hard-ish]
\label{positive FT over R^d}
Using the idea of Exercise \ref{positive FT over R} in Chapter \ref{Fourier analysis basics},
and using the sum of two
indicator functions of balls (with incommensurable radii) in $\mathbb{R}^d$,
show that there exists a compactly supported function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ such that
\[
\hat f(\xi) >0,
\]
for all $\xi \in \mathbb{R}^d$.
}
\end{prob}
\bigskip
\chapter{Shannon sampling, in one and several dimensions}
\label{Chapter:Shannon sampling}
\begin{quote}
``It is easy to argue that real signals must be band-limited. \\
\quad It is also easy to argue that they cannot be so.''
-- David Slepian \index{Slepian, David}
\end{quote}
Sampling theory consists in the reconstruction of a continuous function with only a discrete or finite amount of data and has many applications in signal processing and other engineering applications. At a first glance this task sounds impossible, however it can be done well in practice. One of the reasons for this success comes from the Fourier analysis, which deals with the representation of a function in terms of its ``frequencies", and functions without high frequencies (bandlimited) represents very well the real-world signals.
In one dimension, the classical example is a sound signal, and since typical humans can only hear sounds with frequencies smaller than $20$ kHz, the bandlimited assumption is appropriate.
Examples in higher dimensions include images or MRI exams where higher frequencies are associated with random noises and measurement errors, more connected to the physical apparatus than the object being measured \cite{EpsteinBook}. In this sense it is even desirable to remove the high
frequency information.
More recently, the interest in bandlimited functions increased in the machine learning community, because
it was observed that neural networks learns low frequencies faster and this might explain why they often generalize quickly from the training sets.
On the other hand, by a basic uncertainty principle of Theorem \ref{basic uncertainty principle}, we know that a function with compact support can never be bandlimited, so representing an arbitrary function using
this class of functions is in general not exact. It is therefore desirable to also give
some theoretical results concerning the error of such approximations.
Here we introduce the classical sampling theorem by Shannon and Whittaker
for one dimensional sampling, and then we study some of its generalizations to higher dimensions,
where much less is known.
An excellent introduction to Sampling Theory, from an expository as well as a rigorous perspective,
is the book of J. R. Higgins \cite{Higgins1996}.
\bigskip
\section{The Shannon-Whittaker sampling Theorem}
\label{sec:one-dimensional-stuff}
Claude Shannon~\cite{Shannon1} showed how to reconstruct a complete signal $f$ by sampling it only discretely,
in a classical paper that gave rise to the field of information theory.
To accomplish this, Shannon used an interesting assumption, namely that the Fourier transform of $f$
vanishes outside of some interval.
One of the main characters of this story is our old friend, the ${\rm{sinc}}$ function:
\begin{equation}\label{eq:sinc-def}
{\rm{sinc}}(x):= \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{2\pi i \xi x} d\xi = \begin{cases}
\frac{\sin(\pi x)}{\pi x}, &\mbox{if } x \not= 0 \\
1 & \mbox{if } x= 0,
\end{cases}
\end{equation}
which plays a central role in the sampling theory for functions in $\mathbb{R}$, because it turns out to be a building block for a basis of the Paley-Wiener space $PW_c$, as the Shannon-Whittaker sampling theorem shows.
Reviewing some of the Fourier facts that we learned in Chapter \ref{Fourier analysis basics},
we recall that if $f \in L^1(\mathbb{R})$, then $\hat f$ is uniformly continuous and $\hat f(\xi) \to 0$ as $|\xi| \to \infty$.
So not every function can be the Fourier transform of some other function in $L^1(\mathbb{R})$.
In practice, we are often interested in functions that are not absolutely integrable, and yet possess a (conditionally convergent) Fourier transform, such as the important ${\rm{sinc}}$ function. To resolve this issue, the theory progresses by first defining the transform in the space $L^1(\mathbb{R})$, and then extending the definition of $\hat f$ to all of $L^2(\mathbb{R})$, by taking the limit $\lim_{n \to \infty} \int_{|x| < n}f(x)e^{-2\pi i x \xi}dx$. This unique extension of the Fourier transform,
from the $L^1(\mathbb{R})$ space to the $L^2(\mathbb{R})$ space, is sometimes called the Plancherel-Fourier transform. From now on, we'll follow the usual Fourier convention and simply call both transforms ``the Fourier transform''.
For a given number $c>0$, a function $f\in L^2(\mathbb{R})$ is called {\bf c-bandlimited} if
\[
\hat{f}(x) = 0 \text{ for all } x \not\in [-c, c].
\]
We will sometimes just say `bandlimited' if the $c$ is not contextually important. A bandlimited function $f$ has a Fourier transform that decays at the `best possible rate', in the sense that its Fourier transform is identically zero outside the interval $[-c, c]$.
It's easy to notice that any $c$-bandlimited function $f$ must be equal (almost everywhere)
to an infinitely smooth
function, because by Fourier inversion, we have:
\begin{equation}\label{eq:bandlimit-inversion}
f(x) = \int_{\mathbb{R}} \hat f(\xi) e^{2\pi i \xi x} d\xi
=\int_{-c}^c \hat f(\xi) e^{2\pi i \xi x} d\xi.
\end{equation}
This identity implies that we can differentiate the last expression with respect to $x$ as many times as we like under the integral sign, because the integrand is a smooth function of $x$, and we are integrating over a compact domain. Therefore $f$ is infinitely smooth.
For simplicity, when considering a bandlimited function $f$,
we will always assume that $f$ is also continuous, which is
consistent with the equality in~\eqref{eq:bandlimit-inversion}.
Given any $c>0$, we define the space of all $c$-bandlimited functions in
$L^2(\mathbb{R})$ by
\[
PW_c:= \{ f \in L^2(\mathbb{R}) \mid \, \hat f(\xi) = 0 \text{ for } \xi \notin [-c, c], \text{ and } f
\text{ is continuous} \},
\]
called {\bf the Paley-Wiener space} \cite{Higgins1996}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.6in]{Shannon1}
\end{center}
\caption{We can visualize the isometries of the various spaces of functions, given by the Fourier transform.
First, the Fourier transform $\mathcal{F}$ gives us an isometry from $L^2(\mathbb{R})$ onto itself.
Second, restricting attention to the subspace $PW_c \subset L^2(\mathbb{R})$ of bandlimited functions, $\mathcal{F}$ also gives us an isometry from $PW_c$ to $L^2([-c, c])$, carrying the basis of translated ${\rm{sinc}}$ functions to the basis of exponentials. }
\label{Shannon1}
\end{figure}
\begin{thm}[Shannon-Whittaker] \label{Shannon}
Suppose that $f \in PW_c$. Then we have
\begin{equation}
\label{shannon_sampling}
f(x) = \sum_{n \in \mathbb{Z}} f\left( \frac{n}{2c} \right) {\rm sinc } \left( 2c x - n \right),
\end{equation}
and the series converges absolutely and uniformly over $\mathbb{R}$.
\hfill $\square$
\end{thm}
In other words, if we sample a $c$-bandlimited function $f$ at only the discrete set of points $\{ \frac{n}{2c} \mid n \in \mathbb{Z}\}$, we may reconstruct the whole function $f(x)$ for all $x\in \mathbb{R}$! In the next two sections, we give two different proofs of Theorem \ref{Shannon}.
The quantity $\frac{1}{2c}$ is called the {\bf sample spacing} and its reciprocal $2c$ is the {\bf sampling rate}.
So what the theorem says is that to reconstruct a function with bandlimit $c$, one has to sample at a rate~$2c$. Offhand, it seems rather incredible that some (non-periodic) functions $f:\mathbb{R} \rightarrow \mathbb{C}$ may be completely recovered by knowing only their values $f(n)$ at a discrete set of points.
This phenomenon shows, in a sense, how the Paley-Wiener space is a very special subspace of $L^2(\mathbb{R})$.
\bigskip
\section{The approach of G. H. Hardy}\label{sec:Hardy}
G. H. Hardy's proof \cite{Hardy41} of Theorem \ref{Shannon} is particularly interesting because it also answers the following informal question:
\begin{question}\rm{[Rhetorical]}
How large is the space of bandlimited functions?
\end{question}
Hardy's approach also clarifies some of the underlying structure of bandlimited functions. It relies on the following isometry.
\begin{lem}\label{isometry}
The Fourier transform ${\mathcal F}$ gives a bijection between the following two Hilbert spaces:
\begin{equation*}
{\mathcal F}: PW_{c} \rightarrow
L^2(\left[-c, c\right]).
\end{equation*}
Moreover, this bijection is an isometry.
\end{lem}
\begin{proof}
First, given any $f \in PW_{c} \subset L^2(\mathbb{R})$, we need to show that $\hat f \in L^2(\left[-c, c\right])$.
By definition $f$ is $c$-bandlimited, hence its Fourier transform can be naturally viewed as a function with domain $[-c,c]$.
We need to show that $\hat f$ has a finite $L^2(\left[-c, c\right])$ norm.
The following computation uses Parseval's identity, namely that
$\| f \|^2_{L^2(\mathbb{R})} = \|\hat f \|^2_{L^2(\mathbb{R})}$:
\begin{align}
{\| \hat f \|}^2_{L^2(\left[-c, c\right])}
&:=\int_{-c}^{c}
|\hat f(\xi)|^2 d\xi
=\int_{\mathbb{R}}
|\hat f(\xi)|^2 d\xi
=\int_{\mathbb{R}}
|f(x)|^2 dx
=: \| f \|^2_{L^2(\mathbb{R})}
< \infty,
\end{align}
proving that
$\hat f \in L^2(\left[-c, c\right])$.
Conversely, given any $g \in L^2(\left[-c, c\right])$, we need to show that ${\mathcal F}^{-1}(g) \in PW_c$.
We may extend $g$ to be equal to $0$ outside the interval $\left[-c, c\right]$, so that now $g \in L^2(\mathbb{R})$. By construction of $g$, we also
have $g \in L^1(\mathbb{R})$, so that now Lemma \ref{uniform continuity} guarantees that ${\mathcal F}^{-1}(g)$ is uniformly continuous on $\mathbb{R}$.
Because the Fourier transform is an isometry of $L^2(\mathbb{R})$, and $g \in L^2(\mathbb{R})$, we also have ${\mathcal F}^{-1}(g) \in L^2(\mathbb{R})$. So now we have
${\mathcal F}^{-1}(g) \in PW_c$.
Finally, the Fourier transform is invertible, and since we have
$\|f-g\|_{L^2(\mathbb{R})} = \|\hat f-\hat g\|_{L^2(\mathbb{R})}$ by Parseval again, we have an isometry between the two Hilbert spaces $PW_c$ and $L^2([-c, c])$.
\end{proof}
\medskip
Hardy's insight is to consider an orthonormal basis for $L^2([-c,c])$ and then pull it back to an orthonormal basis for $PW_c$. We recall the classical fact (Theorem \ref{Fourier series for periodic functions}) that the set of exponentials
\[ \Big\{ \frac{1}{\sqrt{2c}}e^{\frac{2\pi i n x}{2c}} \mid n \in \mathbb{Z}\Big\}\]
form a complete orthonormal basis for the Hilbert space $L^2(\left[-c, c\right])$.
Moreover, any $g \in L^2(\left[-c, c\right])$ has a unique representation in this basis (which we called its Fourier series), that converges in the $L^2$-norm on $[-c, c]$:
\begin{equation}\label{good series for now}
g(\xi)
\underset{L^2([-c,c])}{=}
\sum_{m\in \mathbb{Z}} \hat g_m e^{\frac{2\pi i m \xi}{2c}},
\end{equation}
with coefficients equal to
\begin{equation} \label{good series for now: coefficients}
\hat g_m = \frac{1}{2c}\big\langle g(\xi), e^{\frac{2\pi i m \xi}{2c}} \big\rangle
:= \frac{1}{2c}\int_{-c}^c g(\xi) e^{-\frac{2\pi i m \xi}{2c}} d\xi.
\end{equation}
Using the Fourier series~\eqref{good series for now}, we may expand $\|g\|_{L^2([-c,c])} = \langle g, g\rangle^{1/2}$, obtaining
\begin{equation*}
\| g \|_{L^2([-c,c])} = \Big( 2c \sum_{m \in \mathbb{Z}} |\hat g_m|^2 \Big)^{1/2}.
\end{equation*}
Although we've only scratched the surface, we've already scratched it enough in order to prove
Theorem \ref{Shannon}.
\begin{comment}
\begin{figure}
\centering
\fbox{
\includegraphics[width=1.0\textwidth]{images/L^2.png}
}
\caption{Visualizing the isometries of various spaces of functions of one variable, given by the Fourier transform. First, the Fourier transform $\mathcal{F}$ gives us an isometry from $L^2(\mathbb{R})$ onto itself. Next, restricting attention to the subspace $PW_c \subset L^2(\mathbb{R})$ of bandlimited functions, $\mathcal{F}$ also gives us an isometry from $PW_c$ to $L^2([-c, c])$, carrying the basis of translated ${\rm{sinc}}$ functions to the basis of exponentials. \red{[Fabricio: the Fourier transform of $2c{\rm{sinc}}(2cx)$ is $1_{[-c,c]}(\xi)$ and the basis element should be $2c{\rm{sinc}}(2cx-n)$.]}}
\label{fig:L^2 spaces}
\end{figure}
\end{comment}
\bigskip
\begin{proof}[Proof of Theorem~\ref{Shannon}]
For any $f \in PW_{c}$, we know by Lemma~\ref{isometry} that $\hat f \in L^2(\left[-c, c\right])$, so that $\hat f$ has a Fourier series that converges in the $L^2$-norm on $[-c, c]$:
\begin{equation*}
\hat f(\xi)
\underset{L^2([-c,c])}{=}
\sum_{m\in \mathbb{Z}} c_m e^{\frac{2\pi i m \xi}{2c}},
\end{equation*}
with coefficients equal to
$c_m = \frac{1}{2c}\int_{-c}^c \hat f(\xi) e^{-\frac{2\pi i m \xi}{2c}} d\xi = \frac{1}{2c}f(-\frac{m}{2c})$ by the Fourier inversion formula \eqref{eq:bandlimit-inversion}. It follows that
\begin{equation}
\label{eq:fhat-L2series}
\hat f(\xi) \underset{L^2([-c,c])}{=} \frac{1}{2c} \sum_{m\in \mathbb{Z}} f\left(\frac{m}{2c}\right) e^{-\frac{2\pi i m \xi}{2c}}.
\end{equation}
From the orthonormality of the exponentials and Parseval's identity (Lemma~\ref{isometry}) we have
\begin{equation}\label{eq:series-fnorm}
\| f \|_{L^2(\mathbb{R})} = \| \hat f \|_{L^2([-c,c])} = \Big( \frac{1}{2c} \sum_{m \in \mathbb{Z}} \Big|f\left(\frac{m}{2c}\right)\Big|^2 \Big)^{1/2}.
\end{equation}
We recall that the equality in the norm in~\eqref{eq:fhat-L2series} means that
\[
\lim_{N \to \infty} \bigg\| \hat f(\xi) - \frac{1}{2c}\sum_{|m| < N} f\left(\frac{m}{2c}\right) e^{-\frac{2\pi i m \xi}{2c}} \bigg \|_{L^2([-c,c])} = 0.
\]
Using the Fourier inversion and the isometry stated in Lemma~\ref{isometry},
\begin{align*}
0 &= \lim_{N \to \infty} \bigg\| \int_{-c}^{c} \Big( \hat f(\xi) - \frac{1}{2c}\sum_{|m| < N} f\left(\frac{m}{2c}\right) e^{-\frac{2\pi i m \xi}{2c}}\Big) e^{2\pi i \xi x}d\xi \bigg \|_{L^2(\mathbb{R})}\\
&= \lim_{N \to \infty} \bigg\| \int_{-c}^{c} \hat f(\xi)e^{2\pi i \xi x}d\xi - \frac{1}{2c}\sum_{|m| < N} f\left(\frac{m}{2c}\right) \int_{-c}^{c} e^{-\frac{2\pi i m \xi}{2c}} e^{2\pi i \xi x}d\xi \bigg \|_{L^2(\mathbb{R})}\\
&= \lim_{N \to \infty} \bigg\| f(x) - \sum_{|m| < N} f\left(\frac{m}{2c}\right) \frac{\sin(\pi(2cx-m))}{\pi(2cx-m)} \bigg \|_{L^2(\mathbb{R})}\\
&= \lim_{N \to \infty} \bigg\| f(x) - \sum_{|m| < N} f\left(\frac{m}{2c}\right) {\rm{sinc}}(2cx-m) \bigg \|_{L^2(\mathbb{R})},
\end{align*}
and therefore
\begin{equation}\label{eq:formulal2norm}
f(x) \underset{L^2(\mathbb{R})}{=} \sum_{m \in \mathbb{Z}} f\left(\frac{m}{2c}\right) {\rm{sinc}}(2cx-m).
\end{equation}
To pass from the convergence in the norm to pointwise convergence, we need to show that the latter
series converges uniformly, so that we can conclude that it represents a continuous function and hence by Lemma \ref{norm convergence plus absolute convergence implies equality} it is equal to $f$ everywhere.
To prove the uniform convergence, we make use of the Cauchy-Schwartz inequality for infinite series, namely
\begin{equation}\label{C-S for infinite series}
\sum_{m=N}^\infty \Big|f\left(\frac{m}{2c}\right) {\rm{sinc}}(2cx-m)\Big| \leq
\Big(\sum_{m=N}^\infty \Big|f\left(\frac{m}{2c}\right)\Big|^2\Big)^{1/2}
\Big(\sum_{m=N}^\infty {\rm{sinc}}^2(2cx-m)\Big)^{1/2}.
\end{equation}
The rest of the proof consists in showing that the right-hand side of \eqref{C-S for infinite series}
goes to zero as $N \rightarrow \infty$, uniformly for $x \in \mathbb{R}$.
The same proof will also work for the series defined from $-N$ to $-\infty$. Together these results show that the expression in~\eqref{eq:formulal2norm} converges absolutely and uniformly over~$\mathbb{R}$, giving the result stated in the theorem.
From~\eqref{eq:series-fnorm}, we see that
\begin{equation}\label{eq:fgoestozero}
\Big(\sum_{m = N}^\infty \Big|f\left(\frac{m}{2c}\right)\Big|^2\Big)^{1/2} \to 0,\quad \text{as } N \to \infty.
\end{equation}
Clearly
\[
\sum_{m = N}^\infty {\rm{sinc}}^2(2cx - m) \leq \sum_{m \in \mathbb{Z}} {\rm{sinc}}^2(2cx - m),
\]
and since the latter series are periodic function of $x$, with period $\frac{1}{2c}$,
we may assume that $0 \leq x < \frac{1}{2c}$.
For $m = 0$ and $1$, we note that ${\rm{sinc}}^2(2 c x - m) \leq 1$. For $m \geq 2$, we use the estimate
\[
{\rm{sinc}}^2(2cx -m) = \frac{\sin^2(2\pi cx - \pi m)}{(2\pi cx - \pi m)^2} \leq \frac{1}{\pi^2(m-1)^2},
\]
so that
\[
\sum_{m = 2}^\infty {\rm{sinc}}^2(2cx - m) \leq \sum_{m=2}^\infty \frac{1}{\pi^2(m-1)^2}
= \frac{1}{6}.
\]
Similarly, for $m \leq -1$,
\[
\sum_{m = -\infty}^{-1} {\rm{sinc}}^2(2cx - m) \leq \sum_{m=1}^\infty \frac{1}{\pi^2 m^2} =\frac{1}{6}.
\]
We conclude that for all $x \in \mathbb{R}$,
$\sum_{m = N}^\infty {\rm{sinc}}^2(2cx-m) \leq \sum_{m \in \mathbb{Z}} {\rm{sinc}}^2(2cx-m) \leq 2 + \frac{1}{6}+ \frac{1}{6} = \frac{7}{3}$,
and therefore the series in~\eqref{eq:holder} converges uniformly to $0$ as $N\rightarrow \infty$.
\end{proof}
\bigskip
It follows from this approach of G.H. Hardy, that despite being a subspace of $L^2(\mathbb{R})$, the Paley-Wiener space $PW_{c}$ has
a concrete, countable basis, which we record as follows.
\begin{cor}\label{cor:sinc-orthonormal}
The set of translated ${\rm{sinc}}$ functions
\begin{equation}
\{(\sqrt{2c}) \,{\rm sinc}(2cx-n) \mid n \in \mathbb{Z}\}
\end{equation}
is a complete orthonormal basis for the Hilbert space
$PW_{c}$ of $c$-bandlimited functions.
\hfill $\square$
\end{cor}
\noindent
It is also worthwhile recording here the orthonormality of the ${\rm{sinc}}$ functions explicitly. For each
$n, m \in \mathbb{Z}$, we have:
\begin{equation}\label{eq:sinc-orthonormal}
2c\int_{\mathbb{R}} {\rm{sinc}}(2cx-n) {\rm{sinc}}(2cx-m) dx =
\begin{cases}
1 &\mbox{if } n = m,\\
0 &\mbox{otherwise.}
\end{cases}
\end{equation}
\bigskip
\section{An alternative proof, using Poisson summation}\label{sec:proof-Poisson}
Here we give Shannon's proof of the classical Shannon-Whittaker sampling theorem (Theorem~\ref{Shannon}), with some added details.
This proof uses the Poisson summation formula. As we've seen several times before, Poisson summation often simplifies proofs in surprising ways.
To state the formula more precisely, we use $\underset{L^1(\mathbb{R})}{=}$ and $\underset{L^1([-c,c])}{=}$ to denote convergence in the $L^1$-norm, so that equality between functions holds almost everywhere but cannot be assumed at a specific point, unless we have an additional assumption like continuity.
Assuming only that $f \in L^1(\mathbb{R})$, the Poisson summation formula (See \cite{SteinWeiss})
states that the periodized function defined by the series
$
\sum_{n \in \mathbb{Z}} f(x+2cn)
$
converges in the norm of $L^1([-c,c])$ to a function whose Fourier expansion is
\begin{equation}\label{first Poisson summation-one dimension}
\sum_{n \in \mathbb{Z}} f(x+2cn)
\underset{L^1([-c,c])}{=}
\frac{1}{2c} \sum_{m \in \mathbb{Z}} \hat f\left(\frac{m}{2c}\right) e^{\frac{2\pi i m x}{2c} }.
\end{equation}
\begin{proof}[Proof of Theorem~\ref{Shannon}]
We begin with the Fourier series \eqref{first Poisson summation-one dimension}, which converges in the $L^1([-c, c])$ norm.
Step $1$. \ Our first goal will be to
exchange the roles of $f$ and $\hat f$. To justify this,
we begin by noting that our assumption that $f \in PW_c$ implies
$\hat f \in L^2([-c,c])$ by Lemma~\ref{isometry}, and $ L^2([-c,c]) \subset L^1([-c,c])$ by
Lemma \ref{proper containment of L^2 in L^1 for torus}. So we have $\hat f \in L^1([-c,c] \subset L^1(\mathbb{R})$,
allowing us to apply the same Poisson summation formula as above, together with Fourier inversion:
\begin{equation}\label{Poisson strikes again-one dimension}
\sum_{n \in \mathbb{Z}} \hat f(\xi + 2cn)
\underset{L^1([-c,c])}{=}
\frac{1}{2c} \sum_{m \in \mathbb{Z}} f\left( \frac{m}{2c} \right) e^{-\frac{2\pi i m \xi }{2c}}.
\end{equation}
Step $2$. \ Since we are assuming that $f \in PW_c$ and thus $\hat f$ is supported on $[-c, c]$, we may use the indicator function $1_{[-c, c]}$, defined as $1_{[-c, c]}(\xi) = 1$ if $\xi \in [-c,c]$ and $1_{[-c, c]}(\xi) = 0$ otherwise, and write the trivial identity
\begin{equation} \label{step 2}
\hat f( \xi ) = 1_{[-c, c]}(\xi) \sum_{n\in \mathbb{Z}} \hat f(\xi + 2c n ),
\end{equation}
for all real $\xi \not= c, \xi \not= -c$. The reason is that the series on the right-hand-side contains only one term, namely
the $n=0$ term $\hat f(\xi)$.
Step $3$. \ Using the Poisson summation formula \eqref{Poisson strikes again-one dimension} above, together with \eqref{step 2}, we see that
\begin{equation}\label{step 3}
\hat f( \xi ) = 1_{[-c, c]}(\xi) \sum_{n\in \mathbb{Z}} \hat f(\xi + 2c n)
\underset{L^1(\mathbb{R})}{=} \sum_{n \in \mathbb{Z}} f\left( \frac{n}{2c} \right) \Big(
\frac{1}{2c} 1_{[-c, c]}(\xi) e^{ - \frac{2\pi i n \xi }{2c}} \Big).
\end{equation}
We recall that the inverse Fourier transform of the interval $[-c, c]$ is
\[
\int_\mathbb{R} 1_{[-c, c]}(\xi) e^{-2\pi i \xi x} d\xi =2c \, {\rm{sinc}}(2c x),
\]
so that after composing the ${\rm{sinc}}$ function with a translation, we know that the Fourier transform of
${\rm{sinc}} \left( 2c \left(x - \frac{n}{2c} \right)\right)$
is $\frac{1}{2c} 1_{[-c, c]}(\xi)
e^{ - \frac{2\pi i n \xi }{2c}}$.
Multiplying both sides of \eqref{step 3} by $e^{-2\pi i \xi x}$ and integrating term-by-term over $x \in [-c, c]$, we get:
\[
f(x) \underset{L^1(\mathbb{R})}{=} \sum_{n \in \mathbb{Z}} f\left( \frac{n}{2c} \right) {\rm{sinc}} \left( 2c x - n \right),
\]
applying Fourier inversion again on the left-hand side.
Finally, we recall that we are assuming $f$ is continuous, since $f\in PW_c$. So to pass from the convergence in the norm to the pointwise convergence we may apply the same procedure from the first proof to conclude that the series on the right converges uniformly in $\mathbb{R}$ and hence also represents a continuous function.
\end{proof}
\bigskip
There are many different possible extensions of the Shannon-Whittaker sampling theorem to higher dimensions, and below we glimpse some of them below.
\section{Special properties of bandlimited and sinc functions}
\label{sec:special}
Here we focus on some special properties of bandlimited functions. We've already seen in Section~\ref{sec:Hardy} that the space $PW_c$ is isometric to $L^2([-c,c])$,
so many of its special properties comes from $L^2([-c,c])$, and they are then pulled back via inverse Fourier transform.
The special case $c=\frac{1}{2}$ of Theorem \ref{Shannon} is worth pointing out:
\begin{equation}\label{Shannon simplified}
f(x) = \sum_{n \in \mathbb{Z}} f(n) \ {\rm sinc } \left( x - n \right),
\end{equation}
a classical version of the Shannon-Whittaker formula.
The choice of $c = \frac{1}{2}$ means that we begin with the interval $[-\frac{1}{2}, \frac{1}{2}]$ in the frequency space;
this interval is a Voronoi cell of the integer lattice $\mathbb{Z}$.
\begin{example} \label{ex-1}
\rm{
What happens if we apply the Shannon-Whittaker formula
\eqref{Shannon simplified} to the ${\rm{sinc}}$ function itself? Let's try it!
With $f(x):= {\rm{sinc}}(y - x)$, and any fixed $y\in \mathbb{R}$, we have:
\begin{equation}
{\rm{sinc}}(y - x) = \sum_{n \in \mathbb{Z}} {\rm{sinc}}(y - n) \, {\rm{sinc}}( x - n ).
\end{equation}
As a special case, if we let $x=y$, we get:
\begin{equation}
1= {\rm{sinc}}(0) = \sum_{n \in \mathbb{Z}} {\rm{sinc}}^2(x - n).
\end{equation}
\hfill $\square$
}
\end{example}
In Corollary \ref{cor:sinc-orthonormal} we showed that the functions ${\rm{sinc}}(x-n)$ with $n \in \mathbb{Z}$
form an orthonormal basis for the space of bandlimited functions. This has some nice consequences.
\begin{thm} \label{thm:discrete-inner-product}
If $f$ and $g$ are $\tfrac{1}{2}$-bandlimited,
then
\[
\int_\mathbb{R} f(x) \overline{g(x)} dx
= \sum_{n \in \mathbb{Z}} f(n) \overline{g(n)}.
\]
\end{thm}
[{\bf Intuitive proof}] \ If we work formally, then we can use the orthonormality of the sinc functions \eqref{eq:sinc-orthonormal},
together with \eqref{Shannon simplified} to quickly see that:
\begin{align*}
\langle f(x), g(x) \rangle &=
\Big\langle \sum_{m \in \mathbb{Z}} f(m) {\rm{sinc}}(x-m), \sum_{n \in \mathbb{Z}} g(n){\rm{sinc}}(x-n) \Big\rangle \\
&= \sum_{m, n \in \mathbb{Z}}
f(m)\overline{g(n)} \Big\langle {\rm{sinc}}(x-m), {\rm{sinc}}(x-n) \Big\rangle \\
&=\sum_{n \in \mathbb{Z}}f(n) \overline{g(n)},
\end{align*}
and we're done. Although this intuitive proof may seem `fast and loose', these steps can
be made rigorous if we would prove just a bit more about Hilbert spaces,
because the Paley-Wiener space $PW_c$ is a Hilbert space, and the translated
functions ${\rm{sinc}}(x-n)$ are a basis
for this Hilbert space.
\begin{comment}
[{\bf Rigorous proof}] \ Let's redo the steps above, adding rigor and using the theorems that we've already
proved and/or seen. Using the Shannon sampling identity \eqref{Shannon simplified} for both $f$ and $g$,
we have:
\begin{align*}
\int_\mathbb{R} f(x) \overline{g(x)} dx &= \int_\mathbb{R} \sum_{m \in \mathbb{Z}} f(m) {\rm{sinc}}(x-m)
\sum_{n \in \mathbb{Z}} \overline{g(n)} {\rm{sinc}}(x-n) dx \\
&= \int_\mathbb{R} \sum_{(m, n) \in \mathbb{Z}^2} f(m) \overline{g(n)} \, {\rm{sinc}}(x-m) \, {\rm{sinc}}(x-n) dx.
\end{align*}
If we can move the integral past the sum, then we're done, because we would have:
\begin{align*}
\int_\mathbb{R} f(x) \overline{g(x)} dx
&= \sum_{(m, n) \in \mathbb{Z}^2} f(m) \overline{g(n)} \, \int_\mathbb{R} {\rm{sinc}}(x-m) \, {\rm{sinc}}(x-n) dx \\
&= \sum_{n \in \mathbb{Z}} f(n) \overline{g(n)},
\end{align*}
by \eqref{eq:sinc-orthonormal}, the orthogonality relations of the ${\rm{sinc}}$ functions.
To justify moving the integral past the sum, we may use \eqref{Application of dominated convergence} (in the Appendix), once we show that $\sum_{n=1}^\infty \int_{\mathbb{R}^d} | f_n(x) | dx < \infty$, where
$ f_n(x) := $
\hfill $\square$
\end{comment}
\begin{comment}
Let $F$ be a class of functions defined in a Hilbert space $E$.
A function $f:E \times E$ is a Reproducing Kernel of $F$ if
\begin{enumerate}
\item For every $x$, $f(y,x)$ as function of $y$ belongs to $F$.
\item For every $x \in E$, for every $f \in F$, $f(x) = \langle f(y),K(y,x) \rangle$
\end{enumerate}
\end{comment}
One important case of the previous theorem is when $g(x) := {\rm{sinc}}(x - y)$, which combined again with the Shannon-Whittaker formula
\eqref{Shannon simplified} results in the next theorem.
\begin{thm} \label{thm:ReproducingSinc}
The space $PW_{\frac{1}{2}}$ is a space with a reproducing kernel ${\rm{sinc}}(x-y)$, which means by definition
that any $f \in PW_{\frac{1}{2}}$ can be written as
\[
f(x) = \int_\mathbb{R} f(y) {\rm{sinc}}(x-y) d y.
\]
\end{thm}
\begin{proof}
For $x \in \mathbb{R}$, take $g(y) := {\rm{sinc}}(x - y)$ in Theorem~\ref{thm:discrete-inner-product}:
\[
\int_R f(y) {\rm{sinc}}(x-y) d y = \sum_{n \in \mathbb{Z}} f(n) {\rm{sinc}}(x-n) = f(x),
\]
where the second equality follows from the Shannon-Whittaker formula.
\end{proof}
\bigskip
We also have the following somewhat surprising properties of bandlimited functions on $\mathbb{R}$. First we recall
that we called a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ nice if $f, \hat f \in L^1(\mathbb{R}^d)$, and $f$ satisfies the Poisson summation formula:
\begin{equation} \label{Poisson summation again}
\sum_{n \in \mathbb{Z}^d} f(n+ x) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle},
\end{equation}
valid pointwise for each $x \in \mathbb{R}^d$.
\begin{thm}
\label{bandlimited, Poisson summation}
Let $f:\mathbb{R}\rightarrow \mathbb{C}$ be a nice function, such that $f$ is $c$-bandlimited. Then we have:
\begin{enumerate}[(a)]
\item \label{firstpart}
We have, for each $k>c$,
\begin{equation}\label{exact Riemann approximation}
\frac{1}{k}\sum_{n\in\mathbb{Z}}
f\left(\frac{n}{k}\right)
= \int_{\mathbb{R}} f(x) dx,
\end{equation}
\blue{We note that the identity \eqref{exact Riemann approximation}
can be interpreted to mean that the Riemann approximation to the integral is always exact for such an $f$, provided that the step size is $\Delta x:=\frac{1}{k}<\frac{1}{c}$.}
\item \label{secondpart}
For all $a, k$ with $a>c$ and $k > a+c$, we have
\begin{equation}
\sum_{n\in\mathbb{Z}} f\left(\frac{n}{k}\right)
e^{\frac{2\pi i n a}{k}}=0.
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}
To prove \ref{firstpart}, we use Poisson summation \eqref{Poisson summation again}, with ${\mathcal L}:= \frac{1}{k} \mathbb{Z}$:
\begin{equation} \label{equality of sum and integral}
\sum_{n\in\mathbb{Z}}
f\left(\frac{n}{k}\right)=
\sum_{\xi \in {\mathcal L}} f(\xi) =
\frac{1}{\det {\mathcal L}}
\sum_{m \in {\mathcal L}^*} \hat f(m) =
k \sum_{m \in \mathbb{Z}} \hat f(mk) = k\int_\mathbb{R} f(x) dx,
\end{equation}
which is the desired identity. In the last equality we used the assumption that the indices of summation satisfy $ |mk| > c$,
for $m\not=0$,
so that $\hat f(mk) = 0$ because $f$ is $c$-bandlimited by assumption.
We also used the fact that
$\hat f(0) = \int_\mathbb{R} f(x) dx$.
\noindent
To prove \ref{secondpart}, we apply the following small variation of Poisson summation:
\begin{equation}\label{magic2}
\frac{1}{k}
\sum_{n \in \mathbb{Z}} f\left( \frac{n}{k} \right) e^{\frac{2\pi i n a }{k}}
=\sum_{n\in \mathbb{Z}} \hat f(-a+kn),
\end{equation}
which follows quickly from the Poisson summation formula given above in \eqref{Poisson summation again} (Exercise \ref{Exercise:PoissonSummation1}).
But by the assumption that $f$ is $c$-bandlimited, we also have
\begin{equation}
\sum_{n\in \mathbb{Z}} \hat f(-a+kn)=0,
\end{equation}
provided that
\begin{equation}\label{constraint}
|-a+kn|>c, \ \text{ for all } n\in \mathbb{Z}.
\end{equation}
For $n=0$, we see that a necessary condition for \eqref{constraint} is $|a|>c$. Geometrically, \eqref{constraint} tells us that the arithmetic progression $\{ kn -a\}_{n \in \mathbb{Z}}$ does not intersect the interval $[-c, c]$. It is easily checked that the additional constraint $k>c+a$ gives us a sufficient condition for \eqref{constraint} to hold.
\end{proof}
As is easily observed, sums and products of bandlimited functions are again bandlimited. In particular, more precise statements such as the following are possible.
\begin{lem}\label{lemma:product}
Suppose that $f$ is $c$-bandlimited, and $g$ is $d$-bandlimited. \\
Then $fg$ is $(c+d)$-bandlimited.
\end{lem}
\begin{proof} By assumption, $\hat f(\xi) = 0$ outside of $[-c, c]$, and
$\hat g(\xi) = 0$ outside of $[-d, d]$.
We know that the support of the convolution $\hat f(\xi)*\hat g(\xi)$ is contained in the closure of the Minkowski sum \index{Minkowski sum}
of the individual supports of $f$ and $g$ (by Exercise \ref{support of convolution}), which is $[-c, c]+[-d, d] = [-c-d, c+d]$.
This argument justifies the second equality in the chain of equalities below.
Also, the assumption that $f, g \in L^2(\mathbb{R})$ implies $fg \in L^1(\mathbb{R})$, by the Cauchy-Schwartz argument we gave in
\eqref{product of two L^2 functions is L^1}.
We'd like to say that $\widehat{(fg)}(\xi)=(\hat f*\hat g)(\xi)$, so let's brute-force
the computation, using our knowledge of Lemma \ref{isometry}:
\begin{align*}
(\hat f*\hat g)(\xi) &:= \int_{\mathbb{R}} \hat f(x-\xi)\hat g(x) dx\\
&= \int_{-c-d}^{c+d} \hat f(x-\xi)\hat g(x) dx\\
&:= \int_{-c-d}^{c+d} \int_{-c}^c f(u) e^{-2\pi i (x-\xi)u} du \int_{-d}^d g(v) e^{-2\pi i v x} dv \, dx\\
&:= \int_{-c}^c \int_{-d}^d f(u) g(v) e^{2\pi i \xi u}
\left(
\int_{-c-d}^{c+d} e^{-2\pi i x(u + v)}
\right) \, dx \\
& \text{ ... and now using the orthogonality relations for the exponentials ....} \\
& \text{ In other words, ` a miracle happens here ' .... } \\
&= \int_{\mathbb{R}} f(u) g(u) e^{-2\pi i u \xi }du = \widehat{(fg)}(\xi).
\end{align*}
So we conclude that $fg$ is $(c+d)$-bandlimited. Another argument is also possible, using
$L^2[-c, c] \subset L^1[-c, c]$, which we proved in Lemma \ref{proper containment of spaces over the torus}.
\end{proof}
\bigskip
\begin{example}
\rm{
Here are some fun consequences of Theorem \ref{bandlimited, Poisson summation}. Let's fix any $\epsilon >0$.
By Theorem \ref{bandlimited, Poisson summation}, part \rm{\ref{secondpart}} , we can pick $k=1, c=\frac{1}{2}-\epsilon$, and $a=\frac{1}{2}$, all of
which satisfy the hypothesis, so that $f$ is $(\frac{1}{2}-\epsilon)$-bandlimited by definition. We then have
\begin{equation}
0= \sum_{n\in\mathbb{Z}} f(n)
e^{\pi in}
= \sum_{n\in\mathbb{Z}} f(n)
(-1)^n.
\end{equation}
Seperating the lattice sum into $n$ even and $n$ odd, we have
\begin{equation}
\sum_{m\in \mathbb{Z}} f(2m)
=\sum_{m \in \mathbb{Z}} f(2m+1).
\end{equation}
Generalizing the latter identity, we fix any positive integer $N$, and we let $k=\frac{2}{N}, c=\frac{1}{N}-\epsilon$, and $a=\frac{1}{N}$, so that $f$ is $(\frac{1}{N}-\epsilon)$-bandlimited by definition.
By Theorem
\ref{bandlimited, Poisson summation}, part \rm{\ref{secondpart}}:
\begin{equation}
0= \sum_{n\in\mathbb{Z}} f\left(\frac{n}{k}\right)
e^{\frac{2\pi i n a}{k}}
=\sum_{n\in\mathbb{Z}} f\left(\frac{Nn}{2}\right)
(-1)^n,
\end{equation}
so that we get the identity
\begin{equation}
\sum_{m\equiv 0\text{ mod N }} f(m)
=\sum_{m\equiv 0\text{ mod N }}
f\Big(m+ \frac{N}{2}\Big).
\end{equation}
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
With $k=1, c=\frac{1}{3}-\epsilon$, and $a=\frac{1}{3}$, part \rm{\ref{secondpart}} of Theorem
\ref{bandlimited, Poisson summation}
gives:
\begin{align*}
0&=\sum_{n\in\mathbb{Z}} f(n) e^{\frac{2\pi i n}{3}}
=\sum_{n\equiv 0\text{ mod 3 }} f(n)
+ \omega\sum_{n\equiv 1\text{ mod 3 }} f(n)
+ \omega^2\sum_{n\equiv 2\text{ mod 3 }} f(n),
\end{align*}
where $\omega:= e^{2\pi i / 3}$.
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
Consider ${\rm{sinc}}^N(x / \pi)$, which has the bandlimit $c:= \frac{N}{2\pi}$.
By Theorem~\ref{bandlimited, Poisson summation}, the strange relation
\begin{equation} \label{example:strange 1}
\sum_{n \in \mathbb{Z}} {\rm{sinc}}^N\left(\frac{n}{\pi}\right) = \int_{\mathbb{R}} {\rm{sinc}}^N\left(\frac{x}{\pi}\right) dx,
\end{equation}
holds for $N = 2, \dots, 6$, because in this range we have
$c=\frac{N}{2\pi} \leq \frac{6}{2\pi} < 1=: k$. It's also true for $N=1$, with some care:
\[
\lim_{M\rightarrow \infty} \sum_{|n| < M \atop n \in \mathbb{Z}}
{\rm{sinc}} \left(\frac{n}{\pi}\right) = \lim_{M\rightarrow \infty} \int_{-M}^M {\rm{sinc}} \left(\frac{x}{\pi}\right) dx.
\]
It turns out that this identity fails, however, for $N \geq 7$. Indeed, by
Poisson summation~\eqref{Poisson summation again}, for a nice function $f$ we have:
\[
\sum_{n \in \mathbb{Z}}f(n) = \sum_{m \in \mathbb{Z}}\hat f(m) = \int_\mathbb{R} f(x) dx + \sum_{m \in \mathbb{Z} \setminus \{0\}} \hat f(m).
\]
Taking $f(x) := {\rm{sinc}}^N\left( \frac{x}{\pi}\right)$, we see that
that the last sum is zero when $N \leq 6$ and positive when $N \geq 7$, since $\hat f$ has support $[-\frac{N}{2\pi}, \frac{N}{2\pi}]$ and is positive inside this interval.
In a similar manner to eq. \eqref{example:strange 1}, we have:
\begin{equation} \label{example:strange 2}
\sum_{n \in \mathbb{Z}}\, \prod_{k = 0}^N {\rm{sinc}}\left(\frac{n}{(2k+1)\pi}\right) = \int_{\mathbb{R}}\, \prod_{k = 0}^N {\rm{sinc}}\left(\frac{x}{(2k+1)\pi}\right) dx,
\end{equation}
holds for $N = 0, \dots, 40248$, since for these $N$ we have $1 + \frac{1}{3} + \dots + \frac{1}{2N+1} < 2\pi$.
It can be also checked that the equality above fails for $N = 40249$. These facts are easy corollaries of Theorem~\ref{bandlimited, Poisson summation}, but may seem surprising when taken out of this context. The identities \eqref{example:strange 1}
and \eqref{example:strange 2} appeared in \cite{Baillie}.
}
\hfill $\square$
\end{example}
\bigskip
\section{Shannon sampling in higher dimensions}
The first research into higher-dimensional Shannon-type sampling theorems, as far as we know,
was
the work of Petersen and Middleton~\cite{PetersenMiddleton62}. We'll also follow a bit of
Chapter $14$ from Higgins~\cite{Higgins1996}.
For a convex body ${\mathcal P}$, we say that a function $f$ is {\bf ${\mathcal P}$-bandlimited} if $\hat f$ vanishes outside of ${\mathcal P}$.
We note that this does not preclude the possibility that $\hat f$ may only be nonzero on some proper subset of ${\mathcal P}$.
Assuming that $f$ is real-valued, we know that the image of $\hat f$ is symmetric about the origin
(Lemma \ref{symmetric iff FT is real});
so the assumption that ${\mathcal P}$ is symmetric is natural.
By analogy with the $1$-dimensional Paley-Wiener space $PW_c$, we define the
{\bf Paley-Wiener space} of ${\mathcal P}$-bandlimited functions in $L^2(\mathbb{R}^d)$:
\begin{equation}
PW_{\mathcal P}:= \{ f \in L^2(\mathbb{R}^d) \mid
f \text{ is continuous and ${\mathcal P}$-bandlimited} \}.
\end{equation}
A new twist in higher dimensions is the strong distinction between packing and tiling,
so the following question motivates some of these research directions.
\begin{question}
Given a convex $d$-dimensional body ${\mathcal P} \subset \mathbb{R}^d$, suppose we want to have a sampling theorem for functions that are
${\mathcal P}$-bandlimited. Does ${\mathcal P}$ have to tile $\mathbb{R}^d$ by translations, or is it sufficient to consider a packing of ${\mathcal P}$ by some lattice ${\mathcal L}$?
\end{question}
Interestingly, we don't observe this distinction in dimension $1$, because optimal packing and tiling are equivalent. But they are quite different in dimensions $d \geq 2$.
Luckily, our elementary $1$-dimensional Lemma~\ref{isometry} does extend directly to our new $d$-dimensional setting.
\begin{lem}
\label{isometry lemma for R^d}
Let ${\mathcal P}$ be a bounded convex body in $\mathbb{R}^d$. The Fourier transform ${\mathcal F}$ is an isometry between the two Hilbert spaces:
\begin{equation*}
{\mathcal F}: PW_{{\mathcal P}} \rightarrow
L^2({\mathcal P}).
\end{equation*}
\end{lem}
\begin{proof}
Given $f \in PW_{{\mathcal P}} \subset L^2(\mathbb{R}^d)$, by definition $\mathrm{supp}(\hat f) \subseteq {\mathcal P}$,
so using Parseval's identity we have:
\begin{align}
{\| \hat f \|}^2_{L^2({\mathcal P})}
&:=\int_{\mathcal P}
|\hat f(\xi)|^2 d\xi =
\int_{\mathbb{R}^d}
|\hat f(\xi)|^2 d\xi
=\int_{\mathbb{R}^d}
|f(\xi)|^2 d\xi =: \| f \|^2_{L^2(\mathbb{R}^d)}
< \infty,
\end{align}
which shows that $\hat f \in L^2({\mathcal P})$.
Using the fact that the Fourier transform is an isometry of $L^2(\mathbb{R}^d)$, and is invertible, we are done.
\end{proof}
\bigskip
\begin{thm}
\label{Higher dimensional sampling}
Suppose we have a lattice packing for a symmetric convex body ${\mathcal P}$, with a lattice~${\mathcal L}^*\subset \mathbb{R}^d$.
We let $\phi(x):= \hat 1_{\mathcal P}(x)$, our old friend.
If $f \in PW_{\mathcal P}$, then
then we can reconstruct the function $f$ completely by sampling it only at the lattice points of ${\mathcal L}$:
\[
f(x) = \det {\mathcal L} \sum_{n \in {\mathcal L}} f(n) \phi(x-n),
\]
and the series converges absolutely and uniformly over $\mathbb{R}^d$.
\end{thm}
Before diving into the proof, let's see why Theorem \ref{Higher dimensional sampling} indeed generalizes the classical $1$-dimensional
Theorem~\ref{Shannon}. In the one dimensional case, the lattice ${\mathcal L}$ is just the sampling domain $\{\frac{n}{2c} \mid n \in \mathbb{Z}\}$ and
hence $\det {\mathcal L} = \frac{1}{2c}$, while ${\mathcal P}:= [-c,c]$ is simply an interval.
Therefore:
\[
\phi(x) = \int_{-c}^c e^{2\pi i \xi x}d\xi
= \frac{1}{2\pi i x}e^{2\pi i xc} - \frac{1}{2\pi i x}e^{-2\pi i xc}
= \frac{1}{\pi x} \sin(2 \pi c x)
= 2c \, {\rm{sinc}}(2cx).
\]
The formula from Theorem~\ref{Higher dimensional sampling} reduces to the formula from Theorem~\ref{Shannon}:
\begin{align*}
f(x) = \det {\mathcal L} \sum_{n \in {\mathcal L}}f(n)\phi(x-n)
&= \frac{1}{2c} \sum_{n \in \mathbb{Z}} f\Big(\frac{n}{2c}\Big) 2c \, {\rm{sinc}}\Big(2c\big(x-\frac{n}{2c}\big)\Big) \\
&= \sum_{n \in \mathbb{Z}} f\Big(\frac{n}{2c}\Big) {\rm{sinc}}(2cx-n),
\end{align*}
which is the Shannon-Whittaker sampling formula.
\bigskip
\begin{proof}
[Proof of Theorem~\ref{Higher dimensional sampling}]
The assumption that $f \in PW_{\mathcal P}$, together with ${\mathcal P}$ being compact, implies that
$\hat f \in L^2({\mathcal P}) \subseteq L^1({\mathcal P}) \subseteq L^1(\mathbb{R}^d)$. Now we may use the Poisson summation formula
\eqref{Poisson summation again}, but with $f$ replaced by $\hat f$, and with ${\mathcal L}$ replaced by ${\mathcal L}^*$:
\begin{equation}\label{Poisson strikes again}
\sum_{m \in {\mathcal L}^*} \hat f(\xi + m)
\underset{L^1(\mathbb{R}^d/{\mathcal L}^*)}{=}
\det {\mathcal L} \sum_{n \in {\mathcal L}} f\left( n \right) e^{-2\pi i \langle \xi, n \rangle},
\end{equation}
where we also have used that $\hat{\hat{f}}(m) = f(-m)$.
Because $\hat f$ is supported on ${\mathcal P}$, we have by definition $\sum_{m \in {\mathcal L}^*} \hat f(\xi + m) = \hat f(\xi)$,
so that we may write
\begin{equation} \label{step 2.1}
\hat f( \xi ) = 1_{{\mathcal P}}(\xi) \sum_{m \in {\mathcal L}^*} \hat f(\xi + m),
\end{equation}
for all $\xi\in \mathbb{R}^d$ that do not lie on the boundary of ${\mathcal P}$.
Because of our packing assumption, all of the translated supports of $ \hat f(\xi + m)$ are disjoint,
as $m$ varies over the lattice ${\mathcal L}^*$. In other words, these supports are
\begin{equation*}
\{ \supp( \hat f ) + m \mid \, m \in {\mathcal L}^* \} \subseteq \{{\mathcal P}+ m \mid \, m \in {\mathcal L}^*\},
\end{equation*}
a disjoint collection of translates of ${\mathcal P}$.
This means that the latter identity \eqref{step 2.1} holds
because the series on the right-hand-side contains only one term, namely
the $m=0$ term $\hat f(\xi)$.
Next, we combine~\eqref{Poisson strikes again} with~\eqref{step 2.1} to get
\begin{equation}\label{step 3.1}
\hat f( \xi ) \underset{L^1(\mathbb{R}^d)}{=} \det {\mathcal L} \sum_{n \in {\mathcal L}} f\left( n \right) 1_{{\mathcal P}}(\xi) e^{-2\pi i \langle \xi, n \rangle}.
\end{equation}
Now we'd like to take the inverse Fourier transform of both sides of \eqref{step 3.1}.
We'll use the following elementary identity, for a fixed $n$:
\[
{\mathcal F}^{-1} \big( 1_{{\mathcal P}}(\xi) e^{-2\pi i \langle \xi, n \rangle} \big)(x) = {\mathcal F}^{-1}(1_{{\mathcal P}})(x - n)
= \int_{\mathcal P} e^{2\pi i \langle \xi, x-n \rangle}d\xi = \phi(x-n),
\]
Arriving finally at
\begin{align*}
f(x) \underset{L^1(\mathbb{R}^d)}{=} \det {\mathcal L} \sum_{n \in {\mathcal L}} f\left( n \right) \phi(x-n),
\end{align*}
it seems like we're done. We just to pass from the $L^1$-convergence of the latter identity, to its pointwise and uniform convergence.
There is just a slippery issue with uniform convergence that we need to
justify. To finish the argument, we would like to show that the series
$\sum_{n \in {\mathcal L}} f\left( n \right) \phi(x-n)$ converges
uniformly on $\mathbb{R}^d$, which is analogous to the proof in the previous Section, and for which we call on Lemma \ref{nasty uniform convergence}.
\end{proof}
\bigskip
\begin{lem} \label{nasty uniform convergence}
Given $f \in PW_{\mathcal P}$, $\phi:= \hat 1_{\mathcal P}$, and any full-rank lattice ${\mathcal L}$, the series
\[
\sum_{n \in {\mathcal L}} f\left( n \right) \phi(x-n)
\]
converges uniformly on $\mathbb{R}^d$.
\end{lem}
(See \cite{Higgins1996})
\hfill $\square$
Intuitively, if we pick a larger set ${\mathcal P}$, then the vectors from ${\mathcal L}^*$ will have to be more widely spaced in order to satisfy the packing condition
for ${\mathcal P}$.
Therefore our samples, which we always take from the lattice ${\mathcal L}$, will have to be denser due to the reciprocal relation $(\det {\mathcal L} )(\det {\mathcal L}^*) = 1$.
On the other hand, for a given sampling lattice ${\mathcal L}$, in this multidimensional case we can consider infinitely many different
bodies ${\mathcal P}$ that form a packing of $\mathbb{R}^d$ with the lattice ${\mathcal L}$.
One of the most natural choices for such a convex set ${\mathcal P}$ is the Voronoi cell of ${\mathcal L}^*$.
In closing, we note that it is impossible for both $f$ and $\hat f$ to be simultaneously bandlimited, by the basic uncertainty principle,
Theorem \ref{basic uncertainty principle}. However, in practice, if we are given a function $f\in L^2(\mathbb{R}^d)$ that is not bandlimited,
we can form a sequence of bandlimited functions that approach $f$ as $n\rightarrow \infty$, as follows. To make $\hat f$ compactly supported, we'll multiply $\hat f$ by $1_{[-n, n]^d}$, the indicator function of the cube.
Pulling things back to the space domain, we have:
\begin{align}
{\mathcal F}^{-1} \left(
1_{[-n, n]^d} \, \hat f
\right)
&= {\mathcal F}^{-1} \left( 1_{[-n, n]^d} \right) * {\mathcal F}^{-1} (\hat f) \\ \label{inverting sinc^d}
&= {\rm{sinc}}^d* f.
\end{align}
So if we define $f_n:= {\rm{sinc}}^d* f$, then ${\mathcal F}(f_n) = 1_{[-n, n]^d} \, \hat f$, a compactly supported function that is bandlimited to the cube $ [-n, n]^d$.
The careful reader might notice that in \eqref{inverting sinc^d}, we are applying the Fourier inversion formula on $L^2(\mathbb{R}^d)$, and not on
$L^1(\mathbb{R}^d)$. We do this because although ${\rm{sinc}}^d(x) \notin L^1(\mathbb{R}^d)$, we do have ${\rm{sinc}}^d(x) \in L^2(\mathbb{R}^d)$.
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.Shannon}
\begin{enumerate}[(a)]
\item Higgins' book \cite{Higgins1996}, Chapter $6$, has an excellent account of the Paley-Wiener space, and its connections to the Paley-Wiener theorem, which also answers the question: ``how may we analytically continue bandlimited functions of a real variable, to $\mathbb{C}$?'' Moreover, Higgin's book has more mathematical rigor than many other books.
\item For further reading, the following articles are of interest: \cite{Entezari09}, \cite{Unser00}, \cite{Ye12}.
\item Interesting relations between rates of convergence of bandlimited functions,
Nikol'skij type functions spaces, and Plancherel-Polya type inequalities is given in \cite{SchmeisserSickel2000}.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
\end{quote}
\begin{prob}
By using an example, show that a bandlimited function $f\in L^2(\mathbb{R})$ may not be in $L^1(\mathbb{R})$.
\end{prob}
\medskip
\begin{prob}
Let $f \in PW_c$, and fix any $x_0 \in \mathbb{R}$. Prove that $f$ is completely determined
by the samples
\[
\left\{ f \left(x_0 +\frac{\pi n}{c} \right) \mid n \in \mathbb{Z} \right\}.
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Exercise:PoissonSummation1}
Here we give another small variation on Poisson summation, namely that for any $a, k \in \mathbb{R}$, we have
\begin{equation}
\frac{1}{k}
\sum_{n \in \mathbb{Z}} f\left( \frac{n}{k} \right) e^{\frac{2\pi i n a }{k}}
=\sum_{n\in \mathbb{Z}} \hat f(-a+kn),
\end{equation}
where $f:\mathbb{R}\rightarrow \mathbb{C}$ is a nice function.
\end{prob}
Hint: you can use the usual Poisson summation formula given in \eqref{Poisson summation again}.
\bigskip
\chapter{Appendix A: The dominated convergence theorem, and other goodies}
\label{Appendix A}
A frequent question that comes up in proofs is ``when may we take the limit inside the integral''?
A general tool that allows us to do so is the Dominated convergence theorem. Here we remind the reader of
some of the basic results from real analysis, but we skip the proofs and give references for them. For our purposes, we only need these results in Euclidean spaces, although all of these theorems have extensions to arbitrary measure spaces. All functions here are assumed to be measurable.
\begin{thm}[Fatou's lemma] \label{Fatou}
\index{Fatou's lemma}
Fixing any subset $E\subset \mathbb{R}^d$, let $f_n:E \rightarrow [0, \infty)$ be a sequence of nonnegative functions. Then we have:
\begin{equation}
\int_{E} \lim \inf \, f_n(x) dx \leq \lim \inf \int_{E} \, f_n(x) dx.
\end{equation}
\hfill $\square$
\end{thm}
The inherent flexibility in {\bf Fatou's lemma} allows it to be useful in many different contexts, because the lim inf $f_n$ always exists, and are even allowed to be equal to $\pm$ infinity. In fact, Fatou's lemma is the main tool in proving Lebesgue's dominated convergence theorem, below.
Another essential fact for us is {\bf Fubini's theorem}, which allows us to interchange integrals with integrals, and series with integrals, for product spaces.
If we write $\mathbb{R}^d = \mathbb{R}^m \times \mathbb{R}^n$, and we denote a point $z\in \mathbb{R}^d$ by $z:= (x,y)$, then we may also write
$f(z):= f(x, y)$.
\begin{thm}[Fubini] \label{Fubini}
\index{Fubini's theorem}
Let $f \in L^1(\mathbb{R}^d)$. Then:
\begin{equation}
\int_{\mathbb{R}^d} f(z) dz = \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^m} f(x, y) dx \right) dy,
\end{equation}
and
\begin{equation}
\int_{\mathbb{R}^d} f(z) dz = \int_{\mathbb{R}^m} \left( \int_{\mathbb{R}^n} f(x, y) dy \right) dx.
\end{equation}
\hfill $\square$
\end{thm}
There is also a version of Fubini's theorem that uses the counting measure in one of the
factors of $\mathbb{R}^m \times \mathbb{R}^n$, giving us:
\begin{equation}\label{Fubini for sums and integrals}
\sum_{\xi \in \mathbb{Z}^n} \left( \int_{\mathbb{R}^m} f(x, \xi) dx \right) = \int_{\mathbb{R}^m} \left( \sum_{\xi \in \mathbb{Z}^n} f(x, \xi) \right) dx.
\end{equation}
(See \cite{PaulSally1}, p. 220, for a proof of Theorem \ref{Fubini})
\medskip
\section{The Dominated Convergence Theorem}
\begin{thm}[Dominated convergence theorem] \label{Dominated convergence theorem}
\index{Lebesgue dominated convergence theorem}
\ Suppose that we have a sequence of functions $ f_n(x):\mathbb{R}^d \rightarrow \mathbb{C}$, for $n = 1, 2, 3, \dots $, and suppose
there exists a limit function $f(x) =\lim_{n\rightarrow \infty} f_n(x)$, valid for all $x\in \mathbb{R}^d$.
If there exists a function $g \in L^1(\mathbb{R}^d)$ such that for all $x \in \mathbb{R}^d$, we have:
\[
\left| f_n(x) \right| \leq g(x), \quad n = 1, 2, 3, \dots
\]
then:
\begin{enumerate}[(a)]
\item $f \in L^1(\mathbb{R}^d)$.
\item $ \lim_{n\rightarrow \infty} \int_{\mathbb{R}^d} | f_n(x) - f(x) | dx = 0$.
\item And finally, we may interchange limits and integrals:
\[
\lim_{n \rightarrow \infty} \int_{\mathbb{R}^d} f_n(x) dx = \int_{\mathbb{R}^d} f(x) dx.
\]
\end{enumerate}
\hfill $\square$
\end{thm}
Theorem \ref{Dominated convergence theorem} is sometimes called the \emph{Lebesgue dominated convergence theorem}, honoring the work of Lebesgue.
There is a useful application of Lebesgue's dominated convergence theorem, which allows us to interchange summations with integrals as follows.
\medskip
\begin{thm} \label{Application of dominated convergence}
\ Suppose that we have a sequence of functions $ f_n(x):\mathbb{R}^d \rightarrow \mathbb{C}$, such that
\[
\sum_{n=1}^\infty \int_{\mathbb{R}^d} | f_n(x) | dx < \infty.
\]
Then the series $ \sum_{n=1}^\infty f_n(x)$
converges for all $x\in \mathbb{R}^d$, and we have:
\[
\sum_{n=1}^\infty \int_{\mathbb{R}^d} f_n(x) dx = \int_{\mathbb{R}^d} \sum_{n=1}^\infty f_n(x) dx.
\]
\hfill $\square$
\end{thm}
(See \cite{RudinGreenBook}, p. 26 for a proof of Theorem \ref{Dominated convergence theorem}, and
\cite{RudinGreenBook}, p. 29 for a proof of Theorem \ref{Application of dominated convergence})
\section{Big-O}
Very often we'd like to compare, in a quick-and-dirty way that avoids uncountably many details,
how fast two functions grow. We review here two of the most common ways to do this.
Suppose we are given two functions $f, g:\mathbb{R}^d \rightarrow \mathbb{C}$.
We say that $f(x) = O( g(x) )$ (pronounced ``Big o''), as $x \rightarrow x_0$,
if {\bf there exists a positive constant} $C$ such that
\begin{equation}
|f(x)| \leq C |g(x)|,
\end{equation}
for all $x$ that are sufficiently close to $x_0$. Here we allow $x_0$ to be any real vector,
and we also allow the very common case $x_0 = \pm \infty$.
Equivalently, we may say
\[
\left| \frac{f(x)}{g(x)} \right| \text{ is eventually bounded above}.
\]
\begin{example}
\rm{
We write
$e^x = 1 + x + \frac{1}{2} x^2 + O(x^3)$,
as $x \rightarrow 0$. We could, of course, also write $e^x -(1 + x + \frac{1}{2} x^2 ) = O(x^3)$, though the
former way of writing it is much more common.
In this case, we can give a `better' Big-O estimate by adding more terms of the Taylor series:
$e^x = 1 + x + \frac{1}{2} x^2 + \frac{1}{6} x^3 + O(x^4)$,
as $x \rightarrow 0$.
\hfill $\square$
}
\end{example}
\begin{example}
\rm{
Given $f(x) := x \sin \left(\frac{1}{x}
\right)$, and $g(x) := x^2 - 12$, we have
\[
f(x) = O( g(x) ), \text{ as } x \rightarrow \infty.
\]
In other words, for all sufficiently large $x$, $|f(x)| \leq C g(x)$, despite the fact that
this statement is false for these particular functions, for some small positive values of $x$.
\hfill $\square$
}
\end{example}
One of the useful properties of the Big-O notation is transitivity:
if $f = O(g)$, and $g = O(h)$, then $f= O(h)$.
\begin{proof}
For all $x$ sufficiently close to $x_0$, there exists positive constants $C_1, C_2$ such that
$|f(x)| \leq C_1 |g(x)|$ and $|g(x)| \leq C_2 |h(x)|$, implying that
\[
|f(x)| \leq C_1 |g(x)| \leq C_1 C_2 |h(x)|.
\]
\end{proof}
\bigskip
\section{little-o}
We are again given two functions $f, g:\mathbb{R}^d \rightarrow \mathbb{C}$.
We say that $f(x) = o( g(x) )$ (pronounced ``little o''), as $x \rightarrow x_0$,
if {\bf for all positive constants} $C$, we have:
\begin{equation}
|f(x)| \leq C | g(x) |,
\end{equation}
for all $x$ that are sufficiently close to $x_0$. Again
we allow $x_0$ to be any real vector, and we also allow the very common case $x_0 = \pm \infty$.
Equivalently, we may also write
\[
\lim_{x \rightarrow x_0} \left| \frac{ f(x) }{ g(x) } \right| = 0,
\]
which intuitively means that $g$ approaches $x_0$ faster than $f$ does.
\begin{comment}
\begin{example}
\rm{
Given $f(x) := 3^x e^{-x \log x}$, and $g(x) := 2^{-x} $, we have
\[
f(x) = o( g(x) ), \text{ as } x \rightarrow \infty.
\]
Proof. \ We have
\[
\lim_{x \rightarrow \infty} \left| \frac{ f(x) }{ g(x) } \right| =
\lim_{x \rightarrow \infty} \left| \frac{ 3^x }{ x^x } \right| 2^x = 0.
\]
So for large values of $x$, both functions are decreasing, but $g$ is decreasing much faster than $f$.
\hfill $\square$
}
\end{example}
\end{comment}
\begin{example}
\rm{
Given $f(x) := \sqrt x$, and $g(x) :=x$, where we restrict the domain of
both functions to be $(0, +\infty)$.
We claim
$f(x) = o( g(x) ), \text{ as } x \rightarrow 0$.
\begin{proof}
\[
\lim_{x \rightarrow 0} \left| \frac{ f(x) }{ g(x) } \right| =
\lim_{x \rightarrow 0} \left| \frac{ \sqrt x }{ x } \right| =
\lim_{x \rightarrow 0} \left| \frac{ 1}{ \sqrt x } \right|
= 0.
\]
So $g$ approaches $0$ much faster than $f$.
\end{proof}
}
\end{example}
\medskip
Here we have transitivity as well: if $f = o(g)$ and $g=o(h)$, then $f=o(h)$.
\begin{proof}
The two given limits
$\lim_{x \rightarrow x_0} \left| \frac{ f(x) }{ g(x) } \right| = 0$ and
$\lim_{x \rightarrow x_0} \left| \frac{ g(x) }{ h(x) } \right| = 0$ together imply that
\[
\lim_{x \rightarrow x_0} \left| \frac{ f(x) }{ h(x) } \right|
=\lim_{x \rightarrow x_0} \left| \frac{ f(x) }{ g(x) } \right| \left| \frac{ g(x) }{ h(x) } \right|
= 0.
\]
\end{proof}
\bigskip
\chapter{Appendix B: Various forms of convergence}
\label{Appendix B}
\section{Weierstrass M-test}
How can we quickly conclude that certain series converge uniformly? The following criterion, discovered by Karl Weierstrass, comes to the rescue.
\begin{thm} \rm{[Weierstrass M-test]}
Suppose that $f_n(x)$ is a sequence of complex-valued functions defined on a set $E\subset \mathbb{R}$,
such that there exists a sequence of numbers $M_n\geq 0$ satisfying the following conditions:
\begin{enumerate}
\item $ |f_{n}(x)| \leq M_{n}, \ \forall n \in \mathbb{Z} \text{ and all } x \in E$.
\item $\sum_{n \in \mathbb{Z}} M_n < \infty$.
\end{enumerate}
Then the series
$\sum _{n\in \mathbb{Z}} f_n(x) $
converges absolutely and uniformly on $E$.
\hfill $\square$
\end{thm}
In practice, the Weierstrass $M$-test gets used together with the following test, which allows us to partially answer the question:
\begin{question}
When does a series $\sum_{n\in \mathbb{Z}} f_n(x)$ converge to a continuous function of $x$?
\end{question}
\begin{thm} \label{uniform limit test} \rm{[Uniform limit]}
Suppose that $s_n(x):E \rightarrow \mathbb{C}$ is a sequence of continuous functions defined on a set $E\subset \mathbb{R}$,
and that $s_n$ converges uniformly to $s(x)$, on $E$. Then $s(x)$ is continuous on $E$.
\hfill $\square$
\end{thm}
\bigskip
\section{Some things you wanted to know about convergence but were afraid to ask}
It's often useful to pass from $L^2$ convergence to pointwise convergence, under some additional hypothesis on $f$. Throughout, we fix a real number $1\leq p < \infty$. Given a subset $E\subset \mathbb{R}^d$, and a sequence of functions $f_n:E \rightarrow \mathbb{C}$, we say that
$f_n(x) \rightarrow f(x)$ in the $p$-norm} if
\begin{equation} \label{convergence in L^p norm, take 2}
\lim_{n \rightarrow \infty}
\int_{E} \left| f_n(x) - f(x) \right |^p dx =0,
\end{equation}
for which we will also use here the notation $\lim_{n\rightarrow \infty} \| f_n - f \|_{L^p(E)} = 0$.
Sometimes, if the constant $p$ is not specified, it is common to simply call \eqref{convergence in L^p norm, take 2}
{\bf convergence in norm}.
The two most common subsets are $E:= \mathbb{R}^d$, and $E:= [0, 1]^d$.
A natural question arises:
\begin{question}
When can we pass from convergence in norm to pointwise convergence?
\end{question}
Given a series $\sum_{n \in \mathbb{Z}} f_n(x)$, we consider the sequence of partial sums
$S_N(x):= \sum_{|n| < N} f_n(x)$.
By definition, we say the series converges
\begin{enumerate}
\item {\bf pointwise on $E$} if the sequence $\{S_N(x)\}_{N=1}^\infty$ converges, for each $x \in E$.
\item {\bf absolutely on $E$} if the series $\sum_{n \in \mathbb{Z}} |f_n(x)|$ converges pointwise,
for each $x \in E$.
\item {\bf uniformly on $E$} if the sequence of partial sums $S_N(x)$ converge uniformly on $E$.
\item {\bf in the $p$-norm} on $E$ if $\lim_{n\rightarrow \infty} \| f_n - f \|_{L^p(E)} = 0$.\end{enumerate}
\medskip
\begin{lem} \label{technical equality a.e.}
Consider the partial sums
\[
S_N(x):= \sum_{|n| < N \atop n \in \mathbb{Z}^d} f_n(x),
\]
for all $x$ in a given subset $E\subset \mathbb{R}^d$. Suppose we have the following two properties:
\begin{enumerate}
\item There exists a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ such that $S_N(x) \rightarrow f(x)$ in the $p$-norm, on $E$.
\item $S_N(x)$ converges uniformly to the series $S(x):= \sum_{n\in \mathbb{Z}^d} f_n(x)$ on $E$.
\end{enumerate}
Then:
\[
S(x) = f(x) \text{ a.e. on } E.
\]
\hfill $\square$
\end{lem}
\begin{lem} \label{absolute convergence of Fourier series implies continuity}
Let $f\in L^1([-c,c])$, and suppose we already know that its Fourier series converges pointwise:
\begin{equation} \label{Fourier series converging pointwise}
f(x) = \sum_{n\in \mathbb{Z}} \hat f_n e^{\frac{2\pi i n x}{c}},
\end{equation}
If the series \eqref{Fourier series converging pointwise}
converges absolutely, then $f$ is a continuous function on $[-c,c]$, and $f(-c) = f(c)$.
\end{lem}
\begin{proof}
The idea is to use the uniform limit Theorem \ref{uniform limit test}, together with the fact that the summands $\hat f_n e^{\frac{2\pi i n x}{c}}$ are continuous functions of $x$. So it remains to show that the convergence of the series is uniform.
\[
|S_N(x)| :=
\left| \sum_{|n|<N} \hat f_n e^{\frac{2\pi i n x}{c}} \right|
\leq \sum_{|n|<N} \left| \hat f_n e^{\frac{2\pi i n x}{c}} \right|
= \sum_{|n|<N} | \hat f_n | < \infty,
\]
where the penultimate equality holds because $| e^{\frac{2\pi i n x}{c}} |=1$, and the last inequality holds by assumption.
So by the $M$-test, with $M_n := | \hat f_n |$, we have uniform convergence of the series.
Finally, the claim $f(-c) = f(c)$ is trivial, because $f(\pm c) := \sum_{n\in \mathbb{Z}} \hat f_n e^{\pm 2\pi i n} = \sum_{n\in \mathbb{Z}} \hat f_n$.
\end{proof}
In the previous lemma, we could have also used the alternate notation of the circle $ \mathbb{R}/c\mathbb{Z}$, and rewrite everything
in terms of it, which automatically incorporates periodicity.
The following passage from convergence in the $ L^2([-c,c])$ norm, to pointwise convergence, is often useful.
\begin{lem} \label{norm convergence plus absolute convergence implies equality}
Let $f\in L^2([-c,c])$ be a continuous function, and write its Fourier series as
\begin{equation} \label{L^2 convergent series}
f(x) \underset{L^2([-c,c])}{=}
\sum_{n\in \mathbb{Z}} \hat f_n e^{\frac{2\pi i n x}{c}},
\end{equation}
which by definition means that this series converges in the $L^2([-c,c])$-norm. If the series \eqref{L^2 convergent series}
converges absolutely, then it also converges pointwise and uniformly to $f(x)$, for all $x \in [-c,c]$.
\end{lem}
\begin{proof}
Repeating the computation of the previous proof, we have:
\[
|S_N(x)| :=
\left| \sum_{|n|<N} \hat f_n e^{\frac{2\pi i n x}{c}} \right|
\leq \sum_{|n|<N} \left| \hat f_n e^{\frac{2\pi i n x}{c}} \right|
\leq \sum_{|n|<N} | \hat f_n | < \infty,
\]
Therefore by the $M$-test again, the sequence $S_N(x)$ converges
uniformly to the series $S(x):= \sum_{n\in \mathbb{Z}} \hat f_n e^{\frac{2\pi i n x}{c}}$, for all $x \in [-c, c]$.
We also know, by Lemma \ref{absolute convergence of Fourier series implies continuity},
that $S(x)$ is continuous on $[-c, c]$.
We still need to prove that the series converges to $f$, but now we at least know that both hypotheses of Lemma \ref{technical equality a.e.} are satisfied (with $p=2$ and $E:= [-c, c]$),
and therefore $S(x) = f(x) \text{ a.e. on } E$.
To prove that $S(x) = f(x)$ for all $x \in [-c, c]$, we observe that the summands $\hat f_n e^{\frac{2\pi i n x}{c}}$ are continuous functions of $x$, and hence by the uniform limit theorem (Theorem \ref{uniform limit test}), the series $S(x)$ is itself a continuous function of $x$. Since $f$ is also continuous on $[-c, c]$, and $S(x) = f(x)$ almost everywhere, they must agree everywhere.
\end{proof}
\begin{comment}
\begin{lem}
Suppose a continuous function $F:\mathbb{R}^d\rightarrow \mathbb{C}$ is represented by a series
\begin{equation}
F(x) \underset{L^2(\mathbb{R}^d)}{=}
\sum_{n\in \mathbb{Z}} a_n(x),
\end{equation}
where the convergence is in $L^2(\mathbb{R}^d)$-norm, and where each $a_n(x)$ is continuous on $\mathbb{R}^d$. If the series
converges absolutely, then it also converges pointwise and uniformly to $F(x)$, for all $x \in \mathbb{R}^d$.
\end{lem}
\end{comment}
\section{Bump functions}
Perhaps the easiest bump function to define is the function (see \cite{SteinShakarchi}, page 209):
\begin{equation*}
\varphi(x):=
\begin{cases}
ce^{-\frac{1}{1-\|x\|^2}} & \text{ if } \|x\|<1, \\
0 & \text{ if } \|x\| \geq1. \\
\end{cases}
\end{equation*}
By definition, $\varphi$ is compactly supported, on the unit ball. Here the constant $c$ is chosen so that
$\int_{\mathbb{R}^d} \varphi(x) dx = 1$. It turns out that $\varphi$ is infinitely smooth.
As usual, using $\varphi$ we can build a family of integrable functions:
\begin{equation*}
\varphi_{\varepsilon}(x):=\varepsilon^{-d}\varphi(x\varepsilon^{-1}), \text{ for all } 0<\varepsilon\leq1.
\end{equation*}
Thus, the family $\{\varphi_\varepsilon\}$ is an approximate identity.
More generally, a {\bf bump function} is defined to be any infinitely smooth function $\varphi:\mathbb{R}^d\rightarrow \mathbb{C}$ that is compactly supported.
By Lemma \ref{useful Schwartz fact}, we know that any such bump function
$\varphi$ lies in the Schwartz class $S(\mathbb{R}^d)$. Clearly finite linear combinations of bump functions are again bump functions, making the space of bump functions a vector subspace of the space
of Schwartz functions.
\chapter{Solutions and hints}
\begin{quote}
``There are no problems, just pauses between ideas.''
-- David Morrell, Brotherhood of the Rose
\end{quote}
{\bf \Large Chapter \ref{Chapter.Tiling.A.Rectangle}}
\bigskip
\index{tiling}
\bigskip
Exercise \ref{TrivialExponential}
\quad
By Euler, we have $1 = e^{i \theta} = \cos\theta + i\sin\theta$, which holds
if and only if
$\cos\theta = 1$, and $\sin\theta = 0$. The latter two conditions hold simultaneously if and only if
$\theta \in 2 \pi k$, with
$k \in \mathbb{Z}$.
\medskip
Exercise \ref{bound of the exponential function}
\quad Let $z := a + bi$, so that $|e^z| = |e^{a+bi}| =|e^a| | e^{bi} | = e^a \cdot 1
\leq e^{\sqrt{ a^2 + b^2}} = e^{|z|}$.
\bigskip
Exercise \ref{orthogonality for exponentials}
\quad
In case $a \not= b$, we have
\begin{equation*}
\int_0^1 e_a(x) \overline{e_b(x)} dx = \int_0^1 e^{2\pi i (a-b) x} dx
= \frac{e^{2\pi i (a-b)}}{2\pi i(a-b)} - 1=0,
\end{equation*}
because we know that $a-b \in \mathbb{Z}$. In case $a = b$, we have
\begin{equation*}
\int_0^1 e_a(x) \overline{e_a(x)} dx = \int_0^1 dx = 1.
\end{equation*}
\bigskip
Exercise \ref{definition of complex integral} \
\quad By definition,
\begin{align*}
\int_{[0,1]} e^{-2\pi i \xi x} dx &:= \int_{[0,1]} \cos(2\pi \xi x) dx + i \int_{[0,1]} \sin(2\pi \xi x) dx\\
&= \frac{\sin(2\pi \xi)}{2\pi\xi} +i \frac{-\cos(2\pi \xi)+1}{2\pi\xi}\\
&= \frac{i\sin(2\pi \xi)}{2\pi i \xi} + \frac{\cos(2\pi \xi)-1}{2\pi i \xi}\\
&= \frac{e^{2\pi i \xi}-1}{2\pi i \xi}.
\end{align*}
\bigskip
Exercise \ref{SumOfRootsOfUnity} \
\quad Let $S:= \sum_{k = 0}^{N-1} e^{\frac{2\pi i k}{N}}$, and note that we may write
\[
S = \sum_{k\text{ mod } N} e^{\frac{2\pi i k}{N}}.
\]
Now, pick any $m$ such that $e^{\frac{2\pi i m}{N}} \not=1$. Consider
\begin{align*}
e^{\frac{2\pi i m}{N}} S &= \sum_{k\text{ mod } N} e^{\frac{2\pi i (k + m)}{N}} \\
&= \sum_{n\text{ mod } N} e^{\frac{2\pi i n}{N}} = S,
\end{align*}
so that $0=( e^{\frac{2\pi i m}{N}} -1)S$, and since
by assumption $e^{\frac{2\pi i m}{N}} \not=1$, we have $S = 0$.
\bigskip
Exercise \ref{DivisibilityUsingExponentials}
\quad
We use the finite geometric series:
$1+x+ x^2 + \cdots + x^{N-1} = \frac{x^{N} - 1}{x-1}$. Now, if $N \not{\mid} M$, then $x:= e^{\frac{2\pi i M}{N}} \not=1$, so we may substitute this value of $x$ into the finite geometric series to get:
\begin{align*}
\frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i kM}{N}} &=
\frac{ e^{\frac{2\pi i MN}{N}}- 1}{ e^{\frac{2\pi i M}{N}}-1} \\
&= \frac{0}{e^{\frac{2\pi i M}{N}}-1}=0.
\end{align*}
On the other hand, if $N \mid M$, then $ \frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i kM}{N}} =
\frac{1}{N} \sum_{k = 0}^{N-1} 1 = 1$.
\bigskip
Exercise \ref{Orthogonality.for.roots.of.unity} \
\begin{align*}
\frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i ka}{N}} e^{-\frac{2\pi i kb}{N}}
&= \frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i k(a-b)}{N}}. \\
\end{align*}
Therefore, using Exercise \ref{DivisibilityUsingExponentials}, we see that the latter sum equals
$1$ exactly when $N \mid a-b$, and vanishes otherwise.
\bigskip
Exercise \ref{trick-write an integer as a product with roots of unity} \
\quad We begin with the factorization of the polynomial
$x^n-1= \prod_{k=1}^n (x - \zeta^k)$, with $\zeta:= e^{2\pi i /n}$. Dividing both sides by
$x-1$, we obtain $1+ x + x^2 + \cdots + x^{n-1} = \prod_{k=1}^{n-1} (x - \zeta^k)$. Now
substituting $x=1$, we have $n = \prod_{k=1}^{n-1} (1 - \zeta^k)$.
\bigskip
Exercise \ref{PrimitiveRootsOfUnity} \
\quad Suppose to the contrary, that a primitive $N$'th root of unity is of the form $e^{2\pi i m/N}$, where
$\gcd(m,N) > 1$. Let $m_1 := \frac{m}{\gcd(m, N)}$, and $k:=\frac{N}{\gcd(m, N)}$, so that by assumption both
$m_1$ and $k$ are integers. Thus $e^{2\pi i m/N} = e^{2\pi i m_1/k}$, a $k$'th root of unity, with $k < N$, a contradiction.
\bigskip
Exercise \ref{zeros of the sin function} \
\quad We recall Euler's identity:
\[
e^{iw} = \cos w + i \sin w,
\]
which is valid for all $w \in \mathbb C$. Using Euler's identity first with $w:= \pi z$, and then with $w := -\pi z$, we have the two identities $e^{\pi i z} = \cos \pi z + i \sin \pi z$, and $e^{-\pi i z} = \cos \pi z - i \sin \pi z$. Subtracting the second identity from the first, we have
\[
\sin(\pi z) = \frac{1}{2i}\left( e^{\pi i z} - e^{-\pi i z} \right).
\]
Now it's clear that $\sin(\pi z) = 0 \iff e^{\pi i z} = e^{-\pi i z} \iff e^{2\pi i z} = 1 \iff z \in \mathbb{Z}$, by Exercise~\ref{TrivialExponential}.
\bigskip \medskip
Exercise \ref{Erdos lattice partition problem} \
\quad We will assume, to the contrary,
that we only have one arithmetic progression with a common difference of $a_N$, the largest of the common differences.
We hope to obtain a contradiction. To each arithmetic progression $\{ a_k n + b_k \mid n \in \mathbb{Z}\}$, we associate the generating function
\[
f_k(q):= \sum_{a_k n + b_k \geq 0, \ n\in \mathbb{Z}} q^{a_k n + b_k },
\]
where $|q| < 1$, in order to make the series converge. The hypothesis that we have a tiling
\index{tiling}
of the integers by these $N$
arithmetic progressions translates directly into an identity among these generating functions:
\[
\sum_{a_1 n + b_1 \geq 0, \ n\in \mathbb{Z}} q^{a_1 n + b_1 } + \cdots +
\sum_{a_N n + b_N \geq 0, \ n\in \mathbb{Z}} q^{a_N n + b_N } = \sum_{n=0}^\infty q^n.
\]
Next, we use the fact that we may rewrite each generating function in a `closed form' of the following kind, because they are geometric series:
$f_k(q):= \sum_{a_k n + b_k \geq 0, \ n\in \mathbb{Z}} q^{a_k n + b_k } = \frac{q^{b_k}}{1-q^{a_k}}$. Thus, we have:
\[
\frac{q^{b_1}}{1-q^{a_1}} + \cdots + \frac{q^{b_N}}{1-q^{a_N}} = \frac{1}{1-q}.
\]
Now we make a `pole-analysis' by observing that each rational function $f_k(q)$ has poles at precisely all of the $k$'th roots of unity.
The final idea is that the `deepest' pole, namely $e^{ \frac{2\pi i}{N} }$, cannot cancel with any of the other poles. To make this idea precise,
we isolate the only rational function that has this pole (by assumption):
\[
\frac{q^{b_N}}{1-q^{a_N}} = \frac{1}{1-q} - \left( \frac{q^{b_1}}{1-q^{a_1}} + \cdots + \frac{q^{b_{N-1}}}{1-q^{a_{N-1}}} \right).
\]
Finally, we let $q\rightarrow e^{ \frac{2\pi i}{N} }$, to get a finite number on the right-hand-side,
and infinity on the left-hand-side of the latter identity, a contradiction.
\bigskip \bigskip
{\bf \Large Chapter \ref{Chapter.Examples}}
\medskip
Exercise \ref{transform.of.interval.a.to.b} \
\quad If $\xi = 0$, we have $\hat 1_{[a,b]}(0) := \int_a^b e^0 dx = b-a$.
If $\xi \not=0$, we can compute the integral:
\begin{align*}
\hat 1_{[a,b]}(\xi) &:= \int_a^b e^{-2\pi i \xi x} dx \\
&=\frac{e^{-2\pi i \xi b} - e^{-2\pi i \xi a} }{-2\pi i \xi}.
\end{align*}
\medskip
Exercise \ref{transform.of.unit.cube} \
\quad Beginning with the definition of the Fourier transform of the unit cube $[0,1]^d$, we have:
\begin{align*}
\hat 1_{\square}(\xi) &= \int_{\square} e^{2\pi i \langle x, \xi \rangle}dx \\
&= \int_0^1 e^{2\pi i \xi_1 x_1} dx_1 \int_0^1 e^{2\pi i \xi_2 x_2} dx_2 \cdots
\int_0^1 e^{2\pi i \xi_d x_d} dx_d \\
&= \frac{1}{(-2\pi i)^d} \prod_{k=1}^d \frac{ e^{-2\pi i \xi_k} -1 }{ \xi_k },
\end{align*}
valid for all $\xi \in \mathbb{R}^d$, except for the finite union of hyperplanes defined by \\
$H := \{ x \in \mathbb{R}^d \mid \xi_1 = 0 \text{ or } \xi_2 = 0 \dots \text{ or } \xi_d = 0 \}$.
\medskip
Exercise \ref{brute force Bernoulli polys} \
\quad To see that the generating-function definition of the Bernoulli polynomials in fact gives polynomials, we first write the Taylor series of the following two analytic functions:
\[
\frac{t}{e^t - 1} = \sum_{k=0}^\infty \frac{B_k}{k!} t^k
\]
\[
e^{xt} = \sum_{j=0}^\infty \frac{ x^j t^j}{j!}.
\]
Multiplying these series together by brute-force gives us:
\begin{align}
\frac{t}{e^t - 1} e^{xt} &= \left( \sum_{k=0}^\infty \frac{B_k}{k!} t^k \right)
\left( \sum_{j=0}^\infty \frac{ x^j}{j!} t^j \right) \\
&= \sum_{n=0}^\infty \left( \sum_{j+k = n} \frac{B_k}{k!} \frac{ x^j}{j!} \right) t^n \\
&= \sum_{n=0}^\infty \left( \sum_{k = 0}^n \frac{B_k}{k!} \frac{ x^{n-k}}{(n-k)!} \right) t^n.
\end{align}
The coefficient of $t^n$ on the LHS is by definition $\frac{1}{n!} B_n(x)$, and by uniqueness of Taylor series, this must also be the coefficient on the RHS, which is seen here to be a polynomial in $x$. In fact, we see more, namely that
\[
\frac{1}{n!} B_n(x) = \sum_{k = 0}^n \frac{B_k}{k!} \frac{ x^{n-k}}{(n-k)!},
\]
which can be written more cleanly as
$ B_n(x) = \sum_{k = 0}^n {n\choose k} B_k x^{n-k}$.
\medskip
Exercise \ref{Reflection property for B_n(x)} \
\quad Commencing with the generating-function definition of the Bernoulli polynomials,
equation
\ref{generating function for Bernoulli polynomials},
we replace $x$ with $1-x$ in order to observe the coefficients $B_k(1-x)$:
\begin{align*}
\sum_{k=0}^\infty \frac{B_k(1-x)}{k!} t^k &= \frac{te^{t(1-x)}}{e^t - 1} \\
&= \frac{te^t e^{-tx}}{e^t - 1} \\
&= \frac{t e^{-tx}}{1- e^{-t} } \\
&= \frac{-t e^{-tx}}{e^{-t}-1 } \\
&= \sum_{k=0}^\infty \frac{B_k(x)}{k!} (-t)^k,
\end{align*}
where the last equality follows from the definition of the same generating function, namely equation
\ref{generating function for Bernoulli polynomials}, but with the variable $t$ replaced by $-t$.
Comparing the coefficient of $t^k$ on both sides, we have $B_k(1-x) = (-1)^k B_k(x)$.
\medskip
Exercise \ref{difference of Bernoulli polys} \
To show that
$B_n(x+1) - B_n(x) = n x^{n-1}$, we play with:
\begin{align*}
\sum_{k=0}^\infty \left(\frac{B_k(x+1)}{k!} t^k - \frac{B_k(x)}{k!} t^k \right) &=
\frac{te^{t(x+1)}}{e^t - 1} - \frac{te^{t(x)}}{e^t - 1} \\
&= e^t \frac{te^{tx}}{e^t - 1} - \frac{te^{t(x)}}{e^t - 1} \\
&= (e^t - 1) \frac{te^{tx}}{e^t - 1} \\
&= te^{tx} \\
&= \sum_{k=0}^\infty \frac{x^k}{k!} t^{k+1} \\
&= \sum_{k=1}^\infty \frac{x^{k-1}}{(k-1)!} t^{k} \\
&= \sum_{k=1}^\infty \frac{k x^{k-1}}{k!} t^{k}.
\end{align*}
Therefore, again comparing the coefficients of $t^k$ on both sides, we arrive at the required identity.
\medskip
Exercise \ref{derivative of Bernoulli polys} \
\quad We need to show that $\frac{d}{dx} B_n(x) = n B_{n-1}(x)$. Well,
\begin{align*}
\sum_{k=0}^\infty \frac{d}{dx} \frac{B_k(x)}{k!} t^k
&= \frac{d}{dx} \frac{te^{tx}}{e^t - 1} \\
&= t \sum_{k=0}^\infty \frac{B_k(x)}{k!} t^k \\
&= \sum_{k=0}^\infty \frac{B_k(x)}{k!} t^{k+1} \\
&= \sum_{k=1}^\infty \frac{B_{k-1}(x)}{(k-1)!} t^{k} \\
&= \sum_{k=1}^\infty k \frac{B_{k-1}(x)}{k!} t^{k},
\end{align*}
so that comparing the coefficient of $t^n$ on both sides, the proof is complete.
\medskip
Exercise \ref{Dirichlet's convergence test}
\quad Considering the partial sum $S_n:= \sum_{k=1}^n a_k b_k$, we know by Abel summation that
\[
S_n = a_n B_n + \sum_{k=1}^{n-1} B_k(a_k - a_{k+1}),
\]
for each $n \geq 2$, where $B_n := \sum_{k=1}^n b_k$.
By assumption, $|B_n|:= | \sum_{k=1}^n b_k | \leq M$, and the $a_k$'s are going to $0$,
so we see that the first part of the right-hand-side approaches
zero, namely:
$|a_n B_n| := |a_n| | \sum_{k+1}^n b_k| \rightarrow 0$, as $n \rightarrow \infty$.
Next, we have
\[
| \sum_{k=1}^{n-1} B_k(a_k - a_{k+1}) | \leq \sum_{k=1}^{n-1} | B_k| | a_k - a_{k+1} |
\leq M \sum_{k=1}^{n-1} | a_k - a_{k+1} | = M \sum_{k=1}^{n-1} (a_k - a_{k+1}),
\]
where the last equality holds because by assumption the $a_k$'s are decreasing. But the last finite
sum equals $ -M a_{n} + M a_1$, and we have $ \lim_{n\rightarrow \infty} (-M a_{n} +M a_1)= M a_1$, a finite limit.
Therefore $\sum_{k=1}^{n-1} B_k(a_k - a_{k+1})$ converges absolutely, and so $S_n$ converges, as desired.
\medskip
Exercise \ref{exponential sum bound}
\quad We fix $x \in \mathbb{R}-\mathbb{Z}$, and let $z:= e^{2\pi i x}$, which lies on the unit circle, and by assumption $z \not= 1$.
Then
\begin{equation}
\left | \sum_{k= 1}^n e^{2\pi i k x} \right |
= \left | \sum_{k= 1}^n z^k \right | = \left | \frac{z^{n+1} -1}{z-1} \right | \leq \frac{2}{z-1},
\end{equation}
because $|z^{n+1} -1| \leq |z^{n+1}| +1 = 2$. We also have
\begin{align*}
|z-1|^2 &= |e^{2\pi ix}-1||e^{-2\pi ix}-1| = |2-2\cos(2\pi x)| = 4\sin^2(\pi x),
\end{align*}
so that we have the equality
$\left | \frac{2}{z-1} \right | =\left | \frac{1}{\sin(\pi x)} \right |$.
Altogether, we see that
\begin{equation}
\left | \sum_{k= 1}^n e^{2\pi i k x} \right | \leq \frac{1}{ | \sin(\pi x) | }.
\end{equation}
\medskip
Exercise \ref{rigorous convergence of P_1(x)}
\quad We fix $a \in \mathbb{R} - \mathbb{Z}$ and need to prove that $\sum_{m = 1}^\infty \frac{e^{2\pi i m a}}{m}$ converges.
Abel's summation formula \eqref{actual Abel summation} gives us
\[
\sum_{k = 1}^n \frac{e^{2\pi i k a}}{k} = \frac{1}{n}\sum_{r=1}^n e^{2\pi i r a}
+ \sum_{k=1}^{n-1} \Big( \sum_{r=1}^k e^{2\pi i r a} \Big) \frac{1}{k(k+1)},
\]
so that
\[
\sum_{k = 1}^\infty \frac{e^{2\pi i k a}}{k} =
\sum_{k=1}^{\infty} \Big( \sum_{r=1}^k e^{2\pi i r a} \Big) \frac{1}{k(k+1)}.
\]
and the latter series in fact converges absolutely.
\begin{comment}
\medskip
Exercise \ref{rigorous limit formula for sinc}
\quad
The main idea here is to transform everything into exponentials, which we can then explicitly integrate.
For added simplicity, we initially omit the $\pi$'s everywhere. To prove the first part, we compute the integral:
\[
\int_0^\infty e^{-xt} dt = -\frac{1}{x} e^{-xt}\Big|_0^\infty = \frac{1}{x},
\]
valid for all $x >0$.
For the second part, it suffices to prove $\int_0^\infty \frac{\sin x}{ x} dx = \frac{\pi}{2}$.
We will substitute for $\frac{1}{x}$ using part (a), and
then we use Fubini's theorem:
\begin{align*}
\int_{0}^\infty \frac{\sin x}{ x} dx
&= \frac{1}{\pi} \int_{0}^\infty \sin x
\Big(\int_0^\infty e^{-xt} dt \Big) dx \\
&=
\int_0^\infty \int_{0}^\infty \sin x \, e^{-xt} \, dx \, dt.
\end{align*}
To justify the use of Fubini's theorem, we need to check that $ \sin x \, e^{-xt}\in L^1(\mathbb{R}^2)$, which
doesn't seem correct?!?!?
Now we can give an explicit formula for the inner integral, using exponentials. The computation simplifies
if we now let $N\in 2\pi\mathbb{Z}$, and we integrate from $0$ to $N$:
\begin{align*}
\int_0^N \sin x \, e^{-xt} \, dx
&= \frac{1}{2i} \int_0^N \left( e^{ i x} - e^{- i x} \right) e^{-xt} dx \\
&= \frac{1}{2i} \int_0^N \left( e^{-x( t - i ) } - e^{-x( t + i ) } \right) dx \\
&= \frac{1}{2i} \left( \frac{ e^{-x( t - i ) } }{ i -t }
+ \frac{ e^{-x( t + i ) } }{ i + t }
\right) \Big|_{x=0}^{x=N} \\
&= \frac{1}{2i} \left( \frac{ e^{-Nt } }{ i -t }
+ \frac{ e^{-Nt } }{ i + t }
\right)
-
\frac{1}{2i} \left( \frac{ 1 }{ i -t }
+ \frac{ 1 }{ i + t }
\right) \\
&= \frac{ 1- e^{-Nt } }{1+t^2}.
\end{align*}
Letting $N\rightarrow \infty$, we get $\int_0^\infty \sin x \, e^{-xt} \, dx = \frac{1}{1+t^2}.$
Altogether, we have
\[
\int_0^\infty \frac{\sin x}{ x} dx = \int_0^\infty \int_{0}^\infty \sin x \, e^{-xt} \, dx \, dt =
\int_0^\infty \frac{ dt }{1+t^2} = \tan^{-1} (t) \Big|_0^\infty = \frac{\pi}{2}.
\]
\end{comment}
\bigskip \bigskip
{\bf \Large Chapter \ref{Fourier analysis basics}}
\medskip
Exercise \ref{elementary norm relations}
\quad For all four inequalities, we will use an arbitrary vector $a \in \mathbb{R}^d$.
For the first inequality, $a_1^2 + \cdots + a_d^2 \geq \max\{ |a_1|, \dots, |a_d| \}^2 := \|a\|_\infty^2$.
The second inequality $\|a\|_2 \leq \|a\|_1$ means that $\sqrt{a_1^2 + \cdots + a_d^2} \leq |a_1| + \cdots + |a_d|$, which is clear by squaring both sides.
To prove the third and most interesting inequality here, we use the Cauchy-Schwarz inequality, with the two vectors
$x:= (a_1, \dots, a_d)$ and $(1, 1, \dots, 1)$:
\[
\| a \|_1:= |a_1| \cdot 1 + \cdots + |a_d| \cdot 1 \leq \sqrt{ a_1^2 + \cdots + a_d^2} \sqrt{ 1 + \cdots + 1 }
= \sqrt{d} \ \| a \|_2,
\]
which also shows that we obtain equality if and only if $(a_1, \dots, a_d)$ is a
scalar multiple of $(1, 1, \dots, 1)$.
For the fourth inequality, we have:
\[
\sqrt{a_1^2 + \cdots + a_d^2} \leq
\sqrt{ d \max\{ |a_1|, \dots, |a_d| \}^2 }:=
\sqrt{d} \|a\|_\infty.
\]
\medskip
Exercise \ref{exercise:hyperbolic cosine and sine}
\quad
To prove part (a), we compute:
\begin{align*}
\left( \frac{e^{t} + e^{-t}}{2} \right)^2 - \left( \frac{e^{t} - e^{-t}}{2} \right)^2
&= \frac{e^{2t} + 2 + e^{-2t} - \left( e^{2t} - 2 + e^{-2t} \right) }{4} = 1.
\end{align*}
To prove part (b), we begin with the definition of the hyperbolic cotangent:
\begin{align*}
t \coth t &= t\frac{ e^t + e^{-t}}{e^t - e^{-t}} = t \frac{ e^t }{e^t - e^{-t}} + t \frac{ e^{-t}}{e^t - e^{-t}} \\
&= \frac{ t }{1 - e^{-2t}} + \frac{ t }{ e^{2t}-1}.
\end{align*}
Recalling the definition of the Bernoulli numbers, namely
$
\frac{t}{e^t-1} = \sum_{k =0}^\infty B_k \frac{t^k}{k!},
$
we see that
\begin{align*}
t \coth t &= \frac{1}{2} \left( \frac{ -2t }{ e^{-2t}-1} \right) + \frac{1}{2} \left( \frac{ 2t }{ e^{2t}-1} \right) \\
&= \frac{1}{2} \sum_{k =0}^\infty B_k \frac{(-2t)^k}{k!} + \frac{1}{2} \sum_{k =0}^\infty B_k \frac{(2t)^k}{k!} \\
&= \sum_{k =0}^\infty \tfrac{1}{2} \left( (-1)^k + 1 \right) B_k \frac{(2t)^k}{k!},
\end{align*}
so the only surviving terms in the latter series are the terms whose index $k$ is
an even integer. This yields
$
t \coth t = \sum_{n=0}^\infty \frac{2^{2n}}{(2n)!} B_{2n} t^{2n}.
$
\medskip
Exercise \ref{compute FT for exponential of abs value}
\quad
We know, by equation \eqref{FT of the abs value exponential}, that the Fourier transform of
$f(x):=e^{-2\pi t |x|}$ is equal to $\hat f(\xi) = \frac{ t }{\pi (t^2 + \xi^2)}$.
So using Poisson summation, we have:
\index{Poisson summation formula}
\begin{align*}
\sum_{n \in \mathbb{Z}} e^{-2\pi t |n|} = \sum_{n \in \mathbb{Z}} f(n)
= \sum_{\xi \in \mathbb{Z}}\hat f(\xi)
= \frac{t}{\pi} \sum_{\xi \in \mathbb{Z}} \frac{1}{\xi^2 + t^2}.
\end{align*}
\medskip
Exercise \ref{Elementary bounds for sin(x), sinc(x)}
We'll prove part \ref{Elementary trig bounds, part b}. To begin, we have:
\begin{align*}
\big| e^{i\theta} - 1 \big|^2 &= \big| \cos \theta -1 + i \sin \theta \big|^2 = (\cos \theta -1)^2 + \sin^2\theta \\
&= 2 - 2 \cos \theta = 4 \sin^2\left(\frac{\theta}{2}\right).
\end{align*}
So it suffices to show that
$4 \sin^2\left(\frac{\theta}{2}\right) \leq \theta^2$, for all $0 \leq \theta \leq 2\pi$.
In other words, the problem is reduced to the Calculus I problem of showing that
$ \sin\left(\frac{\theta}{2}\right) \leq \frac{\theta}{2}$, for $\theta \in [0, 2\pi]$.
To prove this, we let $y(x)=x- \sin x $, so that it suffices to prove that $y \geq 0$ on $[0, \pi]$.
Computing its derivative,
$y'(x) = 1-\cos x \geq 0$ on $[0, \pi]$, and since $y(0) = 0$, we conclude that $y$ is an increasing function. This proves $y \geq 0$ on $[0, \pi]$.
\medskip
Exercise \ref{positive FT over R}
We need to show that there exist two real numbers $r, s$ such that
\[
f:= 1_{[-r, r]}*1_{[-r, r]} + 1_{[-s, s]}*1_{[-s, s]}
\]
enjoys the property:
\[
\hat f(\xi) >0,
\]
for all $\xi \in \mathbb{R}$. Let's pick any two real numbers $r, s$ that are incommensurable, meaning that
$\frac{r}{s} \notin \mathbb{Q}$. Using \eqref{Stretch lemma for the sinc function}, we compute $\hat f$:
\[
\hat f(\xi):= \Big( \hat 1_{[-r, r]}(\xi) \Big)^2 + \Big( \hat 1_{[-s, s]}(\xi) \Big)^2 =
\left( \frac{ \sin(2r\pi \xi) }{ \pi \xi } \right)^2 + \left( \frac{ \sin(2s\pi \xi) }{ \pi \xi } \right)^2 \geq 0.
\]
To prove positivity, suppose to the contrary that there exists a nonzero $\xi\in \mathbb{R}$ such that $\hat f(\xi)=0$. Then
$\left( \sin(2r\pi \xi) \right)^2 + \left( \sin(2s\pi \xi) \right)^2 = 0$, but the vanishing of a sum of two squares implies that
they must both equal $0$:
\[
\sin(2r\pi \xi) =0, \text{ and } \sin(2s\pi \xi) = 0.
\]
Therefore $2r \pi \xi = m \pi$ and $2s \pi \xi = n \pi$, for some integers $m, n$. We conclude that $\xi = \frac{m}{2r} = \frac{n}{2s}$,
so $\frac{r}{s}=\frac{m}{n} \in \mathbb{Q} $, a contradiction that proves $\hat f(\xi) >0$ for all nonzero real $\xi$.
\bigskip
Exercise \ref{tricky application of Poisson summation}
By assumption, $g:\mathbb{R}^d\rightarrow \mathbb{C}$ is infinitely smooth, and compactly supported.
By Corollary \ref{cor: f smoother implies FT of F decays faster}, $\hat g$ is a rapidly decreasing function. Because $g$ has compact support, we also know that $\hat g$ is infinitely smooth. So $\hat g$ is a Schwartz function (and
$g$ is also a Schwartz function - in fact $g$ is a `bump function', by definition). Therefore we may apply the Poisson summation formula for Schwartz functions (Theorem \ref{Poisson.Summation}) to $\hat g$:
\[
\sum_{\xi \in \mathbb{Z}^d} \hat g(\xi) = \sum_{n \in \mathbb{Z}^d} g(n),
\]
which is a finite sum due to the compact support of $g$.
\bigskip \bigskip
{\bf \Large Chapter \ref{Geometry of numbers}}
\medskip
Exercise \ref{c.s. C equals its symmetrized body}
\quad
For part (a), we suppose that
\begin{equation} \label{symmetrized toy}
\frac{1}{2}C - \frac{1}{2}C = C.
\end{equation}
Let's pick any $x \in C$; we need to
show that $-x \in C$. Since $x \in \frac{1}{2}C - \frac{1}{2}C$,
we know that there must exist $y, z\in C$ such that $x=\frac{1}{2}y - \frac{1}{2}z$.
This implies that $-x = \frac{1}{2}z -\frac{1}{2}y \in \frac{1}{2}C - \frac{1}{2}C\subseteq C$. Therefore $C$ is centrally symmetric.
To show part (b), we note that the convexity of $C$ gets used in the step
$\frac{1}{2}C + \frac{1}{2}C = C$.
First, we suppose that $C$ is centrally symmetric, so that $C = -C$. This implies that
$\frac{1}{2}C - \frac{1}{2}C
= \frac{1}{2}C + \frac{1}{2}(-C) = \frac{1}{2}C + \frac{1}{2}(C) = C$.
Conversely, suppose that $C = \frac{1}{2}C - \frac{1}{2}C$. Then by part (a), we already know that
$C$ is centrally symmetric.
For part (c), consider the counter-example $C:= [-2, -1] \cup [1, 2]$, a nonconvex set in $\mathbb{R}$. Here $C$ is centrally symmetric, yet
$C - C = [-3, 3] \not= [-4, -2] \cup [2, 4] = 2C$.
\bigskip
Exercise \ref{support of convolution}
\quad To prove part (a), we are given two convex bodies $A, B \subset \mathbb{R}^d$, so by definition we have
\[
\supp( 1_A * 1_B) := \closure \left\{ y \in \mathbb{R}^d \bigm | \int_{\mathbb{R}^d} 1_A(x) 1_B(y-x) dx \not= 0 \right\},
\]
and we must prove that $\supp( 1_A * 1_B) = A + B$, their Minkowski sum. \index{Minkowski sum}
In general, we have:
\begin{align} \label{equivalences for Minkowski sums}
1_A(x) 1_B(y-x)>0
& \iff 1_A(x) =1 \text{ and } 1_B(y-x) = 1 \\
& \iff x\in A \text{ and } y-x \in B \\
& \iff y \in A+B.
\end{align}
If we fix any $y \notin \supp( 1_A * 1_B)$,
then $\int_{\mathbb{R}^d} 1_A(x) 1_B(y-x) dx = 0$, which implies that $1_A(x) 1_B(y-x)=0$ for all $x\in \mathbb{R}^d$.
But by the equivalences \eqref{equivalences for Minkowski sums}
above, we see that
$1_A(x) 1_B(y-x)=0 \iff y\notin A+B$, proving that $ A + B \subset \supp( 1_A * 1_B)$.
Conversely, suppose that $y \in \supp( 1_A * 1_B)$, meaning that there exists a sequence $y_n\in \mathbb{R}^d$ with
$ \int_{\mathbb{R}^d} 1_A(x) 1_B(y_n-x) dx \not= 0$. This implies that for each such $y_n$, there exists at least
one $x \in \mathbb{R}^d$ with $1_A(x) 1_B(y_n-x) >0$. This last inequality, using our
our equivalences \eqref{equivalences for Minkowski sums}, implies that the sequence $y_n \in A+B$.
Because $A+B$ is a closed set, we finally have $y := \lim_{n\rightarrow \infty} y_n \in A+B$.
\bigskip
To prove part (b), we must show that
$ \supp(f*g)\subseteq C$, where
\[
C:= \closure\left( \supp(f) + \supp(g) \right).
\]
We'll prove the contrapositive: if $x \notin C$, then $x \notin \supp(f*g)$ . So we suppose $x \notin C$, and we have to prove that $(f*g)(x)=0$.
By our assumption on $x$, for each $y \in \supp(g)$, we have that $x-y \notin \supp(f)$. The last assertion means that $f(x-y) = 0$, so we now know that $f(x-y) g(y)=0$ for all $y\in \mathbb{R}^d$.
Finally, we have $(f*g)(x) := \int_{\mathbb{R}^d} f(x-y) g(y) dy = 0$.
\bigskip
Exercise \ref{non-unimodular but empty simplex}
\quad
Define $\Delta:= \conv\{ (0, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1)\}$, an integer $3$-simplex. It's clear that
$\Delta$ is subset of the unit cube $[0, 1]^3$, and therefore $\Delta$ has no integer points in its interior.
To see that $\Delta$ is not a unimodular simplex, we can consider its tangent $K_0$ cone at the origin, which has primitive integer vectors $(1, 1, 0), (1, 0, 1), (0, 1, 1)$, so that the determinant of $K_0$ is
equal to $\left| \det
\begin{pmatrix}
1 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 1
\end{pmatrix}
\right| = 2 > 1.
$
\medskip
Exercise \ref{FT of a polytope is not Schwartz}
\quad
Suppose to the contrary, that for some polytope ${\mathcal P}$ we have $\hat 1_{{\mathcal P}}(\xi)= g(\xi)$, a Schwartz function. Taking the Fourier transform of both sides of the latter equality, and using the fact that the Fourier transform takes Schwartz functions to Schwartz functions, we conclude that $1_{{\mathcal P}}(-x) = \hat g(-x)$ is a Schwartz function. But this is a contradiction, because the indicator function of a polytope is not even continuous.
\medskip
Exercise \ref{an application of Cauchy-Schwartz 1}
\quad We use the Cauchy-Schwartz inequality:
\[
{\Big \langle \icol{a \\ b} , \icol{ \sin x \\ \cos x} \Big \rangle}^2
:= ( a \sin x+ b \cos x )^2
\leq \big( a^2 + b^2 \big) \big( \sin^2 x+ \cos^2 x \big) = a^2 + b^2.
\]
By the equality condition of Cauchy-Schwartz, we see that the maximum is obtained when
the two vectors are linearly dependent, which gives
$\tan x = \frac{a}{b}$.
\bigskip \bigskip
{\bf \Large Chapter \ref{chapter.lattices}}
\medskip
Exercise \ref{distance between hyperplanes}
\quad We are given the hyperplanes $H_1:= \{ x\in \mathbb{R}^d \mid c_1 x_1 + \cdots + c_d x_d = k_1\}$,
and $H_2:= \{ x\in \mathbb{R}^d \mid c_1 x_1 + \cdots + c_d x_d = k_2\}$. First we'll pick a point $x \in H_1$,
and then we'll `walk along its normal vector', until we get to $H_2$. With this `walk' in mind, we may assume WLOG
that $k_2 > k_1$, and that the normal vector is pointing from $H_1$ towards $H_2$.
For simplicity, we'll let $L:= \sqrt{ c_1^2 + \cdots + c_d^2}$, and with this definition
the unit normal vector to $H_1$ is
$n:= \frac{1}{ L }(c_1, \dots, c_d)^T$, and we want to find $\delta>0$ such that
$x + \delta n \in H_2$.
Unraveling the definition of the latter statement, we must have
\begin{align*}
& c_1 ( x_1 + \delta \tfrac{1}{L} c_1) + \cdots + c_d (x_d + \delta \tfrac{1}{L} c_d) = k_2 \\
\iff & (c_1 x_1 + \cdots + c_d x_d) + \frac{\delta}{L}( c_1^2 + \cdots + c_d^2) = k_2 \\
\iff & k_1 + \delta\sqrt{c_1^2 + \cdots + c_d^2} = k_2 \\
\iff & \delta = \frac{k_2 - k_1}{\sqrt{c_1^2 + \cdots + c_d^2}}.
\end{align*}
\medskip
Exercise \ref{Hadamard's inequality, exercise}
\quad We consider each $k$'th row of $M$ as a vector, call it $v_k$. By assumption,
the norm of $v_k$ is bounded by $\|v\| \leq \sqrt{ B^2 + \cdots B^2 } = B\sqrt d$.
Using Hadamard's inequality \ref{Hadamard inequality}, we have:
\begin{align*}
|\det M| \leq \|v_1\| \cdots \|v_d\| \leq \left(B\sqrt d \right)^d.
\end{align*}
\medskip
Exercise \ref{Ellipsoid problem}
\quad It's easy to see that the inverse matrix for $M$ is
\[
M^{-1} := \begin{pmatrix} | & | & ... & | \\
\frac{1}{c_1} b_1 & \frac{1}{c_2} b_2 & ... & \frac{1}{c_d} b_d \\
| & | & ... & | \\
\end{pmatrix}^T.
\]
The image of the unit sphere under the matrix $M$ is, by definition:
\begin{align*}
M(S^{d-1}) &:= \{ u \in \mathbb{R}^d \mid u = Mx, x \in S^{d-1} \} \\
&= \{ u \in \mathbb{R}^d \mid M^{-1}u \in S^{d-1} \} \\
&= \{ u \in \mathbb{R}^d \mid \frac{1}{c_1^2} \langle b_1, u \rangle^2 + \cdots +
\frac{1}{c_d^2} \langle b_d, u \rangle^2 = 1 \},
\end{align*}
using our description of $M^{-1}$ above.
For part (b), we begin with the definition of volume, and we want to compute the
volume of the region $M(B):= \{ u \in \mathbb{R}^d \mid u = My, \text{ with } \| y \| \leq 1 \}$,
where $B$ is the unit ball in $\mathbb{R}^d$.
\begin{align*}
\vol(Ellipsoid_M) &:= \int_{M(B)} du \\
&= | \det M | \int_{B} dy \\
&= | \det M | \vol(B).
\end{align*}
using the change of variable $u = My$, with $y \in B$. We also used the Jacobian,
which gives $du = | \det M | dy$.
Finally, we note that the matrix $M^T M$ is a diagonal matrix, with diagonal entries $c_k^2$, due
to the fact that the $b_k$'s form an orthonormal basis. Thus we use: $| \det M |^2 = | \det M^T M |
= \prod_{k=1}^d c_k^2$, so taking the positive square root, we arrive at $| \det M | = \prod_{k=1}^d c_k$,
because all of the $c_k$'s are positive by assumption.
\medskip
Exercise \ref{exercise:2by2 positive definite matrix}
\quad Let $A:=
\left(\begin{smallmatrix} a & b \\ b & d \end{smallmatrix}\right) $ be an invertible, symmetric matrix.
Because $A$ is symmetric, we know both of its eigenvalues $\lambda_1, \lambda_2$ are real.
The characteristic polynomial of $A$, namely
$(a-\lambda)(d-\lambda) - b^2 $, may also be factored and rewritten as
\[
\lambda^2 - (a+ d) \lambda + (ad-b^2) = (\lambda - \lambda_1)(\lambda - \lambda_2) = \lambda^2 - (\lambda_1 + \lambda_2)\lambda + \lambda_1 \lambda_2.
\]
Equating coefficients of the latter identity between polynomials, we therefore have $\lambda_1 + \lambda_2 = {\rm Trace } A$, and $\lambda_1 \lambda_2= \det A$.
From these last two relations, we see that if both eigenvalues are positive, then ${\rm Trace } A>0$ and $\det A>0$.
Conversely, suppose that ${\rm Trace } A>0$ and $\det A>0$.
Then $\lambda_1 \lambda_2 >0$, so either both eigenvalues are positive, or both eigenvalues are negative.
But the eigenvalues cannot both be negative, for this would contradict our assumption that $\lambda_1 + \lambda_2>0$.
\medskip
Exercise \ref{translating the Voronoi cell around}
\quad
Given a full rank lattice ${\mathcal L}\subset \mathbb{R}^d$, and any $m \in {\mathcal L}$, we have:
\begin{align}
{\rm Vor}_0({\mathcal L}) + m
&:=
\left\{ x +m \in \mathbb{R}^d \bigm | \|x\| \leq \|x - v\|, \ \text{ for all } v \in {\mathcal L} \right\} \\ \label{last expression for Voronoi}
&= \left\{ y \in \mathbb{R}^d \bigm | \|y-m\| \leq \|y-m - v\|, \ \text{ for all } v \in {\mathcal L} \right\}.
\end{align}
But as $v$ varies over ${\mathcal L}$, so does $m + v$, because $m \in {\mathcal L}$. Hence the expression
\eqref{last expression for Voronoi} above is ${\rm Vor}_m({\mathcal L})$.
\bigskip \bigskip
{\bf \Large Chapter \ref{chapter.Brion}}
\medskip
Exercise \ref{independent of edge vectors}
We are given $\alpha >0$, and a simplicial cone ${\mathcal K}_v$, with edge vectors $w_1, \dots, w_d \in \mathbb{R}^d$.
By definition, $\det {\mathcal K}_v$ is the determinant of the matrix whose columns are the $w_k$'s.
Replacing each $w_k(v)$ by $\alpha_k w_k(v)$, we see that the determinant $|\det {\mathcal K}_v|$
gets multiplied by $\alpha^d$, and so
\[
\frac{ \alpha^d |\det {\mathcal K}_v| }{\prod_{k=1}^d \langle \alpha w_k(v), z \rangle}
= \frac{ |\det {\mathcal K}_v| }{\prod_{k=1}^d \langle w_k(v), z \rangle}.
\]
\medskip
Exercise \ref{polytope from pentagons}
\quad
Euler's formula gives us
\[
V-E+F =2,
\]
and the hypotheses also imply that:
\begin{align}
5F&=2E \\
5F &\geq 3V.
\end{align}
Altogether, we get
\begin{equation*}
2=V-E+F \leq \frac{5}{3} F - \frac{5}{2} F + F = \frac{1}{6} F,
\end{equation*}
so that $F \geq 12$.
\bigskip \bigskip
{\bf \Large Chapter \ref{chapter:Discrete Brion}}
Exercise \ref{unimodular cone, integer point transform}
\quad
The main point here is that at each vertex $v\in V$, the edge vectors form a basis for $\mathbb{Z}^d$, and therefore
the only integer point in the (half-open) fundamental parallelepiped $\Pi_v$ is $v$ itself. So we see that
its integer point transform is $\sigma_{\Pi_v}(x) = e^{\langle v, z \rangle}$.
Now we use Theorem \ref{brion, discrete form}, followed by Theorem \ref{closed form for integer point transform of a cone}:
\begin{equation*}
\sigma_{\mathcal P}(z) = \sum_{v \in V} \sigma_{{\mathcal K}_v}(z) =
\frac{e^{\langle v, z \rangle} }{\prod_{k=1}^d \left( 1 - e^{\langle w_k, z\rangle} \right) }.
\end{equation*}
Exercise \ref{bound for integer point transform}
\quad
Because $|e^{2\pi i \langle x, n \rangle}|=1$ for all $x\in \mathbb{R}^d$, we have:
\[
\left | \sigma_{\mathcal P}(2\pi i x) \right |
\leq \sum_{n\in {\mathcal P} \cap \mathbb{Z}^d } \left| e^{2\pi i \langle x, n \rangle} \right|
= \sum_{n\in {\mathcal P} \cap \mathbb{Z}^d } 1 = \left |\mathbb{Z}^d \cap {\mathcal P} \right |.
\]
\bigskip \bigskip
{\bf \Large Chapter \ref{Ehrhart theory}}
\medskip
Exercise \ref{Bernoulli polynomial as an Ehrhart polynomial}
\quad
Here ${\mathcal P}:= \conv\{ C, {\bf e_d} \}$, where $C$ is the $(d-1)$-dimensional unit cube $[0, 1]^{d-1}$. To compute the Ehrhart
polynomial ${\mathcal L}_{{\mathcal P}}(t)$ here, we use the fact that a `horizontal' slice of ${\mathcal P}$, meaning a slice parallel to $C$, and orthogonal
to $e_d$, is a dilation of $C$. Thus, each of these slices counts the number of points in a $k$-dilate of $C$, as $k$ varies from $0$ to $t+1$. Summing over these integer dilations of $C$, we have
\[
{\mathcal L}_{{\mathcal P}}(t) = \sum_{k=0}^{t+1} (t+1 - k)^{d-1} = \sum_{k=0}^{t+1} k^{d-1} =
\frac{1}{d}(B_d(t+2) - B_d),
\]
where the last step holds thanks to Exercise \ref{historical origin of Bernoulli poly}.
\medskip
Exercise \ref{unimodular triangle}
Using Pick's formula, the unimodular triangle ${\mathcal P}$ has area:
\[
\rm{Area } {\mathcal P} = I + \frac{1}{2} B -1 = 0 + \frac{1}{2} 3 -1 = \frac{1}{2}.
\]
\medskip
Exercise \ref{properties of floor, ceiling, fractional part}
Throughout, we first write $x = n + \alpha$, with $\lfloor x \rfloor := n \in \mathbb{Z}$ and $0 \leq \alpha < 1$.
We prove part \ref{ex:part 1 of fractional parts}, namely that $- \floor{-x} = \left\lceil x \right\rceil$.
Case $1$: $x \in \mathbb{Z}$. Here $\alpha = 0$ and $x=n$, so that $- \floor{-x} = - (-n)= n = \left\lceil x \right\rceil$.
Case $2$: $x \notin \mathbb{Z}$. In this case
$\left\lceil x \right\rceil = n+1$. We have $-x = -n - \alpha = - n-1 + (1-\alpha)$,
from which we see that $- \floor{-x} = - (-n-1) = n + 1 = \left\lceil x \right\rceil$.
To prove part \ref{ex:part 2 of fractional parts}, we need to show that $\floor{x} - \left\lceil x \right\rceil +1=1_{\mathbb{Z}}(x)$.
Case $1$: $x \in \mathbb{Z} \implies \floor{x} - \left\lceil x \right\rceil +1 = n - n + 1 = 1 = 1_{\mathbb{Z}}(x)$.
Case $2$: $x \notin \mathbb{Z} \implies \floor{x} - \left\lceil x \right\rceil +1
= n - (n + 1) + 1 = 0 = 1_{\mathbb{Z}}(x)$.
To prove part \ref{ex:part 3 of fractional parts}, we need to show that
$ \{ x \} + \{-x\} = 1- 1_{\mathbb{Z}}(x)$.
This follows from part \ref{ex:part 2 of fractional parts} if we use the definitions
$\floor{x} := x - \{x\}, \lceil x \rceil:= x + \{x\}$. Using the identity of part \ref{ex:part 2 of fractional parts}, we have
\[
1- 1_{\mathbb{Z}}(x) = \left\lceil x \right\rceil - \floor{x} = x + \{x\} - ( x - \{x\}) = \{x\} + \{x\}.
\]
Finally, for part \ref{ex:part 4 of fractional parts}, we have to prove that
if $m \in \mathbb{Z}_{>0}, n \in \mathbb{Z}$, then $\floor{ \frac{n-1}{m} } + 1 = \left\lceil \frac{n}{m} \right\rceil$.
We begin by using the division algorithm, which gives us $n=qm+r$, with integers $q$ and $0\leq r < m$.
Case $1$: $r=0$. Here $n = qm$, and we have $\floor{ \frac{n-1}{m} } + 1 = \floor{ q- \frac{1}{m} } + 1=q
= \frac{n}{m} = \left\lceil \frac{n}{m} \right\rceil$.
Case $2$: $0 < r < m$. Here
$\floor{ \frac{n-1}{m} } + 1 = \floor{ \frac{qm + r - 1}{m} } + 1
= \floor{ q + \frac{ r - 1}{m} } + 1 = \floor{ \frac{ r - 1}{m} } + 1 = 1$.
On the other hand, $ \left\lceil \frac{n}{m} \right\rceil = \left\lceil \frac{qm + r}{m} \right\rceil
= \left\lceil q + \frac{ r}{m} \right\rceil = \left\lceil \frac{ r}{m} \right\rceil =1 $.
\bigskip
\chapter{Preface}
\chapter*{Acknowledgements}
The famous saying ``no man is an island'' is doubly-true in Mathematics,
and indeed I've had the good fortune to know and learn from many interesting people,
concerning the contents of this book. Special thanks goes to Ricardo Diaz, my first collaborator along these topics.
I would like to thank the following people, from the bottom of my heart, for their valuable input and interesting
discussions about some of these topics over the years:
Ian Alevy, Artur~Andr\'e, Christine~Bachoc, Tamar~Bar,
Imre~B\'ar\'any, Alexander~Barvinok, Matthias~Beck, Dori~Bejleri,
Luca~Brandolini, Michel~Brion, Henry~Cohn, Leonardo~Colzani, Amalia~Culiuc, Pierre~Deligne,
Jes\'us A. De Loera, Michel~Faleiros, Lenny~Fukshansky, Nick~Gravin,
Martin~Henk, Didier~Henrion, Roberto Hirata Junior, Jeffrey~Hoffstein, \\
Alex~Iosevich, Michael Joswig,
Marvin~Knopp, Mihalis~Kolountzakis, Matthias~K\"oppe, Greg~Kuperberg,
Jean~Bernard~Lasserre,
Nhat~Le~Quang, Rafael~Zuolo~Coppini~Lima, Sameer~Iyer,
Fabr\'icio~Caluza~Machado,
Romanos~Malikiosis, M\'at\'e Matolci, \\
Tyrrell~McAllister,
Victor Moll, Mel~Nathanson,
James~Pommersheim, Jim~Propp, \\
Thales~Paiva, Jill~Pipher, Jorge Luis Ram\'irez Alfons\'in, Ethan~Reiner, Bruce~Reznick,
Tiago~Royer, Nicolas~Salter,
Gerv\'asio~Santos, Richard~Schwartz,
Dima~Shiryaev, \\
Joseph~Silverman, Richard~Stanley, Christophe~Vignat, Sergei~Tabachnikov, \\
Giancarlo~Travaglini, Kevin~Woods, Ren~Yi,
G\"unter~Ziegler, Chuanming~Zong.
\mainmatter
\chapter{\blue{Introduction} }
\begin{wrapfigure}{R}{0.4\textwidth}
\centering
\includegraphics[width=0.34\textwidth]{Fourier1}
\caption{Joseph Fourier} \label{Fourier}
\end{wrapfigure}
What is a Fourier transform? Why is it so useful? How can we apply Fourier transforms and Fourier series - which were originally used by Fourier to study heat diffusion - in order to better understand topics in
discrete and combinatorial geometry, number theory, and sampling theory?
To begin, there are some useful analogies: imagine that you are drinking a milk-shake (lactose-free), and you want to know the ingredients of your tasty drink. You would need to filter out the shake into some of its most basic components. This decomposition into its basic ingredients may be thought of as a sort of
``Fourier transform of the milk-shake''. Once we understand each of the ingredients, we will also be able to
restructure these ingredients in new ways, to form many other types of tasty goodies. To move the analogy back into mathematical language, the milkshake represents a function, and each of its basic ingredients represents for us the basis of sines and cosines; we may also think of a basic ingredient more compactly as a complex exponential $e^{2\pi i nx}$, for some $n\in \mathbb{Z}$. Composing these basic ingredients together in a new way represents a Fourier series.
Mathematically, one of the most basic kinds of milk-shakes is the indicator function of the unit interval, and to break it down into its basic components, mathematicians, Engineers, Computer scientists, and Physicists have used the
sinc function (since the $1800$'s):
\[
{\rm{sinc}}(z):= \frac{\sin(\pi z)}{\pi z}
\]
with great success, because it happens to be the Fourier transform of the unit interval
$[-\frac{1}{2}, \frac{1}{2}]$:
\[
\int_{-\frac{1}{2}}^\frac{1}{2} e^{-2\pi i z x} dx = {\rm{sinc}}(z),
\]
as we will compute shortly in identity \eqref{sinc function formula}.
Somewhat surprisingly, comparatively little energy has been given to some of its higher dimensional
extensions, namely those extensions that arise naturally as Fourier transforms of polytopes.
One motivation for this book is to better understand how this $1$-dimensional function -- which has proved to be extremely powerful in applications -- extends to higher dimensions. Namely, we will build various mathematical structures that are motivated by the question:
\[
\text{ {\bf
What is the Fourier transform of a polytope}? }
\]
Of course, we will ask ``how can we apply it"? An alternate title for this book might have been:
\centerline{ {\bf We're taking Poisson summation and Fourier transforms of polytopes}}
\centerline{ {\bf for a very long ride....}}
Historically, sinc functions were used by Shannon (as well as Hardy, Kotelnikov, and Whittaker) when he published his seminal work on sampling theory and information theory.
In the first part of this book, we will learn how to use the technology of Fourier transforms of polytopes in order to build the (Ehrhart) theory of integer point enumeration in polytopes, to prove some of Minkowski's theorems in the geometry of numbers, and to understand when a polytope tiles Euclidean space by translations.
In the second portion of this book, we give some applications to active research areas which are sometimes considered more applied, including
the sphere-packing problem, and the sampling of signals in higher dimensions.
There are also current research developments of the material developed here, to
the learning of deep neural networks.
In many applied scientific areas, in particular radio astronomy, computational tomography, and magnetic resonance imaging, a frequent theme is the reconstruction of a function from knowledge of its Fourier transform. Somewhat surprisingly,
in various applications we only require very partial/sparse knowledge of its Fourier transform in order to reconstruct the required function, which may represent an image or a signal.
There is a rapidly increasing amount of research focused in these directions in recent years, and it is therefore time to put some of these new findings in one place, making them accessible to a general scientific reader.
The fact that the sinc function is indeed the Fourier transform of the $1$-dimensional line segment
$[-\frac{1}{2}, \frac{1}{2}]$, which is a $1$-dimensional polytope, \index{polytope}
gives us a first hint that
there is a deeper link between the geometry of a polytope and the analysis of its Fourier transform.
Indeed one reason that sampling and information theory, as initiated by Claude Shannon, \index{Shannon, Claude}
works so well is precisely because the Fourier transform of the unit interval has this nice form, and even more-so because of the existence of the Poisson summation formula.
The approach we take here is to gain insight into how the Fourier transform of a polytope \index{polytope}
can be used to solve various specific problems in discrete geometry, combinatorics, optimization, approximation theory, and the Shannon-Whittaker sampling theory in higher dimensions:
\begin{enumerate}[(a)]
\item Analyze tilings of Euclidean space by translations of a polytope
\item Give wonderful formulas for volumes of polytopes
\item Compute discrete volumes of polytopes, which are combinatorial approximations to the continuous volume
\item Introduce the geometry of numbers, via Poisson summation
\item Optimize sphere packings, and get bounds on their optimal densities
\item Study the Shannon-Whittaker sampling theorem and its higher-dimensional siblings
\item Recover a polytope by the inverse problem of knowing enough of its moments
\end{enumerate}
\medskip
Let's see at least one direction that quickly motivates the study of Fourier transforms. In particular, we often begin with
simple-sounding problems that arise naturally in combinatorial enumeration, discrete and computational geometry, and number theory.
Throughout, an {\bf integer point} \index{integer point} is any vector $v:=(v_1, \dots, v_d)\in \mathbb{R}^d$, all of whose coordinates $v_j$ are integers. In other words, $v$ belongs to the integer lattice $\mathbb{Z}^d$.
A {\bf rational point}
\index{rational point} is a point $m$ whose coordinates are rational numbers, in other words
$m \in \mathbb{Q}^d$.
We define the {\bf Fourier transform} of a function $f(x)$:
\begin{align} \label{Fourier transform} \index{Fourier transform}
\hat f(\xi) := \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle \xi, x \rangle} dx,
\end{align}
defined for all $\xi \in \mathbb{R}^d$ for which the latter integral converges, and where we use the standard inner product
$\langle a, b \rangle:= a_1 b_1 + \cdots + a_d b_d$. We will also use the notation
${\mathcal F}(f)$ for the Fourier transform of $f$, which is useful in some typographical contexts, for example when considering ${\mathcal F}^{-1}(f)$.
We introduce one of the main objects of study in this book, the {\bf Fourier transform of a polytope}
\index{Fourier transform of a polytope}
${\mathcal P}$, defined by:
\begin{align} \label{Fourier transform of P}
\hat 1_{\mathcal P}(\xi)
:= \int_{\mathbb{R}^d} 1_{\mathcal P}(x) e^{-2\pi i \langle \xi, x \rangle} dx = \int_{{\mathcal P}} e^{-2\pi i \langle \xi, x \rangle} dx,
\end{align}
where the function $1_{\mathcal P}(x)$ is the {\bf indicator function} of ${\mathcal P}$, defined by
\[
1_{\mathcal P}(x):=
\begin{cases}
1 & \mbox{if } x\in {\mathcal P} \\
0 & \mbox{if not}.
\end{cases}
\]
Thus, the words ``Fourier transform of a polytope ${\mathcal P}$'' will always mean the Fourier transform of the
indicator function \index{indicator function} of ${\mathcal P}$.
\begin{wrapfigure}{R}{0.4\textwidth}
\centering
\includegraphics[width=0.32\textwidth]{Poisson2}
\caption{Sim\'eon Denis Poisson} \label{PoissonHimself}
\end{wrapfigure}
The {\bf Poisson summation formula}, named after Sim\'eon Denis Poisson,
\index{Poisson summation formula}
tells us that for any ``sufficiently nice"
function $f : \mathbb{R}^d \rightarrow \mathbb{C}$ we have:
\[
\sum_{n \in \mathbb{Z}^d} f(n) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi).
\]
In particular, if we were to naively set $f(n) := 1_{{\mathcal P}}(n)$, the indicator function of a polytope ${\mathcal P}$, then we would get:
\begin{align} \label{PoissonSummation1}
\sum_{n \in \mathbb{Z}^d} 1_{{\mathcal P}}(n) = \sum_{\xi \in \mathbb{Z}^d} \hat 1_{{\mathcal P}}(\xi),
\end{align}
which is technically false in general due to the fact that the indicator function $1_{\mathcal P}$ is a discontinuous function on $\mathbb{R}^d$.
However, this technically false statement is very useful! We make this claim because it helps us build intuition for the more rigorous statements that are true, and which we study in later chapters.
For applications to discrete geometry, we are interested in the number of integer points in a closed convex polytope ${\mathcal P}$, namely $|{\mathcal P} \cap \mathbb{Z}^d|$.
The combinatorial-geometric quantity $|{\mathcal P} \cap \mathbb{Z}^d|$
may be regarded as a {\bf discrete volume}
\index{discrete volume}
for ${\mathcal P}$.
From the definition of the indicator function of a polytope, the left-hand-side of \eqref{PoissonSummation1}
counts the number of integer points in ${\mathcal P}$, namely
we have by definition
\begin{equation}
\sum_{n \in \mathbb{Z}^d} 1_{{\mathcal P}}(n) = |{\mathcal P} \cap \mathbb{Z}^d|.
\end{equation}
On the other hand, the right-hand-side of \eqref{PoissonSummation1} allows us to compute this discrete volume of ${\mathcal P}$ in a new way. This is great, because it opens a wonderful window of computation for us in the following sense:
\begin{align} \label{PoissonSummation2}
|{\mathcal P} \cap \mathbb{Z}^d| = \sum_{\xi \in \mathbb{Z}^d} \hat 1_{{\mathcal P}}(\xi).
\end{align}
We notice that for the $\xi = 0$ term, we have
\begin{align} \label{Fourier transform at 0}
\hat 1_{\mathcal P}(0) := \int_{\mathbb{R}^d} 1_{{\mathcal P}}(x) e^{-2\pi i \langle 0, x \rangle} dx =
\int_{{\mathcal P}} dx = \vol({\mathcal P}),
\end{align}
and therefore the {\bf discrepancy} \index{discrepancy}
between the continuous volume of ${\mathcal P}$ and the discrete volume of ${\mathcal P}$ is
\begin{align} \label{PoissonSummation3}
|{\mathcal P} \cap \mathbb{Z}^d| - \vol({\mathcal P}) = \sum_{\xi \in \mathbb{Z}^d-\{0\}} \hat 1_{\mathcal P}(\xi),
\end{align}
showing us very quickly that indeed $|{\mathcal P} \cap \mathbb{Z}^d|$ is a discrete approximation to the classical Lebesgue volume $\vol({\mathcal P})$, and pointing us to the task of finding ways to evaluate the transform $\hat 1_P(\xi)$.
From the trivial but often very useful identity
\[
\hat 1_{\mathcal P}(0) = \vol({\mathcal P}),
\]
we see another important motivation for this book: the Fourier transform of a polytope is a very {\bf natural extension of volume}.
\index{volume}
Computing the volume of a polytope ${\mathcal P}$ captures a bit of information about ${\mathcal P}$, but we also lose a lot of information.
On the other hand, computing the Fourier transform of a polytope
$\hat 1_{\mathcal P}(\xi)$ uniquely determines ${\mathcal P}$, so we do not lose any information at all. Another way of saying this is that the Fourier transform of a polytope is a {\bf complete invariant}.
\index{complete invariant}
In other words,
it is a fact of life that
\[
\hat 1_{\mathcal P}(\xi) = \hat 1_{\mathcal Q}(\xi) \text{ for all } \xi \in \mathbb{R}^d \ \iff \ {\mathcal P} = \mathcal Q.
\]
Combinatorially, there are brilliant identities (notably the Brion identities) that emerge between
the Fourier and Laplace transforms of a given polytope, and its facets and vertex tangent cones.
In Statistics, the moment generating function of any probability distribution is given by a Fourier transform of the indicator function of the distribution, hence Fourier transforms arise very naturally in Statistical applications.
At this point, a natural glaring question naturally comes up:
\begin{equation}
\text{ How do we {\bf compute} the Fourier transform of a polytope }
\hat 1_P(\xi)?
\end{equation}
And how do we use such computations to help us understand the important
``error'' term
\[
\sum_{\xi \in \mathbb{Z}^d-\{0\}} \hat 1_{\mathcal P}(\xi)
\]
that came up naturally in \eqref{PoissonSummation3} above?
There are many applications of the theory that we will build-up. Often, we find it instructive to
sometimes give an informal proof first, because it
brings the intuitive ideas to the
foreground, allowing the reader to gain an overview of the steps. Then, later on, we revisit the same intuitive proof again, making it rigorous.
The Poisson summation formula \index{Poisson summation formula} is one of our main stars,
and has a relatively easy proof. But it constitutes a very first step for many of our explorations. It may even be said that, from this perspective, the Poisson summation formula is to combinatorial analysis as a microscope is to our vision. It enhances our ability to see mathematical facts, and often in a surprisingly simple way. So it's a question of what we do with these tools - where do we point them?
A word about {\bf prerequisites} for this book: {\bf Linear Algebra} is always very useful!
A couple of calculus courses are required as well, with a touch of real analysis. In particular, familiarity with infinite series is assumed. We give new proofs for
some of the main theorems in this theory,
including Theorem \ref{Siegel for general lattices}, Theorem \ref{brion, continuous form}, Theorem \ref{brion2}, and Theorem \ref{brion, discrete form}. These new Fourier-type proofs help streamline the theory, unifying sporadic results in the literature. This
unifying thread will hopefully help the reader put the various results, from both the past and the present, into context.
We will assume some familiarity with the basic definitions of polytopes and their faces, although at places we will remind the reader of some of these definitions.
There are many excellent texts that introduce the student to the classical language of polytopes, in particular the two classics: G\"unter Ziegler's ``Lectures on Polytopes" \cite{Ziegler}, and Branko Gr\"unbaum's ``Convex Polytopes" \cite{Grunbaum}.
For an easy introduction to the interactions between polytopes and lattice point enumeration, the reader is invited to consult ``Computing the continuous discretely: integer point enumeration in polytopes", by Beck and Robins \cite{BeckRobins}.
The level of the current book is aimed at {\bf advanced undergraduates} and {\bf beginning graduate students} in various fields, and in particular Mathematics, Computer Science, Electrical Engineering, and Physics.
Because of the large number of exercises, with solutions to many of them in the back, this book can also be used effectively for self-study.
Finally, this book is still in draft form, and in particular Chapters 9, 10, 11, and 13 are still under revision.
We proceed by developing an intuitive understanding first, using many examples and analogies, and this intuition then points us to a rigorous path for the details of the ensuing proofs.
\bigskip
Sinai Robins \hfill December 2021
IME, University of S\~ao Paulo
\chapter{
A motivating problem: \\ tiling a rectangle with rectangles}
\label{Chapter.Tiling.A.Rectangle}
\index{Tiling a rectangle}
\begin{quote}
``Ripping up carpet is easy -- {\it tiling} is the issue''.
-- Douglas Wilson
\end{quote}
\begin{wrapfigure}{R}{0.5\textwidth}
\centering
\includegraphics[width=0.45\textwidth]{rectangle}
\caption{A rectangle tiled by nice rectangles} \label{nice rectangle}
\end{wrapfigure}
\section{Intuition}
To warm up, we begin with a simple tiling problem in the plane. A rectangle will be called
{\bf nice} if at least one of its sides is an integer.
We prove a now-classical fact about tiling a rectangle with nice rectangles, namely Theorem \ref{Integer.Side.Rectangle}, and we focus on the {\bf method} of the straightforward proof.
This proof brings to the foreground an important idea: by simply taking a Fourier transform of a body $B$, we immediately get interesting geometric consequences for $B$.
In particular, we will see throughout this book various ways in which the Fourier transform of a geometric body is a natural extension of its volume, sometimes in a continuous way, and sometimes in a discrete way.
So in order to study relationships between volumes of bodies, it is very natural and useful to play with their Fourier transforms.
\section{Nice rectangles}
The tilings that we focus on, in this small chapter, are tilings that are composed of smaller rectangles, all of which have their sides parallel to the axes, and all of which are nice.
There are at least $14$ different known proofs \cite{StanWagon} of Theorem \ref{Integer.Side.Rectangle}.
Here we give the proof that uses very basic Fourier tools, from first principles, motivating the chapters that follow.
The idea for this proof goes back to Nicolaas Govert De Bruijn \cite{DeBruijn.Book}.
\index{De Bruijn, Nicolaas Govert}
\begin{thm}[De Bruijn] \label{Integer.Side.Rectangle}
Suppose we tile a fixed rectangle $\mathcal{R}$ with smaller, nice rectangles. \\
Then $\mathcal{R}$ is a nice rectangle.
\end{thm}
\begin{proof}
Suppose that the rectangle $\mathcal{R}$ is tiled with smaller rectangles $\mathcal{R}_1, \dots, \mathcal{R}_N$, as in
Figure~\ref{nice rectangle}.
Due to our tiling
\index{tiling}
hypothesis, we have
\begin{equation} \label{first identity of rectangles}
1_{\mathcal{R}}(x) = \sum_{k=1}^N 1_{\mathcal{R}_k}(x) + \sum (\pm \text{ indicator functions of lower-dimensional polytopes}),
\end{equation}
where the notation $1_S(x)$ always means we are using indicator functions.
To ease the reader into the computations, we
recall that the Fourier transform of the indicator function of any rectangle $R:=[a, b] \times [c, d]$ is defined by:
\begin{equation}
\hat 1_{\mathcal{R}}(\xi) := \int_{\mathbb{R}^2} 1_{\mathcal{R}}(x) e^{-2\pi i \langle \xi, x \rangle} dx
=\int_a^b \int_c^d e^{-2\pi i (\xi_1 x_1 + \xi_2 x_2)}dx_1 dx_2.
\end{equation}
Now we may formally take the Fourier transform of both sides of \eqref{first identity of rectangles}.
In other words we
simply multiply both sides of \eqref{first identity of rectangles}
by the exponential function $e^{-2\pi i \langle \xi, x \rangle} $
and then integrate both sides over $\mathbb{R}^2$, to get:
\begin{equation} \label{sum.of.little.transforms}
\hat 1_{\mathcal{R}}(\xi) = \sum_{k=1}^N \hat 1_{\mathcal{R}_k}(\xi).
\end{equation}
In \eqref{sum.of.little.transforms}, we have used the fact that a $2$-dimensional integral over a $1$-dimensional line segment always vanishes, due to the fact that
a line segment has measure $0$ relative to the $2$-dimensional measure of the $2$-dimensional transform.
Let's compute one of these integrals, over a generic rectangle $\mathcal{R}_k := [a_1, a_2] \times [b_1, b_2]$:
\begin{align} \label{transform.of.a.rectangle}
\hat 1_{\mathcal{R}_k}(\xi) &:= \int_{\mathbb{R}^2} 1_{\mathcal{R}_k}(x) e^{-2\pi i \langle x, \xi \rangle} dx =
\int_{\mathcal{R}_k} e^{-2\pi i \langle x, \xi \rangle} dx \\
&= \int_{b_1}^{b_2} \int_{a_1}^{a_2} e^{-2\pi i \langle x, \xi \rangle} dx \\
&= \int_{a_1}^{a_2} e^{-2\pi i \xi_1 x_1} dx_1 \int_{b_1}^{b_2} e^{-2\pi i\xi_2 x_2} dx_2 \\
&= \frac{ e^{-2\pi i \xi_1 a_2} - e^{-2\pi i \xi_1 a_1} }{-2\pi i \xi_1}
\cdot \frac{ e^{-2\pi i \xi_2 b_2} - e^{-2\pi i \xi_2 b_1} }{-2\pi i \xi_2}\\ \label{last one}
&= \frac{1}{(-2\pi i)^2} \frac{ e^{-2\pi i (\xi_1 a_1 + \xi_2 b_1)} }{\xi_1 \xi_2}
(e^{-2\pi i \xi_1 (a_2-a_1)} - 1 ) (e^{-2\pi i \xi_2 (b_2-b_1)} -1),
\end{align}
valid for all $(\xi_1, \xi_2) \in \mathbb{R}^2$ except for the union of the two lines $\xi_1 = 0$ and $\xi_2 = 0$.
Considering the latter formula for the Fourier transform of a rectangle, we make the following leap of faith:
{\bf Claim}. \ Suppose that $\mathcal{R}$ is a rectangle whose sides are parallel to the axes. Then
\begin{equation} \label{First.case.of.Fourier.tiling.criterion}
\index{tiling}
\mathcal{R} \text{ is a nice rectangle } \iff \hat 1_{\mathcal{R}}\Big(\icol{1\{\bf 1}} \Big) = 0.
\end{equation}
Proof of the claim. \ We consider the last equality \eqref{last one}.
We see that
\begin{equation} \label{twofactors}
\hat 1_{\mathcal{R}_k}(\xi) =0 \iff (e^{-2\pi i \xi_1 (a_2-a_1)} - 1 ) (e^{-2\pi i \xi_2 (b_2-b_1)} -1)=0,
\end{equation}
which is equivalent to having either $e^{-2\pi i \xi_1 (a_2-a_1)} =1$, or $e^{-2\pi i \xi_2 (b_2-b_1)} =1$.
But we know that due to Euler,
$e^{2\pi i \theta} = 1$ if and only if $\theta \in \mathbb{Z}$ (Exercise \ref{TrivialExponential}), so we have
\begin{equation} \label{the last bit}
\hat 1_{\mathcal{R}}(\xi) = 0 \ \iff \xi_1 (a_2-a_1) \in \mathbb{Z} \ \text{ or } \ \xi_2 (b_2-b_1) \in \mathbb{Z}.
\end{equation}
Now, if $\mathcal{R}$ is a nice rectangle, then one of its sides is an integer, say $a_1 - a_2 \in \mathbb{Z}$ without loss of generality. Therefore
$\xi_1 (a_2-a_1) \in \mathbb{Z}$ for $\xi_1 = 1$, and by \eqref{the last bit}, we see that
$ \hat 1_{\mathcal{R}}\Big(\icol{1\{\bf 1}} \Big) = 0$.
Conversely, if we assume that $ \hat 1_{\mathcal{R}}\Big(\icol{1\{\bf 1}} \Big) = 0$, then by
\eqref{the last bit} either \\
$1\cdot (a_2-a_1) \in \mathbb{Z} \text{ or } 1\cdot (b_2-b_1) \in \mathbb{Z}$, proving the claim.
\medskip
To finish the proof of the theorem, by hypothesis each little rectangle $\mathcal{R}_k$ is a nice rectangle, so by the claim above it satisfies
$ \hat 1_{\mathcal{R}_k}\Big(\icol{1\{\bf 1}} \Big) = 0$.
Returning to \eqref{sum.of.little.transforms}, we see that therefore
$\hat 1_{\mathcal{R}}(\xi) = \sum_{k=1}^N \hat 1_{\mathcal{R}_k}(\xi) = 0$, for $\xi = \icol{1\{\bf 1}}$, and using the claim again (the converse part of it this time), we see that $\mathcal{R}$ must be nice.
\end{proof}
The proof of Theorem \ref{Integer.Side.Rectangle} was simple and elegant, motivating the use of Fourier transforms of polytopes in the ensuing chapters. The claim, namely equation \eqref{First.case.of.Fourier.tiling.criterion}, offers an intriguing springboard for deeper investigations - it tells us that we can convert a geometric statement about tiling into a purely analytic statement about the vanishing of a certain integral transform. Later, when we learn about
Theorem \ref{zero set of the FT of a polytope}, we will see that this small initial success of \eqref{First.case.of.Fourier.tiling.criterion} is part of a larger theory. This is the beginning of a beautiful friendship.......
\bigskip
\section{Conventions, and quick basics}
We mention some conventions that we use throughout the book.
First, we note that whenever we are given a complex-valued function $f:\mathbb{R}^d \rightarrow \mathbb{C}$, we may write $f$ in terms of its
real and imaginary parts: $f(x):= u(x) + i v(x)$. The {\bf integral of such an $f$} is defined by
\begin{equation}
\int_{\mathbb{R}^d} f(x) dx := \int_{\mathbb{R}^d} u(x) dx + i \int_{\mathbb{R}^d} v(x) dx,
\end{equation}
so that all of our Fourier transforms are really reduced to the usual integration of real-valued functions on Euclidean space
(see Exercise \ref{definition of complex integral}). This is good news for the reader, because even though we see complex functions in the integrand, elementary calculus suffices.
\medskip
Let $S\subset \mathbb{R}^d$ be a set.
For our purposes, we may call $S$ a {\bf measurable} set if the integral
$
\int_S dx \text{ exists},
$
and in this case we define
\[
{\rm measure}(S) := \int_S dx.
\]
Equivalently, we may call $S$ measurable if the indicator function $1_S$ is an integrable function, by definition of the integral.
A set $S$ is said to have {\bf measure zero} if
\[
\int_S dx = 0.
\]
In $\mathbb{R}$, for example, we may also define a set $S$ of measure $0$ by saying that, given any $\varepsilon >0$,
there exists a countable collection of open intervals $I_n$ that cover all of $S$, and whose total length satisfies
$\sum_{n=1}^\infty |I_n|< \varepsilon$. But we will assume the reader knows the definition(s) of an integral
(either the Riemann integral or the Lebesgue integral),
circumventing discussions about $\sigma$-algebras of sets, so that the background required of the reader is kept to a minimum.
The point we want to make here is that most things are in fact easier than the reader may have previously thought.
We say that a statement $A(x)$ concerning points $x\in \mathbb{R}^d$
{\bf holds for almost every} $x\in \mathbb{R}^d$ (we also use the words {\bf almost everywhere})
if the set of $x\in \mathbb{R}^d$ for which $A(x)$ is false is a set of measure $0$.
In this connection, we will assume the following fact from real analysis:
\[
\int_{\mathbb{R}^d} f(x) dx = \int_{\mathbb{R}^d} g(x) dx \ \iff \ f = g \text{ almost everywhere},
\]
which means that $f(x) = g(x)$ for all $x\in \mathbb{R}^d$, except perhaps on a set of measure $0$.
We also mention our convention/notation for some definitions. Whenever we want to define a new object called $N$, in terms of some combination of previously known mathematical objects called $K$, we will use the notation
\[
N:= K.
\]
For any set $A\subset \mathbb{R}^d$, we define the {\bf closure} of $A$ as the
the smallest (w.r.t containment) closed set that contains $A$, written as $\closure A$.
We define the {\bf interior} of $A$ as the set of all points $x \in A$ such that
there exists a ball of some positive radius $\varepsilon$, centered at $x$, with $B_\varepsilon(x) \subset A$.
We define the {\bf boundary} of $A$, written as $\partial A$, by
\[
\partial A:= \closure A \setminus \interior A.
\]
An important concept is that of the support of a function $f:\mathbb{R}^d\rightarrow \mathbb{C}$, defined by
\begin{equation} \label{def of support}
\supp(f):= \closure \{ x \in \mathbb{R}^d \bigm | f(x) \not=0 \}.
\end{equation}
With this definition, we have for example:
\[
\supp(1_{[0, 1]}) = \supp(1_{(0, 1)}) = [0, 1].
\]
We will also say that a function $f$ is {\bf compactly supported} if the support of $f$ is a
compact set $C$. In particular this means that $f$ vanishes outside of $C$.
\bigskip
\section*{Notes}
\begin{enumerate}[(a)]
\item This little chapter was motivated by the lovely article written by Stan Wagon \cite{StanWagon}, which gives $14$ different proofs of Theorem \ref{Integer.Side.Rectangle}.
The article \cite{StanWagon} is important because it shows how tools from one field can leak into another field, and thus may lead to important discoveries in the future.
\item In a related direction, we might wonder which polygons, and more generally which polytopes, tile Euclidean space by translations with a lattice. It turns out (Theorem~\ref{zero set of the FT of a polytope}) that this question is equivalent to the statement that the Fourier transform of ${\mathcal P}$ vanishes on a (dual) lattice.
\item In the context of the Hilbert space of functions $L^2([0,1])$, Exercise \ref{orthogonality for exponentials}
is one step towards showing that the set of exponentials $\{ e_n(x) \}_{n \in \mathbb{Z}}$ is a basis for $L^2([0,1])$. Namely, the identity above shows that these basis elements are orthogonal to each other - their inner product
$\langle e_a, e_b \rangle := \int_0^1 e_a(x) \overline{e_b(x)} dx $ vanishes
for integers $a \not= b$. Thus, the identity of Exercise \ref{orthogonality for exponentials}
is often called the orthogonality relations
for exponentials, over $L^2([0,1])$. To show that they {\it span} the space of functions in $L^2([0,1])$ is a bit harder, but see \cite{Travaglini} for details.
\item The question in Exercise \ref{Erdos lattice partition problem} for $\mathbb{Z}$ was originally asked by Paul Erd\"os
\index{Erd\"os, Paul} in $1951$, and has an affirmative answer. This question also has
higher-dimensional analogues:
\begin{quote}
Suppose we give a partition of the integer lattice $\mathbb{Z}^d$
into a finite, disjoint union of translated sublattices.
Is it always true that at least two of these sublattices are translates of each other?
\end{quote}
The answer is known to be false for $d \geq 3$, but is still unsolved for $d=2$ (see \cite{FeldmanProppRobins},\cite{BorodzikNguyenRobins}).
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
If there is a problem you can't solve, then there is an easier problem you can't solve: find it.
-- George Polya
\end{quote}
\medskip
\begin{prob} $\clubsuit$ \label{TrivialExponential}
Show that if $x \in \mathbb{C}$, then $e^{2\pi i x} = 1$ if and only if $x \in \mathbb{Z}$.
\end{prob}
\medskip
\begin{prob} \label{bound of the exponential function}
Show that $|e^z| \leq e^{|z|}$, for all complex numbers $z \in \mathbb{C}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{orthogonality for exponentials}
\index{orthogonality of exponentials in $L^2([0,1])$}
Here we prove the {\bf orthogonality relations for the exponential functions} defined by
$e_n(x) := e^{2\pi i n x}$, for each integer $n$.
Recall that the complex conjugate of any complex number $x + iy$ is defined by
\[
\overline{x+ iy} := x - iy,
\]
so that $\overline{e^{i \theta}} := e^{-i \theta}$ for real $\theta$.
Prove that for all integers $a,b$:
\begin{equation}
\int_0^1 e_a(x) \overline{e_b(x)} dx =
\begin{cases}
1 & \mbox{if } a=b \\
0 & \mbox{if not}.
\end{cases}
\end{equation}
\end{prob}
\medskip
\begin{prob} \label{definition of complex integral}
Here the reader may gain some practice with the definitions of integrals that use complex-valued integrands $f(x) := u(x) + iv(x)$.
We recall for the reader the following definition:
\begin{equation} \label{real.and.imaginary.parts}
\int_{\mathbb{R}^d} f(x) dx := \int_{\mathbb{R}^d} \left( u(x) + i v(x) \right) dx := \int_{\mathbb{R}^d} u(x) dx + i \int_{\mathbb{R}^d} v(x) dx,
\end{equation}
a linear combination of two real-valued integrals.
Recalling that by definition,
\[
\hat 1_{[0,1]}(\xi) := \int_{[0,1] } e^{-2\pi i \xi x} dx,
\]
show directly from definition \ref{real.and.imaginary.parts} and from Euler's identity
$e^{i\theta} = \cos \theta + i \sin \theta$,
that for any nonzero $\xi \in \mathbb{R}$, we have
\begin{equation*}
\int_{[0,1]} e^{-2\pi i \xi x} dx = \frac{e^{-2\pi i \xi } -1}{ -2\pi i \xi}.
\end{equation*}
{\rm Notes. Another way of thinking about this exercise is that it extends the `Fundamental theorem of calculus' to complex-valued functions in a rather easy way. The anti-derivative of the integrand $f(x):= e^{-2\pi i \xi x}$ is $F(x):= \frac{e^{-2\pi i \xi x}}{-2\pi i \xi}$, and we are saying that it is ok to use it in place of the usual anti-derivative in Calculus $1$ - it is consistent with definition \ref{real.and.imaginary.parts}.
In the future, we generally do not have to break up complex integrals into their real and imaginary parts,
because we can make use of the fact that antiderivatives of complex-valued functions are often simple, such as the one in this example.
We also note that this is {\bf not} calculus with a complex variable, because the {\bf domains of our integrands}, as well as the measures we are using throughout this book, in order to integrate, are always defined over real Euclidean space $\mathbb{R}^d$. This means we are still using basic Calculus.
}
\end{prob}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{6thRootsOfUnity}
\end{center}
\caption{The $6$'th roots of unity, with $\zeta:= e^{\frac{2\pi i}{6}}$. Geometrically, Exercise \ref{SumOfRootsOfUnity} tells
us that their center of mass is the origin. } \label{6th roots of unity}
\end{figure}
\medskip
\begin{prob} $\clubsuit$ \label{SumOfRootsOfUnity}
We recall that the $N$'th roots of unity are by definition the set of $N$ complex solutions to $z^N =1$, and are given by the set
$\{e^{2\pi i k/N} \mid k = 0, 1, 2, \dots, N-1 \}$ of points on the unit circle. Prove that the sum of all of the $N$'th roots of unity vanishes. Precisely, fix any positive integer $N\geq 2$,
and show that
\[
\sum_{k = 0}^{N-1} e^{\frac{2\pi i k}{N}} = 0.
\]
\end{prob}
\medskip
\begin{prob} \label{DivisibilityUsingExponentials}
Prove that, given positive integers $M, N$, we have
\[
\frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i kM}{N}} =
\begin{cases}
1 & \mbox{if } N \mid M \\
0 & \mbox{if not}.
\end{cases}
\]
\end{prob}
Notes. This result is sometimes referred to as {\bf ``the harmonic detector"} for detecting when a rational
number $\frac{M}{N}$ is an integer; that is, it assigns a value of $1$ to the sum if $\frac{M}{N} \in \mathbb{Z}$, and it assigns a value
of $0$ to the sum if
$\frac{M}{N} \not\in \mathbb{Z}$.
\medskip
\begin{prob} $\clubsuit$ \label{Orthogonality.for.roots.of.unity} \index{orthogonality, roots of unity}
Here we prove the {\bf orthogonality relations for roots of unity}. Namely, fix any two nonnegative integers $a,b$,
and prove that
\begin{equation} \label{12345}
\frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i ka}{N}} e^{-\frac{2\pi i kb}{N}} =
\begin{cases}
1 & \mbox{if } a \equiv b \mod N \\
0 & \mbox{if not}.
\end{cases}
\end{equation}
\end{prob}
Notes. In a later chapter on Euclidean lattices (Chapter \ref{chapter.lattices}), we will see that the identity
\ref{12345} is a special case of the
more general orthogonality relations for characters on lattices. From this perspective, this exercise is the orthogonality relations on the finite cyclic group $\mathbb{Z}/{N\mathbb{Z}}$. There are more general orthogonality relations
for characters of group representations, which play an important role in Number Theory.
\medskip
\begin{prob} \label{trick-write an integer as a product with roots of unity}
Show that for any positive integer $n$, we have
\[
n = \prod_{k=1}^{n-1} (1-\zeta^k),
\]
where $\zeta:= e^{2\pi i / n}$.
\end{prob}
\medskip
\begin{prob} \label{PrimitiveRootsOfUnity} \index{root of unity, primitive}
An $N$'th root of unity is called a {\bf primitive root of unity} if it is not a $k$'th root of unity for some smaller
positive integer $k < N$.
Show that the primitive $N$'th roots of unity are precisely the numbers $e^{2\pi i k/N}$ for which
$\gcd(k, N) = 1$.
\end{prob}
\medskip
\begin{prob} \label{SumOfPrimitiveRootsOfUnity}
The M\"obius $\mu$-function
\index{M\"obius $\mu$-function}
is defined by:
\[
\mu(n) := \begin{cases}
(-1)^{\text{ number of distinct prime factors of } n} & \mbox{if } n > 1 \\
1 & \mbox{if } n=1.
\end{cases}
\]
Prove that the sum of all of the primitive $N$'th roots of unity is equal to the
M\"obius $\mu$-function, evaluated at $N$:
\begin{equation}
\sum_{1\leq k < N \atop \gcd(k, N) = 1} e^{\frac{2\pi i k}{N}} = \mu(N).
\end{equation}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{extension of exponential}
We follow the Weierstrassian approach to defining the complex exponential $e^{z}$
for all complex $z \in \mathbb{C}$:
\begin{equation}
e^{z} := \sum_{n=0}^\infty \frac{1}{n!} z^n,
\end{equation}
which converges absolutely for all $z\in \mathbb{C}$. We also have the (Weierstrassian) definitions
of $\cos z$ and $\sin z$:
\[
\cos z:= \sum_{n=0}^\infty \frac{1}{(2n)!} (-1)^n z^{2n}, \quad
\sin z:= \sum_{n=1}^\infty \frac{1}{(2n-1)!} (-1)^{n-1} z^{2n-1},
\]
both converging absolutely again for all $z \in \mathbb{C}$. Prove that Euler's formula
has the extension:
\[
e^{iz} = \cos z + i \sin z,
\]
valid for all $z \in \mathbb{C}$.
\end{prob}
Notes. \ Karl Weierstrass developed a rigorous and beautiful theory of real and complex functions, beginning with such a power series approach.
\medskip
\begin{prob}
Here the reader needs to know a little bit about the quotient of two groups (this is one of the few exercises that assumes group theory). We
prove that the group of `real numbers mod $1$' under addition, is
isomorphic to the unit circle, under multiplication of complex numbers. Precisely, we can define
$h: \mathbb{R} \rightarrow S^1$ by $h(x) := e^{2\pi i x}$.
\begin{enumerate}[(a)]
\item We recall the definition of the kernel of a map, namely $ker(h):= \{ x\in \mathbb{R} \mid h(x) = 1\}$. Show that $ker(h) = \mathbb{Z}$.
\item Using the first isomorphism Theorem for groups, show that
$\mathbb{R}/\mathbb{Z}$ is isomorphic to the unit circle $S^1$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
Using gymnastics with roots of unity, we recall here a very classical solution to the problem of finding the roots of a cubic polynomial.
\begin{enumerate}[(a)]
\item Let $\omega:= e^{2\pi i/3}$, and show that we have the polynomial identity:
\[
(x + a + b)( x + \omega a + \omega^2 b) (x + \omega^2 a + \omega b)
= x^3 - 3abx + a^3 + b^3.
\]
\item Using the latter identity, solve the cubic polynomial:
$x^3 - px + q = 0$ by substituting $p = 3ab$ and $q= a^3 + b^3$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}\label{zeros of the sin function}
Thinking of the function $\sin(\pi z)$ as a function of a complex variable $z\in \mathbb C$, show that its zeros are precisely the set of integers $\mathbb{Z}$.
\end{prob}
\medskip
\begin{prob}
Here we give another equivalent condition for a rectangle in Theorem \ref{Integer.Side.Rectangle} to be a nice rectangle, using the same
definitions as before.
Let's call $\xi \in \mathbb{Z}^2$ a {\bf generic} integer point
if $\xi$ is not orthogonal to any of the edges of $\mathcal{R}$. In other words, a generic integer vector satisfies
$\langle \xi, p \rangle \not=0$, for all $p\in \mathcal{R}$, and in particular $p=0$ is not generic, nor is any
point $p$ on the $x$-axis or the $y$-axis.
Then
\begin{equation}
\mathcal{R} \text{ is a nice rectangle } \iff \hat 1_{\mathcal{R}}(\xi) = 0, \text{ for all generic points } \xi \in \mathbb{Z}^2.
\end{equation}
\end{prob}
\medskip
\begin{prob}[Erd\"os, 1951] \label{Erdos lattice partition problem}
\rm{
Erd\"os asked: ``Can the set $\mathbb{Z}_{>0}$ of all positive integers be partitioned (that is, written as a disjoint union) into a finite number of
arithmetic progressions, such that no two of the arithmetic progressions will have the same common difference?''
Suppose that we have a list of disjoint arithmetic progressions, each with its common difference $a_k$:
\[
\{ a_1 n + b_1 \mid n \in \mathbb{Z}\}, \dots, \{ a_N n + b_N \mid n \in \mathbb{Z}\},
\]
where $a_1 \leq a_2 \leq \cdots \leq a_N$, and $N\geq 2$.
Prove that in any such partitioning of the integers, there are at least two arithmetic progressions that have the same maximal $a_N$.
Notes. For example, if we write $\mathbb{Z} = \{ 4 n + 1 \mid n \in \mathbb{Z}\} \cup \{ 2 n \mid n \in \mathbb{Z}\} \cup \{ 4 n + 3 \mid n \in \mathbb{Z}\}$, a disjoint union of
$3$ arithmetic progressions, then we see
that the common difference of $4$ appears twice. Erd\"os noticed that such a phenomenon must always occur.
(See also Exercise \ref{Extension of Erdos to dimension d} for an extension to lattices in $\mathbb{R}^d$).
}
\end{prob}
\chapter{Examples that nourish the theory} \label{Chapter.Examples}
\begin{quote}
``A pint of example is worth a gallon of advice.''
-- Anonymous
\end{quote}
\smallskip
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{Bernoulli}
\end{center}
\caption{The first periodic Bernoulli polynomial $P_1(x)$, sometimes called the sawtooth function,
which turns out to be one of the building blocks of integer point enumeration in polytopes }
\label{FirstBernoulli}
\end{figure}
\section{Intuition}
One way to think about the Fourier transform of a polytope ${\mathcal P} \subset \mathbb{R}^d$ is that it simultaneously captures all of the moments of ${\mathcal P}$, thereby uniquely defining ${\mathcal P}$. Here we begin concretely by computing some Fourier transforms of various polytopes in dimensions $1$ and $2$, as well as the Fourier transforms of some simple families of polytopes in dimension $d$ as well.
The $2$-dimensional computations will get the reader more comfortable with the basics. In later chapters, once we learn a little more theory, we will return to these families of polytopes and compute some of their Fourier transforms in general.
We also see, from small examples, that the Bernoulli polynomials immediately enter into the picture, forming natural building blocks.
In this chapter we compute Fourier transforms without thinking too much about convergence issues, to let the reader run with the ideas.
But commencing with the next chapter, we will be more rigorous when using Poisson summation, and with convergence issues.
\section{Dimension $1$ - the classical sinc function}
\bigskip
We begin by computing the classical $1$-dimensional example of the Fourier transform \index{Fourier transform}
of the symmetrized unit interval ${\mathcal P}:= [-\frac{1}{2}, \frac{1}{2}]$:
\begin{align} \label{ClassicalExample}
\hat 1_{{\mathcal P}}(\xi) & := \int_{\mathbb{R}} 1_{\mathcal P}(x) \ e^{-2\pi i x \xi } dx \\
&= \int_{[-\frac{1}{2}, \frac{1}{2}]} e^{-2\pi i x \xi } dx \\
& = \frac{e^{-2\pi i \left( \frac{1}{2} \right) \xi} - e^{-2\pi i \left( \frac{-1}{2} \xi\right) } }{-2\pi i \xi} \\ \label{sinc}
& = \frac{ \cos (-\pi \xi) + i \sin(-\pi \xi) - (\cos(\pi \xi) + i \sin(\pi \xi)) }{-2\pi i \xi} \\ \label{sinc function formula}
&= \frac{\sin(\pi \xi)}{\pi \xi},
\end{align}
valid for all $\xi \not= 0$. The latter function is also known as the {\bf sinc function}.
\index{Sinc function} We notice that $\xi = 0$ is a removable singularity, so that we may define the continuous sinc-function by
\begin{equation}\label{SincFunction}
{\rm{sinc}}(x):= \begin{cases}
\frac{\sin(\pi x)}{\pi x}, &\mbox{if } x \not= 0 \\
1 & \mbox{if } x= 0,
\end{cases}
\end{equation}
which is in fact infinitely smooth, via Lemma \ref{FT of a polytope is entire} below.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.6in]{sinc1}
\end{center}
\caption{The function ${\rm{sinc}}(x)$, which is Fourier transform of the $1$-dimensional
polytope ${\mathcal P} = [-\frac{1}{2}, \frac{1}{2}]$. }
\label{sinc.pic} \index{sinc function}
\end{figure}
\bigskip
\section{The Fourier transform of ${\mathcal P}$ as a complete invariant}
\label{Fourier inversion}
The main goal of this section is to state Lemma \ref{complete invariance of the FT}, which tells us that all of the information
about a polytope is contained in its Fourier transform. To that end, we
introduce
the inverse Fourier transform, \index{inverse Fourier transform} often called the {\bf Fourier inversion formula}.
We'd like to see the fundamental fact that under certain conditions,
the Fourier transform is invertible. First, we call a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ absolutely integrable if
$\int_{\mathbb{R}^d} | f(x) | dx < \infty$, and we write this as $f \in L^1(\mathbb{R}^d)$.
\begin{thm} \label{thm:Inverse Fourier transform}
Given a function $f$ such that both $f \in L^1(\mathbb{R}^d)$ and $\hat f \in L^1(\mathbb{R}^d)$, we have
\begin{equation}\label{second version of Fourier inversion}
f(x) = \int_{\mathbb{R}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle} d\xi,
\end{equation}
for all $x \in \mathbb{R}^d$.
\hfill $\square$
\end{thm}
(see \cite{Travaglini} for a proof).
Equation \eqref{second version of Fourier inversion} tells us that the inverse Fourier transform ${\mathcal F}^{-1}$ exists, and is almost equal to ${\mathcal F}$ itself. A moment's thought reveals that we may rewrite \eqref{second version of Fourier inversion} in the following useful form:
\begin{equation} \label{InverseFourierTransform2}
({\mathcal F} \circ {\mathcal F})f (x) = f(-x).
\end{equation}
\bigskip
\begin{example}\label{Integral.of.sinc}
\rm{
A famous and historically somewhat tricky integral formula for the sinc function is the following fact:
\begin{equation} \label{area under sinc}
\int_{-\infty}^\infty {\rm{sinc}}(x) dx := \int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} dx = 1,
\end{equation}
also known as `the Dirichlet integral'. \index{Dirichlet integral}
The careful reader might notice that the latter integrand is not absolutely convergent, which means that
$\int_{-\infty}^\infty \Big| \frac{\sin(\pi x)}{\pi x} \Big| dx = \infty$ (Exercise \ref{divergence of |sinc|}). So we have to specify what we really mean
by the identity \eqref{area under sinc}. The rigorous claim is:
\begin{equation*}
\lim_{N\rightarrow \infty}
\int_{-N}^N \frac{\sin(\pi x)}{\pi x} dx =1.
\end{equation*}
Let's see an intuitive derivation of \eqref{area under sinc}, where we will be fast-and-loose for the moment. Using \eqref{ClassicalExample},
we've seen above that the Fourier transform of the indicator function of the interval ${\mathcal P} := [-\frac{1}{2}, \frac{1}{2}]$ is:
\begin{equation}
{\mathcal F}(1_{{\mathcal P}})(\xi) = \frac{\sin(\pi \xi)}{\pi \xi},
\end{equation}
so that
\begin{equation}
{\mathcal F} \left( \frac{\sin(\pi \xi)}{\pi \xi} \right)
= ({\mathcal F} \circ {\mathcal F})(1_{{\mathcal P}})(\xi)
= 1_{ {\mathcal P} }(-\xi).
\end{equation}
Using the definition of the Fourier transform, the latter identity is:
\begin{equation}
\int_{\mathbb{R}} \frac{\sin(\pi x)}{\pi x} e^{-2\pi i \xi x} dx
= 1_{{\mathcal P}}(\xi),
\end{equation}
and now evaluating both sides at $\xi = 0$ gives us \eqref{area under sinc}.
Although this derivation appears very convincing, it would not make it past the rigor police. So why not? It is because we applied the Fourier inversion formula
to a function that was {\bf not} in $L^1(\mathbb{R})$, namely the sinc function. So we owe it to ourselves to pursue a rigorous
approach by showing that
\begin{equation}
\lim_{N\rightarrow \infty} \int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} e^{-2\pi i \langle \xi, x \rangle} d\xi = 1_ {[-\frac{1}{2}, \frac{1}{2}]}(x),
\end{equation}
whose validity would give us a variation on Fourier inversion, for a function that is not in $L^1(\mathbb{R})$, namely $\hat 1_ {[-\frac{1}{2}, \frac{1}{2}]}(\xi)= {\rm{sinc}}(\xi)$.
This is tricky business, but such an endeavor is taken up in Exercise \ref{rigorous inversion formula for sinc}.
}
\hfill $\square$
\end{example}
We can extend Example \ref{Integral.of.sinc} in a natural way to all Fourier pairs of functions, $\{f(x), \hat f(\xi)\}$, provided
that we may apply Fourier inversion, as follows.
Simply let $x=0$ in \eqref{second version of Fourier inversion}, to get:
\begin{equation} \label{IntegralTrick}
f(0)=\int_{\mathbb{R}^d} \hat f(x) dx.
\end{equation}
To summarize, Example \ref{Integral.of.sinc} is simply identity \eqref{IntegralTrick} with
$f(x) := 1_{[-\frac{1}{2}, \frac{1}{2}]}(x)$.
Another nice - and very useful - fact about the Fourier transform
of a polytope is that it is an entire function, meaning that it is differentiable everywhere. This differentiability
is already observable in the sinc function above, with its removable singularity at the origin.
\begin{lem} \label{FT of a polytope is entire}
Let ${\mathcal P} \subset \mathbb{R}^d$ be a $d$-dimensional polytope. Then $\hat 1_{\mathcal P}(\xi)$ is an entire function of $\xi \in \mathbb{C}^d$.
\end{lem}
\begin{proof}
Because ${\mathcal P}$ is compact, we can safely differentiate under the integral sign (this is a special case of Lebesgue's Dominated Convergence Theorem). Namely, for any coordinate variable $\xi_k$, we have:
$\frac{d}{d\xi_k} \int_{{\mathcal P}} e^{-2\pi i \langle \xi, x \rangle} dx= \int_{{\mathcal P}} \frac{d}{d\xi_k}e^{-2\pi i \langle \xi, x \rangle} dx
= 2\pi i \int_{{\mathcal P}} x_k e^{-2\pi i \langle \xi, x \rangle} dx$, and it is clear that all possible derivatives exist in this manner, because the integrand is infinitely smooth.
\end{proof}
We also have the very fortuitous fact that the Fourier transform of any polytope ${\mathcal P} \subset \mathbb{R}^d$ is a complete invariant, in the following sense.
We recall that by definition a polytope is in particular a closed set.
\begin{lem} \label{complete invariance of the FT}
Let ${\mathcal P}\subset \mathbb{R}^d$ be a polytope. Then
$\hat 1_{\mathcal P}(\xi)$ uniquely determines ${\mathcal P}$. Precisely, given any two $d$-dimensional polytopes $P, Q\subset \mathbb{R}^d$,
we have
\[
\hat 1_{\mathcal P}(\xi) = \hat 1_{Q}(\xi) \text{ for all } \xi \in \mathbb{R}^d \ \iff \ {\mathcal P} = Q.
\]
In other words, for any polytope ${\mathcal P}$, its Fourier transform $\hat 1_{\mathcal P}$ uniquely determines the polytope.
\end{lem} \label{FT.complete invariant}
\begin{proof} (outline)
If ${\mathcal P}=Q$, it is clear that $\hat 1_{\mathcal P}(\xi) = \hat 1_{Q}(\xi)$ for all $\xi \in \mathbb{R}^d$. Conversely, suppose that
$\hat 1_{\mathcal P}(\xi) = \hat 1_{Q}(\xi)$ for all $\xi \in \mathbb{R}^d$. Using Fourier inversion (see \cite{Podkorytov}),
we may take the Fourier transform of both sides of the latter equation to get $1_{\mathcal P}(-\xi) = 1_Q(-\xi)$, for all $\xi \in \mathbb{R}^d$.
\end{proof}
The reason that the proof above is only an outline is because we have applied the Fourier inversion formula to
$\hat 1_{\mathcal P}$, which is not absolutely integrable
(see Exercise \ref{the FT of 1_P is not in L^1} below, in Chapter \ref{Stokes' formula and transforms}).
However, there is a nice version of the Fourier inversion formula, due to Podkorytov and Minh, that
holds for such functions and nicely patches up this hole (see \cite{Podkorytov}).
The reason we've put Lemma \ref{complete invariance of the FT}
so early in the text is because it offers an extremely strong motivation for the study of Fourier transforms of polytopes, showing
that they are complete invariants.
A fascinating consequence of Lemma \ref{complete invariance of the FT} is that when we take the Fourier transform of a polytope, then {\bf all of the combinatorial and geometric information} of ${\mathcal P}$ is contained in the formula of its transform......
So we may begin to create a complete dictionary between the geometry and combinatorics of a polytope in the space domain, and its Fourier transform in the frequency domain.
\bigskip
\section{Bernoulli polynomials} \index{Bernoulli polynomial}
We introduce the Bernoulli polynomials, which turn out to be a sort of ``glue'' between
discrete geometry and number theory, as we will see throughout the book.
The {\bf Bernoulli polynomials} \index{Bernoulli polynomial}
are defined via the following generating function:
\begin{equation} \label{generating function for Bernoulli polynomials}
\frac{te^{xt}}{e^t-1} = \sum_{k =0}^\infty B_k(x) \frac{t^k}{k!}.
\end{equation}
It's fruitful to sometimes restrict the Bernoulli polynomials to the unit interval $[0,1]$,
and then periodize them. In other words, using
\[
\{x\} := x - \lfloor x \rfloor,
\]
the fractional part of $x$, \index{fractional part}
we may define the $n$'th {\bf periodic Bernoulli polynomial}: \index{periodic Bernoulli polynomial}
\begin{equation} \label{definition of periodic Bernoulli polys}
P_n(x) := B_n(\{x\}),
\end{equation}
for $n\geq 2$. Since $P_n(x)$ is periodic on $\mathbb{R}$ with period $1$,
it has a Fourier series, and in fact:
\begin{equation}
P_n(x) = -\frac{n!}{(2\pi i)^n} \sum_{k \in \mathbb{Z} - \{0\}} \frac{e^{2\pi i k x}}{k^n},
\end{equation}
valid for $x \in \mathbb{R}$ (Exercise \ref{Bernoulli Polynomials}).
When $n=1$, we have the first Bernoulli polynomial
\[
P_1(x):= x - \lfloor x \rfloor - \frac{1}{2},
\]
which
is very special (see Figure \ref{FirstBernoulli}). For one thing,
it is the only periodic Bernoulli polynomial that is not continuous everywhere,
and we note that its Fourier series does not converge absolutely, although it is quite appealing:
\begin{equation} \label{FirstBernoulliPolynomial}
P_1(x) = -\frac{1}{2\pi i} \sum_{k \in \mathbb{Z} - \{0\}} \frac{e^{2\pi i k x}}{k},
\end{equation}
valid for all $x \notin \mathbb{Z}$. Hence special care must be taken with $P_1(x)$. Exercises \ref{brute force Bernoulli polys} through
\ref{vanishing identity for Beroulli numbers} illustrate some of the important properties of
these polynomials. Exercise \ref{rigorous convergence of P_1(x)} provides a rigorous proof
of the convergence of \eqref{FirstBernoulliPolynomial}.
\begin{example} \rm{
The first few Bernoulli polynomials are:
\begin{align}
B_0(x) &= 1 \\
B_1(x) &= x - \frac{1}{2} \\
B_2(x) &= x^2 - x + \frac{1}{6} \\
B_3(x) &= x^3 - \frac{3}{2} x^2 +\frac{1}{2} x \\
B_4(x) &= x^4 -2x^3 + x^2 - \frac{1}{30} \\
B_5(x) &= x^5 - \frac{5}{2} x^4 + \frac{5}{3} x^3 - \frac{1}{6} x \\
B_6(x) &= x^6 - 3x^5 + \frac{5}{2} x^4 - \frac{1}{2} x^2 + \frac{1}{42} \label{B_6}
\end{align}
The {\bf Bernoulli numbers} are defined to be the constant terms of the Bernoulli polynomials:
\[
B_k := B_k(0).
\]
The first few Bernoulli numbers are: \\
\[
B_0 = 1, \ B_1 = -\frac{1}{2}, \ B_2 = \frac{1}{6}, \ B_3 = 0, \ B_4 = - \frac{1}{30}, \ B_5 = 0, \ B_6 = \frac{1}{42}.
\]
It follows quickly from
the definition \ref{generating function for Bernoulli polynomials}
above that for odd $k \geq 3$, $B_k = 0$ (Exercise \ref{odd Bernoulli numbers}).
From the generating function \ref{generating function for Bernoulli polynomials}
the Bernoulli numbers are defined via
\begin{equation} \label{Def. of Bernoulli numbers}
\frac{t}{e^t-1} = \sum_{k =0}^\infty B_k \frac{t^k}{k!}.
\end{equation}
}
\hfill $\square$
\end{example}
Historically, the first appearance of the Bernoulli polynomials occurred while Jakob Bernoulli tried to
compute sums of powers of integers. In particular, Bernoulli showed that:
\[
\sum_{k=1}^{n-1} k^{d-1} = \frac{ B_d(n) - B_d }{ d },
\]
for all integers $d \geq 1$ and $n \geq 2$ (Exercise \ref{historical origin of Bernoulli poly}).
An interesting identity that allows us to compute the Bernoulli numbers recursively rather quickly is:
\[
\sum_{k=0}^n {n+1 \choose k}B_k = 0,
\]
valid for all $n \geq 1$ (Exercise \ref{vanishing identity for Beroulli numbers}).
Some of the most natural, and beautiful, Fourier series arise naturally from the periodized Bernoulli polynomials. The following intuitive application of the Poisson summation formula already suggests an initial connection between periodized Bernoulli polynomials and Fourier transforms of polytopes - even in dimension $1$.
\begin{example} [Intuitive Poisson summation] \label{intuitive Poisson example}
\index{Poisson summation formula}
\rm{
In this example we allow ourselves to be completely intuitive, and unrigorous at this moment,
but often such arguments are useful in pointing us to their rigorous counterparts.
Consider the $1$-dimensional polytope ${\mathcal P}:= [a,b]$, and restrict attention to the case
of $a, b \not\in \mathbb{Z}$. If we could use the Poisson summation formula
\[
\sum_{n \in \mathbb{Z}^d} f(n) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi),
\]
applied to the function $f(x):= 1_{\mathcal P}(x)$, then we would get:
\begin{align*}
\sum_{n \in \mathbb{Z}} 1_{\mathcal P}(n) &``=\text{''} \sum_{\xi \in \mathbb{Z}} \hat 1_{\mathcal P}(\xi)\\
&``=\text{''} \
\hat 1_{\mathcal P}(0)+\sum_{\xi \in \mathbb{Z} - \{0\}} \frac{e^{-2\pi i \xi b} - e^{-2\pi i \xi a} }{-2\pi i \xi} \\
&``=\text{''} \
(b-a) -\frac{1}{2\pi i} \sum_{\xi \in \mathbb{Z} - \{0\}} \frac{e^{-2\pi i \xi b}}{\xi}
+ \frac{1}{2\pi i} \sum_{\xi \in \mathbb{Z}-\{0\}} \frac{e^{-2\pi i \xi a}}{\xi} \\
&``=\text{''} \
(b-a) + \frac{1}{2\pi i} \sum_{\xi \in \mathbb{Z} - \{0\}} \frac{e^{2\pi i \xi b}}{\xi}
- \frac{1}{2\pi i} \sum_{\xi \in \mathbb{Z}-\{0\}} \frac{e^{2\pi i \xi a}}{\xi} \\
&``=\text{''} \
(b-a) -\left( \{b\}- \frac{1}{2} \right) + \left( \{a\} - \frac{1}{2} \right) \\
&``=\text{''} \
b - \{b\} - ( a - \{a\} ) = \lfloor b \rfloor - \lfloor a \rfloor.
\end{align*}
Since we already know how to evaluate the LHS of Poisson summation above, namely that
$\sum_{n \in \mathbb{Z}} 1_{\mathcal P}(n) = \#\left\{ \mathbb{Z} \cap {\mathcal P} \right\} = \lfloor b \rfloor - \lfloor a \rfloor$,
we have confirmed that Poisson summation has given us here the correct formula, in spite of the lack of rigor here.
Why is the intuitive argument above not rigorous yet? In order to plug a function $f$ into
Poisson summation, and consider convergence at each point of the domain,
$f$ and its Fourier transform $\hat f$ must both satisfy some growth conditions at infinity, at the very least
ensuring proper convergence of both sides of the Poisson summation formula.
We will see such conditions later, in Chapter~\ref{Fourier analysis basics}, Theorem \ref{nice2}. Once we learn how to use Poisson summation, we will return to this example (see
Example \ref{rigorous example of P_1}).
}
\hfill $\square$
\end{example}
We recall that a series $\sum_{n\in \mathbb{Z}} a_n$ is said to {\bf converge absolutely} if $\sum_{n\in \mathbb{Z}} |a_n|$ converges. It's easy to see that the series in \eqref{FirstBernoulliPolynomial} for $P_1(x)$ does not converge absolutely. Such convergent series that do not converge absolutely are called {\bf conditionally convergent}.
To prove rigorously that the conditionally convergent series
\eqref{FirstBernoulliPolynomial} does in fact converge, see
Exercises \ref{Abel summation by parts}, \ref{Dirichlet's convergence test}, \ref{exponential sum bound}, and
\ref{rigorous convergence of P_1(x)}, which include the Abel summation formula, and the Dirichlet convergence test (although extremely useful, we will not use them
very much in the ensuing chapters).
\bigskip
\section{The cube, and its Fourier transform}
Perhaps the easiest way to extend the Fourier transform of the unit interval is to consider the
$d$-dimensional unit cube
\[
\square := \left[-\frac{1}{2}, \frac{1}{2} \right]^d.
\]
What is its Fourier transform? When we compute a Fourier transform of a function $f$, we will
say that $\{ f, \hat f\}$ is a {\bf Fourier pair}. We have seen that
$\left\{ 1_{[-\frac{1}{2}, \frac{1}{2}]}(x), {\rm{sinc}}(\xi) \right\}$ is a Fourier pair in dimension $1$.
\bigskip
\begin{example} \label{Example, unit cube}
\rm{
Due to the fact that the cube is the direct product of line segments, it follows that the ensuing integral can be separated into a product of integrals, and so it is the product of $1$-dimensional transforms:
\begin{align}
\hat 1_{\square}(\xi) &= \int_{\mathbb{R}^d} 1_\square(x) e^{-2\pi i \langle x, \xi \rangle} dx \\
&= \int_{\square} e^{-2\pi i(x_1 \xi_1 + \cdots + x_d \xi_d)} dx \\
&= \prod_{k=1}^d \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{-2\pi i x_k \xi_k} dx_k \\
&= \prod_{k=1}^d \frac{\sin(\pi \xi_k)}{\pi \xi_k},
\end{align}
valid for all $\xi \in \mathbb{R}^d$ such that none of their coordinates vanishes. So here we have the Fourier pair
\[
\left\{ 1_\square(x), \, \prod_{k=1}^d \frac{\sin(\pi \xi_k)}{\pi \xi_k} \right\}.
\]
In general, though,
polytopes are not a direct product of lower-dimensional polytopes, so we will need to develop more tools to compute their Fourier transforms.
}
\hfill $\square$
\end{example}
\bigskip
\section{The simplex, and its Fourier transform}
Another basic building block for polytopes is the {\bf standard simplex},
\index{standard simplex}
defined by
\begin{equation}
\RightTriangle := \left\{
x \in \mathbb{R}^d \bigm | \, x_1 + \cdots + x_d \leq 1, \text{ and all } x_k \geq 0
\right\} .
\end{equation}
\begin{figure}[!h]
\centering
\begin{tikzpicture}[scale=1]
\draw (0,0) node[below left] {$0$};
\draw[loosely dotted] (-1,-1) grid (2,2);
\draw[->] (-1.25,0) -- (2.25,0) node[right] {$x$};
\draw[->] (0,-1.25) -- (0,2.25) node[above] {$y$};
\draw[thick] (0,0) -- (1,0) -- (0,1) -- cycle;
\filldraw[nearly transparent, blue] (0,0) -- (1,0) -- (0,1) -- cycle;
\end{tikzpicture}
\caption{The standard simplex in $\mathbb{R}^2$}
\label{standard simplex in two dimensions} \index{standard simplex}
\end{figure}
\begin{example}\label{standard simplex FT}
\index{standard simplex}
\rm{
Just for fun, let's compute the Fourier transform of $\triangle$
for $d=2$, via brute-force.
We may use the following parametrization (called a hyperplane description)
for this standard triangle:
\[
\RightTriangle = \left\{ (x, y) \bigm | x+y \leq 1, \text{ and } x\geq 0, y\geq0 \right\}.
\]
Hence, we have:
\begin{align*}
&\hat 1_{\, \rt}(\xi_1, \xi_2) := \int_{\, \rt} e^{-2\pi i \big(x \xi_1 + y \xi_2\big)} dx dy \\
&= \int_0^1 \int_{y=0}^{y=1-x} e^{-2\pi i \big(x \xi_1 + y \xi_2\big)} dy dx \\
&= \int_0^1 e^{-2\pi i x \xi_1}
\left[
\frac{ e^{-2\pi i y \xi_2 } }{-2\pi i \xi_2 } \Big|_{y=0}^{y=1-x}
\right]
dx \\
&= \frac{1}{-2\pi i \xi_2 } \int_0^1 e^{-2\pi i x \xi_1}
\left( e^{-2\pi i (1-x) \xi_2 } - 1 \right) dx \\
&= \frac{1}{-2\pi i \xi_2 } \int_0^1
\left(
e^{-2\pi i x (\xi_1 -\xi_2)} e^{-2\pi i \xi_2 } - e^{-2\pi i x \xi_1}
\right) dx \\
&= \frac{1}{(-2\pi i)^2}
\frac{ e^{-2\pi i \xi_2 } }{ \xi_2(\xi_1-\xi_2) }
(e^{-2\pi i (\xi_1 -\xi_2)} -1)
-
\frac{1}{(-2\pi i)^2} \frac{ e^{-2\pi i \xi_1 } -1 }{ \xi_1 \xi_2 } \\
&= \frac{1}{(-2\pi i)^2} \left[
\frac{ e^{-2\pi i \xi_1} - e^{-2\pi i \xi_2 } }{ \xi_2(\xi_1-\xi_2) }
-
\frac{ e^{-2\pi i \xi_1 } -1 }{ \xi_1 \xi_2 }
\right].
\end{align*}
We may simplify further by noticing the rational function identity
\[
\frac{ e^{-2\pi i \xi_1 } }{ \xi_2 ( \xi_1 - \xi_2 ) }
-\frac{ e^{-2\pi i \xi_1 } }{ \xi_1 \xi_2 }
= \frac{ e^{-2\pi i \xi_1 } }{ \xi_1 ( \xi_1 - \xi_2 ) },
\]
giving us the symmetric function of $(\xi_1, \xi_2)$:
\begin{equation}\label{actual FT of the standard simplex}
\hat 1_{\rt}(\xi_1, \xi_2) =
\frac{1}{(-2\pi i)^2} \left[
\frac{ e^{-2\pi i \xi_1 } }{ \xi_1 ( \xi_1 - \xi_2 ) }
+ \frac{ e^{-2\pi i \xi_2 } }{ \xi_2(\xi_2-\xi_1) }
+
\frac{ 1 }{ \xi_1 \xi_2 }
\right].
\end{equation}
}
\hfill $\square$
\end{example}
We need the concept of a {\bf convex set} $X\subset \mathbb{R}^d$, defined by the property that for any two points
$x, y \in X$, the line segment joining them also lies in $X$. In other words, the line segment
$\left\{ \lambda x + (1-\lambda)y \bigm | 0\leq \lambda \leq 1 \right\} \subset X$, $\forall x, y \in X$.
Given any finite set of points $S:= \{ v_1, v_2, \dots, v_N\} \subset \mathbb{R}^d$, we can also form the set of all
{\bf convex linear combinations} of $S$ by defining
\begin{equation}
\conv(S):=
\left\{
\lambda_1 v_1 + \lambda_2 v_2+ \dots +\lambda_{N} v_{N}
\bigm |
\sum_{k=1}^N \lambda_k = 1, \text{ where all } \lambda_k \geq 0
\right\}.
\end{equation}
Given any set $U\subset \mathbb{R}^d$ (which is not restricted to be finite), we define the {\bf convex hull} of $U$
\index{convex hull}
as the set of convex linear combinations, taken over all finite subsets of $U$, and denoted by $\conv(U)$.
We define a {\bf polytope} as the convex hull of any finite set of points in $\mathbb{R}^d$. This definition of a polytope is called its {\bf vertex description}.
\index{vertex description of a polytope}
We define a {\bf $k$-simplex} \index{simplex}
$\Delta$ as the convex hull of a finite set of vectors
$\{ v_1, v_2, \dots, v_{k+1} \}$:
\[
\Delta := \conv\{ v_1, v_2, \dots, v_{k+1} \},
\]
where $0 \leq k \leq d$, and $v_2-v_1, v_3-v_1, \dots, v_{k+1} - v_1$ are linearly independent vectors in $\mathbb{R}^d$.
The points $v_1, v_2, \dots, v_{k+1}$ are called the vertices of $\Delta$, and this
object is one of the basic building-blocks of polytopes, especially when triangulating a polytope.
The simplex $\Delta$ is a $k$-dimensional polytope, sitting in $\mathbb{R}^d$. When $k=d$, the dimension of $\Delta$ equals the dimension of the ambient space $\mathbb{R}^d$ - see Figure \ref{simplex}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{simplex}
\end{center}
\caption{A $3$-simplex and its faces, which are lower-dimensional simplices as well}
\label{simplex}
\end{figure}
We have already computed the Fourier transform of a particular $2$-simplex,
in \eqref{actual FT of the standard simplex}.
How do we define a face of a polytope ${\mathcal P}$ more precisely? Here we need a new notion.
A {\bf hyperplane}
\[
H:=\{ x \in \mathbb{R}^d \mid \langle x, n \rangle = b \}
\]
is called a {\bf supporting hyperplane for} ${\mathcal P}$ if ${\mathcal P}$ lies on one side of $H$, in the precise sense that:
\[
{\mathcal P} \subset \{ x \in \mathbb{R}^d \mid \langle x, n \rangle \leq b \} \ \text{ or }
{\mathcal P} \subset \{ x \in \mathbb{R}^d \mid \langle x, n \rangle \geq b \}.
\]
We now call $F\subseteq {\mathcal P}$ a {\bf face of } ${\mathcal P}$ if $F= H \cap {\mathcal P}$, for some supporting hyperplane $H$ of ${\mathcal P}$.
${\mathcal P}$ is also considered a face of ${\mathcal P}$ itself (use a degenerate hyperplane $H$ with $n:=0:=b$); for logical consistencies,
the empty set is also defined to be
a face of ${\mathcal P}$ (use a hyperplane $H$ far away from ${\mathcal P}$).
With these preliminaries, we're now ready to compute the Fourier transform of any $2$-simplex in
$\mathbb{R}^2$.
In order to handle a general triangle, let $\Delta$ be any triangle in the plane, with vertices
\[
v_1:= \icol{ a_1 \\ b_1}, v_2:=\icol{ a_2 \\ b_2} , v_3:= \icol{ a_3 \\ b_3}.
\]
Can we reduce the computation of $\hat 1_{\Delta}$ to our already known formula
for $\hat 1_{\rt}$, given by \eqref{actual FT of the standard simplex}?
We first notice (after a brief cup of coffee) that we can map any triangle in the plane to the standard triangle, by using a linear transformation followed by a translation:
\begin{equation}\label{M followed by T}
\Delta = M ( \, \RightTriangle ) + v_3,
\end{equation}
where $M$ is the $2\times2$ matrix whose columns are $v_1-v_3$ and $v_2-v_3$.
We are now ready to compute the Fourier transform of a general triangle $\Delta$:
\[
\hat 1_{\Delta}(\xi) = \int_{\Delta} e^{-2\pi i \langle \xi, x \rangle} dx
= \int_{M(\rt)+ v_3} e^{-2\pi i \langle \xi, x \rangle} dx.
\]
Making the substitution $x := My+v_3$, with $y \in \rt$, we have $dx = |\det M| dy$, and so
\begin{align*}
& \int_{M(\rt)+v_3} e^{-2\pi i \langle \xi, x \rangle} dx
= |\det M| \int_{\, \rt} e^{-2\pi i \langle \xi, M y +v_3 \rangle} dy \\
&= |\det M| e^{-2\pi i \langle \xi, v_3 \rangle}
\int_{\, \rt} e^{-2\pi i \langle M^{T} \xi, y \rangle} dy \\
&= |\det M| e^{-2\pi i \langle \xi, v_3 \rangle} \hat 1_{\, \rt}(M^{T} \xi) \\
&= |\det M| e^{-2\pi i \langle \xi, v_3 \rangle}
\hat 1_{\, \rt} \big( \langle v_1-v_3, \xi \rangle, \langle v_2-v_3, \xi \rangle \big) \\
&=
|\det M| e^{-2\pi i \langle \xi, v_3 \rangle}
\frac{1}{(-2\pi i)^2} \left[
\frac{ e^{-2\pi i z_1 } }{ z_1 ( z_1 - z_2 ) }
+ \frac{ e^{-2\pi i z_2 } }{ z_2(z_2-z_1) }
+
\frac{ 1 }{ z_1 z_2 }
\right],
\end{align*}
where we've used our formula \eqref{actual FT of the standard simplex} for the FT of the standard triangle (thereby bootstrapping out way to the general case) with
$z_1:=\langle v_1-v_3, \xi \rangle$, and $z_2:= \langle v_2-v_3, \xi \rangle$. Substituting these values into the latter expression, we finally arrive at the FT of our general triangle $\Delta$:
\begin{align}\label{FT of a general triangle}
\hat 1_{\Delta}(\xi)
= \tfrac{ |\det M| }{(-2\pi i)^2}
\left[
\frac{ e^{-2\pi i \langle v_1, \xi \rangle } }{ \langle v_1-v_3, \xi \rangle
\langle v_1-v_2, \xi \rangle }
+ \frac{ e^{-2\pi i \langle v_2, \xi \rangle } }{ \langle v_2-v_3, \xi \rangle
\langle v_2-v_1, \xi \rangle }
+
\frac{ e^{-2\pi i \langle \xi, v_3 \rangle} }{ \langle v_3-v_1, \xi \rangle
\langle v_3-v_2, \xi \rangle }
\right].
\end{align}
We can notice in equation \eqref{FT of a general triangle} many of the same patterns that had already occurred in Example
\ref{cross-polytope example in R^2}. Namely,
the Fourier transform of a triangle has denominators that are products of linear forms in $\xi$,
and it is a finite linear combination of rational functions multiplied by complex exponentials.
Also, in the particular case of
equation \eqref{FT of a general triangle}, $ \hat 1_\Delta(\xi) $ is a symmetric function of $v_1, v_2, v_3$, as we might have expected.
Using exactly the same ideas that were used in equation \eqref{FT of a general triangle}, it is possible to prove (by induction on the dimension) that the Fourier transform of a general $d$-dimensional simplex $\Delta \subset \mathbb{R}^d$ is:
\begin{equation}\label{FT of a d-dimensional simplex}
\hat 1_{\Delta}(\xi) = (\vol \Delta) d! \sum_{j=1}^N
\frac{e^{-2\pi i \langle v_j, \xi \rangle}}{\prod_{k=1}^d \langle v_j-v_k, \xi \rangle }[k \not= j],
\end{equation}
where the vertex set of ${\mathcal P}$ is $\{ v_1, \dots, v_N\}$ (Exercise \ref{FT of a general simplex, brute-force}), and in fact the same formula persists for all complex $\xi \in \mathbb{C}^d$ such that the products of linear forms in the denominators do not vanish.
However, looking back at the computation leading to \eqref{FT of a general triangle},
and the corresponding computation which would give \eqref{FT of a d-dimensional simplex},
the curious reader might be thinking:
\medskip
\centerline{ ``There must be an easier way!'' }
But never fear - indeed there is.
So even though at this point the computation of $\hat 1_\Delta(\xi)$ may be a bit laborious (but still interesting),
computing the Fourier transform of a general simplex will become quite easy once we will revisit it in a later chapter (see Theorem \ref{brion, continuous form}).
\bigskip
\section{Stretching and translating}
\bigskip
The perspicacious reader may have noticed that in order to arrive at the formula
\eqref{FT of a general triangle} above for the FT of a general triangle, we exploited the fact that the Fourier transform interacted peacefully with the linear transformation $M$, and with the translation by the vector $v$. Is this true in general?
Indeed it is, and we record these thoughts in the following two lemmas, which will become our bread and butter for future computations.
In general, given any invertible linear transformation $M :\mathbb{R}^d \rightarrow \mathbb{R}^d$, and any function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ whose FT (Fourier transform) exists, we have the following useful interaction between Fourier transforms and linear transformations.
\begin{lem}[Stretch] \index{stretch lemma}
\label{FT under linear maps}
\begin{equation}\label{The FT under streching}
(\widehat{f \circ M})(\xi)= \frac{1}{|\det M|} \hat f\left(M^{-T}\xi \right)
\end{equation}
\end{lem}
\begin{proof}
By definition, we have
$
(\widehat{f \circ M})(\xi) :=\int_{\mathbb{R}^d} f(Mx) e^{-2\pi i \langle \xi, x \rangle} dx.
$
We perform the change of variable $y:= Mx$, implying that $dy = |\det M| dx$, so that:
\begin{align*}
(\widehat{f \circ M})(\xi) &=
\frac{1}{|\det M|} \int_{\mathbb{R}^d} f(y) e^{-2\pi i \langle \xi, M^{-1}y \rangle} dy \\
&=\frac{1}{|\det M|} \int_{\mathbb{R}^d} f(y) e^{-2\pi i \langle M^{-T}\xi, y \rangle} dy \\
&= \frac{1}{|\det M|} \hat f\left(M^{-T} \xi \right).
\end{align*}
\end{proof}
What about translations? They are even simpler.
\begin{lem}[Translate] \index{translate lemma}
\label{FT under translations}
For any translation $T(x):= x + v$, where $v\in \mathbb{R}^d$ is a fixed vector, we have
\begin{equation}\label{The FT under translations}
(\widehat{f \circ T})(\xi)= e^{2\pi i \langle \xi, v \rangle} \hat f(\xi).
\end{equation}
\end{lem}
\begin{proof}
Again, by definition we have
$
(\widehat{f \circ T})(\xi) := \int_{\mathbb{R}^d} f( Tx) e^{-2\pi i \langle \xi, x \rangle} dx,
$
so that performing the simple change of variable $y = Tx := x + v$, we have $dy = dx$.
The latter integral becomes
\begin{align*}
(\widehat{f \circ T})(\xi)
&= \int_{\mathbb{R}^d} f(y) e^{-2\pi i \langle \xi, y-v \rangle} dy \\
&= e^{2\pi i \langle \xi, v \rangle}
\int_{\mathbb{R}^d} f(y) e^{-2\pi i \langle \xi, y \rangle} dy :=
e^{2\pi i \langle \xi, v \rangle} \hat f(\xi).
\end{align*}
\end{proof}
In general, any function $\phi:\mathbb{R}^d \rightarrow \mathbb{C}$ of the form
\begin{equation}
\phi(x) = Mx+v,
\end{equation}
where $M$ is a fixed linear transformation and $v\in \mathbb{R}^d$ is a fixed vector, is
called an {\bf affine transformation}. \index{affine transformation} For example, we've already seen
in \eqref{M followed by T} that the right triangle $\RightTriangle$ was mapped to the more general triangle
$\Delta$ by an affine transformation.
So the latter two lemmas allow us to compose Fourier transforms very easily with affine transformations.
\begin{example}
\rm{
The simplest example of the Stretch Lemma \ref{FT under linear maps} is obtained
in $\mathbb{R}$, where the matrix $M = r$, a positive real number. So we have $M^{-T} = \frac{1}{r}$.
Considering $f(rx)$ as a function of $x \in \mathbb{R}$, we have by \eqref{The FT under translations}:
\begin{equation} \label{simple example of stretching}
\widehat{f(rx)} := (\widehat{f \circ M})(\xi) = \tfrac{1}{r} \hat f\left( \tfrac{1}{r} \xi \right).
\end{equation}
As an interesting sub-example, let's take $f(x) := 1_{\left[-\tfrac{c}{2}, \tfrac{c}{2} \right]}(x) $,
for a fixed constant $c>0$. What's the easy way to use the Stretch lemma to compute $\hat f(\xi)$? First, we have to make a slight conversion: $1_{\left[-\tfrac{c}{2}, \tfrac{c}{2} \right]}(x)
= 1_{\left[-\tfrac{1}{2}, \tfrac{1}{2} \right]}(\tfrac{1}{c} x)$. Using the FT of the unit interval, equation \eqref{sinc function formula},
together with \eqref{simple example of stretching}, we have:
\begin{equation}\label{Stretch lemma for the sinc function}
\hat 1_{\left[-\tfrac{c}{2}, \tfrac{c}{2} \right]}(\xi) = c\, \hat 1_{\left[-\tfrac{1}{2}, \tfrac{1}{2} \right]}(c \xi)
=c\, {\rm{sinc}}(c\xi) = \frac{\sin(c\pi \xi)}{\pi \xi}.
\end{equation}
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
Consider any set $B\subset \mathbb{R}^d$, for which $1_B$ is integrable, and let's translate $B$ by a fixed vector $v \in \mathbb{R}^d$, and compute
$\hat 1_{B+v}(\xi)$.
We note that because $1_{B+v}(\xi) = 1_{B}(\xi-v)$, the translate lemma applies, but with a minus sign. That is, we can use
$T(x):= x-v$ and $f:= 1_B$ to get:
\begin{equation} \label{The FT of a translate of B}
\hat 1_{B+v}(\xi) = \widehat{ (1_B \circ T) }(\xi) = e^{-2\pi i \langle \xi, v \rangle} \hat 1_B(\xi).
\end{equation}
}
\hfill $\square$
\end{example}
\bigskip
\section{The parallelepiped, and its Fourier transform} \index{parallelepiped}
Now that we know how to compose the FT with affine transformations (translations and linear transformations), we can easily find the FT of
any parallelepiped in $\mathbb{R}^d$ by using our formula for the Fourier transform of the unit cube
$\square := \left[-\frac{1}{2}, \frac{1}{2} \right]^d$, which we derived in Example \ref{Example, unit cube}:
\begin{align}\label{first FT of a cube}
\hat 1_{\square}(\xi) = \prod_{k=1}^d \frac{\sin(\pi \xi_k)}{\pi \xi_k},
\end{align}
for all $\xi \in \mathbb{R}^d$ such that all the coordinates of $\xi$ do not vanish.
First, we translate the cube $\square$
by the vector $(\frac{1}{2}, \cdots, \frac{1}{2})$, to obtain
\[
C:= \square + \left(\frac{1}{2}, \, \cdots, \frac{1}{2} \right) = [0, 1]^d.
\]
It's straightforward to compute its FT as well (Exercise \ref{transform.of.unit.cube}), by using Lemma \ref{FT under translations}, the `translate' lemma:
\begin{equation}\label{second appearance of FT of the cube}
\hat 1_{C}(\xi) = \frac{1}{(2\pi i)^d} \prod_{k=1}^d \frac{ 1- e^{-2\pi i \xi_k} }{ \xi_k }.
\end{equation}
Next, we define a $d$-dimensional {\bf parallelepiped} ${\mathcal P} \subset \mathbb{R}^d$ as an affine image of the unit cube. In other words,
any parallelepiped has the description
\[
{\mathcal P} = M(C) + v,
\]
for some linear transformation $M$, and some translation vector $v$. Geometrically, the cube is stretched and translated
into a parallelepiped.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{3Dparallelepiped}
\end{center}
\caption{Mapping the unit cube to a parallelepiped}
\label{3Dparallelepiped}
\end{figure}
For the sake of concreteness, will will first set $v:= 0$ and compute the Fourier transform of ${\mathcal P}:= M(C)$, where we now give $M$ as a $d \times d$ invertible matrix whose columns are
$w_1, w_2, \dots, w_d$. Because the cube $C$ may be written as a convex linear combination of the basis vectors $e_j$,
we see that ${\mathcal P}$ may be written as a convex linear combination of $M e_j = w_j$.
In other words, we see that the parallelepiped ${\mathcal P}$ has the equivalent vertex description:
\[
{\mathcal P} = \left\{ \sum_{k=1}^d \lambda_k w_k \bigm | \text{ all } \lambda_k \in [0, 1] \right\}.
\]
To review the basics, let's compute the FT of our parallelepiped ${\mathcal P}$ from first principles:
\begin{align} \label{cube composed with M}
\hat 1_{\mathcal P}(\xi) &:= \int_{{\mathcal P}} e^{-2\pi i \langle \xi, x \rangle} dx = \int_{M(C)} e^{-2\pi i \langle \xi, x \rangle} dx\\
&= |\det M| \int_{C} e^{-2\pi i \langle \xi, My \rangle} dy\\
&= |\det M| \int_{C} e^{-2\pi i \langle M^T \xi, y \rangle} dy := |\det M| \, \hat 1_C\left(M^T \xi \right) \\ \label{for Q}
&= \frac{ |\det M| }{(2\pi i)^d} \prod_{k=1}^d \frac{ 1- e^{-2\pi i \langle w_k, \xi \rangle} }{ \langle w_k, \xi \rangle }.
\end{align}
where in the third equality we used the substitution $x:= My$, with $y\in C$, yielding $dx = |\det M| dy$. In the last equality, we used our known formula \eqref{second appearance of FT of the cube} for the FT of the cube $C$, together with the elementary linear algebra fact that the $k$'th coordinate of $M^T \xi$ is given by
$\langle w_k, \xi \rangle$.
Finally, for a general parallelepiped, we have $Q:= {\mathcal P} + v$, so that by definition
\[
Q = \left\{ v+ \sum_{k=1}^d \lambda_k w_k \bigm | \text{ all } \lambda_k \in [0, 1] \right\}.
\]
Noting that $1_{{\mathcal P}+ v}(\xi) = 1_{{\mathcal P}}(\xi -v)$,
we compute the Fourier transform of $Q$ by using the `translate lemma' (Lemma \ref{FT under translations}),
together with formula \eqref{for Q} for the Fourier transform of ${\mathcal P}$:
\begin{equation}\label{FT of a general parallelepiped}
\hat 1_Q(\xi) = e^{-2\pi i \langle \xi, v \rangle}
\frac{ |\det M| }{(2\pi i)^d}
\prod_{k=1}^d \frac{ 1- e^{-2\pi i \langle w_k, \xi \rangle} }{ \langle w_k, \xi \rangle },
\end{equation}
for all $\xi \in \mathbb{R}^d$, except for those $\xi$ that are orthogonal to one of the $w_k$ (which are edge vectors for $Q$).
\begin{example}
\rm{
A straightforward computation shows that if we let $v:= -\frac{w_1 + \cdots + w_d}{2}$, then
$Q:= \{ v+ \sum_{k=1}^d \lambda_k w_k \mid \text{ all } \lambda_k \in [0, 1] \}$ is symmetric about the origin, in the sense that
$x \in Q \iff -x \in Q$ (Exercise \ref{symmetrized parallelepiped}). In other words, the center of mass of this new $Q$ is now the origin.
Geometrically, we've translated the previous parallelepiped by using half its `body diagonal'.
For such a parallelepiped $Q$, centered at the origin, formula \eqref{FT of a general parallelepiped} above gives
\begin{align}
\hat 1_Q(\xi) &= e^{2\pi i \langle \xi, \frac{w_1 + \cdots + w_d}{2} \rangle}
\frac{ |\det M| }{(2\pi i)^d}
\prod_{k=1}^d \frac{ 1- e^{-2\pi i \langle w_k, \xi \rangle} }{ \langle w_k, \xi \rangle } \\
&=
\frac{ |\det M| }{(2\pi i)^d}
\prod_{k=1}^d \frac{ e^{\pi i \langle w_k, \xi \rangle}- e^{-\pi i \langle w_k, \xi \rangle} }{ \langle w_k, \xi \rangle } \\
&= \frac{ |\det M| }{(2\pi i)^d}
\prod_{k=1}^d \frac{ (2i) \sin( \pi \langle w_k, \xi \rangle) }{ \langle w_k, \xi \rangle } \\
&= |\det M|
\prod_{k=1}^d \frac{ \sin( \pi \langle w_k, \xi \rangle) }{ \pi \langle w_k, \xi \rangle }.
\end{align}
To summarize, for a parallelepiped that is symmetric about the origin, we have the Fourier pair
\[
\left\{ 1_Q(x), \ \ | \det M |
\prod_{k=1}^d \frac{ \sin( \pi \langle w_k, \xi \rangle) }{ \pi \langle w_k, \xi \rangle } \right\}.
\]
We could have also computed the latter FT by beginning with our known
Fourier transform \eqref{first FT of a cube} of the cube $\square$, composing the FT
with the same linear transformation $M$ of \eqref{cube composed with M}, and using the `stretch' lemma, so everything is consistent.
}
\hfill $\square$
\end{example}
\bigskip
\section{The cross-polytope}
\index{cross-polytope}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.8in]{Octahedron}
\end{center}
\caption{The cross-polytope $\Diamond$ in $\mathbb{R}^3$ (courtesy of David Austin)} \label{crosspic}
\end{figure}
Another natural convex body in $\mathbb{R}^2$ is the cross-polytope
\begin{equation} \label{2dim.crosspolytope}
\Diamond_2 := \left\{ \left( x_1, x_2 \right) \in \mathbb{R}^2 \bigm | \, \left| x_1 \right| + \left| x_2 \right| \leq 1 \right\} .
\end{equation}
In dimension $d$, the {\bf cross-polytope} $\Diamond_d$ \label{cross polytope}
can be defined similarly by its {\bf hyperplane description}
\begin{equation} \label{crosspolytopehyperplanes}
\Diamond_d := \left\{ \left( x_1, x_2, \dots, x_d \right) \in \mathbb{R}^d \bigm | \, \left| x_1 \right| + \left| x_2 \right| + \dots + \left| x_d \right| \leq 1 \right\} .
\end{equation}
The cross-polytope is also, by definition, the unit ball in the $L_1$-norm on Euclidean space, and from this perspective a very natural object. In $\mathbb R^3$,
the cross-polytope $\Diamond_3$ is often called an {\bf {octahedron}}. \index{octahedron}
In this section we only work out the $2$-dimensional case of the Fourier transfrom of the crosspolytope,
In Chapter \ref{chapter.Brion}, we will work out the
Fourier transform of any $d$-dimensional cross-polytope, $\hat 1_{\Diamond_d}$,
because we will have more tools at our disposal.
Nevertheless, it's instructive to compute $\hat 1_{\Diamond_2}$ via brute-force for $d=2$ here,
in order to gain some practice.
\begin{example} \label{cross-polytope example in R^2}
\rm{
Using the definition of the Fourier transform, we first compute the FT of the $2$-dimensional
cross polytope:
\begin{align}
\hat 1_{\Diamond_2}(\xi) &:= \int_{\Diamond_2} e^{-2\pi i \langle \xi, x \rangle} dx.
\end{align}
In $\mathbb{R}^2$, we may write $\Diamond_2$ as a union of the following $4$ triangles:
\begin{align*}
\Delta_1&:= \conv ( \icol{0\{\bf 0}}, \icol{1\{\bf 0}}, \icol{0\{\bf 1}} )\\
\Delta_2&:= \conv ( \icol{0\{\bf 0}}, \icol{-1\{\bf 0}}, \icol{0\{\bf 1}} )\\
\Delta_3&:= \conv ( \icol{0\{\bf 0}}, \icol{-1\{\bf 0}}, \icol{0\\-1} )\\
\Delta_4&:= \conv ( \icol{0\{\bf 0}}, \icol{1\{\bf 0}}, \icol{0\\-1} ).
\end{align*}
Since these four triangles only intersect in lower-dimensional subsets of $\mathbb{R}^2$, the $2$-dimensional integral vanishes on such lower dimensional subsets, and we have:
\begin{equation} \label{transform of 2d crosspolytope}
\hat 1_{\Diamond_2}(\xi) = \hat 1_{\Delta_1}(\xi) + \hat 1_{\Delta_2}(\xi)
+ \hat 1_{\Delta_3}(\xi) + \hat 1_{\Delta_4}(\xi).
\end{equation}
Recalling from equation \eqref{actual FT of the standard simplex}
of example \ref{standard simplex FT}
that the Fourier transform of the standard simplex
\index{standard simplex}
$\Delta_1$ is
\begin{equation}\label{simplex transform}
\hat 1_{\Delta_1}(\xi) = \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{1}{\xi_1 \xi_2}
+ \frac{\ e^{-2\pi i \xi_1} }{(-\xi_1 + \xi_2) \xi_1}
+ \frac{ \ e^{-2\pi i \xi_2} }{( \xi_1 - \xi_2) \xi_2}
\right),
\end{equation}
we can compute $\hat 1_{\Delta_2}(\xi)$, by reflecting $\Delta_2$ about the $x_2-axis$ (the Jacobian of this transformation is $1$), and
using the already-computed transform \eqref{simplex transform} of $\Delta_1$:
\begin{align*}
\hat 1_{\Delta_2}(\xi_1, \xi_2) &:= \int_{\Delta_2} e^{-2\pi i (x_1 \xi_1 + x_2 \xi_2)} dx \\
&= \int_{\Delta_1} e^{-2\pi i (-x_1 \xi_1 + x_2 \xi_2)} dx \\
&= \int_{\Delta_1} e^{-2\pi i (x_1 (-\xi_1) + x_2 \xi_2)} dx \\
&= \hat 1_{\Delta_1}(-\xi_1, \xi_2)).
\end{align*}
Similarly, we have $\hat 1_{\Delta_3}(\xi_1, \xi_2) = \hat 1_{\Delta_1}(-\xi_1, -\xi_2)$, and
$\hat 1_{\Delta_4}(\xi_1, \xi_2) = \hat 1_{\Delta_1}(\xi_1, -\xi_2)$.
Hence we may continue the computation from equation \ref{transform of 2d crosspolytope} above, putting all the pieces back together:
\begin{align} \label{Fourier transform of 2d crosspolytope}
\hat 1_{\Diamond_2}(\xi) &= \hat 1_{\Delta_1}(\xi_1, \xi_2) + \hat 1_{\Delta_1}(-\xi_1, \xi_2)
+ \hat 1_{\Delta_1}(-\xi_1, -\xi_2) + \hat 1_{\Delta_1}(\xi_1, -\xi_2) \\
&= \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{1}{\xi_1 \xi_2}
+ \frac{-\ e^{2\pi i \xi_1} }{(-\xi_1 + \xi_2) \xi_1}
+ \frac{ - \ e^{2\pi i \xi_2} }{( \xi_1 - \xi_2) \xi_2}
\right) \\
&+ \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{-1}{\xi_1 \xi_2}
+ \frac{\ e^{-2\pi i \xi_1} }{(\xi_1 + \xi_2) \xi_1}
+ \frac{ \ e^{2\pi i \xi_2} }{( \xi_1 + \xi_2) \xi_2}
\right) \\
&+ \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{1}{\xi_1 \xi_2}
+ \frac{e^{-2\pi i \xi_1} }{(\xi_1 - \xi_2) \xi_1}
+ \frac{e^{-2\pi i \xi_2} }{( -\xi_1 + \xi_2) \xi_2}
\right) \\
&+ \left( \frac{1}{2\pi i} \right)^2
\left(
\frac{-1}{\xi_1 \xi_2}
+ \frac{e^{2\pi i \xi_1} }{(\xi_1 + \xi_2) \xi_1}
+ \frac{e^{-2\pi i \xi_2} }{( \xi_1 + \xi_2) \xi_2}
\right) \\
&= -\frac{1}{2\pi^2} \left(
\frac{\cos(2\pi \xi_1) }{(\xi_1 - \xi_2) \xi_1}
+ \frac{\cos(2\pi \xi_2) }{(-\xi_1 + \xi_2) \xi_2}
+ \frac{\cos(2\pi \xi_1) }{(\xi_1 + \xi_2) \xi_1}
+ \frac{\cos(2\pi \xi_2) }{(\xi_1 + \xi_2) \xi_2}
\right) \\ \label{formula 1 for the FT of the 2-d crosspolytope}
&= -\frac{1}{\pi^2} \left(
\frac{ \cos(2\pi \xi_1) - \cos(2\pi \xi_2) }{ (\xi_1 + \xi_2)( \xi_1 - \xi_2) }
\right).
\end{align}
}
\hfill $\square$
\end{example}
It's time to mention another important relationship between the cross-polytope $\Diamond$
\index{cross-polytope}
and the cube ${\mathcal P}~:= ~[-1, 1]^d$.
To see this relationship, we define, for any polytope ${\mathcal P} \subset \mathbb{R}^d$, its {\bf dual polytope}:
\begin{equation}\label{dual polytope, definition}
{\mathcal P}^* := \left\{ x\in \mathbb{R}^d \bigm | \, \langle x, y\rangle \leq 1, \text{ for all } y \in {\mathcal P} \right\}.
\end{equation}
It is an easy fact (Exercise \ref{duals of each other}) that in $\mathbb{R}^d$, the cross-polytope $\Diamond$ and the cube ${\mathcal P}:= [-1, 1]^d$ are dual to each other, as in the figure below.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.0in]{duals.jpg}
\end{center}
\caption{Left: a page from
Kepler's book, \emph{Harmonices Mundi} ($1619$), showing the author's interest in various dual polytopes, over $400$ years ago. Right: The cube and the cross-polytope as duals of each other. } \label{duals}
\end{figure}
\bigskip
\section{Observations and questions}
Now we can make several observations about all of the formulas that we found so far,
for the Fourier transforms of various polytopes. For the $2$-dimensional cross-polytope, we found that
\begin{equation} \label{again the FT of a 2d crosspolytope}
\hat 1_{\Diamond_2}(\xi) = -\frac{1}{\pi^2} \left(
\frac{ \cos(2\pi \xi_1) - \cos(2\pi \xi_2) }{ (\xi_1 + \xi_2)( \xi_1 - \xi_2) }
\right).
\end{equation}
\begin{enumerate}[(a)]
\item \ It is real-valued for all $\xi \in \mathbb{R}^2$, and this is due to the fact that $\Diamond_2$ is symmetric about the origin (see section \ref{Centrally symmetric polytopes}).
\begin{question}
Is it true that {\emph any} symmetric property of a polytope ${\mathcal P}$ is somehow mirrored by a corresponding symmetric property of its Fourier transform?
\end{question}
Although this question is not well-defined at the moment (it depends on how we define `symmetric property'),
it does sound exciting, and we can morph it into a few well-defined questions later.
\item \ The only apparent singularities of the FT in \eqref{again the FT of a 2d crosspolytope}
(though they are in fact removable singularities) are the two lines
$\xi_1 - \xi_2=0$ and $\xi_1 + \xi_2=0$, and these two
lines are {\it perpendicular} to the facets of $\Diamond_2$,
which is not a coincidence (see Chapter \ref{Stokes' formula and transforms}).
\item \ It is always true that the Fourier transform of a polytope is an entire function, by
Lemma \ref{FT of a polytope is entire}, so that the singularities
in the denominator $ (\xi_1 + \xi_2)( \xi_1 - \xi_2)$ of
\eqref{again the FT of a 2d crosspolytope} must be removable singularities!
\item The denominators of all of the FT's so far are always products of {\bf linear forms} in $\xi$.
\begin{question} \rm{[Rhetorical]}
Is it true that the Fourier transform of any polytope is always a finite sum of rational functions times an exponential, where the denominators of the rational functions are always products of linear forms?
\end{question}
{\bf Answer}: (spoiler alert) Yes! It's too early to prove this here, but we will do so in Theorem \ref{brion2}.
\item \ We may retrieve the volume of $\Diamond_2$ by letting $\xi_1$ and $\xi_2$ tend to zero
(Exercise \ref{retreiving volume of 2d.crosspolytope}), as always. Doing so, we obtain
$\lim_{\xi \rightarrow 0} \hat 1_{\Diamond_2}(\xi) = 2 = \text{Area}(\Diamond_2)$.
\end{enumerate}
\section*{Notes}
\begin{enumerate}[(a)]
\item Another way to compute $1_{\Diamond}(\xi)$ for the $2$-dimensional cross-polytope $\Diamond$ is by starting with the square $[-\frac{1}{2}, \frac{1}{2}]^2$ and applying a rotation of the plane by $\pi/4$, followed by a simple dilation. Because we know that linear transformations interact in a very elegant way with the FT, this method gives an alternate approach
for the Example \ref{cross-polytope example in R^2} in $\mathbb{R}^2$.
However, this method no longer works for the cross-polytope in dimensions $d \geq 3$,
where it is not (yet) known if there is a simple way to go from the FT of the cube to the FT of the cross-polytope.
\index{cross-polytope}
More generally, one may ask:
\begin{question}
is there a nice relationship between the FT of a polytope ${\mathcal P}$ and the FT of its dual?
\end{question}
\item We note that $P_1({x})$ is defined to be
equal to $0$ at the integers, because its Fourier series naturally converges to the mean of
the discontinuity of the function, at each integer.
\item It has been known since the work of Riemann that the Bernoulli numbers occur as special values of the Riemann zeta function (see Exercise \ref{Riemann zeta function, and Bernoulli numbers}). Similarly, the
Hurwitz zeta function, defined for each fixed $x >0$ by
\[
\zeta(s, x) := \sum_{n =0}^\infty \frac{1}{(n+x)^s},
\]
has a meromorphic continuation to all of $\mathbb{C}$, and its special values at the negative integers are the Bernoulli polynomials $B_n(x)$ (up to a multiplicative constant).
\item There are sometimes very unusual (yet useful) formulations for the Fourier transform of certain functions.
Ramanujan \cite{Ramanujan1} discovered the following remarkable formula
for the Fourier transform of the Gamma function:
\begin{equation}
\int_{\mathbb{R}} |\Gamma(a + iy)| e^{-2\pi i \xi y} dy =
\frac{ \sqrt{\pi} \ \Gamma(a) \Gamma( a + \frac{1}{2})}{\cosh(\pi \xi)^{2a}},
\end{equation}
valid for $a>0$. For example with $a:= \frac{1}{2}$, in the language of this chapter we have the Fourier pair $\{ |\Gamma(\frac{1}{2} + iy)|, \frac{\pi}{\cosh(\pi \xi)} \}$.
\bigskip
\end{enumerate}
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
Problems worthy of attack prove their worth by fighting back.
-- Paul Erd\"os
\end{quote}
\medskip
\begin{prob} \label{transform.of.interval.a.to.b} $\clubsuit$
Show that the Fourier transform of the closed interval $[a, b]$ is:
\[
\hat 1_{[a,b]}(\xi) =\frac{ e^{-2\pi i \xi a} - e^{-2\pi i \xi b} }{2\pi i \xi},
\]
for $\xi \not=0$.
\end{prob}
\medskip
\begin{prob} \label{transform.of.unit.cube}
Show that the Fourier transform of the unit cube $C:= [0,1]^d \subset \mathbb{R}^d$ is:
\begin{equation}
\hat 1_{C}(\xi) = \frac{1}{(2\pi i)^d} \prod_{k=1}^d \frac{ 1- e^{-2\pi i \xi_k} }{ \xi_k },
\end{equation}
valid for all $\xi \in \mathbb{R}^d$, except for the union of hyperplanes defined by \\
$H := \left\{ x \in \mathbb{R}^d \bigm | \xi_1 = 0 \text{ or } \xi_2 = 0 \dots \text{ or } \xi_d = 0 \right\}$.
\end{prob}
\medskip
\begin{prob}
Suppose we are given two polynomials $p(x)$ and $q(x)$, of degree $d$. If there are $d+1$ distinct points
$\{z_1, \dots, z_{d+1}\}$ in the complex plane such that $p(z_k) = q(z_k)$ for $k = 1, \dots, d+1$, show that the two polynomials are identical. (Hint: consider $(p-q)(z_k)$)
\end{prob}
\medskip
\begin{prob} \label{brute force Bernoulli polys}
To gain some facility with generating functions, show by a brute-force computation with Taylor series that
the coefficients on the right-hand-side of equation \eqref{generating function for Bernoulli polynomials}, which are
called $B_n(x)$ by definition, must in fact be polynomials in $x$.
In fact, your direct computations will show that for all $n \geq 1$, we have
\[
B_n(x) = \sum_{k=0}^n {n \choose k} B_{n-k} \ x^k,
\]
where $B_j$ is the $j$'th Bernoulli number. \index{Bernoulli number}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Reflection property for B_n(x)}
Show that for all $n \geq 1$, we have
\[
B_n(1-x) = (-1)^n B_n(x).
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{difference of Bernoulli polys}
Show that for all $n \geq 1$, we have
\[
B_n(x+1) - B_n(x) = n x^{n-1}.
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{derivative of Bernoulli polys}
Show that for all $n \geq 1$, we have
\[
\frac{d}{dx} B_n(x) = n B_{n-1}(x).
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{historical origin of Bernoulli poly}
Prove that:
\[
\sum_{k=1}^{n-1} k^{d-1} = \frac{ B_d(n) - B_d }{ d },
\]
for all integers $d \geq 1$ and $n \geq 2$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{Bernoulli Polynomials} \index{Bernoulli polynomial}
Show that the periodic Bernoulli polynomials $P_n(x) := B_n(\{ x \})$, for all $n\geq 2$, have the following Fourier series:
\begin{equation}
P_n(x) = -\frac{n!}{(2\pi i)^n} \sum_{k \not=0} \frac{e^{2\pi i k x}}{k^n},
\end{equation}
valid for all $x \in \mathbb{R}$. For $n \geq 2$, these series are absolutely convergent. We note that from the definition above, $B_n(x) = P_n(x)$ when $x\in (0,1)$.
\end{prob}
\medskip
\begin{prob} \label{Raabe's identity for { } via Fourier series}
Show that the greatest integer function $\floor{x}$ (often called the `floor function') enjoys the
property:
\[
\sum_{k=0}^{N-1} \floor{ x + \frac{k}{N} } = \floor{Nx},
\]
for all $x\in \mathbb{R}$, and all positive integers $N$, and that in the same range we also have
\[
\sum_{k=0}^{N-1} \left\{ x + \frac{k}{N} \right\} = \left\{ Nx \right\}.
\]
\end{prob}
\medskip
\begin{prob}
Show that the Bernoulli polynomials enjoy the following identity, proved by Joseph Ludwig Raabe in 1851:
\[
B_n(Nx) = N^{n-1} \sum_{k=0}^{N-1} B_n\left( x + \frac{k}{N} \right),
\]
for all $x\in \mathbb{R}$, all positive integers $N$, and for each $n \geq 1$.
Notes. Such formulas, in these last two exercises, are also called ``multiplication Theorems'', and they hold
for many other functions, including the Gamma function, the dilogarithm, the Hurwitz zeta function, and many more.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{another definition for Bernoulli polynomials}
Here we give a different method for defining the Bernoulli polynomials, based on the following three properties that they enjoy:
\begin{enumerate}
\item $B_0(x) = 1$.
\item For all $n \geq 1, \frac{d}{dx} B_n(x) = n B_{n-1}(x)$.
\item For all $n \geq 1$, we have $\int_0^1 B_n(x) dx = 0$.
\end{enumerate}
Show that the latter three properties imply the original defining property of the Bernoulli polynomials
\eqref{generating function for Bernoulli polynomials}.
\end{prob}
\medskip
\begin{prob}
Here is a more explicit, useful recursion for computing the Bernoulli polynomials. Show that
\[
\sum_{k=0}^{n-1} {n \choose k} B_k(x) = n x^{n-1},
\]
for all $n \geq 2$.
\end{prob}
\medskip
\begin{prob} \label{B_7}
Use the previous exercise, together with the known list the first $6$ Bernoulli polynomials that appear
in equation \ref{B_6}, to compute $B_7(x)$.
\end{prob}
\medskip
\begin{prob} \label{odd Bernoulli numbers}
Show that for odd $k \geq 3$, we have $B_k = 0$.
\end{prob}
\medskip
\begin{prob} \label{ Bernoulli numbers alternate in sign}
Show that the even Bernoulli numbers alternate in sign. More precisely, show that
\[
(-1)^{n+1} B_{2n} \geq 0,
\]
for each positive integer $n$.
\end{prob}
\medskip
\begin{prob} \label{vanishing identity for Beroulli numbers}
Show that the Bernoulli numbers enjoy the recursive property:
\[
\sum_{k=0}^n {n+1 \choose k}B_k = 0,
\]
for all $n \geq 1$.
\end{prob}
\medskip
\begin{prob} \label{asymptotics for Beroulli numbers}
Show that the Bernoulli numbers enjoy the following asymptotics:
\[
B_{2n} \sim 2 \frac{(2n)!}{(2\pi)^{2n}}
\]
as $n\rightarrow \infty$. Here we are using the usual notation for asymptotic functions, namely that
$f(n) \sim g(n)$ as $n\rightarrow \infty$ if $\lim_{n\rightarrow \infty} \frac{f(n)}{g(n)} \rightarrow 1$.
\end{prob}
\medskip
\begin{prob} \label{Fresnel} $\clubsuit$
Show that the following integrals converge and have the closed forms:
\begin{align}
\int_{-\infty}^\infty \cos(x^2) dx &= \sqrt{\frac{\pi}{2}}, \\
\int_{-\infty}^\infty \sin(x^2) dx &= \sqrt{\frac{\pi}{2}}.
\end{align}
Notes. These integrals are called Fresnel integrals, and they are related to the Cornu spiral, which was created by Marie Alfred Cornu. Marie used the spiral
as a tool for computing diffraction patterns that arise naturally in optics.
\end{prob}
\medskip
\begin{prob}
Prove the following Gamma function identity, using the sinc function:
\[
\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1+x)\Gamma(1-x)},
\]
for all $x \not\in \mathbb{Z}$.
Notes. This identity is often called Euler's reflection formula. $\Gamma(x):= \int_0^\infty e^{-t} t^{x-1} dt$ is by definition
the Gamma function,
where the integral converges for all $x > 0$ (see Section \ref{Volume of the ball, the Gamma function} for more on the $\Gamma$ function).
\end{prob}
\medskip
\begin{prob}
\label{retreiving volume of 2d.crosspolytope}
$\clubsuit$ Using the formula for the Fourier transform of the $2$-dimensional cross-polytope $\Diamond$, derived in the text,
namely
\[
\hat 1_{\Diamond}(\xi) =
-\frac{1}{\pi^2} \left(
\frac{ \cos(2\pi \xi_1) - \cos(2\pi \xi_2) }{ \xi_1^2 - \xi_2^2}
\right),
\]
find the area of $\Diamond$ by letting $\xi \rightarrow 0$ in the latter formula.
\end{prob}
\medskip
\begin{prob}
\label{Elementary bounds for sin(x), sinc(x)}
Some elementary but very useful bounds for trig functions are developed here.
\begin{enumerate}[(a)]
\item Prove that
\[
\frac{2}{\pi} < \frac{\sin x}{x} \leq 1,
\]
where the left inequality holds for $ 0 < x < \frac{\pi}{2}$, and the right inequality holds for $x\in \mathbb{R}$.
\item \label{Elementary trig bounds, part b}
Prove that
\[
\frac{2x}{\pi} \leq |1-e^{ix} | \leq |x|,
\]
where the left inequality holds for $|x| \leq \pi$, and the right inequality holds for $x\in \mathbb{R}$.
\item Prove that
\[
\frac{2x^2}{\pi^2} \leq |1-\cos x | \leq \frac{x^2}{2},
\]
where the left inequality holds for $|x| \leq \pi$, and the right inequality holds for $x\in \mathbb{R}$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
\label{divergence of |sinc|}
$\clubsuit$
Show that
$\int_{-\infty}^\infty \Big| \frac{\sin(\pi x)}{\pi x} \Big| dx = \infty$.
\end{prob}
\medskip
\begin{prob}
There are (at least) two different ways of periodizing a given function $f:\mathbb{R} \rightarrow \mathbb{C}$
with respect to $\mathbb{Z}$.
First, we can define $F_1(x) := f(\{x\})$, so that $F_1$ is periodic on $\mathbb{R}$ with period~$1$.
Second, we may also define $F_2(x) := \sum_{n\in \mathbb{Z}} f(x+n)$, which is also a periodic function on $\mathbb{R}$ with period $1$.
Find an integrable (meaning that $\int_\mathbb{R} f(x)dx$ converges) function $f$ for which $F_1 \not= F_2$, as functions.
Notes. \ In Chapter \ref{Fourier analysis basics}, we will see that the latter function $F_2(x) := \sum_{n\in \mathbb{Z}} f(x+n)$ captures a lot more information about $f$, and often captures all of $f$ as well.
\end{prob}
\medskip
\begin{prob} \label{symmetrized parallelepiped} \index{parallelepiped}
Given linearly independent vectors $w_1, \dots, w_d \in \mathbb{R}^d$, let $v:= -\frac{w_1 + \cdots + w_d}{2}$, and define
$Q:= \{ v+ \sum_{k=1}^d \lambda_k w_k \mid \text{ all } \lambda_k \in [0, 1] \}$, a parallelepiped.
Show that $Q$ is symmetric about the origin, in the sense that
$x \in Q \iff -x \in Q$.
\end{prob}
\medskip
\begin{prob}
\label{duals of each other}
$\clubsuit$ Show that the $d$-dimensional cross-polytope $\Diamond$ and the cube
$\square:= [-1, 1]^d$ are dual to each other.
\end{prob}
\medskip
\begin{prob}
\label{2-dim'l formula for triangle}
Prove the following $2$-dimensional integral formula:
\begin{align}
\int_{\lambda_1, \lambda_2 \geq 0 \atop \lambda_1 + \lambda_2 \leq 1}
e^{A \lambda_1 } e^{B \lambda_2 } d\lambda_1 d\lambda_2
= \frac{ B e^A - A e^B }{AB(A-B)} +\frac{1}{AB},
\end{align}
valid for all $A, B \in \mathbb{C}$ such that $AB(A-B) \not=0$.
\end{prob}
\medskip
\begin{prob}
\label{FT of a general simplex, brute-force}
Using the ideas of Example \ref{FT of a general triangle}, prove (by induction on the dimension) that the Fourier transform of a general $d$-dimensional simplex $\Delta \subset \mathbb{R}^d$ is given by:
\begin{equation}
\hat 1_{\Delta}(\xi) = (\vol \Delta) d!
\sum_{j=1}^N
\frac{e^{-2\pi i \langle v_j, \xi \rangle}}{\prod_{1\leq k \leq d} \langle v_j-v_k, \xi \rangle }
[k\not=j],
\end{equation}
for all $\xi \in \mathbb{R}^d$, where the vertex set of ${\mathcal P}$ is $\{ v_1, \dots, v_N\}$.
\end{prob}
\medskip
\begin{prob}[Abel summation by parts] $\clubsuit$ \label{Abel summation by parts}
Here we prove the straightforward but very useful technique of Niels Abel, called {\bf Abel summation by parts}.
\index{Abel, Niels} \index{Abel summation by parts}
Suppose we are given two sequences $\{a_n\}_{n=1}^\infty$, and $\{b_n\}_{n=1}^\infty$. We define the finite partial sums
$B_n:= \sum_{k=1}^n b_k$. Then we have
\begin{equation} \label{actual Abel summation}
\sum_{k=1}^n a_k b_k = a_n B_n + \sum_{k=1}^{n-1} B_k(a_k - a_{k+1}),
\end{equation}
for all $n\geq 2$.
\end{prob}
Notes. \ Using the forward difference operator, it's easy to recognize identity \eqref{actual Abel summation}
as a discrete version of integration by parts.
\medskip
\begin{prob}[{\bf Dirichlet's convergence test}] $\clubsuit$ \label{Dirichlet's convergence test}
\index{Dirichlet's convergence test}
Suppose we are given a real sequence $\{a_n\}_{n=1}^\infty$, and a complex sequence
$\{b_n\}_{n=1}^\infty$,
such that
\begin{enumerate}[(a)]
\item $\{a_n\}$ is monotonically decreasing to $0$, and
\item $| \sum_{k=1}^n b_k | \leq M$, for some positive constant $M$, and all $n \geq 1$.
\end{enumerate}
Then $\sum_{k=1}^\infty a_k b_k$ converges.
\end{prob}
\medskip
\begin{prob} \label{first Dirichlet kernel}
Prove that for all $x \in \mathbb{R}-\mathbb{Z}$, we have the following important identity, called the ``Dirichlet kernel'',
\index{Dirichlet kernel}
named after Peter Gustav Lejeune Dirichlet:
\begin{equation}
\sum_{k= -n}^n e^{2\pi i k x} = \frac{\sin \left( 2\pi x(n + \frac{1}{2}) \right) }{\sin(\pi x)}.
\end{equation}
\end{prob}
\medskip
\begin{prob} \label{exponential sum bound}
For any fixed $x \in \mathbb{R}-\mathbb{Z}$, show that we have the bound on the following exponential sum:
\begin{equation}
\left | \sum_{k= 1}^n e^{2\pi i k x} \right | \leq \frac{1}{ | \sin(\pi x) | }.
\end{equation}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{rigorous convergence of P_1(x)}
Prove that $\sum_{m = 1}^\infty \frac{e^{2\pi i m a}}{m}$ converges, given any fixed $a \in \mathbb{R} - \mathbb{Z}$.
Notes. \ We see that, although $\sum_{m = 1}^\infty \frac{e^{2\pi i m a}}{m}$ does not converge absolutely,
Abel's summation formula \eqref{actual Abel summation} gives us
\[
\sum_{k = 1}^n \frac{e^{2\pi i k a}}{k} = \frac{1}{n}\sum_{r=1}^n e^{2\pi i r a}
+ \sum_{k=1}^{n-1} \Big( \sum_{r=1}^k e^{2\pi i r a} \Big) \frac{1}{k(k+1)},
\]
and the latter series {\bf does converge absolutely}, as $n \rightarrow + \infty$. So we see that Abel summation transforms one series (that barely converges at all) into
another series that converges more rapidly.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{integral of sinc is 1}
Here we'll prove that
\begin{equation} \label{exercise:the Dirichlet integral}
\int_{-\infty}^\infty \frac{ \sin(\pi t) }{\pi t} dt = 1,
\end{equation}
and this integral is known as ``the Dirichlet integral''.
We notice that due to the evenness of the integrand, after a change of variable
it suffices to prove that
$\int_0^ \infty \frac{ \sin t }{ t} dt =\frac{\pi}{2}$. Comparing this Dirichlet integral with
Exercise \ref{divergence of |sinc|}, we see that there is something very subtle going on here.
\begin{enumerate}[(a)]
\item Define
\begin{equation} \label{Laplace transform of sinc}
F(s):= \int_0^\infty e^{-st} \frac{ \sin t }{ t} dt,
\end{equation}
for each $s>0$. Justify differentiation under the integral sign, and show that
\[
\frac{dF}{ds} = -\int_0^\infty e^{-st} \sin t dt,
\]
\item Show that $\int_0^\infty e^{-st} \sin t dt = \frac{1}{1+s^2}$.
\item Show that $F(s) = C - \tan^{-1} s$, and then show that the constant $C= \frac{\pi}{2}$.
\item Prove that $F$ is a continuous function of $s\in \mathbb{R}_{>0}$, and finally prove that
\[
\lim_{s\rightarrow 0} F(s) = \frac{\pi}{2},
\]
which is the desired result (Here you might want to integrate by parts first, and then use the Dominated convergence theorem).
\end{enumerate}
\end{prob}
Notes. There are many proofs of this famous identity \eqref{exercise:the Dirichlet integral}; here we are only assuming
knowledge of some real analysis. The expression in \eqref{Laplace transform of sinc} is also known as the Laplace transform of the sinc function, and it is a variation of the Fourier transform that we will return to when studying similar transforms of cones in
Section \ref{Fourier Laplace transforms of cones}.
\medskip
\begin{prob} $\clubsuit$
\label{rigorous inversion formula for sinc}
\rm{
Here we give a rigorous proof of the tricky business that for all $x\in \mathbb{R}$, we have
\begin{equation*}
\lim_{N\rightarrow \infty} \int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} e^{-2\pi i \xi x} d\xi = 1_ {[-\frac{1}{2}, \frac{1}{2}]}(x),
\end{equation*}
following an approach of S. Bochner \cite{BochnerBook}.
We begin by noticing that this integral can be easily reduced to a real-valued integral:
\begin{equation*}
\int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} e^{-2\pi i \xi x} d\xi
= \int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} \cos( 2\pi \xi x) d\xi,
\end{equation*}
because for each $x\in \mathbb{R}$, $\int_{-N}^N \frac{\sin(\pi \xi)}{\pi \xi} \sin( 2\pi \xi x) d\xi = 0$ owing to the oddness of the integrand.
\begin{enumerate}[(a)]
\item \label{tricky middle part with alpha}
Using the result from Exercise \ref{integral of sinc is 1}, prove that
\[
\lim_{N\rightarrow \infty} \int_{-N}^N \frac{\sin(\pi \alpha t)}{\pi t} dt=
\begin{cases}
\ \ 1 & \mbox{if } \alpha >0, \\
\ \ 0 & \mbox{if } \alpha =0, \\
-1 & \mbox{if } \alpha <0.
\end{cases}
\]
\item Finish up by using \ $2\sin t \cos(\alpha t) = \sin(1- \alpha)t + \sin(1+\alpha)t$, thereby showing that the desired
integral
\[
\lim_{N\rightarrow \infty} \int_{-N}^N \frac{\sin(\pi t)}{\pi t} \cos( 2\pi t x) dt
\]
reduces to part \ref{tricky middle part with alpha}.
\end{enumerate}
}
\end{prob}
\chapter{The basics of Fourier analysis}
\label{Fourier analysis basics}
\index{Fourier analysis}
\begin{quote}
``If a function is periodic, then we should try to expand it into its Fourier series, and wonderful things will begin to happen....."
-- Erich Hecke
\end{quote}
\begin{quote}
``. . . Fourier's great mathematical poem.''
[Referring to Fourier's mathematical theory of the conduction of heat]
-- William Thomson Kelvin \index{Kelvin, William Thomson}
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.4in]{cube}
\end{center}
\caption{The unit cube $\square:= [0,1]^3$, in $\mathbb{R}^3$, which tiles the space by translations.
Which other polytopes tile by translations? How can we make mathematical use of such tilings?
In particular, can we give an explicit basis of exponentials for functions defined on $\square$?} \label{crosspic}
\end{figure}
\bigskip
\section{Intuition}
Because we will use tools from Fourier analysis throughout, we introduce them here as an outline of the field,
with the goal of {\it applying} them to the discrete geometry of polytopes, lattices, and their interactions.
We will sometimes introduce a concept by using an intuitive argument, which we call ``fast and loose", but after such
an intuitive argument, we state the precise version of the corresponding theorem.
In this chapter, we will sometimes point to the literature for some of the proofs.
Our goal is to use the necessary tools of Fourier analysis in order to tackle problems in the enumerative combinatorics of polytopes,
in number theory, discrete geometry, and in some other fields.
We emphasize that the Poisson summation formula allows us to {\bf discretize integrals},
in a sense that will be made precise in later chapters.
One pattern that the reader may have already noticed, among all of the examples of Fourier transforms of polytopes computed thus far, is that each of them is a linear combination of a very special kind of rational function of $\xi$, multiplied by a complex exponential that involves a vertex of the polytope:
\begin{equation}\label{structure 1}
\hat 1_{\mathcal P}(\xi) =
\sum_{k=1}^M \frac{1}{\prod_{j=1}^d \left\langle \omega_{j,k}(v_k), \xi \right\rangle} \,
e^{2\pi i \langle v_k, \xi \rangle},
\end{equation}
where the vertices of ${\mathcal P}$ are $v_1, \dots, v_N$, and where $M \geq N$. We observed that in all of our examples thus far, the denominators
are in fact products of linear forms, as in \eqref{structure 1}.
We will be able to see some of the more precise geometric structure for these products of linear forms, which
come from the edges of the polytope, once we learn more about Fourier-Laplace transforms of cones.
It is rather astounding that every single fact about a given polytope ${\mathcal P}$ is somehow
hiding inside these {\bf rational-exponential} functions given by \eqref{structure 1}, due to the fact that the Fourier transform $\hat 1_{\mathcal P}$ is a complete invariant (Lemma \ref{complete invariance of the FT}).
\bigskip
\section{Introducing the Fourier transform on $L^1(\mathbb{R}^d)$}
In the spirit of bringing the reader very quickly up to speed, regarding the applications of Fourier analytic tools, we outline the basics
of the field, and prove some of them. Nowadays, there are many good texts on Fourier analysis, and the reader is encouraged to peruse some of these books (see Note \ref{Fourier books}).
One of the most useful tools for us is the Poisson summation formula. We provide several versions of Poisson summation, each of
which uses a different set of sufficient conditions.
As we will see, the Fourier transform \index{Fourier transform} is a very friendly creature, allowing us to travel back and forth between the ``space domain'' and the ``frequency domain'' to obtain many useful results.
The readers who are already familiar with basics of Fourier analysis may easily skip
this chapter without impeding their understanding of the rest of the book.
Although we enjoy thinking about the warm and cozy Hilbert spaces $L^2(\mathbb{R}^d)$ and $L^2([0, 1]^d)$, there are many subtle convergence issues (and divergence issues) of Fourier series, a whole field onto itself. We won't go there.
However, the very basic convergence issues are still important for us as well, and we want to get the reader up and running.
The function space that immediately come up very naturally is the
the space of {\bf absolutely integrable functions} on $\mathbb{R}^d$:
\[
L^1(\mathbb{R}^d) :=\left\{ f: \mathbb{R}^d \rightarrow \mathbb{C} \bigm | \ \int_{\mathbb{R}^d} |f(x)| dx < \infty \right\}.
\]
Secondly, the space of
{\bf square-integrable functions} on $\mathbb{R}^d$ is defined by:
\[
L^2(\mathbb{R}^d) := \left\{ f: \mathbb{R}^d \rightarrow \mathbb{C} \bigm | \ \int_{\mathbb{R}^d} |f(x)|^2 dx < \infty \right\}.
\]
The usual theory of Fourier transforms progresses by first defining the Fourier transform for functions belonging to $L^1(\mathbb{R}^d)$, which is quite a natural condition, and then later extending the Fourier transform to the $L^2(\mathbb{R}^d)$ space by taking appropriate limits. We initially restrict attention to functions $f \in L^1(\mathbb{R}^d)$.
There are many fascinating facts about all of these functions spaces. For practice, let's ask:
\begin{question} \rm{[Rhetorical]}
Given two functions $f, g \in L^2(\mathbb{R}^d)$, is their product always in $L^1(\mathbb{R}^d)$?'
\end{question}
Well, by the Cauchy-Schwartz inequality for the
Hilbert space $L^2(\mathbb{R}^d)$, we have:
\begin{equation}\label{product of two L^2 functions is L^1}
\int_{\mathbb{R}^d} |f(x) g(x)|dx \leq
\left(\int_{\mathbb{R}^d} |f(x)|^2 dx \right)^{\frac{1}{2}} \left(\int_{\mathbb{R}^d} |g(x)|^2 dx\right)^{\frac{1}{2}} < \infty,
\end{equation}
the latter inequality holding by the assumption $f, g \in L^2(\mathbb{R}^d)$. So the product $f(x) g(x)$ is indeed in $L^1(\mathbb{R}^d)$. This is the first sign that there
are fascinating links between $L^1$ functions and $L^2$ functions.
Moreover, the utility of the Cauchy-Schwarz inequality should never be underestimated.
It's interesting that $L^1(\mathbb{R}^d)$ is not a Hilbert space, as we can easily show by exhibiting a counter-example
to the Cauchy-Schwarz inequality, as follows.
\bigskip
\begin{example} \label{cool CS counterexample}
\rm{
We claim that the Cauchy-Schwarz inequality
is false in $L^1(\mathbb{R})$. If the Cauchy-Schwarz inequality was true here, then \eqref{product of two L^2 functions is L^1} would be valid
for all functions
$f, g \in L^1(\mathbb{R})$. But as a counterexample, let
\[
f(x):= 1_{(0,1)}(x) \frac{1}{\sqrt x}.
\]
It's easy to see that $f \in L^1(\mathbb{R})$:
\[
\int_{\mathbb{R}} 1_{(0,1)}(x) \frac{1}{\sqrt x} dx = \int_0^1 \frac{1}{\sqrt x} dx = \frac{1}{2}.
\]
But $ \int_{\mathbb{R}} f(x) \cdot f(x) dx = \int_0^1 \frac{1}{x} dx$ diverges, so that we do not have a Cauchy-Schwarz inequality in $L^1(\mathbb{R})$, because here both the left-hand-side and the right-hand-side of such an inequality do not even converge.
However, if two functions $f, g$ are bounded on $\mathbb{R}$, and absolutely integrable on $\mathbb{R}$, then we do have a Cauchy-Schwartz inequality
for the pair $f, g$, and we let the reader enjoy its verification.
}
\hfill $\square$
\end{example}
An easy but extremely important inequality is the {\bf triangle inequality for integrals}, as follows.
\begin{thm} \label{triangle inequality for integrals}
For any $f\in L^1(\mathbb{R}^d)$, and any measurable subset $S \subset \mathbb{R}^d$,
we have:
\begin{equation} \index{triangle inequality for integrals}
\Big| \int_S f(x) dx \Big| \leq \int_S | f(x) | dx.
\end{equation}
\end{thm}
\begin{proof}
Letting $z:= \int_S f(x) dx \in \mathbb{C}$, we may write $|z| = \alpha z$, for a (unique) complex $\alpha$ on the unit circle.
We let $u$ be the real part of $\alpha f:= u + iv $, so that $u \leq \sqrt{u^2 + v^2} = |\alpha f| = |f|$.
Altogether, we have:
\begin{equation*}
\Big| \int_S f(x) dx \Big| = \alpha \int_S f(x) dx = \int_S \alpha f(x) dx = \int_S u(x) dx
\leq \int_S |f(x)| dx.
\end{equation*}
In the third equality, we used the fact that $\int_S \alpha f(x) dx$ is real, which follows
from the first two equalities:
$\int_S \alpha f(x) dx = \Big| \int_S f(x) dx \Big|$.
\end{proof}
\medskip
\begin{cor} If $f$ is bounded on a measurable set $S\subset \mathbb{R}^d$ by a constant $M>0$, then:
\begin{equation}
\Big| \int_S f(x) dx \Big| \leq {\rm measure} (S) \cdot M.
\end{equation}
\begin{proof}
\[
\Big| \int_S f(x) dx \Big| \leq \int_S | f(x) | dx \leq \int_S M dx = {\rm measure} (S) \cdot M,
\]
where the first inequality uses the triangle inequality for integrals, namely Theorem \eqref{triangle inequality for integrals}, and the second inequality uses
the boundedness assumption on $f$.
\end{proof}
\end{cor}
\medskip
We've defined the Fourier transform before, and we remind the reader that for any function $f\in L^1(\mathbb{R}^d)$,
the {\bf Fourier transform} of $f$ \index{Fourier transform} is
\begin{equation}
\hat f(\xi) := \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, \xi \rangle} dx.
\end{equation}
Where does this definition really come from? One motivation comes from the inner product for functions
(in $L^2(\mathbb{R}^d)$),
where we project a function $f$ onto each exponential function:
\[
\langle f, e^{2\pi i \langle x, \xi \rangle} \rangle := \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, \xi \rangle} dx.
\]
Another motivation comes from the proof of the Poisson summation formula - eq. \eqref{finally, the FT} below, which
shows a crucial connection between the Fourier transform of $f$ and the Fourier coefficients of the periodized function
$\sum_{n\in \mathbb{Z}^d} f(x+n)$.
One of the first things we might notice is:
\begin{claim}
The Fourier transform is a bounded linear operator.
\end{claim}
The Fourier transform is a linear operator, by the linearity
of the integral: $\widehat{(f+g)} = \hat f + \hat g$, and it is a bounded operator
due to the elementary estimate in \eqref{boundedness of FT} below.
A natural question is: where does the Fourier transform take a function $f\in L^1(\mathbb{R}^d)$?
An immediate partial answer is that for any $f\in L^1(\mathbb{R}^d)$, we have:
\[
\hat f \in B(\mathbb{R}^d),
\]
where $B(\mathbb{R}^d):=
\{f:\mathbb{R}^d \rightarrow \mathbb{C} \bigm | \, \exists M>0 \text{ such that } |f(x)| < M, \text{ for all } x \in \mathbb{R}^d \}$ is the space of bounded functions on $\mathbb{R}^d$. Here the constant $M$ depends only on $f$.
To see this, consider:
\begin{align}
| \hat f(\xi) | &:= \left | \int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, \xi \rangle} dx \right |
\leq
\int_{\mathbb{R}^d} \left | f(x) e^{-2\pi i \langle x, \xi \rangle} \right | dx \\ \label{boundedness of FT}
&= \int_{\mathbb{R}^d} \left | f(x) \right | dx := \| f \|_{L^1(\mathbb{R}^d)},
\end{align}
where we used Theorem \ref{triangle inequality for integrals}, the triangle inequality for integrals, together with the fact that
$ \left | e^{-2\pi i \langle x, \xi \rangle} \right | =1$.
\begin{example}
\rm{
Let's bound the Fourier transform of an integrable indicator function $1_S$, for any measurable set $S\subset \mathbb{R}^d$:
\[
| \hat 1_S(\xi) |:= \left | \int_{S} e^{-2\pi i \langle x, \xi \rangle} dx \right |
\leq \int_{S} \left | e^{-2\pi i \langle x, \xi \rangle} \right | dx = \int_{S} dx = {\rm measure} (S).
\]
In particular, for any polytope ${\mathcal P}\subset \mathbb{R}^d$,
\[
|\hat 1_{\mathcal P}(\xi) | \leq \vol {\mathcal P}, \text{ for all } \xi \in \mathbb{R}^d.
\]
We already know that $\hat 1_{\mathcal P}(0) = \vol {\mathcal P}$, so it's natural to ask whether the maximum allowed value of $\vol {\mathcal P}$ can also be achieved by a nonzero $\xi \in \mathbb{R}^d$; or perhaps it may be the case that we always have the strict inequality
$|\hat 1_{\mathcal P}(\xi) | < \vol {\mathcal P}, \text{ for all nonzero } \xi \in \mathbb{R}^d$?
(Exercise \ref{strictly less than the FT at zero}).
}
\hfill $\square$
\end{example}
But a lot more is true for absolutely integrable functions.
\medskip
\begin{lem} \label{uniform continuity}
If $f\in L^1(\mathbb{R}^d)$, then $\hat f$ is uniformly continuous on $\mathbb{R}^d$.
\begin{proof}
We fix any $\xi \in \mathbb{R}^d$, and $h \in \mathbb{R}^d$, and we compute:
\begin{align*}
\hat f(\xi + h) - f(\xi) &:=
\int_{\mathbb{R}^d} f(x) \Big( e^{-2\pi i \langle x, \xi + h \rangle} - e^{-2\pi i \langle x, \xi \rangle} \Big) dx \\
&= \int_{\mathbb{R}^d} f(x)e^{-2\pi i \langle x, \xi \rangle} \Big( e^{-2\pi i \langle x, h \rangle} -1 \Big) dx,
\end{align*}
so by the triangle inequality for integrals, we have
\begin{equation} \label{pre-uniform conv}
| \hat f(\xi + h) - f(\xi)| \leq \int_{\mathbb{R}^d} |f(x)| | e^{-2\pi i \langle x, h \rangle} -1 | dx.
\end{equation}
Letting $g_h(x):= f(x) \Big( e^{-2\pi i \langle x, h \rangle} -1 \Big)$, we see that
\[
|g_h(x)| \leq 2 |f(x)|, \text{ and } \lim_{h \rightarrow 0} |g_h(x)| =0,
\]
using $ |e^{-2\pi i \langle x, h \rangle} -1| \leq 2$. We may now use the dominated convergence theorem, because the functions
$g_h$ are dominated by the absolutely integrable function $2 f$. So we get:
\[
\lim_{h\rightarrow 0} \int_{\mathbb{R}^d} |f(x)| | e^{-2\pi i \langle x, h \rangle} -1 | dx = \int_{\mathbb{R}^d} \lim_{h\rightarrow 0} |f(x)| | e^{-2\pi i \langle x, h \rangle} -1 | dx
= 0.
\]
Because the latter limit is independent of $\xi$, \eqref{pre-uniform conv} tells us that $| \hat f(\xi + h) - f(\xi)| \rightarrow 0$, as $h\rightarrow 0$, uniformly
in $\xi \in \mathbb{R}^d$.
\end{proof}
\end{lem}
It turns out that sometimes we need to measure distance between functions in a manner different than just pointwise convergence. We therefore introduce convergence in the $L^2$~norm. We say that a sequence of functions $f_n:\mathbb{R}^d \rightarrow \mathbb{C}$ converges to a function $f$
{\bf in the $L^2$ norm} if
\begin{equation} \label{convergence in the L^2 norm}
\index{convergence in the $L^2$ norm}
\int_{\mathbb{R}^d} \left| f_n(x) - f(x) \right |^2 dx \rightarrow 0, \text{ as } n \rightarrow \infty,
\end{equation}
for which we also use the notation $\lim_{n\rightarrow \infty} \| f_n - f\|_2 =0$.
It is also very useful to define the $L^p(\mathbb{R}^d)$ spaces, for each $1\leq p < \infty$:
\begin{equation}
L^p(\mathbb{R}^d):= \{ f:\mathbb{R}^d \rightarrow \mathbb{C} \bigm | \int_{\mathbb{R}^d} |f(x)|^p dx < \infty \},
\end{equation}
which naturally extend the $L^1$ and $L^2$ spaces. It is well-known that among all of the $L^p(\mathbb{R}^d)$ spaces, the only one that is a Hilbert space is $L^2(\mathbb{R}^d)$. For the curious reader, the other $L^p(\mathbb{R}^d)$ spaces, for $p \not=2$, also possess some additional structure, namely they are Banach algebras, after identifying two functions that are equal a.e. (see \cite{EinsiedlerWardBook} for details). The development of $L^p$ spaces is very important for Fourier analysis; for the sake of simplicity of exposition, here we will mostly work with $p=1$ and $p=2$.
Similarly to \eqref{convergence in the L^2 norm}, we define {\bf convergence in the $L^p$ norm},
for $1\leq p < \infty$ by
\begin{equation} \label{convergence in the L^p norm}
\int_{\mathbb{R}^d} \left| f_n(x) - f(x) \right |^p dx \rightarrow 0, \text{ as } n \rightarrow \infty,
\end{equation}
for which we also use the notation
\[
\lim_{n\rightarrow \infty} \| f_n - f \|_p = 0.
\]
For a review of some of these various forms of convergence, see the Appendix - Chapter \ref{Appendix A}.
The celebrated {\bf Riemann--Lebesgue lemma}
gives us the basic decay property of the Fourier transform $\hat f(\xi)$ as $|\xi| \rightarrow \infty$.
To prove it, we will use the fact that we can approximate any function $f\in L^1(\mathbb{R}^d)$ with arbitrary precision by using
`step functions' in $\mathbb{R}^d$. More precisely, let a {\bf box in $\mathbb{R}^d$} \index{box}
be defined by ${\mathcal P}:= [a_1, b_1]\times \cdots \times [a_d, b_d]$, and consider
the indicator function $1_{\mathcal P}$ of this box. If we consider the set of all finite sums, taken over all such indicator functions (varying over all boxes), with arbitrary real coefficients, then this set turns out to be dense in $L^1(\mathbb{R}^d)$, in the $L^1$ norm. We record this fact as a lemma.
\begin{lem} \label{box functions dense in L^1}
If $f \in L^1(\mathbb{R}^d)$, then there is a finite sum of indicator functions of boxes that approaches $f$, in the $L^1$ norm.
\hfill $\square$
\end{lem}
\index{Riemann-Lebesgue lemma}
\begin{lem}[Riemann-Lebesgue] \label{Riemann--Lebesgue lemma}
If $f \in L^1(\mathbb{R}^d)$, then:
\[
\lim_{|\xi| \rightarrow \infty} \hat f(\xi) = 0.
\]
\begin{proof}
We first show the result in the case that $f$ is the indicator function of a box. We already know, via
Exercise \ref{transform.of.interval.a.to.b}, that if
${\mathcal P}:= [a_1, b_1]\times \cdots \times [a_d, b_d]$, then
\begin{equation} \label{FT of boxes...}
\hat 1_{\mathcal P}(\xi) = \prod_{k=1}^d
\frac{ e^{-2\pi i \xi_k a_k} - e^{-2\pi i \xi_k b_k} }{2\pi i \xi_k}.
\end{equation}
As $|\xi| \rightarrow \infty$, $\prod_{k=1}^d \xi_k \rightarrow \infty$, while the numerator of \eqref{FT of boxes...} stays bounded, so we've proved
the lemma for indicator functions of boxes. Since $f \in L^1(\mathbb{R}^d)$, we know by Lemma \ref{box functions dense in L^1} that
there exists a sequence of functions $g_n \in L^1(\mathbb{R}^d)$ such that $\|f -g_n\|_1 \rightarrow 0$, as $n\rightarrow \infty$.
Also, by \eqref{FT of boxes...} we know that this sequence already satisfies $\lim_{|\xi| \rightarrow \infty} \hat g_n(\xi) =0$.
Using the elementary inequality \eqref{boundedness of FT},
we get:
\[
\big| \hat f(\xi) - \hat g_n(\xi) \big|
= \big| \widehat{(f - g_n)}(\xi) \big|
\leq \|f -g_n\|_1 \rightarrow 0,
\]
as $n\rightarrow \infty$. Therefore $\lim_{|\xi| \rightarrow \infty} \hat f(\xi) = 0$.
\end{proof}
\end{lem}
With all of the above properties, it is now natural to consider the space of all uniformly continuous functions on $\mathbb{R}^d$ that go to $0$ at infinity:
\begin{equation}
C_0(\mathbb{R}^d):= \{ f: \mathbb{R}^d \rightarrow \mathbb{C} \bigm | f \text{ is uniformly continuous on } \mathbb{R}^d, \text{ and } \lim_{|x| \rightarrow \infty} |f| = 0 \}.
\end{equation}
So although the Fourier transform does not map the space $L^1(\mathbb{R}^d)$ into itself, all of the above results may be summarized as follows.
\begin{lem}
If $f \in L^1(\mathbb{R}^d)$, then $\hat f \in C_0(\mathbb{R}^d)$.
\end{lem}
\begin{proof}
The boundedness of $\hat f$ was given by the inequality $ | \hat f(\xi) | \leq \|f\|_1$ \eqref{boundedness of FT}, the uniform continuity by Lemma \ref{uniform continuity},
and the decay to zero at infinity by Lemma \ref{Riemann--Lebesgue lemma}.
\index{Riemann-Lebesgue lemma}
\end{proof}
Interestingly, there exists an even more precise statement (using convolutions) that involves the $L^2(\mathbb{R}^d)$ space, for the image of the space $L^1(\mathbb{R}^d)$ under the Fourier transform; we state it in \eqref{RudinAmazingConvolutions} below. This
dance beween the $L^1$ and $L^2$ spaces has more to offer.
\begin{lem} \label{both f and its FT in L^1 implies L^2}
If $f \in L^1(\mathbb{R}^d)$ and $\hat f \in L^1(\mathbb{R}^d)$, then both $f, \hat f \in L^2(\mathbb{R}^d)$.
\end{lem}
\begin{proof}
Because $\hat f \in L^1(\mathbb{R}^d)$, we know by the basic inequality \eqref{boundedness of FT} that $f$ must be bounded on $\mathbb{R}^d$: $|f(x)| \leq M$ for some $M>0$. We now compute:
\[
\int_{\mathbb{R}^d} |f(x)|^2 dx \leq \int_{\mathbb{R}^d} M |f(x)| dx= M \int_{\mathbb{R}^d} |f(x)| dx < \infty,
\]
where the last inequality holds because $f \in L^1(\mathbb{R}^d)$ by assumption. So $f \in L^2(\mathbb{R}^d)$.
Finally, because both $f, \hat f \in L^1(\mathbb{R}^d)$, we can invoke Fourier inversion,
namely Theorem \ref{thm:Inverse Fourier transform}, and therefore exactly the same reasoning applies to $\hat f$.
\end{proof}
\section{The torus $\mathbb{R}^d/\mathbb{Z}^d$}
Suppose a function $f:\mathbb{R} \rightarrow \mathbb{C}$ is {\bf periodic on the real line}, with period $1$: $f(x + 1) = f(x)$, for all $x\in \mathbb{R}$. Then we may think
of $f$ as `living' on the unit circle, via the map $x\rightarrow e^{2\pi i x}$ which wraps the real line onto the circle. In this setting, we may also think of the circle as the quotient group $\mathbb{R}/\mathbb{Z}$ (though as we promised, group theory will not be assumed of the reader here).
We may travel along these ideas in the other direction: commencing with any function $g$ whose domain is just $[0, 1)$, we can always extend $g$ by periodicity to the whole real line by
defining $G(x):= \{ x\}$, the fractional part of $x$, for all $x \in \mathbb{R}$. Then $G(x) = g(x)$ for all $x \in \mathbb T$, $G$ is periodic on $\mathbb{R}$,
and therefore we may think of
$g$ as living on the circle $\mathbb T$.
More generally,
we may think of a {\bf periodic function $f:\mathbb{R}^d \rightarrow \mathbb{C}$} as living on the cube $\square := [0,1]^d$,
if we insist that $f$ is periodic in the following sense:
\[
f(x) = f(x + e_k), \text{ for all } x\in \square, \text{ and all } 1\leq k \leq d.
\]
In this case, the $1$-dimensional circle is replaced by the $d$-dimensional torus
\[
{\mathbb T^d}:= \mathbb{R}^d/\mathbb{Z}^d,
\]
which we may also think of as the unit cube $[0, 1]^d$, but with opposite facets `glued together'.
Here we define another infinite-dimensional vector space, namely:
\begin{equation}
L^2({\mathbb T^d}):= \{ f:{\mathbb T^d} \rightarrow \mathbb{C} \bigm | \int_{[0, 1]^d} |f(x)|^2 dx < \infty \}.
\end{equation}
We notice that the domains of the integrals in $L^2({\mathbb T^d})$ are cubes, and hence always compact. So we may therefore expect nicer phenomena to occur in this space.
We also have the space of functions
\begin{equation}
L^1({\mathbb T^d}):= \{ f:{\mathbb T^d} \rightarrow \mathbb{C} \bigm | \int_{[0, 1]^d} |f(x)| dx < \infty \},
\end{equation}
which plays a simpler role than the analogous $L^1(\mathbb{R}^d)$ space we had before. And finally
we also define the useful space of $k$-differentiable functions on the torus:
\begin{equation} \label{k derivatives on the torus}
C^k({\mathbb T^d}):= \{ f:{\mathbb T^d} \rightarrow \mathbb{C} \bigm | f \text{ has $k$ continuous derivatives} \}.
\end{equation}
As a special case, we'll simply denote by $C({\mathbb T^d})$ the space of all continuous functions on the torus.
We emphasize that by definition, all of the latter function spaces consist
of \emph{periodic functions} on the cube $[0, 1]^d$.
Similarly to the inner product on $L^2(\mathbb{R}^d)$, we also have in this new context a natural inner product for the space of square-integrable functions
$f\in L^2({\mathbb T^d})$, defined by:
\begin{equation}
\langle f, g \rangle := \int_{[0,1]^d} f(x) \overline{g(x)} dx,
\end{equation}
making $L^2({\mathbb T^d})$ a Hilbert space.
For each $n\in \mathbb{Z}^d$, we define $e_n: \mathbb{R}^d \rightarrow \mathbb{C}$ by:
\begin{equation} \label{basis for Hilbert space}
e_n(x):= e^{2\pi i \langle n, x\rangle}.
\end{equation}
This countable collection of exponentials turns out to form a complete orthonormal basis for $L^2({\mathbb T^d})$.
The orthogonality is the first step, which we prove next. For the proof that the exponentials span $L^2({\mathbb T^d})$ and are complete,
we refer the reader to \cite{EinsiedlerWardBook}.
\bigskip
\begin{thm} [{\bf Orthogonality relations for the exponentials $e_n(x)$ on the torus}]
\index{orthogonality relations for the exponentials $e_n(x)$}
\label{Orthogonality relations for the exponentials $e_n(x)$}
\begin{equation}
\int_{[0,1]^d} e_n(x) \overline{e_m(x)} dx =
\begin{cases}
1 & \mbox{if } n=m \\
0 & \mbox{if not}.
\end{cases}
\end{equation}
\end{thm}
\begin{proof}
Because of the geometry of the cube, we can proceed in this case by separating the variables. If $n \not= m$, then there is at least one index $k$ for which $n_k \not= m_k$. We compute:
\begin{align*}
\int_{[0,1]^d} e_n(x) \overline{e_m(x)} dx
&= \int_{[0,1]^d} e^{2\pi i \langle n-m, x \rangle} dx \\
&= \int_0^1 e^{2\pi i (n_k -m_k)x_k} dx \int_{[0,1]^{d-1}} \prod_{j\not=k} e^{2\pi i (n_j -m_j)x_j} dx \\
&= \int_0^1 e^{2\pi i (n_k -m_k)x_k} dx \int_{[0,1]^{d-1}} \prod_{j\not=k} e^{2\pi i (n_j -m_j)x_j} dx \\
&= \left( \frac{e^{2\pi i (n_k-m_k)}-1}{2\pi i (n_k-m_k)} \right) \int_{[0,1]^{d-1}} \prod_{j\not=k} e^{2\pi i (n_j -m_j)x_j} dx =0,
\end{align*}
because $n_k -m_k$ is a nonzero integer.
\end{proof}
Because $L^2({\mathbb T^d})$ is also an inner product space, it still enjoys the Cauchy-Schwartz inequality.
Intuitively, the space $L^2({\mathbb T^d})$ should be a cozier little space than $L^1({\mathbb T^d})$. This intuition can be made more
rigorous by the following Lemma, despite the fact that $L^2(\mathbb{R}^d) \not\subset L^1(\mathbb{R}^d)$.
\begin{lem} \label{proper containment of L^2 in L^1 for torus}
We have the proper containment of spaces:
\begin{equation}\label{proper containment of spaces over the torus}
L^2({\mathbb T^d}) \subset L^1({\mathbb T^d}).
\end{equation}
\end{lem}
\begin{proof}
Given $f \in L^2({\mathbb T^d})$, we must show that $f \in L^1({\mathbb T^d})$. Using the Cauchy-Schwartz inequality for
$ L^2({\mathbb T^d})$, with $f$ and the constant function $h(x) \equiv 1$, we have:
\begin{align*}
\int_{\mathbb T^d} |f(x)| dx &= \int_{\mathbb T^d} |f(x) h(x) |dx \\
& \leq
\left( \int_{\mathbb T^d} |f(x)|^2 dx \right)^{\frac{1}{2}}
\left( \int_{\mathbb T^d} |h(x)|^2 dx \right)^{\frac{1}{2}} \\
&= \left( \int_{\mathbb T^d} |f(x)|^2 dx \right)^{\frac{1}{2}},
\end{align*}
so we see that $f$ is absolutely integrable over the torus ${\mathbb T^d}$. To show that the containment in \eqref{proper containment of spaces over the torus} is proper, we can consider the following function on
$[0, 1]$:
\[
f(x):=
\begin{cases}
\frac{1}{\sqrt x} & \text{ if } x \in (0, 1], \\
0 & \text{ if } x=0.
\end{cases}
\]
So $\int_0^1 f(x) dx = x^{\frac{1}{2}} \Big|_0^1=1$, but $\int_0^1 |f(x)|^2 dx =
\int_0^1 \frac{1}{x} dx = \infty$.
\end{proof}
\bigskip
\subsection{Fourier series: fast and loose}
Let's see how we can expand (certain) functions in a Fourier series, as well as find a formula for their series coefficients, in a foot-loose and carefree way - i.e. abandoning all rigor for the moment.
Given that the sequence of exponential functions $\{e_n(x) \}_{n \in \mathbb{Z}^d}$ forms a basis
for the infinite dimensional vector space $V:=L^2({\mathbb T^d})$,
we know from Linear Algebra that any function $f \in V$ may be written in terms of this basis:
\begin{equation}
f(x) = \sum_{n \in \mathbb{Z}^d} a_n e_n(x).
\end{equation}
How do we compute the Fourier coefficients $a_n$? Let's go through the intuitive process here, ignoring convergence issues. Well, again by Linear Algebra, we take the inner product of both sides with a fixed basis element $e_k(x)$:
\begin{align*}
\langle f(x), e_k(x) \rangle &= \langle \sum_{n \in \mathbb{Z}^d} a_n e_n(x), e_k(x) \rangle \\
&= \sum_{n \in \mathbb{Z}^d} a_n \langle e_n(x), e_k(x) \rangle \\
&= \sum_{n \in \mathbb{Z}^d} a_n \, \delta(n,k) \\
&= a_k
\end{align*}
where we've used the orthogonality relations, Theorem \ref{Orthogonality relations for the exponentials $e_n(x)$} above, in the third equality. We also used the standard notation $\delta(n,k) := 0$ if $n\not=k$, and $\delta(n,k):=1$ if $n=k$.
Therefore, it must be the case that
\begin{align*}
a_k &= \langle f(x), e_k(x) \rangle \\
&:= \int_{[0,1]^d} f(x) \overline{ e^{2\pi i \langle k, x \rangle} } dx \\
&= \int_{[0,1]^d} f(x) e^{-2\pi i \langle k, x \rangle} dx,
\end{align*}
also called the
{\bf Fourier coefficients} of $f$.
\bigskip
\subsection{Fourier series: slow and rigorous}
Let's record now the rigorous statements of the intuitive arguments that we constructed in the previous section. We may think of a periodic function on $\mathbb{R}^d$ as a function belonging to $L^2({\mathbb T^d})$.
\begin{thm}[{\bf Fourier series for functions on ${\mathbb T^d}$}]
\label{Fourier series for periodic functions}
\index{Fourier series for periodic functions}
The set of exponentials
\[
\{ e_n(x) \bigm | n \in \mathbb{Z}^d \}
\]
form a {\bf complete orthonormal basis} for $L^2({\mathbb T^d})$.
Moreover, we have the following:
\begin{enumerate}[(a)]
\item Every function $g \in L^2({\mathbb T^d})$ has a {\bf Fourier series}
\begin{equation} \label{The Fourier series}
g(x) = \sum_{n\in \mathbb{Z}^d} c_n e^{2\pi i \langle n, x \rangle},
\end{equation}
where the convergence in \eqref{The Fourier series} takes place in the $L^2$ norm on the torus ${\mathbb T^d}$.
\item The {\bf Fourier coefficients} $c_n$ may be computed via the formula:
\begin{equation} \label{Fourier coefficients}
c_n = \int_{[0,1]^d} g(t) e^{-2\pi i \langle n, t\rangle} dt,
\end{equation}
for all $n\in \mathbb{Z}^d$.
\item \rm{(\bf The Parseval identity})
The function $g\in L^2({\mathbb T^d})$ in \eqref{The Fourier series} satisfies
\begin{equation} \label{True Parseval identity}
\int_{[0, 1]^d} | g(x)|^2 dx = \sum_{n \in \mathbb{Z}^d} |c_n|^2.
\end{equation}
\end{enumerate}
\hfill $\square$
\end{thm}
(For a proof, see \cite{EinsiedlerWardBook}, p. 96)
At the risk of overstating the obvious, we note that the equality in \eqref{True Parseval identity} is simply equality between real numbers.
We also note that the Fourier coefficients above are integrals over the unit cube
$[0,1]^d$, and may also be thought of as $c_n = \langle g, e_n \rangle$, the projection of $g$ onto each basis element. To summarize, we've encountered the following types of transforms so far:
\begin{equation} \label{both integrals}
\int_{[0,1]^d} g(t) e^{-2\pi i \langle n, t\rangle} dt, \text{ and } \int_{\mathbb{R}^d} g(t) e^{-2\pi i \langle n, t\rangle} dt.
\end{equation}
To disambiguate, the first integral in \eqref{both integrals} arises from periodic functions on $\mathbb{R}^d$, and it appears as a Fourier coefficient in
Theorem \ref{Fourier series for periodic functions}.
The second integral is our old friend the Fourier transform. How are the two integrals related to each other? This is exactly the magic of the Poisson summation formula, Theorem \ref{Poisson.Summation}.
In the pretty proof of Poisson summation, we begin with a Fourier series of a periodized version of $f$,
and end up showing that its Fourier coefficients, by a small miracle of nature, turn out to also be Fourier transforms of $f$.
A natural question is:
\begin{question}
Which functions have a pointwise convergent Fourier series?
\end{question}
But this question turns out to be rather difficult, and many lifetimes have been devoted to related questions.
It is a fact of life that the Fourier series of an arbitrary continuous function on $\mathbb{R}^d$ may fail to converge uniformly, or even pointwise.
However, there is some good news. As it turns out, if we impose some smoothness conditions on $f$, then $f$ does have a Fourier series which converges pointwise. The next theorem gives
a useful criterion of this type in dimension $1$.
For the real line, we have the following refined version of Theorem \ref{Fourier series for periodic functions}.
We use the standard notation
$f(x_0^+):= \lim_{\varepsilon \rightarrow 0} f(x_0 + \varepsilon)$, and
$f(x_0^-):= \lim_{\varepsilon \rightarrow 0} f(x_0 - \varepsilon)$, where $\varepsilon$ is always chosen to be positive.
We call $f$ piecewise smooth on $[0, 1]$ if
$f'$ is a piecewise continuous function on $[0, 1]$.
\bigskip
\begin{thm} \label{theorem:Fourier series convergence to the mean}
Let $f:\mathbb{R} \rightarrow \mathbb{C}$ be a periodic function, with domain $[0 , 1]$, and piecewise smooth on $\mathbb{R}$.
Then, for each $t \in \mathbb{R}$, we have
\begin{equation} \label{convergence of Fourier series to the mean}
\lim_{N\rightarrow \infty} \sum_{n= -N}^N c_n e^{2\pi i n t} =
\frac{ f(t^+) + f(t^-) }{2},
\end{equation}
where $c_n:= \int_0^1 f(x) e^{-2\pi i x n} dx$ are the Fourier coefficients of $f$.
\rm{(For a proof of Theorem \ref{theorem:Fourier series convergence to the mean} see \cite{Travaglini})}.
\hfill $\square$
\end{thm}
We will come back to these {\bf partial Fourier sums}, \index{partial Fourier sums}
occurring in Theorem \ref{theorem:Fourier series convergence to the mean},
and defined by
\begin{equation} \label{partial sums}
S_N f(t):= \sum_{n= -N}^N c_n e^{2\pi i n t}.
\end{equation}
There is also a natural and easy extension of Parseval's identity \eqref{True Parseval identity}.
Given any two functions $f, g\in L^2({\mathbb T^d})$, we've seen in \eqref{The Fourier series}
that
\[
f(x) = \sum_{n\in \mathbb{Z}^d} a_n \, e^{2\pi i \langle n, x \rangle}, \text{ and }
g(x) = \sum_{n\in \mathbb{Z}^d} b_n \, e^{2\pi i \langle n, x \rangle},
\]
both converging in the $L^2({\mathbb T^d})$ norm.
\begin{thm}
If $f, g, \in L^2({\mathbb T^d})$, then with the notation above we have
\[
\int_{{\mathbb T^d}} f(x) \overline{g(x)} dx = \sum_{n\in \mathbb{Z}^d} a_n \overline{b_n}.
\]
\hfill $\square$
\end{thm}
\subsection{The first periodic Bernoulli polynomial}
To see a concrete instance of Theorem \ref{Fourier series for periodic functions}, we study the
function $P_1(x)$, which we've briefly encountered before, as the first periodic Bernoulli polynomial. This function turns out to be so important that it deserves its own section here. We recall its definition:
\begin{equation} \label{def of P_1 again}
P_1(x):=
\begin{cases}
\{ x \} - \frac{1}{2} &\mbox{if } x \notin \mathbb{Z}, \\
0 & \mbox{if } x \in \mathbb{Z}.
\end{cases}
\end{equation}
It's easy to see that
$P_1 \in L^1(\mathbb T)$, so it has a Fourier series, by Theorem \ref{Fourier series for periodic functions}, part (a):
\begin{equation}
P_1(x) = \sum_{n \in \mathbb{Z}} c_n e^{2\pi i n x},
\end{equation}
and the equality here means equality in the $L^2(\mathbb T)$ norm.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.8in]{Bernoulli}
\end{center}
\caption{The first periodic Bernoulli polynomial $P_1(x)$ }
\end{figure}
Let's compute the Fourier coefficients of $P_1$, according to
Theorem \ref{Fourier series for periodic functions}, part (b). We will use integration by parts:
\begin{align*}
c_n&= \int_0^1 \left( \{ x \} - \tfrac{1}{2} \right) e^{-2\pi i n x} dx
= \int_0^1 x e^{-2\pi i n x} dx - \tfrac{1}{2} \int_0^1 e^{-2\pi i n x} dx \\
&= x \frac{ e^{-2\pi i n x} }{-2\pi i n } \Big|_0^1 - \int_0^1 \frac{ e^{-2\pi i n x} }{-2\pi i n } dx
= \frac{ 1 }{-2\pi i n } - 0 = \frac{ 1 }{-2\pi i n },
\end{align*}
when $n \not=0$. For $n=0$, we have $c_0 = \int_0^1 (x - \tfrac{1}{2})dx = 0$.
Hence we have the Fourier series
\begin{equation} \label{Fourier series of P_1 in the norm}
P_1(x) =\{ x \} -\frac{1}{2}= - \frac{ 1 }{2\pi i } \sum_{n \in \mathbb{Z} \atop n\not=0} \frac{ 1 }{ n } e^{2\pi i n x},
\end{equation}
where the latter equality means convergence in the $L^2({\mathbb T^d})$ norm.
If we want to get pointwise convergence of this Fourier series, we may apply
Theorem \ref{theorem:Fourier series convergence to the mean}, which allows us to conclude that we have pointwise convergent sums:
\begin{align} \label{example of theorem on pointwise convergence}
\lim_{N\rightarrow \infty} -\frac{1}{2\pi i} \sum_{-N \leq n \leq N \atop n\not=0} \frac{1}{n} e^{2\pi i n x}
&= \frac{ P_1(x^+) + P_1(x^-) }{2} \\
&= \{ x \} -\frac{1}{2},
\end{align}
when $x\notin \mathbb{Z}$. For $x \in \mathbb{Z}$, we can also check that the equality
\eqref{example of theorem on pointwise convergence} holds by observing that
\[
\sum_{-N \leq n \leq N \atop n\not=0} \frac{1}{n} e^{2\pi i n x} =
\sum_{-N \leq n \leq N \atop n\not=0} \frac{1}{n} = 0,
\]
while $\frac{ P_1(x^+) + P_1(x^-) }{2} =\tfrac{1}{2}\left(- \frac{1}{2} + \frac{1}{2}\right) =0 $ as well, which is consistent with the definition
\eqref{def of P_1 again}
of $P_1(x)$ at the integers.
Next, we can give a classical application of the Fourier series \eqref{Fourier series of P_1 in the norm}
using Parseval's identity \eqref{True Parseval identity}:
\[
\int_0^1 |P_1(u)|^2 du = \sum_{n \in \mathbb{Z}} |a_n|^2.
\]
Let's simplify both sides:
\begin{align*}
\sum_{n \in \mathbb{Z}} |a_n|^2 &= \frac{1}{4\pi^2} \sum_{n \in \mathbb{Z}-\{0\}} \frac{1}{n^2} =
\frac{1}{2\pi^2} \sum_{n \geq 1} \frac{1}{n^2},
\end{align*}
while
\begin{align*}
\int_0^1 |P_1(u)|^2 du = \int_0^1 \left( \{ x \} - \frac{1}{2} \right)^2 dx
&=
\int_0^1 \left(x - \frac{1}{2}\right)^2 dx
= \frac{1}{12}.
\end{align*}
Therefore
\[
\sum_{n \geq 1} \frac{1}{n^2} = \frac{\pi^2}{6},
\]
a number-theoretic identity that goes back to Euler.
In a similar manner one can evaluate the Riemann zeta function at all positive even integers,
using the cotangent function
(Exercise \ref{Riemann zeta function, and Bernoulli numbers}).
\begin{comment}
Now that we have the Fourier series for $P_1(x)$, let's see what happens if we integrate both sides
of \eqref{Fourier series of P_1 in the norm}, assuming for the moment that we may integrate the RHS of \eqref{Fourier series of P_1 in the norm} term-by-term:
\begin{align*}
-\frac{ 1 }{(2\pi i)^2 } \sum_{n \in \mathbb{Z} \atop n\not=0} \frac{ e^{2\pi i n t} }{ n^2 } =
\int_0^t P_1(x) dx =
\end{align*}
using the fact that $\frac{d}{dx} B_n(x) = n B_{n-1}(x)$ for all $n \geq 1$
(Exercise \ref{derivative of Bernoulli polys}), and hence the same is true for $P_n(x)$.
\end{comment}
\bigskip
Another natural question arises.
\begin{question}
What sort of functions $f$ are uniquely determined by all of their Fourier coefficients?
\end{question}
To describe a partial answer, we recall the space of all continuous functions on the torus:
\begin{equation}
C({\mathbb T^d}):= \{ f: {\mathbb T^d} \rightarrow \mathbb{C} \bigm | f \text{ is continuous on } {\mathbb T^d} \}.
\end{equation}
\begin{thm}
Let $f \in C({\mathbb T^d})$, and suppose that $\hat f(n) = 0$ for all $n \in \mathbb{Z}^d$. Then $f(x)=0$,
for all $ x \in [0, 1]^d$.
In particular, if $f, g \in C({\mathbb T^d})$ and $\hat f(n) =\hat g(n)$ for all $n \in \mathbb{Z}^d$,
then $f(x) = g(x)$ for all $x \in [0, 1]^d$.
\hfill $\square$
\end{thm}
In other words, a continuous function on the torus is uniquely determined by its Fourier coefficients
(see \cite{EinsiedlerWardBook} for a proof).
\begin{comment}
We might hope that the same is true for a function that is piecewise continuous on the torus ${\mathbb T^d}$.
\begin{align}
\hat F(n) &= \int_0^{1/2} e^{-2\pi i n x} dx - \int_{1/2}^1 e^{-2\pi i n x} dx =
\frac{e^{-2\pi i n x}}{-2\pi i n}\Big|_0^{1/2}- \frac{e^{-2\pi i n x}}{-2\pi i n}\Big|_{1/2}^{0} \\
&= \frac{1}{-2\pi i n}
\Big(
e^{-\pi i n} - 1 - ( 1 - e^{-\pi i n} )
\Big)
=\frac{1 - e^{-\pi i n}}{\pi i n}=\frac{1 - (-1)^n}{\pi i n}.
\end{align}
\end{comment}
\bigskip
\section{As $f$ gets smoother, $\hat f$ decays faster}
\label{As f gets smoother, the FT decays faster}
There is a very basic and important relationship between the level of smoothness of $f$,
and the speed with which $\hat f$ tends to $0$ as $x \rightarrow \infty$. To capture this relation very concretely, let's compute things on the real line,
to see how the FT interacts with the derivative.
\begin{lem} \label{FT interacts with derivative}
Let $f \in L^1(\mathbb{R})$.
\begin{enumerate}[(a)]
\item \label{FT of derivative}
If $f$ is piecewise smooth, and also enjoys $f' \in L^1(\mathbb{R})$, then:
\[
\widehat{ f' }(\xi) = (2\pi i) \xi \hat f(\xi).
\]
\item \label{explicit decay rate}
More generally, let $k\geq 0$, suppose that $f$ has $k$ derivatives, $f^{(k)}$ is piecewise smooth,
and that
we also have $f^{(k+1)} \in L^1(\mathbb{R})$.
Then:
\[
\widehat{ f^{(k+1)} }(\xi) = (2\pi i \xi)^{k+1} \hat f(\xi).
\]
\item \label{derivative of the FT}
Now we suppose that $x f(x) \in L^1(\mathbb{R})$. Then:
\[
\frac{d}{d\xi} {\mathcal F}(f)(\xi) = (-2\pi i) \, {\mathcal F}(x f(x))(\xi).
\]
\end{enumerate}
\end{lem}
\begin{proof}
To prove part \ref{FT of derivative}, we notice that $\lim_{x\rightarrow \infty} f(x) = f(0) + \int_0^\infty f'(x) dx$, using the hypothesis $f' \in L^1(\mathbb{R})$.
Using the hypothesis $f \in L^1(\mathbb{R})$, we know that the Riemann-Lebesgue Lemma \ref{Riemann--Lebesgue lemma} implies that $\lim_{x\rightarrow \infty} f(x)=0$.
Similarly, $\lim_{x\rightarrow -\infty} f(x) =0$. Integration by parts now gives us:
\begin{align*}
\widehat{ f' }(\xi) &= \int_{\mathbb{R}} f'(x) e^{-2\pi i x \xi} dx
= f(x)e^{-2\pi i x \xi}\Big |_{-\infty}^\infty - \int_{\mathbb{R}} f(x) (-2\pi i \xi) e^{-2\pi i x \xi} dx \\
&= 2\pi i \xi \int_{\mathbb{R}} f(x) e^{-2\pi i x \xi} dx := 2\pi i \xi \hat f(\xi).
\end{align*}
Part \ref{explicit decay rate} follows from \ref{FT of derivative} by induction on $k$. To prove part \ref{derivative of the FT}, we have:
\begin{align*}
{\mathcal F}(x f(x))(\xi) &:= \int_{\mathbb{R}} xf(x) e^{-2\pi i x \xi} dx = \frac{1}{-2\pi i} \int_{\mathbb{R}} \frac{d}{d\xi} f(x) e^{-2\pi i x \xi} dx\\
&=-\frac{1}{2\pi i} \frac{d}{d\xi} \int_{\mathbb{R}} f(x) e^{-2\pi i x \xi} dx
=-\frac{1}{2\pi i} \frac{d}{d\xi} \hat f(\xi).
\end{align*}
\end{proof}
It follows from Theorem \ref{FT interacts with derivative}, part \ref{explicit decay rate}, that we have an explicit decay rate for the Fourier coefficients of a periodic function $f$,
assuming that $f$ is sufficiently smooth. To obtain the following Corollary, we can simply use the fact that $f^{(k+1)} \in L^1(\mathbb{R})$ implies that $ \widehat{ f^{(k+1)}}$ is uniformly bounded:
$\Big| \frac{1}{(2\pi )^{k+1}} \widehat{ f^{(k+1)} }(\xi) \Big| < C$, for a positive constant $C$.
\begin{cor} \label{cor: f smoother implies FT of F decays faster}
If $f$ has $k$ continuous derivatives, and we also have $f^{(k+1)} \in L^1(\mathbb{R})$,
then there is a constant $C>0$ such that:
\begin{equation}
| \hat f(\xi) | < C \frac{1}{ | \xi|^{k+1}},
\end{equation}
for all $\xi \not=0$.
\hfill $\square$
\end{cor}
In other words, we now understand
the dictum ``as $f$ gets smoother, $\hat f$ decays faster'' in a precise quantitative manner:
if $f$ has $k$ derivatives, then $\hat f$ decays faster than a polynomial of degree $k$.
\bigskip
\section{How fast do Fourier coefficients decay?}
In a manner completely analogous to the previous Section \ref{As f gets smoother, the FT decays faster},
we can repeat the important idea of integration by parts to see how fast
Fourier coefficients decay, and here we may expect even better results, because we will integrate over the compact unit cube, rather than over the non-compact space $\mathbb{R}^d$. We first work things out in dimension $1$, recalling that
the Fourier coefficients of $f$ are defined by $c_n:= \int_0^1 f(x) e^{-2\pi i n x} dx$, for all $n \in \mathbb{Z}$.
For the sake of the reader, we recall the space of functions $C^k(\mathbb T)$ from
\ref{k derivatives on the torus}, which have $k$ continuous derivatives. We also recall that
$f \in L^1(\mathbb T)$ means $\int_0^1f(x) dx$ is finite, and that $f(x+1) = f(x)$,
for all $x\in [0, 1]$. Finally, we note that the same conclusion of the Riemann-Lebesgue lemma \ref{Riemann--Lebesgue lemma} also holds for
functions $f \in L^1({\mathbb T^d})$, with exactly the same proof that we gave in Lemma \ref{Riemann--Lebesgue lemma}.
\begin{thm} \label{decay of Fourier coefficients}
Let $f\in L^1(\mathbb T)$.
\begin{enumerate}[(a)]
\item \label{decay of Fourier coefficients for C^1}
If $f \in C^1(\mathbb T)$, then its Fourier coefficients satisfy
\begin{equation}
\lim_{|n|\rightarrow \infty} |n c_n| = 0.
\end{equation}
In other words, $|c_n| = o\left( \frac{1}{n} \right)$.
\item \label{decay of Fourier coefficients for C^k}
More generally, fix an integer $k\geq 1$. If $f \in C^k(\mathbb T)$, then its Fourier coefficients satisfy
\begin{equation}
\lim_{|n|\rightarrow \infty} |n^k c_n| = 0.
\end{equation}
In other words, $|c_n| = o\left( \frac{1}{n^k} \right)$.
\end{enumerate}
\end{thm}
\begin{proof}
We compute the Fourier coefficients using integration by parts. For each $n \not=0$, we have:
\begin{align*}
c_n &:= \int_0^1 f(x) e^{-2\pi i n x} dx
= \left[ f(x) \frac{ e^{-2\pi i n x} }{-2\pi i n}\right] \Big|_0^1 + \frac{ 1}{2\pi i n}\int_0^1 f'(x) e^{-2\pi i n x} dx \\
&= \frac{ f(1)-f(0) }{ -2\pi i n } + \frac{ 1}{2\pi i n}\int_0^1 f'(x) e^{-2\pi i n x} dx \\
&= \frac{ 1}{2\pi i n}\int_0^1 f'(x) e^{-2\pi i n x} dx,
\end{align*}
using the periodicity of $f$. Because $f'$ is continuous, the Riemann-Lebesgue lemma on $L^1(\mathbb T)$ gives us
$\lim_{|n|\rightarrow \infty} \int_0^1 f'(x) e^{-2\pi i n x} dx = 0$. So we see that
\[
|n c_n| \rightarrow 0, \ \text{ as } |n| \rightarrow \infty,
\]
completing part \ref{decay of Fourier coefficients for C^1}. Part \ref{decay of Fourier coefficients for C^k} follows easily by induction on $k$, repeating exactly the same integration by parts computation above.
\end{proof}
We note that the same proof works with even weaker hypotheses in part \ref{decay of Fourier coefficients for C^k}. Namely, given an integer $k\geq 1$, all we require is that
$f^{(j)}$ is continuous on $\mathbb T$, for $0\leq j < k$, and $f^{(k)} \in L^1(\mathbb T)$.
Let's see a concrete application of these ideas (see Note \ref{Note:GregKuperberg}).
\begin{thm} \label{Euler-Maclaurin type identity}
Suppose that $f\in C^k(\mathbb T)$, for a fixed integer $k\geq 1$. Then:
\begin{equation}
\int_0^1 f(x) dx = \frac{1}{N} \sum_{m=0}^{N-1} f\left(\tfrac{m}{N}\right) + o\left(\tfrac{1}{N^{k}}\right),
\end{equation}
as $N\rightarrow \infty$.
\end{thm}
\begin{proof}
Because $f$ is periodic on $\mathbb{R}$, we follow ``Hecke's dictum''; namely, we first expand $f$ into its Fourier series, which is guaranteed by
Theorem \ref{Fourier series for periodic functions}:
\[
f(x) = \sum_{n\in \mathbb{Z}} c_n e^{2\pi i n x}.
\]
Since this Fourier series converges absolutely, we may interchange the finite sum with the series:
\begin{align*} \label{first step of finite sum approximation}
\frac{1}{N} \sum_{m=0}^{N-1} f\left(\tfrac{m}{N}\right) &= \frac{1}{N} \sum_{m=0}^{N-1}
\sum_{n\in \mathbb{Z}} c_n e^{2\pi i n \tfrac{m}{N}} \\
&= \sum_{n\in \mathbb{Z}} c_n
\Big( \tfrac{1}{N} \sum_{m=0}^{N-1} e^{2\pi i n \tfrac{m}{N}} \Big) \\
&=\sum_{n\in \mathbb{Z}} c_{Nn},
\end{align*}
using Exercise \ref{DivisibilityUsingExponentials} (the harmonic detector for divisibility).
Next, we recall that the constant term is $c_0= \int_0^1 f(x) dx$, and we separate out this term from the latter series:
\begin{align*}
\frac{1}{N} \sum_{m=0}^{N-1} f\left(\tfrac{m}{N}\right)
= \int_0^1 f(x) dx + \sum_{n\in \mathbb{Z} \atop {n \not=0 } } c_{Nn},
\end{align*}
Now we can use the (little-o) rate of decay of the Fourier coefficients, given by
Theorem \ref{decay of Fourier coefficients}, part \ref{decay of Fourier coefficients for C^k}, to write
$|c_{Nn}| < \frac{C}{(Nn)^k}$ for \emph{all} constants $C>0$. We conclude that
\[
\sum_{n\in \mathbb{Z} \atop {n \not=0 } } |c_{Nn}|
< C \sum_{n\in \mathbb{Z} \atop {n \not=0 }} \frac{1}{ N^{k} |n|^{k}}
< 2 C\, \zeta(k) \frac{1}{ N^{k}},
\]
for all constants $C>0$. So as $N\rightarrow \infty$, the error term
$\sum_{n\in \mathbb{Z} \atop {n \not=0 } } c_{Nn}$ is $o\left( \frac{1}{ N^k} \right)$, as claimed.
\end{proof}
\begin{comment}
\begin{example}
\rm{
Let's study the sum $ \frac{1}{N} \sum_{m=0}^{N-1} f\left(\tfrac{m}{N}\right) $, where
$f(x) := \sqrt{\{x\}+\frac{1}{N}}$, and $N$ is a fixed positive integer, and $\{x\}$ is the fractional part of $x$. Because $f$ is infinitely smooth
on $\mathbb T$, and periodic on $\mathbb{R}$,
the error term in Theorem \ref{Euler-Maclaurin type identity} holds for all positive integers $k$,
giving us super-exponential decay in $N$ (and the latter conclusion holds for any $C^\infty$ function on the circle $\mathbb T$).
By Theorem \ref{Euler-Maclaurin type identity}, we have
\begin{align*}
\frac{1}{N} \sum_{m=0}^{N-1} \sqrt{ \frac{m}{N} +\frac{1}{N} }
&=\int_0^1 \sqrt{x +\frac{1}{N}} \, dx + o\left(\frac{1}{N^{k}}\right)
=\tfrac{2}{3} \left(1+ \frac{1}{N} \right)^{\frac{3}{2}} - \tfrac{2}{3} \left( \frac{1}{N} \right)^{\frac{3}{2}} + o\left(\frac{1}{N^{k}}\right),
\end{align*}
or equivalently
\begin{equation}
1+ \sqrt 2 + \cdots + \sqrt N
=\tfrac{2}{3} \left(N+1 \right)^{\frac{3}{2}} - \tfrac{2}{3} + o\left(\frac{1}{N^{k-\frac{3}{2}}}\right),
\end{equation}
which is valid for all $k \geq 1$.
Although this might seem `too good to be true',
by definition we may equivalently write:
\begin{align*}
\lim_{N \rightarrow \infty} N^{k-\frac{3}{2}}
\left | \sum_{m=1}^{N} \sqrt m - \left(N+1 \right)^{\frac{3}{2}} + \tfrac{2}{3} \right | =0,
\end{align*}
for each $k\geq 1$.
}
\hfill $\square$
\end{example}
\end{comment}
It is worth mentioning that although our proof of Theorem \ref{Euler-Maclaurin type identity}
does not cover the case `$k=0$', this case
is also true because it represents the Riemann sum approximation to the integral.
\section{The Schwartz space} \label{nice functions}
\index{Schwartz space}
We saw in Section \ref{As f gets smoother, the FT decays faster} that a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ in the space domain, that has $k$ derivatives, corresponds to a function $\hat f$ in the Fourier transform domain.
If we `take this idea to the limit', so to speak, What does that last adjective mean?
Following Laurent Schwartz, we can make rigorous sense of the words `rapidly decreasing', as follows.
We recall that our definition of a `nice function' was any function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ for which the Poisson summation formula holds. Here we give our first family of sufficient conditions for a function $f$ to be nice.
A {\bf Schwartz function} $f: \mathbb{R} \rightarrow \mathbb{C}$ is defined as any infinitely smooth function ($f \in C^\infty(\mathbb{R})$)
that satisfies the following
growth condition:
\begin{equation}
|x^a \frac{d}{dx^k} f(x)| \text{ is bounded on } \mathbb{R},
\end{equation}
for all integers $a, k \geq 0$.
In particular, a Schwartz function decreases faster than any polynomial function, as $|x|$ tends to infinity.
\medskip
\begin{example}
\rm{
The Gaussian function $G_t(x) := e^{-t ||x||^2}$ is a Schwartz function, for each fixed
$t>0$. To see this, we first consider $\mathbb{R}^1$, where we note that the $1$-dimensional Gaussian
is a Schwartz function, as follows. We observe that for all positive integers $k$,
$ \frac{d}{dx^k} G_t(x) = H_n(x) G_t(x)$, where $H_n(x)$ is a univariate polynomial in $x$ (which also depends on the parameter $t$, but we think of $t$ as a constant).
Since $\lim_{x\rightarrow \infty} \frac{x^a \, H_n(x)}{ e^{t ||x||^2}} =0$, for all positive integers $a$, we see that $G_t(x)$ is a Schwartz function.
Now we note that the product of Schwartz functions is again a Schwartz function; hence the $d$-dimensional Gaussian, $G_t(x) := e^{-t ||x||^2} = \prod_{k=1}^d e^{-t x_k^2} $, a product of $1$-dimensional Gaussians, is a Schwartz function.
Some might say the Gaussian is the quintessential Schwartz function, partly because it is also an eigenfunction of the Fourier transform, as we'll see below.
}
\hfill $\square$
\end{example}
\medskip
\begin{example} \label{example: the abs value exponential}
\rm{
We define $f(x) := e^{-2\pi t |x|}$ on the real line, for a fixed $t >0$.
To see that $f$ is \emph{not} a Schwarz function, we merely have to observe that $f$ is not differentiable at $x=0$.
To be a Schwartz function, $f$ would have to be infinitely differentiable everywhere on $\mathbb{R}$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{exponential}
\end{center}
\caption{The function $e^{-|x|}$. }
\end{figure}
Interestingly, we can also see that $f$ is not a Schwartz function in another way - by computing its Fourier transform and observing that it is not rapidly decreasing:
\begin{align*}
\hat f(\xi) &:= \int_\mathbb{R} e^{-2\pi t |x| -2\pi i x \xi }dx \\
&= \int_{-\infty}^0 e^{2\pi t x -2\pi i x \xi } dx
+ \int_0^{+\infty} e^{-2\pi t x -2\pi i x \xi } dx \\
&= \int_{-\infty}^0 e^{2\pi x(t - i \xi) } dx
+ \int_0^{+\infty} e^{-2\pi x (t+ i \xi) } dx \\
&= \frac{ e^{2\pi x(t - i \xi)} }{2\pi (t-i\xi)}\Big|_{x=-\infty}^{x=0} +
\frac{ e^{-2\pi x(t + i \xi)} }{-2\pi (t+i\xi)}\Big|_{x=0}^{x=\infty} \\
&= \frac{ 1 }{2\pi (t-i\xi)} +
\frac{ 1 }{2\pi (t+i\xi)} \\
&=
\frac{ t }{\pi (t^2 + \xi^2)},
\end{align*}
valid for all $\xi \in \mathbb{R}$. Because the Fourier transform
\begin{equation} \label{FT of the abs value exponential}
\frac{ t }{\pi (t^2 + \xi^2)}
\end{equation}
is not a rapidly decreasing function, we have another proof that $f$ is not a Schwartz function.
This example is interesting in that $f$ is infinitely differentiable everywhere, except at one point, namely
$x =0$. Yet this local lack of smoothness - at only a single point - is enough to cause a global change in decay for its Fourier transform.
}
\hfill $\square$
\end{example}
It is just as easy to define Schwartz functions on $\mathbb{R}^d$ as well.
For any $k:= (k_1, \dots, k_d) \in \mathbb{Z}_{\geq 0}^d$, we can define the multivariable differential operator
\[
D_k := \frac{\partial}{\partial x_1^{k_1} \cdots \partial x_d^{k_d}}.
\]
\begin{example}
In $\mathbb{R}^1$, this is the usual $k$'th derivative, namely
$D_k f(x) := \frac{d}{dx^k}f(x)$.
In $\mathbb{R}^2$, for example, we have $D_{(1,7)} f(x) := \frac{\partial}{\partial x_1 \partial x_2^7}f(x)$.
\hfill $\square$
\end{example}
The {\bf order} of the differential operator
$D_k$ is by definition $|k|:= k_1+ \cdots + k_d$.
To define spaces of differentiable functions, we call a function $f : \mathbb{R}^d \rightarrow \mathbb{C}$
a $C^m$-function if all partial derivatives
$D_k f$ of order $|k| \leq m$ exists and are continuous. We denote the collection of all such
$C^m$-functions on Euclidean space by $C^m(\mathbb{R}^d)$.
When considering {\bf infinitely-differentiable functions on Euclidean space}, we denote this space by $C^\infty(\mathbb{R}^d)$.
So we see that in $\mathbb{R}^d$, we can define {\bf Schwartz functions} \index{Schwartz function}
similarly to our previous definition: they are infinitely differentiable
functions $f:\mathbb{R}^d \rightarrow \mathbb{C}$ such that for all vectors $a, k \in \mathbb{Z}_{\geq 0}^d$
we have:
\begin{equation}
|x^a D_k f(x)| \text{ is bounded on } \mathbb{R}^d,
\end{equation}
where $x^a:= x_1^{a_1} \cdots x_d^{a_d}$ is the standard multi-index notation.
We also define the {\bf Schwartz space} $S(\mathbb{R}^d)$ to be set of all Schwartz functions $f:\mathbb{R}^d \rightarrow \mathbb{C}$.
\begin{thm} \label{Schwartz goes to Schwartz}
The Fourier transform maps the Schwartz space $S(\mathbb{R}^d)$ one-to-one, onto itself.
(See Exercise \ref{Schwartz space convolution invariance})
\end{thm}
In fact, more is true: the mapping $f \rightarrow \hat f$ from $S(\mathbb{R}^d)$ to itself is an isometry.
The proof of this
fact uses the Parseval relation below.
And now that we know the definition of rapid decay, we see that an obvious consequence of
Corollary \ref{cor: f smoother implies FT of F decays faster}
is the following:
\begin{equation} \label{smooth implies FT is rapidly decreasing}
\text{ If } f \text{ is infinitely smooth, then } \hat f \text{ is rapidly decreasing}.
\end{equation}
In fact, we can combine some of the ideas above to record another useful fact.
\begin{lem} \label{useful Schwartz fact}
Let $\phi:\mathbb{R}^d \rightarrow \mathbb{C}$ be compactly supported and infinitely smooth.
Then
\[
\phi \in {\mathcal S}(\mathbb{R}^d).
\]
\end{lem}
\begin{proof}
Because $\phi$ is compactly supported, we know that $\hat \phi$ is infinitely smooth (differentiation under the integral). Moreover, the assumption that $\phi$ is infinitely smooth implies that
$\hat \phi$ is rapidly descreasing, by \eqref{smooth implies FT is rapidly decreasing}. So now we know that
$\hat \phi$ is both rapidly decreasing and infinitely smooth - i.e. a Schwartz function.
Applying Theorem \ref{Schwartz goes to Schwartz}, we see that its Fourier transform is also a Schwartz function. Namely, using Fourier inversion, we conclude that $\hat{\hat \phi}(-x) = \phi(x) \in S(\mathbb{R}^d)$.
\end{proof}
The functions satisfying the conditions of Lemma \ref{useful Schwartz fact} are also called {\bf bump functions}.
\section{Poisson Summation I} \label{Poisson Summation section}
\index{Poisson summation formula}
We introduce the Poisson summation formula, one of the most useful tools in analytic number theory, and in discrete / combinatorial geometry. This version of Poisson summation holds for Schwartz functions.
There are many different families of sufficient conditions that a function $f$ can satisfy, in order for Poisson summation to be applicable to $f$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.8in]{NiceFunctions}
\end{center}
\caption{Spaces of functions for Poisson summation}
\label{nice functions, containment}
\end{figure}
\bigskip
\begin{thm}[Poisson summation formula, I]
\index{Poisson summation formula} \index{Poisson summation formula}
\label{Poisson.Summation}
Given a {\it Schwartz function} $f~:~\mathbb{R}^d \rightarrow \mathbb{C}$, we have
\begin{equation} \label{Poisson.summation1}
\sum_{n \in \mathbb{Z}^d} f(n+ x) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle},
\end{equation}
valid for all $x \in \mathbb{R}^d$.
In particular, we have:
\begin{equation} \label{Poisson.summation2}
\sum_{n \in \mathbb{Z}^d} f(n) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi).
\end{equation}
Both sides of \eqref{Poisson.summation1} converge absolutely, and are continuous functions on $\mathbb{R}^d$.
\end{thm}
\begin{proof}
If we let $F(x):= \sum_{n \in \mathbb{Z}^d} f(n+ x)$, then we notice that $F$ is periodic on $\mathbb{R}^d$, with the cube $[0,1)^d$ as a fundamental domain. The argument is easy: fix any $m \in \mathbb{Z}^d$. Then
$F(x + m) = \sum_{n \in \mathbb{Z}^d} f(n+ x + m) = \sum_{k \in \mathbb{Z}^d} f(x + k)$, because $\mathbb{Z}^d + m = \mathbb{Z}^d$.
By Theorem \ref{Fourier series for periodic functions}, $F$ has a fourier series, so let's compute it:
\[
F(x) := \sum_{k \in \mathbb{Z}^d} a_k e^{2\pi i \langle k, x \rangle},
\]
where $a_k = \int_{[0,1)^d} F(u) e^{2\pi i \langle k, u \rangle}du$ for each fixed $k\in \mathbb{Z}^d$.
Let's see what happens if we massage $a_k$ a bit:
\begin{align}
a_k &:= \int_{[0,1)^d} F(u) e^{-2\pi i \langle k, u \rangle} du \\
&=\int_{[0,1)^d} \sum_{n \in \mathbb{Z}^d} f(n+ u)
e^{-2\pi i \langle k, u \rangle} du \\
&=\sum_{n \in \mathbb{Z}^d} \int_{[0,1)^d} f(n+ u) \label{outersum}
e^{-2\pi i \langle k, u \rangle} du.
\end{align}
The interchange of summation and integral in the latter step is allowed by Theorem \ref{Application of dominated convergence}, which is an application of the dominated convergence theorem,
because the integrand satisfies
$ | f(n+ u) e^{-2\pi i \langle k, u \rangle} | = | f(n+ u) | \in L^1(\mathbb{R}^d)$. The latter absolute integrability of $f$ is
due to the fact that $f$ is a Schwartz function.
Now we fix an $n\in\mathbb{Z}^d$ in the outer sum of \eqref{outersum}, and make the change of variable in the integral:
$n+u := w$, so that $du = dw$.
A critical step in this proof is the fact that as $u$ varies over the cube ${[0,1)^d}$, $w:= n+u$ varies
over all of $\mathbb{R}^d$ because we have a tiling
\index{tiling}
of Euclidean space by the unit cube: ${[0,1)^d} + \mathbb{Z}^d = \mathbb{R}^d$.
We note that under this change of variable,
$e^{-2\pi i \langle k, u \rangle} = e^{-2\pi i \langle k, w-n \rangle} =
e^{-2\pi i \langle k, w \rangle}$, because $k, n \in \mathbb{Z}^d$ and hence
$e^{2\pi i \langle k, n \rangle} =1$.
Therefore, we finally have:
\begin{align}\label{finally, the FT}
a_k = \sum_{n \in \mathbb{Z}^d} \int_{n+ [0,1)^d} f(w)
e^{-2\pi i \langle k, w \rangle} dw
= \int_{\mathbb{R}^d} f(w)
e^{-2\pi i \langle k, w \rangle} dw := \hat f(k),
\end{align}
so that $F(x) = \sum_{k \in \mathbb{Z}^d} a_k e^{2\pi i \langle k, x \rangle}
= \sum_{k \in \mathbb{Z}^d} \hat f(k) e^{2\pi i \langle k, x \rangle}$.
\end{proof}
We define a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ to be a {\bf nice function}
if both $f, \hat f \in L^1(\mathbb{R}^d)$, and if the Poisson summation formula
\begin{equation} \label{nice functions}
\sum_{n \in \mathbb{Z}^d} f(n+ x) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle}
\end{equation}
holds for $f$ pointwise, for each $x \in \mathbb{R}^d$.
We will give various different sets of sufficient conditions for a
function $f$ to be nice.
Figure \ref{nice functions, containment} suggests a simple containment relation between some of these function spaces,
as we will easily prove.
There are a few things to notice about the classical, and pretty proof of Theorem \ref{Poisson.summation1}. The first is that we began with any square-integrable function $f$ defined on all of $\mathbb{R}^d$,
and forced a periodization of it, which was by definition $F$. This is known as the ``folding'' part of the proof.
Then, at the end of the proof, there was the ``unfolding'' process, where we summed an integral over a lattice, and because the cube tiles $\mathbb{R}^d$, the sum of the integrals transformed into a single integral over $\mathbb{R}^d$.
The second thing we notice is that the integral $\int_{\mathbb{R}^d} f(x) e^{-2\pi i \langle x, \xi \rangle} dx$,
which is by definition the Fourier transform of $f$,
appears quite naturally due to the tiling of $\mathbb{R}^d$ by the unit cube $[0,1)^d$.
Hopefully there will now be no confusion as to the difference between the integral over the cube, and the integral over $\mathbb{R}^d$, both appearing together in this proof.
\section{Useful convergence lemmas, in preparation for Poisson summation II}
To prepare ourselves for Poisson's original summation formula, which we give in the next section, we will see here Poisson's hypotheses for the growth of $f$ and $\hat f$, together with the immediate convergence consequences they carry.
\begin{lem} \label{Poisson bound implies L^1}
Let $f:\mathbb R^d \rightarrow \mathbb{C}$ be a function that enjoys the bound
\[
|f(x)|\leq\frac{C}{(1+||x||)^{d+\delta}},
\]
for all $x\in \mathbb R^d$, and for constants $C, \delta >0$ that are independent of $x$.
Then $f\in L^1(\mathbb{R}^d)$.
\end{lem}
\begin{proof}
Consider the cube
$Q_n:= [-n,n]^d$ and let $D_n:=Q_{n+1} - Q_n$ denote the set difference; in other words, $D_n$ is the cubical shell between the cube $Q_n$ and the cube $Q_{n+1}$.
We have $\mathbb{R}^d=\bigcup_{n\geq 0} D_n$, and $D_0=Q_1$. Also, we note that on each shell $D_n$, $\frac{1}{\|x\|} \leq \frac{1}{n}$, so that:
\begin{align}
\int_{\mathbb{R}^d}|f(x)|dx
&=\sum_{n\geq0}\int_{D_n}|f(x)|dx\\
&= \int_{D_0}|f(x)|dx +
\sum_{n\geq 1}\int_{D_n}|f(x)|dx \\
&\leq\frac{C}{2^{d+\delta}}+\sum_{n\geq1}\int_{D_n}\frac{C}{(1+n)^{d+\delta}}dx \\
&=\frac{C}{2^{d+\delta}}+\sum_{n\geq1} \frac{C}{(1+n)^{d+\delta}} \int_{D_n} dx \\
&=\frac{C}{2^{d+\delta}}+\sum_{n\geq1}\frac{C}{(1+n)^{d+\delta}}\left((2n+2)^{d}-(2n)^{d}\right)\\
&=\frac{C}{2^{d+\delta}}+2^dC\sum_{n\geq1}\frac{1}{(1+n)^{d+\delta}}\left((n+1)^{d}-n^{d}\right)\\ \label{Big-O}
&=\frac{C}{2^{d+\delta}}+\sum_{n\geq1}\frac{O(n^{d-1})}{(1+n)^{d+\delta}}\\
&=\frac{C}{2^{d+\delta}}+\sum_{n\geq1}O\left(\frac{1}{n^{1+\delta}}\right)\\
&=\frac{C}{2^{d+\delta}}+O\left(\sum_{n\geq1}\frac{1}{n^{1+\delta}}\right)<\infty,
\end{align}
where we've used the fact that the constant in the Big-O of equation \eqref{Big-O} is independent of $n$, so that we can move the series inside.
\end{proof}
For the absolute summability of functions satisfying the same growth condition of the previous lemma, we have the following.
\begin{lem}
\label{Poisson bound implies absolutely summable}
Let $f:\mathbb R^d \rightarrow \mathbb{C}$ be a function that enjoys the bound
\[
|f(x)|\leq\frac{C}{(1+||x||)^{d+\delta}},
\]
for all $x\in \mathbb R^d$, and for constants $C, \delta >0$ that are independent of $x$.
Then the series
\[
\sum_{k\in \mathbb{Z}^d} f(x+k)
\]
converges uniformly and absolutely, for $x\in\mathbb{R}^d$.
\end{lem}
\begin{proof}
We will restrict attention to $x \in [0, 1)^d$, because the function $F(x):= \sum_{k\in \mathbb{Z}^d} f(k+x)$, if convergent, forms
a periodic function of $x \in \mathbb{R}^d$, with the unit cube $[0, 1)^d$ being a period. We also note for all $x \in [0, 1)^d$,
we have the bound $\|x\| \leq \sqrt d$.
We consider the tail of the series, for any given $N>0$:
\begin{align}
| \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} f(k+x) |
& \leq \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \left | f(k+x) \right |
\leq C \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \frac{1}{(1+\|k+x\|)^{d+\delta}} \\
& \leq \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \frac{1}{ \left(1+\frac{\|k+x\|}{1+\sqrt d} \right)^{d+\delta}}
\\ \label{second equality here}
&=\sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \frac{\left(1+\sqrt{d}\right)^{d+\delta}}{\left(1+\sqrt{d}+\|k+x\| \right)^{d+\delta}}
\\ \label{third line here}
&\leq C_{d, \delta} \sum_{k \in \mathbb{Z}^{d} \atop \|k\| > N} \frac{1}{(1+\|k\|)^{d+\delta}}
\\ \label{surface area of sphere approximation}
&= C_{d, \delta} \sum_{n \geq N}\frac{1}{(1+n)^{d+\delta}} O \left( n^{d-1} \right) \\
&=\sum_{n \geq N}O\left(\frac{1}{n^{1+\delta}} \right)\\
&=O\left(\sum_{n \geq N}\frac{1}{n^{1+\delta}} \right) \rightarrow 0, \text{ as } N\rightarrow \infty,
\end{align}
and the last bound is independent of $x$. In passing from \eqref{second equality here} to \eqref{third line here},
we used the estimate
$\|k + x \| \geq \|k \| - \| x\| \geq \|k\|-\sqrt{d}$, and $C_{d, \delta}:= \left(1+\sqrt{d}\right)^{d+\delta}$. The equality in \eqref{surface area of sphere approximation} is due to the fact that the number of integer points $k \in \mathbb{Z}^d$ that lie on a sphere of radius $n$ is $O\left(\text{surface area of } nS^{d-1} \right)
= O \left( n^{d-1} \right) $. We've shown that the series $\sum_{k\in \mathbb{Z}^d} f(x+k)$ converges uniformly on $\mathbb{R}^d$.
\end{proof}
We note that the only reason for having $(1+ \|x\|)^{d+ \delta}$ in the denominators of the bounds, instead of simply $\|x\|^{d+ \delta}$,
is to give simultaneously a bound at the origin, as well as any nonzero $x$.
\bigskip
\section{Poisson summation II, \'a la Poisson}
There are various different families of functions for which the adjective `nice' applies, in \eqref{nice functions},
and one of the simplest to understand is the Schwartz class of functions.
But there is a more general family of nice functions that is extremely useful, given by Poisson himself, as follows.
\begin{thm}[Poisson summation formula, II] \index{Poisson summation formula}
\label{nice2}
Suppose that for some positive constants $\delta$, $C$, and for all $x \in \mathbb{R}^d$, we have the bounds:
\begin{align} \label{growth conditions for Poisson}
|f(x)| < \frac{C}{ (1+\|x\|)^{d+\delta} }
\text{ \, and \, } |\hat{f}(x)| < \frac{C}{ ( 1+\|x\|)^{d+\delta} }.
\end{align}
Then we have the pointwise equality:
\begin{equation} \label{Poisson summation, take 2}
\sum_{n \in \mathbb{Z}^d} f(n+ x) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle},
\end{equation}
for each $x\in \mathbb{R}^d$. In addition, both sides of \eqref{Poisson summation, take 2}
converge absolutely, and are continuous functions on $\mathbb{R}^d$.
\end{thm}
\begin{proof}
Step $1$. \ The growth conditions \eqref{growth conditions for Poisson} allow us to conclude that both $f, \hat f \in L^1(\mathbb{R}^d)$, by Lemma \ref{Poisson bound implies L^1}.
This implies that both $f, \hat f \in L^2(\mathbb{R}^d)$, by the elementary Lemma \ref{both f and its FT in L^1 implies L^2}.
We also know that the Fourier transform of an $L^1$ function must be uniformly continuous on $\mathbb{R}^d$, and so both $f$ and $\hat f$ are uniformly continuous (Lemma \ref{uniform continuity}).
Step $2$. \ The hypothesis regarding the growth conditions \eqref{growth conditions for Poisson} implies that the series defined by
$F(x):= \sum_{n \in \mathbb{Z}^d} f(n+ x)$ converges uniformly on $[0, 1]^d$, as we showed in Lemma \ref{Poisson bound implies absolutely summable}.
It follows that this series must also converge in the $L^2$-norm on $[0, 1]^d$. So $F \in L^2({\mathbb T^d})$, and it must therefore
possess a Fourier series, which converges to it in the $L^2$-norm:
\begin{equation}
F(x) = \sum_{n \in \mathbb{Z}^d} a_n \, e^{2\pi i \langle n, x \rangle}.
\end{equation}
Step $3$. \ Next, we compute the Fourier coefficients $a_k$.
This is almost the same step that already appeared in the proof of Theorem \ref{Poisson.Summation}, but we repeat it for completeness, and also because the interchange of sum and integral below is justified in a different way.
\begin{align}
a_k &:= \int_{[0,1)^d} F(u) e^{-2\pi i \langle k, u \rangle} du \\
&=\int_{[0,1)^d} \sum_{n \in \mathbb{Z}^d} f(n+ u)
e^{-2\pi i \langle k, u \rangle} du \\
&=\sum_{n \in \mathbb{Z}^d} \int_{[0,1)^d} f(n+ u) \label{outersum}
e^{-2\pi i \langle k, u \rangle} du.
\end{align}
The interchange of summation and integral in the latter step is allowed by the uniform convergence of the series
$ \sum_{n \in \mathbb{Z}^d} f(n+ x)$.
We fix an $n\in\mathbb{Z}^d$ in the outer sum of \eqref{outersum}, and make the change of variable in the integral:
$n+u := w$. As $u$ varies over the cube ${[0,1)^d}$, $w:= n+u$ varies
over all of $\mathbb{R}^d$ because the unit cube tiles the whole space:
\[
{[0,1)^d} + \mathbb{Z}^d = \mathbb{R}^d.
\]
We also have
$e^{-2\pi i \langle k, u \rangle} = e^{-2\pi i \langle k, w-n \rangle} =
e^{-2\pi i \langle k, w \rangle}$, because $k, n \in \mathbb{Z}^d$ and hence
$e^{2\pi i \langle k, n \rangle} =1$.
Finally:
$a_k = \sum_{n \in \mathbb{Z}^d} \int_{n+ [0,1)^d} f(w)
e^{-2\pi i \langle k, w \rangle} dw
= \int_{\mathbb{R}^d} f(w)
e^{-2\pi i \langle k, w \rangle} dw := \hat f(k)$.
Step $4$. \ Since each summand $f(n+x)$ is a continuous function of $x$, and since the convergence is uniform, the function $F(x)$ must also be continuous.
Finally, we'd like to pass from the convergence of the Fourier series in the $L^2$-norm, to pointwise and uniform convergence. For this task we can use
Lemma \ref{norm convergence plus absolute convergence implies equality},
assuming that we can show the absolute convergence of the Fourier series
$\sum_{n \in \mathbb{Z}^d} a_n \, e^{2\pi i \langle n, x \rangle} =
\sum_{n \in \mathbb{Z}^d} \hat f(n) e^{2\pi i \langle n, x \rangle}$.
But this absolute convergence follows from the same Lemma \ref{Poisson bound implies absolutely summable}, with $f$ replaced by $\hat f$, because the same growth bounds \eqref{growth conditions for Poisson}
are also assumed for $\hat f$.
To summarize this last step, we know that $F$ is continuous, and the previous remarks allow us to use Lemma \ref{norm convergence plus absolute convergence implies equality} to conclude that the Fourier series converges pointwise and uniformly to $F(x)$.
\end{proof}
We call a function that enjoys property \eqref{growth conditions for Poisson} a {\bf Poisson function},
because Sim\'eon Denis Poisson proved Theorem \ref{nice2}
between $1823$ and $1827$ \cite{Travaglini}.
Poisson's Theorem \ref{nice2} is a {\bf stronger} version of Poisson summation
than Theorem \ref{Poisson.Summation} above.
To justify this latter claim, we need to show that any Schwartz function also satisfies the growth conditions \eqref{growth conditions for Poisson}, but this is clear because Schwartz functions (and their transforms) decay faster than any polynomial, hence faster than the bounds given by \eqref{growth conditions for Poisson}.
We call the space of functions that satisfy the hypotheses of Theorem \ref{nice2}, the {\bf Poisson space} of functions, in honor of the mathematician that discovered this class. As we've just seen, the suggestion of Figure \ref{nice functions, containment} is correct, showing that the Schwartz space is contained in the Poisson space.
\begin{question}
Are there some natural necessary and sufficient conditions for Poisson summation?
\end{question}
This is an important open question.
In other words, we may ask what are the inherent limitations of functions that satisfy Poisson summation? Although there are well
over $20$ different versions of sufficient conditions in the literature on Poisson summation, there are currently no known necessary and sufficient conditions for Poisson summation to hold.
It is natural to wonder if `nice' functions might include all functions $f:\mathbb{R}^d\rightarrow \mathbb{C}$ such that
\[
f\in L^1(\mathbb{R}^d) \text{ and } \hat f\in L^1(\mathbb{R}^d)?
\]
Sadly, the answer is ``no'' in general, and there is an important counterexample,
by Yitzhak Katznelson (\cite{Katznelson}, Ch. VI, p. 143, Exercise 15).
There are many other families of nice functions in the literature, which include hypotheses such as `functions of bounded variation', and `absolutely continuous' functions. We'll not delve into these other families here, but the reader may glance at
Figure \ref{Refined nice functions} for a slightly more refined relationship between nice functions and the $L^1$ and $L^2$ spaces.
To justify the new containments that is suggested by Figure \ref{Refined nice functions}, we recall that a nice function $f$ was defined in \eqref{nice functions} to include the property that both $f, \hat f \in L^1(\mathbb{R}^d)$.
By Lemma \ref{both f and its FT in L^1 implies L^2}, we know that therefore both
$f, \hat f \in L^2(\mathbb{R}^d)$ as well, so the Figure is correct.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.8in]{RefinedPoisson}
\end{center}
\caption{A slightly more detailed Venn diagram for function spaces related to nice functions}
\label{Refined nice functions}
\end{figure}
We will use a slightly more general version of the Poisson summation formula, which holds for any lattice, and which follows rather quickly from the Poisson summation formula above.
We define a (full-rank) lattice ${\mathcal L}:= M(\mathbb{Z}^d) \subset \mathbb{R}^d$, the image of the integer lattice under an invertible linear transformation $M$.
The {\bf dual lattice} \index{dual lattice} of ${\mathcal L}$ is defined by
${\mathcal L}^* := M^{-T}(\mathbb{Z}^d)$, where $M^{-T}$ is the inverse transpose matrix of the real matrix $M$ (see Section \ref{dual lattice} for more on dual lattices).
As we've seen in Lemma \ref{FT under linear maps}, Fourier Transforms behave beautifully under compositions with any linear transformation.
We will use this fact again in the proof of the following extension of Poisson summation, which holds for
all lattices ${\mathcal L}$ and is quite standard. We recall that a Poisson function $f$ by definition
satisfies the growth conditions \eqref{growth conditions for Poisson}.
\begin{thm} [Poisson summation formula, III]
\label{The Poisson Summation Formula, for lattices}
\index{Poisson summation formula for lattices}
Given a full-rank lattice ${\mathcal L}\subset \mathbb{R}^d$, and a Poisson function $f: \mathbb{R}^d \rightarrow \mathbb{C}$, we have
\begin{equation} \label{PoissonSummationForLattices}
\sum_{n \in {\mathcal L}} f(n+x) = \frac{1}{\det {\mathcal L}} \sum_{m \in {\mathcal L}^*}
\hat f(m) e^{2\pi i \langle x, m \rangle},
\end{equation}
valid for all $x \in \mathbb{R}^d$. In particular, we have
\begin{equation} \label{Poisson.summation3} \index{Poisson summation formula}
\sum_{n \in {\mathcal L}} f(n) = \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi).
\end{equation}
Both sides of \eqref{PoissonSummationForLattices} converge absolutely and are continuous functions on $\mathbb{R}^d$.
\end{thm}
\begin{proof}
Any lattice (full-rank) may be written as ${\mathcal L} := M(\mathbb{Z}^d)$, so that $\det {\mathcal L} := |\det M|$.
Using the Poisson summation formula \eqref{Poisson.summation1}, with the change of variable $n = Mk$,
with $k \in \mathbb{Z}^d$, we have:
\begin{align*}
\sum_{n \in {\mathcal L}} f(n) &= \sum_{k \in \mathbb{Z}^d} (f\circ M)(k) \\
&= \sum_{\xi \in \mathbb{Z}^d} \widehat{(f\circ M)}(\xi) \\
&= \frac{1}{|\det M|} \sum_{\xi \in \mathbb{Z}^d} \hat f\left(M^{-T} \xi \right) \\
&=\frac{1}{\det {\mathcal L}} \sum_{m \in {\mathcal L}^*} \hat f(m).
\end{align*}
where in the third equality we used the elementary `Stretch' Lemma \ref{FT under linear maps}, and in the fourth equality we used the definition of the dual lattice ${\mathcal L}^*:= M^{-T} \mathbb{Z}^d$.
\end{proof}
As an afterthought, it turns out that the special case \eqref{Poisson.summation3}
also easily implies the general case, namely \eqref{PoissonSummationForLattices}
(Exercise \ref{going backwards in Poisson summation}).
A traditional application of the Poisson summation formula is the quick derivation of
the functional equation of the theta function.
We first define the Gaussian function by:
\begin{equation} \index{Gaussian}
G_t (x) := t^{-\frac{d}{2}} e^{ -\frac{\pi}{t} || x ||^2 },
\end{equation}
for each fixed $t >0$, and for all $x \in \mathbb{R}^d$, as depicted in Figure \ref{pic of Gaussians}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.4in]{Gaussians}
\end{center}
\caption{The Gaussian family of functions $G_t(x)$ with $t = 1, t=.5, t= .3, $ and $t=.1$ respectively. }
\label{pic of Gaussians}
\end{figure}
Two immediately interesting properties of the Gaussian are:
\begin{equation}
\int_{\mathbb{R}^d} G_{t} (x) dx = 1,
\end{equation}
for each $t>0$, and
\begin{equation}\label{transform of the Gaussian}
\hat G_t( m ) = e^{ -\pi t || m ||^2 },
\end{equation}
properties which are important in Statistics as well
(Exercises \ref{Gaussian1} and \ref{Gaussian2}). Each fixed $\varepsilon$ gives us one Gaussian function and intuitively, as $\varepsilon \rightarrow 0$, this sequence of Gaussians approaches the ``Dirac delta function'' at the origin, which is really known as a ``generalized function'', or ``distribution'' (Note \ref{Dirac delta}).
\begin{example}
\rm{
The classical theta function \index{theta function} (for the integer lattice) is defined by:
\begin{equation} \label{theta function for the integer lattice}
\theta(t) = \sum_{n\in \mathbb{Z}^d} e^{ -\pi t || n ||^2 }.
\end{equation}
This function plays a major role in analytic number theory. One of its first historical applications was carried out
by Riemann himself, who proved its functional equation (eq. \eqref{theta functional equation} below) and then applied a ``Mellin transform'' to it, to prove the functional equation of the Riemann zeta function $\zeta(s):= \sum_{n=1}^\infty \frac{1}{n^s}$.
We claim that the theta function has the functional equation
\begin{equation} \label{theta functional equation}
\theta\left( \frac{1}{t} \right) = t^{\frac{d}{2}} \theta(t),
\end{equation}
for all $t>0$. This will follow immediately from the Poisson summation formula \ref{Poisson.summation2} by using \index{Poisson summation formula}
$f(x):= G_t(x)$. Using our knowledge of its FT, from
\ref{transform of the Gaussian}, we have:
\begin{align*}
\sum_{n\in \mathbb{Z}^d} G_t(n) &= \sum_{\xi \in \mathbb{Z}^d} \hat G_t(\xi) \\
&= \sum_{\xi \in \mathbb{Z}^d} e^{ -\pi t || \xi ||^2 } := \theta(t).
\end{align*}
Since by definition $\sum_{n\in \mathbb{Z}^d} G_t(n) :=
t^{-\frac{d}{2}} \sum_{n\in \mathbb{Z}^d} e^{ -\frac{\pi}{t} ||n||^2 }
:= t^{-\frac{d}{2}} \theta\left(\frac{1}{t}\right)$, \eqref{theta functional equation} is proved.
}
\hfill $\square$
\end{example}
\bigskip
\section{The convolution operation} \label{* is born}
\index{convolution}
For $f,g \in L^1(\mathbb{R}^d)$, their {\bf convolution} is defined by
\begin{equation} \label{def of convolution}
(f * g)(x) = \int_{\mathbb{R}^d} f(x-y) g(y) dy,
\end{equation}
We will also use definition \eqref{def of convolution} to include more general functions $f, g$, for which the latter
integral still converges (see Examples \ref{ex heaviside}, \ref{ex ramp} below).
It is possible to think intuitively of this analogue of multiplication as: ``this is how waves like to multiply", via Lemma \ref{convolution theorem} \ref{convolution under FT}. We have the following basic relations for the convolution operation.
\begin{lem} \label{convolution theorem}
For all $f, g, h \in L^1(\mathbb{R}^d)$, we have:
\begin{enumerate}[(a)]
\item $f*g \in L^1(\mathbb{R}^d)$.
\label{part 1:convolution theorem}
\item $ \widehat{(f * g)}(\xi) = {\hat f}(\xi) {\hat g}(\xi)$.
\label{convolution under FT}
\item $ f*g = g*f, \ \ f*(g*h)= (f*g)*h $, and $ \ f*(g+h) = f*g + f*h$.
\label{part 3:convolution theorem}
\item $\|f*g\|_1 \leq \|f\|_1 \|g\|_1$.
\label{part 4:convolution theorem}
\item More generally, when $f\in L^p(\mathbb{R}^d), \ \ g \in L^1(\mathbb{R}^d)$, with $1\leq p <\infty$, then we have
$f*g \in L^p(\mathbb{R}^d)$ and
\[
\|f*g\|_p \leq \|f\|_p \|g\|_1.
\]
\end{enumerate}
\end{lem}
\begin{proof}
To prove part \ref{convolution under FT}, we use Fubini's Theorem
(Theorem \ref{Fubini} in the Appendix):
\begin{align*}
\widehat{(f * g)}(\xi) &:= \int_{\mathbb{R}^d} e^{-2\pi i \langle x, \xi \rangle} \left(\int_{\mathbb{R}^d}f(x-y) g(y) dy \right) dx \\
&= \int_{\mathbb{R}^d} g(y) e^{-2\pi i \langle y, \xi \rangle} dy \int_{\mathbb{R}^d}f(x-y) e^{-2\pi i \langle x-y, \xi \rangle} dx \\
&= \int_{\mathbb{R}^d} g(y) e^{-2\pi i \langle y, \xi \rangle} dy \int_{\mathbb{R}^d}f(x) e^{-2\pi i \langle x, \xi \rangle} dx \\
&:= {\hat f}(\xi) {\hat g}(\xi),
\end{align*}
where we've used the translation invariance of the measure, in the penultimate equality.
To prove part \ref{part 4:convolution theorem}, we use Fubini's theorem again, and the triangle inequality for integrals:
\begin{align*}
\| f * g\|_1&:= \int_{\mathbb{R}^d} \left | \int_{\mathbb{R}^d}f(x-y) g(y) dy \right | dx \\
&\leq \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \left | f(x-y) g(y) \right | dy dx \\
&=\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \left | f(y) g(y) \right | dy dx \\
&= \int_{\mathbb{R}^d} \left | f(y) \right | dy \int_{\mathbb{R}^d} \left | g(y) \right | dx \\
&:= \|f\|_1 \|g\|_1.
\end{align*}
For the proofs of the remaining parts, we recommend Rudin's book \cite{RudinGreenBook}.
\end{proof}
Lemma \ref{convolution theorem} \ref{convolution under FT}
means that convolution of functions in the space domain corresponds to
the usual multiplication of functions in the frequency domain (and vice-versa).
\begin{example}
\rm{
When ${\mathcal P} := [-\frac{1}{2}, \frac{1}{2}]$, the convolution of $1_{\mathcal P}$ with itself is drawn in Figure \ref{pic of convolution of indicator}. We can already see that this convolution is a continuous function,
hence a little smoother than the discontinuous function $1_{\mathcal P}$.
Using Lemma \ref{convolution theorem} we have
\[
\widehat{ (1_{\mathcal P} * 1_{\mathcal P}) }(\xi) = \hat{1}_{\mathcal P}(\xi) \hat{1}_{\mathcal P}(\xi) =
\left( \frac{\sin(\pi \xi)}{\pi \xi} \right)^2.
\]
We've used equation \ref{ClassicalExample} in the last equality, for the Fourier transform of our interval ${\mathcal P}$ here. Considering the graph in Figure \ref{pic of sinc2}, for the Fourier transform of the convolution $(1_{\mathcal P} * 1_{\mathcal P})$, we see that this positive function is already much more tightly concentrated near the origin, as compared with
$\rm{sinc}(x):= \hat 1_{\mathcal P}(\xi)$. We work out all of the details for this $1$-dimensional function, and generalize it,
in Example \ref{hat function} below.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.4in]{ConvolutionOfIndicator}
\end{center}
\caption{The function $\left( 1_{\mathcal P} * 1_{\mathcal P} \right) (x)$, with ${\mathcal P}:= \left[ -\frac{1}{2}, \frac{1}{2} \right]$ }
\label{pic of convolution of indicator}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.5in]{sinc2}
\end{center}
\caption{The Fourier transform $ \widehat{ \left( 1_{\mathcal P} * 1_{\mathcal P} \right) } (\xi)$,
which is equal to the infinitely smooth, nonnegative function
$\left( \frac{\sin(\pi \xi)}{\pi \xi} \right)^2 := \rm{sinc}^2(\xi)$. }
\label{pic of sinc2}
\end{figure}
}
\hfill $\square$
\end{example}
Another useful bit of intuition about convolutions is that they are a kind of averaging process, and that
the convolution of two functions becomes smoother than either one of them.
For our applications, when we consider the indicator function $1_{\mathcal P}(x)$ for a polytope ${\mathcal P}$,
then this function is not continuous on $\mathbb{R}^d$, so that the Poisson summation formula does not
necessarily hold for it. But if we consider the convolution of $1_{\mathcal P}(x)$ with a Gaussian, for example,
then we arrive at the $C^\infty$ function
\[
(1_{\mathcal P} * G_t)(x),
\]
for which the Poisson summation
does hold. In the sequel, we will use the latter convolved function in tandem with Poisson summation to study ``solid angles".
\medskip
\begin{example} \label{convolution of general bodies}
\rm{
For any bounded measurable sets $K, L \subset \mathbb{R}^d$, we have
\begin{align}
(1_K * 1_L)(y) &:= \int_{\mathbb{R}^d} 1_{K}(x) 1_{L}(y-x) dx \\
&= \int_{\mathbb{R}^d} 1_{K \cap (-L + y)}(x) dx \\
&= \int_{K \cap (-L + y)} dx \\ \label{volume formula for convolution}
& = \vol\left( K \cap (-L + y) \right),
\end{align}
so that the convolution of indicator functions gives volumes, and this simple connection is one of the entry points into convex geometry.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.5in]{HatFunction}
\end{center}
\caption{The hat function $ 1_{\left[ -r, r \right]} * 1_{\left[ -r, r \right]}$ of Example \ref{hat function}, with $r = 3.5$.}
\label{pic of hat function, take 2}
\end{figure}
\medskip
\begin{example} \label{hat function}
\rm{
As a special case of Example \ref{convolution of general bodies}, consider the case
$K= L := \left[ -r, r \right] \subset \mathbb{R}$. So we now know,
by \eqref{volume formula for convolution}, that
\begin{align}
g(x):=\left(1_{\left[ -r, r \right]} * 1_{\left[ -r, r \right]} \right)(x) =
\vol\Big( \left[ -r, r \right] \cap (\left[ -r, r \right] + x) \Big),
\end{align}
making it clear that for $x \leq -2r$ and $x \geq 2r$, we have
$\vol\Big( \left[ -r, r \right] \cap (\left[ -r, r \right] + x) \Big)=0$.
Precisely, when $x \in [-2r, 0]$, we have the function
\[
g(x):= \vol\Big( \left[ -r, r \right] \cap (\left[ -r, r \right] + x) \Big) = | x-2r | = x+ 2r,
\]
Finally, when $x \in [0, 2r]$, we have the function
$g(x):= \vol\Big( \left[ -r, r \right] \cap (\left[ -r, r \right] + x) \Big) = | x-2r | = 2r - x $. To summarize, we have
\[
g(x) =
\begin{cases}
2r-|x| & \text{ if } x \in [-2r, 2r] \\
0 & \text{ if not.}
\end{cases}
\]
Due to its shape, $g$ is sometimes called the {\bf hat function}.
The hat function is extremely useful in many applications. For example, we can use it to build up functions that are compactly supported on $\mathbb{R}$, and yet whose Fourier transform is
\emph{strictly positive} on $\mathbb{R}$ - see Exercise \ref{positive FT over R}.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.4in]{HeavisideFunction}
\end{center}
\caption{The heaviside function $H_0(x)$ }
\label{Heaviside function}
\end{figure}
\bigskip
\begin{example} \label{ex heaviside}
\rm{
The {\bf Heaviside function} is defined by
\begin{equation} \label{def of heaviside}
H_a(x):= \begin{cases}
1 &\mbox{if } x \geq a \\
0 & \mbox{if } x < a,
\end{cases}
\end{equation}
where $a$ is any fixed real number. Although the Heaviside function is clearly not absolutely integrable over $\mathbb{R}$,
we may still use the same definition \eqref{def of convolution} for its convolution with a function $f\in L^1(\mathbb{R})$:
\begin{equation}\label{Heaviside convolution}
(f*H_0)(x):= \int_{\mathbb{R}} f(x-y) H_0(y) dy=\int_{0}^\infty f(x-y) dy=\int_{-\infty}^x f(t) dt,
\end{equation}
a convergent integral.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.4in]{RampFunction}
\end{center}
\caption{The ramp function $r_5(x)$ }
\label{Ramp function}
\end{figure}
\bigskip
\begin{example} \label{ex ramp}
\rm{
The {\bf ramp function} is defined by
\begin{equation} \label{def of ramp}
r_a(x):= \begin{cases}
x &\mbox{if } x \geq a \\
0 & \mbox{if } x < a,
\end{cases}
\end{equation}
where $a$ is any fixed real number. It is evident that we also have
$r_0(x) = \max\{ x, 0\}$. It is also clear that $r'_a(x) = H_a(x)$. The ramp function is ubiquitous in the analysis of machine learning algorithms, where it is called
the ReLu (Rectified Linear Unit) function.
There is an elegant relationship between the ramp function and the Heaviside function:
\begin{equation} \label{claim for two heavisides}
H_0*H_0 = r_0,
\end{equation}
so we see that convolution makes sense here despite the fact that none of these functions are in $L^1(\mathbb{R})$!
To check the latter claim \eqref{claim for two heavisides}, we use \eqref{Heaviside convolution} above:
\begin{align*}
H_0*H_0(x)&:= \int_{-\infty}^x H_0(t) dt =
\begin{cases}
\int_0^x dx &\mbox{if } x \geq 0 \\
0 & \mbox{if } x<0
\end{cases} \\
&=
\begin{cases}
x &\mbox{if } x \geq 0 \\
0 & \mbox{if } x<0
\end{cases}
:= r_0(x).
\end{align*}
There is also a straightforward extension: $H_a*H_b = r_{a+b}$ (Exercise \ref{Heaviside and ramp}).
}
\hfill $\square$
\end{example}
Having seen convolutions, with various examples, we can now return to the question:
\begin{question}
What is the image of the space $L^1(\mathbb{R}^d)$ under the Fourier transform?
\end{question}
It seems that there is no known `complete' answer to this question yet; however, an apparently lesser-known
but elegant result, due to W. Rudin, is the following correspondence.
\begin{thm}[Rudin] \label{RudinAmazingConvolutions}
\begin{equation}
f \in L^1(\mathbb{R}^d) \iff \hat f = g*h, \text{ with } g, h \in L^2(\mathbb{R}^d).
\end{equation}
\hfill $\square$
\end{thm}
In words, Theorem \ref{RudinAmazingConvolutions} tells us that
the image of $L^1(\mathbb{R}^d)$ under the Fourier transform consists precisely of the set of convolutions
$g*h$, where $g, h \in L^2(\mathbb{R}^d)$
(See \cite{RudinGroups}, Theorem 1.6.3, p.~27).
Here is an outline of a proof for the easy direction: suppose that $g, h \in L^2(\mathbb{R}^d)$. Because we want to find
a solution in $f$, to the equation $\hat f = g*h$, it's natural to try $f := \widehat{g*h} = \hat g \cdot \hat h$. Let's try it, by defining
\[
f:= \hat g \cdot \hat h.
\]
Because the Fourier transform acting on $L^2(\mathbb{R}^d)$ is an isometry,
we have $\hat g, \hat h \in L^2(\mathbb{R}^d)$. Also, the product of two $L^2$ functions in an $L^1$ function
(eq. \eqref{product of two L^2 functions is L^1}), so we conclude that
$f:= \hat g \cdot \hat h \in L^1(\mathbb{R}^d)$, as required.
\bigskip
\subsection{How natural is the Fourier transform?}
We close by thinking a bit about another natural question. We've seen that if $f, \hat f \in L^1(\mathbb{R}^d)$, then the map
\[
\Phi_\xi: f \rightarrow \hat f(\xi),
\]
for each fixed $\xi \in \mathbb{R}^d$,
is a homomorphism from $L^1(\mathbb{R}^d)$ to $\mathbb{C}$, due to
Lemma \ref{convolution theorem} \ref{convolution under FT}.
Are there other transforms that act on $L^1(\mathbb{R}^d)$ as a homomorphism? It turns out there are not!
The Fourier transform is the unique homomorphism here, and we record this fact as a lemma, whose proof appears in
\cite{RudinGroups}, Theorem 1.2.2, p.~7).
\begin{lem}
Suppose $\phi: L^1(\mathbb{R}^d) \rightarrow \mathbb{C}$ is a nonzero complex homomorphism.
Then $\phi = \Phi_\xi$, for some $\xi \in \mathbb{R}^d$.
\hfill $\square$
\end{lem}
\begin{comment}
Sometimes it is not necessary for $f$ or $g$ to be in $L^1(\mathbb{R}^d)$ in order for their convolution to make sense, as we've already seen in Examples \ref{ex heaviside} and \ref{ex ramp}. Here is another case where such a construction is sometimes useful.
\begin{example}
\rm{
Suppose we have two functions $\psi, f:\mathbb{R}^d\rightarrow \mathbb{C}^d$ that are both compactly supported,
and $\psi(x) = 1$ for all $x \in \supp(f)$. Here we notice that by construction we have $f(x) = \psi(x) f(x), \forall x \in \mathbb{R}^d$.
Even in the case that we're out of luck, and
$\hat f \notin L^1(\mathbb{R}^d)$, let's ask if we might still have the identity
\begin{equation}
\widehat{(\psi \cdot f)} = (\hat \psi * \hat f) \,?
\end{equation}
We can compute:
\begin{align*}
(\hat \psi * \hat f)(t) &:= \int_{\mathbb{R}^d} \hat \psi(t-x) \hat f(x) dx \\
&= \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \psi(u) e^{-2\pi i \langle u, t-x \rangle} du
\int_{\mathbb{R}^d} f(v) e^{-2\pi i \langle v, x \rangle} dv dx \\
\end{align*}
}
\end{example}
\end{comment}
\bigskip
\subsection{The Dirichlet Kernel}
Using convolutions, we may now also go back to the partial sums of a Fourier series, which we have defined in \eqref{partial sums}
by
\begin{equation}
S_N f(t):= \sum_{n= -N}^N \hat f(n) e^{2\pi i n t}.
\end{equation}
We compute:
\begin{align*}
S_N f(t)
&:=
\sum_{n= -N}^N \hat f(n) e^{2\pi i n t} =
\sum_{n= -N}^N \int_0^1 f(x) e^{-2\pi i x n} dx \, e^{2\pi i n t} \\
&=
\int_0^1 f(x) \sum_{n= -N}^N e^{2\pi i (t-x) n} dx\\
&:= (f*D_N)(t),
\end{align*}
where this convolution is defined on the $1$-Torus (the circle), and where we introduced the important definition
\begin{equation}
D_N(x):= \sum_{n= -N}^N e^{2\pi i x n},
\end{equation}
known as the {\bf Dirichlet kernel}.
But look how naturally another convolution came up! We've just proved the following elementary Lemma.
\begin{lem}
If $f \in L^2(\mathbb T)$, then
\[
S_N f(t) = (f*D_N)(t),
\]
where this convolution is taken over $[0, 1]$.
\hfill $\square$
\end{lem}
It's therefore very natural to study the behavior of the Dirichlet kernel on its own. In Exercise \ref{first Dirichlet kernel}, we showed that
the Dirichlet kernel has the closed form
\[
D_N(x) = \frac{\sin \left( \pi x(2N + 1) \right) }{\sin(\pi x)}.
\]
\index{Dirichlet kernel}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.4in]{DirichletKernel}
\end{center}
\caption{The Dirichlet Kernel $D_{20}(x)$, restricted to the interval $[-1, 1]$ }
\label{pic of Dirichlet Kernel}
\end{figure}
It's clear from the definition of $D_N(x)$ that it is a periodic function of $x$, with period $1$, and if we restrict our attention to the interval $[-1, 1]$, then its graph appears in Figure \ref{pic of Dirichlet Kernel}. It turns out the the $L^1$ norm of the Dirichlet kernel becomes unbounded as $N\rightarrow \infty$, and this phenomenon is responsible for a lot of results about pointwise divergence of Fourier series, a very delicate subject that is replete with technical subtleties.
There are even examples of continuous functions $f$ whose partial Fourier sums
\index{partial Fourier sums}
$S_N f ( x )$ do not converge anywhere (\cite{Travaglini}, Theorem 4.19).
However, the Dirichlet kernel is also useful for proving pointwise
convergence theorems, such as the important
Theorem \ref{theorem:Fourier series convergence to the mean}.
\bigskip
\section{Plancherel} \index{Plancherel Theorem}
One of the main results in Fourier analysis is the {\bf Plancherel Theorem},
which tells us that the Fourier transform, acting on $L^2(\mathbb{R}^d)$, is an isometry.
In other words, we'll show that the Fourier transform preserves norms of functions: $\| \hat f \|_2 = \| f \|_2$.
\begin{thm}[Plancherel]
\label{thm:Plancherel}
Let $f, g \in L^2(\mathbb{R}^d)$. Then we have:
\begin{enumerate}[(a)]
\item
\begin{equation}
\int_{\mathbb{R}^d} |\hat f(\xi)|^2 d\xi = \int_{\mathbb{R}^d} |f(x)|^2 dx.
\end{equation}
\item More generally, we have:
\begin{equation} \label{Plancherel identity}
\int_{\mathbb{R}^d} f(x) \overline{g(x)} dx = \int_{\mathbb{R}^d} \hat f(x) \overline{\hat g(x)} dx.
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}
We prove a slightly weaker claim, assuming that we have the additional hypothesis
$f, g \in L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$ as well, so that we may use Lemma \ref{convolution theorem}.
In this way we may get to used to some of the ideas involved without all of the machinery that is required in the case of the
strict $L^2(\mathbb{R}^d)$ assumption
(for a proof under the more general hypothesis of the functions belonging to $L^2(\mathbb{R}^d)$, see \cite{Folland}, for example).
We let $g(x) := \overline{f(-x)}$, so that
\begin{align*}
\hat g(\xi) &= \int_{\mathbb{R}^d} \overline{f(-x)} e^{-2\pi i \langle x, \xi \rangle} dx \\
&= \overline{ \int_{\mathbb{R}^d} f(-x) e^{2\pi i \langle x, \xi \rangle} dx } \\
&= \overline{ \hat f(\xi)}.
\end{align*}
We define $h := f*g$, and by Lemma \ref{convolution theorem} we have $\hat h(\xi) = \hat f(\xi)\hat g(\xi)$,
so that $\hat h(\xi) = \|\hat f(\xi)\|^2$.
Now, $h(0) := \int_{\mathbb{R}^d} f(0-x) g(x)dx = \int_{\mathbb{R}^d} f(-x) \overline{f(-x)} dx =
\int_{\mathbb{R}^d} |f(x)|^2$.
On the other hand, $h(0) = \int_{\mathbb{R}^d} \hat h(\xi) d\xi = \int_{\mathbb{R}^d} |\hat f(\xi)|^2 d\xi $. We therefore have
\[
\int_{\mathbb{R}^d} |\hat f(\xi)|^2 d\xi = \int_{\mathbb{R}^d} |f(x)|^2 dx.
\]
The proof of part (b) is quite similar, and we do not want to deprive the reader of the pleasure (Exercise \ref{Plancherel extended}).
\end{proof}
\medskip
\begin{example}
\rm{
Let's fix any compact set $Q\subset \mathbb{R}^d$.
Because $1_Q \in L^2(\mathbb{R}^d)$, we also have $\hat 1_Q \in L^2(\mathbb{R}^d)$, by Plancherel. In other words, Theorem \ref{thm:Plancherel} gives us
\begin{equation} \label{consequence of Plancherel for indicator}
\int_{\mathbb{R}^d} |\hat 1_Q (\xi)|^2 d\xi = \int_{\mathbb{R}^d} |1_Q (x)|^2 dx = \int_Q dx = \vol Q < \infty.
\end{equation}
Let's ask a question:
\begin{question} \rm{[Rhetorical]}
Does the function $g:= 1_Q * 1_{-Q}$ belong to $L^1(\mathbb{R}^d)$?
\end{question}
Well, we also know that trivially $1_Q, 1_{-Q} \in L^1(\mathbb{R}^d)$, which implies that
$\hat g := {\mathcal F}\left( 1_Q * 1_{-Q} \right) = \hat 1_Q \hat 1_{-Q}$, via Lemma \ref{convolution theorem} \ref{convolution under FT}.
The Cauchy-Schwarz inequality gives us:
\[
\int_{\mathbb{R}^d} | \hat g(\xi) | \, d \xi
= \int_{\mathbb{R}^d} | \hat 1_Q(\xi) | |\hat 1_{-Q}(\xi) | \, d \xi
\leq \left( \int_{\mathbb{R}^d} | \hat 1_Q (\xi) |^2 \,d \xi \right)^{1/2}
\left( \int_{\mathbb{R}^d} | \hat 1_{-Q} (\xi) |^2 \,d \xi \right)^{1/2}
< \infty,
\]
the last inequality owing itself to \eqref{consequence of Plancherel for indicator}.
}
\hfill $\square$
\end{example}
\medskip
\begin{example}
\rm{
As we recall, the sinc function, defined by
\begin{equation*}
{\rm{sinc}}(x):= \begin{cases}
\frac{\sin(\pi x)}{\pi x}, &\mbox{if } x \not= 0 \\
1 & \mbox{if } x= 0,
\end{cases}
\end{equation*}
plays an important role (in many fields).
Here we'll glimpse another aspect of its importance, as an application of Plancherel's identity
\eqref{Plancherel identity} above. Let's show that
\begin{equation}
\int_\mathbb{R} {\rm{sinc}}(x-n) {\rm{sinc}}(x-m) dx =
\begin{cases}
1 & \mbox{if } n=m \\
0 & \mbox{if } n\not=m.
\end{cases}
\end{equation}
Although ${\rm{sinc}}(x) \notin L^1(\mathbb{R})$, it is true that ${\rm{sinc}}(x) \in L^2(\mathbb{R})$.
Using Plancherel, we have
\begin{align*}
\int_\mathbb{R} {\rm{sinc}}(x-n) {\rm{sinc}}(x-m) dx & = \int_\mathbb{R} {\mathcal F}({\rm{sinc}}(x-n))(\xi) \overline{ {\mathcal F}( {\rm{sinc}}(x-m) )(\xi) } d\xi \\
&= \int_\mathbb{R} 1_{\mathcal P}(\xi) e^{2\pi i \xi n} 1_{\mathcal P}(\xi) \overline{ e^{2\pi i \xi m} }d\xi\\
&= \int_{\mathcal P} e^{2\pi i \xi (n-m)} d\xi \\
&= \delta(n, m),
\end{align*}
where ${\mathcal P}:= [-\frac{1}{2}, \frac{1}{2}]$, and where we've used
the orthogonality of the exponentials over ${\mathcal P}:= [-\frac{1}{2}, \frac{1}{2}]$
(Exercise \ref{orthogonality for exponentials}).
So we see that the collection of functions
\[
\left\{ {\rm{sinc}}(x - n) \bigm | n \in \mathbb{Z} \right\}
\]
forms an orthonormal collection of functions in the Hilbert space $L^2([-\tfrac{1}{2}, \tfrac{1}{2}] )$, relative to its norm. It turns out that when one studies
Shannon's sampling theorem, these translated sinc functions are in fact a complete orthonormal basis for the Hilbert subspace
of $L^2(\mathbb{R})$ that consists of `bandlimited functions' (see Theorem \ref{Shannon}).
}
\hfill $\square$
\end{example}
\section{Approximate identities}
It is a sad fact of life that there is no identity in $L^1(\mathbb{R}^d)$ for the convolution product - in other words, there is no function
$h \in L^1(\mathbb{R}^d)$ such that
\begin{equation}\label{if there was an identity}
f*h = f
\end{equation}
for all $f \in L^1(\mathbb{R}^d)$.
Why is that? Suppose there was such a function $h\in L^1(\mathbb{R}^d)$. Then taking the Fourier transform of both sides of \eqref{if there was an identity}, we would also
have
\begin{equation} \label{taking FT's...}
\hat f \ \hat h= \widehat{f*h} = \hat f,
\end{equation}
for all $f\in L^1(\mathbb{R}^d)$. Picking an $f$ whose transform is nowhere zero, we can divide both sides of \eqref{taking FT's...}
by $\hat f$, to conclude that $\hat h \equiv 1$, the constant function. But by the Riemann-Lebesgue Lemma \ref{Riemann--Lebesgue lemma}, we know that $\hat h$ must go to $0$ as $|x| \rightarrow \infty$, which is a contradiction.
\index{Riemann-Lebesgue lemma}
Nevertheless, it is still interesting to think about what would happen if we were able to apply the
inverse Fourier transform to $\hat h$, formally applying the Fourier transform to the
equation $\hat h = 1$ to get:
\begin{equation}
h(x) = \int_{\mathbb{R}^d} e^{2\pi i \langle x, \xi \rangle} dx,
\end{equation}
an extremely interesting integral that unfortunately diverges. In note \ref{Dirac delta}, we mention briefly that such observations became
critically important for the development of generalized functions that do play the role of the identity for convolutions, and much more.
Although there is no identity element for convolutions, it turns out that using sequences of functions we can get close! Here is how we may do it, and as a consequence we will be able to rigorously apply the Poisson summation formula to a wider class of functions, including smoothed versions of the indicator function of a polytope.
Fix a function $\phi \in L^1(\mathbb{R}^d)$, such that $\int_{\mathbb{R}^d} \phi(x) dx = 1$.
Beginning with any such function $\phi$, we construct an {\bf approximate identity} by defining the sequence of functions
\begin{equation}\label{approximate identity}
\phi_n(x):= n^d \phi(n x),
\end{equation}
for each $n = 1, 2, 3, \dots$.
It's easy to check that we also have $\int_{\mathbb{R}^d}\phi_n(x) dx = 1$, for all $n\geq~1$ (Exercise \ref{total mass 1}).
So scaling $\phi$ by these $n$'s has the effect of
squeezing $\phi$ so that it is becomes concentrated near the origin, while maintaining a total mass of $1$.
Then intuitively a sequence of
such $\phi_n$ functions approach the ``Dirac delta-function" at the origin (which is a distribution, not a function).
There are many families of functions that give an approximate identity.
In practice, we will seldom have to specify exactly which sequence $\phi_n$ we pick, because we will merely use the existence of such a sequence to facilitate the use of Poisson summation.
Returning now to the motivation of this section, we can recover the next-best-thing to an identity for the convolution product, as follows.
\begin{thm}\label{approximate identity convolution}
Suppose we are given a function $f \in L^1(\mathbb{R}^d)$, such that $p \in \mathbb{R}^d$ is a point of continuity for $f$.
Fix an approximate identity $\phi_n(x)$, and assume $f*\phi$ exists. Then we have:
\begin{equation}\label{basic smoothing}
\lim_{n \rightarrow \infty} \left(f * \phi_n \right)(p) = f(p).
\end{equation}
\end{thm}
\begin{proof}
We begin by massaging the convolution product:
\begin{align*}
(\phi_n*f)(p) &:= \int_{\mathbb{R}^d} \phi_n(x) f(p-x) dx \\
&= \int_{\mathbb{R}^d} \phi_n(x) \Big(f(p-x) - f(p) + f(p) \Big) dx \\
&= \int_{\mathbb{R}^d} \phi_n(x) \Big(f(p-x) - f(p) \Big) dx +
f(p) \int_{\mathbb{R}^d} \phi_n(x) dx \\
&= f(p) + \int_{\mathbb{R}^d} \phi_n(x) \Big(f(p-x) - f(p) \Big) dx,
\end{align*}
using the assumption that $\int_{\mathbb{R}^d} \phi_n(x) dx=1$.
Using the definition of
$\phi_n(x):= n^{d} \phi(n x)$, and making a change of variable
$u= n x$ in the latter integral, we have:
\begin{align*}
(\phi_n*f)(p) &:= f(p) + \int_{\mathbb{R}^d} \phi(u) \Big( f\left(p- \frac{1}{n} u\right) - f(p) \Big) du.
\end{align*}
In the second part of the proof, we will show that as $n \rightarrow \infty$, the latter integral tends to zero.
We will do this in two steps, first bounding the tails of the integral in a neighborhood of infinity, and then bounding the integral in a neighborhood of the origin.
Step $1$. \ Given any $\varepsilon >0$, we note that the latter integral converges, so the `tails are arbitrarily small'. In other words, there exists an $r > 0$ such that
\[
\left| \int_{\| u \| > r} \phi(u) \left(f\left(p- \frac{1}{n} u\right) - f(p) \right) du \right| < \varepsilon.
\]
Step $2$. \ Now we want to bound $\int_{\| u \| < r} \phi(u) \left( f\left(p-\frac{1}{n} u\right) - f(p) \right) du$.
We will use the fact that $ \int_{\mathbb{R}^d} | \phi(u) | du = M$, a constant. Also, by continuity of $f$ at $p$,
we can pick an $n$ sufficiently large, such that:
\[
\left| f\left(p-\frac{1}{n} u\right) - f(p) \right| < \frac{\varepsilon}{M},
\]
when $\| \frac{1}{n} u \| < r$. Putting all of this together, and using the triangle inequality for integrals,
we have the bound
\begin{align*}
\Big| \int_{\| u \| < r} \phi(u) \left(f\left(p-\frac{1}{n} u\right) - f(p) \right) du \Big|
&\leq \int_{\| u \| < r} | \phi(u) | \left| f\left(p-\frac{1}{n} u\right) - f(p) \right| du < \varepsilon.
\end{align*}
Therefore, as $n \rightarrow \infty $, we have $(\phi_n*f)(p) \longrightarrow f(p)$.
\end{proof}
We note that a point of discontinuity of $f$, Theorem \ref{approximate identity convolution}
may be false even in dimension $1$, as the next example shows.
\begin{example}
Let $f(x):= 1_{[0,1]}(x)$, which is discontinuous at $x=0$ and $x=1$. We claim that
for $p=1$, for example, we have
\[
\lim_{n \rightarrow \infty} (f * \phi_n)(p) = \frac{1}{2} f(p),
\]
so that the result of Theorem \ref{approximate identity convolution}
does not hold at this particular $p$, because $p$ lies on the boundary
of the $1$-dimensional polytope $[0,1]$. When $p \in \interior([0,1])$, however, Theorem
\ref{approximate identity convolution} does hold.
\hfill $\square$
\end{example}
\section{A practical Poisson summation formula}
In practice, we want to apply Poisson summation to indicator functions $1_{\mathcal P}$ of polytopes and general convex bodies.
With this in mind, it's useful for us to have our own, home-cooked version of Poisson summation that is made for this culinary purpose.
Throughout this section, we fix any compactly supported, nonnegative function $\varphi \in L^2(\mathbb{R}^d)$, with
$\int_{\mathbb{R}^d} \varphi(x) dx = 1$,
and we
set $\varphi_\varepsilon(x) := \frac{1}{\varepsilon^d} \varphi \left( \frac{x}{\varepsilon} \right)$, for each $\varepsilon > 0$.
\bigskip
\begin{thm}[Poisson summation formula IV] \label{PracticalPoisson}
\index{Poisson summation formula}
Let $f(x) \in L^2(\mathbb{R}^{d})$ be a compactly supported function, and suppose that for each $x\in \mathbb{R}^d$,
we have:
\begin{equation} \label{hypothesis, practical Poisson}
f(x) =\lim_{\varepsilon\rightarrow0^{+}}
\left(
\varphi_{\varepsilon}\ast f
\right)(x).
\end{equation}
Then the following hold:
\begin{enumerate}[(a)]
\item
For each $\varepsilon>0$, we have absolute convergence:
$
\sum_{m\in\mathbb{Z}^d} \left| \widehat{\varphi}\left( \varepsilon
m\right) \widehat{f}\left( m\right) \right| <+\infty.
$
\item
For all sufficiently small $\varepsilon >0$, and for each fixed $x\in \mathbb{R}^d$, we have the
pointwise equality:
\begin{equation} \label{practical Poisson summation, first version}
\sum_{n\in\mathbb{Z}^{d}}
\left(
\varphi_{\varepsilon}\ast f
\right)\left( n+x\right)
=
\sum_{m\in\mathbb{Z}^d} \widehat{\varphi}\left(
\varepsilon m\right) \widehat{f}\left( m\right)
e^{2\pi i \langle m, x \rangle}.
\end{equation}
\item
\begin{equation} \label{practical Poisson summation, second version}
\sum_{n\in\mathbb{Z}^{d}}f\left( n+x\right)
=
\lim_{\varepsilon \rightarrow 0}
\sum_{m\in\mathbb{Z}^d} \widehat{\varphi}\left(
\varepsilon m\right) \widehat{f}\left( m\right)
e^{2\pi i \langle m, x \rangle}.
\end{equation}
\end{enumerate}
\end{thm}
Because both $f$ and $\varphi_\varepsilon$ are compactly supported,
the left-hand-sides of equations
\eqref{practical Poisson summation, first version} and
\eqref{practical Poisson summation, second version}
are finite sums.
\hfill $\square$
For a detailed proof of Theorem \ref{PracticalPoisson}, see \cite{BrandoliniColzaniTravagliniRobins1}.
An interesting aspect of this version of Poisson summation is that it can sometimes even apply to functions $f$
that are only piecewise continuous on $\mathbb{R}^d$, as long as \eqref{hypothesis, practical Poisson} holds. Our prime example is of course
\[
f(x):= 1_{\mathcal P}(x),
\]
the indicator function of a polytope ${\mathcal P}$, and more generally $1_Q$ for a compact set $Q$ with reasonable behavior, such as a convex body.
In Chapter \ref{Chapter.Minkowski}, we will use this version of Poisson summation, Theorem \ref{PracticalPoisson},
to prove Theorem \ref{zero set of the FT of a polytope}.
An interesting tool that gets used in the proof of Theorem \ref{PracticalPoisson} is a
{\bf Plancherel-Polya type inequality}, as follows.
\begin{lem}
Suppose that $ f \in L^1(\mathbb{R}^d), \hat f \in L^1(\mathbb{R}^d)$, and $f$ is compactly supported. Then there exists a constant $c > 0$, depending on the support of $f$, such that
\begin{equation}\label{Plancherel-Polya inequality}
\sum_{n \in \mathbb{Z}^d}
| \hat f(n) | \leq c \int_{\mathbb{R}^d} | \hat f(\xi) | d \xi.
\end{equation}
\end{lem}
\begin{proof}
We define a new function $\psi$, which is infinitely smooth, and compactly supported, with $\psi(x) = 1$
for all $x$ in the support of $f$. So we have $f(x) = \psi(x) f(x), \forall x \in \mathbb{R}^d$, and
therefore $\hat f(\xi) =(\hat\psi * \hat f)(\xi)$ (using $\hat f \in L^1(\mathbb{R}^d)$). Because $\psi$ is smooth, we know that $\hat \psi$ is rapidly decreasing (by Corollary \ref{cor: f smoother implies FT of F decays faster}),
and we have
\begin{align}
\sum_{n \in \mathbb{Z}^d} | \hat f(n) | &=
\sum_{n \in \mathbb{Z}^d} \left | \int_{\mathbb{R}^d} \hat \psi( n - \xi) \hat f(\xi) d\xi \right | \\
&\leq \sum_{n \in \mathbb{Z}^d} \int_{\mathbb{R}^d} \left | \hat \psi( n - \xi) \hat f(\xi) \right | d\xi \\
&= \int_{\mathbb{R}^d} \sum_{n \in \mathbb{Z}^d} \left | \hat \psi( n - \xi) \right | \left | \hat f(\xi) \right | d\xi \\
& \leq \sup_{\xi \in \mathbb{R}^d} \left( \sum_{n \in \mathbb{Z}^d} \left | \hat \psi( n - \xi) \right | \right)
\int_{\mathbb{R}^d} | \hat f(\xi) | d\xi \\
&\leq c \int_{\mathbb{R}^d} | \hat f(\xi) | d\xi.
\end{align}
The constant $c$ depends on $\psi$, and hence on the support of $f$. To justify the last step, we note that
$g(\xi):= \sum_{n \in \mathbb{Z}^d} | \hat \psi( n - \xi) |$
is a periodic function of $\xi$, with the unit cube $[0, 1]^d$ being a fundamental domain, so it suffices to show that $g$ is bounded on the unit cube.
But due to the rapid decay of $\hat \psi$, we may apply the Weierstrass $M$-test to conclude that
the series $g$ is a uniformly convergent sum of continuous functions; hence $g$ is itself a continuous function on a compact set (the cube), and in fact achieves its maximum there.
\end{proof}
The reader may consult \cite{SchmeisserSickel2000}, for example, for more information about related Plancherel-Polya type inequalities.
In general, there are many functions $f \in L^1(\mathbb{R})$ such that $\sum_{n \in \mathbb{Z}} | \hat f(n) | $ diverges, yet
$ \int_{\mathbb{R}} | \hat f(\xi) | d \xi$ converges, so that \eqref{Plancherel-Polya inequality} is false for these functions
(Exercise \ref{exercise:Plancherel-Polya type inequalities}).
\bigskip
\section{Uncertainty principles}
\label{section:uncertainy principles}
Perhaps the most basic type of \emph{uncertainy principle} is the fact that if a function $f$ is compactly supported,
then its Fourier transform $\hat f$ cannot be compactly supported - Theorem \ref{basic uncertainty principle} below.
Similar impossible constraints, placed simultaneously on both $f$ and $\hat f$,
have become known as {\bf uncertainty principles}. Perhaps the most famous of these, originating in quantum mechanics, is Heisenberg's
discovery, as follows.
\begin{thm}[Heisenberg uncertainty principle]
\index{uncertainty principle, Heisenberg}
Let $f\in L^2(\mathbb{R}^d)$, with the normalization assumption that $\int_{\mathbb{R}^d} |f(x)|^2 dx=1$. Then:
\begin{equation}
\int_{\mathbb{R}^d} \|x\|^2 |f(x)|^2 dx \int_{\mathbb{R}^d} \|x\|^2 |\hat f(x)|^2 dx \geq \frac{1}{16 \pi^2},
\end{equation}
with equality holding if and only if $f$ is equal to a Gaussian.
\rm{(For a proof see \cite{OsgoodBook}, or \cite{DymMcKean}. )}
\hfill $\square$
\end{thm}
\bigskip
\begin{thm}[Hardy uncertainty principle]
\label{Hardy uncertainty principle} \index{uncertainty principle, Hardy}
Let $f\in L^1(\mathbb{R}^d)$ be a function that enjoys the property that
\begin{equation*}
|f(x)| \leq A e^{-\pi c x^2 } \text{ and } \ |\hat f(\xi) | \leq B e^{-\pi \xi^2/c},
\end{equation*}
for all $x, \xi\in \mathbb{R}^d$, and for some constants $A, B, c >0$.
Then $f(x)$ is a scalar multiple of the Gaussian $e^{-\pi c x^2}$.
\rm{(For a proof see \cite{Hardy.uncertainty})}
\hfill $\square$
\end{thm}
\bigskip
\begin{thm} \label{basic uncertainty principle}
Let $f\in L^1(\mathbb{R}^d)$ be a function that is supported on a compact set in $\mathbb{R}^d$.
Then $\hat f$ is not supported on any compact set in $\mathbb{R}^d$.
\rm{(For a proof see \cite{EpsteinBook})}
\hfill $\square$
\end{thm}
\section*{Notes}
\begin{enumerate}[(a)]
\item \label{Fourier books}
There are some wonderful introductory books that develop Fourier analysis from first principles, such as the books by
Stein and Shakarchi \cite{SteinShakarchi} and Giancarlo Travaglini \cite{Travaglini}.
The reader is also encouraged to read more advanced but fundamental introductions to Fourier analysis, in particular the books by Mark Pinsky \cite{MarkPinsky}, Edward Charles Titchmarsh \cite{Titchmarsh},
Einsiedler and Ward \cite{EinsiedlerWardBook}, Dym and McKean \cite{DymMcKean},
and of course the classic: Stein and Weiss \cite{SteinWeiss}. In addition, the book \cite{Terras} by Audrey Terras is a good introduction to Fourier analysis
on finite groups, with applications.
A more informal introduction to Fourier analysis, focusing on various applications, is given by Brad Osgood \cite{OsgoodBook}.
\item There are some ``elementary'' techniques that we will use, from the calculus of a complex variable, but which require essentially no previous knowledge in this field. In particular, suppose we have two analytic functions
$f:\mathbb{C} \rightarrow \mathbb{C}$ and $g:\mathbb{C} \rightarrow \mathbb{C}$, such that $f(z_k) = g(z_k)$ for a convergent sequence of complex numbers $z_k \rightarrow L$, where $L$ is any fixed complex number. Then $f(z) = g(z)$ for all
$z \in \mathbb{C}$.
The same conclusion is true even if the hypothesis is relaxed to the assumption that
both $f$ and $g$ are meromorphic functions, as long as the sequence and its limit stay away from the poles of $f$ and $g$.
\item \label{Dirac delta}
The ``Dirac delta function" is part of the theory of ``generalized functions'' and may be intuitively defined by the full sequence
of Gaussians $G_t (x) := t^{-\frac{d}{2}} e^{ -\frac{\pi}{t} || x ||^2 }$, taken over all $t>0$. The observation that there is no identity for the convolution product
on $\mathbb{R}^d$ is a clear motivation for a theory of generalized functions, beginning with the Dirac delta function.
Another intuitive way of ``defining'' the Dirac delta function is:
\[
\delta_0(x) :=
\begin{cases}
\infty & \mbox{if } x=0 \\
0 & \mbox{if not},
\end{cases}
\]
even though this is not a function. But in the sense of distributions (i.e. generalized functions), we have
$\lim_{\rightarrow 0} G_t(x) = \delta_0(x)$.
More rigorously, the $\delta$-function belongs to a theory of distributions that was developed by Laurent Schwartz
\index{Schwartz, Laurent}
in the 1950's and by S.L. Sobolev in 1936, where we can think of generalized functions as linear functionals on the space of all bump functions on $\mathbb{R}^d$ (see the book by Lighthill \cite{Lighthill} for a nice introduction to generalized functions).
Such generalized functions were originally used by the Physicist Paul Dirac in 1920, before the rigorous mathematical theory was even created for it, in order to better understand quantum mechanics. \index{Dirac, Paul}
\item \label{Note:GregKuperberg} I'd like to thank Greg Kuperberg for very helpeful comments, and in particular for introducing
me to Theorem \ref{Euler-Maclaurin type identity}, for which we still cannot find a published reference.
\item
It is sometimes interesting to derive analogues between norms in $\mathbb{R}^d$ and norms in an infinite dimensional function space.
Among the many norm relations in $\mathbb{R}^d$, we mention one elementary but interesting relation:
\[
\| x \|_1 \leq \sqrt{n} \ \| x \|_2,
\]
for all vectors $ x \in \mathbb{R}^d$, where $\|x\|_1:= |x_1|+\cdots + |x_d|$, and $\|x\|_2:= \sqrt{x_1^2+\cdots + x_d^2}$.
(see Exercise \ref{elementary norm relations} for more practice with related norm relations).
At this point the curious reader might wonder ``are there any other inner products on $\mathbb{R}^d$, besides the usual inner product
$\langle x, y \rangle:= \sum_{k=1}^d x_k y_k$?" A classification of all inner products that exist on $\mathbb{R}^d$ is given in Exercise \ref{All norms on Euclidean space}.
\item
Of great practical importance, and historical significance, a {\bf bump function} is defined as any infinitely smooth function
on $\mathbb{R}^d$, which is compactly supported. In other words, a bump function enjoys the following properties:
\begin{itemize}
\item $\phi$ has compact support on $\mathbb{R}^d$.
\item $\phi \in C^\infty(\mathbb{R}^d)$.
\end{itemize}
Bump functions are also called {\bf test functions}, and if we consider the set of all bump functions
on $\mathbb{R}^d$, under addition, we get a vector space $V$, whose dual vector space is called the space of {\bf distributions on $\mathbb{R}^d$}.
\item The cotangent function, appearing in some of the exercises below, is the unique {\it meromorphic function} that has a simple pole at every integer, with residue 1 (up to multiplication by an entire function with the same residues). The cotangent function also forms an entry point for Eisenstein series in number theory, through the corresponding partial fraction expansion of its derivatives.
\item A deeper exploration into projections and sections of the unit cube in $\mathbb{R}^d$ can be found in
``The cube - a window to convex and discrete geometry'', by Chuangming~Zong \cite{Zong.book}.
In \cite{KoldobskyBook}, Alexander Koldobsky gives a thorough introduction to sections of convex bodies, intersection bodies, and the Busemann-Petty problem.
\item There are numerous other identities throughout mathematics that are equivalent to special cases of Poisson summation, such as the Euler-MacLaurin summation formula, the Abel-Plana
formula, and the Approximate sampling formula of signal analysis
(see \cite{Butzer.etal} for a nice treatment of such equivalences for functions of $1$ real variable, and functions of $1$ complex variable).
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
``In theory, there is no difference between theory and practice; but in practice, there is! '' \ \ \ \ -- Walter J. Savitch \\
\end{quote}
\medskip
\begin{prob} $\clubsuit$ \label{elementary norm relations}
On $\mathbb{R}^d$ the $L^2$-norm is defined by
$\|x\|_2:= \sqrt{ x_1^2 + \cdots + x_d^2}$, the $L^1$-norm is defined by
$\|x\|_1:= |x_1| + \cdots + |x_d|$, and the $L^\infty$-norm is defined by
$\|x\|_\infty:= \max\{ |x_1|, \dots, |x_d| \}$.
Prove the following four norm relations:
\[
\|x\|_\infty \leq \| x \|_2 \leq \| x \|_1 \leq \sqrt{d} \, \| x \|_2 \leq d \, \|x\|_\infty,
\]
for all $x \in \mathbb{R}^d$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{CS inequality for integrals}
Show that the Cauchy-Schwarz inequality holds in the Hilbert space $L^2({\mathbb T^d})$:
\begin{equation}
\int_{{\mathbb T^d}} f(x)\overline{g(x)} dx \leq
\left(\int_{{\mathbb T^d}} |f(x)|^2 dx \right)^{\frac{1}{2}} \left(\int_{{\mathbb T^d}} |g(x)|^2 dx\right)^{\frac{1}{2}},
\end{equation}
for all $f, g \in L^2({\mathbb T^d})$, with equality if and only if $f(x) = C g(x)$ for some constant $C$.
\end{prob}
\medskip
\begin{prob} \label{exercise:hyperbolic cosine and sine}
We know that the functions $u(t) := \cos t = \frac{e^{it} + e^{-it}}{2}$ and $v(t) := \sin t =
\frac{e^{it} - e^{-it}}{2i}$ are natural, partly because they parametrize the unit circle: $u^2 + v^2 = 1$. Here we see that there are other similarly natural functions, parametrizing the hyperbola.
\begin{enumerate}[(a)]
\item Show that the following functions parametrize the hyperbola $u^2 - v^2 = 1$:
\[
u(t) := \frac{e^t + e^{-t}}{2}, \ \ \ v(t) := \frac{e^t - e^{-t}}{2}.
\]
(This is the reason that the function $\cosh t:= \frac{e^t + e^{-t}}{2}$ is called the hyperbolic cosine, and the function
$\sinh t := \frac{e^t - e^{-t}}{2}$ is called the hyperbolic sine)
\item The hyperbolic cotangent is defined as $\coth t := \frac{ \cosh t }{ \sinh t} =
\frac{ e^t + e^{-t}}{e^t - e^{-t}}$. Using Bernoulli numbers, show that $t \coth t$ has the Taylor series:
\[
t \coth t = \sum_{n=0}^\infty \frac{2^{2n}}{(2n)!} B_{2n} t^{2n}.
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{compute FT for exponential of abs value}
Prove that:
\[
\frac{t}{\pi} \sum_{n \in \mathbb{Z}} \frac{1}{n^2 + t^2} = \sum_{m \in \mathbb{Z}} e^{-2\pi t |m|}.
\]
Hint. \ Think of Poisson summation, applied to the function $f(x):= e^{-2\pi t |x|}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Riemann zeta function, and Bernoulli numbers}
Here we evaluate the Riemann zeta function at the positive even integers.
\begin{enumerate}[(a)]
\item Show that
\[
\sum_{n\in \mathbb{Z}} e^{-2 \pi t |n|} =
\frac{ 1 + e^{-2\pi t} }{ 1-e^{-2\pi t} } := \coth(\pi t),
\]
for all $t>0$.
\bigskip
\item Show that the cotangent function has the following well-known partial fraction expansion:
\[
\pi \cot(\pi x) = \frac{1}{x} + 2x \sum_{n=1}^\infty \frac{1}{ x^2 - n^2},
\]
valid for any $x \in \mathbb{R} - \mathbb{Z}$.
\item Let $0 < t < 1$. Show that
\[
\frac{t}{\pi} \sum_{n\in \mathbb{Z}} \frac{1}{n^2 + t^2} = \frac{1}{\pi t} +
\frac{2}{\pi} \sum_{m=1}^\infty (-1)^{m+1} \zeta(2m) \ t^{2m-1},
\]
where $\zeta(s):= \sum_{n=1}^\infty \frac{1}{n^s}$ is the Riemann zeta function, initially defined by the latter series, which is valid for all $s \in \mathbb{C}$ with $Re(s) >1$.
\item Here we show that we may quickly evaluate the Riemann zeta function at all even integers, as follows. We recall the definition of the Bernoulli numbers, namely:
\[
\frac{z}{e^z - 1} = 1 - \frac{z}{2} + \sum_{m \geq 1} \frac{B_{2m}}{2m!} z^{2m}.
\]
Prove that for all $m \geq 1$,
\[
\zeta(2m) = \frac{(-1)^{m+1} }{2} \frac{ (2\pi)^{2m}}{ (2m)!} B_{2m}.
\]
Thus, for example, using the first $3$ Bernoulli numbers, we have: $\zeta(2) = \frac{\pi^2}{6}$, $\zeta(4) = \frac{\pi^4}{90}$, and $\zeta(6) = \frac{\pi^6}{945}$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{strictly less than the FT at zero}
Given a $d$-dimensional polytope ${\mathcal P} \subset \mathbb{R}^d$, prove the strict inequality
\[
|\hat 1_{\mathcal P}(\xi) | < \vol {\mathcal P}, \text{ for all nonzero } \xi \in \mathbb{R}^d.
\]
\end{prob}
\medskip
\begin{prob} \label{Chebyshev polys}
For each $n\geq 1$, let $T_n(x) = \cos(nx)$.
For example, $T_2(x) = \cos(2x) =2 \cos^2(x) - 1$, so $T_2(x) = 2u^2 -1$, a polynomial in $u:= \cos x$.
\begin{enumerate}[(a)]
\item Show that for all $n\geq 1$, $T_n(x)$ is a polynomial in $\cos x$.
\item Can you write $x^n + \frac{1}{x^n}$ as a polynomial in the variable $x + \frac{1}{x}$?
Would your answer be related to the polynomial $T_n(x)$? What's the relationship in general?
For example, $x^2 + \frac{1}{x^2} = \Big( x + \frac{1}{x}\Big)^2 - 2$.
\end{enumerate}
\end{prob}
Notes. The polynomials $T_n(x)$ are very important in applied fields such as approximation theory, and optimization,
because they have many useful extremal properties.
They are called Chebyshev polynomials. \index{Chebyshev polynomials}
\medskip
\begin{prob} \label{sec - its own Fourier transform}
The hyperbolic secant is defined by
\[
{\rm sech}(\pi x) := \frac{2}{e^{\pi x} + e^{-\pi x}}, \text{ for } x \in \mathbb{R}.
\]
\begin{enumerate}[(a)]
\item \label{eigenfunction of FT}
Show that ${\rm sech}(\pi x)$ is its own Fourier transform:
\[
{\mathcal F}({\rm sech})(\xi) = {\rm sech}(\xi),
\]
for all $\xi \in \mathbb{R}$.
\item \label{bounded above by Gaussian}
Show that ${\rm sech}(\pi x)$ can never be bounded above by any Gaussian, in the precise sense that the following claim is impossible: there exists a constant $c>0$ such that for all $x \in \mathbb{R}$ we have:
\[
{\rm sech}(\pi x) \leq e^{-cx^2}.
\]
\end{enumerate}
Notes. For part \ref{eigenfunction of FT}, the reader may need some background in complex analysis for this exercise. For part \ref{bounded above by Gaussian}, it may be helpful to look at Hardy's uncertainty principle, Theorem \ref{Hardy uncertainty principle}. We can also conclude from Hardy's uncertainty principle that any eigenfunction $f$ of the Fourier transform cannot be bounded above by a Gaussian, aside from the case that $f$ is itself a Gaussian.
\end{prob}
\medskip
\begin{prob}
Using the previous exercise, conclude that
\[
\int_{\mathbb{R}} \frac{1}{e^{\pi x} + e^{-\pi x}} dx = \frac{1}{2}.
\]
\end{prob}
\medskip
The following exercises give more practice in computing/handling general Fourier transforms and their important properties. Throughout, we assume that the Fourier transform of $f$ exists, where
$f:\mathbb{R} \rightarrow \mathbb{C}$ is any integrable function.
\medskip
\begin{prob} $\clubsuit$
Prove that:
\[
\int_0^1 P_1(ax) P_1(bx) dx = \frac{ 1 }{12 \ \rm{gcd}^2(a, b)}.
\]
for all positive integers $a, b$. Here $P_1(x) := x - \{x\} - \frac{1}{2}$ is the first periodic Bernoulli polynomial.
Notes. This integral is called a {\bf Franel integral}, and there is a substantial literature about related integrals.
In 1924, J\'er\^ome Franel related this integral to the Riemann hypothesis, and to Farey fractions.
\end{prob}
\medskip
\begin{prob}
Using Theorem \ref{Euler-Maclaurin type identity}, obtain the little-o asymptotics (with $N\rightarrow \infty$)
for the finite sums
\[
\sum_{m=0}^{N-1} \sin \left(\tfrac{m \pi }{N}\right),
\]
which is simple enough that it also offers an independent verification.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Schwartz space convolution invariance}
Let $f: \mathbb{R} \rightarrow \mathbb{C}$ belong to the Schwarz class of functions on $\mathbb{R}$, denoted by $S(\mathbb{R})$.
Show that $\hat f \in S(\mathbb{R})$ as well.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{All norms on Euclidean space}
Here we answer the very natural question ``What are the other inner products on $\mathbb{R}^d$, besides the usual inner product $\langle x, y \rangle:= \sum_{k=1}^d x_k y_k$ ?"
The fact is that all inner products are related to each other via positive definite matrices, as follows. We recall from Linear Algebra that a symmetric
matrix is called positive definite if all of its eigenvalues are positive.
Prove that the following two conditions are equivalent:
\begin{enumerate}
\item $ \langle x, y \rangle$ is an inner product on $\mathbb{R}^d$.
\item $ \langle x, y \rangle := x^T M y$, for some positive definite matrix $M$.
\end{enumerate}
\end{prob}
\bigskip
\begin{prob}
For any positive real numbers $a < b < c < d$, define
\[
f(x) := 1_{[a, b]}(x) + 1_{[c, d]}(x).
\]
Can you find $a,b,c,d$ such that $\hat f(\xi)$ is nonzero for all $\xi \in \mathbb{R}$?
\end{prob}
\medskip
\begin{prob}
\end{prob}
\medskip
\begin{prob} $\clubsuit$
Show that for $f, \hat f \in L^1(\mathbb{R}^d)$, the only eigenvalues of the linear operator
\[
f \rightarrow \hat f
\]
are $\{ 1, -1, i, -i \}$, and show that each of these eigenvalues is achieved by some function $f \in L^1(\mathbb{R}^d)$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{going backwards in Poisson summation}
Show that the special case of Poisson summation, \ref{Poisson.summation3}, implies the general case, Theorem \ref{The Poisson Summation Formula, for lattices}. \index{Poisson summation formula}
\end{prob}
\medskip
\begin{prob} \label{Gaussian1} $\clubsuit$
We define the Gaussian, for each fixed $\varepsilon >0$, and for all $x \in \mathbb{R}^d$, by
\begin{equation} \index{Gaussian}
G_{\varepsilon} (x) := \frac{1}{\varepsilon^{\frac{d}{2}}} e^{ -\frac{\pi}{\varepsilon} || x ||^2 }.
\end{equation}
Show that:
\[
\int_{\mathbb{R}^d} G_{\varepsilon} (x) dx = 1.
\]
\end{prob}
\medskip
\begin{prob} \label{Gaussian2} $\clubsuit$
Show that, for all $m \in \mathbb{R}^d$, the Fourier transform of the Gaussian
$G_{\varepsilon}(x)$ is:
\[
\hat G_\varepsilon( m ) = e^{ -\pi \varepsilon || m ||^2 }.
\]
\end{prob}
\medskip
\begin{prob}
Prove that if $f, g \in L^1(\mathbb{R}^d)$ are bounded functions, then $f*g$ is continuous on $\mathbb{R}^d$.
\end{prob}
Notes. In particular, this exercise shows that if $A, B \subset \mathbb{R}^d$ are convex bodies, then \\
$(1_A*1_B)(x)= \vol\left( A \cap (-B + x) \right)$ is a continuous function of $x \in \mathbb{R}^d$.
\medskip
\begin{prob} \label{Plancherel extended} $\clubsuit$
For all $f, g \in S(\mathbb{R}^d)$ (the Schwartz space), show that $\langle f, g\rangle = \langle \hat f, \hat g \rangle$.
\end{prob}
\medskip
\begin{prob} \label{total mass 1} $\clubsuit$
Given any approximate identity sequence $\phi_\varepsilon$, as defined in \eqref{approximate identity},
show that for each~$\varepsilon~>~0$,
\[
\int_{\mathbb{R}^d}\phi_\varepsilon(x) dx = 1.
\]
\end{prob}
\medskip
\begin{prob}
Show that the ramp function, defined in \eqref{def of ramp}, also has the representation:
\begin{equation}
r_0(x) = \frac{ x + |x| }{2},
\end{equation}
for all $x\in \mathbb{R}$.
Notes. \ Some books, particularly in approximation theory, use the notation $r_0(x) := x_+$.
\end{prob}
\medskip
\begin{prob}
\label{positive FT over R} $\clubsuit$
\rm{
Here we show that there exist compactly supported functions $f:\mathbb{R}\rightarrow \mathbb{C}$
whose Fourier transform is {\bf strictly positive} on all of $\mathbb{R}$.
Fix any two incommensurable real numbers $r, s$ (meaning that $\frac{r}{s} \notin \mathbb{Q}$),
and define
\[
f:= 1_{[-r, r]}*1_{[-r, r]} + 1_{[-s, s]}*1_{[-s, s]},
\]
which is a sum of two hat functions, as depicted in Figure \ref{A sum of two hat functions}.
Prove that for all $\xi \in \mathbb{R}$, we have $ \hat f(\xi) >0$.
}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{SumOfTwoHatFunctions}
\end{center}
\caption{The function $f$ of Exercise \ref{positive FT over R}, a sum of two hat functions,
with $s = \sqrt{\frac{2}{3}}$, and $r= 1.9$}
\label{A sum of two hat functions}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.5in]{StrictlyPositive}
\end{center}
\caption{The \emph{strictly positive} Fourier transform $\hat f(\xi)$ of Exercise \ref{positive FT over R}, with the two incommensurable numbers $s = \sqrt{\frac{2}{3}}$, and $r= 1.9$}
\label{strictly positive FT on R}
\end{figure}
Notes. This construction can be extended to higher dimensions, once we know more about the Fourier transforms of balls in $\mathbb{R}^d$ - see Exercise \ref{positive FT over R^d}.
\end{prob}
\medskip
\begin{prob}
\label{Heaviside and ramp}
$\clubsuit$
Show that for all $a, b \in \mathbb{R}$, we have:
\[
H_a*H_b = r_{a+b},
\]
where $H_a$ is the heaviside function of \eqref{def of heaviside}, and $r_a$ is the ramp function of \eqref{def of ramp}.
\end{prob}
\medskip
\begin{prob}
\label{exercise:Plancherel-Polya type inequalities} $\clubsuit$
Here we show that the absolute convergence of a series, and the absolute convergence of the corresponding integral, are independent of each other.
\begin{enumerate} [(a)]
\item
Find a function $f :\mathbb{R} \rightarrow \mathbb{C}$ such that $\sum_{n \in \mathbb{Z}} | \hat f(n) | $ diverges, yet
$ \int_{\mathbb{R}} | \hat f(\xi) | d \xi$ converges.
\item On the other hand, find a function $f :\mathbb{R} \rightarrow \mathbb{C}$ such that $ \int_{\mathbb{R}} | \hat f(\xi) | d \xi$ diverges, yet
$\sum_{n \in \mathbb{Z}} | \hat f(n) | $ converges.
\end{enumerate}
\end{prob}
Notes. This exercise shows that there the Plancherel-Polya inequality holds only for a special class of functions.
\medskip
\begin{prob} \label{tricky application of Poisson summation}
Here is a tiny variation on Poisson summation. If $g:\mathbb{R}^d\rightarrow \mathbb{C}$ is infinitely smooth, and compactly supported, prove that
\[
\sum_{n \in \mathbb{Z}^d} \hat g(n) = \sum_{n \in \mathbb{Z}^d} g(n),
\]
and the right-hand-side is a finite sum.
\end{prob}
\chapter{The geometry of numbers - \\
Minkowski's first theorem, and Siegel's extension} \label{Chapter.Minkowski}
\begin{wrapfigure}{R}{0.51\textwidth}
\centering
\includegraphics[width=0.24\textwidth]{Minkowski}
\caption{Hermann Minkowski}
\end{wrapfigure}
\label{Geometry of numbers}
\index{Siegel's formula} \index{Minkowski}
\begin{quote}
``Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.''
-- Hermann Minkowski \index{Minkowski, Hermann}
\end{quote}
\section{Intuition}
To see a wonderful and fun application of Poisson summation, we give a relatively easy proof of Minkowski's first theorem, in the Geometry of Numbers.
Minkowski's theorem gives the existence of an integer point inside symmetric
bodies in $\mathbb{R}^d$, once we know their volume is sufficiently large.
In fact we first prove a more powerful identity which
is a classical result of Carl Ludwig Siegel (Theorem \ref{Siegel}), yielding an identity between Fourier transforms of convex bodies and their volume. Our proof of this identity of Siegel uses Poisson summation, applied to the convolution of an indicator function with itself. \index{Poisson summation formula}
The geometry of numbers is an incredibly beautiful field, and too vast to encompass in just one chapter (see note \ref{new books, geometry of numbers}). We hope this chapter, a small bite of a giant fruit, gives the reader motivation to pursue the interactions between convex bodies and lattices even further.
\bigskip
\section{Minkowski's convex body Theorem}
\begin{wrapfigure}{L}{0.55\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{convexbody}
\caption{A convex, symmetric body in $\mathbb{R}^2$, with area bigger than $4$, containing two nonzero integer points.} \label{convex body}
\end{wrapfigure}
Minkowski initiated the field that we call today `the geometry of numbers', around 1890. To begin, we define a {\bf body} ${\mathcal P}$ in $\mathbb R^d$ as a compact set. In other words, ${\mathcal P}$ is a bounded, closed set. Most of the time, it is useful to work with convex bodies that enjoy the following symmetry.
We call a body ${\mathcal P}$ {\bf centrally symmetric}, also called {\bf symmetric about the origin}, if for all ${\bf x} \in \mathbb{R}^d$ we have
\begin{equation} \label{definition of symmetric body}
{\bf x} \in {\mathcal P} \iff -{\bf x} \in {\mathcal P}.
\end{equation}
\bigskip
A body ${\mathcal P}$ is called {\bf symmetric} if some translation of ${\mathcal P}$ is symmetric about the origin.
For example, the ball
$\{ x\in \mathbb{R}^d \mid \| x\| \leq 1\}$ is centrally symmetric, and the translated ball
$
\{ x\in \mathbb{R}^d \mid \| x- w\| \leq 1\}
$
is symmetric, but not centrally symmetric. An initial, motivating question in the geometry of numbers is:
\begin{question}\label{Rhetorical question, centrally symmetric}
{\rm[Rhetorical]} How large does a convex body ${\mathcal P}$ have to be in order to contain a nonzero integer point?
\end{question}
However, if we are not careful, then Figure \ref{nonexample}, for example, shows that ${\mathcal P}$ can be as large as we like, and yet never contain an integer point. So without further hypotheses, there are no positive answers to Question \ref{Rhetorical question, centrally symmetric}. Therefore, it is natural to assume that our body ${\mathcal P}$ is positioned in a `nice' way relative to the integer lattice, and centrally symmetry is a natural assumption in this respect.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{BodyCounterexample}
\end{center}
\caption{A convex symmetric body in $\mathbb{R}^2$, which is not centered at the origin, may be constructed with arbitrarily large volume and simultaneously with
no integer points.} \label{nonexample}
\end{figure}
\bigskip
\begin{thm}[Minkowski's convex body Theorem for $\mathbb{Z}^d$]
\label{Minkowski's convex body theorem, for Z^d}
Let $B$ be a $d$-dimensional convex body in $\mathbb R^d$, symmetric about the origin.
\begin{equation}
\text{ If } \vol B > 2^d, \text{ then } B
\text{ must contain a nonzero integer point in its interior}.
\end{equation}
\hfill $\square$
\end{thm}
Sometimes this classical and very useful result of Minkowski is stated in its contrapositive form:
Let $B \subset \mathbb{R}^d$ be any convex body, symmetric about the origin.
\begin{equation}
\text{ If the only integer point
in the interior of } B \text{ is the origin, then } \vol B \leq 2^d.
\end{equation}
It is natural, and straightforward, to extend this result to any lattice ${\mathcal L}:= M(\mathbb{Z}^d)$, by simply applying the linear transformation $M$ to both the integer lattice, and to the convex body $B$. The conclusion is the following, which is the version that we will prove as a consequence of Siegel's Theorem \ref{Siegel}.
\begin{thm}[Minkowski's convex body Theorem for a lattice ${\mathcal L}$]
\label{Minkowski convex body Theorem for L}
Let $B$ be a $d$-dimensional convex body in $\mathbb R^d$, symmetric about the origin, and let
${\mathcal L}$ be a (full rank) lattice in $\mathbb{R}^d$.
\begin{equation} \label{Minkowski 2}
\text{ If } \vol B > 2^d (\det {\mathcal L}), \text{ then } B
\text{ must contain a nonzero point of } {\mathcal L} \text{ in its interior}.
\end{equation}
\end{thm}
\begin{proof}
The proof appears below - see \eqref{ACTUAL proof of Minkowski's convex body theorem for lattices}.
\end{proof}
These very important initial results of Minkowski \cite{Minkowski} have found applications in algebraic number theory, diophantine analysis, combinatorial optimization, and other fields.
In the next section we show that Minkowski's result \eqref{Minkowski 2} follows as a special case of Siegel's formula.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.8in]{Rhombicdodecahedron}
\end{center}
\caption{The Rhombic dodecahedron, a $3$-dimensional symmetric polytope that tiles $\mathbb{R}^3$ by translations, and is another extremal body for Minkowski's convex body Theorem. }
\label{The Rhombic dodecahedron}
\end{figure}
\section{Siegel's extension of Minkowski: \\
a Fourier transform identity for convex bodies}
An important construction in the geometry of number is the {\bf Minkowski sum} of convex bodies. \index{Minkowski sum}
Given two convex bodies $K, L \subset \mathbb{R}^d$, their Minkowski sum is defined by
\[
K + L := \{ x + y \mid x \in K, y \in L\}.
\]
Another related construction, appearing in some of the results below, is
\[
K- L := \{ x-y \mid x \in K, y \in L\},
\]
the Minkowski difference of $K$ and $L$.
A very useful special case is the gadget known as the
{\bf Minkowski symmetrized body} of $K$, \index{symmetrized body}
defined by
\begin{equation}
\frac{1}{2} K - \frac{1}{2} K,
\end{equation}
and often also called the {\bf difference body} of $\frac{1}{2}K$.
Given any set $K\subset \mathbb{R}^d$, the difference body $K-K$ is centrally symmetric. To see this, suppose $x\in K-K$, so we may write
$x= y-z$, with $y, z \in K$. Then $-x = z-y \in K-K$.
In addition, we have the fortuitous and easy fact that a convex set $K\subset \mathbb{R}^d$ is centrally symmetric if and only if we have the equality
\begin{equation} \label{centrally symmetric set}
\frac{1}{2} K - \frac{1}{2} K=K.
\end{equation}
(Exercise \ref{c.s. C equals its symmetrized body}). Now suppose we are given two convex bodies $K, L\subset \mathbb{R}^d$. Then the resulting bodies $K+L$, $K-L$ turn out to also be convex (Exercise \ref{convexity of K-K}).
Another important geometric notion is the dilation of a convex body by a positive real number~$t$:
\[
tB := \{ t x \mid x\in B\},
\]
The most basic version of Siegel's theorem is the following identity, which assumes that a convex body $K$ is symmetric about the origin.
\bigskip
\begin{thm} [Siegel] \label{Siegel} \index{Siegel's formula}
Let $B$ be any $d$-dimensional convex body in $\mathbb R^d$, symmetric about the origin, and suppose that the only integer point in the interior of $B$ is the origin. Then
\begin{align} \label{Siegel, version 1}
2^d
&= \vol B + \frac{4^d}{ \vol B } \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} B}(\xi) \right|^2.
\end{align}
\hfill $\square$
\end{thm}
We now prove the following extension of Siegel's Theorem \eqref{Siegel}, namely \eqref{Siegel.formula} below, which applies to bodies that are not necessarily convex, nor necessarily symmetric about the origin.
Our proof of Theorem \ref{Siegel for general lattices} below
consists of yet another application of Poisson summation. \index{Poisson summation formula}
It turns out that if $K$ is any convex body, then $f:= 1_{\frac{1}{2} K}*1_{-\frac{1}{2} K}$ is a nice function (Exercise \ref{convolution of indicators is a nice function}),
in the sense that Poisson summation \eqref{nice functions} holds for $f$.
So Theorem \ref{Siegel} is a consequence of the following extension to bodies that are
not necessarily convex or symmetric.
\medskip
\begin{thm}[Siegel's formula, for a general body $K$, and a lattice ${\mathcal L}$]
\label{Siegel for general lattices}
\index{Siegel's formula}
Let $K\subset \mathbb{R}^d$ be a body (compact set) for which
the convolution $1_{\frac{1}{2} K}*1_{-\frac{1}{2} K}$ is a nice function.
If the only integer point in the interior of the difference body
$ \frac{1}{2}K - \frac{1}{2}K$ is the origin, then
\begin{equation}\label{Siegel.formula}
2^d
= \vol K + \frac{4^d}{ \vol K } \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{equation}
More generally, if we replace the lattice $\mathbb{Z}^d$ by any full-rank lattice ${\mathcal L}$, and assume that the only lattice point of ${\mathcal L}$ in the interior of $ \frac{1}{2}K - \frac{1}{2}K$ is the origin, then we have:
\begin{equation}\label{Siegel formula 2}
2^d \det {\mathcal L}
= \vol K + \frac{4^d}{ \vol K } \sum_{\xi \in {\mathcal L}^* - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{equation}
\end{thm}
\begin{proof}
We start with the function
\begin{equation}
f(x):= \left( 1_{\frac{1}{2} K}*1_{-\frac{1}{2} K} \right) (x),
\end{equation}
which is continuous on $\mathbb R^d$, and we plug $f$ into Poisson summation
\eqref{Poisson.summation2}:
\index{Poisson summation formula}
\begin{align}
\sum_{n \in \mathbb{Z}^d} f(n) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi).
\end{align}
We first compute the left-hand-side of Poisson summation, using the definition of $f$:
\begin{align}
\sum_{n \in \mathbb{Z}^d} f(n) &= \sum_{n \in \mathbb{Z}^d} \int_{\mathbb{R}^d} 1_{\frac{1}{2} K}(y) 1_{-\frac{1}{2} K}(n - y) dy \\
&= \sum_{n \in \mathbb{Z}^d} \int_{\mathbb{R}^d} 1_{\frac{1}{2} \rm{int} K}(y) 1_{-\frac{1}{2} \rm{int} K}(n - y) dy,
\end{align}
where the last step follows from the fact that the integral does not distinguish between a convex set or its closure.
Now we follow the definition of containment: $y \in \frac{1}{2} K$ and $n - y \in -\frac{1}{2} K$ imply that the integer point $n \in \frac{1}{2} K -\frac{1}{2} K$. But by hypothesis $ \frac{1}{2} K -\frac{1}{2} K$ contains the origin as its {\em only} interior integer point, so the left-hand-side of the
Poisson summation formula contains only one term, namely the $n=0$ term:
\begin{align}
\sum_{n \in \mathbb{Z}^d} f(n) &= \sum_{n \in \mathbb{Z}^d} \int_{\mathbb{R}^d} 1_{\frac{1}{2} K}(y) 1_{-\frac{1}{2} K}(n - y) dy \\
&= \int_{\mathbb{R}^d} 1_{\frac{1}{2} K}(y) 1_{-\frac{1}{2} K}(- y) dy \\
&= \int_{\mathbb{R}^d} 1_{\frac{1}{2} K}(y) dy \\
&= \vol \left( {\frac{1}{2} K} \right) = \frac{\vol K}{2^d}.
\end{align}
\noindent
On the other hand, the right-hand-side of Poisson summation gives us:
\begin{align}
\sum_{\xi \in \mathbb{Z}^d} \hat f(\xi)
&= \sum_{\xi \in \mathbb{Z}^d} {\hat 1}_{\frac{1}{2} K}(\xi) {\hat 1}_{-\frac{1}{2} K}(\xi) \\
&= \sum_{\xi \in \mathbb{Z}^d} \int_{\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx
\int_{-\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx \\
&= \sum_{\xi \in \mathbb{Z}^d} \int_{\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx
\int_{\frac{1}{2} K} e^{2\pi i \langle -\xi, x \rangle} dx \\
&= \sum_{\xi \in \mathbb{Z}^d} \int_{\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx
\ \overline{
\int_{\frac{1}{2} K} e^{2\pi i \langle \xi, x \rangle} dx
} \\ \label{pulling out the zero term}
&= \sum_{\xi \in \mathbb{Z}^d} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2 \\
&= \left| \hat 1_{\frac{1}{2} K}(0) \right|^2 + \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2 \\
&= \frac{\vol^2 K}{4^d} + \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2,
\end{align}
where we have pulled out the $\xi=0$ term from the series \eqref{pulling out the zero term}.
So we've arrived at
\begin{align*}
\frac{\vol K}{2^d}
&= \frac{\vol^2 K}{4^d} + \sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2,
\end{align*}
yielding the required identity:
\begin{align*}
2^d &= \vol K + \frac{4^d}{\vol K}\sum_{\xi \in \mathbb{Z}^d - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{align*}
Finally, to prove the stated extension to all lattices ${\mathcal L}$, we use the slightly more general
form of Poisson summation, Theorem \ref{The Poisson Summation Formula, for lattices},
valid for any lattice ${\mathcal L}$:
\begin{align}
\sum_{n \in {\mathcal L}} f(n) = \frac{1}{\det {\mathcal L}}\sum_{\xi \in {\mathcal L}^*} \hat f(\xi).
\end{align}
All the steps of the proof above are identical, except for the factor of $\frac{1}{\det {\mathcal L}}$, so that we arrive at
the required identity of Siegel for arbitrary lattices:
\begin{align}\label{Siegel, take 2}
\frac{\vol K}{2^d}
&= \frac{\vol^2 K}{4^d \det {\mathcal L}} + \frac{1}{\det {\mathcal L}}\sum_{\xi \in {\mathcal L}^* - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{align}
\end{proof}
The proof of Minkowski's convex body Theorem for lattices, namely Theorem
\ref{Minkowski convex body Theorem for L} above, now follows immediately.
\begin{proof}[Proof of Theorem \ref{Minkowski convex body Theorem for L}]
\rm{[Minkowski's convex body Theorem for a lattice} ${\mathcal L}$]
Applying Siegel's Theorem \ref{Siegel for general lattices} to the centrally symmetric body $B:=K$,
we see that the lattice sum on the right-hand-side of identity \eqref{Siegel.formula}
contains only non-negative terms. It follows that we immediately get the analogue of Minkowski's result for a given cenetrally symmetric body $B$ and a lattice ${\mathcal L}$, in its contrapositive form:
\begin{align} \label{ACTUAL proof of Minkowski's convex body theorem for lattices}
&\text{If the only lattice point of ${\mathcal L}$ in the interior of $B$ is the origin, then } 2^d \det {\mathcal L} \geq \vol B.
\end{align}
\end{proof}
In fact, we can easily extend Minkowski's Theorem \ref{Minkowski convex body Theorem for L}, using the same ideas of the latter proof, by using Siegel's Theorem \ref{Siegel for general lattices}
so that it applies to non-symmetric bodies as well (but there's a small `catch' - see Exercise \ref{Extending Minkowski to nonconvex bodies}).
\bigskip
\section{Tiling and multi-tiling Euclidean space by translations of polytopes}
\index{tiling}
First, we give a `spectral' equivalence for being able to tile Euclidean space by a single polytope, using only translations by a lattice. It will turn out that the case of equality in Minkowski's convex body Theorem is characterized precisely by the polytopes that tile $\mathbb{R}^d$ by translations. These bodies are called extremal bodies.
More generally, we would like to also consider the notion of multi-tiling, as follows. We say that a polytope
${\mathcal P}$ {\bf $k$-tiles $\mathbb{R}^d$ by using a set of translations ${\mathcal L}$} if
\begin{equation}
\sum_{n \in {\mathcal L}} 1_{{\mathcal P} + n}(x) = k,
\end{equation}
for all $x \in \mathbb{R}^d$, except those points $x$ that lie on the boundary of ${\mathcal P}$ or its translates under ${\mathcal L}$ (and of course these exceptions form a set of measure $0$ in $\mathbb{R}^d$). In other words,
${\mathcal P}$ is a $k$-tiling body if almost every $x \in \mathbb{R}^d$ is covered by exactly $k$ translates of ${\mathcal P}$.
Other synonyms for $k$-tilings in the literature are {\bf multi-tilings} of
$\mathbb{R}^d$, or {\bf tiling at level $k$}.
When ${\mathcal L}$ is a lattice, we will say that such a $k$-tiling is {\bf periodic}.
A common research theme is to search for tilings which are not necessarily periodic, but this is a difficult problem in general.
The classical notion of tiling, such that there are no overlaps between the interiors of any two tiles, corresponds here to the case $k=1$. We have the following dictionary between multi-tiling or Euclidean space by translations of a convex body ${\mathcal P}$, and a property of the zero set of the Fourier transform of ${\mathcal P}$, due to Kolountzakis \cite{Kolountzakis1}.
\bigskip
\begin{thm} \label{zero set of the FT of a polytope}
Suppose that ${\mathcal P}\subset \mathbb{R}^d$ is a compact set. The following two properties are equivalent:
\begin{enumerate}[(a)]
\item ${\mathcal P}$ $k$-tiles $\mathbb{R}^d$ by translations with a lattice ${\mathcal L}$.
\item
$\hat 1_{\mathcal P}(\xi) = 0$ for all nonzero $\xi \in {\mathcal L}^*$, the dual lattice.
\end{enumerate}
Either of these conditions also implies that $k = \frac{\vol {\mathcal P}}{ \det {\mathcal L}}$, an integer.
\end{thm}
\begin{proof}
We begin with the definition of multi-tiling, so that by assumption
\begin{equation} \label{by definition of k-tiling}
\sum_{n \in {\mathcal L}} 1_{{\mathcal P} + n}(x) = k,
\end{equation}
for all $x \in \mathbb{R}^d$ except those points $x$ that lie on the boundary of ${\mathcal P}$ or its translates under ${\mathcal L}$ (and of course these exceptions form a set of measure $0$ in $\mathbb{R}^d$). A trivial but useful
observation is that
\[
1_{{\mathcal P} + n}(x) =1 \iff 1_{{\mathcal P}}(x-n) =1,
\]
so we can rewrite the defining identity \eqref{by definition of k-tiling} as $\sum_{n \in {\mathcal L}} 1_{{\mathcal P}}(x-n) = k$. Now we notice that the left-hand-side is a periodic function of $x$, namely
\[
F(x) := \sum_{n \in {\mathcal L}} 1_{{\mathcal P}}(x-n)
\]
is periodic in $x$ with ${\mathcal L}$ as its set of periods. This is easy to see: if we let $l \in {\mathcal L}$, then $F(x + l) = \sum_{n \in {\mathcal L}} 1_{{\mathcal P}}(x+ l -n) = \sum_{m \in {\mathcal L}} 1_{{\mathcal P}}(x+ m) = F(x)$, because the lattice ${\mathcal L}$ is invariant under a translation by any vector that belongs to it.
The following `intuitive proof' would in fact be rigorous if we were allowed to use `generalized functions',
but since we do not use them in this book, we label this part of the proof as `intuitive', and we then give a rigorous proof, using functions rather than generalized functions.
[{\bf Intuitive proof}] \ By Theorem \ref{Fourier series for periodic functions}, we may expand $F$ into its Fourier series, because it is a periodic function on $\mathbb{R}^d$. Now by Poisson summation, namely
Theorem \ref{The Poisson Summation Formula, for lattices}, we know that its Fourier coefficients are the following:
\begin{equation}\label{Poisson version of k-tiling}
\sum_{m \in {\mathcal L}} 1_{{\mathcal P}}(x+m) = \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*} \hat 1_{{\mathcal P}}(\xi) e^{2\pi i \langle \xi, x \rangle},
\end{equation}
If we now make the assumption that $\hat 1_{{\mathcal P}}(\xi)=0$ for all nonzero $\xi \in {\mathcal L}^*$, then by \eqref{Poisson version of k-tiling} this assumption is equivalent to
\[
\sum_{m \in {\mathcal L}} 1_{{\mathcal P}}(x+m) = \frac{\hat 1_{\mathcal P}(0)}{\det {\mathcal L}} = \frac{\vol {\mathcal P}}{\det {\mathcal L}}.
\]
This relation means that we have a $k$-tiling, where $k:= \frac{\vol {\mathcal P}}{\det {\mathcal L}}$.
Now we replace the intuitive portion of the proof with a rigorous proof.
[{\bf Rigorous proof}] \ In order to apply Poisson summation, \index{Poisson summation formula}
it is technically necessary to replace $1_P(x)$ by
a smoothed version of it, in \eqref{Poisson version of k-tiling}. Because this process is so common and useful in applications, this proof is instructive.
We pick an approximate identity $\phi_n$, which is also compactly supported and continuous. Applying the Poisson summation formula of Theorem \ref{PracticalPoisson} to the smoothed function $1_P*\phi_n$, we get:
\begin{align}\label{rigorous limit for multi-tiling}
\sum_{m \in {\mathcal L}} \left( 1_{\mathcal P}*\phi_n \right)(x+m) &= \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*} \hat 1_{{\mathcal P}}(\xi) \hat \phi_n(\xi) e^{2\pi i \langle \xi, x \rangle}.
\end{align}
Using the fact that the convolution of two compactly supported functions is itself compactly supported, we see that $1_{\mathcal P}*\phi_n $ is again compactly supported. Thus the sum on the LHS of \eqref{rigorous limit for multi-tiling} is a finite sum.
Performing a separate computation, we take the limit as $n\rightarrow \infty$ inside this finite sum, and using
Theorem \ref{approximate identity convolution} (due to the continuity of $1_{\mathcal P}*\phi_n$), we obtain
\[
\lim_{n\rightarrow \infty} \sum_{m \in {\mathcal L}} \left( 1_{\mathcal P}*\phi_n \right)(x+m)
= \sum_{m \in {\mathcal L}} \lim_{n\rightarrow \infty} \left( 1_{\mathcal P}*\phi_n \right)(x+m)
=\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m),
\]
and moreover by \eqref{practical Poisson summation, first version}, we have
\begin{align} \label{smoothed-out RHS}
\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) = \frac{1}{\det {\mathcal L}}
\sum_{\xi \in {\mathcal L}^*} \hat 1_{{\mathcal P}}(\xi) \hat \phi_n(\xi)
\, e^{2\pi i \langle \xi, x \rangle}.
\end{align}
for all sufficiently large values of $n$. Separating the term $\xi=0$ on the RHS of this Poisson summation formula, we have:
\begin{align}
\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) &= \frac{\hat 1_{\mathcal P}(0)}{\det {\mathcal L}}
+ \sum_{\xi \in {\mathcal L}^*-\{0\}} \hat 1_{{\mathcal P}}(\xi) \hat \phi_n(\xi)
\, e^{2\pi i \langle \xi, x \rangle}\\ \label{equivalent condition for tiling}
&=
\frac{\vol {\mathcal P}}{\det {\mathcal L}}
+ \sum_{\xi \in {\mathcal L}^*-\{0\}} \hat 1_{{\mathcal P}}(\xi) \hat \phi_n(\xi)
\, e^{2\pi i \langle \xi, x \rangle}.
\end{align}
Now, $\hat 1_{{\mathcal P}}(\xi) =0$ for all $\xi \in {\mathcal L}^*-\{0\}$ in
\eqref{equivalent condition for tiling} will hold
\begin{align*}
&\iff
\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) = \frac{\vol {\mathcal P}}{\det {\mathcal L}},
\end{align*}
an equivalent condition which we may write as $\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) = k$, where necessarily
$k:= \frac{\vol {\mathcal P}}{\det {\mathcal L}}$. The condition $\sum_{m \in {\mathcal L}} 1_{\mathcal P}(x+m) = k$ means that ${\mathcal P}$ $k$-tiles
$\mathbb{R}^d$ by translations with the lattice ${\mathcal L}$, and also implies that $k$ must be an integer.
\end{proof}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.7in]{ExtremalBody}
\end{center}
\caption{An extremal body in $\mathbb{R}^2$, relative to the integer lattice, which is a hexagon. It has
area $4$, and no integer points in its interior. We also get a $2$-parameter family of such extremal bodies, parametrized by the point $p\in \mathbb{R}^2$ in the figure. It is clear from the picture
that this family of extremal bodies consists of either symmetric hexagons, or symmetric quadrilaterals.}
\label{Extremal body in R^2}
\end{figure}
In 1905, Minkowski gave necessary conditions for a polytope ${\mathcal P}$ to tile $\mathbb{R}^d$ by translations. Later, Venkov and independently McMullen found sufficient conditions as well, culminating in the following fundamental result.
\begin{thm}[Minkowski-Venkov-McMullen] \label{Minkowski-Venkov-McMullen}
A polytope ${\mathcal P}$ tiles $\mathbb{R}^d$ by translations if and only if the following $3$ conditions hold:
\begin{enumerate}
\item ${\mathcal P}$ is a symmetric polytope.
\item The facets of ${\mathcal P}$ are symmetric polytopes.
\item Fix any face $F\subset {\mathcal P}$ of codimension $2$, and project ${\mathcal P}$ onto the $2$-dimensional plane that is orthogonal to the $(d-2)$-dimensional affine span of $F$. Then this projection is either a parallelogram, or a centrally symmetric hexagon.
\end{enumerate}
\end{thm}
\bigskip
\section{Extremal bodies} \label{extremal bodies}
\index{extremal body}
An {\bf extremal body} is a convex, symmetric body $K$ for which we have equality in Minkowski's convex body Theorem:
\[
\vol K = 2^d(\det {\mathcal L}).
\]
If we just look at equation \ref{Siegel.formula} a bit more closely, we quickly get a nice corollary that
arises by combining Theorem \ref{zero set of the FT of a polytope} and Siegel's Theorem \ref{Siegel}.
Namely, equality occurs in Minkowski's convex body theorem if and only if
$K$ tiles \index{tiling} $\mathbb{R}^d$ by translations.
Let's prove this.
\begin{thm}[Extremal bodies] \label{thm:extremal bodies}
Let $K$ be any convex, centrally symmetric subset of $\mathbb{R}^d$, and fix a full-rank lattice ${\mathcal L} \subset \mathbb{R}^d$.
Suppose that the only point of ${\mathcal L}$ in the interior of $K$ is the origin.
Then:
\begin{quote}
$2^d \det {\mathcal L}= \vol K \ \iff \ \frac{1}{2}K$ tiles $\mathbb{R}^d$ by translations with the lattice ${\mathcal L}$.
\end{quote}
\end{thm}
\begin{proof}
By Siegel's formula \eqref{Siegel formula 2}, we have
\begin{equation}
2^d \det {\mathcal L}
= \vol K + \frac{4^d}{ \vol K } \sum_{\xi \in {\mathcal L}^* - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2.
\end{equation}
Therefore, the assumption $2^d \det {\mathcal L} = \vol K$ holds $\iff$
\begin{equation}
0 = \frac{4^d}{ \vol K } \sum_{\xi \in {\mathcal L}^* - \{0\}} \left| \hat 1_{\frac{1}{2} K}(\xi) \right|^2,
\end{equation}
$\iff$ all of the non-negative summands $ \hat 1_{\frac{1}{2} K}(\xi) =0$, for all nonzero $\xi \in {\mathcal L}^*$. Now we would like to
use Theorem \ref{zero set of the FT of a polytope} to show the required tiling equivalence, namely that
$\frac{1}{2} K$ tiles $\mathbb{R}^d$ by translations with the lattice ${\mathcal L}$. We have already verified condition (a) of Theorem \ref{zero set of the FT of a polytope}, applied to the body $\frac{1}{2} K$, namely that
$\hat 1_{\frac{1}{2} K}(\xi) =0$, for all nonzero $\xi \in {\mathcal L}^*$.
To verify condition (b) of Theorem
\ref{zero set of the FT of a polytope}, we notice that because $\vol\left( \frac{1}{2} K \right)= \frac{1}{2^d}\vol K$, it follows that
$2^d \det {\mathcal L}= \vol K$ is equivalent to
$1 = \frac{ \vol \left( \frac{1}{2} K \right)}{ \det {\mathcal L}}$, so that we may apply Theorem
\ref{zero set of the FT of a polytope} with ${\mathcal P}:= \frac{1}{2} K$, and with the multiplicity $k:=1$.
\end{proof}
\medskip
There is an extension of theorem \ref{Minkowski-Venkov-McMullen}, the Minkowski-Venkov-McMullen result, to multi-tilings.
\begin{thm}\cite{GravinShiryaevRobins} \label{k-tiling theorem, GravinShiryaevRobins}
If a polytope ${\mathcal P}$ multi-tiles $\mathbb{R}^d$ by translations with a discrete set of vectors, then
\begin{enumerate}
\item ${\mathcal P}$ is a symmetric polytope.
\item The facets of ${\mathcal P}$ are symmetric polytopes.
\end{enumerate}
\end{thm}
In the case that ${\mathcal P}\subset \mathbb{R}^d$ is a rational polytope, meaning that all the vertices of ${\mathcal P}$ have rational coordinates, the latter two necessary conditions for multi-tiling become sufficient conditions as well \cite{GravinShiryaevRobins}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{TruncatedOctahedron}
\end{center}
\caption{The truncated Octahedron, one of the $3$-dimensional polytopes that tiles $\mathbb{R}^3$ by translations. }
\label{TruncatedOctahedron}
\end{figure}
\section{More about centrally symmetric polytopes} \label{Centrally symmetric polytopes}
\index{centrally symmetric polytope}
It's both fun and instructive to begin by seeing how very simple Fourier methods can give us deeper insight into the geometry of symmetric polytopes. The reader may glance at the definitions above, in
\eqref{definition of symmetric body}.
\bigskip
\begin{example}
\rm{
Consider the cross-polytope $\Diamond \subset \mathbb{R}^3$,
\index{cross-polytope}
defined in Chapter \ref{Chapter.Examples}. This is a centrally symmetric polytope, but each of its facets is {\em not} a symmetric polytope, because its facets are triangles.
}
\hfill $\square$
\end{example}
If \emph{all} of the $k$-dimensional faces of a polytope ${\mathcal P}$ are symmetric, for each $1\leq k \leq d$,
then ${\mathcal P}$ is called a {\bf zonotope}.
\index{zonotope}
Zonotopes form an extremely important class of polytopes, and have various equivalent formulations.
\begin{lem}
A polytope ${\mathcal P} \subset \mathbb{R}^d$ is a zonotope $\iff$ ${\mathcal P}$ has one of the following properties.
\begin{enumerate}[(a)]
\item ${\mathcal P}$ is a projection of some $n$-dimensional cube.
\item ${\mathcal P}$ is the Minkowski sum of a finite number of line segments.
\end{enumerate}
\end{lem}
A projection here means any affine transformation of ${\mathcal P}$, where the rank of the associated matrix may be less than $d$.
Zonotopes
have been very useful in the study of tilings (\cite{Ziegler}, \cite{BeckRobins}).
\index{tiling}
For instance, in dimension $3$, the only polytopes that tile $\mathbb{R}^3$ by translations with a lattice are zonotopes, and there is a list of $5$ of them (up to an isomorphism of their face posets), called the {\bf Fedorov solids}, and drawn in Figure \ref{Fedorov solids} (also see our Note \ref{Fedorov Note} below). \index{Fedorov solids}
By definition, any zonotope is a symmetric polytope, but the converse is not true; for example,
the cross-polytope
\index{cross-polytope}
is symmetric, but it has triangular faces, which are not symmetric, so the crosspolytope is not a zonotope.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{rhombic}
\end{center}
\caption{A $3$-dimensional zonotope, called the rhombic dodecahedron, showing in bold its $4$ line segments whose Minkowski sum generate the object. }
\label{first 3d zonotope pic}
\end{figure}
\index{Minkowski sum}
\bigskip
\begin{example}
\rm{
Consider the following $3$ line segments in $\mathbb{R}^2$:
$\conv\{ \icol{0\{\bf 0}}, \icol{1 \{\bf 0}} \}, \conv\{ \icol{0\{\bf 0}}, \icol{2 \{\bf 1}} \}$, and
$\conv\{ \icol{0\{\bf 0}}, \icol{1 \\ 3} \}$.
The Minkowski sum of these three line segments, by definition a zonotope in $\mathbb{R}^2$, is
the symmetric hexagon whose vertices are $\icol{0\{\bf 0}}, \icol{1 \{\bf 0}}, \icol{2 \{\bf 1}}, \icol{3 \\3}, \icol{3 \{\bf 1}}, \icol{4 \\3}$.
Notice that once we graph it, in Figure \ref{a zonotope}, the graph is hinting to us
that this body is a projection of a $3$-dimensional cube, and indeed this turns
out to be always true for Minkowski sums of line segments.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.4in]{zonotope1}
\end{center}
\caption{The Minkowski sum of $3$ line segments in the plane, forming a $2$-dimensional zonotope. }
\label{a zonotope}
\end{figure}
\index{Minkowski sum}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.1in]{Fedorov}
\end{center}
\caption{The Fedorov solids, the only $3$-dimensional polytopes that tile $\mathbb{R}^3$ by translations. All $5$ of them are zonotopes, and they are also extreme bodies for Minkowski's convex body theorem.
The top three, from left to right, are: the Truncated octahedron, the Rhombic dodecahedron, and the Hexarhombic dodecahedron. The bottom two are the cube and the hexagonal prism. }
\label{Fedorov solids} \index{Fedorov solids}
\end{figure}
\begin{example} \label{truncated octahedron}
\rm{
A particular embedding of the truncated octahedron ${\mathcal P}$, drawn in Figure \ref{TruncatedOctahedron},
is given by the convex hull of the set of $24$ vertices defined by all permutations of $(0, \pm 1, \pm 2)$.
We note that this set of vertices can also be thought of as the orbit of just the one point $(0, 1, 2)\in \mathbb{R}^3$ under the hyperoctahedral group (see
\cite{BillChen} for more on the hyperoctahedral group). It turns out that this truncated octahedron ${\mathcal P}$ tiles $\mathbb{R}^3$ by translations with a lattice
(Exercise \ref{tiling using the truncated octrahedron}).
}
\hfill $\square$
\end{example}
As the following Lemma shows, it is easy to detect/prove whether or not $S$ is centrally symmetric by just observing whether or not its Fourier transform is real-valued.
To make the proof go through more easily, we will assume that $\hat 1_S$ is absolutely integrable, so that the usual inverse Fourier transform applies, and we call such a set admissible. But the curious reader might consider extensions to more general sets.
\bigskip
\begin{lem}\label{symmetric iff FT is real}
An admissible set $S \subset \mathbb{R}^d$ is symmetric about the origin $\iff$
\[
\hat 1_{S}(\xi) \in \mathbb{R},
\]
\text{ for all } $\xi \in \mathbb{R}^d$.
\end{lem}
\begin{proof}
Suppose that the set $S$ is centrally symmetric. Then we have
\begin{align}
\overline{ \hat 1_S(\xi)} := \overline{ \int_{S} e^{2\pi i \langle \xi, x \rangle} dx}
&= \int_{S} e^{-2\pi i \langle \xi, x \rangle} dx \\
&= \int_{-S} e^{2\pi i \langle \xi, x \rangle} dx \\
&= \int_{S} e^{2\pi i \langle \xi, x \rangle} dx := \hat 1_S(\xi), \\
\end{align}
showing that the complex conjugate of $\hat 1_S$ is itself, hence that it is real-valued.
Conversely, suppose that $\hat 1_{S}(\xi) \in \mathbb{R}$, for all $\xi \in \mathbb{R}^d$. We use the fact that the Fourier transform $\hat 1_S$ is invertible, so that by Theorem \ref{thm:Inverse Fourier transform} we have:
\begin{equation} \label{Fourier inversion of indicator}
({\mathcal F} \circ {\mathcal F})(1_S)(x) = 1_S(-x),
\end{equation}
for all $x \in \mathbb{R}^d$.
To show that $S$ is centrally symmetric, we need to show that $1_{-S}(x) = 1_{S}(x)$, for all $x \in \mathbb{R}^d$. Further, by \ref{Fourier inversion of indicator}, it now suffices to show that
$\hat 1_{-S}(\xi) = \hat 1_{S}(\xi)$, for all $\xi \in \mathbb{R}^d$.
We therefore compute:
\begin{align}
\hat 1_{-S}(\xi) := \int_{-S} e^{2\pi i \langle \xi, x \rangle} dx
&= \int_{S} e^{-2\pi i \langle \xi, x \rangle} dx \\
&= \overline{ \int_{S} e^{2\pi i \langle \xi, y \rangle} dy } \\
&:= \overline{ \hat 1_S(\xi) } \\
&= \hat 1_S(\xi),
\end{align}
for all $\xi \in \mathbb{R}^d$, where we have used the assumption that $ \hat 1_S(\xi)$ is real-valued in the last equality.
\end{proof}
\bigskip
\begin{example} \rm{
The interval ${\mathcal P}:= [-\frac{1}{2}, \frac{1}{2}]$ is a symmetric polytope, and indeed we can see that
its Fourier transform
$\hat 1_{\mathcal P}(\xi)$ is real-valued, namely we have $\hat 1_{\mathcal P}(\xi) = {\rm{sinc}}(\xi)$, as we saw in equation
\eqref{SincFunction}.}
\hfill $\square$
\end{example}
\bigskip
\begin{example} \rm{
The cross-polytope $\Diamond_2$ is a symmetric polytope, and as we verified in dimension $2$,
equation \eqref {Fourier transform of 2d crosspolytope},
its Fourier transform
$1_{\Diamond_2}(\xi)$ is real-valued. }
\hfill $\square$
\end{example}
Alexandrov \cite{Alexandrov}, and independently Shephard \cite{ShephardSymmetricPolytopes}, proved the following remarkable fact.
\begin{thm}[Alexandrov and Shephard] \label{cs1} \index{Alexandrov, A. D. } \index{Shephard}
\label{Alexandrov-Shepard thm}
Let $P$ be any real, $d$-dimensional polytope, with $d \geq 3$. If all of the facets of $P$
are centrally symmetric, then $P$ is centrally symmetric.
\hfill $\square$
\end{thm}
\begin{example}
\rm{
The converse to the latter result is clearly false, as demonstrated by the cross-polytope in dimension $d > 2$: it is centrally symmetric, but its facets are not symmetric because they are simplices and we know that
no simplex (of dimension $\geq 2$) is symmetric
(Exercise \ref{no simplex is symmetric}).
}
\hfill $\square$
\end{example}
\begin{wrapfigure}{R}{0.49\textwidth}
\centering
\includegraphics[width=0.20\textwidth]{zonotope2}
\caption{A $3$-dimensional zonotope that does not tile $\mathbb{R}^3$ by translations. }
\label{complex zonotope}
\end{wrapfigure}
Suppose we consider $3$-dimensional polytopes ${\mathcal P}$, and ask which ones enjoy the property that all of their $2$-dimensional faces are symmetric? Because $1$-dimensional faces are always symmetric, and because Theorem \ref{Alexandrov-Shepard thm} tells us that ${\mathcal P}$ itself must also be symmetric, the answer is that ${\mathcal P}$ must be a zonotope - in other words all of its faces are symmetric.
Moving up to $4$-dimensional polytopes, our curiosity might take the next step:
which $4$-dimensional polytopes enjoy the property that all of their $3$-dimensional faces are symmetric? Must they also be zonotopes? The $24$-cell is a good counterexample, because it has triangular $2$-dimensional faces, and hence is not a zonotope. On the other hand, the $24$-cell tiles $\mathbb{R}^4$ by translations with a lattice (it is the Voronoi cell of the D$4$ lattice), and therefore by Theorem \ref{Minkowski-Venkov-McMullen} its $3$-dimensional faces must be symmetric.
What if we ask which $4$-dimensional polytopes enjoy the property that all of their $2$-dimensional faces are symmetric?
Peter McMullen \cite{McMullen4}
discovered the wonderful conclusion that all of their faces must be symmetric - in other words they must be zonotopes - and that much more is true.
\begin{thm}[McMullen] \label{McMullen extension to Alexandrov}
Let $P$ be any real, $d$-dimensional polytope, with $d \geq 4$.
Fix any positive integer $k$ with
$2 \leq k \leq d-2$.
If the $k$-dimensional faces of ${\mathcal P}$ are symmetric, then ${\mathcal P}$ is a zonotope.
\hfill $\square$
\end{thm}
\bigskip
One might wonder what happens if we `discretize the volume' of a symmetric body $K$, by counting integer points, and then ask for an analogue of Minkowski Theorem \ref{Minkowski's convex body theorem, for Z^d}.
In fact,
Minkowski already had a result about this too
(and he had so many beautiful ideas that it's hard to put them all in one place!). We give Minkowski's own elegant and short proof.
\begin{thm}[Minkowski, 1910] \label{Minkowski's 3^d theorem}
Let $K\subset \mathbb{R}^d$ be any $d$-dimensional, convex, centrally symmetric set.
If the only integer point in the interior of $K$ is the origin, then
\begin{equation} \label{Minkowski, $3^d$}
\left | K \cap \mathbb{Z}^d \right | \leq 3^d.
\end{equation}
\end{thm}
\begin{proof}
We define the map $\phi: \mathbb{Z}^d \rightarrow \left( \mathbb{Z}/3\mathbb{Z} \right)^d$, by reducing each coordinate modulo $3$.
Now we claim that when restricted to the set $K\cap \mathbb{Z}^d$, our map $\phi$ is $1-1$. The statement of the theorem follows directly from this claim. So let $x,y \in K\cap \mathbb{Z}^d$, and suppose $\phi(x) = \phi(y)$. Then, by definition of the map $\phi$, we have
\begin{equation} \label{z point}
n:= \frac{1}{3}(x-y) \in \mathbb{Z}^d,
\end{equation}
Now we define $C$ to be the {\bf interior} of the convex hull of $x, -y$, and $0$. Because $K$ is symmetric, and $x, y\in K$, we know that $-y \in K$ as well, so that $C \subset \text{int}(K)$. Now using the convexity of $C$, we also see that $n \in C$, because $n$ is a non-trivial convex linear combination of $0, x, -y$.
Therefore $n \in \text{int}(K)$ as well. Altogether, $n \in \text{int}(K) \cap \mathbb{Z}^d = \{0\}$, which forces $n =0$.
Hence $x-y=0$.
\end{proof}
Theorem \ref{Minkowski's 3^d theorem} is often called { \bf Minkowski's $3^d$ theorem}. \index{Minkowski's $3^d$ theorem}
An immediate and natural question is: which bodies account for the `equality case'? One direction is easy to see: if $K$ is the integer cube $[-1, 1]^d$, then it is clear that $K$ is symmetric about the origin, and the only integer point in its interior is the origin.
In addition, $\vol K = 2^d$, and $K$ contains precisely $3^d$ integer points. It is a bit surprising, perhaps, that only in 2012 was it proved that this integer cube is the only case of equality in Minkowski's $3^d$ theorem \cite{DraismaMcAllisterNill}.
In a different direction, it turns out that the volume of the difference body
\index{symmetrized body}
$\frac{1}{2} K - \frac{1}{2} K$, which appeared quite naturally in some of the proofs above, can be related in a rather precise manner to the volume of $K$ itself.
The consequence is the following inequality, known as the {\bf Rogers-Shephard inequality}
\cite{RogersShephard},
\begin{equation} \label{Rogers-Shephard inequality}
\vol K \leq \vol \left( \frac{1}{2} K - \frac{1}{2} K \right) \leq {2d \choose d} \vol K,
\end{equation}
where equality on the left holds $\iff$ $K$ is a symmetric body, and equality on the right
holds $\iff$ $K$ is a simplex (see Cassels \cite{CasselsBook}). There is also an extension of the Rogers-Shephard inequality to two distinct convex bodies $K, L\subset \mathbb{R}^d$:
\begin{equation}
\vol \left( K - L \right) \vol \left( K \cap L \right) \leq {2d \choose d} \vol K \vol L.
\end{equation}
(\cite{RogersShephard} and \cite{Gutierrez.Jimenez.Villa}).
A quick way of proving \eqref{Rogers-Shephard inequality} is by using the ubiquitous
{\bf Brunn-Minkowski inequality} (\cite{Schneider.book}, section $7.1$)
\index{Brunn-Minkowski inequality}
which tells us the following. Two sets $A, B\subset \mathbb{R}^d$ are called homothetic if $A = \lambda B + v$, for some
fixed $v\in \mathbb{R}^d$, and some $\lambda >0$ (or either $A$ or $B$ consist of just one point).
\begin{thm}
If $K$ and $L$ are convex subsets of $\mathbb{R}^d$, then
\begin{equation}
\vol(K+L)^\frac{1}{d} \geq \vol(K)^\frac{1}{d} + \vol(L)^\frac{1}{d},
\end{equation}
with equality if and only if $K$ and $L$ lie in parallel hyperplanes or are homothetic to each other.
\hfill $\square$
\end{thm}
\bigskip
\section*{Notes}
\begin{enumerate}[(a)]
\item Siegel's original proof of Theorem \ref{Siegel} used Parseval's identity, but the spirit of the two proofs is similar.
\item In Exercise \ref{equivalent statements for unimodular triangles} below, we see three equivalent conditions for a $2$-simplex to be unimodular. In higher dimensions, a $d$-simplex will not satisfy all
three conditions, and hence this exercise shows one important `breaking point' between $2$-dimensional and $3$-dimensional discrete geometry.
\item \label{new books, geometry of numbers}
There are a growing number of interesting books on the geometry of numbers. One encyclopedic text
that contains many other connections to the geometry of numbers is Peter Gruber's book \cite{GruberBook}.
Two other excellent and classic introductions are Siegel's book \cite{SiegelBook}, and Cassels' book \cite{CasselsBook}. An expository introduction to some of the elements of the Geometry of numbers, at a level that is even appropriate for high school students, is given by Olds, Lax, and Davidoff \cite{OldsBook}.
For upcoming books, the reader may also consult Martin Henk's lecture notes `Introduction to geometry of numbers' \cite{Henk3}, and
the book by Lenny Fukshansky and Stephan Ramon Garcia, `Geometry of Numbers' \cite{FukshanskyBook}.
\item \label{Brunn-Minkowski} The Brunn-Minkowski inequality is fundamental to many branches of mathematics, including the geometry of numbers. A wonderful and encyclopedic treatment of the
Brunn-Minkowski inequality, with its many interconnections, appears in \cite{Schneider.book}.
\item \label{Fedorov Note}
The Fedorov solids are depicted, and explained via the modern ideas of Conway and Sloan, in an excellent
expository article by David Austin \cite{DavidAustin}.
For a view into the life and work of Evgraf Stepanovich Fedorov, as well as
a fascinating account of how Fedorov himself thought about the $5$ parallelohedra,
the reader may
consult the article by Marjorie Senechal and R. V. Galiulin \cite{SenechalGaliulin}. The authors of
\cite{SenechalGaliulin} also discuss the original book of Fedorov, called
\emph{An Introduction to the Theory of Figures}, published in 1885, which is now considered a pinnacle of modern crystallography.
Fedorov later became one of the great crystallographers of his time.
In $\mathbb{R}^4$, it is known that there are $52$ different combinatorial types of $4$-dimensional parallelohedra. In $\mathbb{R}^5$, the complete classification of all the combinatorial types of
$5$-dimensional paralellohedra was completed in 2016 \cite{Dutour et al.parallelohedra}, where the authors found $110, 244$ of them.
\item The field of multi-tiling is still growing. One of the first important papers in this field was by Mihalis Koloutzakis \cite{Kolountzakis1}, who related the multi-tiling problem to a famous technique known as the idempotent theorem, and thereby proved
that if we have a multi-tiling in $\mathbb{R}^2$ with any discrete set of translations, then we also have a multi-tiling with a finite union of lattices.
A recent advance is an equivalence between multi-tiling and
certain Hadwiger-type invariants, given by Nir Lev and Bochen Liu \cite{LevLiu}. Here the authors show as well that
for a generalized polytope ${\mathcal P} \subset \mathbb{R}^d$ (not necessarily convex or connected), if ${\mathcal P}$ is spectral, then ${\mathcal P}$
is equidecomposable by translations to a cube of equal volume.
Another natural question in multi-tiling, which is still open, is the following:
\begin{question} \label{multi-tiling - what is the discrete set of translations}
Suppose that ${\mathcal P}$ multi-tiles with a discrete set of translations $D$. Do we really need the set $D$ of translates of ${\mathcal P}$ to be a very complicated discrete set,
or is it true that just a finite union of lattices suffices?
Even better, perhaps one lattice always suffices?
\end{question}
In this direction, Liu proved recently that if we assume that ${\mathcal P}$ multi-tiles with a finite union of lattice, then ${\mathcal P}$ also multi-tiles with a single lattice \cite{Liu}. This is big step in the direction of answering Question
\ref{multi-tiling - what is the discrete set of translations} in general. An earlier, and smaller step, was taken in \cite{GravinKolountzakisRobinsShiryaev}, where the authors answered part of Question \ref{multi-tiling - what is the discrete set of translations} in $\mathbb{R}^3$, reducing the search from an arbitrary discrete set of translations, to translations by a finite union of lattices. Taken together, the latter two steps imply that in $\mathbb{R}^3$ (and in $\mathbb{R}^2$), any multi-tiling with a discrete set of translations also occurs with just a one lattice.
In a different direction, the work of Gennadiy Averkov \cite{Averkov} analyzes the equality cases for an extension of Minkowski's theorem, relating those extremal bodies to multi-tilers.
In \cite{YangZong}, Qi Yang and Chuanming Zong show that
the smallest $k$ for which we can obtain a nontrivial $k$-tiling in $\mathbb{R}^2$ is $k=5$, and the authors characterize
those $5$-tiling bodies, showing in particular that if a convex polygon is a $5$-tiler, then it must be either an octagon, or a decagon.
\begin{question}\label{smallest k}
In $\mathbb{R}^d$, what is the smallest integer $k$ such that there exists a $d$-dimensional polytope
${\mathcal P}$ that $k$-tiles $\mathbb{R}^d$ by translations?
\end{question}
\item \label{Fuglede conjecture}
We say that a body ${\mathcal P}$ (any compact subset of $\mathbb{R}^d$)
is `spectral' if the function space $L^2({\mathcal P})$ possesses an orthonormal, complete basis of exponentials.
There is a fascinating and vast literature about such spectral bodies, relating them to tiling,
\index{tiling}
and multi-tiling problems.
One of the most interesting and natural questions in this direction is the following conjecture, by Bent Fuglede \cite{Fuglede74}.
The Fuglede conjecture asks whether the following is true.
\begin{question}\label{Fuglede}
${\mathcal P}$ tiles $\mathbb{R}^d$ by translations $\iff$ ${\mathcal P}$ is spectral?
\end{question}
Terry Tao disproved the Fuglede conjecture for some nonconvex bodies, but in the case that ${\mathcal P}$ is convex one might hope that more is true. Indeed, in 2003
Alex Iosevich, Nets Katz, and Terry Tao \cite{IosevichKatzTao} proved that
the Fuglede conjecture is true for all convex domains in $\mathbb{R}^2$.
In 2021, this conjecture was proved for all convex domains (which must necessarily be polytopes by
an additional simple argument), in the work of Nir Lev and M\'at\'e Matolcsi \cite{LevMatolcsi}.
In a related direction, Sigrid Grepstad and Nir Lev \cite{GrepstadLev} showed that for any bounded, measurable subset $S\subset \mathbb{R}^d$, if $S$ multi-tiles by translations with a discrete set, then
$S$ has a Riesz basis of exponentials.
\item We have seen
that the zero set of the Fourier transform of a polytope is very important, in that Theorem \ref{zero set of the FT of a polytope}
gave us a necessary and sufficient condition for multi-tiling. But the zero set of the FT also gives more information, and an interesting application of the information content in the zero set is the Pompeiu problem. \index{Pompeiu problem}
The Pompeiu problem is an ancient problem (defined in 1929 by Pompeiu) that asks the following:
which bodies ${\mathcal P} \in \mathbb{R}^d$ are uniquely characterized by the collection of their integrals over ${\mathcal P}$, and over all rigid motions of ${\mathcal P}$?
An equivalent formulation is the following.
\begin{question} \label{Pompeiu conjecture}
Given a body ${\mathcal P}$ with nonempty interior, does there exist a nonzero continuous function $f$ that allows for the
the vanishing of all of the integrals
\begin{equation}\label{Pompeiu question}
\int_{M({\mathcal P})} f(x) dx = 0,
\end{equation}
taken over all rigid motions $M$, including translations?
\end{question}
A body ${\mathcal P} \subset \mathbb{R}^d$, for which the answer to the question above is affirmative, is said to have the Pompeiu property.
Even for convex bodies ${\mathcal P}$, it is still an open problem in general dimension whether
${\mathcal P}$ has the Pompeiu property.
It is known, by the work of Brown, Schreiber, and Taylor \cite{BrownSchreiberTaylor} that ${\mathcal P}$ has the Pompeiu property $\iff$
the collection of Fourier transforms $\hat 1_{\sigma({\mathcal P})}(z)$, taken over all rigid motions
$\sigma$ of $\mathbb{R}^d$,
have a common zero $z$. It was also known that all polytopes have the Pompeiu property.
Recently, in \cite{FabricioSinai1}, Fabricio Machado and SR showed that the zero set of the FT does not contain (almost all) circles, and as a consequence we get a simple new proof that all polytopes have the `Pompeiu property'.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
``Every problem has a creative solution''.
-- Folklore
\end{quote}
\begin{quote}
``Every problem has a solution that is simple, neat, and wrong''.
-- Mark Twain
\end{quote}
\medskip
\medskip
\begin{prob}
Suppose that in $\mathbb{R}^2$, we are given a symmetric, convex body $K$ of area $4$, which contains only the origin.
Prove that $B$ must tile $\mathbb{R}^2$ by translations.
\end{prob}
\medskip
\begin{prob} \label{convexity of K-K} $\clubsuit$
Given convex $d$-dimensional bodies $K, L \subset \mathbb{R}^d$, prove that $K+L$ is convex, and that $K-L$ is convex.
\end{prob}
\medskip
\begin{prob} \label{c.s. C equals its symmetrized body}
$\clubsuit$
Suppose initially that $C \subset \mathbb{R}^d$ is any set.
\begin{enumerate}[(a)]
\item Show that
\begin{equation}
\frac{1}{2}C - \frac{1}{2}C = C
\ \implies \text{ $C$ is centrally symmetric}.
\end{equation}
\item
Now suppose that $C$ is convex. Show that
\begin{equation} \label{equivalence for a convex set}
C \text{ is centrally symmetric } \iff \frac{1}{2} C - \frac{1}{2} C = C.
\end{equation}
\item Find an example of a centrally symmetric set $C$ that is not convex, and satisfies
\[
\frac{1}{2} C - \frac{1}{2} C \not= C.
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob}\label{support of convolution} $\clubsuit$
Recalling the definition of the support of a function $f$ from \eqref{def of support}, show that:
\begin{enumerate}[(a)]
\item
Suppose that we are given two closed, convex bodies $A, B \subset \mathbb{R}^d$. Show that
\[
\supp ( 1_A * 1_B) = A + B,
\]
where the addition is the Minkowski addition of sets.
\index{Minkowski sum}
\item
More generally, if two functions $f, g:\mathbb{R}^d \rightarrow \mathbb{C}$ are compactly supported,
show that
\[
\supp(f*g)\subseteq \closure\left( \supp(f) + \supp(g) \right),
\]
the closure of the Minkowski sum of their individual supports.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{equivalent statements for unimodular triangles}
\rm{
Suppose we have a triangle $\Delta$ whose vertices $v_1, v_2, v_3$ are integer points. Prove that the following
properties are equivalent:
\begin{enumerate}[(a)]
\item $\Delta$ has no other integer points inside or on its boundary (besides its vertices).
\item $Area(\Delta) = \frac{1}{2}$.
\item $\Delta$ is a unimodular triangle, which in this case means that $v_3 - v_1$ and $v_2- v_1$ form a basis for $\mathbb{Z}^2$.
\end{enumerate}
(Hint: You might begin by ``doubling'' the triangle to form a parallelogram.)
}
\end{prob}
\medskip
\begin{prob}
Show that in $\mathbb{R}^d$, an integer simplex $\Delta$ is unimodular if and only if $\vol(\Delta) = \frac{1}{d!}$.
\end{prob}
\medskip
\begin{prob} \label{non-unimodular but empty simplex}
In $\mathbb{R}^3$, find an integer simplex $\Delta$ that has no other integer points inside or on its boundary (other than its vertices of course), but
such that $\Delta$ is not a unimodular simplex.
\end{prob}
\medskip
\begin{prob} \label{FT of a polytope is not Schwartz}
Prove that for any polytope ${\mathcal P}$, $\hat 1_{{\mathcal P}}$ is not a Schwartz function.
\end{prob}
\medskip
\begin{prob} \label{convolution of indicators is a nice function}
$\clubsuit$
(hard-ish) Show that if $K$ is any convex body, then $1_K*1_{-K}$ is a nice function, in the sense of \eqref{nice functions}.
In other words, show that the Poisson summation formula holds for the function $f(x):= \left( 1_K*1_{-K} \right)(x)$.
Hint. Use the Parseval identity, valid for functions $f \in L^2(\mathbb{R}^d)$. For this particular exercise, feel free to use the results of all of the later sections (though in general we refrain from such a `look ahead').
\end{prob}
\medskip
\begin{prob} \label{Cantor set}
We first define the following sets recursively:
\[
C_0 := [0, 1], \ C_1 := [0, \tfrac{1}{3}] \cup [\tfrac{2}{3}, 1], \dots ,
C_n:= \tfrac{1}{3} C_{n-1} \cup \left\{ \tfrac{1}{3} C_{n-1} + \tfrac{2}{3} \right\},
\]
and now the {\bf Cantor set} is defined by their infinite intersection:
\[
\mathcal C:= \cap_{n=0}^\infty C_n.
\]
It is a standard fact (which you may assume here) that the Cantor set $\mathcal C$ is compact, uncountable, and has measure $0$. Despite these facts, show that its difference body satisfies the somewhat surprising identity:
\[
\mathcal C-\mathcal C = [-1, 1].
\]
\end{prob}
\medskip
\begin{prob} \label{tiling using the truncated octrahedron}
Show that the truncated octahedron, defined in Example \ref{truncated octahedron}, tiles $\mathbb{R}^3$ by using only
translations with a lattice. Which lattice can you use for this tiling?
\end{prob}
\medskip
\begin{prob} \label{an application of Cauchy-Schwartz 1}
Define $f(x):= a \sin x + b \cos x$, for constants $a,b\in \mathbb{R}$.
Show that the maximum value of $f$ is $ \sqrt{ a^ 2 + b ^2 } $, and occurs
when $\tan x = \frac{ a}{b}$.
\end{prob}
\medskip
\begin{prob}
\rm{
Find an example of a symmetric polygon ${\mathcal P} \subset \mathbb{R}^2$ that multi-tiles (nontrivially) with multiplicity $k = 5$.
Notes. A trivial multi-tiling for ${\mathcal P}$ is by definition a multi-tiling that uses ${\mathcal P}$, with some multiplicity $k>1$, but such that
there also exists a $1$-tiling (classical) using the same ${\mathcal P}$.
}
\end{prob}
\medskip
\begin{prob}
Let $K\subset \mathbb{R}^d$ be centrally symmetric. Show that
\begin{equation*}
\frac{1}{2}K \cap \left( \frac{1}{2}K + n \right) \not= \phi \iff n \in K.
\end{equation*}
\end{prob}
\medskip
\begin{prob} \label{Extending Minkowski to nonconvex bodies}
$\clubsuit$
\rm{
Here we use Siegel's theorem \ref{Siegel for general lattices} to give the following extension of
Minkowski's classical Theorem \ref{Minkowski convex body Theorem for L}, but
for bodies $K$ that are not necessarily symmetric, nor necessarily convex.
Namely, let $K$ be any bounded, measurable subset of $\mathbb{R}^d$, with positive $d$-dimensional volume.
Let $B:= \frac{1}{2}K - \frac{1}{2}K$ be the symmetrized body of $K$ (hence $B$ is a centrally symmetric set containing the origin).
Let ${\mathcal L}$ be a (full rank) lattice in $\mathbb{R}^d$. Prove the following statement:
\begin{equation*}
\text{ If } \vol K > 2^d (\det {\mathcal L}), \text{ then } B
\text{ must contain a nonzero point of } {\mathcal L} \text{ in its interior}.
\end{equation*}
Notes. We note that the positive conclusion of the existence of a nonzero integer point holds only for the symmetrized body $B$,
with no guarantees for any integer points in $K$.
}
\end{prob}
\chapter{An introduction to Euclidean lattices} \label{chapter.lattices}
\index{lattice} \index{integer lattice}
\begin{quote}
``Lattices quantify the idea of periodic structures.''
-- Anonymous
\end{quote}
\begin{quote}
``Less is more........more or less.''
-- Ludwig Mies van der Rohe
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.6in]{parallelepiped1}
\end{center}
\caption{A fundamental parallelepiped (half-open), for a lattice ${\mathcal L}$, generated by the vectors $v_1$ and $v_2$.} \label{parallelepiped1}
\end{figure}
\bigskip
\section{Intuition}
We introduce Euclidean lattices here, which may be thought of intuitively as regularly-spaced points in
$\mathbb{R}^d$, with some hidden number-theoretic structure. Another intuitive way to think of lattices is that they are one of the most natural ways to {\bf discretize Euclidean space}.
A lattice in $\mathbb{R}^d$ is also the most natural extension of an infinite set of equally-spaced points on the real line. In the real-world, lattices come up very naturally when we study crystals, for example.
It is perhaps not surprising that number theory comes in through
study of the integer lattice $\mathbb{Z}^d$, as it is the $d$-dimensional extension of the integers $\mathbb{Z}$. Moreover, whenever we
study almost any periodic behavior, lattices naturally come up, essentially from the
definition of {\bf periodicity} in Euclidean space. \index{periodicity} And of course, where there are lattices, there are Fourier series, as we also saw in Chapter \ref{Fourier analysis basics}.
\section{Introduction to lattices}
\begin{defi}
A {\bf lattice} \index{lattice} is defined by the integer linear span of a fixed set of linearly independent vectors $\{ v_1, \dots, v_m \} \subset \mathbb{R}^d$:
\begin{equation}\label{def.lattice}
{\mathcal L} := \left\{ n_1 v_1 + \cdots + n_m v_m \in \mathbb{R}^d \bigm | \text{ all } n_j \in \mathbb{Z} \right\}.
\end{equation}
\end{defi}
The most common lattice is the {\bf integer lattice} \index{integer lattice}
\[
\mathbb{Z}^d:= \left\{ (x_1, \dots, x_d) \in \mathbb{R}^d \bigm | \text{ all } x_j \in \mathbb{Z} \right\}.
\]
However, we often encounter different types of lattices, occurring very naturally in practice, and it is natural to ask how they are related to each other.
The first thing we might notice is that, by Definition \ref{def.lattice}, a lattice may also be written as follows:
\begin{equation}
{\mathcal L} := \left\{
\begin{pmatrix} | & | & ... & | \\
v_1 & v_2 & ...& v_m \\
| & | & ... & | \\
\end{pmatrix}
\begin{pmatrix}
n_1 \\
\vdots \\
n_m \\
\end{pmatrix}
\ \biggm | \
\begin{pmatrix}
n_1 \\
\vdots \\
n_m \\
\end{pmatrix}
\in \mathbb{Z}^m
\right\}
:= M(\mathbb{Z}^m),
\end{equation}
where by definition, $M$ is the $d \times m$ matrix whose columns are the vectors $v_1, \dots, v_m$. This set of basis vectors
$\{ v_1, \dots, v_m\}$ is called a {\bf basis} \index{lattice basis}
for the lattice ${\mathcal L}$, and $m$ is called the {\bf rank} of the lattice ${\mathcal L}$.
In this context, we also use the notation ${\rm rank}({\mathcal L}) = m$.
We will call $M$ a {\bf basis matrix} \index{basis matrix}
for the lattice ${\mathcal L}$.
But there are always infinitely many other bases for ${\mathcal L}$ as well, and Lemma \ref{changing basis matrices} below shows how they are related to each other.
Most of the time, we will be interested in {\bf full-rank} \index{full rank lattice}
lattices, which means that $m=d$; however, sometimes we will also be interested in lattices that have lower rank, and it is important to understand them.
The {\bf determinant} of a full-rank lattice ${\mathcal L} := M(\mathbb{Z}^d)$ is defined by
\[
\det {\mathcal L} := |\det M|.
\]
It is easy to prove that this definition is independent of the choice of basis matrix $M$, which is the content of Lemma
\ref{changing basis matrices} below.
\begin{example}
\rm{
In $\mathbb{R}^1$, we have the integer lattice $\mathbb{Z}$, but we also have lattices of the form $r\mathbb{Z}$, for any real number $r$.
It's easy to show that any lattice in $\mathbb{R}^1$ is of this latter type (Exercise \ref{lattices in R^1}).
For example, if $r = \sqrt 2$, then all integer multiples of $\sqrt 2$ form a $1$-dimensional lattice. }
\hfill $\square$
\end{example}
\begin{example}
\rm{
In $\mathbb{R}^2$, consider the lattice ${\mathcal L}$ generated by the two integer vectors
$v_1:=\icol{-1\\3}$
and $v_2:= \icol{-4\\ 1}$, drawn in Figure \ref{parallelepiped1}.
A different choice of basis for the same lattice ${\mathcal L}$ is $\{ \icol{-3\\-2}, \icol{-8\\ -9} \}$, drawn in
Figure \ref{parallelepiped2}. We note that $\det {\mathcal L} = 11$, and indeed the areas of both half-open parallelepipeds equals $11$. }
\hfill $\square$
\end{example}
\medskip
A {\bf fundamental parallelepiped} \index{fundamental parallelepiped}
for a lattice ${\mathcal L}$ with basis $\{ v_1, \dots, v_m \}$ is:
\begin{equation} \label{def:half-open parallelepiped}
D:= \left\{ \lambda_1 v_1 + \cdots + \lambda_m v_m \bigm | \text{ all } 0 \leq \lambda_k < 1 \right\},
\end{equation}
also known as a {\bf half-open parallelepiped}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{parallelepiped2}
\end{center}
\caption{A second fundamental parallelepiped for the same lattice ${\mathcal L}$ as in
Figure \ref{parallelepiped1}} \label{parallelepiped2}
\end{figure}
We have the pleasant property
that D tiles $\mathbb{R}^d$ by translations with vectors
from ${\mathcal L}$, and with no overlaps. Let's make this intuition more precise, in the following lemma.
We'll use the standard notation that for any real $\alpha$, $\lfloor \alpha \rfloor$ is the greatest integer not exceeding $\alpha$, and $\{ \alpha\}$ is the fractional part of $\alpha$.
\begin{lem}
Suppose we are given a full rank lattice ${\mathcal L}\subset \mathbb{R}^d$, and a fundamental parallelepiped $D$ for ${\mathcal L}$,
as in Definition \eqref{def:half-open parallelepiped}.
Then any $x\in \mathbb{R}^d$ may be written uniquely as
\[
x = n + y
\]
where $n \in {\mathcal L}$, and $y \in D$.
\end{lem}
\begin{proof}
We know that $D$ is formed by a basis for the lattice ${\mathcal L}$, and we can label the basis elements by
$v_1, \dots, v_d$. These $d$ vectors also form a basis for $\mathbb{R}^d$,
so in particular any $x\in \mathbb{R}^d$ may be written as
\[
x = \sum_{j=1}^d \alpha_j v_j.
\]
Writing each $\alpha_j:= \lfloor \alpha_j \rfloor + \{ \alpha_j \}$, we have
\[
x = \sum_{j=1}^d \lfloor \alpha_j \rfloor v_j + \sum_{j=1}^d \{ \alpha_j \} v_j := n + y,
\]
where we've defined $n := \sum_{j=1}^d \lfloor \alpha_j \rfloor v_j$, and
$y:= \sum_{j=1}^d \{ \alpha_j \} v_j$. Since $\lfloor \alpha_j \rfloor \in \mathbb{Z}$, we see that
$n \in {\mathcal L}$. Since $0\leq \{ \alpha_j \} < 1$, we see that $y \in D$.
To prove uniqueness, suppose we are given $x:= n_1 + y_1 = n_2 + y_2$, where $n_1, n_2 \in {\mathcal L}$ and $y_1, y_2 \in D$. So by definition
$y_1 = \sum_{j=1}^d \{ \alpha_{j, 1} \} v_j$ and $y_2=\sum_{j=1}^d \{ \alpha_{j, 2} \} v_j$.
Then $y_1 - y_2 = n_2 - n_1 \in {\mathcal L}$, which means that $ \alpha_{j, 1} - \alpha_{j, 2} \in \mathbb{Z}$. But $0 \leq \alpha_{j, 1}<1$ and $0 \leq \alpha_{j, 2}<1$ implies that $ \alpha_{j, 1} - \alpha_{j, 2}=0$. Therefore $y_1 = y_2$, and so $n_1 = n_2$.
\end{proof}
How do we define the determinant of a general lattice ${\mathcal L}\subset \mathbb{R}^d$ of rank $r$?
We can start by observing how the squared lengths of vectors in ${\mathcal L}$ behave w.r.t. a given basis of ${\mathcal L}$:
\begin{equation}
\| x \|^2 = \left\langle \sum_{j=1}^r c_j v_j, \, \sum_{k=1}^r c_k v_k \right\rangle =
\sum_{1\leq j, k \leq r} c_j c_k \langle v_j, \, v_k \rangle := c^T M^T M c,
\end{equation}
where $M^TM$ is an $r\times r$ matrix whose columns are basis vectors of ${\mathcal L}$.
With this as motivation, we define:
\begin{equation}\label{def. of sublattice determinant}
\det {\mathcal L} := \sqrt{ M^T M},
\end{equation}
called the {\bf determinant of the lattice} ${\mathcal L}$. \index{determinant of a general lattice}
This definition coincides, as it turns out,
with the Lebesgue measure of any fundamental parallelepiped of ${\mathcal L}$ (Exercise \ref{equivalence between determinants of a sublattice}).
\bigskip
\section{Sublattices}
Given two lattices ${\mathcal L}\subset \mathbb{R}^d$, and ${\mathcal M} \subset \mathbb{R}^d$, such that
\[
{\mathcal L} \subseteq {\mathcal M},
\]
we say that {\bf ${\mathcal L}$ is a sublattice of ${\mathcal M}$}. \index{sublattice}
For example,
Figure \ref{sublattice, rank 1} shows a rank $1$ sublattice of the integer lattice $\mathbb{Z}^2$, together with its determinant.
On the other hand, sublattices that have the same rank are very interesting, and quite useful in applications. Given a sublattice ${\mathcal L}$ of ${\mathcal M}$, both of the same rank, a crucial idea is to think of all of the translates
of ${\mathcal L}$ by an element of the coarser lattice ${\mathcal M}$, which we call:
\begin{equation}
{\mathcal M} / {\mathcal L} := \left\{ {\mathcal L} + m \bigm | m \in {\mathcal M} \right\}.
\end{equation}
Each such translate ${\mathcal L} + m$ is called a {\bf coset} \index{coset}
of ${\mathcal L}$ in ${\mathcal M}$, and the collection of all of these cosets, namely ${\mathcal M} / {\mathcal L}$, is called the
{\bf quotient lattice} (or quotient group) \index{quotient lattice}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.2in]{sublattice1}
\end{center}
\caption{
A sublattice ${\mathcal L} \subset \mathbb{Z}^2$ of rank $1$, which has just one basis vector.
Here ${\mathcal L}$ has a $1$-dimensional fundamental parallelepiped, showing that
$\det {\mathcal L} = \sqrt{v^T v} = \sqrt 5$, consistent with Definition \ref{def. of sublattice determinant}.
}
\label{sublattice, rank 1}
\end{figure}
\begin{thm} \label{sublattice index}
Let ${\mathcal L} \subseteq {\mathcal M}$ be any two lattices of the same rank. Then
\begin{enumerate}
\item $\frac{ \det {\mathcal L} }{ \det {\mathcal M} }$ is an integer.
\item The positive integer $\frac{ \det {\mathcal L} }{ \det {\mathcal M} }$ is equal to the number of
cosets of ${\mathcal L}$ in ${\mathcal M}$. In other words, $\left| {\mathcal M} / {\mathcal L} \right| = \frac{ \det {\mathcal L} }{ \det {\mathcal M} }$.
\end{enumerate}
\end{thm}
For a proof of Theorem \ref{sublattice index}, see \cite{FukshanskyBook}.
\begin{example}
{\rm
Let ${\mathcal M}:= \mathbb{Z}^d$, and ${\mathcal L}:= 2\mathbb{Z}^d$, the sublattice consisting of vectors all of whose coordinates are even integers. So ${\mathcal L} \subset {\mathcal M}$, and the quotient lattice ${\mathcal M}/ {\mathcal L}$ consists of the sets
$\left\{ 2\mathbb{Z}^d + n \bigm | n \in \mathbb{Z}^d \right\}$. It is (almost) apparent that the number of elements of the latter set is exactly $2^d$, so in our new notation we have $\left| \mathbb{Z}^d / 2\mathbb{Z}^d \right| = 2^d$.
We may also think of this quotient lattice $ \mathbb{Z}^d / 2\mathbb{Z}^d $ as the discrete unit cube, namely
$\left\{ 0, 1 \right\}^d$, a common object in theoretical computer science.
\hfill $\square$
}
\end{example}
\section{Discrete subgroups - \\ an alternate definition of a lattice}
The goal here is to give another useful way to define a lattice. The reader does not need any background in group theory, because the ideas here are self-contained, given some background in basic linear algebra.
\smallskip
\begin{defi} \label{discrete subgroup}
\begin{enumerate}[(a)]
We define a {\bf discrete subgroup} \index{discrete subgroup}
of $\mathbb{R}^d$ as a set $S \subset \mathbb{R}^d$, together with the operation of vector addition between all of its elements, which enjoys the following two properties.
\item {\bf [The subgroup property]} If $x, y \in S$, then $ x-y \in S$. \\
\label{discrete subgroup.first part}
\item {\bf [The discrete property]} There exists a positive real number $\delta >0$,
such that \\
the distance between any two distinct points of $S$
is at least $\delta$. \\ \label{discrete subgroup.second part}
\end{enumerate}
\end{defi}
In particular, it follows from Definition \ref{discrete subgroup} \ref{discrete subgroup.first part}
that the zero vector must be in $S$, because for any $x \in S$, it must be the case that $x - x \in S$.
The distance function that we alluded to in Definition \ref{discrete subgroup} \ref{discrete subgroup.second part}
is the usual Euclidean distance function, which we denote here by
\[
\|x-y\|_2:= \sqrt{ \sum_{k=1}^d (x_k - y_k)^2}.
\]
\begin{example}
\rm{
The lattice $\mathbb{Z}^d$ is a discrete subgroup of $\mathbb{R}^d$. In dimension $1$, the lattice
$r\mathbb{Z}$ is a discrete subgroup of $\mathbb{R}$, for any fixed $r>0$. Can we think of discrete subgroups that are not lattices? The answer is given by Lemma \ref{discrete subgroup equivalence} below.
}
\hfill $\square$
\end{example}
The magic here is the following very useful way of going back and forth between this new notion of a discrete subgroup of $\mathbb{R}^d$, and our Definition \ref{def.lattice} of a lattice. The idea of using this alternate Definition \ref{discrete subgroup}, as opposed to our previous Definition \ref{def.lattice} of a lattice, is that it gives us a {\bf basis-free} \index{basis-free} way of proving and discovering facts about lattices.
\bigskip
\begin{lem}\label{discrete subgroup equivalence}
${\mathcal L} \subset \mathbb{R}^d$ is a lattice $\iff$ ${\mathcal L}$ is a discrete subgroup of $\mathbb{R}^d$.
\hfill $\square$
\end{lem}
(For a proof see \cite{GruberBook}).
\bigskip
\begin{example}
\rm{
Given any two lattices ${\mathcal L}_1, {\mathcal L}_2 \subset \mathbb{R}^d$, let's show that $S := {\mathcal L}_1 \cap {\mathcal L}_2$ is also a lattice.
First, any lattice contains the zero vector, and it may be the case that their intersection consists of only the zero vector. For any vectors $x, y \in S$, we also have $x,y \in {\mathcal L}_1$, and $x, y \in {\mathcal L}_2$, hence
by the subgroup property of ${\mathcal L}_1 $ and of ${\mathcal L}_2$, we know that both $x-y \in {\mathcal L}_1$, and $x-y \in {\mathcal L}_2$. In other words, $x-y \in {\mathcal L}_1 \cap {\mathcal L}_2:= S$. To see why the discrete property of
Definition \ref{discrete subgroup} holds here, we just notice that since $x-y \in {\mathcal L}_1$, we already know that
$| x-y | > \delta_1$, for some $\delta_1>0$; similarly, because $x-y \in {\mathcal L}_2$, we know that
$| x-y | > \delta_2$ for some $\delta_2>0$. So we let $\delta:= \min(\delta_1, \delta_2\}$, and we have shown that $S$ is a discrete subgroup of $\mathbb{R}^d$. By Lemma \ref{discrete subgroup equivalence}, we see that $S$ is a lattice.
If we had used Definition \ref{def.lattice} of a lattice to show that $S$ is indeed a lattice, it would require us to work with bases, and this proof would be longer and less transparent.
}
\hfill $\square$
\end{example}
\bigskip
\begin{example} \label{A_d example}
\rm{
Consider the following discrete set of points in $\mathbb{R}^d$:
\[
A_{d-1}:= \left\{ x \in \mathbb{Z}^d \bigm | \sum_{k=1}^d x_k =0 \right\},
\]
for any $d\geq 2$, as depicted in Figure \ref{A_d}. Is $A_d$ a lattice? Using the definition
\ref{def.lattice} of a lattice, it is not obvious that $A_d$ is a lattice, because we would have to exhibit a basis, but
it turns out that the following set of vectors may be shown to be a basis: $ \left\{e_2 - e_1, e_3 - e_1, \cdots e_d - e_1 \right\}$, and hence $A_d$ is a sublattice
of $\mathbb{Z}^d$, of rank $d-1$ (Exercise \ref{basis for A_d}).
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.8in]{A_d}
\end{center}
\caption{The lattice $A_1$, and
the lattice $A_2$, with basis $\left\{ v_1, v_2 \right\}$} \label{A_d}
\end{figure}
Just for fun, we will use Lemma \ref{discrete subgroup equivalence} to show that
$A_d$ is indeed a lattice. To verify the subgroup property of
Definition \ref{discrete subgroup} \ref{discrete subgroup.first part}
suppose that $x, y \in A_d$. Then by definition we have
$\sum_{k=1}^d x_k =0$ and $ \sum_{k=1}^d y_k =0$. So $\sum_{k=1}^d (x_k - y_k)=0$, implying that $x-y \in A_d$.
To verify the discrete property of
Definition \ref{discrete subgroup} \ref{discrete subgroup.second part}
suppose we are given two distinct points $x, y \in A_d$. We can first compute their ``cab metric'' distance function, in other
words the $L^1$-norm defined by
\[
\| x-y \|_1:= |x_1 - y_1| + \cdots + |x_d - y_d|,
\]
By assumption, there is at least one coordinate where $x$ and $y$ differ, say the $k$'th coordinate. Then
$ \| x-y \|_1 := |x_1 - y_1| + \cdots + |x_d - y_d| \geq 1$, because all of the coordinates are integers, and $x_k \not= y_k$ by assumption. Since the $L^1$-norm and the $L^2$-norm
are only off
by $\sqrt d$ (by Exercise \ref{elementary norm relations}), we have:
\[
\sqrt{d} \| x-y \|_2 \geq \| x-y \|_1 \geq 1,
\]
so the property \ref{discrete subgroup} \ref{discrete subgroup.second part}
is satisfied with $\delta := \frac{1}{\sqrt{d}}$, and we've shown that $A_d$ is a lattice.
\hfill $\square$
}
\end{example}
We note that the lattices $A_d$ defined in Example \ref{A_d example} are very important in many fields of Mathematics, including Lie algebras (root lattices), Combinatorial geometry, and Number theory.
\section{Lattices defined by congruences}
In this section we develop some of the theory in a concrete manner. A classic example of a lattice defined by an auxiliary algebraic construction is the following.
Suppose we are given a constant integer vector $(c_1, \dots, c_d) \in \mathbb{Z}^d$, where we further assume
that $\gcd(c_1, \dots, c_d) = 1$. Let
\begin{equation}\label{Lattice from congruences}
C := \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d \equiv 0 \mod N \right\},
\end{equation}
where $N$ is a fixed positive integer.
Is $C$ a lattice?
Indeed, we can see that $C$ is a lattice by first checking
Definition \ref{discrete subgroup} \ref{discrete subgroup.first part}. For any $x, y \in C$, we have
$c_1 x_1 + \cdots + c_d x_d \equiv 0 \mod N$ and $c_1 y_1 + \cdots + c_d y_d \equiv 0 \mod N$.
Subtracting these two congruences gives us
$c_1 (x_1-y_1) + \cdots + c_d (x_d-y_d) \equiv 0 \mod N$, so that $x-y \in C$. The verification
of Definition \ref{discrete subgroup} \ref{discrete subgroup.second part} if left to the reader, and its logic is similar to Example \ref{A_d example}.
There is even a simple formula for the volume of a fundamental parallelepiped for $C$:
\begin{equation}
\det C = N,
\end{equation}
as we prove below, in \eqref{proof of det C = N}.
Perhaps we can solve an easier problem first. Consider the {\bf discrete hyperplane} defined by:
\[
H:= \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =0 \right\},
\]
Is $H$ a lattice? We claim that $H$ itself is indeed a sublattice of $\mathbb{Z}^d$, and has rank $d-1$.
Since this verification is quite similar to the arguments above, we leave this as Exercise \ref{hyperplane lattice}.
The fundamental parallelepiped (which is $(d-1)$-dimensional) of $H$ also has a wonderful formula, as follows. First, we recall
a general fact (from Calculus/analytic geometry) about hyperplanes,
namely that the distance $\delta$ between any two parallel hyperplanes \\
$c_1 x_1 + \cdots + c_d x_d = k_1$ and
$c_1 x_1 + \cdots + c_d x_d = k_2$ is given by
\begin{equation} \label{distance between two hyperplanes}
\delta = \frac{ |k_1 - k_2|}{\sqrt{ c_1^2 + \cdots + c_d^2}}.
\end{equation}
(see Exercise \ref{distance between hyperplanes})
\medskip
\begin{lem} \label{wonderful hyperplane determinant formula}
For any latttice defined by a discrete hyperplane \\
$H:= \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =0 \right\}$, with
$\gcd(c_1, \dots, c_d) = 1$, we have:
\begin{equation} \label{tricky volume of sublattice}
\det H = \sqrt{ c_1^2 + \cdots + c_d^2}.
\end{equation}
\end{lem}
\begin{proof}
We first fix a basis $\{v_1, \dots, v_{d-1}\}$ for the $(d-1)$-dimensional sublattice defined by
$H:= \left\{ x \in \mathbb{Z}^d \mid c_1 x_1 + \cdots + c_d x_d =0 \right\}$. We adjoin to this basis one new vector, namely any integer vector $w$ that translates $H$ to its `hyperplane companion'
$H + w$, which we define by
\[
H+w:= ~\left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =1 \right\}.
\]
It's easy - and fun - to see that there are no integer points strictly between these two hyperplanes $H$ and $H+w$ (Exercise \ref{tiling the integer lattice with hyperplanes}), and so
the parallelepiped ${\mathcal P}$ formed by the edge vectors $v_1, \dots, v_{d-1}, w$ is a fundamental domain for $\mathbb{Z}^d$, hence has volume $1$.
On the other hand, we may also calculate the volume of ${\mathcal P}$ by multiplying the volume of its base times its height, using \eqref{distance between two hyperplanes}:
\begin{align}
1= \vol {\mathcal P} &= (\text{volume of the base of } {\mathcal P})(\text{height of } {\mathcal P}) \\
&= (\det H)\cdot \delta \\
&= (\det H) \frac{1}{\sqrt{c_1^2 + \cdots + c_d^2}},
\end{align}
and so $\det H = \sqrt{ c_1^2 + \cdots + c_d^2}$.
\end{proof}
It follows directly from the definition \ref{Lattice from congruences} of $C$ that we may write the lattice $C$ as a countable, disjoint union of translates of $H$:
\begin{equation}
C := \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d = kN, \text{ where } k = 1, 2, 3, \dots \right\}.
\end{equation}
To be concrete, let's work out some examples.
\begin{example}
\rm{
Using Lemma \ref{wonderful hyperplane determinant formula}, we can easily compute the determinant of the $A_d$ lattice from Example \ref{A_d example}:
\[
\det A_d = \sqrt{1 + 1 + \cdots + 1} = \sqrt d.
\]
}
\end{example}
\medskip
\begin{example}\label{Congruence lattice 2d}
\rm{
As in Figure \ref{lattice 2d},
consider the set of all integer points $(m, n) \in \mathbb{R}^2$ that satisfy
\[
2m + 3n \equiv 0 \mod 4.
\]
In this case the related hyperplane is the line $2x+3y = 0$, and the solutions to the latter congruence may be thought of as a union of discrete lines:
\[
C = \left\{ {x\choose y} \in \mathbb{Z}^2 \bigm | 2x+3y = 4k, \text{ and } k \in \mathbb{Z} \right\}.
\]
}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.7in]{LatticeCongruence}
\end{center}
\caption{The lattice of Example \ref{Congruence lattice 2d}} \label{lattice 2d}
\end{figure}
\rm{
In other words, our lattice $C$, a special case of \eqref{Lattice from congruences}, can in this case be visualized
in Figure \ref{lattice 2d} as a disjoint union of discrete lines. If we denote the distance between any two of these
adjacent discrete lines by $\delta$, then using \eqref{distance between two hyperplanes} we have:
\begin{equation}
\delta = \frac{4}{\sqrt{3^2 + 2^2}}.
\end{equation}
Finally, the determinant of our lattice $C$ here is the area of the
shaded parallelepiped:
\begin{equation}
\det C = \delta \sqrt{3^2 + 2^2} = 4.
\end{equation}
}
\hfill $\square$
\end{example}
Eager to prove the volume relation $ \det C = N$,
we can use the ideas of Example \ref{Congruence lattice 2d} as a springboard for this generalization.
Indeed, Example \ref{Congruence lattice 2d} and the proof of Lemma
\ref{wonderful hyperplane determinant formula} both
suggest that we should compute the volume of a fundamental parallelepiped ${\mathcal P}$, for the lattice $C$ (as opposed to the lattice $\mathbb{Z}^d$),
by using a fundamental domain for its base, and then by multiplying its volume by the height of ${\mathcal P}$.
\begin{lem}\label{lemma:lattice defined by congruence}
Given a constant integer vector $(c_1, \dots, c_d) \in \mathbb{Z}^d$, with $\gcd(c_1, \dots, c_d) = 1$,
let
\begin{equation}
C := \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d \equiv 0 \mod N \right\},
\end{equation}
where $N$ is a fixed positive integer. Then $C$ is a lattice, and
\[
\det C = N.
\]
\end{lem}
\begin{proof}
We fix a basis $\{v_1, \dots, v_{d-1}\}$ for the $(d-1)$-dimensional sublattice defined by
$H:= \left\{ x \in \mathbb{Z}^d \mid c_1 x_1 + \cdots + c_d x_d =0 \right\}$, and we adjoin to this basis one new vector, namely any integer vector $w$ that translates $H$ to its nearest discrete hyperplane companion
\[
H+w:= \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =N \right\}.
\]
Together, the set of vectors
$ \{ v_1, \dots, v_{d-1}, w \} $
form the edge vectors of a fundamental parallelepiped ${\mathcal P}$ for the lattice $C$, whose hight $\delta$ is the distance between these two parallel
hyperplanes $H$ and $H+w$. Using \eqref{distance between two hyperplanes}, we can
may calculate the volume of ${\mathcal P}$ (which is by definition equal to $\det C$) by multiplying the volume of its `base' times its `height':
\begin{align}
\det C &= (\text{volume of the base of } {\mathcal P})(\text{height of } {\mathcal P}) = (\det H)\delta \\ \label{proof of det C = N}
&= (\det H) \frac{N}{\sqrt{c_1^2 + \cdots + c_d^2}} = N,
\end{align}
using the fact that $\det H = \sqrt{c_1^2 + \cdots + c_d^2}$ from
Lemma \ref{wonderful hyperplane determinant formula}.
\end{proof}
\bigskip
\section{The Gram matrix}
There is another very natural matrix that we may use to study lattices, which we can motivate as follows.
Suppose we are given any basis for a lattice ${\mathcal L}\subset \mathbb{R}^d$, say $\beta:= \{ v_1, \dots, v_r \}$,
where $1 \leq r \leq d$. By definition ${\mathcal L} = M(\mathbb{Z}^d)$, and ${\rm rank}({\mathcal L}) = r$,
where the columns of $M$ are defined by the basis vectors from $\beta$, and so $M$ is a $d\times r$ matrix.
We can therefore represent any $x\in \mathbb{R}^d$ uniquely in terms of the basis $\beta$ like this:
\begin{equation} \label{writing a vector in terms of a basis}
x = c_1 v_1 + \cdots + c_r v_r,
\end{equation}
and the squared length of $x$ is:
\begin{equation} \label{Gram matrix positive semidefinite}
\| x \|^2 = \left\langle \sum_{j=1}^r c_j v_j, \, \sum_{k=1}^r c_k v_k \right\rangle =
\sum_{1\leq j, k \leq r} c_j c_k \langle v_j, \, v_k \rangle := c^T M^T M c,
\end{equation}
where $c:= (c_1, \dots, c_r)^T$ is the coefficient vector defined by
\eqref{writing a vector in terms of a basis}.
It's therefore very natural to focus on the matrix $M^T M$, whose entries are the inner products
$\langle v_j, v_k \rangle$ of all the basis vectors of the lattice ${\mathcal L}$, so
we define
\[
G:= M^TM,
\]
a {\bf Gram matrix} for ${\mathcal L}$. It's clear from the computation above in
\eqref{Gram matrix positive semidefinite} that $G$ is positive definite. Although
$G$ does depend on which basis of ${\mathcal L}$ we choose, it is an elementary fact that $\det G$ is independent of the basis of ${\mathcal L}$.
Because we are always feeling the urge to learn more Linear Algebra, we would like to see
why any real symmetric matrix $B$ is the Gram matrix of some set of vectors. To see this, we apply the
Spectral Theorem: $B = P D P^T$, for some orthogonal matrix $P$ and a diagonal matrix $D$ with nonnegative diagonal elements. So we can write $B = (P \sqrt D) (P \sqrt D)^T := M^T M$, where we defined the matrix $M:= (P \sqrt D)^T$, so that the columns of $M$ are the vectors whose corresponding dot products form the symmetric matrix $B$, and now $B$ is a Gram matrix.
To review some more linear algebra, suppose we are given a real symmetric matrix $A$.
We recall that such a matrix is called {\bf positive definite} if in addition we have the positivity condition
\[
x^T A x > 0,
\]
for all $x \in \mathbb{R}^d$. Equivalently, all of the eigenvalues of $A$ are positive. The reason is easy:
$Ax = \lambda x$ for a non-zero vector $x \in \mathbb{R}^d$
implies that
\[
x^T A x := \langle x, Ax \rangle = \langle x, \lambda x \rangle = \lambda \| x \|^2,
\]
so that $x^T A x >0 $ if and only if $\lambda > 0$. In the sequel, if we only require a symmetric matrix
$A$ that enjoys the property $x^T A x \geq 0$ for all
$x\in \mathbb{R}^d$, then we call such a matrix {\bf positive semidefinite}.
Also, for a full-rank lattice, we see that $B:= M^T M$ will be positive definite if and only if $M$ is invertible, so that the columns of $M$ are a basis. Since a positive definite matrix is symmetric by definition, we've proved:
\begin{lem} Suppose we are given a real symmetric matrix $B$. Then:
\begin{enumerate}[(a)]
\item $B$ is positive definite if and only if it is the Gram matrix of a basis.
\item $B$ is positive semidefinite if and only if it is the Gram matrix of some set of vectors.
\end{enumerate}
\hfill $\square$
\end{lem}
What about reconstructing a lattice, knowing only one of its Gram matrices? This is almost possible to accomplish, up to an orthogonal transformation, as follows.
\begin{lem}\label{reconstructing a lattice basis from its Gram matrix}
Suppose that $G$ is an invertible matrix, whose spectral decomposition is
\[
G = P D P^T.
\]
Then
\begin{equation}
G = X^T X \quad \iff \quad X = Q (\sqrt{D} P^T),
\end{equation}
for some orthogonal matrix $Q$.
\end{lem}
\begin{proof}
The assumption $G = X^T X$ guarantees that $G$ is symmetric and has positive eigenvalues, so by the Spectral Theorem we have:
\[
G = P D P^T,
\]
where $D$ is a diagonal matrix consisting of the positive eigenvalues of $G$, and $P$ is an orthogonal matrix consisting of eigenvectors of $G$. Setting $X^T X = P D P^T$, we must have
\begin{equation} \label{technical orthogonal identity}
I = X^{-T} PDP^T X^{-1} = (X^{-T} P \sqrt D) (X^{-T} P \sqrt D)^T,
\end{equation}
where we define $\sqrt D$ to be the diagonal matrix whose diagonal elements are the positive square roots of the eigenvalues of $G$. From \ref{technical orthogonal identity}, it follows that
$X^{-T} P \sqrt D$ is an orthogonal matrix, let's call it $Q^{-T}$. Finally,
$X^{-T} P \sqrt D = Q^{-T}$ implies that $X = Q \sqrt D P^T$.
\end{proof}
So Lemma \ref{reconstructing a lattice basis from its Gram matrix} allows us to reconstruct a lattice ${\mathcal L}$, up to an orthogonal transformation, by only knowing one of its Gram matrices.
To better understand lattices, we need the {\bf unimodular group}, which we write as
${\rm SL}_d(\mathbb{Z})$, \index{unimodular matrix} under matrix multiplication:
\begin{equation}\label{Definition of the unimodular group}
{\rm SL}_d(\mathbb{Z}) :=
\left\{ M \bigm | M \text{ is a }
d \times d \text{ integer matrix, with} \ |\det M| = 1
\right\}.
\end{equation}
\index{unimodular group}
\noindent
The elements of $\rm{SL_d}(\mathbb{Z})$ are called {\bf unimodular matrices}. \index{unimodular matrix}
\begin{example}
\rm{
Some typical elements of $\rm{SL_2}(\mathbb{Z})$ are
\[
S = \big(\begin{smallmatrix}
\ 0 & 1 \\
-1 & 0
\end{smallmatrix}
\big),
T:= \big(\begin{smallmatrix}
1 & 1 \\
1 & 0
\end{smallmatrix}
\big),
\text{ and \ }
-I := \big(\begin{smallmatrix}
-1 & \ 0 \\
\ 0 & -1
\end{smallmatrix}
\big),
\]
so we include matrices with determinant equal to $-1$ as well as $1$.}
\hfill $\square$
\end{example}
Any lattice ${\mathcal L}$ has infinitely many fundamental parallelepipeds and (Exercise \ref{fundamental domains})
it is a nice fact that they are all images of one another by the unimodular group.
Now, suppose a lattice ${\mathcal L}$ is defined by two different basis matrices: ${\mathcal L} = M_1(\mathbb{Z}^d)$ and
${\mathcal L} = M_2(\mathbb{Z}^d)$. Is there a nice relationship between $M_1$ and $M_2$?
\begin{lem}\label{changing basis matrices}
If a full-rank lattice ${\mathcal L} \subset \mathbb{R}^d$ is defined by two different basis matrices $M_1$, and $M_2$,
then
\[
M_1 = M_2 U,
\]
where $U \in \rm{SL_d}(\mathbb{Z})$, a unimodular matrix.
In particular, $\det {\mathcal L}$ is independent of the
choice of basis matrix $M$.
\end{lem}
\begin{proof}
By hypothesis, we know that the columns of $M_1$, say $v_1, \dots, v_d$, form a basis of ${\mathcal L}$, and that the columns of $M_2$, say $w_1, \dots, w_d$, also form a basis of ${\mathcal L}$. So we can begin by writing each fixed basis vector $v_j$ in terms of all the basis vectors $w_k$:
\[
v_j = \sum_{k=1}^d c_{j,k} w_k,
\]
for each $j = 1, \dots, d$, and for some $c_{j,k} \in \mathbb{Z}$. We may collect all $d$ of these identities into matrix form:
\[
M_1 = M_2 C,
\]
where $C$ is the integer matrix whose entries are defined by the integer coefficients $c_{j,k}$ above.
Conversely, we may also write each basis vector $w_j$ in terms of the basis vectors $v_k$:
$w_j = \sum_{k=1}^d d_{j,k} v_k$, for some $d_{j,k}\in\mathbb{Z}$, getting another matrix identity:
\[
M_2 = M_1 D.
\]
Altogether we have
\[
M_1 = M_2 C = (M_1 D) C,
\]
and since $M_1^{-1}$ exists by assumption, we get $DC= I$, the identity matrix. Taking determinants, we see that
\[
| \det D | | \det C | = 1,
\]
and since both $C$ and $D$ are integer matrices, they must belong to $\rm{SL_d}(\mathbb{Z})$, by definition.
Finally, because, because a unimodular matrix $U$ has $|\det U|=1$, we see that any two basis $M_1, M_2$ matrices satisfy
$|\det M_1 | = |\det M_2 |$.
\end{proof}
Using similar techniques, it is not hard to show the following fact
(Exercise \ref{Theorem: Automorphisms of lattices}).
\begin{thm} \label{Automorphisms of lattices}
Fix a full-rank lattice ${\mathcal L} \subset \mathbb{R}^d$. The group of one-to-one, onto, linear transformations from ${\mathcal L}$ to itself
is equal to
the unimodular group $SL_d(\mathbb{Z})$. \index{unimodular group}
\end{thm}
\bigskip
\section{Dual lattices} \index{dual lattice} \label{dual lattice}
Every lattice ${\mathcal L}:= M(\mathbb{Z}^d)$ has a {\bf dual lattice}, which we have already encountered in the Poisson summation formula
for arbitrary lattices. The dual lattice of a full-rank lattice ${\mathcal L}$ was defined by:
\begin{equation}\label{first definition of dual lattice}
{\mathcal L} ^*= M^{-T}(\mathbb{Z}^d).
\end{equation}
But there is another way to define the dual lattice of a lattice ${\mathcal L} \subset \mathbb{R}^d$ (of any rank), which is coordinate-free:
\begin{equation}\label{second definition of dual lattice}
{\mathcal L}^* := \left\{ x \in \mathbb{R}^d \mid \langle x, n \rangle \in \mathbb{Z}, \text{ for all } n \in {\mathcal L} \right\}.
\end{equation}
\begin{lem}
The two definitions above, \eqref{first definition of dual lattice} and \eqref{second definition of dual lattice}, are equivalent.
\begin{proof}
We let $A := {\mathcal L}^*:= M^{-T}(\mathbb{Z}^d)$, and
$B:= \left\{ x \in \mathbb{R}^d \bigm | \langle x, n \rangle \in \mathbb{Z}, \text{ for all } n \in {\mathcal L} \right\}$.
We first fix any $x \in A$. To show $x \in B$, we fix any
$n\in {\mathcal L}$, and we now have to verify that $\langle x, n \rangle \in \mathbb{Z}$.
By assumption, $x = M^{-T}m$ for some $m \in \mathbb{Z}^d$, and $n= M k$, for some $k\in\mathbb{Z}^d$. Therefore
\[
\langle x, n \rangle =\langle M^{-T}m, n \rangle = \langle m, M^{-1}n \rangle
= \langle m, k \rangle \in \mathbb{Z},
\]
because both $m,k \in \mathbb{Z}^d$. So we have $A \subset B$.
For the other direction, suppose that $y\in B$, so by definition
\begin{equation*}\label{def. of B}
\langle y, n \rangle \in \mathbb{Z}, \text{ for all } n \in {\mathcal L}.
\end{equation*}
We need to show that $y = M^{-T} k$ for some $k\in \mathbb{Z}^d$, which is equivalent to $M^T y \in \mathbb{Z}^d$.
Noticing that the $k$'th element of $M^T y$ is
$\langle n, y \rangle$ with $n$ belonging to a basis of ${\mathcal L}$, we are done, by \eqref{def. of B}.
Therefore $A=B$.
\end{proof}
\end{lem}
\bigskip
\begin{example}
\rm{
Let ${\mathcal L} := r \mathbb{Z}^d$, the integer lattice dilated by a positive real number $r$. It's dual lattice is ${\mathcal L}^* = \frac{1}{r} {\mathcal L}$, because a basis for ${\mathcal L}$ is $M := rI$, implying that a basis matrix for ${\mathcal L}^*$ is $M^{-T} = \frac{1}{r} I$.
We also notice that $\det {\mathcal L} = r^d$, while $\det {\mathcal L}^* = \frac{1}{r^d}$.
}
\hfill $\square$
\end{example}
A fundamental relation between a full-rank lattice and its dual follows immediately from Definition
\ref{first definition of dual lattice}: $\det({\mathcal L}^*) := \det(M^{-T}) = \frac{1}{\det M}= \frac{1}{\det {\mathcal L}}$,
which we record as:
\begin{equation} \label{FundamentalDualRelation}
(\det {\mathcal L} )(\det {\mathcal L}^*) = 1.
\end{equation}
If we consider any integer sublattice of $\mathbb{Z}^d$, say ${\mathcal L} \subset \mathbb{Z}^d$, together with its dual lattice ${\mathcal L}^*$ in the same space, some interesting relations unfold between them. Let's consider an example.
\bigskip
\begin{example} \label{dual lattice example}
\rm{
In $\mathbb{R}^2$, let
${\mathcal L} := \left\{m \icol{1\{\bf 1}} + n \icol{1\\ 4} \bigm | m,n \in \mathbb{Z} \right\}$,
a lattice with $\det {\mathcal L} = 3$ that is depicted in Figure \ref{DualLattice} by the larger green balls.
Its dual lattice is
\[
{\mathcal L}^*:= \left\{
\frac{1}{3} \left( a\icol{\ 4\\-1} +b \icol{ \ \, -1\\ \ \ \ 1} \right) \bigm | a, b \in \mathbb{Z}
\right\},
\]
whose determinant equals $\frac{1}{3}$, and is depicted in Figure
\ref{DualLattice} by the smaller orange balls. So ${\mathcal L}$ is a coarser lattice than ${\mathcal L}^*$.
}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.3in]{DualLattice}
\end{center}
\caption{The lattice of Example \ref{dual lattice example}, together with its dual lattice}
\label{DualLattice}
\end{figure}
We can verify that the relation \eqref{FundamentalDualRelation} works for this example: $\det {\mathcal L}^* = \frac{1}{3} = \frac{1}{\det {\mathcal L}}$. We also notice that ${\mathcal L}$ is a sublattice of ${\mathcal L}^*$.
We may notice here that
${\mathcal L}^*/{\mathcal L}$ forms a finite group of order $9 = (\det {\mathcal L})^2$, which is equal to the number of cosets of
the coarser lattice ${\mathcal L}$ in the finer lattice ${\mathcal L}^*$.
\hfill $\square$
\end{example}
The dual lattice also appears as the kernel of a certain map,
as follows. Suppose that for each point $n \in {\mathcal L}$, we define a function called a {\bf character} of ${\mathcal L}$:
\begin{equation}
\chi_n(y) := e^{2\pi i \langle n, y \rangle},
\end{equation}
whose domain is the whole space $\mathbb{R}^d$. To see a connection with the dual lattice ${\mathcal L}^*$, we may consider the simultaneous kernel of all of these functions taken together:
\[
\rm{Ker}:= \{ x \in \mathbb{R}^d \mid \chi_n(x)=1, \text{ for all } n \in {\mathcal L}\}.
\]
It's clear that $\rm{Ker} = {\mathcal L}^*$,
because $e^{2\pi i z} = 1$ if and only if $z \in \mathbb{Z}$.
Now let's consider the collection of all of these characters:
\begin{equation}
G_{\mathcal L} := \{ \chi_n \mid n \in {\mathcal L} \}.
\end{equation}
If we multiply these character together by defining $\chi_n \chi_m:= \chi_{n+m}$, then
$G_{\mathcal L}$ forms a group, called the {\bf group of characters} of ${\mathcal L}$. To see that this multiplication makes sense, we can compute:
\[
(\chi_n \chi_m) (x) =
e^{2\pi i \langle n, x \rangle} e^{2\pi i \langle m, x \rangle}
=e^{2\pi i \langle n+m, x \rangle} =
\chi_{n+m}(x).
\]
Even more is true: $G_{\mathcal L}$ is isomorphic, as a group, to the lattice ${\mathcal L}$ (Exercise \ref{character group}) via
the map $n \rightarrow \chi_n$.
Intuitively, one of the great benefits of group characters is that by using the magic of just two-dimensional complex numbers, we can study high-dimensional lattices.
\begin{example}
For the integer lattice $\mathbb{Z}^d$, its group of characters is composed of the following functions, by definition:
\[
\chi_n(x):= e^{2\pi i \langle n, x \rangle},
\]
for each $n \in \mathbb{Z}^d$.
\hfill $\square$
\end{example}
\bigskip
\section{The successive minima of a lattice: \\
more geometry of numbers}
To warm up, we recall a very classical inequality of Hadamard, giving a bound on determinants.
Intuitively, Hadamard's inequality tells us that if we keep all the lengths of the sides of a parallelepiped constant, and consider all possible parallelepipeds ${\mathcal P}$
with these fixed side lengths, then the volume of ${\mathcal P}$ is maximized exactly when ${\mathcal P}$ is rectangular.
\begin{thm}[Hadamard's inequality] \label{Hadamard inequality} \index{Hadamard's inequality}
Given a non-singular matrix $M$, over the reals, whose column vectors are
$v_1, \dots, v_d$, we have:
\[
|\det M| \leq \| v_1 \| \|v_2\| \cdots \|v_d\|,
\]
\end{thm}
with equality if and only if all of the $v_k$'s are pairwise orthogonal.
\begin{proof}
We use the following matrix decomposition from Linear Algebra: $M = QR$, where $Q$ is an orthogonal
matrix, $R:= [r_{i,j}]$ is an upper-triangular matrix, and $r_{kk} > 0$ (this decomposition is a well-known consequence of the Gram-Schmidt process applied to the columns of M). So now we know that $|\det Q| = 1$, and $\det R = \prod_{k=1}^d r_{kk}$, and it follows that
\[
|\det M| = |\det Q \det R| = \det R = \prod_{k=1}^d r_{kk}.
\]
Let's label the columns of $Q$ by $Q_k$, and the columns of $R$ by $R_k$. We now consider the matrix
$M^T M = R^T Q^T Q R = R^T R$. Comparing the diagonal elements on both sides of $M^T M = R^T R$,
we see that $\| Q_k\|^2 = \| R_K \|^2$. But we also have $\| R_K \|^2 \geq r_{kk}^2$, so that
$\|Q_k \| \geq r_{kk}$.
Altogether we have
\begin{equation} \label{product formula}
|\det M| = \prod_{k=1}^d r_{kk} \leq \prod_{k=1}^d \|Q_k \|.
\end{equation}
The case of equality occurs if and only if $\| R_K \|^2 = r_{kk}^2$ for all $1\leq k\leq d$, and this case of equality would mean
that $R$ is a diagonal matrix. Thus, we have equality in inequality \eqref{product formula} if and only if
$M^T M = R^T R$ is a diagonal matrix, which means that the columns of $M$ are mutually orthogonal.
\end{proof}
A very important characteristic of a lattice ${\mathcal L}$ is the {\bf length of its shortest nonzero vector}:
\index{shortest nonzero vector in a lattice}
\[
\lambda_1({\mathcal L}):=\min \left\{ \| v \| \biggm | v \in {\mathcal L}-\{0\} \right\}.
\]
Every lattice has at least two shortest nonzero vectors, because if $v \in {\mathcal L}$, then $-v \in {\mathcal L}$. Thus, when we use the words `its shortest vector', we always mean that we are free to make a choice between any of its vectors that have the same shortest, nonzero length. A natural question, which has many applications, is ``how short is the shortest nonzero vector in ${\mathcal L}$, as we somehow
vary over all normalized lattices ${\mathcal L}$?"
\begin{example}
\rm{
We define the following lattice in $\mathbb{R}^2$:
\[
{\mathcal L} := \left\{ m \icol{102 \\ 11 } + n \icol{200\\16} \bigm | m,n\in\mathbb{Z} \right\}.
\]
What is the shortest nonzero vector in this lattice ${\mathcal L}$? Without using any fancy
Theorems, we might still try simple subtraction, sort of
mimicking the Euclidean algorithm. So for example, we might try $\icol{200\\16} - 2 \icol{102 \\ 11 } = \icol{-4 \\ -6 }$, which is pretty short. So we seem to have gotten lucky - we found a relatively short vector. But here
comes the impending question:
\begin{question}
How do we know whether or not this is really the shortest nonzero vector in our lattice ${\mathcal L}$?
Can we find an even shorter vector in ${\mathcal L}$?
\end{question}
This is not easy to answer in general, and we need to learn a bit more theory even to approach it in $\mathbb{R}^2$.
Moreover, in dimensions $d\geq 3$, the corresponding problem of finding a shortest nonzero vector in any given lattice is terribly difficult - it is considered to be one of the most difficult problems in computational number theory.
}
\hfill $\square$
\end{example}
To capture the notion of the second-smallest vector in a lattice, and third-smallest vector, etc, we begin by
imagining balls of increasing radii, centered at the origin, and we can (at least theoretically) keep track of how
they intersect ${\mathcal L}$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.6in]{SuccesiveMinima}
\end{center}
\caption{The two successive minima for this lattice ${\mathcal L}$ are $\lambda_1({\mathcal L}) = \sqrt 2$, and $\lambda_2({\mathcal L}) = \sqrt 5$.} \label{Successive Minima}
\end{figure}
Let $B^d(r)$ be the closed ball of radius $r$, centered at the origin.
For each fixed $j$, with $1 \leq j \leq d$, let $r$ be the smallest positive real number such that
$B^d(r)$ contains at least $j$ linearly independent lattice points of ${\mathcal L}$.
This value of $r$ is called $\lambda_j({\mathcal L})$,
the {\bf $j$'th successive minima} of the lattice.
Another way of saying the same thing is:
\begin{equation}
\lambda_j({\mathcal L}) := \min \left\{ r >0 \bigm |
\dim \big( \text{span }({\mathcal L} \cap {B^d(r)}) \big) \geq j \right\}.
\end{equation}
By definition, we have $| \lambda_1({\mathcal L}) | \leq | \lambda_2({\mathcal L}) | \leq \cdots \leq | \lambda_d({\mathcal L}) |$.
\medskip
\begin{example}
\rm{
For ${\mathcal L}:= \mathbb{Z}^d$, the shortest nonzero vector has length $\lambda_1(\mathbb{Z}^d) = 1$, and the successive minima for $\mathbb{Z}^d$ all have the same value. One choice for their corresponding vectors is
$v_1:= {\bf e_1}, \dots, v_d:= {\bf e_d}$, the standard basis vectors.
}
\hfill $\square$
\end{example}
\medskip
\begin{example}
\rm{
In $\mathbb{R}^2$, there is a very special
lattice, sometimes called the {\bf hexagonal lattice} (also known as the {\bf Eisenstein lattice}):
\[
{\mathcal L} := \left\{ m { \frac{\sqrt{3}}{2} \choose \frac{1}{2} } + n {1 \choose 0} \mid m,n\in\mathbb{Z} \right\}.
\]
This lattice has $\det {\mathcal L} = \frac{\sqrt{3}}{2}$ and is generated by the $6$'th roots of unity (Exercise \ref{Eisenstein lattice}). Given the basis above, we see that here we have $\lambda_1({\mathcal L}) = \lambda_2({\mathcal L}) =1$.
It also turns out to be an
extremal lattice in the sense that it (more precisely a dilate of it) is the lattice that achieves Hermite's constant $\gamma_2$, below, over all lattices in $\mathbb{R}^2$.
(Exercise \ref{minimal lattice in R^2}).
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.0in]{EisensteinLattice}
\end{center}
\caption{The hexagonal lattice, also known as the Eisenstein lattice} \label{Eisenstein Lattice}
\end{figure}
\medskip
\begin{example} \label{a curve in the space of lattice}
\rm{
Let's define the following family of $2$-dimensional lattices. For each $t > 0$, we let
\[
M:= \begin{pmatrix}
e^t & 0 \\
& e^{-t} \end{pmatrix}, \text{ and we let } {\mathcal L}_t:= M(\mathbb{Z}^d),
\]
so that we get a parametrized family of lattices. While all of the lattices in this family have $\det {\mathcal L}=1$, their shortest nonzero vectors approach $0$ as $t\rightarrow \infty$, since $\lambda_1({\mathcal L}_t) = e^{-t}$.
So we see that it does not necessarily make sense to talk about the shortest nonzero vector among a collection of lattices, but it will make sense to consider a ``max-min problem'' of this type
(Hermite's constant \eqref{Hermite's constant} below).
}
\hfill $\square$
\end{example}
For each dimension $d$, we define {\bf Hermite's constant} as follows:
\begin{equation}\label{Hermite's constant}
\gamma_d := \max \left\{ \lambda_1({\mathcal L})^2 \bigm | {\mathcal L} \text{ is a full-rank lattice in $\mathbb{R}^d$, with } \det {\mathcal L} =1 \right\}.
\end{equation}
In words, Hermite's constant is retrieved by varying over all normalized lattices in $\mathbb{R}^d$, which have determinant $1$, picking out the smallest squared norm of any nonzero vector in each lattice, and then
taking the maximum of these smallest norms. In a later chapter, on sphere packings, we will see an interesting interpretation of Hermite's constant in terms of the densest lattice packing of spheres.
We next give a simple bound, in Theorem \ref{First successive minima bound} below, for the shortest nonzero
vector in a lattice and hence for Hermite's constant. But first we need to give
a simple lower bound for the volume of the unit ball, in Lemma \ref{volume bound for the ball}.
Curiously, Hermite's constant $\gamma_d$ is only known
precisely for $1 \leq d \leq 8$, and $d=24$, as of this writing.
\bigskip
\begin{lem}\label{volume bound for the ball}
\[
\vol B^d(r) \geq \left( \frac{2r}{\sqrt d} \right)^d.
\]
\end{lem}
\begin{proof}
The cube $C:= \left\{ x\in \mathbb{R}^d \bigm |
\text{ all } |x_k| \leq \frac{r}{\sqrt d} \right\}$ is contained in the ball $B^d(r)$:
if $x \in C$ then $\sum_{k=1}^d x_k^2 \leq d \left( \frac{r}{\sqrt d} \right)^2 = r^2$. So the volume of the ball $B^d(r)$ is greater than the volume of the cube, which is equal to $\left( \frac{2r}{\sqrt d} \right)^d$.
\end{proof}
\bigskip
The following result of Minkowski gives a bound for the shortest nonzero vector in a lattice.
\begin{thm}[Minkowski] \label{First successive minima bound}
Suppose that ${\mathcal L} \subset \mathbb{R}^d$ is a full-rank lattice. Then the shortest nonzero vector $v \in {\mathcal L}$ satisfies
\begin{equation}\label{shortest vector in a lattice bound}
\|v\| \leq \sqrt{d} (\det {\mathcal L})^{\frac{1}{d}}.
\end{equation}
Equivalently, we will often write
\[
\lambda_1({\mathcal L}) \leq \sqrt{d} (\det {\mathcal L})^{\frac{1}{d}}.
\]
\end{thm}
\begin{proof}
The idea is to apply Minkowski's convex body Theorem \ref{Minkowski convex body Theorem for L}
to a ball of sufficiently large radius.
Let $r:= \lambda_1({\mathcal L})$ be the length of the shortest nonzero vector in ${\mathcal L}$, and consider the
ball $B^d(r)$ of radius $r$. By definition, $B^d(r)$ does not contain any lattice points of ${\mathcal L}$ in its interior.
So by Minkowski's convex body Theorem, and Lemma \ref{volume bound for the ball},
\[
\left( \frac{2 \lambda_1({\mathcal L})}{\sqrt d} \right)^d
\leq
\vol B^d(r)
\leq
2^d \det {\mathcal L}.
\]
It follows that $\lambda_1({\mathcal L}) \leq \sqrt{d} \left( \det {\mathcal L} \right)^{\frac{1}{d}}$, proving the claim. \\
\end{proof}
Despite the bound \eqref{shortest vector in a lattice bound} on the shortest nonzero vector in a lattice, there are currently no known efficient algorithms to find such a vector for an arbitrary lattice, and it is thought to be one of the most difficult problems we face today. In practice, researchers often use the LLL algorithm
to find a `relatively short' vector in a given lattice, and the same algorithm even finds a relatively short basis for ${\mathcal L}$.
We already have enough knowledge to relate the length of a shortest nonzero vector of a lattice ${\mathcal L}$ to the length of a
shortest nonzero vector of its dual lattice ${\mathcal L}^*$, as follows.
\begin{cor}
Let ${\mathcal L}\subset \mathbb{R}^d$ be a full-rank lattice, and let ${\mathcal L}^*$ be its dual lattice. Then
\[
\lambda_1({\mathcal L}) \lambda_1({\mathcal L}^*) \leq d.
\]
\end{cor}
\begin{proof}
By Minkowski's bound, namely Theorem \ref{First successive minima bound}, applied to both ${\mathcal L}$ and ${\mathcal L}^*$, we have:
\[
\lambda_1({\mathcal L}) \lambda_1({\mathcal L}^*) \leq \sqrt{d} (\det {\mathcal L})^{\frac{1}{d}} \sqrt{d} (\det {\mathcal L}^*)^{\frac{1}{d}} = d,
\]
using the relation $(\det {\mathcal L})( \det {\mathcal L}^*) = 1$.
\end{proof}
Such relations are called {\bf transference theorems}, as they can transfer the complexity of computing a lattice parameter to the complexity of computing a (perhaps different) parameter in the dual lattice.
While we may not know explicitly all of the short vectors in a given lattice, it is often still useful to construct
an ellipsoid that is based on the successive minima of a lattice, as we do below. In the spirit of reviewing basic concepts from Linear Algebra, an
{\bf ellipsoid boundary} \index{ellipsoid}
centered at the origin is defined by the $(d-1)$-dimensional body
\begin{equation} \label{ellipsoid}
\left\{ x\in \mathbb{R}^d \bigm | \sum_{j=1}^d \frac{{\langle x, b_j\rangle}^2}{c_j^2} =1 \right\},
\end{equation}
for some fixed orthonormal basis
$\{ b_1, \dots, b_d \}$ of $\mathbb{R}^d$. Here the vectors $b_j$ are called the {\bf principal axes} of the ellipsoid, and
the $c_j$'s are the lengths along the principal axes of the ellipsoid.
A more geometric way of
defining an ellipsoid (which turns out to be equivalent to our definition above) is attained
by applying a linear transformation $M$ to the unit sphere
$S^{d-1} \subset \mathbb{R}^d$ (Exercise \ref{Ellipsoid problem}). For the next couple of lemmas, we follow the approach taken by Regev.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{Ellipsoid}
\end{center}
\caption{An ellipsoid in $\mathbb{R}^3$.} \label{Ellipsoid}
\end{figure}
\bigskip
\begin{lem}\label{Empty Ellipsoid}
\rm{
Corresponding to the successive minima of a full-rank lattice ${\mathcal L}$, we have
$d$ linearly independent vectors
$v_1, \dots, v_d$, so that by definition $\| v_k \| := \lambda_k({\mathcal L})$. We apply the
Gram-Schmidt algorithm to this set of vectors $\{v_1, \dots, v_d\}$, obtaining a corresponding orthonormal basis
$\{b_1, \dots, b_d\}$ for $\mathbb{R}^d$.
Now, we define the following open {\bf ellipsoid} by:
\begin{equation}\label{open ellipsoid}
E:=\left\{ x\in \mathbb{R}^d \bigm | \sum_{k=1}^d \frac{{\langle x, b_k \rangle}^2}{{\lambda_k}^2}
< 1
\right\},
\end{equation}
whose axes are the $b_k$'s, and whose radii are the $\lambda_k:= \lambda_k({\mathcal L})$.
We claim that $E$ does not contain any lattice points of ${\mathcal L}$.
}
\end{lem}
\begin{proof}
We fix any vector $v \in {\mathcal L}$.
Let $1\leq k \leq d$ be the maximal index such that $\lambda_k({\mathcal L}) \leq \|v\|$.
We may write $v = \sum_{j=1}^d \langle v, b_j \rangle b_j$, so that
$\|v\|^2 = \sum_{j=1}^d {\langle v, b_j \rangle}^2$.
Now $v$ must lie in $\text{span}\{v_1, \dots v_k\} = \text{span}\{b_1, \dots b_k\}$, for some $1\leq k\leq d$.
Hence we may write $v = \sum_{j=1}^d \langle v, b_j \rangle b_j = \sum_{j=1}^k \langle v, b_j \rangle b_j $,
so that
$\|v\|^2 = \sum_{j=1}^k | \langle v, b_j \rangle |^2$.
We now check if $v$ is contained in $E$:
\[
\sum_{j=1}^d \frac{{\langle v, b_j \rangle}^2}{{\lambda_j}^2} =
\sum_{j=1}^k \frac{{\langle v, b_j \rangle}^2}{{\lambda_j}^2} \geq
\frac{1}{ {\lambda_k}^2 } \sum_{j=1}^k {\langle v, b_j \rangle}^2
= \frac{ \|v\|^2 }{ {\lambda_k}^2 } \geq 1,
\]
so that $v\not\in E$.
\end{proof}
More generally, we have the following refinement of Theorem \ref{First successive minima bound}, which
gives us a bound on the first $d$ shortest (nonzero) vectors in a lattice.
\begin{thm}[Minkowski's second theorem] \label{successive minima bound}
The successive minima of a full-rank lattice ${\mathcal L}$ enjoy the property:
\[
\Big( \| \lambda_1({\mathcal L}) \| \cdots \|\lambda_d({\mathcal L}) \| \Big)^{\frac{1}{d}}
\leq \sqrt{d} \Big(\det {\mathcal L}\Big)^{\frac{1}{d}}.
\]
\end{thm}
\begin{proof} Using Lemma \ref{Empty Ellipsoid}, the ellipsoid $E$ contains no lattice points belonging to
${\mathcal L}$, so that by Minkowski's convex body Theorem, we have $\vol E \leq 2^d \det {\mathcal L}$.
We also know that
\[
\vol E = \left( \prod_{j=1}^d \lambda_j \right) \vol B_1 \geq
\left( \prod_{j=1}^d \lambda_j \right) \left(\frac{2}{\sqrt{d}} \right)^d.
\]
Altogether, we have
\[
2^d \det {\mathcal L} \geq \vol E \geq \left( \prod_{j=1}^d \lambda_j \right) \left(\frac{2}{\sqrt{d}} \right)^d,
\]
arriving at the desired inequality.
\end{proof}
We notice that
$ \Big(\| \lambda_1({\mathcal L}) \| \cdots \|\lambda_d({\mathcal L}) \| \Big)^{\frac{1}{d}}
\geq \| \lambda_1({\mathcal L}) \|$, because $\| \lambda_1({\mathcal L}) \| \leq \| \lambda_k({\mathcal L}) \| $
for all indices $1 < k \leq d$. We therefore see that Theorem \ref{successive minima bound} is indeed a refinement
of Theorem \ref{First successive minima bound}.
\medskip
\begin{example}
\rm{
The $E_8$ lattice is defined by
\begin{equation} \label{E_8}
E_8 := \left\{ (x_1, x_2, \cdots x_8) \in \mathbb{Z}^8 \cup \Big(\mathbb{Z} + \frac{1}{2} \Big)^8 \ \bigm | \ \sum_{k=1}^8 x_k
\equiv 0 \mod 2
\right\}.
\end{equation}
It turns out that the $E_8$ lattice gives the optimal solution to the sphere packing problem, as well as the optimal solution for the kissing number problem in $\mathbb{R}^8$.
}
\hfill $\square$
\end{example}
\begin{comment}
\section{The packing radius, and the covering radius}
Suppose we are given a convex body $K \subset \mathbb{R}^d$, containing the origin, and a full-rank lattice ${\mathcal L}\subset \mathbb{R}^d$.
We define
the {\bf packing radius of the lattice ${\mathcal L}$}, relative to $K$, written as $\mu(K, {\mathcal L})$,
as the largest $r>0$ such that
$(r K + l_1) \cap (r K + l_2) \not= \phi$, for all $l_1, l_2 \in {\mathcal L}$. In other words, the packing radius is the largest dilate of $K$ such that after translating $K$ by all elements of the lattice ${\mathcal L}$, we do not have any overlapping bodies.
More compactly, we may also give the following description for the covering radius:
\begin{align}
\mu(K, \mathcal L)
&:= \min \{ r \geq 0 \mid r K + \mathcal L = \mathbb{R}^d \} \\
&= \min \{ r \geq 0 \mid r K + \mathcal L \text{ is a covering} \}
\end{align}
In a of dual manner, the packing radius of $K \subset \mathbb{R}^d$ with respect to a lattice ${\mathcal L}$ is defined by
In a somewhat dual fashion, we also define the {\bf covering radius of the lattice ${\mathcal L}$}, relative to $K$, as the
largest $r>0$ such that every point
When $K:= B(r)$, the open ball of radius $r$, it is traditional to omit $K$ in the notation, and
we simply write the packing radius in this case as $\mu(K, {\mathcal L}):= \mu({\mathcal L})$. In words, when $K$ is an open ball,
the packing radius is the largest $r > 0$ such that the collection of
open balls $B(v, r)$ centered at all lattice points of ${\mathcal L}$ do not intersect.
It is immediate to see that the packing radius of a lattice equals precisely half the minimum distance.
The covering radius $\mu({\mathcal L})$ is defined by the smallest $\mu >0$ s.t. the collection of open balls
centered at all lattice points of ${\mathcal L}$ cover $\mathbb{R}^d$.
The packing radius $\mu(L)$ equals the inradius of $V(L)$, the covering radius............. equals
the circumradius of $V(L)$, and the quantizer constant is $G(L) = (\det {\mathcal L})^{1+ 2/n} \int_{V(L)} \|x\|^2 dx$.
\end{comment}
\section{Hermite normal form}
We call a lattice ${\mathcal L}$ an {\bf integral lattice} if ${\mathcal L} \subset \mathbb{Z}^d$.
Further, we may recall that any lattice ${\mathcal L} \subset \mathbb{R}^d$ has infinitely many bases, so it may seem impossible at first to associate a single matrix with a given lattice. However, there is an elegant way to do this, as follows.
\begin{example}\label{first ex. of HNF}
{\rm
Suppose we are given a lattice ${\mathcal L}$ as the integral span of the vectors
\[
v_1:= \icol{3\{\bf 1}}, v_2:= \icol{-2\\ \ 2},
\]
which clearly has determinant $8$.
Then any integer linear combinations of $v_1$ and $v_2$ is still in ${\mathcal L}$. In particular, mimicking Gaussian elimination, we place $v_1$ and $v_2$ as rows of a matrix, and row-reduce over the integers:
\begin{align*}
\begin{pmatrix}
\ \ 3 & \ 1 \\
-2 & \ 2
\end{pmatrix}
\rightarrow
\begin{pmatrix}
3 & \ 1 \\
1 & \ 3
\end{pmatrix}
\rightarrow
\begin{pmatrix}
0 & \ -8 \\
1 & \ \ \ 3
\end{pmatrix}
\rightarrow
\begin{pmatrix}
1 & \ \ 3 \\
0 & -8
\end{pmatrix}
\rightarrow
\begin{pmatrix}
1 & 3 \\
0 & 8
\end{pmatrix},
\end{align*}
where at each step we performed row operations (over $\mathbb{Z}$) that did not change the lattice. Hence we have a reduced basis for ${\mathcal L}$, consisting of $\icol{1\\3}$ and $\icol{0\\8}$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.2in]{HNF}
\end{center}
\caption{
The lattice ${\mathcal L}$ of Example \ref{first ex. of HNF}, depicted by the bold green points, and showing the
original basis $\{ v_1, v_2\}$ of ${\mathcal L}$, and the Hermite-reduced basis of ${\mathcal L}$
}
\label{HNF.pic}
\end{figure}
We notice that the resulting matrix is upper-triangular, with positive integers on the diagonal, nonnegative integers elsewhere, and in each column the diagonal element is the largest element in that column.
There is another way to interpret the matrix reductions above, by using unimodular matrices, as follows. The first reduction step can be accomplished by the multiplication on the left by a unimodular matrix:
\[
\begin{pmatrix}
1 & \ 0 \\
1 & \ 1
\end{pmatrix}
\begin{pmatrix}
\ 3 & \ 1 \\
-2 & \ 2
\end{pmatrix}
=
\begin{pmatrix}
3 & \ 1 \\
1 & \ 3
\end{pmatrix}
\]
Similarly, each step in the reduction process can be interpreted by multiplying on the left by some new unimodular matrix, so that at the end of the process we have a product of unimodular matrices times
our original matrix $\begin{pmatrix}
\ 3 & \ 1 \\
-2 & \ 2
\end{pmatrix}$. Because a product of unimodular matrices is yet another unimodular matrix, we can see that we arrived at a reduction of the form:
\[
U
\begin{pmatrix}
\ 3 & \ 1 \\
-2 & \ 2
\end{pmatrix}
=
\begin{pmatrix}
1 & \ 3 \\
0 & 8
\end{pmatrix},
\]
where $U$ is a unimodular matrix.
\hfill $\square$
}
\end{example}
The point of Example \ref{first ex. of HNF} is that a similar matrix reduction persists for all integer lattices, culminating in the following result, which just hinges on the fact that $\mathbb{Z}$ has a division algorithm.
\begin{thm} \label{theorem.HNF}
Given an invertible integer $d\times d$ matrix $M$, there exists a unimodular matrix $U$ with
$UM = H$, such that $H$ satisfies the following conditions:
\begin{enumerate}
\item $[H]_{i, j} = 0$ if $i>j$.
\item $[H]_{i, i} > 0$, for each $1\leq i \leq d$.
\item $0 \leq [H]_{i, j} < [H]_{i, i} $, for each $i >j $. \label{third property}
\end{enumerate}
Property \ref{third property} tells us that each diagonal element $[H]_{i, i}$ in the $i$'th column of $H$ is the largest element in the $i$'th column.
Moreover, the matrix $H$ is the unique integer matrix that satisfies the above conditions.
\hfill $\square$
\end{thm}
The matrix $H$ in Theorem \ref{theorem.HNF} is called the {\bf Hermite normal form} of $M$.
To associate a unique matrix to a given integral full-rank lattice ${\mathcal L} \subset \mathbb{R}^d$, we first choose
any basis of ${\mathcal L}$, and we then construct a $d\times d$ integer matrix $M$ whose rows are the basis vectors that we chose. We then apply Theorem \ref{theorem.HNF} to $M$, arriving at an integer matrix $H$ whose rows are another basis of ${\mathcal L}$, called the {\bf Hermite-reduced basis}.
\begin{cor}
There is a one-to-one correspondence between full-rank integral lattices in $\mathbb{R}^d$ and
integer $d \times d$ matrices in their Hermite Normal Form.
\hfill $\square$
\end{cor}
\section{The Voronoi cell of a lattice}
The {\bf Voronoi cell} of a lattice ${\mathcal L}$, at the origin, is defined by
\begin{equation}
\text{Vor}_0({\mathcal L}) := \left\{ x \in \mathbb{R}^d \bigm | \|x\| \leq \|x - v\|, \ \text{ for all } v \in {\mathcal L} \right\}.
\end{equation}
In other words, the Voronoi cell $\text{Vor}_0({\mathcal L})$ of a lattice ${\mathcal L}$ is the set of all point in space that are closer to the origin than to any other lattice point in ${\mathcal L}$. Because the origin wins the battle of minimizing this particular distance function, it is also possible to construct the Voronoi cell by using half-spaces. Namely, for each $v\in {\mathcal L}$, we define the half-space
\[
H_v:= \left\{ x \in \mathbb{R}^d \bigm | \langle x, v \rangle \leq \tfrac{1}{2} \|v\| \right\},
\]
and we observe that the Voronoi cell may also be given by
\[
{\rm Vor}_0({\mathcal L}) = \bigcap_{v\in {\mathcal L}- \{0\}} H_v,
\]
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.2in]{VoronoiConstruction}
\end{center}
\caption{Top left: a sublattice ${\mathcal L}$ of $\mathbb{Z}^2$, of index $3$. Top right: $v \in {\mathcal L}$ is one of the $6$ relevant vectors, with its corresponding half-plane $H_v$, helping to define the Voronoi cell at the origin.
Bottom: The Voronoi cell $\text{Vor}_0({\mathcal L})$, \index{Voronoi cell} a symmetric hexagon of area $3$, with its $6$ relevant (heavy blue) lattice points of ${\mathcal L}$.
}
\label{VoronoiConstruction}
\end{figure}
as drawn in Figure \ref{VoronoiConstruction}.
It is easy to observe that the Voronoi cell of a lattice is symmetric about the origin, convex, and compact (Exercise \ref{facts about Vor cell}). So we may expect that Minkowski's theorems apply to $\text{Vor}_0({\mathcal L})$, as we see in the proof of Lemma \ref{basic Voronoi lemma} below.
It's also useful to define an analogous Voronoi cell located at each lattice point $m \in {\mathcal L}$:
\begin{equation}
\text{Vor}_m({\mathcal L}) := \left\{ x \in \mathbb{R}^d \bigm | \|x - m \| \leq \|x - v\|, \ \text{ for all } v \in {\mathcal L} \right\}.
\end{equation}
A moment's thought (but this is good practice - Exercise \ref{translating the Voronoi cell around})
reveals that a translation of the Voronoi cell at the origin is exactly the Voronoi cell at another lattice point of ${\mathcal L}$, namely:
\begin{equation} \label{translated Voronoi cells}
\text{Vor}_0({\mathcal L}) + m = \text{Vor}_m({\mathcal L}).
\end{equation}
\begin{lem} \label{basic Voronoi lemma}
Given a full-rank lattice ${\mathcal L}\subset \mathbb{R}^d$, whose Voronoi cell at the origin is $K$, we have:
\begin{enumerate}[(a)]
\item $K$ tiles $\mathbb{R}^d$ by translations with ${\mathcal L}$. \label{part 1 of Voronoi}
\item
$
\vol (K) = \det {\mathcal L}. \label{part 2 of Voronoi}
$
\end{enumerate}
\end{lem}
\begin{proof}
Part \ref{part 1 of Voronoi} follows from the observation that
any $x \in \mathbb{R}^d$, there exists a lattice point $m \in {\mathcal L}$ that is at least as close to $x$ as it is to any other lattice point of ${\mathcal L}$.
In other words, $ \|x - m\| \leq \|x - v\|, \forall v\in {\mathcal L}$, and so
$x \in \text{Vor}_m({\mathcal L})$.
From \eqref{translated Voronoi cells} we see that $x$ is covered by the translate $\text{Vor}_0({\mathcal L}) + m$. It's also clear that as $n$ varies over ${\mathcal L}$, all of the interiors of the translates $\text{Vor}_0({\mathcal L}) + n$ are disjoint, so that
$K:= \text{Vor}_0({\mathcal L})$ tiles $\mathbb{R}^d$ by translations with ${\mathcal L}$.
To prove part \ref{part 2 of Voronoi}, we let $B:= 2K$.
By Theorem \ref{thm:extremal bodies} (regarding extremal bodies), we know that
$\frac{1}{2}B = K$ tiles $\mathbb{R}^d$ with the lattice ${\mathcal L}$ if and only if $ \vol(B) = 2^d \det {\mathcal L}$. Since
\ref{part 1 of Voronoi} tells us that
$K=\frac{1}{2}B$ tiles with the lattice ${\mathcal L}$,
we see that
$\vol K = \vol\Big(\frac{1}{2}B\Big) = \frac{1}{2^d} \vol B = \det {\mathcal L}$.
\end{proof}
The proof above shows that the Voronoi cell of ${\mathcal L}$ is also an extremal body for ${\mathcal L}$, according to Theorem \ref{thm:extremal bodies}.
\medskip
\begin{example}\label{D_n lattices}
\rm{
The $D_n$ lattice is defined by
\[
D_n:= \left\{ x\in \mathbb{Z}^n \bigm | \sum_{k=1}^n x_k \equiv 0 \mod 2 \right\}.
\]
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{D2andD3}
\end{center}
\caption{Left: the $D_2$ lattice. Right: the $12$ shortest nonzero vectors of the $D_3$ lattice, inscribed in the cube $[-1, 1]^3$.}
\label{The $D_2$ and $D_3$ lattices}
\end{figure}
In $\mathbb{R}^4$, the $D_4$ lattice turns out to be a fascinating object of study.
The Voronoi cell $\text{Vor}_0(D_4)$ is called the {\bf $24$-cell}, \index{$24$-cell}
and is depicted in Figure \ref{24-cell}. It is a $4$-dimensional polytope with some wonderful properties - for example, it is one of the few polytopes that is self-dual. It is also an example of a polytope ${\mathcal P}$ in the lowest possible dimension $d$ (namely $d=4$) such that ${\mathcal P}$ tiles $\mathbb{R}^d$ by translations, and yet ${\mathcal P}$ is not a zonotope.
By Lemma \ref{lemma:lattice defined by congruence},
we see that $\det D_n = 2$. }
The lattice $D_n$ is often called the ``checkerboard'' lattice, because $\det D_n = 2$ means there are exactly two cosets
in $\mathbb{Z}^d / D_n$. Finally, the dual lattice $D_n^*$ is equal to the lattice
\[
\\mathbb{Z}^d \cup \left( \mathbb{Z}^d + \left( \tfrac{1}{2}, \cdots, \tfrac{1}{2} \right)^T \right),
\]
which we leave for the pleasure of the reader (Exercise \ref{dual of D_n}).
\hfill $\square$
\end{example}
A fascinating open problem is the Voronoi conjecture, named after
the Ukrainian mathematician Georgy Voronoi, who formulated it in 1908.
Two polytopes ${\mathcal P}, Q$ are called {\bf affinely equivalent} \index{affinely equivalent} if
${\mathcal P} = M(Q) + v$, where $M \in GL_d(\mathbb{R})$, and $v\in \mathbb{R}^d$.
\medskip
\begin{conjecture}[Voronoi] \index{Voronoi conjecture}
A polytope ${\mathcal P}$ tiles $\mathbb{R}^d$ by translations if and only if ${\mathcal P}$ is the Voronoi cell of some lattice ${\mathcal L}$, or ${\mathcal P}$ is affinely equivalent
to such a Voronoi cell.
\end{conjecture}
The main difficulty in the Voronoi conjecture is the apriori search among all of the (infinitely many) possible affinely equivalent images of such a Voronoi cell.
\medskip
\begin{example}
\rm{
For the lattice $A_n \subset \mathbb{R}^{n+1}$ defined in Example \ref{A_d example}, its Voronoi cell turns out to have beautiful and important properties: $A_2 \subset \mathbb{R}^3$ is a hexagon, $A_3 \subset \mathbb{R}^4$ is a truncated octahedron (one of the Fedorov solids), and so on (see Conway and Sloane \cite{ConwaySloan.book}).
}
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{24-cell}
\end{center}
\caption{The Voronoi cell of the $D_4$ lattice in $\mathbb{R}^4$, known as the $24$-cell.} \label{24-cell}
\end{figure}
\bigskip
\section{Quadratic forms and lattices} \label{quadratic forms}
The study of lattices is in a strong sense equivalent to the study of positive definite quadratic forms,
over integer point
inputs, for the following
simple reason. Any positive definite quadratic form $f:\mathbb{R}^d \rightarrow \mathbb{R}$
is defined by $f(x):= x^T A x$, where
$A$ is a positive definite matrix, so the image of the integer lattice under $f$ is
\[
\{ x^T A x \mid x \in \mathbb{Z}^d\}.
\]
On the other hand, any full-rank lattice in $\mathbb{R}^d$ is given by
${\mathcal L} := M(\mathbb{Z}^d)$, for some real non-singular matrix $M$. By definition, this implies that the square of the norm of any vector
in ${\mathcal L}$ has the following shape: $\| v\|^2 = v^T v = x^T M^T M x$, for some $x \in \mathbb{Z}^d$. We notice that
$M^T M$ in the last identity is positive definite.
We may summarize this discussion as follows. Given any lattice ${\mathcal L}:= M(\mathbb{Z}^d)$, we have
\[
\left\{ \| v\|^2 \bigm | v \in {\mathcal L} \} = \{ x^T A x \bigm | x \in \mathbb{Z}^d \right\},
\]
where $A:= M^T M$ is positive definite.
So the distribution of the (squared) norms of all vectors in a given lattice is equivalent to the image of $\mathbb{Z}^d$
under a positive
definite quadratic form.
Interestingly, despite this equivalence, for an arbitrary given lattice ${\mathcal L}$ it is not known in general whether the knowledge of
the norms of all vectors in ${\mathcal L}$ uniquely determines the lattice ${\mathcal L}$. In very small dimensions it is true,
but for dimensions $\geq 4$ there are some counterexamples due to Alexander Schiemann (\cite{Schiemann1}, \cite{Schiemann2}).
The above equivalence between lattices in $\mathbb{R}^d$ and quadratic forms is straightforward but often useful, because it allows both algebraic and analytic methods to come to bear on important problems involving lattices.
Gauss initiated the systematic study of finding the minimum value of positive definite, binary quadratic forms $f(x, y) := a x^2 + 2b xy + cy^2$, over
all integer inputs $(x, y) \in \mathbb{Z}^2$. Gauss' theory is also known as a reduction theory for positive definite binary quadratic forms, and is now a popular topic that can be found in many standard Number Theory books. By the discussion of this short section,
it is clear that minimizing positive definite quadratic forms is essentially equivalent
to finding a vector of smallest nonzero length in a lattice.
So far we worked with one lattice at a time, but it turns out to be fruitful to work with infinite sets of lattices simultaneously.
\begin{thm}[Mahler] \label{Mahler}
Fix $\rho >0, C>0$. Then any infinite sequence of lattices
${\mathcal L} \subset \mathbb{R}^d$ such that
\[
\min \left\{ \|x\| \bigm | x \in {\mathcal L}-\{0\} \right\} \geq \rho, \text{ and } \det {\mathcal L} \leq C,
\]
has an infinite convergent subsequence of lattices.
\end{thm}
In other words, Mahler realized that among all lattices of volume $1$,
if a sequence of lattices diverges, then it must be true that the lengths of the shortest nonzero vectors
of these lattices tend to zero.
To complete the story, we should define what it means for a sequence of lattices $\{ {\mathcal L}_n \}_{n=1}^\infty$ to converge to a fixed lattice $L$. One way to define this convergence is to say that there exists a
sequence of bases $\beta_n$ of the lattices ${\mathcal L}_n$ that converge to a basis $\beta$ of $L$, in the sense that the $j$'th basis vector of $\beta_n$ converges to the $j$'th basis vector of $\beta$.
\bigskip
\section*{Notes}
\begin{enumerate}[(a)]
\item Kurt Mahler \index{Mahler, Kurt} was one of the main contributors to the development of the Geometry of Numbers.
His Theorem \ref{Mahler} is often called Mahler's compactness theorem (also known as Mahler's selection theorem).
\item There is a well-known meme in Mathematics: ``Can one hear the shape of a drum?", which is the title of Mark Kac's famous paper regarding the desire to discern the shape of a drum from its `frequencies'.
An analogous question for lattices, studied by John Conway, is
``which properties of quadratic forms are determined by their representation numbers?''.
For further reading, there is the lovely little book by Conway called ``The sensual quadratic form'', which draws connections between quadratic forms and
many different fields of Mathematics \cite{Conway.Book.SensualForm}.
Of course, no library is complete without the important and biblical ``Sphere Packings, Lattices and Groups", by John H. Conway and Neil Sloane \cite{ConwaySloan.book}.
\item The idea of periodicity, as embodied by any lattice in $\mathbb{R}^d$, also occurs on other manifolds, besides Euclidean space. If we consider a closed geodesic on a manifold, then it's intuitively clear that
as we flow along that geodesic, we have a periodic orbit along that geodesic.
One important family of manifolds where this type of periodicity occurs naturally is the family of Hyperbolic manifolds. Following the philosophy that `if we have periodicity, then we have Fourier-like series',
we discover that there is also an hyperbolic analogue of the Poisson summation formula, known as the Selberg trace formula, and this type of number theory has proved extremely fruitful.
\item A strong bound for Hermite's constant in dimension $d$ was given by Blichfeldt \cite{Blichfeldt1}: \index{Blichfeldt}
\[
\gamma_d \leq \left( \frac{2}{\pi} \right) \Gamma \left( 2 + \frac{ d}{2} \right)^{\frac{2}{d}}.
\]
\item Related to the Hermite Normal Form is another extremely important reduction, called the Smith Normal Form, which we will cover in depth in a future version of this book (see \cite{MorrisNewman}).
\item The family of diagonal matrices in Example \ref{a curve in the space of lattice} is very important in the study of homogeneous dynamics, because it acts by multiplication on the left,
on the space of all lattices that have $\det {\mathcal L} = 1$. This fascinating action is sometimes
called the ``modular flow'', and was studied intensively by Etienne Ghys.
A beautiful result in this direction is that the periodic orbits of the modular flow are in bijection with the conjugacy classes of hyperbolic elements in the modular group $SL_2(\mathbb{Z})$, and furthermore that
these periodic orbits
produce incredible knots in the complement of the trefoil knot.
\item
It is clear that because lattices offer a very natural way to discretize $\mathbb{R}^d$, they continue to be of paramount importance to modern research. In particular, the theory of modular forms, with linear (Hecke) operators that are defined using lattices and their fixed finite index sublattices, is crucial for modern number theory. Euclidean lattices are also the bread-and-butter of crystallographers.
\end{enumerate}
\bigskip \bigskip \bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
My dear Watson, once you eliminate the impossible, then whatever remains - no matter how improbable - must be the truth.
-- Arthur Conan Doyle (in his book Sherlock Holmes)
\end{quote}
\medskip
\begin{prob} \label{Dual of the integer lattice} $\clubsuit$
We say that a lattice ${\mathcal L}$ is {\bf self dual} if ${\mathcal L}^* = {\mathcal L}$.
\begin{enumerate}[(a)]
\item Prove that the integer lattice is self dual: $(\mathbb{Z}^d)^* = \mathbb{Z}^d$.
\item Prove that for any lattice ${\mathcal L} \subset \mathbb{R}^d$, we have $({\mathcal L}^*)^* = {\mathcal L}$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{distance between hyperplanes}
Show that the distance $\delta$ between any two parallel hyperplanes, described by
$c_1 x_1 + \cdots + c_d x_d = k_1$ and
$c_1 x_1 + \cdots + c_d x_d = k_2$, is equal to:
\begin{equation*}
\delta = \frac{ |k_1 - k_2|}{\sqrt{ c_1^2 + \cdots + c_d^2}}.
\end{equation*}
\end{prob}
\medskip
\begin{prob}
Suppose we are given a full-rank sublattice of the integer lattice: ${\mathcal L} \subset \mathbb{Z}^d$.
Prove that there is point of ${\mathcal L}$ on the $x$-axis.
\end{prob}
\medskip
\begin{prob} \label{lattices in R^1} $\clubsuit$
Let ${\mathcal L}$ be a lattice in $\mathbb{R}^1$. Show that ${\mathcal L} = r\mathbb{Z}$ for some real number $r$.
\end{prob}
\medskip
\begin{prob} \label{dual of E^8}
Show that the $8$-dimensional lattice $E_8$, defined in \eqref{E_8}, is self-dual:
$(E_8)^* = E_8$.
\end{prob}
\medskip
\begin{prob} \label{matrix form for det of dual lattice}
Suppose we are given a rank $k$ lattice ${\mathcal L}\subset \mathbb{R}^d$, with $1\leq k \leq d$.
If $M$ is a basis matrix for ${\mathcal L}$, then prove that the matrix $ M(M^TM)^{-1}$ gives a basis for the dual lattice ${\mathcal L}^*$.
\end{prob}
\medskip
\begin{prob}
Show that for any two lattices $L, M\subset \mathbb{R}^d$, we have $L\subseteq M \iff M^* \subseteq L^*$.
\end{prob}
\medskip
\begin{prob} \label{dual of D_n}
Prove that we have the following description for the dual lattice of $D_n$:
\[
D_n^* = \mathbb{Z}^d \cup \left( \mathbb{Z}^d + \left( \tfrac{1}{2}, \cdots, \tfrac{1}{2} \right)^T \right).
\]
\end{prob}
\medskip
\begin{prob} \label{Eisenstein lattice}
The {\bf hexagonal lattice} is the $2$-dimensional lattice defined by
\[
{\mathcal L} := \{ m + n \omega \mid m,n \in \mathbb{Z}\}, \text{ where } \omega:= e^{2\pi i/3}.
\]
Prove that $\det {\mathcal L} = \frac{\sqrt 3}{2}$, and give a description of the dual lattice to the hexagonal lattice.
\end{prob}
\medskip
\begin{prob} [hard] \label{minimal lattice in R^2}
Show that the hexagonal lattice attains the minimal value for Hermite's constant
in $\mathbb{R}^2$, namely $\gamma_2^2 = \frac{2}{\sqrt{3}}$.
\end{prob}
\medskip
\begin{prob} \label{special basis in R^2}
Let ${\mathcal L} \subset \mathbb{R}^2$ be any rank $2$ lattice. Show that there exists a basis $\beta:= \{ v, w \}$ of ${\mathcal L}$ such that
the angle $\theta_\beta$ between $v$ and $w$ satisfies
\[
\frac{\pi}{3} \leq \theta_\beta \leq \frac{\pi}{2}.
\]
\end{prob}
\medskip
\begin{prob} \label{Hadamard's inequality, exercise}
Suppose that $M$ is a $d\times d$ matrix, all of whose $d^2$ elements are bounded by $B$.
Show that $|\det M| \leq B^d d^{\frac{d}{2}} $.
\end{prob}
(Hint: consider Hadamard's inequality \ref{Hadamard inequality})
Notes. It follows from this exercise that if all of the elements of $M$ are $\pm 1$, then
$|\det M| \leq d^{\frac{d}{2}} $. Such matrices are important in combinatorics and are called Hadamard matrices.
It is known that if $d > 2$, then Hadamard matrices can only possibly exist when $4 \mid d$. But for each $d = 4m$, it is not known whether a
$d\times d$ Hadamard matrix exists, except for very small cases.
\medskip
\begin{prob} $\clubsuit$ \label{basis for A_d}
Show that the following set of vectors is a basis for $A_d$:
\[
\left\{e_2 - e_1, \ e_3 - e_1, \cdots , \ e_d - e_1 \right\},
\]
where the $e_j$ are the standard basis vectors. Hence $A_d$ is a rank-$(d-1)$
sublattice of $\mathbb{Z}^d$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{character group}
Recall that $G_{\mathcal L}$ is the group of characters of the lattice ${\mathcal L}$, under the usual multiplication of complex numbers, and that the lattice ${\mathcal L}$ is a group under the usual operation of vector addition.
Show that they are isomoprhic as groups: $G_{\mathcal L} \simeq {\mathcal L}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Orthogonality.For.Characters.Of.A.sublattice}
\index{orthogonality relations for lattices}
Here we prove the {\bf orthogonality relations for characters of a lattice ${\mathcal L}$}. We will do it for any sublattice ${\mathcal L} \subset \mathbb{Z}^d$. Let $D$ be a fundamental parallelepiped for ${\mathcal L}$.
Using the notation in Exercise \ref{character group}, prove that for any two characters $\chi_a, \chi_b \in G_{\mathcal L}$,
we have:
\begin{equation}
\frac{1}{\det {\mathcal L}} \sum_{n \in D \cap \mathbb{Z}^d } \chi_a(n) \overline{\chi_b(n)} =
\begin{cases}
1 & \mbox{if } \chi_a = \chi_b \\
0 & \mbox{if not}.
\end{cases}
\end{equation}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Theorem: Automorphisms of lattices}
Prove Theorem \ref{Automorphisms of lattices}.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{fundamental domains}
Prove that any two fundamental parallelepipeds (as defined in the text) of ${\mathcal L}$, say $D_1$ and $D_2$, must be related to each other by an element of the unimodular group:
\[
D_1 = M(D_2),
\]
for some $M \in SL_d(\mathbb{Z})$.
\end{prob}
\medskip
\begin{prob} \label{number of integer sublattices of index n, R^2}
Let $f(n)$ be the number of distinct integer sublattices of index $n$ in $\mathbb{Z}^2$.
We recall from elementary number theory the function $\sigma(n) := \sum_{d | n} d$, the sum of the divisors of $n$.
Show that
\[
f(n) = \sigma(n).
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{equivalence between determinants of a sublattice}
Given a sublattice ${\mathcal L}\subset \mathbb{R}^d$ of rank $r$, show that our definition of its determinant,
namely $\det {\mathcal L} := \sqrt{M^T M}$, conincides with the Lebesgue measure of any of its fundamental parallelepipeds.
(Here $M$ is a $d\times r$ matrix whose columns are basis vectors of ${\mathcal L}$)
\end{prob}
\medskip
\begin{prob}
Show that a set of vectors $v_1, \dots, v_m \in \mathbb{R}^d$, where $1\leq m \leq d$, are linearly independent
$\iff$ their Gram matrix is nonsingular.
\end{prob}
\medskip
\begin{prob}
Prove that for any given lattice ${\mathcal L} \subset \mathbb{R}^2$, any two(nonzero) shortest linearly independent
vectors for ${\mathcal L}$ generate the lattice ${\mathcal L}$.
Note. \ As a reminder, the first two shortest nonzero vectors of ${\mathcal L}$ may have equal length.
We note that in dimensions $d \geq 5$, such a claim is false in general, as problem \ref{counterexample in RË5} below shows.
\end{prob}
\medskip
\begin{prob} \label{counterexample in RË5}
Find a lattice ${\mathcal L} \subset \mathbb{R}^5$ such that any set of five shortest nonzero vectors of ${\mathcal L}$ do not generate ${\mathcal L}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{hyperplane lattice}
Consider the {\bf discrete hyperplane} defined by:
\[
H:= \left\{ x \in \mathbb{Z}^d \bigm | c_1 x_1 + \cdots + c_d x_d =0 \right\},
\]
Show that $H$ is a sublattice of $\mathbb{Z}^d$, and has rank $d-1$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{tiling the integer lattice with hyperplanes}
\index{discrete hyperplane}
Suppose we are given a discrete hyperplane $H$, as in Exercise \ref{hyperplane lattice}.
\begin{enumerate}[(a)]
\item Prove there exists a vector $w\in \mathbb{R}^d$ such that
\[
\{ H+ kw \bigm | k \in \mathbb{Z} \} = \mathbb{Z}^d.
\]
\item Prove that there are no integer points strictly between $H$ and $H + w$.
\end{enumerate}
\end{prob}
Notes.
You may assume Bezout's identity. \index{Bezout's identity}
Namely, if $\gcd(c_1, \dots, c_d)=1$
then there exists
an integer vector $(m_1, \dots, m_d)$ such that $c_1 m_1 + \cdots + c_d m_d = 1$.
This exercise shows that we can tile the integer lattice with discrete
translates of a discrete hyperplane.
\medskip
\begin{prob}\label{Ellipsoid problem}
Here we give the details for \eqref{ellipsoid}, the definition of an ellipsoid in $\mathbb{R}^d$. Starting over again,
we fix an orthonormal basis $\{ b_1, \dots, b_d\}$ for $\mathbb{R}^d$, and
we define the following matrix:
\[
M := \begin{pmatrix} | & | & ... & | \\
c_1 b_1 & c_2 b_2 & ...& c_d b_d \\
| & | & ... & | \\
\end{pmatrix},
\]
where the $c_k$'s are positive scalars.
We now apply the linear transformation $M$ to the unit sphere $S^{d-1}:= \{ x \in \mathbb{R}^d \mid \| x \|^2 = 1\}$ in
$\mathbb{R}^d$, and we recall what this means. Now we define the $\text{Ellipsoid}_M:=
M(S^{d-1})$, a $(d-1)$-dimensional object. In the spirit of review, we recall the definition
$M(S^{d-1}) := \{ u \in \mathbb{R}^d \mid u = Mx, x \in S^{d-1} \}$.
\begin{enumerate}[(a)]
\item Show that
\begin{equation} \label{equation of ellipsoid}
\text{Ellipsoid}_M =
\left\{ x\in \mathbb{R}^d \bigm | \sum_{j=1}^d \frac{{\langle x, b_j\rangle}^2}{c_j^2} =1 \right\}.
\end{equation}
\item We recall that the unit ball in $\mathbb{R}^d$ is defined by
$B := \left\{ x \in \mathbb{R}^d \bigm | \| x \|^2 \leq 1\right\}$.
Show that for the open ellipsoid body $E$ (a $d$-dimensional object), as defined in \eqref{open ellipsoid}, we have the
$d$-dimensional volume formula:
\[
\vol(E) = \vol B \prod_{j=1}^d c_j.
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
We will use the equation \eqref{equation of ellipsoid} definition of an ellipsoid, from above.
We can extend the previous exercise in the following way. Let $A$ be {\bf any} $d \times d$ real matrix,
and look at the action of $A$ on the unit sphere
$S^{d-1} \subset \mathbb{R}^d$.
Suppose that $rank(A) = r$. Show:
(a) If $r = d$, then $A(S^{d-1})$ is a $d$-dimensional ellipsoid, defined by an equation of the form
\eqref{equation of ellipsoid}.
(b) If $r < d$, then $A(S^{d-1})$ is an $r$-dimensional ellipsoid.
\end{prob}
\medskip
\begin{prob} \label{square root of a matrix}
Suppose that $A$ is a positive definite, real matrix.
Solve for (i.e. characterize) all matrices $X$ that are the `square roots' of $A$:
\[
A = X^2.
\]
\end{prob}
\medskip
\begin{prob}
Suppose that a certain $2$-dimensional lattice ${\mathcal L}$ has a Gram matrix
\[
G := \begin{pmatrix}
\ 2 & -1 \\
-1 & \ 2 \end{pmatrix} .
\]
Reconstruct ${\mathcal L}$ (i.e. find a basis for ${\mathcal L}$), up to an orthogonal transformation.
\end{prob}
\medskip
\begin{prob}
Find a $2$ by $2$ matrix $M$ that enjoys one of the properties of a positive semidefinite
matrix, namely that $x^T M x \geq 0$, for all $x\in \mathbb{R}^2$, but such that $M$ is not symmetric.
\end{prob}
\medskip
\begin{prob} \label{exercise:2by2 positive definite matrix}
Show that any real $2$ by $2$ matrix $A$ is positive definite if and only if both
$\rm{trace}(A) >0$ and
$\det A > 0$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{facts about Vor cell}
Show that $\rm{Vor}_0({\mathcal L})$ is symmetric about the origin, convex, and compact.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{translating the Voronoi cell around}
Given a full rank lattice ${\mathcal L}\subset \mathbb{R}^d$, and any $m \in {\mathcal L}$, show that
\[
{\rm Vor}_0({\mathcal L}) + m = {\rm Vor}_m({\mathcal L}).
\]
\end{prob}
\medskip
\begin{prob} \label{Extension of Erdos to dimension d}
\rm{
(hard) Erd\"os' question, given in Exercise \ref{Erdos lattice partition problem}, possesses a natural extension to dimension $d$.
\begin{question}\label{tiling the lattice with translated sublattices}
Suppose that the integer lattice $\mathbb{Z}^d$ is partitioned into a disjoint union of a finite number of translates of integer
sublattices, say:
\[
\mathbb{Z}^d = \{ {\mathcal L}_1 +v_1 \} \cup \{ {\mathcal L}_2 + v_2 \} \cup \dots \cup \{ {\mathcal L}_N +v_N\}.
\]
Is it true that there are at least two integer sublattices, say ${\mathcal L}_j, {\mathcal L}_k$,
that enjoy the property that ${\mathcal L}_k = {\mathcal L}_j + w$, for some integer vector $w$?
\end{question}
Here we prove that in $\mathbb{R}^3$, Question \ref{tiling the lattice with translated sublattices}
has a negative answer. In particular, find a partition of $\mathbb{Z}^3$ into $4$ integer sublattices, such that no two of them are integer translates of one another. Using an easy extension to $d >3$, also show that the answer to the question above is `no', if $d \geq 3$.
Notes. Question \ref{tiling the lattice with translated sublattices} remains unsolved in dimension $d=2$ \cite{FeldmanProppRobins}.
}
\end{prob}
\chapter{The Fourier transform of a polytope via its vertex description: \\ The Brion theorems} \label{chapter.Brion}
\begin{quote}
``See in {\bf nature} the cylinder, the sphere, the cone.''
-- Paul C\'ezanne
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{Dodecahedron}
\end{center}
\caption{The Dodecahedron in $\mathbb{R}^3$, an example of a simple polytope.
In Exercise \ref{FT of a Dodecahedron}, we compute its Fourier-Laplace
transform by using Theorem \ref{brion, continuous form} below.} \label{Dodecahedron}
\end{figure}
\bigskip
\section{Intuition}
Here we introduce the basic tools for computing precise expressions for the Fourier transform of a polytope. To compute transforms here, we assume that we
are given the vertices of a polytope ${\mathcal P}$ , together with the local geometric information at each vertex of ${\mathcal P}$, namely its neighboring vertices
in ${\mathcal P} \subset \mathbb{R}^d$. It turns out that computing the Fourier-Laplace transform of the tangent cone at each vertex of ${\mathcal P}$ completely characterizes the Fourier transform of ${\mathcal P}$.
One of the basic results here, called the discrete version of Brion's Theorem (\ref{brion, discrete form}),
may be viewed as an extension of the finite geometric sum in dimension $1$, to sums in integer cones, in dimension $d$.
Some basic families of polytopes are introduced, including simple polytopes and their duals, which are simplicial polytopes. These families of polytopes play an important role in the development of Fourier analysis on polytopes.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{C60}
\end{center}
\caption{The C60 Carbon molecule, also known as a buckeyball, is another example of a simple polytope. The nickname ``buckeyball' came from Buckminster Fuller, who
used this molecule as a model for many other tensegrity structures.
\index{simple polytope}
(the graphic is used with permission from
Nanografi, at https://phys.org/news/2015-07-scientists-advance-tunable-carbon-capture-materials.html) } \label{C60}
\end{figure}
\bigskip
\section{Cones, simple polytopes, and simplicial polytopes}
One of the most important concepts in combinatorial geometry is the definition of a {\bf cone ${\mathcal K} \subset \mathbb{R}^d$, with an apex $v$}, defined by;
\begin{equation} \label{def of a cone}
{\mathcal K}:= \left\{ v+ \sum_{k=1}^N \lambda_k w_k \mid \lambda _k \geq 0 \right \}.
\end{equation}
The {\bf edge vectors} of ${\mathcal K}$ are those vectors among the $w_1, \dots, w_N$ (not necessarily all of them) which belong to the boundary
$\partial {\mathcal K}$ of ${\mathcal K}$.
A fun exercise is to show that the following two conditions are equivalent:
\begin{enumerate}[(a)]
\item A cone ${\mathcal K}$ has an apex at the origin.
\item ${\mathcal K}$ is a cone that enjoys the property $\lambda {\mathcal K} = {\mathcal K}$, for all $\lambda >0$.
\end{enumerate}
(Exercise \ref{cone equivalence}).
We note that according to definition \eqref{def of a cone}, an apex need not be unique - in Figure \ref{Cones, pointed and unpointed}, the cone on the left has a unique apex, while the cone on the right has infinitely many apices.
If the vectors $w_1, \dots, w_N$ span a $k$-dimensional subspace of $\mathbb{R}^d$, we say that the
cone ${\mathcal K}$ has {{\bf dimension $k$}.
When a $k$-dimensional cone ${\mathcal K} \subset \mathbb{R}^d$ has exactly $k$ linearly independent edge vectors $w_1, \dots w_k \in \mathbb{R}^d$,
we call such a cone a {\bf simplicial cone}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.6in]{Cones1}
\end{center}
\caption{The cone on the left is pointed, and has edges $w_1, w_2$. The cone on the right, with edges $w_1, w_2$, is also a half-space and it is not pointed.
\index{cones} } \label{Cones, pointed and unpointed}
\end{figure}
A {\bf pointed cone}
\index{cone, pointed}
is a cone ${\mathcal K}\subset \mathbb{R}^d$ with apex $v$, such that its edge vectors $w_1, \dots w_N$ are linearly independent. The following $4$ conditions give equivalent characterizations of a pointed cone~${\mathcal K}$:
\begin{enumerate}[(a)]
\item There exists a hyperplane $H$ such that $H\cap {\mathcal K} = v$.
\item The translated cone $C:= {\mathcal K}-v$, with apex at the origin, enjoys $C \cap (-C) = \{0\}$.
\item ${\mathcal K}$ has a unique apex.
\item ${\mathcal K}$ does not contain an entire line. \label{non-pointed cone contains a line}
\end{enumerate}
(Exercise \ref{pointed cone equivalence}). Part \ref{non-pointed cone contains a line} is equivalent to the statement that for a non-pointed cone ${\mathcal K}$,
there exists a vector $u\in \mathbb{R}^d$ such that ${\mathcal K} + u = {\mathcal K}$. We note that every cone has an apex, it's just that the apex may not be unique, for example when ${\mathcal K}$ is a half-space. All cones are unbounded regions, by definition, so some care will have to be taken when integrating over them. On the other hand, they are `almost linear', because for a cone with apex at the origin, we have
\[
x, y \in {\mathcal K} \ \implies \ x+y \in {\mathcal K}.
\]
This closure property, which does not exist for polytopes, makes cones extremely helpful in the analysis of polytopes (for example, Section \ref{section.Brianchon-Gram}).
An $n$-dimensional polytope ${\mathcal P}\subset \mathbb{R}^d$ is called a {\bf simplicial polytope} if every facet of ${\mathcal P}$ is a simplex. Equivalently:
\begin{enumerate}[(a)]
\item Each facet of ${\mathcal P}$ has exactly $n$ vertices.
\item Each $k$-dimensional face of ${\mathcal P}$ has exactly $k+1$ vertices, for $0\leq k \leq n-1$.
\end{enumerate}
It is a fun exercise to show that any simplicial cone is always a pointed cone (Exercise \ref{simplicial implies pointed}), but the converse is clearly false.
By contrast with the notion of a simplicial polytope, we have the following `dual' family of polytopes.
An $n$-dimensional polytope ${\mathcal P}\subset \mathbb{R}^d$ is called a {\bf simple} polytope if every vertex is contained in exactly $n$ edges of ${\mathcal P}$.
Equivalently:
\begin{enumerate}[(a)]
\item Each vertex of ${\mathcal P}$ is contained in exactly $n$ of its facets.
\item Each $k$-dimensional face of ${\mathcal P}$ is contained in exactly $d-k$ facets, for all $k \geq 0$.
\end{enumerate}
\medskip
\begin{example}
\rm{
Any $d$-dimensional simplex $\Delta$ is a simple polytope. In fact, any $k$-dimensional
face of the simplex $\Delta$ is also a simplex, and hence a simple polytope of lower dimension.
The $3$-dimensional dodecahedron, in Figure \ref{Dodecahedron}, is also a simple polytope. Its edge graph, which
is always a planar graph for a convex polytope, in this case consists of $20$ vertices, $30$ edges, and $12$ faces.
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
A $d$-dimensional simplex also happens to be a simplicial polytope. The $3$-dimensional icosahedron is a simplicial polytope.
}
\hfill $\square$
\end{example}
It is a nice exercise to show that the only polytopes which are both simple and simplicial are either
simplices, or $2$-dimensional polygons (Exercise \ref{simplicial AND simple}).
\begin{example}
\rm{
The $d$-dimensional cube $[0,1]^d$ is a simple polytope. Its dual polytope, which is the cross-polytope
$\Diamond$ (see \eqref{cross polytope}), is a simplicial polytope.
}
\hfill $\square$
\end{example}
One might ask: are the facets of a simple polytope necessarily simplicial polytopes?
Again, an example helps here.
\begin{example}
\rm{
The $120$-cell is a $4$-dimensional polytope whose $3$-dimensional boundary is composed of $120$ dodecahedra \cite{SchleimerSegerman}.
The $120$-cell is a simple polytope, but because all of its facets are dodecahedra, it does not have any simplicial facets.
}
\hfill $\square$
\end{example}
As becomes apparent after comparing the notion of a simple polytope with that of a simplicial polytope, these two types of polytopes are indeed dual to each other, in the sense of duality that we've already encountered in definition \eqref{dual polytope, definition}
\begin{lem}
${\mathcal P}\subset \mathbb{R}^d$ is a simple polytope $\iff$ ${\mathcal P}^*$ is a simplicial polytope.
\end{lem}
(see Gr\"unbaum \cite{Grunbaum} for a thorough study of this duality).
This duality between simple and simplicial polytopes suggests a stronger connection between our geometric structures thus far, and the combinatorics inherent in the partially ordered set of faces of ${\mathcal P}$. Indeed, Gr\"unbaum put it elegantly:
\begin{quote}
``In my opinion, the most satisfying way to approach the definition of polyhedra
is to distinguish between the combinatorial structure of a polyhedron,
and the geometric realizations of this combinatorial structure.'' \cite{Grunbaum2}
\end{quote}
\bigskip
\section{Tangent cones, and the Fourier transform of a simple polytope}
An important step for us is to work with the Fourier-Laplace transform of a cone, and then build some
theorems that allow us to simplify many geometric computations, by using the frequency domain on the Fourier transform side.
We may define the {\bf tangent cone} \index{tangent cone} of each face ${\mathcal F} \subset {\mathcal P}$ as follows:
\begin{equation}\label{tangentcone}
{\mathcal K}_{{\mathcal F}} = \left\{ q+ \lambda(p-q) \mid q\in {\mathcal F}, p\in {\mathcal P}, \lambda \in \mathbb{R}_{\geq 0} \right\}.
\end{equation}
We note that in general ${\mathcal K}_{{\mathcal F}}$ does not necessarily contain the origin.
The tangent cone is also known as the {\bf cone of feasible directions}. Intuitively, we can imagine standing at the point
$q\in {\mathcal F}$, and looking in the direction of
all points that belong to $P$. Then we take the union of all of these directions.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{TangentCones1}
\end{center}
\caption{The triangle ${\mathcal P}$ has three vertex tangent cones: ${\mathcal K}_{v_1}, {\mathcal K}_{v_2}, {\mathcal K}_{v_3}$. The picture is meant to signify that
these cones are, of course, unbounded.
\index{tangent cones} }
\label{Tangent Cones1}
\end{figure}
In the case that the face $F$ is a vertex of ${\mathcal P}$, we call this tangent cone a {\bf vertex tangent cone}.
The vertex tangent cone ${\mathcal K}_v$, which is a cone with apex $v$, may also be
generated by the edge vectors $v_k - v$, where $ [v_k, v]$ is an edge of ${\mathcal P}$:
\begin{equation} \label{main definition of tangent cone}
{\mathcal K}_v = \{ v+ \sum_{k=1}^N \lambda_k (v_k-v) \mid \text{ all } \lambda_k \geq 0, \text{ and the }
v_k \text{ are the neighboring vertices of } v\},
\end{equation}
a construction we will often use in practice.
The tangent cone of an edge of a $3$-dimensional convex polytope is an infinite wedge containing the whole line passing through that edge, while the tangent cone of a vertex (for a convex polytope) never contains a whole line (Exercise \ref{Exercise.tangent cone of a vertex}).
For non-convex polytopes, there are many competing definition for the vertices, and not all of them agree.
One definition for the vertices of non-convex polytopes appears in \cite{BaranyAkopyanRobins}, using Fourier transforms of cones. But in this chapter we focus mainly on convex polytopes.
\begin{example}
\rm{
For the unit cube $\square := [0,1]^d$, the tangent cone at the vertex $v=0$ is
\[
{\mathcal K}_0 = \left\{ \lambda_1 {\bf e_1}+ \lambda_2 {\bf e_2} +
\lambda_3 {\bf e_3} + \cdots + \lambda_d {\bf e_d} \mid \lambda_k \geq 0
\right\},
\]
which also happens to be the {\bf positive orthant} $\mathbb{R}^d_{\geq 0}$.
On the other hand, the tangent cone of $\square$ at the vertex $v =(1,0, \dots, 0)$ is:
\[
{\mathcal K}_v = v + \left\{ \lambda_1 (-{\bf e_1})+ \lambda_2 {\bf e_2} +
\lambda_3 {\bf e_3} + \cdots + \lambda_d {\bf e_d} \mid \lambda_k \geq 0\right\} ,
\]
where ${\bf e_j}$ is the standard unit vector along the $j$'th axis.
}
\hfill $\square$
\end{example}
\begin{example}
\rm{ To relate some of these definitions, consider a $d$-dimensional simplex $\Delta\subset \mathbb{R}^d$.
Located at each of its vertices $v \in \Delta$, we have a tangent cone $K_v$, as in \eqref{main definition of tangent cone}, and here $K_v$ is a simplicial cone. The simplex $\Delta$ is both a simple polytope and a simplicial polytope. }
\hfill $\square$
\end{example}
\bigskip
\section{The Brianchon-Gram identity} \label{section.Brianchon-Gram}
\bigskip
The following combinatorial identity, called the Brianchon-Gram identity, may be thought of as a geometric inclusion-exclusion principle. This identity is quite general, holding true for any convex polytope, simple or not. For a proof of the following result see, for example, \cite{BarvinokEhrhartbook} or \cite{BeckRobins}.
\begin{thm}[Brianchon-Gram identity]\label{Brianchon}
\index{Brianchon-Gram identity}
Let ${\mathcal P}$ be any convex polytope. Then
\begin{equation}\label{BG}
1_{\mathcal P} = \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{dim {\mathcal F}} 1_{{\mathcal K}_F},
\end{equation}
where the sum takes place over all faces of ${\mathcal P}$, including ${\mathcal P}$ itself.
\hfill $\square$
\end{thm}
\medskip
It turns out that the Brianchon-Gram relations \eqref{BG} can be shown to be equivalent (in the sense that one easily implies the other)
to the {\bf Euler-Poincare relation} \index{Euler-Poincare relation} (Exercise \ref{Euler equivalent to Brianchon-Gram})
for the face-numbers \index{face-numbers}
of a convex polytope, which says that
\begin{equation}\label{Euler}
f_0 - f_1 + f_2 - \cdots + (-1)^{d-1} f_{d-1} + (-1)^{d} f_{d}= 1.
\end{equation}
Here $f_k$ is the number of faces of ${\mathcal P}$ of dimension $k$.
\medskip
\begin{example}
\rm{
If we let ${\mathcal P}$ be a $2$-dimensional polygon (including its interior of course) with $V$ vertices, then if must also have $V$ edges, and exactly $1$ face, so that \eqref{Euler} tells us that $V - V + 1 = 1$, which is
not very enlightening, but true.
}
\hfill $\square$
\end{example}
\medskip
\begin{example}
\rm{
If we let ${\mathcal P}$ be a $3$-dimensional polytope with $V$ vertices, $E$ edge, and $F$ facets, then \eqref{Euler} tells us that $f_0 - f_1 + f_2 - f_3 = 1$, which means that $V - E + F - 1 = 1$. So we've retrieved Euler's well known formula
\[
V-E+F=2
\] for the Euler characteristic of $3$-dimensional polytopes.
}
\hfill $\square$
\end{example}
\medskip
\begin{example}
\rm{
To gain some facility with the Euler characteristic, we consider if it is possible to construct a polytope in $\mathbb{R}^3$ all of whose facets are hexagons (which are not necessarily regular). We claim that this is impossible:
{\bf Claim}. \ There can be no convex polytope ${\mathcal P}\subset \mathbb{R}^3$ with only hexagonal facets.
\begin{proof}
By assumption, all the faces of ${\mathcal P}$ are hexagons (not necessarily regular), and of course each edge bounds exactly two facets. To relate the facets to the edges, consider that
each facet contains $6$ edges, giving us $6F=2E$. Combining this latter identity with Euler's formula, we obtain
$V-E+F= V-2F$.
Now we relate the facets to the vertices. Each vertex meets at least three facets, and each hexagonal facet contains exactly six vertices. From the perspective of the facets towards the vertices, we get
$6F \geq 3V$, so that $V \leq 2F$. Putting things together, we arrive at
\[
2= V-E+F= V-2F \leq 0,
\]
and this contradiction finishes the proof.
\end{proof}
}
\end{example}
\section{Brion's formula for the Fourier transform
\\ of a simple polytope}
Brion \index{Brion} proved the following extremely useful result, Theorem \ref{brion, continuous form},
concerning the Fourier-Laplace transform
of a {\it simple polytope} ${\mathcal P}$. To describe the result, we consider each vertex $v$ of ${\mathcal P}$, and we fix the $d$
edge vectors $w_1(v), \dots, w_d(v)$ that emanate from $v$. We recall that the nonnegative real span of the edge vectors $w_k(v)$ generate the vertex
tangent cone ${\mathcal K}_v$, and that these
edge vectors are not necessarily required to be unit vectors. Placing these edge vectors
as columns of a matrix $M_v$, we define
\[
\det {\mathcal K}_v := | \det M_v |,
\]
the absolute value of the determinant of the ensuing matrix.
\medskip
\begin{thm}[{\bf Brion's theorem - the continuous form, 1988}]
\label{brion, continuous form}
\index{Brion's theorem - the continuous form}
Let ${\mathcal P} \subset \mathbb{R}^d$ be a simple, $d$-dimensional real polytope.
Then
\begin{equation}\label{transform formula for a simple polytope}
\int_{\mathcal P} e^{-2\pi i \langle u, \xi \rangle} \, du =
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ e^{-2\pi i \langle v, \xi \rangle} \det {\mathcal K}_v}
{ \prod_{ k=1 }^d \langle w_k(v), \xi \rangle }
\end{equation}
for all $\xi \in \mathbb{R}^d$ such that the denominators on the right-hand side do not vanish.
\hfill $\square$
\end{thm}
Brion's Theorem \ref{brion, continuous form} is one of the cornerstones of Fourier transforms of polytopes.
We note that the determinant $\det {\mathcal K}_v$ clearly depends on our choice of edge vectors $w_1, \dots, w_d$ for the cone ${\mathcal K}_v$, but it is straightforward (and interesting)
that the quotient $ \frac{ \det {\mathcal K}_v}{ \prod_{ k=1 }^d \langle w_k(v), \xi \rangle }$ does not depend on the choice of edge vectors (Exercise \ref{independent of edge vectors}).
This new proof of Brion's theorem uses some of the Fourier techniques that we've developed so far.
Because we promised a friendly approach, we first give a short outline of the relatively simple ideas of the proof:
{\bf Step $1$}. \ We begin with the Brianchon-Gram identity (a standard first step) involving the indicator functions of all of
the tangent cones of ${\mathcal P}$.
\medskip
{\bf Step $2$}. \ We now multiply both sides of the Brianchon-Gram identity \eqref{BG} with the function
$e^{2\pi i \langle x, \xi \rangle - \varepsilon \| x \|^2}$, where we fix an $\varepsilon >0$, and then we will
integrate over all $x \in \mathbb R^d$. Using these integrals, due to the damped Gaussians for each fixed $\varepsilon >0$, we are able to keep the {\it same domain of convergence} for all of our ensuing functions.
\medskip
{\bf Step $3$}. \ Now we let $\varepsilon \rightarrow 0$ and prove that the limit of each integral gives us something meaningful.
Using integration by parts, we prove that for any vertex tangent cone ${\mathcal K}$
the corresponding integral
$\int_{{\mathcal K}} e^{-2\pi i \langle x, \xi \rangle - \varepsilon \| x \|^2} dx$
converges, as $\varepsilon \rightarrow 0$, to the desired exponential-rational function.
In an analogous but easier manner, we will also prove that the corresponding integral over a
non-pointed cone (which includes all faces of positive dimension) converges to zero, completing the proof.
\medskip \noindent
In many of the traditional proofs of Theorem \ref{brion, continuous form}, the relevant Fourier-Laplace integrals over the vertex tangent cones have disjoint domains of convergence, lending the feeling that something magical is going on with the disjoint domains of convergence. Getting around this problem by defining functions that have the same domain of convergence (throughout the proof) was exactly the motivation for this proof.
We favor a slightly longer but clearer expositional proof over a shorter, more obscure proof. The reader familiar with some physics might notice that this proof idea resembles simulated annealing with a Gaussian.
We also note that throughout the proof we will work over $\xi \in \mathbb{R}^d$, and we don't require any analytic continuation.
Onto the rigorous details of the proof. First, a technical but crucial Lemma.
\begin{lem}\label{IntegByParts}
Let ${\mathcal K}_v$ be a $d$-dim'l simplicial pointed cone, with apex $v$, and edge vectors $w_1, \dots, w_d \in \mathbb{R}^d$.
Then
\begin{equation}\label{LimitDim.d}
\lim_{\varepsilon \rightarrow 0}
\int_{{\mathcal K}_v} e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
=
\ \left( \frac{1}{2\pi i} \right)^d \frac{ e^{-2\pi i \langle v, \xi \rangle} \det {\mathcal K}_v }
{ \prod_{ k=1 }^d \langle w_k(v) , \xi \rangle },
\end{equation}
for all $\xi \in \mathbb{R}^d$ such that $ \prod_{ k=1 }^d \langle w_k(v) , \xi \rangle \not=0$.
\end{lem}
\begin{proof}
We begin by noticing that we may prove the conclusion in the case that $v=0$, the origin, and for simplicity write ${\mathcal K}_v := {\mathcal K}$ in this case.
First we make a change of variables, mapping the simplicial cone ${\mathcal K}$ to the nonnegative orthant $\mathbb{R}^d_{\geq 0}$ by the matrix $M^{-1}$, where $M$ is the $d$ by $d$ matrix whose columns are precisely the vectors $w_k$. Thus, in the integral of \eqref{LimitDim.d}, we let $x:= My$,
with $y \in \mathbb{R}_{\geq 0}^d$, so that $dx = \left| \det M \right| dy$. Recalling that by definition $\det {\mathcal K} =| \det M |$, we have
\begin{equation}
\int_{{\mathcal K}} e^{-2\pi i \langle x, \xi \rangle - \varepsilon ||x||^2} dx
=
\left| \det {\mathcal K} \right|
\int_{\mathbb{R}_{\geq 0}^d} e^{-2\pi i \langle My, \xi \rangle - \varepsilon ||M y||^2} dy.
\end{equation}
It is sufficient to therefore show the following limiting identity:
\begin{equation}\label{SimplerLimit}
\lim_{\varepsilon \rightarrow 0}
\int_{\mathbb{R}_{\geq 0}^d} e^{-2\pi i \langle My, \xi \rangle - \varepsilon ||My||^2} dy
=
\ \left( \frac{1}{2\pi i} \right)^d \frac{ 1}{ \prod_{ k=1 }^d \langle w_k(v) , \xi \rangle }.
\end{equation}
To see things very clearly, we first prove the $d=1$ case. Here we must show that
\begin{equation}\label{LimitDim.1}
\lim_{\varepsilon \rightarrow 0}
\int_{0}^\infty e^{-2\pi i x \xi - \varepsilon x^2} dx
=
\frac{1}{2\pi i \xi},
\end{equation}
for all $\xi \in \mathbb{R}-\{0\}$, and we see that even this $1$-dimensional case is interesting. We proceed with integration by parts by letting
$dv:= e^{-2\pi i x \xi}dx$ and $u:= e^{ - \varepsilon x^2}$, to get
\begin{align}
\int_{0}^\infty e^{-2\pi i x \xi - \varepsilon x^2} dx
&= e^{ - \varepsilon x^2} \frac{e^{-2\pi i x \xi}}{-2\pi i \xi} \Big |_{x=0}^{x=+\infty} -
\int_0^\infty \frac{e^{-2\pi i x \xi}}{-2\pi i \xi} (-2\varepsilon x) e^{ - \varepsilon x^2} dx \\
&= \frac{1}{2\pi i \xi} - \frac{\varepsilon}{\pi i \xi}
\int_{0}^\infty x e^{-2\pi i x \xi - \varepsilon x^2} dx \\ \label{last nasty integral}
&= \frac{1}{2\pi i \xi} - \frac{1 }{\pi i \xi}
\int_{0}^{\infty} e^{-2\pi i \frac{u}{\sqrt{\varepsilon}} \xi } u e^{-u^2} du
\end{align}
where we've used the substitution $u:= \sqrt{\varepsilon} x$ in the last equality \eqref{last nasty integral}.
We now notice that
\[
\lim_{\varepsilon \rightarrow 0} \int_{0}^{\infty} e^{-2\pi i \frac{u}{\sqrt{\varepsilon}} \xi } u e^{-u^2} du
=\lim_{\epsilon \rightarrow 0} \hat g\Big(\frac{\xi}{\sqrt\epsilon}\Big),
\]
where $g(u):=u e^{-u^2}1_{[0, +\infty]}(u)$ is an absolutely integrable function.
Luckily, we know by the Riemann--Lebesgue lemma \ref{Riemann--Lebesgue lemma}
\index{Riemann-Lebesgue lemma}
that
\[
\lim_{w\rightarrow \infty} \hat g(w) =0,
\]
and so we arrive at the desired limit \eqref{LimitDim.1}.
We now proceed with the general case, which just uses the $1$-dimensional idea above several times. To prove \eqref{SimplerLimit}, we first fix the variables $y_2, \dots, y_d$ and perform integration by parts on $y_1$ first. Thus, we let
\begin{align}
dv_1 &:= e^{-2\pi i \langle My, \xi \rangle} dy_1=
e^{-2\pi i \langle y, M^t \xi \rangle} dy_1= e^{-2\pi i \Big( y_1 \langle w_1, \xi \rangle + \cdots +
y_d \langle w_d, \xi \rangle \Big)}dy_1,
\end{align}
thought of as a function of only $y_1$.
Carrying out the integration in the variable $y_1$, we have
$v_1 = e^{-2\pi i \langle y, M^t \xi \rangle} / \left(- 2\pi i \langle w_1, \xi \rangle \right)$.
We let $u_1:= e^{- \varepsilon ||My||^2} $, also thought of as a function of $y_1$ alone.
We have $du_1 = -\varepsilon L(y) e^{- \varepsilon ||My||^2} dy_1$, where $L(y)$ is a real polynomial in $y$, whose coefficients come from the entries of $M$.
Integrating by parts in the variable $y_1$ now gives us
\begin{align}\label{SimplerLimitProof}
& \int_{\mathbb{R}_{\geq 0}^d} e^{-2\pi i \langle My, \xi \rangle - \varepsilon ||My||^2} dy
=
\int_{\mathbb{R}_{\geq 0}^{d-1}} dy_2 \cdots dy_d \left[ u_1 v_1 \Big |_0^\infty - \int_0^\infty v_1 du_1 \right] \\
& = \int_{\mathbb{R}_{\geq 0}^{d-1}} dy_2 \cdots dy_d \left[
\frac{ e^{-2\pi i \langle y, M^t \xi \rangle - \varepsilon ||M y||^2} }{ -2\pi i \langle w_1, \xi \rangle }
\Big |_{y_1=0}^{y_1 = \infty} + \frac{\varepsilon}{-2\pi i \langle w_1, \xi \rangle}
\int_0^\infty L(y) e^{-2\pi i \langle y, M^t \xi \rangle - \varepsilon ||My||^2} dy_1
\right] \\
& = \int_{\mathbb{R}_{\geq 0}^{d-1}}
\frac{ e^{2\pi i \langle t, M^t \xi \rangle - \varepsilon ||M t||^2} }{ 2\pi i \langle w_1, \xi \rangle }dt
- \frac{\varepsilon}{2\pi i \langle w_1, \xi \rangle}
\int_{\mathbb{R}_{\geq 0}^{d}} L(y) e^{-2\pi i \langle y, M^t \xi \rangle - \varepsilon || M y ||^2} dy \\
& = \frac{1}{ 2\pi i \langle w_1, \xi \rangle } \int_{\mathbb{R}_{\geq 0}^{d-1}}
e^{-2\pi i \langle t, M^t \xi \rangle - \varepsilon || M t ||^2} dt
- \frac{\varepsilon}{2\pi i \langle w_1, \xi \rangle}
\int_{\mathbb{R}_{\geq 0}^{d}} L(y) e^{-2\pi i \langle y, M^t \xi \rangle - \varepsilon || M y ||^2} dy,
\end{align}
where we've used $t:= (y_2, \dots, y_d)$ in the $3$'rd equality. We repeat exactly the same process of integration by parts as in \eqref{last nasty integral}, one variable at a time. We observe that after $d$ iterations we get a sum of $d$ terms, where the first term does not contain any $\varepsilon$ factors, while
all the other terms do contain $\varepsilon$ factors in the exponents. Therefore, when we complete the $d$-many integration by parts iteratively, and
finally let $\varepsilon$ tend to zero, only the leading term remains, namely
$\left( \frac{-1}{2\pi i} \right)^d \frac{ 1}{ \prod_{ k=1 }^d \langle w_k, \xi \rangle } $. We've shown that \eqref{SimplerLimit} is true.
\end{proof}
\bigskip
\begin{proof}(of Theorem \ref{brion, continuous form})
We begin with the Brianchon Gram identity:
\begin{equation}\label{BG2}
1_{\mathcal P} = \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{dim {\mathcal F}} 1_{K_F}.
\end{equation}
We fix any $\xi \in \mathbb{R}^d$, and any $\varepsilon > 0$. Multiplying both sides of \eqref{BG2} by
$e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2}$, and integrate over all $x \in \mathbb{R}^d$, we have:
\begin{equation}
\int_{\mathbb{R}^d} 1_{\mathcal P}(x) e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
= \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{dim {\mathcal F}} \int_{\mathbb{R}^d} 1_{K_F}(x)
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx.
\end{equation}
\noindent
Equivalently,
\begin{equation}\label{IntegratingBrianchon}
\int_{{\mathcal P}} e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
= \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{dim {\mathcal F}}
\int_{{\mathcal K}_F}
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx.
\end{equation}
For each fixed $\varepsilon > 0$, all integrands in \eqref{IntegratingBrianchon}
are Schwartz functions, and so all of the integrals in the latter identity now converge absolutely (and rapidly).
We identify two types of tangent cones that may occur on the right-hand side of \eqref{IntegratingBrianchon}, for each face ${\mathcal F} \subseteq {\mathcal P}$.
{\bf Case $1$}. When ${\mathcal F} = v$, a vertex, we have the vertex tangent cone ${\mathcal K}_v$: these are the tangent cones that exist for each vertex of ${\mathcal P}$.
It is a standard fact that all of these vertex tangent cones are pointed cones. By hypothesis, all of our vertex tangent cones are simplicial cones, so letting $\varepsilon \rightarrow 0$ and calling on
Lemma \ref{IntegByParts}, we obtain the required limit for
$\int_{{\mathcal K}_v} e^{2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx$.
\smallskip
{\bf Case $2$}. When ${\mathcal F}$ is not a vertex, we have the tangent cone ${\mathcal K}_{\mathcal F}$, and it is a standard fact
that in this case ${\mathcal K}_{\mathcal F}$ always contains a line.
Another standard fact in the land of polytopes is that each tangent cone in this case
may be written as ${\mathcal K}_{{\mathcal F}} = \mathbb{R}^k \oplus {\mathcal K}_p$, the direct sum of a copy of Euclidean space with a pointed cone ${\mathcal K}_p$ for any point $p \in {\mathcal F}$. (as a side-note, it is also true that $\dim {\mathcal F} = k + \dim( {\mathcal K}_p ))$.
We would like to show that for all faces ${\mathcal F}$ that are not vertices of ${\mathcal P}$, the associated integrals tend to $0$:
\[
\int_{{\mathcal K}_F} e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx \rightarrow 0,
\]
as $\varepsilon \rightarrow 0$. Indeed,
\begin{align}
\int_{{\mathcal K}_F}
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx & =
\int_{ \mathbb{R}^k \oplus {\mathcal K}_p } e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx \\
& = \int_{ \mathbb{R}^k } e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
\int_{ {\mathcal K}_p } e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx. \label{product}
\end{align}
The integral $ \int_{ \mathbb{R}^k } e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx$
is precisely the usual Fourier transform of a Gaussian, which is known to be the Gaussian
$G_{\varepsilon}(x):= \varepsilon^{-k/2} e^{-\frac{\pi}{\varepsilon} \|x\|^2}$ by Exercise \ref{Gaussian2}.
It is apparent that for any fixed nonzero value of $x\in \mathbb{R}^k$, we have
$\lim_{\varepsilon \rightarrow 0}G_{\varepsilon}(x)=0$.
Finally, by Lemma \ref{IntegByParts} again, the limit
$\lim_{\varepsilon \rightarrow 0} \int_{ {\mathcal K}_p }
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx$ is finite, because ${\mathcal K}_p$ is another pointed cone.
Therefore the product of the integrals in
\eqref{product} tends to zero, completing the proof.
\end{proof}
Brion's theorem is particularly useful whenever we are given a polytope in terms of its local data at the vertices - including the edge vectors for each vertex tangent cone. We can then easily write down the Fourier transform of a simple polytope, by Theorem \ref{brion, continuous form}.
What happens, though, for non-simple polytopes?
There is the following natural extension of Brion's Theorem \ref{brion, continuous form} to all real polytopes.
\medskip
\begin{thm}[{\bf Fourier-Laplace transform of any real polytope}]
\label{brion2}
Let ${\mathcal P} \subset \mathbb{R}^d$ be any $d$-dimensional polytope. Then:
\begin{equation}
\int_{\mathcal P} e^{-2\pi i \langle u, \xi \rangle} \, du =
\sum_{v \in V}
\frac{e^{-2\pi i \langle v, \xi \rangle} }{(2\pi i)^d}
\sum_{j=1}^{M(v)} \frac{\det {\mathcal K}_j(v) }{\prod_{k=1}^d \langle w_{j, k}(v), \xi \rangle},
\end{equation}
for all $\xi \in \mathbb{R}^d$ such that all of the denominators
$ \prod_{k=1}^d \langle w_{j, k}(v), \xi \rangle \not=0$.
\end{thm}
\begin{proof}
The proof here is identical in almost every aspect to the proof of Theorem \ref{brion, continuous form}, except for {\bf Case} $1$ of its proof, above. By constrast with the proof above of {\bf Case} $1$, here our vertex tangent cones ${\mathcal K}_v$ need not be simplicial. However, we may triangulate each vertex tangent cone ${\mathcal K}_v$ into simplicial cones ${\mathcal K}_{1, v}$, \dots ${\mathcal K}_{M(v), v}$, so that we have the disjoint union
${\mathcal K}_v = {\mathcal K}_{1, v}\cup \dots \cup {\mathcal K}_{M(v), v}$. Therefore
\begin{align*}
\lim_{\varepsilon \rightarrow 0}
\int_{{\mathcal K}_v}
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx
&= \lim_{\varepsilon \rightarrow 0}
\sum_{j=1}^{M(v)}
\int_{ {\mathcal K}_{j,v} }
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx \\
&= \sum_{j=1}^{M(v)}
\lim_{\varepsilon \rightarrow 0}
\int_{ {\mathcal K}_{j,v} }
e^{-2\pi i \langle x, \xi \rangle - \varepsilon \|x\|^2} dx \\
&= \ \left( \frac{-1}{2\pi i} \right)^d
\sum_{j=1}^{M(v)}
\frac{ e^{-2\pi i \langle v, \xi \rangle} \det {\mathcal K}_{j, v} }
{ \prod_{ k=1 }^d \langle w_{j, k}(v) , \xi \rangle },
\end{align*}
where we've used Lemma \ref{IntegByParts} in the last equality, owing to the fact that all of the cones
${\mathcal K}_{j, v}$ are simplicial.
The calculation above is valid
for each $\xi \in \mathbb{R}^d$ such that $ \prod_{ k=1 }^d \langle w_{j, k}(v) , \xi \rangle \not=0$ for all vertices $v$ and all $j = 1, \dots, M(v)$.
\end{proof}
\bigskip
\section{Fourier-Laplace transforms of cones}
\label{Fourier Laplace transforms of cones}
What about the Fourier transform of a cone? Well, if we naively try to use the same integrand over a cone, the integral will diverge.
But there is a way to fix this divergence by replacing the real vector
$\xi \in \mathbb{R}^d$ by a complex vector $z \in \mathbb{C}^d$.
Let's consider what would happen if we formally replace the variable $\xi \in \mathbb{R}^d$ by a
complex vector $z := x+iy \in \mathbb{C}^d$, to obtain the transform:
\[
1_{\mathcal P}(z):= \int_{\mathcal P} e^{-2\pi i \langle u, z\rangle} \, du.
\]
Our inner product $\langle u, z \rangle := u_1 z_1 + \cdots + u_d z_d$ is always
the usual inner product on $\mathbb{R}^d$, defined
without using the Hermitian inner product here. In other words, we simply use the usual inner product on $\mathbb{R}^d$, and then formally substitute complex numbers $z_k$ into it.
This means, by definition, that
\begin{align}
\int_{\mathcal P} e^{-2\pi i \langle u, z\rangle} \, du &= \int_{\mathcal P} e^{-2\pi i \langle u, x+iy \rangle} \\
&= \int_{\mathcal P} e^{-2\pi i \langle u, x\rangle} e^{2\pi \langle u, y\rangle} \, du,
\end{align}
so that we have an extra useful real factor of $e^{2\pi \langle u, y\rangle}$ that makes the integral
converge quite rapidly over unbounded domains, provided that
$ \langle u, y \rangle < 0$.
If we set $y=0$, then it's clear that we retrieve the usual Fourier transform of ${\mathcal P}$, while if we set $x=0$, we get a new integral, which we call the {\bf Laplace transform} of ${\mathcal P}$. Finally, the {\bf Fourier-Laplace transform} \index{Fourier-Laplace transform}
of ${\mathcal P}$ is defined by:
\[
\hat 1_{\mathcal P}(z) := \int_{\mathcal P} e^{-2\pi i \langle u, z\rangle} \, du
\]
valid for any $z \in \mathbb{C}^d$ for which the integral converges.
One clear reason for the use and flexibility of the full Fourier-Laplace transform is the fact that for a cone ${\mathcal K}$,
its usual Fourier transform diverges. But if we allow a complex variable $z\in \mathbb{C}^d$, then the integral does converge on a restricted domain. Namely, the Fourier-Laplace transform of
a cone ${\mathcal K}$ is defined by:
\[
\hat 1_{\mathcal K}(z) := \int_{\mathcal K} e^{-2\pi i \langle u, z\rangle} \, du,
\]
for a certain set of $z\in \mathbb{C}^d$, but we can easily understand its precise domain of convergence.
For an arbitrary cone ${\mathcal K} \subset \mathbb{R}^d$, we define its {\bf polar cone}
\index{polar cone} by:
\[
{\mathcal K}^o := \{ y \in \mathbb{R}^d \mid \langle y, u \rangle < 0 \text{ for all } u\in {\mathcal K} \},
\]
which is an open cone. As one might expect, there is the following duality.
If ${\mathcal K}_1 \subset {\mathcal K}_2$, then ${\mathcal K}_2^o \subset {\mathcal K}_1^o$ (Exercise \ref{duality of polar cone}).
\bigskip
\begin{example}
\rm{
Given the $1$-dimensional cone ${\mathcal K}_0 := \mathbb{R}_{\geq 0}$, we compute its Fourier-Laplace transform:
\begin{align*}
\int_{{\mathcal K}_0} e^{-2\pi i u z} \, du = \int_0^\infty e^{-2\pi i u z} \, du = &= \frac{1}{-2\pi i z} e^{-2\pi i u (x+iy)}\Big|_{u=0}^{u=\infty} \\
&= \frac{1}{-2\pi i z} e^{-2\pi i ux} e^{2\pi uy}\Big|_{u=0}^{u=\infty} \\
&= \frac{1}{-2\pi i z} (0-1)
= \frac{1}{2\pi i} \frac{1 }{ z },
\end{align*}
valid for all $z:= x + iy\in \mathbb{C}$ such that $y < 0$. We note that for such a fixed complex
$z$, $| e^{-2\pi i u z} | = e^{2\pi u y }$ is a rapidly decreasing function of $u\in \mathbb{R}_{>0}$,
because $y<0$.
}
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{cone1}
\end{center}
\caption{A simplicial, pointed cone in $\mathbb{R}^3$, with apex $v$ and edge vectors $w_1, w_2, w_3$} \label{Simplicial cone}
\end{figure}
Now let's work out the Fourier-Laplace transform of a $d$-dimensional cone whose apex is the origin.
\begin{lem} \label{F-L transform of a simplicial cone, apex at o}
Let ${\mathcal K} \subset \mathbb{R}^d$ be a simplicial, $d$-dimensional cone, with apex at the origin.
If the edges of ${\mathcal K}$ are labelled $w_1, \dots, w_d$, then
\[
\hat 1_K(z) := \int_{\mathcal K} e^{-2\pi i \langle u, z\rangle} \, du = \frac{1}{(2\pi i)^d} \frac{\det {\mathcal K} }{\prod_{k=1}^d \langle w_k, z \rangle}.
\]
Furthermore, the {\bf domain of convergence} for the latter integral is naturally associated with the polar cone, \index{polar cone} and it is given by:
\[
\{ z:= x + iy \in \mathbb{C}^d \mid \ y \in {\mathcal K}^o \}.
\]
\end{lem}
\begin{proof}
We first compute the Fourier-Laplace transform of the positive orthant ${\mathcal K}_0 := \mathbb{R}_{\geq 0}^d$, with a complex vector $z = x + i y \in \mathbb{C}^d$:
\begin{align} \label{transform of a cone}
\hat 1_{{\mathcal K}_0}(z) &:= \int_{{\mathcal K}_0} e^{-2\pi i \langle z, u \rangle} du \\
&= \int_{\mathbb{R}_{\geq 0}} e^{-2\pi i z_1 u_1} du_1 \cdots \int_{\mathbb{R}_{\geq 0}} e^{-2\pi i z_d u_d} d u_d \\
&= \prod_{k=1}^d \frac{ 0- 1}{-2\pi i z_k}
= \left( \frac{1}{2\pi i}\right)^d \frac{1}{ z_1 z_2 \cdots z_d}. \label{trick2}
\end{align}
Next, the positive orthant ${\mathcal K}_0$ may be mapped to the cone ${\mathcal K}$ by a linear transformation. Namely, we may use the matrix $M$ whose columns are defined to be the edges of ${\mathcal K}$, so that by definition ${\mathcal K} = M({\mathcal K}_0)$. Using this mapping, we have:
\begin{align*}
\hat 1_{{\mathcal K}}(z) &:= \int_{{\mathcal K}} e^{-2\pi i \langle z, u \rangle} du \\
&= |\det M| \int_{{\mathcal K}_0} e^{-2\pi i \langle z, M t \rangle} dt \\
&= |\det M| \int_{{\mathcal K}_0} e^{-2\pi i \langle M^T z, t \rangle} dt \\
&= \left( \frac{1}{2\pi i}\right)^d \frac{|\det M| }{\prod_{k=1}^d \langle w_k, z \rangle}.
\end{align*}
where in the second equality we've made the substitution $u = Mt$, with $t\in {\mathcal K}_0, u \in {\mathcal K}$, and
$du = |\det M| dt$. In the final equality, we used equation \eqref{trick2} above, noting that the $k$'th
element of the vector $M^Tz$ is $\langle w_k, z \rangle$, and we note that by definition $|\det M| = \det {\mathcal K}$.
For the domain of convergence of the integral, we observe that
\[
e^{-2\pi i \langle u, z\rangle} = e^{-2\pi i \langle u, x+iy\rangle} =
e^{-2\pi i \langle u, x\rangle} e^{2\pi \langle u, y\rangle},
\]
and because $ \left| e^{-2\pi i \langle u, x\rangle} \right| = 1$, the integral
$\int_{\mathcal K} e^{-2\pi i \langle u, z\rangle} du$
converges $\iff \langle u, y \rangle < 0$ for all $u\in {\mathcal K}$. But by definition of the polar cone, this means that $y \in {\mathcal K}^o$.
\end{proof}
\bigskip
\begin{example}
\rm{
Given the $2$-dimensional cone
${\mathcal K} := \{
\lambda_1 \big(\begin{smallmatrix}
1 \\
5 \\
\end{smallmatrix}
\big) +
\lambda_2 \big(\begin{smallmatrix}
-3 \\
\ 2 \\
\end{smallmatrix}
\big) \mid \lambda_1, \lambda_2 \in \mathbb{R}_{\geq 0} \}$, we compute its Fourier-Laplace transform, and find its domain of convergence. By Lemma \ref{F-L transform of a simplicial cone, apex at o},
\begin{align*}
\hat 1_{{\mathcal K}}(z):= \int_{\mathcal K} e^{-2\pi i \langle u, z \rangle} \, du &=
\frac{1}{(2\pi i)^2} \frac{17}{(z_1 + 5z_2)(-3z_1 + 2z_2)},
\end{align*}
valid for all $z = \icol{z_1\{\bf z}_2} := x + iy$ such that $ y \in {\mathcal K}^o$. Here the polar cone is
given here by \\
${\mathcal K}^o = \interior\{
\lambda_1 \big(\begin{smallmatrix}
\ 5 \\
-1 \\
\end{smallmatrix}
\big) +
\lambda_1 \big(\begin{smallmatrix}
-2 \\
-3 \\
\end{smallmatrix}
\big) \mid \lambda_1, \lambda_2 \in \mathbb{R}_{\geq 0} \}$.
}
\hfill $\square$
\end{example}
To compute the Fourier-Laplace transform of a simplicial cone ${\mathcal K}$ whose apex is $v \in \mathbb{R}^d$, we may first compute the transform of the translated
cone ${\mathcal K}_0:= {\mathcal K} - v$, whose apex is at the origin, using the previous lemma.
We can then use the fact that the Fourier transform behaves in a simple way under translations, namely
\[
\hat 1_{K + v}(z) = e^{2\pi i \langle z, v \rangle} \hat 1_K(z),
\]
to obtain the following result (Exercise \ref{translating a cone}).
\bigskip
\begin{cor} \label{transform of a translated cone}
Let ${\mathcal K}_v \subset \mathbb{R}^d$ be a simplicial $d$-dimensional cone, whose apex is $v \in \mathbb{R}^d$. Then
\begin{equation} \label{IMPORTANT cone transform}
{\hat 1}_{{\mathcal K}_v}(z) := \int_{{\mathcal K}_v} e^{-2\pi i \langle u, z\rangle} \, du = \frac{1}{(2\pi i)^d}
\frac{ e^{-2\pi i \langle v, z \rangle} \det {\mathcal K}_v }{\prod_{k=1}^d \langle w_k, z \rangle},
\end{equation}
a rational-exponential function.
More generally, for any $d$-dimensional cone ${\mathcal K}_v\subset \mathbb{R}^d$ with apex $v$, we can
always triangulate ${\mathcal K}_v$ into $M(v)$ simplicial subcones ${\mathcal K}_j(v)$ \cite{DRS}, and apply the previous result to each simplicial
subcone, obtaining:
\begin{equation} \label{general cone transform}
{\hat 1}_{{\mathcal K}_v}(z) := \int_{{\mathcal K}_v} e^{-2\pi i \langle u, z\rangle} \, du =
\frac{e^{-2\pi i \langle v, z \rangle} }{(2\pi i)^d}
\sum_{j=1}^{M(v)} \frac{\det {\mathcal K}_j(v) }{\prod_{k=1}^d \langle w_{j, k}(v), z \rangle},
\end{equation}
a rational-exponential function.
\hfill $\square$
\end{cor}
For a non-simple polytope, the question of computing efficiently the Fourier-Laplace transforms of all of its tangent cones becomes unwieldy, as far as we know (this problem is related to the $P \not= NP$ problem). In fact, even computing the volume of a polytope is already known to be NP-hard in general, and the volume is just the Fourier transform evaluated at one point: $\vol {\mathcal P} = 1_{\mathcal P}(0)$.
\medskip
\begin{example}
\rm{
Let's work out a $2$-dim'l example of Brion's Theorem \ref{brion, continuous form}, using Fourier-Laplace transforms of tangent cones.
We will find the rational-exponential function for the Fourier-Laplace transform of the triangle $\Delta$, whose vertices are defined by
$v_1:= \icol{0\{\bf 0}}$, $v_2:= \icol{a\{\bf 0}}$, and $v_3:= \icol{0\{\bf b}}$, with $a>0, b>0$.
First, the tangent cone at the vertex $v_1:= \icol{0\{\bf 0}} $ is simply the nonnegative orthant in this case, with edge vectors $w_1 = \icol{1\{\bf 0}}$ and $w_2 = \icol{0\{\bf 1}}$. Its determinant, given these two edge vectors, is equal to $1$. Its Fourier-Laplace transform is
\begin{equation}
\int_{{\mathcal K}_{v_1}} e^{-2\pi i \langle x, z\rangle} \, dx =
\ \frac{1}{(2\pi i)^2} \, \frac{1}{z_1 z_2},
\end{equation}
and note that here we must have both $\Im(z_1)>0$ and $\Im( z_2)>0$ in order to make the integral converge. Here we use the standard notation $\Im(z)$ is the imaginary part of $z$.
The second tangent cone at vertex $v_2$ has edges $w_1 = \icol{-a\\ \ b}$ and
$w_2 = \icol{ \ 0\\ -b}$ (recall that we don't have to normalize the edge vectors at all). Its determinant has absolute value equal to $ab$, and its
Fourier-Laplace transform is
\begin{equation}
\int_{{\mathcal K}_{v_2}} e^{-2\pi i \langle x, z\rangle} \, dx =
\left( \frac{1}{2\pi i} \right)^2 \frac{(ab) e^{-2\pi i a z_1} }{(-a z_1 + b z_2)(-a z_1)},
\end{equation}
and here the integral converges only for those $z$ for which $\Im( -az_1 + bz_2) >0$ and $\Im( -a z_1 ) >0$.
Finally, the third tangent cone at vertex $v_3$ has edges $w_1 = \icol{ \ a\\ -b}$ and
$w_2 =\icol{\ 0\\ -b}$. Its determinant has absolute value equal to $ab$,
and its Fourier-Laplace transform is
\begin{equation}
\int_{{\mathcal K}_{v_3}} e^{-2\pi i \langle x, z\rangle} \, dx =
\left( \frac{1}{2\pi i} \right)^2 \frac{(ab) e^{-2\pi i b z_2} }{(a z_1 - b z_2)(-b z_2)}.
\end{equation}
and here the integral converges only for those $z$ for which $\Im( az_1 - bz_2) >0$ and $\Im( -b z_2 ) >0$.
We can again see quite explicitly the disjoint domains of convergence in this example, so that there is not even one value of $z \in \mathbb{C}^2$ for which all three Fourier-Laplace transforms of all the tangent cones converge simultaneously. Despite this apparent shortcoming, Brion's identity \eqref{brion, continuous form} still tells us that we may somehow still add these local contributions of the integrals at the vertices combine to give us a formula for the Fourier-Laplace transform of the triangle:
\begin{equation}
\hat 1_{\Delta}(z) := \int_{\Delta} e^{-2\pi i \langle x, z\rangle} dx =
\left( \frac{1}{2\pi i} \right)^2
\left(
\frac{1}{z_1 z_2}
+ \frac{-b\ e^{-2\pi i a z_1} }{(-a z_1 + b z_2) z_1}
+ \frac{ -a \ e^{-2\pi i b z_2} }{(a z_1 - b z_2) z_2}
\right),
\end{equation}
which is \emph{now} magically valid for all generic $(z_1, z_2) \in \mathbb{C}^2$; in other words, it is now valid for all
$(z_1, z_2) \in \mathbb{C}^2$ except those values which make the denominators vanish.
}
\hfill $\square$
\end{example}
\begin{example}\label{FT of a symmetric hexagon}
\rm{
What is the Fourier transform of a hexagon?
Suppose we have a hexagon $H$ that is symmetric about the origin; then we know that its Fourier transform is real-valued, by Lemma \ref{symmetric iff FT is real}. In this case it makes sense to form a $3$-dimensional graph of the points
$(x, y, \hat 1_H(x,y))$, as in Figure \ref{HexagonPic}.
To be concrete, let's define a (parametrized) hexagon $H$ with the following vertices:
\[
v_1 = \Big(\frac{2c}{\sqrt{3}}, 0\Big),\ \
v_2 = \Big(\frac{c}{\sqrt{3}}, c\Big),\ \
v_3 = \Big(\frac{-c}{\sqrt{3}}, c\Big),\ \
v_4 = -v_1,\ \ v_5 = -v_2,\ \ v_6 = -v_3,
\]
for each fixed parameter $c>0$.
Just for fun, our hexagon is scaled so that it has an inscribed circle of radius $c$, which may be useful in future applications.
To use Brion's theorem, we compute the Fourier Transforms of the $6$ vertex tangent cones of $H$.
For $v_1$, the two rays defining $K_{v_1}$ are $w_1 := v_2 - v_1 = (-\frac{c}{\sqrt{3}}, c)$ and $w_2 := v_6-v_1 = (-\frac{c}{\sqrt{3}}, -c)$, so the Fourier Transform of $K_{v_1}$ is:
\[
\hat 1_{K_{v_1}}(z)
= \frac{e^{-2\pi i \frac{2c}{\sqrt{3}}z_1}}{(-2\pi i)^2} \frac{\frac{2c^2}{\sqrt{3}} } {(-\frac{c}{\sqrt{3}}z_1 + c z_2)(-\frac{c}{\sqrt{3}}z_1 - c z_2)}
= \frac{2\sqrt 3 }{(2\pi )^2} \frac{ e^{ -\frac{4\pi i c }{\sqrt{3}} z_1}}
{(-z_1 + \sqrt 3 z_2)(z_1 + \sqrt 3 z_2)}.
\]
For $v_2$, the two rays are $w_1 := v_3 - v_2 = (-\frac{2c}{\sqrt{3}}, 0)$ and $w_2 := v_1 - v_2 = (\frac{c}{\sqrt{3}}, -c)$, giving us:
\[
\hat 1_{K_{v_2}}(z)
= \frac{e^{-2\pi i (\frac{c}{\sqrt{3}}z_1+cz_2)}}{(-2\pi i)^2} \frac{\frac{2c^2}{\sqrt{3}} } {\frac{-2c}{\sqrt{3}}z_1(\frac{c}{\sqrt{3}}z_1 - c z_2)}
= \frac{\sqrt 3}{(2\pi )^2} \frac{ e^{-2\pi c i(\frac{1}{\sqrt{3}} z_1+ z_2)}}
{z_1( z_1 - \sqrt 3 z_2)}.
\]
For $v_3$, the two rays are $w_1 := v_4 - v_3 = (-\frac{c}{\sqrt{3}}, -c)$ and $w_2 := v_2 - v_3 = (\frac{2c}{\sqrt{3}}, 0)$, giving us:
\[
\hat 1_{K_{v_3}}(z)
= \frac{e^{-2\pi i (-\frac{c}{\sqrt{3}}z_1+ cz_2)}}{(-2\pi i)^2} \frac{\frac{c}{\sqrt{3}} } {(-\frac{c}{\sqrt{3}}z_1 - c z_2) \frac{2c}{\sqrt{3}}z_1}
= \frac{\sqrt 3}{(2\pi )^2} \frac{ e^{-2\pi c i(-\frac{1}{\sqrt{3}}z_1+ z_2)}}
{z_1(z_1 +\sqrt 3z_2)}.
\]
By the inherent symmetry of our hexagon $H$, the computations for the other tangent cones are just $\hat 1_{K_{-v}}(z) = 1_{K_{v}}(-z)$, so we have:
\begin{equation}\label{eq:sinc-hexagonal}
\begin{aligned}
&\hat 1_H(z_1, z_2) := \int_{H}e^{-2\pi i \langle \xi, z \rangle} d\xi \\
&= \hat 1_{K_{v_1}}(z) + \hat 1_{K_{v_1}}(-z)
+ \hat 1_{K_{v_2}}(z) + \hat 1_{K_{v_2}}(-z)
+ \hat 1_{K_{v_3}}(z) + \hat 1_{K_{v_3}}(-z) \\
&=
\frac{\sqrt{3}}{2\pi^2}\left(
\frac{2 \cos(\frac{4\pi c}{\sqrt{3}} z_1)}{(-z_1 + \sqrt 3 z_2)(z_1 + \sqrt 3 z_2)}
+
\frac{\cos\big(\frac{2\pi c}{\sqrt{3}} z_1 + 2\pi c z_2 \big)}{z_1(z_1 - \sqrt 3 z_2)}
+
\frac{\cos\big(\frac{2\pi c}{\sqrt{3}} z_1 - 2\pi c z_2 \big)}{z_1(z_1 + \sqrt 3 z_2)} \right).
\end{aligned}
\end{equation}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{HexagonPic}
\end{center}
\caption{A graph of the Fourier transform $\hat 1_H(x,y)$ of the symmetric hexagon $H$
in Example \ref{FT of a symmetric hexagon} }
\label{HexagonPic}
\end{figure}
}
\hfill $\square$
\end{example}
\section{An application of transforms to the volume of a simple polytope, and for its moments} \index{moments}
\index{simple polytope}
\index{volume}
The following somewhat surprising formula for the volume of a simple polytope gives us a very rapid algorithm
for computing volumes of simple polytopes. We note that it is an NP-hard problem \cite{Barany}
to compute volumes of general polytopes, without fixing the dimension.
Nevertheless, there are various other families of polytopes whose volumes possess tractable algorithms.
\bigskip
\begin{thm}[Lawrence \cite{LawrenceVolume}]
\label{volume of a simple polytope}
Suppose ${\mathcal P} \subset \mathbb{R}^d$ is a simple, $d$-dimensional polytope.
For a vertex tangent cone ${\mathcal K}_v$ of ${\mathcal P}$, fix a set of edges of the cone, say
$w_1(v), w_2(v), \dots, w_d(v) \in \mathbb{R}^d$.
Then \index{volume of a simple polytope}
\begin{equation} \label{formula for the volume of simple polytope}
\vol {\mathcal P} = \frac{ (-1)^d }{ d! } \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ {\langle v, z \rangle}^{d} \det {\mathcal K}_v }
{ \prod_{ k=1 }^d \langle w_k(v), z \rangle}
\end{equation}
for all $z\in \mathbb{C}^d$ such that the denominators on the right-hand side do not vanish.
More generally, for any integer $k \ge 0$, we have the {\bf moment formulas}:
\index{moment formulas}
\begin{equation}
\int_{\mathcal P} {\langle x, z \rangle}^{k} dx = \frac{ (-1)^d k! }{ (k +d)! } \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ {\langle v, z \rangle}^{k+d}
\det {\mathcal K}_v }{ \prod_{ m=1 }^d \langle w_m(v), z \rangle} \, .
\end{equation}
\end{thm}
\bigskip
\begin{proof}
We begin with Brion's identity \eqref{transform formula for a simple polytope}, and we substitute $z := t z_0$ for a fixed
complex vector $z_0\in \mathbb{C}^d$, and any positive real value of $t$:
\begin{equation*}\label{transform formula for a simple polytope}
\int_{\mathcal P} e^{-2\pi i \langle u, z_0\rangle t} \, du =
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ e^{-2\pi i \langle v, z_0\rangle t} \det {\mathcal K}_v}
{ t^d \prod_{ m=1 }^d \langle w_m(v), z_0 \rangle }.
\end{equation*}
Now we expand both sides in their Taylor series about $t=0$. The left-hand-side becomes:
\begin{align*}
\int_{\mathcal P} \sum_{k=0}^\infty \frac{1}{k!} (-2\pi i \langle u, z_0\rangle t)^k \, du
&=
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ \sum_{j=0}^\infty \frac{1}{j!} (-2\pi i \langle v, z_0\rangle t)^j \det {\mathcal K}_v}
{ t^d \prod_{ m=1 }^d \langle w_m(v), z_0 \rangle }
\end{align*}
Integrating term-by-term on the left-hand-side, we get:
\begin{align*}
\sum_{k=0}^\infty \frac{t^k}{k!} (-2\pi i )^k \int_{\mathcal P} \langle u, z_0\rangle^k \, du
&=
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ \det {\mathcal K}_v}{\prod_{ m=1 }^d \langle w_m(v), z_0 \rangle}
\sum_{j=0}^\infty \frac{t^{j-d}}{j!} (-2\pi i )^j {\langle v, z_0\rangle}^j.
\end{align*}
Comparing the coefficients of $t^k$ on both sides, we have:
\begin{equation*}
\frac{(-2\pi i )^k}{k!} \int_{\mathcal P} \langle u, z_0\rangle^k \, du =
\left( \frac{1}{2\pi i} \right)^d \sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ \det {\mathcal K}_v}{\prod_{ m=1 }^d \langle w_m(v), z_0 \rangle}
\frac{1}{(k+d)!} (-2\pi i )^{k+d} {\langle v, z_0 \rangle}^{k+d},
\end{equation*}
and simplifying, we arrive at the moment formulas:
\begin{equation*}
\int_{\mathcal P} \langle u, z_0\rangle^k \, du = (-1)^d \frac{k!}{(k+d)!}
\sum_{ v \text{ {\rm a vertex of }} {\mathcal P} }
\frac{ {\langle v, z_0 \rangle}^{k+d} \det {\mathcal K}_v }{\prod_{ m=1 }^d \langle w_m(v), z_0 \rangle }.
\end{equation*}
In particular, when $k=0$, we get the volume formula \eqref{formula for the volume of simple polytope}.
\end{proof}
Now that we know about the Fourier-Laplace transforms of cones, we can reinterpret
Theorem \ref{brion2} by meromorphically continuing the real vector $\xi$ as follows.
Suppose we are given any $d$-dimensional polytope
${\mathcal P} \subset \mathbb{R}^d$. Using the notation of Theorem \ref{brion2},
we define
\begin{equation} \label{meromorphic continuation identity}
F_{K_v}(z) :=
\frac{e^{-2\pi i \langle v, z \rangle} }{(2\pi i)^d}
\sum_{j=1}^{M(v)} \frac{\det {\mathcal K}_j(v) }{\prod_{k=1}^d \langle w_{j, k}(v), z \rangle},
\end{equation}
for all $z \in \mathbb{C}^d$ such that all of the denominators
$ \prod_{k=1}^d \langle w_{j, k}(v), z \rangle \not=0$.
Because the function on the right-hand-side of \eqref{meromorphic continuation identity}
is a meromorphic function of $z$, we see that
$ F_{K_v}(z)$ is the meromorphic continuation of the Fourier-Laplace transform of the
vertex tangent cone $K_v$, and we know by Corollary
\ref{general cone transform} that
\begin{equation} \label{rewriting the continuous theorem of Brion}
F_{K_v}(z) = {\hat 1}_{{\mathcal K}_v}(z),
\end{equation}
on the restricted domain
\[
\{ z:= x + iy \in \mathbb{C}^d \mid \ y \in {\mathcal K}_v^o \}.
\]
With this notation we may rewrite Theorem \ref{brion2} as follows:
\begin{equation}
\int_{\mathcal P} e^{-2\pi i \langle u, z\rangle} \, du =
\sum_{v\in V} F_{K_v}(z),
\end{equation}
valid for almost all $z \in \mathbb{C}^d$.
\bigskip
\section*{Notes} \label{Notes.chapter.Brion}
\begin{enumerate}[(a)]
\item There is a large literature devoted to triangulations of cones, polytopes, and general point-sets,
and the reader is invited to consult the excellent and encyclopedic book on triangulations, by
Jes\'us de Loera, J\"org Rambau, and Francisco Santos \cite{DRS}.
\item The notion of a {\bf random polytope} has a large literature as well, and although we do not go into this topic here, one classic survey paper is by Imre B\'ar\'any \cite{Barany}.
\item The attempt to extend Ehrhart theory to non-rational polytopes, whose vertices have some irrational coordinates, is ongoing.
The pioneering papers of Burton Randol \cite{Randol1} \cite{Randol2} extended integer point counting to algebraic polytopes, meaning that their vertices are allowed to have coordinates that are algebraic numbers. Recently, a growing number of papers are considering all real dilates of a rational polytope, which is still rather close to the Ehrhart theory of rational polytopes.
In this direction, it is natural to ask how much more of the geometry of a given polytope ${\mathcal P}$ can be captured by
counting integer points in all of its positive real dilates. Suppose we translate a
$d$-dimensional integer polytope ${\mathcal P} \subset \mathbb{R}^d$ by an integer vector $n \in \mathbb{Z}^d$.
The standard Ehrhart theory gives us an invariance principle, namely the equality of the Ehrhart polynomials for ${\mathcal P}$ and ${\mathcal P} + n$:
\[
L_{{\mathcal P}+n}(t) = L_{\mathcal P}(t),
\]
for all \emph{integer} dilates $t>0$.
However, when we allow $t$ to be a positive real number, then it is in general {\bf false} that
\[
L_{{\mathcal P}+n}(t) = L_{\mathcal P}(t) \text{ for all } t > 0.
\]
In fact, these two Ehrhart functions are so different in general, that by the very recent breakthrough of Tiago Royer \cite{Tiago1},
it's even possible to uniquely reconstruct the polytope ${\mathcal P}$ if we know all the counting quasi-polynomials
$L_{{\mathcal P}+n}(t)$, for all integer translates $n \in \mathbb{Z}^d$. In other words, the work of \cite{Tiago1} shows that for two
rational polytopes ${\mathcal P}, Q \subset \mathbb{R}^d$, the equality $L_{{\mathcal P}+n}(t) = L_{Q+n}(t)$ holds for all integer translates $n \in \mathbb{Z}^d \iff {\mathcal P} = Q$.
It is rather astounding that just by counting integer points in sufficiently many translates of ${\mathcal P}$, we may completely reconstruct the whole polytope ${\mathcal P}$ uniquely. Royer further demonstrated \cite{Tiago2} that such an idea also works if we replace a polytope by any symmetric convex body. It is now natural to try to prove the following extended question.
\begin{question}
\rm{Suppose we are given polytopes ${\mathcal P}, Q \subset \mathbb{R}^d$. Can we always find a finite subset
$S\subset \mathbb{Z}^d$ (which may depend on ${\mathcal P}$ and Q)
such that
\[
L_{{\mathcal P}+n}(t) = L_{Q+n}(t) \text{ for all } n \in S, \text{ and all } t >0 \ \iff \ {\mathcal P} = Q?
\]
}
\end{question}
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
``It is better to solve one problem five different ways, than to solve five problems one way.''
-- George P\'olya
\end{quote}
\medskip
\begin{prob} $\clubsuit$ \label{independent of edge vectors}
\rm{
Although $\det {\mathcal K}_v$ depends on the choice of the length of each edge of ${\mathcal K}_v$, show that the ratio
$ \frac{ |\det {\mathcal K}_v| }{\prod_{k=1}^d \langle w_k(v), z \rangle}$
remains invariant if we replace each edge $w_k(v)$ of a simplicial cone by a constant
positive multiple of it, say $\alpha_k w_k(v)$ with $\alpha_k>0$.
(Here $z$ is any generic complex vector, meaning that $\langle w_k(v), z \rangle \not=0$).
}
\end{prob}
\medskip
\begin{prob}
Consider the regular hexagon ${\mathcal P} \subset \mathbb{R}^2$, whose vertices are the $6$'th roots of unity.
\begin{enumerate}[(a)]
\item Compute the area of ${\mathcal P}$ using Theorem \ref{volume of a simple polytope}.
\item Compute all of the moments of ${\mathcal P}$, as in Theorem \ref{volume of a simple polytope}.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
Compute the Fourier transform of the triangle $\Delta$ whose vertices are given by
\[
(1, 0), (0, 1), (-c, -c),
\]
where $c>0$.
\end{prob}
\medskip
\begin{prob} \label{translating a cone}
$\clubsuit$
Prove Corollary \ref{transform of a translated cone} for a simplicial cone ${\mathcal K}_v$, whose apex is $v$, by
translating a cone whose vertex is at the origin, to get:
\[
{\hat 1}_{{\mathcal K}_v}(z) := \int_{{\mathcal K}_v} e^{-2\pi i \langle u, z\rangle} \, du = \frac{1}{(2\pi i)^d}
\frac{ e^{-2\pi i \langle v, z \rangle} \det {\mathcal K}_v }{\prod_{k=1}^d \langle w_k, z \rangle}.
\]
\end{prob}
\medskip
\begin{prob}
Using some of the idea in Lemma \ref{LimitDim.d}, prove the following:
\begin{enumerate}[(a)]
\item For all nonzero $\alpha \in \mathbb{R}$,
\[
\lim_{\varepsilon \rightarrow 0} \int_0^\infty \cos(\alpha x) \, e^{-\varepsilon |x|^2} dx = 0.
\]
\item For all nonzero $\alpha \in \mathbb{R}$,
\[
\lim_{\varepsilon \rightarrow 0} \int_0^\infty \sin(\alpha x) \, e^{-\varepsilon |x|^2} dx = \frac{1}{\alpha}.
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{Pyramid over a square}
Consider the following $3$-dimensional polytope ${\mathcal P}$, whose vertices are as follows:
\[
\{ (0, 0, 0), \ (1, 0, 0), \ (0, 1, 0), \ (1, 1, 0), \ (0, 0, 1) \}.
\]
``a pyramid over a square". Compute its Fourier-Laplace transform $\hat 1_{\mathcal P}(z)$.
\end{prob}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.1in]{pyramid}
\end{center}
\caption{The pyramid over a square, in Exercise \ref{Pyramid over a square} }
\label{pyramid over a square}
\end{figure}
\medskip
\begin{prob}
We recall that the $3$-dimensional cross-polytope (also called an octahedron)
was defined by $\Diamond:=\left\{ \left( x_1, x_2, x_3 \right) \in \mathbb{R}^d \mid
\, \left| x_1 \right| + \left| x_2 \right| + \left| x_3 \right| \leq 1 \right\}$.
Compute the Fourier-Laplace transform of $\Diamond$ by using
Theorem \ref{brion2}.
\index{cross-polytope}
(Here not all of the tangent cones are simplicial cones, but we may triangulate each vertex tangent cones into simplicial cones).
\end{prob}
\medskip
\begin{prob}[hard-ish] \label{FT of a Dodecahedron}
Here we will find the Fourier transform of a dodecahedron ${\mathcal P}$, centered at the origin.
Suppose we fix the following $20$ vertices of ${\mathcal P}$:
\[
\{ (\pm 1,\ \pm 1, \ \pm 1), \
(0, \ \pm \phi, \ \pm \frac{1}{\phi}), \ (\pm \frac{1}{\phi}, \ 0, \ \pm \phi), \
( \pm \phi, \ \pm \frac{1}{\phi}, \ 0)
\},
\]
where $\phi:= \frac{1+\sqrt{5}}{2}$. It turns out that ${\mathcal P}$ is a simple polytope. Compute its Fourier-Laplace transform using Theorem \ref{brion, continuous form}.
Notes. All of the vertices of ${\mathcal P}$ given here can easily be seen to lie on a sphere $S$ of radius $\sqrt{3}$, and
this is a regular embedding of the dodecahedron. It is also true (though a more difficult fact) that these $20$ points maximize the volume of any polytope whose $20$ vertices lie on the surface of this sphere $S$.
\end{prob}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.3in]{Dodecahedron_climbing_wall}
\end{center}
\caption{A climbing wall in Sweden, made up of Dodecahedrons, showing one of their real-life applications} \label{Dodecahedron}
\end{figure}
\medskip
\begin{prob}
Define the $3$-dimensional polytope
${\mathcal P} := \rm{ conv }\{ (0,0,0), (1,0,0), (0,1,0), (0,0, 1), (a,b,c) \}$, where we fix real the positive real numbers $a,b,c$. Compute $\hat 1_{\mathcal P}(z)$, by computing the Fourier-Laplace transforms of its tangent cones.
(Note. Here, not all of the tangent cones are simplicial cones).
\end{prob}
\medskip
\begin{prob} \label{Pyramid over a cube}
This exercise extends Exercise \ref{Pyramid over a square} to $\mathbb{R}^d$, as follows.
Consider the $d$-dimensional polytope ${\mathcal P}$, called a ``pyramid over a cube", defined by the convex
hull of the unit cube $[0,1]^{d-1} \subset \mathbb{R}^{d-1}$, with the point $(0, 0, \dots, 0, 1) \in \mathbb{R}^{d}$.
Compute its Fourier-Laplace transform $\hat 1_{\mathcal P}(z)$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{cone equivalence}
Show the following two conditions are equivalent:
\begin{enumerate}[(a)]
\item A cone ${\mathcal K}$ has an apex at the origin.
\item ${\mathcal K}$ is a cone that enjoys the property $\lambda {\mathcal K} = {\mathcal K}$, for all $\lambda >0$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{simplicial implies pointed}
Suppose we are given a $d$-dimensional simplicial cone ${\mathcal K} \subset \mathbb{R}^d$ (so be definition ${\mathcal K}$ has exactly $d$ edges).
Show that ${\mathcal K}$ must be pointed.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Exercise.tangent cone of a vertex}
Show that for any polytope ${\mathcal P}\subset \mathbb{R}^d$, a vertex tangent cone ${\mathcal K}_v$ never contains a whole line.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{pointed cone equivalence}
\index{cone, pointed}
Show that if ${\mathcal K}$ is a cone with an apex $v$ (not necessarily a unique apex), the
following conditions are equivalent:
\begin{enumerate}[(a)]
\item ${\mathcal K}$ is a pointed cone.
\item There exists a hyperplane $H$ such that $H\cap {\mathcal K} = v$.
\item The translated cone $C:= {\mathcal K}-v$, with apex at the origin, enjoys $C \cap (-C) = \{0\}$.
\item ${\mathcal K}$ has a unique apex.
\item ${\mathcal K}$ does not contain an entire line.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{simplicial AND simple}
Show that the only polytopes that are both simple and simplicial are either
simplices, or $2$-dimensional polygons.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{duality of polar cone}
Show that if we have reverse inclusions for polar cones. Namely, if we have
cones ${\mathcal K}_1 \subset {\mathcal K}_2$, then ${\mathcal K}_2^o \subset {\mathcal K}_1^o$.
\end{prob}
\medskip
\begin{prob} \label{polar cones and Minkowski sums}
\index{Minkowski sum}
Show that if we take the Minkowski sum $K_1 + K_2$ of two cones ${\mathcal K}_1, {\mathcal K}_2 \subset \mathbb{R}^d$, then
polarity interacts with Minkowski sums in the following pleasant way:
\[
\left({\mathcal K}_1 +{\mathcal K}_2\right)^o = {\mathcal K}_1^o \cap {\mathcal K}_2^o.
\]
\end{prob}
\medskip
\begin{prob} \label{polytope from pentagons}
Suppose we try to construct a polytope ${\mathcal P} \subset \mathbb{R}^3$ all of whose facets are pentagons (not necessarily regular).
Show that
$
F\geq 12,
$
where $F$ is the number of facets of ${\mathcal P}$.
\end{prob}
\medskip
\begin{prob} \label{Euler equivalent to Brianchon-Gram} $\clubsuit$
\rm{
\begin{enumerate}[(a)]
\item Show that the Brianchon-Gram relations \eqref{BG} imply
the Euler-Poincare relation for the face-numbers of a convex polytope ${\mathcal P}$:
\begin{equation}
f_0 - f_1 + f_2 - \cdots + (-1)^{d-1} f_{d-1} + (-1)^{d} f_{d}= 1,
\end{equation}
where $f_k$ is the number of faces of ${\mathcal P}$ of dimension $k$.
\item \label{b} (hard) \ Conversely, given a $d$-dimensional polytope ${\mathcal P}\subset \mathbb{R}^d$, show that the Euler-Poincare relation above implies the Brianchon-Gram relations:
\[
1_{\mathcal P}(x) = \sum_{{\mathcal F} \subset {\mathcal P}} (-1)^{dim {\mathcal F}} 1_{{\mathcal K}_F}(x),
\]
for all $x\in \mathbb{R}^d$.
\end{enumerate}
Notes. Interestingly, even though the above two conditions are equivalent, condition \ref{b} is often more useful in practice, because we have a free variable $x$, over which we may sum or integrate.
}
\end{prob}
\chapter{The discrete Brion theorem: Poisson summation strikes again}
\label{chapter:Discrete Brion}
\index{Poisson summation} \index{discrete Brion theorem}
\begin{quote}
``Everything you've learned in school as `obvious' becomes less and less obvious as you begin to study the universe.
For example, there are no solids in the universe.
There's not even a suggestion of a solid.
There are no absolute continuums.
There are no surfaces.
There are no straight lines.''
-- Buckminster Fuller
\end{quote}
\bigskip
\section{Intuition}
As we saw in Theorem \ref{brion, continuous form}, there exists a wonderful way to decompose the Fourier transform of a polytope in terms of the Fourier-Laplace transforms of its vertex tangent cones.
We can now ask:
\begin{question}{\rm [Rhetorical]
\label{question:discrete Brion}
Is there a natural way to {\bf discretize} this continuous identity for the FT of a polytope?}
\end{question}
Another basic question we could ask is:
\begin{question} {\rm [Rhetorical]
How does the finite geometric sum in dimension $1$ extend to dimension $d$? }
\end{question}
As we'll see, these two questions are intertwined, and one answers the other.
One useful way to make sense of Question \ref{question:discrete Brion} is to replace integrals with sums over the integer lattice:
\begin{equation}
\int_{\mathcal P} e^{-2\pi i \langle u, z \rangle} \, du \longrightarrow
\sum_{n\in \mathbb{Z}^d} e^{2\pi i \langle z, n \rangle}.
\end{equation}
Such a descretization will lead us to a discrete version of Brion's Theorem, namely Theorem \ref{brion, discrete form} below.
\bigskip
\section{Discretizing the Fourier-Laplace transform of a cone}
We may also replace the integer lattice by any lattice ${\mathcal L}$, and the ensuing function is very similar.
But since this is only a cosmetic change of variable, we can simplify life and work with the integer lattice.
To this discrete end, we define the {\bf integer point transform}
\index{integer point transform} of a rational polytope ${\mathcal P}$ by
\[
\sigma_{\mathcal P}(z) := \sum_{n \in {\mathcal P} \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle},
\]
a discretization of the Fourier transform of ${\mathcal P}$.
We may also think of the discretized sum $ \sum_{n\in \mathbb{Z}^d} e^{2\pi i \langle z, n \rangle}$
more combinatorially by making the change of variable $q_1:= e^{2\pi i z_1}, \dots, q_d:= e^{2\pi i z_d}$, so that
we have $q_1^{n_1} q_2^{n_2} \cdots q_d^{n_d} = e^{2\pi i n_1 z_1 + \cdots + 2\pi i n_d z_d} := e^{2\pi i \langle n, z \rangle}$.
with this notation in mind, we define the {\bf multinomial notation} for a monomial in several variables:
\[
q^n:= q_1^{n_1} q_2^{n_2} \cdots q_d^{n_d}.
\]
We will therefore sometimes use the equivalent definition
\[
\sigma_{\mathcal P}(q):= \sum_{n \in {\mathcal P} \cap \mathbb{Z}^d} q^n.
\]
We similarly define the {\bf integer point transform of a rational cone} ${\mathcal K}_v$ by the series
\begin{equation}\label{def of integer point transform of a cone}
\sigma_{{\mathcal K}_v}(z) := \sum_{n \in {\mathcal K}_v \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle}.
\end{equation}
But even in dimension $1$ things can get interesting, so let's see an example.
\begin{example}[Finite geometric sums]
\rm{
Consider the $1$-dimensional polytope ${\mathcal P} := [a,b]$, where $a, b\in \mathbb{Z}$. The problem is to compute the finite
geometric series:
\begin{align*}
\sum_{n \in {\mathcal P} \cap \mathbb{Z}} e^{ 2\pi i n z} &= \sum_{a \leq n \leq b} q^n,
\end{align*}
where we've set $q:= e^{2\pi i z}$. Of course, we already know that it possesses a
`closed form' of the type:
\begin{align} \label{verifying1}
\sum_{a \leq n \leq b} q^n &= \frac{q^{b+1} -q^{a} }{q - 1} \\
&= \frac{q^{b+1}}{q-1} - \frac{ q^{a} }{q - 1}, \label{verifying2}
\end{align}
because we already recognize this formula for a {\bf finite geometric sum}.
On the other hand, anticipating the discrete form of Brion's theorem below, we first compute the discrete
sum corresponding to the vertex tangent cone at the vertex $a$, namely $\sum_{a \leq n} q^n$:
\begin{equation}
\label{cone identity1}
q^a+ q^{a+1} + \cdots = \frac{q^a}{1-q}.
\end{equation}
Now we compute the the sum corresponding to the vertex tangent cone at vertex $b$, namely $\sum_{n \leq b} q^n$:
\begin{equation} \label{cone identity2}
q^b+ q^{b-1} + \cdots = \frac{q^b}{1-q^{-1}} =
\frac{q^{b+1}}{q-1} .
\end{equation}
Summing these two contributions, one from each vertex tangent cone, we get:
\begin{align*}
\frac{q^a}{1-q} + \frac{q^{b+1}}{q-1} = \sum_{a \leq n \leq b} q^n,
\end{align*}
by the finite geometric sum identity, thereby
verifying Theorem \ref{brion, discrete form} for this example.
This example shows that Brion's Theorem \ref{brion, discrete form} (the discrete version) may be thought of as a $d$-dimensional extension of the finite geometric sum.
But something is still very wrong here - namely, identity \eqref{cone identity1} converges for $|q|< 1$,
while identity \eqref{cone identity2} converges only for $|q|>1$, so there is not even one value of $q$ for which the required identity \eqref{verifying2} is true. So how can we make sense of these completely {\bf disjoint domains of convergence} ?!
}
\hfill $\square$
\end{example}
\bigskip
To resolve these conundrums, the very useful result of
Michel Brion \cite{Brion} comes to the rescue.
Our proof of Theorem \ref{brion, discrete form} discretizes the continuous form of Brion's Theorem \ref{brion, continuous form},
using the Poisson summation formula, to arrive at a discrete form of Brion's Theorem.
First, we need a slightly technical but easy Lemma.
\begin{lem}\label{technical lemma1}
Let ${\mathcal K}_v$ be a rational cone, with apex at $v$. We pick any compactly supported and smooth
approximate identity $\phi_\varepsilon$. Then:
\begin{align} \label{lem:technical claim for limiting series}
\lim_{\varepsilon \rightarrow 0}
\sum_{n \in \mathbb{Z}^d }
\left( 1_{\interior {\mathcal K}_v}(x) e^{2\pi i \langle x, z\rangle}*\phi_\varepsilon \right)(n)
= \sum_{n \in \mathbb{Z}^d \cap \interior {\mathcal K}_v } e^{2\pi i \langle n, z\rangle} := \sigma_{ \interior {\mathcal K}_v}(z).
\end{align}
\end{lem}
\begin{proof}
We first note that by our assumptions on $\phi_\varepsilon$, it lies in the Schwartz space $S(\mathbb{R}^d)$, by
Lemma \ref{useful Schwartz fact}.
So $\phi_\varepsilon$ is rapidly decreasing. Using the Weierstrass $M$-test,
we see that the series
$\sum_{n \in \mathbb{Z}^d }
\left( 1_{\interior {\mathcal K}_v}(x) e^{2\pi i \langle x, z\rangle}*\phi_\varepsilon \right)(n)$
converges uniformly in $\epsilon$, and because the summands are continuous functions of $\epsilon$, so is the whole series. So we may take the limit as $\epsilon \rightarrow 0$ inside the series.
Finally, using Lemma \ref{approximate identity convolution}, and the continuity of the function
$1_{\interior {\mathcal K}_v}(x) e^{2\pi i \langle x, z\rangle}*\phi_\varepsilon$ at all $x \in \mathbb{R}^d$,
we have
$\lim_{\varepsilon \rightarrow 0}
\left( 1_{\interior {\mathcal K}_v}(x) e^{2\pi i \langle x, z\rangle}*\phi_\varepsilon \right)(n)
= 1_{\interior {\mathcal K}_v}(n) e^{2\pi i \langle n, z\rangle}$, from which
\eqref{lem:technical claim for limiting series} follows.
\end{proof}
It turns out that the continuous form of Brion's theorem, namely Theorem \ref{brion, continuous form},
can be used to prove the discrete form of Brion's theorem, namely Theorem \ref{brion, discrete form} below.
\bigskip
\begin{thm}[{\bf Brion's theorem - the discrete form, 1988}] \label{brion, discrete form}
\index{Brion's theorem - the discrete form}
Let ${\mathcal P} \subset \mathbb{R}^d$ be a rational, $d$-dimensional polytope, and let $N$ be the number of vertices of ${\mathcal P}$.
For each vertex $v$ of ${\mathcal P}$, we consider the open vertex tangent cone $\interior {\mathcal K}_v$ of $ \interior {\mathcal P}$, the interior of ${\mathcal P}$.
Then
\begin{equation} \label{Discrete formula, Brion's theorem}
\sigma_{ \interior {\mathcal P}}(z) = \sigma_{ \interior {\mathcal K}_{v_1}}(z) + \cdots + \sigma_{ \interior {\mathcal K}_{v_N}}(z).
\end{equation}
for all $z \in \mathbb{C}^d - S$, where $S$ is the hyperplane arrangement defined by the (removable) singularities of all of the transforms $\hat 1_{{\mathcal K}_{v_j}}(z)$.
\end{thm}
\begin{proof} We will use the continuous version of Brion, namely Theorem \ref{brion, continuous form}, together with
the Poisson summation formula, to deduce the discrete version here. In a sense, the Poisson summation formula allows us to discretize the integrals. \index{Poisson summation formula}
Step $1$. [{\bf Intuition - fast and loose}] \ To begin, in order to motivate the rigorous proof that follows, we will use Poisson summation on a function $1_{{\mathcal P}}(n) e^{2\pi i \langle n, z\rangle}$
that ``doesn't have the right"
to be used in Poisson summation, because $\hat 1_{{\mathcal P}} \notin L^1(\mathbb{R}^d)$ . But this first step brings the intuition to the foreground.
Then, in Step $2$, we will literally ``smooth'' out the lack of rigor in Step 1, by smoothing $1_{\mathcal P}$ with an approximate identity.
\begin{align*}
\sum_{n \in {\mathcal P} \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle} &:= \sum_{n \in \mathbb{Z}^d} 1_{{\mathcal P}}(n) e^{2\pi i \langle n, z\rangle} \\
&= \sum_{\xi \in \mathbb{Z}^d} \hat 1_{\mathcal P}(z+ \xi) \\
&= \sum_{\xi \in \mathbb{Z}^d} \left( \hat 1_{K_{v_1}} (z+ \xi) + \cdots + \hat 1_{K_{v_1}} (z+ \xi) \right) \\
&= \sum_{\xi \in \mathbb{Z}^d} \hat 1_{K_{v_1}} (z+ \xi) + \cdots + \sum_{\xi \in \mathbb{Z}^d} \hat 1_{K_{v_N}} (z+ \xi) \\
&= \sum_{n \in \mathbb{Z}^d} 1_{K_{v_1}} (n) e^{2\pi i \langle n, z\rangle} + \cdots +
\sum_{n\in \mathbb{Z}^d} 1_{K_{v_N}} (n) e^{2\pi i \langle n, z\rangle} \\
&:= \sum_{n \in \mathbb{Z}^d \cap K_{v_1} } e^{2\pi i \langle n, z\rangle} + \cdots +
\sum_{n \in \mathbb{Z}^d \cap K_{v_N} } e^{2\pi i \langle n, z\rangle},
\end{align*}
where we have used the Poisson summation formula in the second and fifth equalities.
The third equality uses Brion's Theorem \ref{brion, continuous form} for the Fourier transform of ${\mathcal P}$.
Step $2$ [{\bf Rigorous proof}]. \ To make Step $1$ rigorous, we pick any compactly supported approximate identity
$\phi_\varepsilon$, and form a smoothed
version of the function in step $1$. Namely we let
\[
f_\varepsilon(x):= (1_{{\mathcal P}}(x) e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon(x),
\]
so that now we are allowed to apply Poisson summation \index{Poisson summation formula}
to $f_\varepsilon$, because our choice of a smooth and compactly supported $\phi_\varepsilon$ implies that $f_\varepsilon$ is a Schwartz function.
Recalling Theorem \ref{approximate identity convolution}, we know that at a point $x\in \mathbb{R}^d$ of continuity of
$1_{{\mathcal P}}(x) e^{2\pi i \langle x, z\rangle}$, we have
\[
\lim_{\varepsilon \rightarrow 0} f_\varepsilon(x) = 1_{{\mathcal P}}(x) e^{2\pi i \langle x, z\rangle}.
\]
To proceed further,
it is therefore natural to consider points $x \in \interior {\mathcal P}$, the interior of ${\mathcal P}$, because $1_{\mathcal P}$ is continuous there, while it is not continuous on the boundary of ${\mathcal P}$. To recap, we have so far the equalities
\begin{align*}
\sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle} &:=
\sum_{n \in \mathbb{Z}^d} 1_{ \interior {\mathcal P}}(x) e^{2\pi i \langle x, z\rangle} \\
&= \sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} \lim_{\varepsilon \rightarrow 0} f_\varepsilon(n) \\
&= \lim_{\varepsilon \rightarrow 0} \sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} f_\varepsilon(n),
\end{align*}
where we've used the fact that $f_\varepsilon$ is compactly supported, because it is the convolution of two compactly supported functions. So the exchange above, of the sum with the limit, is trivial because the sum is finite.
With this in mind, the Poisson summation formula, applied to the Schwarz function $f_\varepsilon$,
gives us:
\begin{align*}
&\sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} e^{2\pi i \langle n, z\rangle} =
\lim_{\varepsilon \rightarrow 0} \sum_{n \in \interior {\mathcal P} \cap \mathbb{Z}^d} f_\varepsilon(n)
= \lim_{\varepsilon \rightarrow 0} \sum_{n \in \mathbb{Z}^d} \left(1_{ \interior {\mathcal P}} \
e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon\right) (n) \\
&=\lim_{\varepsilon \rightarrow 0} \sum_{n \in \mathbb{Z}^d} {\mathcal F}{ \big( (1_{ \interior {\mathcal P}} \
e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon \big) }(\xi) \\
&= \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in \mathbb{Z}^d} \hat 1_{ \interior {\mathcal P}}(z+ \xi) \hat \phi_\varepsilon(\xi) \\
&= \lim_{\varepsilon \rightarrow 0}
\sum_{\xi \in \mathbb{Z}^d} \left( \hat 1_{ \interior {\mathcal K}_{v_1}} (z+ \xi) + \cdots +
\hat 1_{ \interior {\mathcal K}_{v_1}} (z+ \xi) \right)
\hat \phi_\varepsilon(\xi) \\
&= \lim_{\varepsilon \rightarrow 0}
\sum_{\xi \in \mathbb{Z}^d} {\mathcal F}{ \big( (1_{ \interior {\mathcal K}_{v_1}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon \big) }(\xi)
+ \cdots +
\lim_{\varepsilon \rightarrow 0} \sum_{\xi \in \mathbb{Z}^d}
{\mathcal F}{ \big( (1_{ \interior {\mathcal K}_{v_N}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon \big) }(\xi) \\
&= \lim_{\varepsilon \rightarrow 0}
\sum_{n \in \mathbb{Z}^d} (1_{ \interior {\mathcal K}_{v_1}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon (n)
+ \cdots +
\lim_{\varepsilon \rightarrow 0}
\sum_{\xi \in \mathbb{Z}^d} (1_{ \interior {\mathcal K}_{v_N}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon (n) \\
&= \sigma_{ \interior {\mathcal K}_{v_1}}(z) + \cdots + \sigma_{ \interior {\mathcal K}_{v_N}}(z),
\end{align*}
We've applied Theorem \ref{approximate identity convolution} to $f(n) := 1_{\interior {\mathcal K}_v}(n)$,
for each $n \in \interior {\mathcal K}_v$, because $f$ is continuous at all such points.
The conclusion of Theorem \ref{approximate identity convolution} is that
\[
\lim_{\varepsilon \rightarrow 0}
\Big( (1_{ \interior {\mathcal K}_{v_1}} \ e^{2\pi i \langle x, z\rangle})*\phi_\varepsilon \Big)(n) =
1_{ \interior {\mathcal K}_{v_1}}(n) \ e^{2\pi i \langle n, z\rangle},
\]
and by Lemma \ref{technical lemma1} the last equality, in the long string of equalities above, is justified.
\end{proof}
\bigskip
\begin{example} \label{example:standard triangle integer point transform}
\rm{
We compute the integer point transform of the {\bf standard triangle} in the plane, using Brion's Theorem \ref{brion, discrete form}. Namely,
for the standard triangle
\[
\Delta:= \conv( \icol{0\{\bf 0}}, \icol{1\{\bf 0}}, \icol{0\{\bf 1}}),
\]
as depicted in Figure \ref{standard triangle}, we find $\sigma_{\Delta}(z)$.
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.7in]{triangle}
\end{center}
\caption{The standard triangle, with its vertex tangent cones} \label{standard triangle}
\end{figure}
By definition, the integer point transform of its vertex tangent cone ${\mathcal K}_{v_1}$ is
\begin{align*}
\sigma_{{\mathcal K}_{v_1}}(z) &:= \sum_{n \in {\mathcal K}_{v_1} \cap \mathbb{Z}^d} e^{\langle n, z\rangle}
=
\sum_{n_1 \geq 0, n_2 \geq 0}
e^{ \langle n_1 \icol{1\{\bf 0}} + n_2\icol{0\{\bf 1}} ,
z\rangle} \\
& = \sum_{n_1 \geq 0} e^{ n_1 z_1} \sum_{n_2 \geq 0} e^{ n_2 z_2} \\
& = \frac{1}{(1- e^{z_1} )(1- e^{z_2})}.
\end{align*}
For the vertex tangent cone ${\mathcal K}_{v_2}$, we have
\begin{align*}
\sigma_{{\mathcal K}_{v_2}}(z) &:= \sum_{n \in {\mathcal K}_{v_2} \cap \mathbb{Z}^d} e^{\langle n, z\rangle}
=
\sum_{n_1 \geq 0, n_2 \geq 0}
e^{\langle \icol{1\{\bf 0}} + n_1 \icol{-1\\ \ 0} + n_2\icol{-1\\ \ 1} ,
z\rangle} \\
& = e^{z_1} \sum_{n_1 \geq 0} e^{ n_1 (-z_1)}
\sum_{n_2 \geq 0} e^{ n_2 (-z_1+z_2)} \\
& = \frac{e^{z_1}}{(1- e^{-z_1} )(1- e^{-z_1+z_2})}.
\end{align*}
Finally, for the vertex tangent cone ${\mathcal K}_{v_3}$, we have
\begin{align*}
\sigma_{{\mathcal K}_{v_3}}(z) &:= \sum_{n_1 \geq 0, n_2 \geq 0}
e^{\langle \icol{0\{\bf 1}} + n_1 \icol{ \ 0\\-1} + n_2\icol{\ 1\\-1}, z \rangle} \\
& = e^{z_2} \sum_{n_1 \geq 0} e^{ n_1 (-z_2)}
\sum_{n_2 \geq 0} e^{ n_2 (z_1-z_2)} \\
& = \frac{e^{z_2}}{(1- e^{-z_2} )(1- e^{z_1-z_2})}.
\end{align*}
Altogether, using \ref{simplified discrete Brion identity} we have
\begin{align}
\sigma_{{\mathcal P}}(z) &= \sigma_{{\mathcal K}_{v_1}}(z) + \sigma_{{\mathcal K}_{v_2}}(z) +\sigma_{{\mathcal K}_{v_3}}(z) \\
&= \frac{1}{(1- e^{z_1} )(1- e^{z_2})} + \frac{e^{z_1}}{(1- e^{-z_1} )(1- e^{-z_1+z_2})}
+\frac{e^{z_2}}{(1- e^{-z_2} )(1- e^{z_1-z_2})}. \label{last line}
\end{align}
}
\hfill $\square$
\end{example}
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.3in]{Example2}
\end{center}
\caption{A triangle with vertices $v_1, v_2, v_3$, and its vertex tangent cones} \label{triangle}
\end{figure}
\bigskip
\section{Examples, examples, examples}
\bigskip
\begin{example}\label{triangle1}
Here we will compute the integer point transform of the triangle $\Delta$ defined by the convex hull of the points $\icol{0\{\bf 0}}, \icol{3\{\bf 1}}, \icol{3\\6}$, as shown in Figure \ref{triangle}.
We first compute the integer point transforms of all of its tangent cones. For the vertex $v_1$, we already computed the integer point transform of its tangent cone in the previous example.
For the vertex $v_2$, we notice that its vertex tangent cone is a unimodular cone, because
$| \det
\big(\begin{smallmatrix}
0 & -1 \\
-1 & -2
\end{smallmatrix}
\big) | = 1$. Its integer point transform is:
\begin{align*}
\sigma_{{\mathcal K}_{v_2}}(z) &:= \sum_{n \in {\mathcal K}_{v_2} \cap \mathbb{Z}^d} e^{\langle n, \ z\rangle}
=
\sum_{n_1 \geq 0, n_2 \geq 0}
e^{ \langle
\icol{3\\6} + n_1 \icol{0\\-1} + n_2\icol{-1\\-2} ,
z\rangle} \\
& = e^{ 3z_1 + 6z_2)} \sum_{n_1 \geq 0, n_2 \geq 0}
e^{ n_1 (-z_2)} e^{ n_2(-z_1 -2z_2)} \\
& = \frac{ e^{3z_1 + 6z_2} }
{
(1- e^{-z_2} )(1- e^{ -z_1 -2z_2} )
}.
\end{align*}
Equivalently, using the notation from Example \ref{integer point transform of cone} above,
\[
\sigma_{{\mathcal K}_{v_2}}(z) := \sum_{n \in {\mathcal K}_{v_2} \cap \mathbb{Z}^d} q^n
=\frac{ q_1^3 q_2^6}
{
(1- q_2^{-1} ) (1- q_1^{-1} q_2^{-2}).
}
\]
For vertex $v_3$, the computation is similar to vertex tangent cone ${\mathcal K}_{v_1}$, and
we have:
\begin{align*}
\sigma_{{\mathcal K}_{v_3}}(z) &:= \sum_{n \in {\mathcal K}_{v_3} \cap \mathbb{Z}^d} e^{\langle n, \ z\rangle}
= \sum_{n_1 \geq 0, n_2 \geq 0} e^{\langle
\icol{3\{\bf 1}} + n_1 \icol{-3\\-1} + n_2\icol{0\{\bf 1}}, \
z\rangle} \\
&= e^{3z_1 + z_2} \sum_{n_1 \geq 0, n_2 \geq 0} e^{(-3z_1-z_2) n_1}
e^{2\pi i (z_2) n_2} \\
&= e^{3z_1 + z_2} \frac{ 1+ e^{-z_1} + e^{-2z_1} }
{
(1- e^{ 3z_1 + z_2} )(1- e^{ z_2} )
} \\
&= \frac{ e^{3z_1 + z_2} + e^{2z_1 + z_2} + e^{z_1 + z_2} }
{
(1- e^{3z_1 + z_2} )(1- e^{z_2} )
} \\
&= \frac{ q_1^3 q_2 + q_1^2 q_2 + q_1q_2 }
{
(1- q_1^{-3} q_2^{-1} )(1- q_2)
}.
\end{align*}
Finally, putting all of the three vertex tangent cone contributions together,
Theorem \ref{brion, discrete form} gives us:
\begin{align*}
\sigma_{\Delta}(z) &= \sigma_{{\mathcal K}_{v_1}}(z) + \sigma_{{\mathcal K}_{v_2}}(z) + \sigma_{{\mathcal K}_{v_3}}(z) \\
&= \frac{ 1+ q_1 q_2 + {q_1}^2 q_2 + {q_1}^2 {q_2}^2 + q_1^3 q_2^2
}
{ (1- q_1^3 q_2 ) (1- q_1 q_2^2)
}
+
\frac{ q_1^3 q_2^6}
{
(1- q_2^{-1} ) (1- q_1^{-1} q_2^{-2})
}
+
\frac{ q_1^3 q_2 + q_1^2 q_2 + q_1q_2 }
{
(1- q_1^{-3} q_2^{-1} )(1- q_2)
}.
\end{align*}
\hfill $\square$
\end{example}
\bigskip
\begin{example}\label{integer point transform of cone}
\rm{
We work out the integer point transform $\sigma_{\mathcal K}(z)$ of the cone
\[
{\mathcal K} := \{
\lambda_1 \big(\begin{smallmatrix}
3 \\
1 \\
\end{smallmatrix}
\big) +
\lambda_2 \big(\begin{smallmatrix}
1 \\
2 \\
\end{smallmatrix}
\big) \mid \lambda_1, \lambda_2 \in \mathbb{R}_{\geq 0} \},
\]
Drawn in the figures below. We note that here $\det {\mathcal K} = 5$, and that there are indeed $5$ integer points in
$D$, its half-open fundamental parallelepiped.
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{slide2}
\end{center}
\caption{The $5$ integer points in a fundamental parallelepiped $D$ of the cone ${\mathcal K}$.} \label{cone2}
\end{figure}
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{slide3}
\end{center}
\caption{The point ${1\choose 1}$ in $D$, with its images in ${\mathcal K}$ under translations by the edge vectors of ${\mathcal K}$.}
\label{cone3}
\end{figure}
We may `divide and conquer' the integer point transform $\sigma_{\mathcal K}(z)$,
by breaking it up into $5$ infinite series,
one for each integer point in $D$, as follows:
\[
\sigma_{\mathcal K}(z) := \sum_{n \in {\mathcal K}\cap \mathbb{Z}^d} e^{\langle n, z\rangle}
:= \sum_{\icol{0\{\bf 0}}} + \sum_{\icol{1\{\bf 1}}} + \sum_{\icol{2\{\bf 1}}} + \sum_{\icol{2\\2}} + \sum_{\icol{3\\2}},
\]
where
\begin{align*}
\sum_{\icol{1\{\bf 1}}} &:= \sum_{n_1 \geq 0, n_2 \geq 0}
e^{\langle \big(\begin{smallmatrix}
1 \\
1 \\
\end{smallmatrix} \big) +
n_1 \big(\begin{smallmatrix}
3 \\
1 \\
\end{smallmatrix}
\big) +
n_2 \big(\begin{smallmatrix}
1 \\
2 \\
\end{smallmatrix}
\big) , z \rangle} \\
&=
e^{ \langle \big(\begin{smallmatrix}
1 \\
1 \\
\end{smallmatrix} \big) , z \rangle}
\sum_{n_1 \geq 0, n_2 \geq 0}
e^{ \langle
n_1 \big(\begin{smallmatrix}
3 \\
1 \\
\end{smallmatrix}
\big) +
n_2 \big(\begin{smallmatrix}
1 \\
2 \\
\end{smallmatrix}
\big) , z\rangle} \\
&=
e^{ \langle \big(\begin{smallmatrix}
1 \\
1 \\
\end{smallmatrix} \big) , z \rangle}
\sum_{n_1 \geq 0}
e^{n_1 \langle
\big(\begin{smallmatrix}
3 \\
1 \\
\end{smallmatrix}
\big) , z\rangle}
\sum_{n_2 \geq 0}
e^{n_2 \langle
\big(\begin{smallmatrix}
1 \\
2 \\
\end{smallmatrix}
\big) , z\rangle} \\
&=
\frac{e^{ z_1 + z_2}
}
{ (1-e^{3z_1 + z_2})(1-e^{z_1 + 2z_2})
},
\end{align*}
and similarly we have
\[
\sum_{\icol{2\{\bf 1}}}=
\frac{e^{2z_1 + z_2}
}
{ (1-e^{ 3z_1 + z_2})(1-e^{z_1 + 2z_2})
},
\]
\[
\sum_{\icol{2\\2}} =
\frac{e^{ 2z_1 + 2z_2}
}
{ (1-e^{ 3z_1 + z_2})(1-e^{z_1 + 2z_2})
},
\]
\[
\sum_{\icol{3\\2}} =
\frac{e^{ 3z_1 + 2z_2}
}
{ (1-e^{ 3z_1 + z_2})(1-e^{z_1 + 2z_2})
},
\]
and finally
\[
\sum_{\icol{0\{\bf 0}}} =
\frac{1}
{ (1-e^{3z_1 + z_2})(1-e^{z_1 + 2z_2})
}.
\]
To summarize, we have the following expression:
\[
\sum_{n \in {\mathcal K}\cap \mathbb{Z}^d} e^{ \langle n, z\rangle}
=\frac{ 1+ e^{z_1 + z_2} + e^{ 2z_1 + z_2)} + e^{2z_1 + 2z_2} + e^{3z_1 + 2z_2 }
}
{ (1-e^{3z_1 + z_2})(1-e^{z_1 + 2z_2})
}.
\]
Equivalently, using our multinomial notation $q_j:= e^{ z_j}$, we have
\[
\sum_{n \in {\mathcal K}\cap \mathbb{Z}^d} q^n
=\frac{ 1+ q_1 q_2 + {q_1}^2 q_2 + {q_1}^2 {q_2}^2 + q_1^3 q_2^2
}
{ (1- q_1^3 q_2 ) (1- q_1 q_2^2)
}.
\]
}
\hfill $\square$
\end{example}
\section{Integer point transforms of cones in general: \\ rational functions}
The Examples \ref{triangle1} and \ref{integer point transform of cone} above suggest a general pattern, namely that integer point transforms are always rational functions, and that their numerators are polynomials that encode the integer points inside a fundamental parallelepiped $\Pi$
that sits at the vertex of each vertex tangent cone. The proof of this general fact will be fairly easy - we only need to put several geometric series together, as in Figure \ref{cone3}. Now that we've seen some examples, we can prove things in general.
First, given any $d$-dimensional simplicial rational cone ${\mathcal K}\subset \mathbb{R}^d$, with integer edge vectors
$w_1, \dots, w_d \in \mathbb{Z}^d$, and apex $v\in \mathbb{R}^d$, we define the {\bf fundamental parallelepiped} of ${\mathcal K}$ by:
\begin{equation}
\Pi := \{ \lambda_1 w_1 + \cdots + \lambda_d w_d \mid
\text{ all } 0 \leq \lambda_j < 1 \},
\end{equation}
a half-open, integer parallelepiped.
In the same way that we've encoded integer points in polytopes using
$\sigma_{\mathcal P}(z)$, we can encode the integer points in $\Pi$ by defining
\[
\sigma_\Pi(z) := \sum_{n \in \mathbb{Z}^d \cap \Pi} e^{\langle z, n \rangle}.
\]
For a rational simplicial cone $K_v$, it turns out that its integer point transform
\[
\sigma_{K_v}(z):= \sum_{n \in {\mathcal K}_v \cap \mathbb{Z}^d} e^{ \langle n, z \rangle}
\]
has a pretty structure theorem - it is a rational function of the variables $e^{z_1}, \dots, e^{z_d}$, as follows.
\begin{thm} \label{closed form for integer point transform of a cone}
Given a $d$-dimensional simplicial cone ${\mathcal K}_v \subset \mathbb{R}^d$, with apex $v \in \mathbb{R}^d$, and with
$d$ linearly independent {\bf integer} edge vectors
$w_1(v), w_2(v), \dots, w_d(v) \in \mathbb{Z}^d$. Then:
\begin{equation}
\sigma_{K_v}(z)
=
\frac{ \sigma_{ \Pi + v}(z) }
{ \prod_{k=1}^d \left( 1 - e^{ \langle w_k , z \rangle} \right) }.
\end{equation}
\end{thm}
\begin{proof}
We claim that we can parametrize all of the integer points in the cone ${\mathcal K}_v$ precisely by
\begin{equation}\label{claim:the cone integer points}
{\mathcal K}_v \cap \mathbb{Z}^d = \{ p + m_1 w_1 + \cdots + m_d w_d \mid p \in (\Pi+ v)\cap \mathbb{Z}^d, \text{ and all } m_j \in \mathbb{Z}_{\geq 0}\}.
\end{equation}
To prove \eqref{claim:the cone integer points}, we begin by writing each $m\in {\mathcal K}_v \cap \mathbb{Z}^d$, by definition of the cone ${\mathcal K}_v$,
as follows:
\[
m = v+ \lambda_1 w_1 + \cdots + \lambda_d w_d,
\]
with the $\lambda_k \geq 0$. This representation of $m$ is unique,
because $w_1, \dots, w_d$ is a basis for $\mathbb{R}^d$. Now we use the fact that each
$\lambda_k = \lfloor \lambda_k \rfloor + \{ \lambda_k \}$, where $\{x\}$ is the fractional part of $x$:
\begin{align*}
m &= v+ \Big( \{ \lambda_1 \} w_1 + \cdots + \{ \lambda_d \} w_d \Big) +
\lfloor \lambda_1 \rfloor w_1 + \cdots + \lfloor \lambda_d \rfloor w_d \\
&:=p + \lfloor \lambda_1 \rfloor w_1 + \cdots + \lfloor \lambda_d \rfloor w_d,
\end{align*}
where we've defined $p:= v+ \Big( \{ \lambda_1 \} w_1 + \cdots + \{ \lambda_d \} w_d \Big)$. We now notice that
$p \in v+ \Pi$, and in fact $p \in \mathbb{Z}^d$, because $m, w_1, \dots, w_d \in \mathbb{Z}^d$.
Since $\Pi$ tiles the cone ${\mathcal K}_v$ precisely by the translation vectors $w_1, \dots, w_d$, we see
that the set of all integer points in ${\mathcal K}_v$ is precisely
the disjoint union of the sets
\begin{equation} \label{typical integer points in the cone}
\{ p + k_1 w_1 + \dots + k_d w_d \mid \, k_1, \dots, k_d \in \mathbb{Z}_{\geq 0} \}
\end{equation}
(which we may think of as `multidimensional arithmetic progressions') , as $p$ varies over the integer points of $\Pi$ .
Finally, we expand each denominator in the following rational function as a geometric series to get:
\[
\frac{ \sigma_{ \Pi + v}(q) }
{ \prod_{j=1}^d \left( 1 - q^{ w_j} \right) } =
\left(
\sum_{p \in (\Pi + v) \cap \mathbb{Z}^d} q^p
\right)
\left(
\sum_{k_1 \geq 0} q^{k_1 w_1}
\right)
\cdots
\left(
\sum_{k_d \geq 0} q^{k_d w_d}
\right).
\]
Multiplying out all of these geometric series together, we see that the exponents look like
the points in \eqref{typical integer points in the cone}.
\end{proof}
\bigskip
\section*{Notes} \label{Notes.chapter.Brion}
\begin{enumerate}[(a)]
\item In the development of the current book, we saw that the discrete version of Brion's theorem
(Theorem \ref{brion, discrete form}) followed from the continuous version of Brion's theorem
(Theorem \ref{brion, continuous form}). The tool we used in order to discretize Theorem \ref{brion, continuous form} was the Poisson summation formula.
In our previous book \cite{BeckRobins}, the ideas developed in exactly the opposite direction: in that context we first proved the discrete Brion theorem, and then derived the continuous version from it.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\medskip
\begin{prob} \label{unimodular cone, integer point transform}
Suppose that ${\mathcal P}\subset \mathbb{R}^d$ is a unimodular polytope, with vertex set $V$. Using Theorem \ref{brion, discrete form}, show that its integer point transform is:
\begin{equation}
\sigma_{\mathcal P}(z) = \sum_{v \in V}
\frac{e^{\langle v, z \rangle} }{\prod_{k=1}^d \left( 1 - e^{\langle w_k, z\rangle} \right) }.
\end{equation}
\end{prob}
\medskip
\begin{prob}
Fix a positive integer $m > 1$, and let ${\mathcal P}$ be the $2$-dimensional triangle
whose vertices are given by
$(0, 0), (0, 1)$, and $(m, 0)$. First compute the integer point transforms
$\sigma_{K_v}(z)$ for its three vertex tangent cones, and then compute the integer point transform $\sigma_{\mathcal P}(z)$.
\end{prob}
\medskip
\begin{prob} \label{bound for integer point transform}
Let ${\mathcal P} \subset \mathbb{R}^d$ be the $d$-dimensional polytope. We recall that for $x \in \mathbb{R}^d$, we have the definition
$\sigma_{\mathcal P}(2\pi i x) := \sum_{n\in {\mathcal P} \cap \mathbb{Z}^d } e^{2\pi i \langle x, n \rangle}$.
Prove that
\[
\left | \sigma_{\mathcal P}(2\pi i x) \right | \leq \left |\mathbb{Z}^d \cap {\mathcal P} \right |,
\]
for all $x \in \mathbb{R}^d$.
\end{prob}
\medskip
\begin{prob}
Let ${\mathcal P}$ be the $3$-dimensional simplex whose vertices are given by
$(0, 0, 0), (1, 1, 0), (1, 0, 1)$, and $(0, 1, 1)$. Compute its integer point transforms
$\sigma_{K_v}(z)$ for all of its four vertex tangent cones, and then compute its integer point transform $\sigma_{\mathcal P}(z)$.
\end{prob}
\medskip
\begin{prob}
Suppose we are given a $2$-dimensional simplicial integer cone ${\mathcal K} \subset \mathbb{R}^2$, together with its polar cone ${\mathcal K}^o$.
Is there a simple relationship between
the integer point transforms $\sigma_{{\mathcal K}}(z)$ and $\sigma_{{\mathcal K}^o}(z)$?
\end{prob}
Notes. It's worth thinking about the relationship between the edge vectors of the
fundamental parallelepipeds for ${\mathcal K}$ and ${\mathcal K}^o$.
\chapter{Counting integer points in polytopes - the Ehrhart theory} \label{Ehrhart theory}
\index{Ehrhart theory}
\begin{quote}
``How wonderful that we have met with a paradox. Now we have some hope of making progress. ''
-- Niels Bohr \index{Niels Bohr}
\end{quote}
\begin{wrapfigure}{R}{0.58\textwidth}
\centering
\includegraphics[width=0.62\textwidth]{DiscreteArea}
\end{wrapfigure}
\section{Intuition}
A basic question in discrete geometry is ``how do we discretize volume?"
One method of discretizing the volume of ${\mathcal P}$ is to count the number of integer points in ${\mathcal P}$.
Even in $\mathbb{R}^2$, this question may be highly non-trivial, depending on the arithmetic properties of the vertices of ${\mathcal P}$. Ehrhart first considered integer dilations of a fixed, integer polytope ${\mathcal P}$, and
defined:
\begin{equation} \label{combinatorial discrete volume}
L_{{\mathcal P}}(t):= | \mathbb{Z}^d \cap t{\mathcal P} |,
\end{equation}
where $t{\mathcal P}$ is the $t$'th dilate of ${\mathcal P}$, and $t$ is a positive integer. Ehrhart showed
that $L_{{\mathcal P}}(t)$ is a polynomial in the positive integer parameter $t$, known as the
{\bf Ehrhart polynomial} of ${\mathcal P}$.
Viewed from the lens of Fourier analysis, Ehrhart polynomials may be computed by `averaging'
the Fourier transform of a polytope over the full integer lattice:
\begin{equation}
L_{{\mathcal P}}(t):= | \mathbb{Z}^d \cap t{\mathcal P} | = \sum_{n\in \mathbb{Z}^d} 1_{t{\mathcal P}}(n) = \sum_{\xi \in \mathbb{Z}^d} \hat 1_{t{\mathcal P}}(\xi),
\end{equation}
where the latter identity uses Poisson summation, but because we may not use indicator functions directly in
Poisson summation, \index{Poisson summation formula}
some care is required and the process of smoothing may be applied to $1_{\mathcal P}$.
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.5in]{Ehrhart1}
\end{center}
\caption{Here the polytope ${\mathcal P}$ is the unit square, and we also have its $5$'th dilate $5{\mathcal P}$.} \label{Ehrhart1}
\end{figure}
We can also compare this combinatorial method to discretize volume, namely equation \eqref{combinatorial discrete volume},
and the discrete volumes of the previous chapter which used solid angles.
More generally, given a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$, we may sum the values of $f$
at all integer points and observe how close this sum gets to the integral of $f$ over ${\mathcal P}$. This approach is known as Euler-Maclaurin summation \index{Euler-Maclaurin summation}
over polytopes, and is a current and exciting topic of a growing literature (see Note \ref{EM summation note} below).
\bigskip
\section{Computing integer points in polytopes via the discrete Brion Theorem}
\begin{example}
\rm{
Probably the simplest example in $\mathbb{R}^2$ is the unit square ${\mathcal P}:= [0, 1]^2$.
As Figure \ref{Ehrhart1} suggests,
the $t$-dilate $t{\mathcal P}$ here contains $(t+1)^2 = t^2 + 2t + 1$ points of the integer lattice $\mathbb{Z}^2$. Here it was easy to conclude that
$L_{\mathcal P}(t)$ was a polynomial function of $t \in \mathbb{Z}_{>0}$, but by a small miracle of nature
a similar phenomenon occurs for \emph{all integer polytopes} in $\mathbb{R}^d$.
\hfill $\square$
}
\end{example}
If all of the vertices of ${\mathcal P}$ have integer coordinates, we call ${\mathcal P}$ an {\bf integer polytope}.
On the other hand, if all of the vertices of a polytope ${\mathcal P}$ have rational coordinates, we call ${\mathcal P}$ a {\bf rational polytope} \index{rational polytope}.
Let ${\mathcal P} \subset \mathbb{R}^d$ be a rational, $d$-dimensional polytope, and let $N$ be the number of its vertices.
For each vertex $v$ of ${\mathcal P}$, we consider the vertex tangent cone ${\mathcal K}_v$ of $ {\mathcal P}$. Once we dilate ${\mathcal P}$ by $t$, each
vertex $v$ of ${\mathcal P}$ gets dilated to become $tv$, and so each of the vertex tangent cones ${\mathcal K}_v$ of ${\mathcal P}$ simply get shifted to the
corresponding vertex tangent cone ${\mathcal K}_{tv}$ of $t{\mathcal P}$. Using the discrete Brion theorem (Theorem \ref{brion, discrete form}), we have
\begin{equation} \label{simplified discrete Brion identity}
\sum_{n \in t{\mathcal P} \cap \mathbb{Z}^d} e^{ \langle n, z\rangle} =
\sum_{n \in {\mathcal K}_{tv_1} \cap \mathbb{Z}^d} e^{ \langle n, z\rangle} + \cdots +
\sum_{n \in {\mathcal K}_{tv_N} \cap \mathbb{Z}^d} e^{\langle n, z\rangle},
\end{equation}
for all $z \in \mathbb{C}^d - S$, where $S$ is the hyperplane arrangement defined by the (removable) singularities of all of the transforms $\hat 1_{{\mathcal K}_{v_j}}(z)$.
To simplify notation, we have absorbed the constant $-2\pi i$ into the complex vector $z$ by replacing
$z$ by $-\frac{1}{2\pi i } z$.
We recall that we rewrote \eqref{simplified discrete Brion identity} by using the notation:
\begin{equation}\label{discrete Brion, with rational functions}
\sigma_{\mathcal P}(z) = \sigma_{{\mathcal K}_{v_1}}(z) + \cdots + \sigma_{{\mathcal K}_{v_N}}(z).
\end{equation}
And now we notice that when $z = 0$, the left-hand-side gives us precisely
\[
\sum_{n \in t{\mathcal P} \cap \mathbb{Z}^d} 1 := | \mathbb{Z}^d \cap t{\mathcal P} |,
\]
which is good news - it is the Ehrhart polynomial $L_{{\mathcal P}}(t)$, by definition. The bad news is that $z=0$ is a singularity of the right-hand-side of
\eqref{discrete Brion, with rational functions}. But then again, there is still more good news - we already saw in the previous chapter
that it is a removable singularity. So we may let $z\rightarrow 0$, and discover what happens.
\bigskip
\begin{example} \label{Ehrhart poly for the standard triangle}
\rm{
We can find a formula for the Ehrhart polynomial $L_{{\mathcal P}}(t) := | \mathbb{Z}^2 \cap t{\mathcal P} |$ of the standard triangle, continuing Example \ref{example:standard triangle integer point transform}. It turns out
that the method we use in this example is universal - it can always be used to find the Ehrhart polynomial of any rational polytope. We will formalize this statement in the ensuing section.
In this example we are lucky in that we may use brute-force to compute it, since the number of integer points in the $t$-dilate of ${\mathcal P}$ may be computed along the diagonals:
\[
L_{{\mathcal P}}(t) = 1 + 2 + 3 + \cdots + (t+1) = \frac{(t+1)( t+2)}{2} =
\frac{1}{2} t^2 + \frac{3}{2}t + 1.
\]
Now we can confirm this lucky answer with our brand new machine, as follows.
Using \eqref{simplified discrete Brion identity},
and the formulation \eqref{last line} from
Example \ref{example:standard triangle integer point transform}.,
we have the integer point transform for the dilates of ${\mathcal P}$:
\begin{align}
& \sum_{n \in t\Delta \cap \mathbb{Z}^d} e^{ \langle n, z\rangle} =
\sum_{n \in {\mathcal K}_{tv_1} \cap \mathbb{Z}^d} e^{ \langle n, z\rangle}
+ \sum_{n \in {\mathcal K}_{tv_1} \cap \mathbb{Z}^d} e^{ \langle n, z\rangle}
+ \sum_{n \in {\mathcal K}_{tv_3} \cap \mathbb{Z}^d} e^{\langle n, z\rangle} \\
\label{integer point transform for standard triangle}
&= \frac{1}{(1- e^{z_1} )(1- e^{z_2})} + \frac{e^{t z_1}}{(e^{-z_1}-1 ) ( e^{-z_1+z_2} -1)}
+\frac{e^{t z_2}}{(e^{-z_2}-1)(e^{z_1-z_2}-1)} \\ \label{symmetric rationals}
& := F_1(z) + F_2(z) + F_3(z),
\end{align}
where we have defined $F_1, F_2, F_3$ by the last equality. We can let $z\rightarrow 0$ along almost any direction, but it turns out that we can simplify our computations by taking advantage of the symmetry of this polytope, so we will pick $z = \icol{ \ x \\ -x} $, which will simplify our computations
(see Note \ref{Michel Faleiros}). Here is our plan:
\begin{enumerate}[(a)]
\item We pick $z:= \icol{ \ x\\ -x} $.
\item We expand all three meromorphic functions $F_1, F_2, F_ 3$
in terms of their Laurent series in $x$, giving us Bernoulli numbers.
\item Finally, we let $x \rightarrow 0$, to retrieve the constant term (which will be
a polynomial function of $t$) of the resulting Laurent series.
\end{enumerate}
To expand $F_1(z), F_2(z), F_3(z)$ in their Laurent series, we recall the definition
\ref{Def. of Bernoulli numbers}
of the Bernoulli numbers in terms of their generating function, namely
$ \frac{t}{e^t-1} = \sum_{k =0}^\infty B_k \frac{t^k}{k!}$:
\begin{align*}
F_1(x, -x) &= \frac{-1}{x^2} \sum_{m \geq 0} B_m \frac{x^m}{m!}
\ \sum_{n \geq 0} B_n \frac{(-x)^n}{n!} \\
&= \frac{-1}{x^2} \left( 1 - \frac{x}{2} + \frac{x^2}{12} + O(x^3)\right)
\left( 1 + \frac{x}{2} + \frac{x^2}{12} + O(x^3)\right) \\
&= \frac{-1}{x^2} - \frac{1}{3} + O(x)
\end{align*}
Similarly, we have
\begin{align*}
F_2(x, -x) &= \frac{1 + t x + \frac{t^2}{2!} x^2 + O(x^3) }{2x^2}
\left(1 + \frac{x}{2} + \frac{x^2}{12} + O(x^3) \right)
\left(1 + \frac{(2x)}{2} + \frac{(2x)^2}{12} + O(x^3) \right) \\
&= \frac{1}{2x^2} + \frac{3}{4x} + \frac{2}{3} + \frac{t}{2x} + \frac{3t}{4} + \frac{t^2}{4} + O(x)
\end{align*}
Now, by symmetry we see that $F_3(x,-x) = F_2(-x, x)$, so that by \eqref{symmetric rationals}
and the latter expansions, we finally have:
\[
\sum_{n \in t\Delta \cap \mathbb{Z}^d} e^{ \langle n, \icol{ \ x\\ -x} \rangle} =
F_1(x, -x) + F_2(x, -x) + F_2(-x, x) = 1 + \frac{3}{2} t + \frac{1}{2} t^2 + O(x).
\]
Letting $z:= \icol{ \ x \\ -x} \rightarrow 0$ in the latter
computation, we retrieve the (Ehrhart) polynomial:
\[
\sum_{n \in t\Delta \cap \mathbb{Z}^d} 1 = L_\Delta(t) = 1 + \frac{3}{2} t + \frac{1}{2} t^2,
\]
as desired.
}
\hfill $\square$
\end{example}
\bigskip
\section{The Ehrhart polynomial of an integer polytope, and the Ehrhart quasi-polynomial of a rational polytope}
\begin{wrapfigure}{R}{0.39\textwidth}
\centering
\includegraphics[width=0.30\textwidth]{Ehrhart}
\caption{Eugene Ehrhart} \label{Ehrhart}
\end{wrapfigure}
Eugene Ehrhart initiated a systematic study of the integer point enumerator
\[
L_{\mathcal P}(t):= \left| t{\mathcal P} \cap \mathbb{Z}^d \right|,
\]
for an integer polytope ${\mathcal P}$, which Ehrhart proved was always a polynomial function of the positive integer dilation parameter $t$.
Ehrhart also proved
that for a rational polytope ${\mathcal P} \subset \mathbb{R}^d$, the integer point enumerator
$L_{\mathcal P}(t)$ is a {\bf quasi-polynomial} in the positive integer parameter $t$, which means by definition that
\begin{equation}
L_{\mathcal P}(t) = c_d t^d + c_{d-1}(t) t^{d-1} + \cdots + c_1(t) t + c_0(t),
\end{equation}
where each $c_j(t)$ is a periodic function of $t \in \mathbb{Z}_{>0}$.
The study of Ehrhart polynomials and Ehrhart quasi-polynomials has enjoyed a renaissance in recent years (\cite{BarvinokEhrhartbook}, \cite{BeckRobins}), and has some suprising connections to many branches of science, and even to voting theory, for example.
\bigskip
\begin{thm}[Ehrhart] \label{Ehrhart's main result}
For an integer polytope \\
${\mathcal P} \subset \mathbb{R}^d$, its discrete volume $L_{\mathcal P}(t)$
is a polynomial functions of $t$, for all positive integer values of the dilation parameter $t$. Moreover, we have
\begin{equation}
L_{\mathcal P}(t) = (\vol {\mathcal P}) t^d + c_{d-1} t^{d-1} + \cdots + c_1 t + 1.
\end{equation}
\end{thm}
\hfill $\square$
Ehrhart's Theorem \ref{Ehrhart's main result} has an extension to rational polytopes, as follows.
We will derive the more general Theorem \ref{Ehrhart's rational polytope theorem} of Ehrhart, by using
the discrete Brion Theorem \ref{brion, discrete form}.
\bigskip
\begin{thm}[Ehrhart] \label{Ehrhart's rational polytope theorem}
For a rational polytope \\
${\mathcal P} \subset \mathbb{R}^d$, its discrete volume $L_{\mathcal P}(t)$
is a quasi-polynomial function of $t$, for all positive integer values of the dilation parameter $t$.
In particular, we have
\begin{equation}
L_{\mathcal P}(t) = (\vol {\mathcal P}) t^d + c_{d-1}(t) t^{d-1} + \cdots + c_1(t) t + c_0(t),
\end{equation}
where each {\bf quasi-coefficient} $c_k(t)$ is a periodic function of $t\in \mathbb{Z}_{>0}$.
\end{thm}
\noindent
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.3in]{Ehrhart2}
\end{center}
\caption{Left: A triangle ${\mathcal P}$, and its dilate $2{\mathcal P}$. \, Right: the vertex tangent cones ${\mathcal K}_v$ and ${\mathcal K}_{2v}$ have
the same edge vectors $w_1, w_2$. } \label{Ehrhart2}
\end{figure}
\begin{proof}
To begin, suppose that $p$ the least common denominator of the coordinates of all the rational vertices of ${\mathcal P}$.
We need to show that, for each fixed $0\leq r < p$, the integer point enumerator $L_{\mathcal P}(r+ pk)$ is a polynomial in the parameter $k\in \mathbb{Z}_{>0}$.
By definition of a quasi-polynomial, this will prove that $L_{\mathcal P}(t)$ is a quasi-polynomial in $t \in \mathbb{Z}_{>0}$. In other words, we restrict attention to each fixed arithmetic progression of dilations in $t:= r + pk$. Now, from the discrete Brion Theorem \ref{brion, discrete form}, we know that
\begin{equation}\label{Brion, for this proof}
\sigma_{ {\mathcal P}}(z) = \sigma_{ {\mathcal K}_{v_1}}(z) + \cdots + \sigma_{ {\mathcal K}_{v_N}}(z),
\end{equation}
and we also know the elementary relation
\[
\sigma_{ t{\mathcal P}}(0) := \sum_{n \in \mathbb{Z}^d \cap t{\mathcal P}} 1 = L_{\mathcal P}(t).
\]
So we'd like to let $z \rightarrow 0$ on both sides of Brion's discrete identity \eqref{Brion, for this proof}:
\begin{equation}\label{first step of Ehrhart in terms of integer point transforms}
L_{\mathcal P}(t) = \lim_{z\rightarrow 0}
\Big(
\sigma_{ t{\mathcal K}_{v_1}}(z) + \cdots + \sigma_{ t {\mathcal K}_{v_N}}(z)
\Big).
\end{equation}
The bad news is that
the right-hand-side of \eqref{Brion, for this proof}
introduces local singularities in the denominators of each
rational-exponential function
\[
\sigma_{ {\mathcal K}_{v_j}}(z) = \sum_{n \in {\mathcal K}_v \cap \mathbb{Z}^d} e^{ \langle n, z\rangle}.
\]
But there is good news too! These singularities must be removable singularities. The reason is easy -
$ \sigma_{{\mathcal P}}(z)$ is a finite sum of exponentials (by compactness of ${\mathcal P}$), and is therefore an analytic function of $z$, so any singularities on the right-hand side of \eqref{Brion, for this proof} must be removable singularities.
To proceed further, we'll begin by writing each vertex tangent cone ${\mathcal K}_v$ in terms of its vertex $v$, and edge vectors $w_j$:
\[
{\mathcal K}_v := \left\{ v + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\}.
\]
Now we consider the dilates of ${\mathcal K}_v$ a bit more carefully, and we will use the fact that the edge
vectors $w_1, \dots, w_{M_v}$ of any dilate of a vertex tangent cone ${\mathcal K}_{tv}$ remain invariant, as in Figure \ref{Ehrhart2}:
\[
{\mathcal K}_{tv} := \left\{t v + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\}.
\]
{\bf Case $1$.} Suppose $r \not=0$. Then:
\begin{align*}
t{\mathcal K}_v &:= (r+pk){\mathcal K}_v = \left\{ (r+pk)v + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\} \\
&= k(pv) + \left\{ rv + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\} \\
&= k(pv) + r {\mathcal K}_v.
\end{align*}
The salient feature of this computation is that $pv$ is an integer vector, by definition of $p$. This implies that
\begin{align*}
\sigma_{ t {\mathcal K}_{v}}(z) &:=\sigma_{ (r+pk) {\mathcal K}_{v}}(z) :=
\sum_{n \in \Big( k(pv) + r {\mathcal K}_v \Big) \cap \mathbb{Z}^d} e^{ \langle n, z\rangle} \\
&= \sum_{m \in r {\mathcal K}_v \cap \mathbb{Z}^d} e^{ \langle k(pv) + m, z\rangle} \\
&= e^{ \langle k(pv), z\rangle} \sum_{m \in r {\mathcal K}_v \cap \mathbb{Z}^d} e^{ \langle m, z\rangle} \\
&:= e^{ \langle k(pv), z\rangle} \sigma_{rK_v}(z) \\
\end{align*}
Summarizing, \eqref{first step of Ehrhart in terms of integer point transforms} gives us:
\[
L_{\mathcal P}(t)= \lim_{z\rightarrow 0}
\sum_{v \in V} e^{ \langle k(pv), z\rangle} \sigma_{rK_v}(z),
\]
and giving a common denominator to all of the rational functions (of $e^{z_j}$) $\sigma_{rK_v}(z)$,
we may apply L'Hospital's rule a finite number of times. Because the integer variable $k$ only appears in the exponents
$e^{ \langle k(pv), z\rangle}$, we see that each time we apply L'Hospital, an extra factor of $k$ comes down, giving us a polynomial function of $k$.
{\bf Case $2$.} Suppose $r =0$. Here the situation is slightly easier: $t = pk$, so
\begin{align*}
t{\mathcal K}_v &:= pk {\mathcal K}_v = \left\{ pk v + \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\} \\
&= k(pv) + \left\{ \sum_{j=1}^{M_v} \lambda_j w_j \mid \text{ all } \lambda_j \geq 0 \right\} \\
&= k(pv) + ({\mathcal K}_v - v),
\end{align*}
which is an integer cone because $pv$ is an integer vector, and ${\mathcal K}_v - v$ is an integer cone with apex at the origin.
Similarly to the computation above, we have
\begin{align*}
\sigma_{ t {\mathcal K}_{v}}(z)
&= \sum_{n \in \Big( kpv + \left({\mathcal K}_v-v \right) \Big) \cap \mathbb{Z}^d} e^{ \langle n, z\rangle}
= \sum_{m \in \left({\mathcal K}_v-v \right) \cap \mathbb{Z}^d} e^{ \langle kpv + m, z\rangle} \\
&= e^{ \langle kpv, z\rangle} \sum_{m \in \left({\mathcal K}_v-v \right)\cap \mathbb{Z}^d} e^{ \langle m, z\rangle} \\
&:= e^{ \langle kpv, z\rangle} \sigma_{ \left({\mathcal K}_v-v \right)}(z),
\end{align*}
and the remaining steps are identital to Case $1$.
\end{proof}
We note that for an integer polytope ${\mathcal P}$, the same proof gives us Theorem \ref{Ehrhart's main result}, namely
that $L_{\mathcal P}(t)$ is a polynomial for positive integer dilations $t$;
here we just need Case $2$, with $t:=k$, so that $r=0$ and $p=1$.
We emphasize again that one of the important steps in the latter computation was the fact that in both cases of the proof above, $k(pv)$ was an integer vector, allowing us to rewrite the integer point transform of the cone in a simpler way.
As a first application of Theorem \ref{Ehrhart's main result}, we show that the discrete volume of a (half-open) parallelepiped has a particularly elegant and useful form.
\begin{lem}
Let $D$ be any half-open integer parallelepiped in $\mathbb{R}^d$, defined by
\[
D:= \left\{
\lambda_1 w_1 + \cdots + \lambda_d w_d \mid 0 \leq \lambda_1, \dots, \lambda_d < 1
\right\},
\]
where $w_1, \cdots w_d \in \mathbb{Z}^d$ are linearly independent. Then:
\[
\#\{ \mathbb{Z}^d \cap D\} = \vol D,
\]
and for each positive integer $t$, we also have
\[
\#\{ \mathbb{Z}^d \cap tD\} = \left(\vol D\right) t^d.
\]
\end{lem}
\begin{proof} We can tile $tD$ by using $t^d$ translates of $D$, because $D$ is half-open. Therefore
\[
\#\{ \mathbb{Z}^d \cap tD\} = \#\{ \mathbb{Z}^d \cap D\} t^d,
\]
and by definition $\#\{ \mathbb{Z}^d \cap tD\} = L_D(t)$. On the other hand, we also know by Ehrhart's Theorem \ref{Ehrhart's main result} that $L_D(t)$ is a polynomial for integer values of $t$, whose leading coefficient is $\vol D$. Since $L_D(t)=\#\{ \mathbb{Z}^d \cap D\} t^d$ for all positive integer values of $t$, we conclude that
\[
\#\{ \mathbb{Z}^d \cap D\}= \vol D.
\]
\end{proof}
\bigskip
\section{Unimodular polytopes}
A $d$-dimensional integer simplex $\Delta$ is called a {\bf unimodular simplex} if $\Delta$ is the modular image of the
standard simplex $\Delta_{\rm standard}$, \index{standard simplex}
the convex hull of the points $\{ 0, {\bf e_1}, \dots, {\bf e_d} \} \subset \mathbb{R}^d$, where
${\bf e_k}:= (0, \dots, 0, 1, 0, \dots, 0)$ is the standard unit vector pointing in the direction of the positive axis $x_k$.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3in]{upolygon}
\end{center}
\caption{A unimodular polygon - each vertex tangent cone is a unimodular cone. It is clear from the construction in the Figure that we can form arbitrarily large unimodular polygons.}
\label{unimodular polygon}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.3in]{ucone}
\end{center}
\caption{A unimodular cone at $v$, appearing as one of the vertex tangent cones in
Figure \ref{unimodular polygon}. We notice that its half-open fundamental parallelepiped, with vertex
at $v$, does not contain any integer points other than $v$.} \label{unimodular cone}
\end{figure}
\begin{example}
\rm{
Let $\Delta := {\rm conv}\left( \icol{0\{\bf 0}\{\bf 0}}, \icol{1\{\bf 0}\{\bf 0}}, \icol{1\{\bf 1}\{\bf 0}}, \icol{1\{\bf 1} \\ 1}
\right)$, their convex hull. Then $\Delta$ is a unimodular simplex, because the unimodular matrix
$\left(
\begin{smallmatrix}
1 & 1 & 1\\
0 & 1 & 1 \\
0 & 0 & 1
\end{smallmatrix}
\right)$ maps the standard simplex $\Delta_{\rm standard}$ to $\Delta$.
}
\hfill $\square$
\end{example}
It is not difficult to show that the tangent cone of a unimodular simplex possesses edge vectors that
form a lattice basis for $\mathbb{Z}^d$. Thus, it is natural to define a {\bf unimodular cone} ${\mathcal K} \subset \mathbb{R}^d$ as a simplicial cone, possessing the additional property that its $d$ edge vectors form a lattice basis for $\mathbb{Z}^d$.
\begin{example}
\rm{
We consider the polygon ${\mathcal P}$ in Figure \ref{unimodular polygon}. An easy verification shows that
each of its vertex tangent cones is unimodular. For example, focusing on the vertex $v$, we
see from Figure \ref{unimodular cone}, that its vertex tangent cone is
${\mathcal K}_v:= v + \{ \lambda_1 \icol{ \ 1\\-2} + \lambda_2 \icol{-1\\ \ 1} \mid
\lambda_1, \lambda_2 \geq 0 \}$.
${\mathcal K}_v$ is a unimodular cone, because the matrix formed by the its two edges
$\icol{ \ 1\\ -2}$ and $\icol{-1\\ \ 1}$ is a unimodular matrix.
}
\hfill $\square$
\end{example}
More generally, a simple, integer polytope is called a
{\bf unimodular polytope} if each of its vertex tangent cones is a unimodular cone.
Unimodular polytopes are the first testing ground for many conjectures in discrete geometry and number theoery. Indeed, we will see later that the number of integer points in a unimodular polytope, namely
$|\mathbb{Z}^d \cap {\mathcal P}|$, admits a simple and computable formula, if we are given the local tangent cone information at each vertex. By contrast, it is in general thought to be quite difficult to compute the number of integer points $|\mathbb{Z}^d \cap {\mathcal P}|$, even for (general) simple polytopes, a problem that belongs to the NP-hard class of problems (if the dimension $d$ is not fixed).
\begin{lem}
Suppose we have two integer polytopes ${\mathcal P}, \mathcal Q \subset \mathbb{R}^d$, which are unimodular images of each other:
\[
{\mathcal P} = U \mathcal Q,
\]
for some unimodular matrix $U$.
Then $L_{\mathcal P}(t) = L_{\mathcal Q}(t)$, for all $t \in \mathbb{Z}_{\geq 0}$.
\end{lem}
\bigskip
\begin{lem}\label{NEW: unimodular simplex equivalence}
Suppose that $\Delta \subset \mathbb{R}^d$ is a $d$-dimensional integer simplex. Then $\Delta$ is a unimodular simplex $\iff$ $(d-1)\Delta$ does not contain any interior integer points.
\end{lem}
\section{More examples of rational polytopes and quasi-polynomials}
The following properties for the floor function, the ceiling function, and the fractional part function
are often useful.
It's convenient to include the following indicator function, for the full set of integers, as well:
\[
1_{\mathbb{Z}}(x) :=
\begin{cases}
1 & \text{if } x \in \mathbb{Z} \\
0 & \text{if } x \notin \mathbb{Z} \\
\end{cases},
\]
the indicator function for $\mathbb{Z}$. For all $x\in \mathbb{R}$, we have:
\begin{enumerate}[(a)]
\item $\left\lceil x \right\rceil = - \floor{-x}$ \label{fractional part property a}
\item $1_{\mathbb{Z}}(x)= \floor{x} - \left\lceil x \right\rceil +1$
\item $ \{ x \} + \{-x\} = 1- 1_{\mathbb{Z}}(x)$
\item Let $m \in \mathbb{Z}_{>0}, n \in \mathbb{Z}$. Then $\floor{ \frac{n-1}{m} } + 1 = \left\lceil \frac{n}{m} \right\rceil$.
\end{enumerate}
(Exercise \ref{properties of floor, ceiling, fractional part})
\bigskip
\begin{example}
\rm{
Let's find the integer point enumerator $L_{\mathcal P}(t) := | \mathbb{Z} \cap t{\mathcal P} |$ of the rational line segment
$ {\mathcal P} := [\frac{1}{3}, \ 1 ]$. Proceeding by brute-force, for $t \in \mathbb{Z}_{>0}$ we have
\begin{align}
L_{\mathcal P}(t) &= \left| \left[\frac{t}{3}, \ t \right] \cap \mathbb{Z} \ \right| \label{answer comparison}
=\floor{t} - \left\lceil \frac{t}{3} \right\rceil + 1 \\
&= t + \floor{ -\frac{t}{3} } +1 \\
&= t + -\frac{t}{3} - \left\{-\frac{t}{3} \right\} +1\\
&= \frac{2}{3} t - \left\{ -\frac{t}{3} \right\} +1,
\end{align}
a periodic function on $\mathbb{Z}$ with period $3$. Here we used property \ref{fractional part property a} in the third equality.
In fact, here we may let $t$ be any positive real number, and we still obtain the same answer, in this $1$-dimensional case.
Now we will compare this to a new computation, but this time from the perspective of the vertex tangent cones.
For the cone ${\mathcal K}_{tv_1} := [\frac{t}{3}, + \infty)$, we can parametrize the integer points in this cone by
${\mathcal K}_{tv_1} \cap \mathbb{Z} =
\{ \left\lceil \frac{t}{3} \right\rceil , \left\lceil \frac{t}{3} \right\rceil +1, \dots \}$, so that
\begin{align*}
\sigma_{{\mathcal K}_{tv_1}}(z) = e^{ \left\lceil \frac{t}{3} \right\rceil z} \sum_{ n \geq 0 } e^{n z} =
e^{ \left\lceil \frac{t}{3} \right\rceil z} \frac{1}{1-e^{z}}.
\end{align*}
For the cone ${\mathcal K}_{tv_2} := (-\infty , t]$, we can parametrize the integer points in this cone by
${\mathcal K}_{tv_2} \cap \mathbb{Z} = \{ t, t-1, \dots \}$, so that
\begin{align*}
\sigma_{{\mathcal K}_{tv_2}}(z) = e^{ t \cdot z} \sum_{ n \leq 0 } e^{n z} =
e^{tz} \frac{1}{1-e^{-z}}.
\end{align*}
So by the discrete Brion Theorem (which is here essentially a finite geometric sum), we get:
\begin{align*}
\sum_{n \in [\frac{t}{3}, t] } e^{nz} &= e^{ \left\lceil \frac{t}{3} \right\rceil z}
\frac{1}{1-e^{z}} + e^{tz} \frac{1}{1-e^{-z}} \\
& =-\left(1 + \left\lceil \frac{t}{3} \right\rceil z + \left\lceil \frac{t}{3} \right\rceil^2 \frac{z^2}{2!} + \cdots \right)
\left(\frac{1}{z} -\frac{1}{2} + \frac{1}{12} z + \cdots \right) \\
&+\left( 1 + (t+1)z + (t+1)^2 \frac{z^2}{2!} + \cdots \right)
\left(\frac{1}{z} -\frac{1}{2} + \frac{1}{12} z + \cdots \right) \\
&= \frac{1}{2} - \left\lceil \frac{t}{3} \right\rceil + (t+1) -\frac{1}{2} +O(z) \longrightarrow \
t - \left\lceil \frac{t}{3} \right\rceil +1,
\end{align*}
as $z\rightarrow 0$, recovering the same answer \ref{answer comparison} above.
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
Let's find the integer point enumerator $L_{\mathcal P}(t) := | \mathbb{Z}^2 \cap t{\mathcal P} |$ of the rational triangle
\[
{\mathcal P}:= \conv\left( \icol{0\{\bf 0}}, \icol{\frac{1}{2} \{\bf 0}}, \icol{0\\ \frac{1}{2} } \right).
\]
First we will proceed by brute-force (which does not always work well), and then we will use the machinery of \eqref{simplified discrete Brion identity}.
For the brute-force method, we need to consider separately the even integer dilates and the odd integer dilates. Letting $t=2n$ be a positive even integer, it's clear geometrically
that
\begin{align*}
L_{\mathcal P}(t) &:= | \mathbb{Z}^2 \cap 2n{\mathcal P} | = 1+ 2 + \cdots + n \\
&= \frac{n(n+1)}{2} = \frac{ \frac{t}{2} ( \frac{t}{2} +1)}{2} \\
&= \frac{1}{8} t^2 + \frac{1}{4} t.
\end{align*}
On the other hand, if $t= 2n-1$, then we
notice that we never have an integer point on the diagonal face of ${\mathcal P}$, so that in this case
we get:
\[
L_{\mathcal P}(t) := | \mathbb{Z}^2 \cap (2n-1){\mathcal P} | = 1+ 2 + \cdots + n = \frac{ \frac{t+1}{2} ( \frac{t+1}{2} +1)}{2}
= \frac{1}{8}t^2 + \frac{1}{2} t + \frac{3}{8}.
\]
Alternatively, we may also rederive the same answer by using the Brion identity
\eqref{simplified discrete Brion identity}. We can proceed as in
Example \ref{Ehrhart poly for the standard triangle}. The only difference now is that the vertex tangent cones have rational apices. So although we may still use the same edge vectors to parametrize the integer points in
${\mathcal K}_{tv_3} \cap \mathbb{Z}^d$, we now have a new problem:
the rational vertex $v_3 = \icol{0\\ \frac{1}{2}}$.
But in any case, we get:
${\mathcal K}_{tv_3} \cap \mathbb{Z}^d= \left\{
\icol{0 \\ \frac{t}{2} } + n_1 \icol{ \ 0 \\ -1} + n_2\icol{ \ 1\\-1} \mid n_1, n_2 \in \mathbb{Z}_{\geq 0} \right\}$.
}
We invite the reader to complete this alternate derivation of the Ehrhart quasi-polynomial $L_{\mathcal P}(t)$ in this case.
\hfill $\square$
\end{example}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.7in]{rationaltriangle1}
\end{center}
\caption{A rational triangle, which happens to be a {\bf rational dilate} of the standard simplex.} \label{rational triangle 1}
\end{figure}
\section{Ehrhart reciprocity} \index{Ehrhart reciprocity}
There is a wonderful, and somewhat mysterious, relation between the Ehrhart polynomial of the (closed) polytope ${\mathcal P}$, and the Ehrhart polynomial
of its interior, called $\interior {\mathcal P}$. We recall our convention that all polytopes are, by definition, closed polytopes.
We first compute $L_{\mathcal P}(t)$, for positive integers $t$, and once we have this polynomial in $t$, we formally replace $t$ by $-t$. So by definition, we form $L_{\mathcal P}(-t)$ algebraically, and then embark on a search for its new combinatorial meaning.
\bigskip \bigskip
\begin{thm}[Ehrhart reciprocity]
Given a $d$-dimensional rational polytope ${\mathcal P}\subset \mathbb{R}^d$, let
$
L_{\interior {\mathcal P}}(t) := | \mathbb{Z}^d \cap \interior {\mathcal P} |,
$
the integer point enumerator of its interior. Then
\begin{equation}
L_{ {\mathcal P}}(-t)= (-1)^d L_{\interior {\mathcal P}}(t),
\end{equation}
for all $t\in \mathbb{Z}$.
\end{thm}
Offhand, it seems like `a kind of magic', and indeed Ehrhart reciprocity is one of the most elegant geometric inclusion-exclusion principles we have. Some examples are in order.
\begin{example}
\rm{
For the unit cube $\square:= [0, 1]^d$, we can easily compute from first principles
$L_\square(t) = (t+1)^d= \sum_{k=0}^d {d\choose k} t^k$.
For the open cube $\interior \square$, we can also easily compute
\begin{align*}
L_{\interior \square}(t) &= (t-1)^d = \sum_{k=0}^d {d\choose k} t^k(-1)^{d-k} \\
&= (-1)^d \sum_{k=0}^d {d\choose k} (-t)^k \\
&= (-1)^d L_\square(-t),
\end{align*}
using our known polynomial $L_\square(t) = (t+1)^d$.
}
\hfill $\square$
\end{example}
\begin{example}
\rm{
For the standard simplex
\index{standard simplex}
$\Delta$, we consider its $t$-dilate, given by
\[
t\Delta := \{ (x_1, \dots, x_d) \in \mathbb{R}^d \mid \sum_{k=1}^d x_i \leq t, \text{ and all } x_k \geq 0\}.
\]
We can easily compute its Ehrhart polynomial, by using combinatorics. We need to find the number of nonnegative
integer solutions to
\[
x_1 + \cdots + x_d \leq t,
\]
which is equal to $L_\Delta(t)$, for a fixed positive integer $t$. We can introduce a `slack variable'
\index{slack variable}
$z$, to transform the
latter inequality to an equality: $x_1 + \cdots + x_d + z = t$, where $ 0 \leq z \leq t$. By a very classical and pretty argument, (involving placing $t$ balls into urns that are separated by $d$ walls) this number is equal to
${t+d \choose d}$ (Exercise \ref{Ehrhart poly for closure of standard simplex}).
So we found that
\begin{equation}\label{Ehrhart poly for closed standard simplex}
L_\Delta = {t+d \choose d} = \frac{ (t+d) (t+d-1) \cdots (t+1)}{d!},
\end{equation}
a degree $d$ polynomial, valid for all positive integers $t$.
What about the interior of $\Delta$? Here we need to find the number of {\bf positive} integer solutions to
$x_1 + \cdots + x_d < t$, for each positive integer $t$.
It turns out that by a very similar argument as above (Exercise \ref{Ehrhart poly for interior of standard simplex}), the number of positive integer solutions is ${t-1 \choose d} = L_{\interior \Delta}(t)$.
So is it really true that
\[
(-1)^d {d-t \choose d} = {t-1 \choose d} \ ?
\]
Let's compute, substituting $-t$ for $t$ in \eqref{Ehrhart poly for closed standard simplex} to get:
\begin{align*}
L_\Delta(-t) = {-t+d \choose d} &= \frac{ (-t+d) (-t+d-1) \cdots (-t+1)}{d!} \\
&= (-1)^d\frac{ (t-d) (t-d+1) \cdots (t-1)}{d!} \\
&= (-1)^d {t-1 \choose d} = (-1)^d L_{\interior \Delta}(t),
\end{align*}
confirming that Ehrhart reciprocity works here as well.
}
\hfill $\square$
\end{example}
\bigskip
\section{The M\"obius inversion formula for the face poset}
\bigskip
Given a polytope ${\mathcal P}\subset \mathbb{R}^d$,
the collection of all faces $F$ of ${\mathcal P}$ - including the empty set and ${\mathcal P}$ itself - is ordered by inclusion.
This ordering forms a partially ordered set, and is called the {\bf face poset}. \index{face poset}
There is a particularly useful inversion formula on this face poset.
\begin{thm}[M\"obius inversion formula for the face poset]
\label{Mobius inversion}
\index{M\"obius inversion formula}
Given any function $g: {\mathcal P}\rightarrow \mathbb{C}$, we may define a sum over the face poset of ${\mathcal P}$:
\begin{equation}
h({\mathcal P}) := \sum_{F\subseteq {\mathcal P}} g(F).
\end{equation}
We then have the following inversion formula:
\begin{equation} \label{Mobius inversion formula}
g({\mathcal P}) := \sum_{F\subseteq {\mathcal P}} (-1)^{\dim F} h(F).
\end{equation}
\end{thm}
\hfill $\square$
To prove (again) that for positive integer values of $t$, the angle polynomial $A_{\mathcal P}(t)$ is indeed a polynomial in $t$, we
may use the following useful little relation between solid angle sums and integer point sums.
We recall that for any polytope ${\mathcal F}$, the integer point enumerator for the relative interior of ${\mathcal F}$ was defined by $L_{\interior {\mathcal F}}(t):= | \mathbb{Z}^d \cap \interior {\mathcal F} |$.
For each face ${\mathcal F} \subseteq {\mathcal P}$, we define the $d$-dimensional {\bf solid angle of the face} ${\mathcal F}$ by picking any point $x$ inside the relative interior of ${\mathcal F}$ and denoting
\[
\omega_{\mathcal P}({\mathcal F}) := \omega_{\mathcal P}(x).
\]
\begin{thm}\label{lemma: relation between angle sum and integer sum}
Let ${\mathcal P}$ be a $d$-dimensional polytope in $\mathbb{R}^d$. Then we have
\begin{equation}
A_{\mathcal P}(t) = \sum_{{\mathcal F} \subseteq {\mathcal P}} \omega_{\mathcal P}({\mathcal F}) L_{\interior {\mathcal F}}(t).
\end{equation}
\end{thm}
\begin{proof}
The polytope ${\mathcal P}$ is the disjoint union of its relatively open faces ${\mathcal F} \subseteq {\mathcal P}$, and similarly
the dilated polytope $t{\mathcal P}$ is the disjoint union of its relatively open faces $t{\mathcal F} \subseteq t{\mathcal P}$. We therefore have:
\[
A_{\mathcal P}(t)=\sum_{n \in \mathbb{Z}^d} \omega_{t{\mathcal P}}(n) = \sum_{{\mathcal F} \subseteq{\mathcal P}} \sum_{n \in \mathbb{Z}^d}
\omega_{t{\mathcal P}}(n) 1_{\interior(t{\mathcal F})}(n).
\]
But by definition each $\omega_{t{\mathcal P}}(n)$ is constant on the relatively open face $\interior(t{\mathcal F})$ of $t{\mathcal P}$, and we denoted this constant by $\omega_{\mathcal P}({\mathcal F})$. Altogether, we have:
\[
A_{\mathcal P}(t)= \sum_{{\mathcal F} \subseteq{\mathcal P}} \omega_{\mathcal P}({\mathcal F}) \sum_{n \in \mathbb{Z}^d} 1_{\interior(t{\mathcal F})}(n)
:= \sum_{{\mathcal F} \subseteq {\mathcal P}} \omega_{\mathcal P}({\mathcal F}) L_{\interior {\mathcal F}}(t).
\]
\end{proof}
\begin{thm}
Given an integer polytope ${\mathcal P} \subset \mathbb{R}^d$, the discrete volume $A_{\mathcal P}(t)$ is a polynomial in $t$, for integer values of the dilation parameter $t$.
\end{thm}
\begin{proof}
\ \ By Ehrhart's Theorem \ref{Ehrhart's main result}, we know that for each face
${\mathcal F}~\subseteq~{\mathcal P}$,
$L_{\interior {\mathcal F}}(t)$ is a polynomial function of $t$, for positive integers $t$. By
Theorem \ref{lemma: relation between angle sum and integer sum}, we see that $A_{\mathcal P}(t)$ is a finite linear combination of polynomials, with constant coefficients, and is therefore a polynomial in $t$.
\end{proof}
We may apply Theorem \ref{Mobius inversion} to invert the
relationship in Theorem
\ref{lemma: relation between angle sum and integer sum}
between solid angle sums and local Ehrhart polynomials, to get the following consequence of the M\"obius inversion formula.
\begin{cor}\label{lemma: Mobius mu-function for angle sum and integer sum}
Let ${\mathcal P} \subset \mathbb{R}^d$ be a $d$-dimensional polytope. Then we have
\begin{equation}
L_{\interior {\mathcal P}}(t) = \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{\dim F} A_F(t).
\end{equation}
\end{cor}
\begin{proof}
We begin with the identity of
Theorem \ref{lemma: relation between angle sum and integer sum}:
\begin{equation}
A_{\mathcal P}(t) = \sum_{{\mathcal F} \subseteq {\mathcal P}} \omega_{\mathcal P}({\mathcal F}) \, L_{\interior {\mathcal F}}(t),
\end{equation}
and we use the M\"obius inversion formula \eqref{Mobius inversion formula}
to get:
\begin{equation}
\omega_{\mathcal P}({\mathcal P}) L_{\interior {\mathcal P}}(t) = \sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{\dim F} A_F(t).
\end{equation}
But $ \omega_{\mathcal P}({\mathcal P})=1$, by definition, and so we are done.
\end{proof}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.7in]{Triangle2}
\end{center}
\caption{An integer triangle, for which we compute $A_F(1)$ for each face $F\subset {\mathcal P}$ in Example \ref{Inverting a solid angle sum using Mobius inversion}, and use M\"obius inversion to find
$L_{int {\mathcal P}}(1)$}
\label{Triangle2}
\end{figure}
\begin{example} \label{Inverting a solid angle sum using Mobius inversion}
\rm{
Let's work out a special case of Corollary \ref{lemma: Mobius mu-function for angle sum and integer sum},
in $\mathbb{R}^2$, for the triangle ${\mathcal P}$ appearing in Figure \ref{Triangle2}, with $t=1$.
${\mathcal P}$ has vertices $v_1:= \icol{-1\\ \ 3}, v_2:=\icol{ \ 2\\ -1}, v_3:= \icol{4\{\bf 1}}$, and edges $E_1, E_2, E_3$.
We have to compute $A_F(1)$ for each face $F \subset {\mathcal P}$.
At the vertices, we have $A_{v_1}(1) = \theta_1$, $A_{v_2}(1) = \theta_2$, and $A_{v_3}(1) = \theta_3$.
For the edges of ${\mathcal P}$, we have:
\[
A_{E_1}(1) = \theta_{v_2} + \tfrac{1}{2} + \theta_{v_3},
\]
\[
A_{E_2}(1) = \theta_{v_3} + \theta_{v_1},
\]
\[
A_{E_3}(1) = \theta_{v_1} + \theta_{v_2}.
\]
Finally, for ${\mathcal P}$ itself, we have
\[
A_{\mathcal P}(1) = 6 + \tfrac{1}{2} + \theta_{v_1} + \theta_{v_2} + \theta_{v_3}
= 7.
\]
Putting everything together, we have:
\begin{align*}
\sum_{{\mathcal F} \subseteq {\mathcal P}} (-1)^{\dim F} A_F(1) &=
\Big( A_{v_1}(1) + A_{v_2}(1) + A_{v_3}(1) \Big)
- \Big( A_{E_1}(1) + A_{E_2}(1) + A_{E_3}(1) \Big)
+ A_{{\mathcal P}}(1) \\
&= \Big( \theta_{v_1} + \theta_{v_2} + \theta_{v_3} \Big)
- \Big(
\theta_{v_2} + \tfrac{1}{2} + \theta_{v_3} + \theta_{v_3} + \theta_{v_1} + \theta_{v_1} + \theta_{v_2}
\Big) +7 \\
&= \frac{1}{2} - \frac{3}{2} + 7 = 6 = L_{\interior {\mathcal P}}(1),
\end{align*}
the number of interior integer points in ${\mathcal P}$.
}
\hfill $\square$
\end{example}
Finally, we mention a fascinating open problem by Ehrhart.
\index{Ehrhart conjecture}
\begin{question}[Ehrhart, 1964] \label{Ehrhart conjecture}
Let $B \subset \mathbb{R}^d$ be a d-dimensional convex body with the origin as its
barycenter. If the origin is the only interior integer point in $B$, then
\[
\vol B \leq \frac{ (d+1)d}{d!},
\]
and futhermore the equality holds if and only if $B$ is unimodularly equivalent to $(d + 1)\Delta$,
where $\Delta$ is the $d$-dimensional standard simplex.
\index{standard simplex}
\end{question}
Ehrhart proved the upper bound for all $d$-dimensional simplices, and also for all convex bodies in dimension $2$. But Question \ref{Ehrhart conjecture} remains open in general
(see \cite{Nill.and.Paffenholz} for more details).
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.Ehrhart}
\begin{enumerate}[(a)]
\item Ehrhart theory has a fascinating history, commencing with the fundamental work of Ehrhart \cite{Ehrhart1}, \cite{Ehrhart2},
\cite{Ehrhartbook}, in the 1960's. Danilov \cite{Danilov} made a strong contribution to the field, but after that
the field of Ehrhart theory lay more-or-less dormant, until it was rekindled by Jamie Pommersheim in 1993 \cite{Pommersheim}, giving it strong connections to Toric varieties. Using the Todd operators to discretize certain volume deformations of polytopes, Khovanskii and Pukhlikov discovered a wonderful result that helped develop the theory further (see Theorem 12.6 of \cite{BeckRobins}).
In 1993, Alexander Barvinok \cite{Barvinok.algorithm} \index{Barvinok, Alexander}
gave the first polynomial-time algorithm for counting integer points in polytopes in fixed dimension.
In recent years, Ehrhart theory has enjoyed an enthusiastic renaissance (for example, the books \cite{BarvinokEhrhartbook}, \cite{BeckRobins}, \cite{FultonBook}). For more relations with combinatorics, the reader may enjoy reading
Chapter $4$ of the classic book ``Enumerative Combinatorics'', \cite{StanleyBook} by Richard Stanley.
\item Regarding the computational complexity of counting integer points in polytopes, Alexander Barvinok settled the problem
in \cite{Barvinok.algorithm} by showing that for a fixed dimension $d$, there is a polynomial-time algorithm, as a function of the
`bit capacity' of any given
rational polytope ${\mathcal P} \subset \mathbb{R}^d$, for counting the number of integer points in ${\mathcal P}$.
\item It is also true that for integer polytopes which are not necessarily convex (for example simplicial complexes),
the integer point enumerator makes sense as well. In this more general context,
the constant term of the corresponding integer point enumerator equals the (reduced) Euler characteristic of the simplicial complex.
\item For more information about the rapidly expanding field of Euler-MacLaurin summation over polytopes, a brief (and by no means complete) list of paper in this direction consists of the work by Berligne and Vergne \cite{BerlineVergne}, Baldoni, Berline, and Vergne \cite{BaldoniBerlineVergne}, Garoufalidis and Pommersheim \cite{GaroufalidisPommersheim}, Brandolini, Colzani, Travaglini, and Robins \cite{BrandoliniColzaniTravagliniRobins2},
Karshon, Sternberg, and Weitsman (\cite{KarshonSternbergWeitsman1}, \cite{KarshonSternbergWeitsman2}), and very recently Fischer and Pommersheim \cite{FischerPommersheim}.
\item There are some fascinating relations between an integer polytope ${\mathcal P}$ and its dual polytope ${\mathcal P}^*$. In particular, let ${\mathcal P} \subset \mathbb{R}^2$ be an integer polygon (convex) whose only interior integer point is the origin.
Such polygons are called reflexive polygons, and up to unimodular transformations there are only a finite number of them in each dimension. If we let $B({\mathcal P})$ be the number of integer points on the boundary of ${\mathcal P}$, then Bjorn Poonen and Fernando Villegas proved \cite{PoonenVillegas} that
\[
B({\mathcal P}) + B({\mathcal P}^*) = 12.
\]
One way to see why we get the number ``12'' is to consider Bernoulli numbers and Dedekind sums, but in \cite{PoonenVillegas} the authors give 4 different proofs, including Toric varieties and modular forms.
\item \label{Michel Faleiros} The trick used in Example \ref{Ehrhart poly for the standard triangle}
of picking the particular vector $z := (x, -x)$, which turns out to simplify the computations a lot, is due to Michel Faleiros.
\item \label{EM summation note}
In a future version of this book, we will also delve into Dedekind sums, which arise very naturally when considering the Fourier series of certain rational-exponential functions. To define a general version of these sums, let
${\mathcal L}$ be a $d$-dimensional lattice in $\mathbb{R}^d$, let $w_1, \dots, w_d$ be linearly independent vectors from
${\mathcal L}^*$, and let $W$ be a matrix with the $w_j$'s as columns. For any $d$-tuple $e = (e_1, \dots, e_d)$ of positive integers $e_j$,
define $|e| := \sum_{j = 1}^k e_j$. Then, for all $x \in \mathbb{R}^d$, a lattice Dedekind sum is defined by
\begin{equation}
L_{\mathcal L}(W, e; x) :=
\lim_{\varepsilon \rightarrow 0}
\frac{1}{(2\pi i)^{|e|}}
\sum_{\substack{\xi \in {\mathcal L} \\ \langle w_j, \xi \rangle \neq 0, \forall j}}
\frac{e^{-2\pi i \langle x, \xi \rangle }}{\prod_{j = 1}^k \langle w_j, \xi \rangle^{e_j}} e^{-\pi \varepsilon \|\xi\|^2}.
\end{equation}
Gunnells and Sczech \cite{GunnellsSczech} have an interesting reduction theorem for these sums,
giving a polynomial-time complexity algorithm for them, for fixed dimension $d$.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\medskip
\begin{prob} \label{exercise:warm up dual lattice}
In $\mathbb{R}$, consider the $1$-dimensional polytope ${\mathcal P}:= [a,b]$, for any
$a,b \in \mathbb{Z}$.
\begin{enumerate}[(a)]
\item Show that the Ehrhart polynomial of ${\mathcal P}$ is $L_{\mathcal P}(t) = (b-a)t + 1$.
\item Find the Ehrhart quasi-polynomial $L_{\mathcal P}(t)$
for the rational segment $\mathcal Q:= [\frac{1}{3}, \frac{1}{2}]$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
Fix positive integers $a, b$.
Working in $\mathbb{R}^2$, show that the closed line segment ${\mathcal P} \subset \mathbb{R}^2$, whose vertices are the origin and $(a, b)$,
contains exactly $\gcd(a, b) + 1$ integer points of $\mathbb{Z}^2$. Conclude that we have the lower-dimensional
Ehrhart polynomial $L_{\mathcal P}(t) = \gcd(a, b) t + 1$.
\end{prob}
\medskip
\begin{prob} \label{2-d cross polytope Ehrhart}
We recall that the $d$-dimensional cross-polytope was defined by
\[
\Diamond:=\left\{ \left( x_1, x_2, \dots, x_d \right) \in \mathbb{R}^d \mid
\, \left| x_1 \right| + \left| x_2 \right| + \cdots + \left| x_d \right| \leq 1 \right\}.
\]
For $d=2$, find the Ehrhart polynomial $L_\Diamond(t)$.
\end{prob}
\medskip
\begin{prob}
Extending Exercise \ref{2-d cross polytope Ehrhart}, show that the Ehrhart polynomial of $\Diamond$ in $\mathbb{R}^d$ is
\[
L_{\Diamond}(t) = \sum_{k=0}^d \binom{d}{k} \binom{t-k+d}{d},
\]
for all $t \in \mathbb{Z}_{>0}$.
\end{prob}
\medskip
\begin{prob}
Let $d=2$, and consider the cross-polytope $\Diamond \subset \mathbb{R}^2$. Find the Ehrhart quasi-polynomial $L_{\mathcal P}(t)$ for the rational polygon ${\mathcal P} := \frac{1}{2} \Diamond$.
\end{prob}
\medskip
\begin{prob}
Suppose $\Delta$ is the standard simplex in $\mathbb{R}^d$. Show that the first $d$ dilations of $\Delta$ do not contain
any integer points in their interior:
\[
t(\interior \Delta) \cap \mathbb{Z}^d = \phi,
\]
for $t = 1, 2, \dots, d$.
In other words, show that $L_{\interior {\mathcal P}}(1) = L_{\interior {\mathcal P}}(2) = \cdots =
L_{\interior {\mathcal P}}(d) = 0$. Conclude that the same statement is true for any unimodular simplex.
\end{prob}
\medskip
\begin{prob} \label{Bernoulli polynomial as an Ehrhart polynomial}
Here we show that the Bernoulli polynomial $B_d(t)$, is essentially equal to the Ehrhart polynomial
$L_{\mathcal P}(t)$ for the ``Pyramid over a cube" (as defined in Exercise
\ref{Pyramid over a square}). We recall the definition: let $C:=[0,1]^{d-1}$ be the $d-1$-dimensional cube,
considered as a subset of $\mathbb{R}^d$, and let ${\bf e_d}$ be the unit vector pointing in the $x_d$-direction.
Now we define ${\mathcal P}:= \conv\{ C, {\bf e_d} \}$, a pyramid over the unit cube.
Show that its Ehrhart polynomial is
\[
L_{{\mathcal P}}(t) = \frac{1}{d}(B_d(t+2) - B_d),
\]
for $t \in \mathbb{Z}_{>0}$.
\end{prob}
\medskip
\begin{prob} \label{Pick's formula, generalization to d dimensions}
For any integer $d$-dimensional (convex) polytope ${\mathcal P} \subset \mathbb{R}^d$, show that
\begin{equation} \label{Volume in terms of forward differences}
\vol {\mathcal P} = \frac{(-1)^d}{d!} \left( 1 + \sum_{k=1}^d {d\choose k} (-1)^k L_{\mathcal P}(k) \right),
\end{equation}
which can be thought of as a generalization of Pick's formula to $\mathbb{R}^d$.
\index{Pick's formula, generalization}
Note. \ Using iterations of the forward difference operator
\[
\Delta f(n):= f(n+1) - f(n),
\]
the latter identity
may be thought of a {\bf discrete analogue} of the $d$'th derivative of the Ehrhart polynomial. This idea in fact gives another method of proving \eqref{Volume in terms of forward differences}.
\end{prob}
\medskip
\begin{prob} \label{Pick's formula from general Ehrhart exercise}
Show that Pick's formula is the special case of
Exercise \ref{Pick's formula, generalization to d dimensions} when the dimension $d=2$.
That is, given an integer polygon ${\mathcal P} \subset \mathbb{R}^2$, we have
\[
\rm{Area } {\mathcal P} = I + \frac{1}{2} B -1,
\]
where $I$ is the number of interior integer points in ${\mathcal P}$, and B is the number of boundary integer points of ${\mathcal P}$.
\end{prob}
\medskip
\begin{prob} \label{convolution of the indicator function with a Gaussian}
Fix $\epsilon >0$. Show that the convolution of the indicator function $1_{\mathcal P}$
with the heat kernel $G_{\varepsilon}$, as in equation \eqref{Gaussian smoothing},
is a Schwartz function (of $x \in \mathbb{R}^d$).
\end{prob}
\medskip
\begin{prob} \label{unimodular triangle}
Show that any unimodular triangle has area equal to $\frac{1}{2}$.
\end{prob}
\medskip
\begin{prob} \label{unimodular triangle, Ehrhart poly}
Show that the Ehrhart polynomial of the standard simplex
\index{standard simplex}
$\Delta \subset \mathbb{R}^d$ is
\[
L_{\Delta}(t) = \binom{t+d}{d}.
\]
\end{prob}
\medskip
\begin{prob}
Consulting Figure \ref{unimodular polygon}:
\begin{enumerate}[(a)]
\item Find the integer point transform of the unimodular polygon in the Figure.
\item Find the Ehrhart polynomial $L_{{\mathcal P}}(t)$ of the integer polygon ${\mathcal P}$ from part (a).
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{two definitions for a solid angle}
Show that \eqref{def. of solid angle} is equivalent to the following definition, using balls instead of spheres.
Recall that the unit ball in $\mathbb{R}^d$ is define by $B^d:= \{ x\in \mathbb{R}^d \mid \| x\| \leq 1 \}$, and similarly
the ball of radius $\varepsilon$, centered at $x\in \mathbb{R}^d$, is denoted by $B^d(x, \varepsilon)$. Show that
for all sufficiently small $\varepsilon$, we have
\begin{equation*}
\frac{\vol(S^{d-1}(x,\varepsilon) \cap {\mathcal P})}{\vol(S^{d-1}(x,\varepsilon))} =
\frac{\vol(B^{d}(x,\varepsilon) \cap {\mathcal P})}{\vol(B^{d}(x,\varepsilon))}.
\end{equation*}
\end{prob}
\medskip
\begin{prob} \label{properties of floor, ceiling, fractional part}
Here we gain some practice with `floors', `ceilings', and `fractional parts'.
First, we recall that by definition, the fractional part of any real number $x$ is $\{x\} := x - \floor{x}$.
Next, we recall the indicator function of $\mathbb{Z}$, defined by:
$
1_{\mathbb{Z}}(x) :=
\begin{cases}
1 & \text{if } x \in \mathbb{Z} \\
0 & \text{if } x \notin \mathbb{Z} \\
\end{cases}.
$
Show that:
\begin{enumerate}[(a)]
\item $\left\lceil x \right\rceil = - \floor{-x}$
\label{ex:part 1 of fractional parts}
\item $1_{\mathbb{Z}}(x)= \floor{x} - \left\lceil x \right\rceil +1$
\label{ex:part 2 of fractional parts}
\item $ \{ x \} + \{-x\} = 1- 1_{\mathbb{Z}}(x)$
\label{ex:part 3 of fractional parts}
\item Let $m \in \mathbb{Z}_{>0}, n \in \mathbb{Z}$. Then $\floor{ \frac{n-1}{m} } + 1 = \left\lceil \frac{n}{m} \right\rceil$.
\label{ex:part 4 of fractional parts}
\end{enumerate}
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Ehrhart poly for closure of standard simplex}
\index{standard simplex}
Show that the number of nonnegative integer solutions $x_1, \dots, x_d, z \in \mathbb{Z}_{\geq 0}$
to
\[
x_1 + \cdots + x_d + z = t,
\]
with $ 0 \leq z \leq t$, equals ${t+d \choose d}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Ehrhart poly for interior of standard simplex}
Show that for each positive integer $t$, the number of {\bf positive} integer solutions to
$x_1 + \cdots + x_d < t$ is equal to ${t-1 \choose d}$.
\end{prob}
\medskip
\begin{prob}
We define the rational triangle whose vertices are $(0, 0), (1, \frac{N-1}{N}), (N, 0)$, where $N \geq 2$ is a fixed integer.
Prove that the Ehrhart quasi-polynomial is in this case
\[
L_{\mathcal P}(t) = \frac{p-1}{2} t^2 + \frac{p+1}{2} t + 1,
\]
for all $t \in \mathbb{Z}_>0$.
Notes. So we see here a phenomenon known as `period collapse', where we expect a quasi-polynomial behavior, with some nontrivial period, but in fact we observe a strict polynomial.
\end{prob}
\medskip
\begin{prob}
Here we show that the Ehrhart polymomial $L_{\mathcal P}(t)$ remains invariant under the full unimodular group $SL_d(\mathbb{Z})$.
In particular, recalling definition \ref{Definition of the unimodular group}, of a unimodular matrix, show that:
\begin{enumerate}[(a)]
\item Every element of $SL_d(\mathbb{Z})$ acts on the integer lattice $\mathbb{Z}^d$ bijectively.
\item \label{invariance of Ehrhart under the unimodular group}
Let ${\mathcal P}$ be an integral polytope, and let $Q := A({\mathcal P})$, where $A \in SL_d(\mathbb{Z})$. Thus, by definition ${\mathcal P}$ and $Q$
are unimodular images of each other. Prove that
\[
L_{{\mathcal P}}(t) = L_Q(t),
\]
for all $t \in \mathbb{Z}_{>0}$.
\item Is the converse of part \ref{invariance of Ehrhart under the unimodular group} true? In other words, given integer polytopes ${\mathcal P}, Q$, suppose that $L_{{\mathcal P}}(t) = L_Q(t)$, for all positive integers $t$. Does it necessarily follow that
$Q := A({\mathcal P})$, for some unimodular matrix $A \in SL_d(\mathbb{Z})$?
\end{enumerate}
\end{prob}
\chapter{The Fourier transform of a polytope via its hyperplane description: \\
the divergence Theorem}
\label{Stokes' formula and transforms}
\index{Stoke's formula} \index{face poset}
\begin{quote}
``Like a zen koan, Stokes' Theorem tells us that in the end, what happens on the outside
is purely a function of the change within.''
--Keenan Crane
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.2in]{VectorField}
\end{center}
\caption{A real vector field in $\mathbb{R}^2$} \label{vector field}
\end{figure}
\section{Intuition}
The divergence theorem is a multi-dimensional version of ``integration by parts'', a very useful tool in
$1$-dimensional calculus. When we apply the divergence theorem, described below, to a polytope, we obtain
a kind of combinatorial version of the divergence theorem, allowing us to transfer some of the complexity of computing the Fourier transform of a polytope to the complexity of computing corresponding Fourier transforms of its facets. This kind of game can be iterated, yielding interesting geometric identities and results for polytopes, as well as for discrete volumes of polytopes.
In the process, we also obtain another useful way to compute the Fourier transform of a polytope in its own right.
\section{The divergence theorem, and a combinatorial \\ divergence theorem for polytopes}
\index{combinatorial divergence theorem}
To warm up, we recall the divergence theorem, with some initial examples.
A {\bf vector field} on Euclidean space is a
function $F:\mathbb{R}^d \rightarrow \mathbb{C}^d$ that assigns to each point in $\mathbb{R}^d$ another vector in $\mathbb{C}^d$, which we will denote by
\[
F(x) := (F_1(x), F_2(x), \dots, F_d(x)) \in \mathbb{C}^d.
\]
If $F$ is a continuous (respectively, smooth) function, we say that $F$ is a
{\bf continuous vector field} (respectively, {\bf smooth vector field}).
If all of the coordinate functions $F_j$ are real-valued functions, we say that we have a {\bf real vector field}.
We define the {\bf divergence} of $F$ at each $x := (x_1, \dots, x_d) \in \mathbb{R}^d$ by
\[
\rm{div} F(x) := \frac{\partial F_1}{{\partial} x_1} + \cdots + \frac{\partial F_d}{\partial x_d},
\]
assuming that $F$ is a smooth (or at least once-differentiable) vector field. This divergence of $F$ is a measure of the local change (sink versus source) of the vector field at each point $x\in \mathbb{R}^d$.
Given a surface $S \subset \mathbb{R}^d$, and an outward pointing unit normal vector ${\bf n}$, defined at each point $x\in S$, we also define the {\bf flux} of the vector field $F$ across the surface $S$ by
\[
\int_S F\cdot {\bf n} \ dS,
\]
where $dS$ denotes the Lebesgue measure of the surface $S$, and where the dot product
$F\cdot {\bf n}$ is the usual inner product $\langle F, {\bf n} \rangle := \sum_{k=1}^d F_k n_k$. We will apply the divergence theorem (which is technically a special case of Stokes' Theorem)
to a polytope ${\mathcal P}\subset \mathbb{R}^d$, and its $(d-1)$-dimensional bounding surface $\partial {\mathcal P}$. Intuitively,
the divergence theorem tells us that the total divergence of a vector field $F$ inside a manifold is equal to the total
flux of $F$ across its boundary.
\begin{thm}[The Divergence Theorem]
Let $M \subset \mathbb{R}^d$ be a piecewise smooth manifold, and let $F$ be a smooth vector field.
Then
\begin{equation}\label{Divergence Theorem} \index{divergence Theorem}
\int_M \rm{div}F(x) dx = \int_S F\cdot {\bf n} \ dS.
\end{equation}
\end{thm}
\begin{example} \label{Pyramid formula via the divergence theorem}
\rm{
Let ${\mathcal P} \subset \mathbb{R}^d$ be a $d$-dimensional polytope, containing the origin, with defining facets $G_1, \dots, G_N$.
Define the real vector field
\[
F(x):= x,
\]
for all $x\in \mathbb{R}^d$. First, we can easily compute
here the divergence of $F$, which turns out to be constant:
\begin{align*}
\rm{div } F(x) &= \frac{\partial F_1}{{\partial} x_1} + \cdots + \frac{\partial F_d}{\partial x_d} =
\frac{\partial x_1}{{\partial} x_1} + \cdots + \frac{\partial x_d}{\partial x_d} = d.
\end{align*}
If we fix any facet $G$ of ${\mathcal P}$ then, due to the piecewise linear structure of the polytope,
every point $x \in G$ has the same constant outward pointing
normal vector to $F$, which we call ${\bf n}_G$.
Computing first the left-hand-side of the divergence theorem, we see that
\begin{equation}
\int_P \rm{div } F(x) dx = d \int_P dx = (\vol {\mathcal P})d.
\end{equation}
Computing now the right-hand-side of the divergence theorem, we get
\begin{align*}
\int_S F\cdot {\bf n} \ dS = \int_{\partial {\mathcal P}} \langle x, {\bf n} \rangle \ dS = \sum_{k=1}^N \int_{G_k} \langle x, {\bf n}_G \rangle \ dS.
\end{align*}
Now it's easy to see that the inner product $ \langle x, n_G \rangle$ is constant on each facet $G \subset {\mathcal P}$, namely
it is the distance from the origin to $G$ (Exercise \ref{distance to a facet}), denoted by $\rm{dist}(G)$.
So we now have
\begin{align*}
\int_{\partial {\mathcal P}} F \cdot n \ dS &= \sum_{k=1}^N \int_{G_k} \langle x, {\bf n}_{G_k} \rangle dS \\
&= \sum_{k=1}^N \rm{dist}(G_k) \int_{G_k} dS = \sum_{k=1}^N \rm{dist}(G_k) \vol G_k,
\end{align*}
so that altogether we the following conclusion from the divergence theorem:
\begin{equation}\label{pyramid formula} \index{pyramid formula}
\vol {\mathcal P} = \frac{1}{d} \sum_{k=1}^N \rm{dist}(G_k) \vol G_k.
\end{equation}
known as ``the pyramid formula'' for a polytope, a classical result in Geometry, which also has a very easy
geometrical proof (Exercise \ref{Pyramid formula}).
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
Let ${\mathcal P} \subset \mathbb{R}^d$ be a $d$-dimensional polytope with defining facets $G_1, \dots, G_N$, and outward pointing unit vectors $n_{G_1}, \dots, n_{G_N}$. We fix any constant vector $\lambda \in \mathbb{C}^d$, and we consider the {\bf constant vector field}
\[
F(x):= \lambda,
\]
defined for all $x\in \mathbb{R}^d$. Here the divergence of
$F$ is $\rm{div} F(x) = 0$, because $F$ is constant, and so the left-hand-side of Theorem
\ref{Divergence Theorem} gives us
\begin{align*}
\int_P \rm{div } F(x) dx = 0.
\end{align*}
Altogether, the divergence theorem gives us:
\begin{align*}
0 = \int_{\partial {\mathcal P}} F \cdot {\bf n} \ dS &= \sum_{k=1}^N \int_{G_k} \langle \lambda, {\bf n}_{G_k} \rangle dS \\
&= \sum_{k=1}^N \langle \lambda, {\bf n}_{G_k} \rangle \int_{G_k} dS \\
&= \langle \lambda, \sum_{k=1}^N \vol G_k {\bf n}_{G_k} \rangle,
\end{align*}
and because this holds for any constant vector $\lambda$, we can conclude that
\begin{equation}\label{Minkowski relation}
\sum_{k=1}^N \vol G_k {\bf n}_{G_k}= 0.
\end{equation}
Identity \eqref{Minkowski relation} is widely known as the {\bf Minkowski relation} for polytopes. There is a marvelous converse to the latter relation, given by Minkowski as well,
for any convex polytope.
[See Theorem \ref{Minkowski's problem for polytopes}]
}
\hfill $\square$
\end{example}
\bigskip
Now we fix $\xi \in \mathbb{R}^d$, and we want to see how to apply the divergence theorem to the vector-field
\begin{equation}\label{First vector field}
F(x) := e^{- 2\pi i \langle x, \xi \rangle} \xi.
\end{equation}
Taking the divergence of the vector field $F(x)$, we have:
\begin{align*}
\rm{div } F(x) &= \frac{\partial \left( e^{- 2\pi i \langle x, \xi \rangle} \xi_1 \right) }{{\partial} x_1} + \cdots +
\frac{\partial ( e^{- 2\pi i \langle x, \xi \rangle} \xi_d )}{\partial x_d} \\
&= (-2\pi i \xi_1^2) e^{- 2\pi i \langle x, \xi \rangle} + \cdots + (-2\pi i \xi_d^2) e^{- 2\pi i \langle x, \xi \rangle} \\
&= -2\pi i \| \xi \|^2 e^{- 2\pi i \langle x, \xi \rangle}.
\end{align*}
So by the divergence theorem we have
\begin{align} \label{initial divergence}
\int_{x\in P} - 2\pi i ||\xi||^2 e^{- 2\pi i \langle x, \xi \rangle} dx
= \int_{x\in P} \text{div} F(x) dx = \int_{\partial P} e^{- 2\pi i \langle x, \xi \rangle}
\langle \xi, {\bf n} \rangle \ dS,
\end{align}
where ${\bf n}$ is the outward-pointing unit normal vector at each point $x\in \partial {\mathcal P}$. When ${\mathcal P}$ is a polytope, these arguments quickly give the following conclusion.
\begin{thm} \label{FT of a polytope, first iteration of divergence}
Given any $d$-dimensional polytope ${\mathcal P}\subset \mathbb{R}^d$, with outward pointing normal vector $n_G$ to each facet $G$ of ${\mathcal P}$, its Fourier transform has the form
\begin{equation} \label{the first step of divergence}
\hat 1_{\mathcal P}(\xi) =\frac{1}{-2\pi i } \sum_{G\subset \partial P} \frac{ \langle \xi, {\bf n}_G \rangle}{ ||\xi ||^2} \hat 1_G(\xi),
\end{equation}
for all nonzero $\xi \in \mathbb{C}^d$. Here the integral that defines each $\hat 1_G$ is taken with respect to Lebesgue measure that matches the dimension of the facet $G \subset \partial P$.
\end{thm}
\begin{proof}
\begin{align*}
\hat 1_{\mathcal P}(\xi) &:= \int_{x\in P} e^{- 2\pi i \langle x, \xi \rangle} dx \\
&= \frac{ 1 }{-2\pi i \|\xi\|^2} \int_{\partial P} \langle \xi, {\bf n} \rangle e^{- 2\pi i \langle x, \xi \rangle} dS
\quad (\text{using} \, \eqref{initial divergence}) \\
&= \frac{ 1}{-2\pi i \|\xi\|^2}
\int_{G_1} \langle \xi, {\bf n}_{G_1} \rangle e^{- 2\pi i \langle x, \xi \rangle} dS
+ \cdots +
\frac{ 1}{-2\pi i \|\xi\|^2}
\int_{G_N} \langle \xi, {\bf n}_{G_N} \rangle e^{- 2\pi i \langle x, \xi \rangle} dS \\
&= \frac{ \langle \xi, {\bf n}_{G_1} \rangle }{-2\pi i \|\xi\|^2} \hat 1_{G_1}(\xi)
+ \cdots +
\frac{ \langle \xi, {\bf n}_{G_N} \rangle }{-2\pi i \|\xi\|^2} \hat 1_{G_N}(\xi),
\end{align*}
where in the third equality we used the fact that the boundary $\partial {\mathcal P}$ of a polytope is a finite union of $(d-1)$-dimensional polytopes (its facets), and hence $\int_{\partial P} = \int_{G_1} + \cdots + \int_{G_N}$, a sum of integrals over the $N$ facets of ${\mathcal P}$.
\end{proof}
This result allows us to reduce the Fourier transform of ${\mathcal P}$ to a finite sum of Fourier transforms of the facets of ${\mathcal P}$. This process can clearly be iterated, until we arrive at the vertices of ${\mathcal P}$. But we will need a few book-keeping devices first.
To simplify the notation that will follow, we can also the {\bf Iverson bracket} notation, defined as follows.
Suppose we have any
boolean property $P(n)$, where $n\in \mathbb{Z}^d$; that is, $P(n)$ is either true or false.
Then the Iverson bracket $[ P ]$ is defined by:
\begin{equation}\label{Iverson bracket}
[P] =
\begin{cases}
1 & \mbox{if P is true } \\
0 & \mbox{if P is false }
\end{cases}
\end{equation}
Now we may rewrite the identity of Theorem \ref{FT of a polytope, first iteration of divergence}
as follows:
\begin{equation}
\hat 1_{\mathcal P}(\xi) =\vol {\mathcal P} \ [\xi = 0] +
\frac{1}{-2\pi i } \sum_{G\subset \partial P} \frac{ \langle \xi, {\bf n}_G \rangle}{ ||\xi ||^2} \hat 1_G(\xi)\
[\xi \not= 0].
\end{equation}
Later, after Theorem \ref{Combinatorial divergence theorem} below, we will return to the Iverson bracket, and be able to use it efficiently.
To proceed further, we need to define the {\bf affine span} \index{affine span} of a face $F$ of ${\mathcal P}$:
\begin{equation} \label{affine span of F}
\rm{aff}(F) := \left\{ \sum_{j=1}^k \lambda_j v_j \mid k>0, v_j \in F,
\lambda_j \in \mathbb{R}, \text{ and } \sum_{j=1}^k \lambda_j = 1 \right\}.
\end{equation}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=3.3in]{AffineSpan}
\end{center}
\caption{The affine span of a face $F$, its linear span , and the projection of $\xi$ onto $F$. Here
we note that the distance from the origin to $F$ is $\sqrt{20}$.} \label{affine span}
\end{figure}
In other words, we may think of the affine span of a face $F$ of ${\mathcal P}$ as follows. We first translate $F$ by any element $x_0 \in F$. So this translate, call if $F_0:= F - x_0$, contains the origin. Then we take all real linear combinations of points of $F_0$, obtaining a vector subspace of $\mathbb{R}^d$, which we call the {\bf linear span } of $F$. Another way to describe the linear span of a face $F$ of ${\mathcal P}$ is:
\[
{\rm lin}(F):= \left\{ x - y \mid x, y \in F \right\}.
\]
\index{linear span}
Finally, we may translate this subspace $\rm{lin}(F)$ back using the same translation vector $x_0$, to obtain $\rm{aff}(F):= \rm{lin}(F) + x_0$ (see Figure \ref{affine span}).
\begin{example}
\rm{
The affine span of two distinct points in $\mathbb{R}^d$ is the unique line in $\mathbb{R}^d$ passing through them. The affine span of three points in $\mathbb{R}^d$ is the unique $2$-dimensional plane passing through them.
The affine span of a $k$-dimensional polytope $F \subset \mathbb{R}^d$ is a translate of a $k$-dimensional vector subspace of $\mathbb{R}^d$.
Finally, the affine span of a whole $d$-dimensional polytope ${\mathcal P} \subset \mathbb{R}^d$ is all of $\mathbb{R}^d$.
}
\hfill $\square$
\end{example}
In formalizing \eqref{the first step of divergence} further, we will require the notion of the projection of any point
$\xi \in \mathbb{R}^d$ onto the linear span of any face $F\subseteq {\mathcal P}$, which we abbreviate by $\rm{Proj}_F \xi$:
\begin{equation}
\rm{Proj}_F \xi := \rm{Proj}_{\rm{lin}(F)}(\xi).
\end{equation}
(see Figure \ref{affine span})
We will also need the following elementary fact. Let $F$ be any $k$-dimensional polytope in
$\mathbb{R}^d$, and fix the outward-pointing unit normal to $F$, calling it ${\bf n}_F$.
It is straightforward to show that if we take any point $x_F \in F$, then $\langle x_F, {\bf n}_F \rangle$ is the distance from the origin to $F$. Therefore, if $\rm{Proj}_F \xi = 0$, then a straightforward computation shows that
$\langle \xi, x_F \rangle = \| \xi \| \rm{dist}(F)$ (Exercise \ref{distance to a facet}).
We can now extend \eqref{the first step of divergence} to lower-dimensional polytopes, as follows.
\begin{thm}[Combinatorial Divergence Theorem] \label{Combinatorial divergence theorem}
Let $F$ be a polytope in $\mathbb{R}^d$, where $1 \leq \dim F \leq d$. For each facet $G \subseteq F$, we let
${\bf n}(G, F)$ be the unit normal vector to $G$, with respect to $\rm{lin}(F)$.
Then for each $\xi \in \mathbb{R}^d$, we have:
\begin{enumerate}[(a)]
\item If $\rm{Proj}_F \xi = 0$, then
\begin{equation}
\hat 1_F(\xi) = (\vol F) e^{-2\pi i \| \xi \| \rm{dist}(F) }.
\end{equation}
\item If $\rm{Proj}_F \xi \not= 0$, then
\begin{equation}
\hat 1_F(\xi) = \frac{1}{-2\pi i } \sum_{G\subset \partial F}
\frac{ \langle \rm{Proj}_F \xi, {\bf n}(G, F) \rangle}{ ||\rm{Proj}_F \xi ||^2} \hat 1_G(\xi).
\end{equation}
\end{enumerate}
\end{thm}
\hfill $\square$
\bigskip
We notice that, as before, we are getting rational-exponential functions for the Fourier transform of a polytope. But Theorem \ref{Combinatorial divergence theorem} gives us the extra freedom to begin with a lower-dimensional polytope $F$, and then find its Fourier transform in terms of its facets.
We are now set up to iterate this process, defined by Theorem \ref{Combinatorial divergence theorem},
reapplying it to each facet $G \subset \partial {\mathcal P}$. Let's use the Iverson bracket,
defined in \eqref{Iverson bracket},
and apply the combinatorial divergence Theorem \ref{Combinatorial divergence theorem} to ${\mathcal P}$ twice:
\begin{align*}
\hat 1_{\mathcal P}(\xi) &=\vol {\mathcal P} \ [\xi = 0] +
\frac{1}{-2\pi i } \sum_{F_1 \subset \partial P} \frac{ \langle \xi, {\bf n}_{F_1} \rangle}{ ||\xi ||^2} \ [\xi \not= 0] \ \hat 1_{F_1}(\xi) \\
&=\vol {\mathcal P} \ [\xi = 0] +
\frac{1}{-2\pi i } \sum_{{F_1}\subset \partial P} \frac{ \langle \xi, {\bf n}_{F_1} \rangle}{ ||\xi ||^2} [\xi \not= 0] \\
& \cdot \Big( (\vol {F_1}) e^{-2\pi i \langle \xi, x \rangle} \ [\rm{Proj}_{F_1} \xi = 0 ] +
\frac{1}{-2\pi i } \sum_{F_2 \subset \partial {F_1}}
\frac{ \langle \rm{Proj}_{F_2} \xi, {\bf n}(F_2, F_1) \rangle}{ ||\rm{Proj}_{F_2} \xi ||^2} \hat 1_{F_2}(\xi) [\rm{Proj}_{F_1} \xi \not= 0] \Big)
\\
&= \vol {\mathcal P} \ [\xi = 0] +
\frac{1}{-2\pi i } \sum_{F_1 \subset \partial P}
\frac{ \langle \xi, {\bf n}_{F_1} \rangle (\vol F_1) e^{-2\pi i \langle \xi, x \rangle} }{ ||\xi ||^2}
\ [\xi \not= 0][\rm{Proj}_{F_1} \xi = 0 ] \\
& + \frac{1}{(-2\pi i )^2} \sum_{F_1 \subset \partial P} \sum_{F_2 \subset \partial F_1}
\frac{ \langle \xi, {\bf n}_{F_1} \rangle}{ ||\xi ||^2}
\frac{ \langle \rm{Proj}_{F_2} \xi, {\bf n}(F_2, F_1) \rangle}{ ||\rm{Proj}_{F_2} \xi ||^2}
\hat 1_{F_2}(\xi) \ [\xi \not= 0] [\rm{Proj}_{F_1} \xi \not= 0]
\end{align*}
It is an easy fact that the product of two Iverson brackets is the Iverson bracket of their intersection:
$[ P ] [ Q ] = [ P \text{ and } Q ]$ (Exercise \ref{Exercise Iverson bracket}). Hence, if we define
\[
F^\perp := \{ x \in \mathbb{R}^d \mid \langle x, y \rangle = 0 \text{ for all } y \in \rm{lin}F \},
\]
Then we see that ${\mathcal P}^\perp = \{ 0 \}$, and we can rewrite the latter identity as
\begin{align*}
\hat 1_{\mathcal P}(\xi) &= \vol {\mathcal P} \ [\xi \in {\mathcal P}^\perp ] +
\frac{1}{-2\pi i } \sum_{F_1 \subset \partial P}
\frac{ \langle \xi, {\bf n}_{F_1} \rangle (\vol F_1) e^{-2\pi i \langle \xi, x \rangle} }{ ||\xi ||^2}
\ [ \xi \in F_1^\perp - {\mathcal P}^\perp] \\
& + \frac{1}{(-2\pi i )^2} \sum_{F_1 \subset \partial P} \sum_{F_2 \subset \partial F_1}
\frac{ \langle \xi, {\bf n}_{F_1} \rangle}{ ||\xi ||^2}
\frac{ \langle \rm{Proj}_{F_2} \xi, {\bf n}(F_2, F_1) \rangle}{ ||\rm{Proj}_{F_2} \xi ||^2}
\hat 1_{F_2}(\xi) \ [ \xi \not\in F_1^\perp].
\end{align*}
In order to keep track of the iteration process, we will introduce another book-keeping device.
The {\bf face poset} of a polytope ${\mathcal P}$ \index{face poset} is defined to be the partially ordered set (poset) of all faces of ${\mathcal P}$, ordered by inclusion, including ${\mathcal P}$ and the empty set.
\bigskip
\begin{example}
\rm{
Consider a $2$-dimensional polytope ${\mathcal P}$ that is a triangle. We have the following picture for the face poset ${\frak F}_P$ of ${\mathcal P}$, as in Figure \ref{FacePoset1}. It turns out that if we consider a $d$-simplex ${\mathcal P}$, then its face poset ${\frak F}_P$ has the structure of a ``Boolean poset'', which is isomorphic to the edge graph of a $(d+1)$-dimensional cube.
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{FacePoset1}
\caption{The face poset of a triangle} \label{FacePoset1}
\end{figure}
}
\end{example}
We only have to consider rooted chains in the face poset ${\frak F}_P$, which means chains whose root is $P$. The only appearance of non-rooted chains are in the following definition.
If $G$ is a facet of $F$, we attach the following weight to any (local) chain $(F,G)$, of length $1$, in the face poset of $P$:
\begin{equation}\label{weight}
W_{(F,G)}(\xi):=\frac{-1}{2 \pi i} \frac{\langle \proj_{F} (\xi), {\bf n}(G, F) \rangle }{\| \proj_{F} (\xi) \|^2}.
\end{equation}
Note that these weights are functions of $\xi$ rather than constants. Moreover, they are all homogeneous of degree $-1$. Let $\mathbf{T}$ be any rooted chain in ${\frak F}_P$, given by
\[
T:= (P \to F_1 \to F_2, \dots, \to F_{k-1} \to F_k),
\]
so that by definition $\dim(F_j) = d-j$.
We define the {\bf admissible set} $S(\mathbf{T})$ of the rooted chain
$\mathbf{T}$ to be the set of all vectors $\xi\in \mathbb{R}^d$ that are orthogonal to the linear span of $F_k$ but not orthogonal to the linear span of $F_{k-1}$. In other words,
\begin{align*}
S(\mathbf{T}) &:= \{ \xi \in \mathbb{R}^d \mid \xi \perp \rm{lin}(F_k), \text{ but }
\xi \not\perp \rm{lin}(F_{k-1}) \} \\
& = \{ \xi \in \mathbb{R}^d \mid \xi \in F_k^\perp - F_{k-1}^\perp \}.
\end{align*}
Finally, we define the following weights associated to any such rooted chain $\mathbf{T}$:
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{graphG}
\caption{A symbolic depiction of the face poset ${\frak F}_P$, where $P$ is a $3$-dimensional tetrahedron. Here the points and arrows are drawn suggestively, as a directed graph. We can see all the rooted chains, beginning from a symbolic vertex in the center, marked with the color purple. The rooted chains that terminate with the yellow vertices have length $1$, those that terminate with the green vertices have length $2$, and those that terminate with the blue vertices have length $3$. }
\end{figure}
\begin{enumerate}[(a)]
\item The rational weight $\mathcal{R}_{\mathbf{T}}(\xi) = \mathcal{R}_{(P \to ... \to F_{k-1} \to F_k)}(\xi)$ is defined to be the product of weights associated to all the rooted chains $\mathbf{T}$
of length $1$, times the Hausdorff volume of $F_k$ (the last node of the chain $\mathbf{T}$). It is clear from this definition that $\mathcal{R}_{\mathbf{T}}(\xi)$ is a homogenous rational function of $\xi$.
\bigskip
\item The exponential weight
$\mathcal{E}_{\mathbf{T}}(\xi) = \mathcal{E}_{(P \to ... \to F_{k-1} \to F_k)}(\xi)$
is defined to be the evaluation of $e^{-2\pi i\langle\xi,x\rangle}$ at any point $x$ on the face $F_k$:
\begin{equation} \label{exponential.weight}
\mathcal{E}_{\mathbf{T}}(\xi) := e^{-2\pi i\langle\xi,x_0\rangle},
\end{equation}
for any $x_0 \in F_k$. We note that the inner product $\langle\xi,x_0 \rangle$ does not depend on the position of $x_0 \in F_k$.
\bigskip
\item The total weight of a rooted chain $T$ is defined to be the rational-exponential function
\begin{equation}
W_{\mathbf{T}}(\xi) = W_{(P \to ... \to F_{k-1} \to F_k)}(\xi):= \mathcal{R}_{\mathbf{T}}(\xi) \mathcal{E}_{\mathbf{T}}(\xi) \mathbf{1}_{S(\mathbf{T})}(\xi),
\end{equation}
\noindent
where $\mathbf{1}_{S(\mathbf{T})}(\xi)$ is the indicator function of the admissible set $S(\mathbf{T})$ of $\mathbf{T}$.
\end{enumerate}
\bigskip
\noindent
By repeated applications of the combinatorial divergence
Theorem \ref{Combinatorial divergence theorem},
we arrive at a description of the Fourier transform of $P$ as the sum of weights of all the rooted chains of
the face poset ${\frak F}_P$, as follows.
\begin{thm} \label{ingredient1}
\begin{align} \label{explicit Fourier transform of a polytope}
\hat 1_{P}(\xi) = \sum_{\mathbf{T}} W_{\mathbf{T}}(\xi) = \sum_{\mathbf{T}} \mathcal{R}_{\mathbf{T}}(\xi) \mathcal{E}_{\mathbf{T}}(\xi) \mathbf{1}_{S(\mathbf{T})}(\xi),
\end{align}
valid for any fixed $\xi \in \mathbb{R}^d$.
\end{thm}
For a detailed proof of Theorem \ref{ingredient1}, see \cite{RicardoNhatSinai}.
Using this explicit description of the Fourier transform of a polytope, we will see an application of it in the following
section, for the coefficients of Macdonald's angle quasi-polynomial. In the process, equation
\eqref{explicit Fourier transform of a polytope}, which gives an explicit description of the Fourier transform of a polytope, using the facets of ${\mathcal P}$ as well as lower-dimensional faces of ${\mathcal P}$, will become even more explicit with some examples.
\section{Generic frequencies versus special frequencies}
Given a polytope ${\mathcal P} \subset \mathbb{R}^d$, we call a vector $\xi \in \mathbb{R}^d$ a {\bf generic frequency} (relative to ${\mathcal P}$) \index{generic frequencies}
if $\xi$ is not orthogonal to any face of ${\mathcal P}$. All other $\xi \in \mathbb{R}^d$ are orthogonal to some face $F$ of ${\mathcal P}$, and are called {\bf special frequencies}. Let's define the following hyperplane arrangement, given by the finite collection of hyperplanes orthogonal to any edge of ${\mathcal P}$:
\[
\mathcal H := \{ x \in \mathbb{R}^d \mid \langle x, F_1 \rangle = 0, \text{ for any $1$-dimensional edge $F_1$ of ${\mathcal P}$ } \}.
\]
Then it is clear that the special frequencies are exactly those vectors that lie in the hyperplane arrangement
$\mathcal H$. So we see from Theorem \ref{ingredient1} that for a generic frequency $\xi$, we have
\begin{align}
\hat 1_{P}(\xi) = \sum_{\mathbf{T}: P \to ... \to F_{1} \to F_0} \mathcal{R}_{\mathbf{T}}(\xi) e^{-2\pi i \langle \xi, F_0 \rangle},
\end{align}
where the $F_0$ faces are the vertices of ${\mathcal P}$. In other words, for generic frequencies, all of our rooted chains in the face poset of ${\mathcal P}$ go all the way to the vertices. The special frequencies, however, are more complex. But we can collect the special frequencies in `packets', giving
us the following result.
\bigskip
\begin{thm} [Coefficients for Macdonald's angle quasi-polynomial] \cite{RicardoNhatSinai}
\label{thm:main}
Let $P$ be a $d$-dimensional rational polytope in $\mathbb{R}^d$, and let $t$ be a positive real number.
Then we have the quasi-polynomial
\[ A_P(t) =\sum_{i = 0}^d a_i(t)t^i, \]
where, for $0 \leq i \leq d$,
\begin{equation}\label{complicatedcoeff}
a_i(t) := \lim_{\varepsilon\to 0^+} \sum_{\xi\in\mathbb{Z}^d \cap S(\mathbf{T})}
\sum_{l(\mathbf{T}) = d-i} \mathcal{R}_{\mathbf{T}}(\xi) \mathcal{E}_{\mathbf{T}}(t\xi) \ e^{-\pi\varepsilon\|\xi\|^2},
\end{equation}
\end{thm}
\noindent
where $l(\mathbf{T})$ is the length of the rooted chain $\mathbf{T}$ in the face poset of $P$,
$\mathcal{R}_{\mathbf{T}}(\xi)$ is the rational function of $\xi$ defined above, $\mathcal{E}_{\mathbf{T}}(t\xi) $ is the complex exponential defined in \eqref{exponential.weight} above, and
$\mathbb{Z}^d \cap S(\mathbf{T})$ is the set of all integer points that are orthogonal to the last node in the chain $T$,
but not to any of its previous nodes.
See \cite{RicardoNhatSinai} for the detailed proof of Theorem \ref{thm:main}.
We call the coefficients $a_i(t)$ the {\bf quasi-coefficients} of the solid angle sum $A_P(t)$. \index{quasi-coefficients}
As a consequence of Theorem \ref{thm:main}, it turns out that there is a closed form for the codimension-$1$ quasi-coefficient, which extends previous special cases of this coefficient.
We recall our first periodic Bernoulli polynomial, from \eqref{definition of periodic Bernoulli polys}:
\begin{equation}
P_1 (x):=
\begin{cases}
x - \lfloor x \rfloor- \frac{1}{2} & \mbox{if } x \notin \mathbb{Z} \\
0 & \mbox{if } x \in \mathbb{Z},
\end{cases}
\end{equation}
where $\lfloor x \rfloor$ is the integer part of $x$.
\begin{thm}\cite{RicardoNhatSinai}
\label{codim1coeff}
Let $P$ be any real polytope. Then the codimension-1 quasi-coefficient of the solid angle sum $A_P(t)$ has the following closed form:
\begin{equation}
a_{d-1} (t) =
-\sum_{\substack{F \textup{ a facet of } P \\ with \ v_F \neq 0}} \frac{\vol F}{\|v_F\|}
P_1 (\langle v_F, x_F \rangle t),
\end{equation}
where $v_F$ is the primitive integer vector which is an outward-pointing normal vector to $F$, $x_F$ is
any point lying in the affine span of $F$, and $t$ is any positive real number.
\end{thm}
\hfill $\square$
\bigskip
We note that, rather surprisingly,
the latter formula shows in particular that for any real polytope ${\mathcal P}$, the quasi-coefficient $a_{d-1}(t)$ is always a periodic function of $t > 0$, with a period of $1$.
Although it is not necessarily true that for any real polytope the rest of the quasi-coefficients $a_k(t)$
are periodic functions of $t$, it is true that in the case of rational polytopes, the quasi-coefficients
are periodic functions of all real dilations $t$, as we show below.
We recall that zonotopes are projections of cubes or, equivalently, polytopes whose faces (of all dimensions) are symmetric. We also recall the result of Alexandrov and Shephard (Theorem \ref{Alexandrov-Shepard thm}) from chapter \ref{Geometry of numbers}: If all the facets of ${\mathcal P}$ are symmetric, then ${\mathcal P}$ must be symmetric as well. The following result appeared in \cite{BarvinokPommersheim}, and here we give a different proof, using the methods of this chapter.
\begin{thm} \label{cs.facets} \index{Barvinok, Alexander}
Suppose $P$ is a $d$-dimensional integer polytope in $\mathbb{R}^d$
all of whose facets are centrally symmetric. Then
\[
A_{\mathcal P}(t) = (\vol {\mathcal P}) t^d,
\]
for all positive integers $t$.
\end{thm}
\begin{proof}
We recall the formula for the solid angle polynomial $A_{\mathcal P}(t)$.
\begin{equation}\label{eq:APtsum}
A_{\mathcal P}(t) = \lim_{\varepsilon \to 0^+} \sum_{\xi \in \mathbb{Z}^d} \hat{1}_{t{\mathcal P}}(\xi)e^{-\pi\varepsilon\|\xi\|^2}.
\end{equation}
The Fourier transform of the indicator function of a polytope may be written as follows, after one application of the combinatorial divergence formula:
\begin{align}
\hat 1_{t{\mathcal P}}(\xi) = t^d \vol {\mathcal P} \, [\xi =0]
+ \left( \frac{-1}{2 \pi i} \right) t^{d-1}
\sum_{{\substack{F \subseteq {\mathcal P} \\ \dim F = d-1}}}
\frac{\langle \xi, {\bf n}_F \rangle }{\| \xi \|^2} \hat 1_F (t \xi) [\xi \not= 0],
\end{align}
where we sum over all facets $F$ of ${\mathcal P}$. Plugging this into~\eqref{eq:APtsum} we get
\begin{align} \label{imaginary}
A_{\mathcal P}(t) - t^d \vol {\mathcal P}
= \left( \frac{-1}{2 \pi i} \right) t^{d-1}
\lim_{\varepsilon \to 0^+} \sum_{\xi \in \mathbb{Z}^d \setminus\{0\}}
\frac{e^{-\pi\varepsilon\|\xi\|^2}}{\| \xi \|^2} \sum_{{\substack{F \subseteq {\mathcal P} \\ \dim F = d-1}}}
\langle \xi, {\bf n}_F \rangle \hat 1_F(t \xi),
\end{align}
so that it is sufficient to show that the latter sum over the facets vanishes.
The assumption that all facets of ${\mathcal P}$ are centrally symmetric implies that ${\mathcal P}$ itself is also centrally symmetric, by Theorem~\ref{cs1}.
We may therefore combine the facets of ${\mathcal P}$ in pairs of opposite facets $F$ and $F'$. We know that $F' = F + c$, where $c$ is an integer vector, using the fact that the facets are centrally symmetric.
Therefore, since ${\bf n}_F' = - {\bf n}_F $, we have
\begin{align*}
\langle \xi, {\bf n}_F \rangle &\hat 1_{F}(t \xi)
+ \langle \xi, -{\bf n}_F \rangle \hat 1_{F+ c}(t \xi)\\
&= \langle \xi, {\bf n}_F \rangle \hat 1_{F}(t \xi)
- \langle \xi, {\bf n}_F \rangle \hat 1_{F}(t \xi)
e^{-2\pi i\langle t\xi, c \rangle} \\
&= \langle \xi, {\bf n}_F \rangle \hat 1_{F}(t \xi)
\left( 1 - e^{-2\pi i\langle t\xi, c \rangle} \right) = 0,
\end{align*}
because $\langle t\xi, c \rangle \in \mathbb{Z}$ when both $\xi \in \mathbb{Z}^d$ and $t \in \mathbb{Z}$. We conclude that the entire right-hand side of \eqref{imaginary} vanishes, and we are done.
\end{proof}
Fourier analysis can also be used to give yet more general classes of polytopes that satisfy the formula
$A_P(t) = (\vol {\mathcal P}) t^d$, for positive integer values of $t$ (See also \cite{FabricioSinai2}, \cite{DeligneTabachnikovRobins}).
\bigskip
There is a wonderful result of Minkowski that gives a converse to the relation
\eqref{Minkowski relation}, as follows.
\begin{thm}[The Minkowski problem for polytopes] \label{Minkowski's problem for polytopes}
\index{Minkowski problem for polytopes}
Suppose that $u_1, \dots, u_k\in \mathbb{R}^d$ are unit vectors that do not lie in a hyperplane.
Suppose further that we are given positive numbers $\alpha_1, \alpha_2, \dots, \alpha_k >0$ that
satisfy the relation
\[
\alpha_1 u_1 + \cdots + \alpha_k u_k =0.
\]
Then there exists a polytope ${\mathcal P}\subset R^d$, with facet normals $u_1, \dots, u_k\in \mathbb{R}^d$, and
facet areas $\alpha_1, \alpha_2, \dots, \alpha_k$.
Moreover, this polytope ${\mathcal P}$ is unique, up to translations.
\hfill $\square$
\end{thm}
There is a large body of work, since the time of Minkowski, that is devoted to extensions of
Minkowski's Theorem \ref{Minkowski's problem for polytopes}, to other convex bodies, as well as to other manifolds.
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.Divergence}
\begin{enumerate}[(a)]
\item We could also define another useful vector field, for our combinatorial divergence theorem, besides
our vector field in equation \eqref{First vector field}. Namely, if we define
$F(x):= e^{2\pi i \langle x, \xi \rangle} \lambda$,
for a fixed $\lambda\in\mathbb{C}^d$, then we would get the analogous combinatorial divergence formula as shown below in
(Exercise \ref{alternate combinatorial divergence Theorem}),
and such vector fields
have been used, for example, by Alexander Barvinok \cite{Barvinok1} in an effective way.
\index{Barvinok, Alexander}
To the best of our knowledge,
the first researcher to use iterations of Stokes' formula to obtain lattice point asymptotics was Burton Randol \cite{Randol3}, \cite{Randol4}.
\item The Minkowski problem for polytopes can also be related directly to generalized isoperimetric inequalities for mixed volumes, as well as the Brunn-Minkowski inequality for polytopes,
as done by Daniel Klain in \cite{Klain1}.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\medskip
\begin{prob} $\clubsuit$ \label{Pyramid formula}
We define the distance from the origin to $F$, denoted by $\rm{dist}(F)$, as the length of the shortest vector of translation between
$\rm{aff}(F)$ and $\rm{lin}(F)$ (resp. the affine span of $F$ and the linear span of $F$, defined in
\eqref{affine span of F}). Figure \ref{affine span} shows what can happen in such a scenario.
\begin{enumerate}[(a)]
\item Suppose that we consider a facet $F$ of a given polytope ${\mathcal P} \subset \mathbb{R}^d$, and we
let ${\bf n}_F$ be the unit normal vector to $F$. Show that the function
\[
x_F \rightarrow \langle x_F, {\bf n}_F \rangle
\]
is constant for $x_F \in F$, and is in fact equal to the distance from the origin to $F$. In other words, show that
\[
\langle x, {\bf n}_F \rangle = \rm{dist}(F).
\]
\item Show that if $\rm{Proj}_F \xi = 0$, then $\langle \xi, x_F \rangle = \| \xi \| \rm{dist} F$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob}
Here we prove the elementary geometric formula for a pyramid over a polytope. Namely, suppose we are
given a
$(d-1)$-dimensional polytope ${\mathcal P}$, lying in the vector space defined by the first $d-1$ coordinates.
We define a pyramid over ${\mathcal P}$, of height $h > 0$, as the $d$-dimensional polytope defined by
\[
\rm{Pyr}({\mathcal P}) := \conv\{ {\mathcal P}, \ h \cdot e_{d} \},
\]
where $e_d := (0, 0, \dots, 0, 1) \in \mathbb{R}^d$. Show that
\[
\vol \rm{Pyr}({\mathcal P}) = \frac{h}{d} \vol {\mathcal P}.
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{distance to a facet}
Prove the Pyramid formula, \eqref{pyramid formula} in
Example \ref{Pyramid formula via the divergence theorem},
for a $d$-dimensional polytope ${\mathcal P}$ which contains the origin, but now using just elementary geometry:
\begin{equation}
\vol {\mathcal P} = \frac{1}{d} \sum_{k=1}^N \rm{dist}(G_k) \vol G_k,
\end{equation}
where the $G_k$'s are the facets of ${\mathcal P}$, and $\rm{dist}(G_k)$ is the distance from the origin to $G_k$.
\end{prob}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.2in]{ExerciseDivergence}
\end{center}
\caption{The meaning of Minkowski's relation in dimension $2$ - see Exercise
\ref{Geometric interpretation of Minkowski's relation for d=2} }
\label{Divergence Exercise}
\end{figure}
\medskip
\begin{prob} $\clubsuit$ \label{alternate combinatorial divergence Theorem}
Show that if we replace the vector field in equation \eqref{First vector field} by the
alternative vector field $F(x):= e^{-2\pi i \langle x, \xi \rangle} \lambda$, with a constant nonzero vector $\lambda \in \mathbb{C}^d$, then we get:
\begin{equation} \label{our alternate formula for the transform}
\hat 1_{\mathcal P}(\xi) =\frac{1}{-2\pi i } \sum_{G\subset \partial P}
\frac{ \langle \lambda, {\bf n}_G \rangle}{ \langle \lambda, \xi \rangle}
\hat 1_G(\xi),
\end{equation}
valid for all nonzero $\xi \in \mathbb{R}^d$. Note that one advantage of this formulation
of the Fourier transform
of ${\mathcal P}$ is that each summand in the right-hand-side of
\eqref{our alternate formula for the transform}
is free of singularities, assuming the vector
$\lambda$ has a nonzero imaginary part.
\end{prob}
\medskip
\begin{prob} \label{equivalent identity to the alternate vector field}
Show that the identity \eqref{our alternate formula for the transform} of Exercise \ref{alternate combinatorial divergence Theorem} is equivalent to the vector identity:
\[
\xi \hat 1_{\mathcal P}(\xi) = \frac{1}{-2\pi i } \sum_{G\subset \partial P} {\bf n}_G \hat 1_G(\xi),
\]
valid for all $\xi \in \mathbb{R}^d$.
\end{prob}
\medskip
\begin{prob}
Show that the result of Exercise \ref{equivalent identity to the alternate vector field}
quickly gives us
the Minkowski relation \eqref{Minkowski relation}:
\[
\sum_{ \text{facets } G \text{ of } P} \vol G {\bf n}_{G}= 0.
\]
\end{prob}
\medskip
\begin{prob}
Continuing Exercise \ref{alternate combinatorial divergence Theorem}, show that by iterating this particular version of the Fourier transform of a polytope ${\mathcal P}$, $k$ times, we get:
\begin{equation}
\hat 1_{\mathcal P}(\xi) =\frac{1}{(-2\pi i )^k} \sum_{G_k \subset G_{k-1} \subset \cdots G_1 \subset \partial P}
\prod_{j=1}^k
\frac{
\langle \lambda, {\bf n}_{G_{j}, G_{j-1}} \rangle}{ \langle \lambda, \rm{Proj}_{G_{j-1}} \xi \rangle }
\hat 1_{G_k}(\xi),
\end{equation}
valid for all nonzero $\xi \in \mathbb{R}^d$, and where we sum over all chains $G_k \subset G_{k-1} \subset \cdots G_1$ of length $k$ in the face poset of ${\mathcal P}$, with \rm{codim}$(G_j) = j$.
\end{prob}
\medskip
\begin{prob} \label{Geometric interpretation of Minkowski's relation for d=2}
Show that in the case of polygons in $\mathbb{R}^2$, the Minkowski relation \eqref{Minkowski relation}
has the meaning that
the sum of the pink vectors in Figure \ref{Divergence Exercise} sum to zero. In other words, the geometric interpretation
of the Minkowski relation in dimension $2$ is that the sum of the boundary (pink) vectors
wind around the boundary and close up perfectly.
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{no simplex is symmetric}
Let's consider a simplex $\Delta \subset \mathbb{R}^d$ whose dimension satisfies $2 \leq \dim \Delta \leq d$.
Show that $\Delta$ is not a symmetric body.
\end{prob}
\medskip
\begin{prob}
Let $F \subset \mathbb{R}^d$ be a centrally symmetric, integer polytope of dimension $k$. Show that the distance from the origin to $F$ is always a half-integer or an integer. In other words, show that
\[
\rm{dist}(F) \in \frac{1}{2} \mathbb{Z}.
\]
(See Exercise \ref{Pyramid formula} above for the definition of distance of $F$ to the origin)
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Exercise Iverson bracket}
To get more practice with the Iverson bracket, defined in \eqref{Iverson bracket}, show that for all logical statements $P$, we have:
\begin{enumerate}[(a)]
\item $ [ P \rm{ \ and \ } Q ] = [ P ] [ Q ]$.
\item $[ P \rm{ \ or \ } Q ] = [P] + [Q] - [P][Q]$.
\item $[\neg P] = 1 - [P]$, where $\neg P$ means the logical negation of $P$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{the FT of 1_P is not in L^1}
Show that for any polytope ${\mathcal P}\subset \mathbb{R}^d$, its Fourier transform $\hat 1_{\mathcal P}(\xi)$ is not in $L^1(\mathbb{R}^d)$.
\end{prob}
\chapter{The angle polynomial of a polytope}
\label{Angle polynomial}
\index{angle polynomial} \index{solid angle}
\begin{figure}[!h]
\centering
\begin{tikzpicture}[scale=.75]
\draw (0,0) node[below left] {$0$};
\draw[loosely dotted] (-1,-1) grid (7,5);
\draw[->] (-1.25,0) -- (7.25,0) node[right] {$x$};
\draw[->] (0,-1.25) -- (0,5.25) node[above] {$y$};
\draw[fill = green] (3,4) circle (.1cm);
\draw[fill = green] (4,3) circle (.1cm);
\draw[fill = green] (3,3) circle (.1cm);
\draw[fill = green] (3,2) circle (.25cm);
\draw[fill = green] (2,2) circle (.1cm);
\draw[fill = green] (2,3) circle (.25cm);
\draw[fill = green] (4,4) circle (.25cm);
\draw[fill = green] (5,3) circle (.5cm);
\draw[fill = green] (3,5) circle (.5cm);
\draw[fill = green] (1,1) circle (.5cm);
\draw[thick] (1,1) -- (5,3) -- (3,5) -- cycle;
\filldraw[nearly transparent, blue] (1,1) -- (5,3) -- (3,5) -- cycle;
\end{tikzpicture}
\caption{A discrete volume of the triange ${\mathcal P}$, called the angle polynomial of ${\mathcal P}$.
Here we sum local angle weights, relative to ${\mathcal P}$, at all integer points.}
\label{triangle solid angle sum}
\end{figure}
\section{Intuition}
There are infinitely many ways to discretize the classical notion of volume, and here we offer a second path, using `local solid angles'. Given a rational polytope ${\mathcal P}$, we will place small spheres at all integer points in $\mathbb{Z}^d$, and compute the proportion of the local intersection of each small sphere with ${\mathcal P}$. This discrete, finite sum, gives us a new method of discretizing the volume of a polytope, and it turns out to be a more symmetric way of doing so. To go forward, we first discuss how to extend the usual notion of `angle' to higher dimensions, and then use Poisson summation again to pursue the fine detail of this new discrete volume.
\bigskip
\section{What is an angle in higher dimensions?} \label{Chapter.solid.angles}
The question of how an angle in two dimensions extends to higher dimensions is a basic one
in discrete geometry. A natural way to extend the notion of an angle is to consider
a cone ${\mathcal K} \subset \mathbb{R}^d$, place a sphere centered at the apex of ${\mathcal K}$, and then compute the proportion
of the sphere that intersects ${\mathcal K}$. This intuition is captured more rigorously by the following integral:
\begin{equation} \label{integral def. of solid angle}
\omega_{\mathcal K} = \int_{\mathcal K} e^{-\pi \| x \|^2} dx.
\end{equation}
called the {\bf solid angle of the cone} ${\mathcal K}$. \index{solid angle}
The literature has other synonyms for solid angles, arising in different fields,
including the {\bf volumetric moduli} \cite{GourionSeeger},
\index{volumetric moduli}
and the {\bf volume of a spherical polytope}
\index{volume of a spherical polytope}
\cite{BeckRobins}, \cite{DesarioRobins}, \cite{RicardoNhatSinai}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2.3in]{solidangle}
\end{center}
\caption{A solid angle in $\mathbb{R}^3$ - note the equivalence with the area of the geodesic triangle on the sphere.}
\end{figure}
We can easily show that the latter definition of a solid angle
is equivalent to the volume of a spherical polytope,
\index{volume of a spherical polytope}
using polar coordinates in $\mathbb{R}^d$, as follows. We denote the
unit sphere by
$S^{d-1}:= \{ x\in \mathbb{R}^d \mid \| x\| = 1 \}$. Then using the fact that the Gaussians give a probability distribution, namely $\int_{\mathbb{R}^d} e^{-\pi ||x||^2} dx = 1$ (which we know by Exercise \ref{Gaussian1}), we have
\begin{align}
\omega_{\mathcal K} &= \frac{\int_{\mathcal K} e^{-\pi\|x\|^2}dx}{\int_{\mathbb{R}^d} e^{-\pi\|x\|^2}dx} \label{second equality}
\ = \ \frac{\int_0^{\infty} e^{-\pi r^2} r^{d-1} dr \int_{S^{d-1} \cap {\mathcal K}} d\theta}{\int_0^{\infty}
e^{-\pi r^2} r^{d-1} dr \int_{S^{d-1}} d\theta} \\
&= \ \frac{\int_{S^{d-1} \cap {\mathcal K}} d\theta}{\int_{S^{d-1}} d\theta} \\ \label{normalized spherical volume}
&= \ \frac{\vol_{d-1} \left({\mathcal K} \cap S^{d-1}\right)}{\vol_{d-1} \left( S^{d-1} \right)},
\end{align}
where $\vol_{d-1}$ denotes the volume measure on the surface of the $(d-1)$-dimensional sphere $S^{d-1}$.
We may think of \eqref{normalized spherical volume} as the {\bf normalized volume} of a spherical polytope defined by the intersection of the cone ${\mathcal K}$
with the unit sphere. Thus for any cone ${\mathcal K}\subset \mathbb{R}^d$, we have
\[
0 \leq \omega_{\mathcal K} \leq 1.
\]
We used polar coordinates in the second equality \eqref{second equality} above:
$x= (r, \theta)$, with $r \geq 0, \ \theta \in {\mathcal S}^{d-1}$.
The Jacobian in the change of variables is $dx = r^{d-1} dr d\theta$.
We note that when ${\mathcal K}= \mathbb{R}^d$, so that the cone is \emph{all} of Euclidean space,
the integral \eqref{integral def. of solid angle}
becomes
\[
\int_{\mathbb{R}^d} e^{-\pi ||x||^2} dx = 1,
\]
by Exercise \ref{Gaussian1}. This computation confirms that we do indeed have the proper normalization
with $\omega_{\mathcal K} = 1$ if and only if ${\mathcal K} = \mathbb{R}^d$.
\begin{example}
\rm{
If ${\mathcal K}\subset \mathbb{R}^d$ is a half-space, then
$\omega_{\mathcal K} = \frac{1}{2}$. If ${\mathcal K}:= \mathbb{R}_{\geq 0}^d$, the positive orthant, then
\begin{align*}
\omega_{\mathcal K} &= \int_{\mathbb{R}^d_{\ge 0}} e^{-\pi ||x||^2} dx =
\left( \int_{\mathbb{R}_{\ge 0}} e^{-\pi u^2} du \right)^d = \frac{1}{2^d}.
\end{align*}
}
\hfill $\square$
\end{example}
\section{Local solid angles for a polytope, and Gaussian smoothing}
Here we want to define solid angles relative to a fixed polytope.
So given any polytope ${\mathcal P} \subset \mathbb{R}^d$, we fix any point $x \in \mathbb{R}^d$ and define a local solid angle relative to ${\mathcal P}$ as follows.
The normalized {\bf solid angle} \index{solid angle} fraction that a
$d$-dimensional polytope ${\mathcal P}$ subtends at any point $x \in \mathbb{R}^d$ is defined by
\begin{equation}\label{def. of solid angle}
\omega_{\mathcal P}(x)=\lim_{\varepsilon \to 0} \frac{\vol(S^{d-1}(x,\varepsilon) \cap {\mathcal P})}{\vol\left(S^{d-1}(x,\varepsilon)\right)}.
\end{equation}
Here, $\omega_{{\mathcal P}}(x)$ measures the fraction of a small $(d-1)$-dimensional sphere
$S^{d-1}(x,\varepsilon)$
centered at $x$, that intersects the polytope ${\mathcal P}$.
We will use the standard notation for the interior of a convex body, namely $\interior({\mathcal P})$, and for the boundary of a convex body, namely $\partial {\mathcal P}$. As a side-note, we mention that balls and spheres can be used interchangeably in this definition, meaning that the fractional weight given by \eqref{def. of solid angle}
is the same using either method (see Exercise \ref{two definitions for a solid angle}).
It follows from the definition of a solid angle that $0 \leq \omega_{\mathcal P}(x) \leq 1$, for all $x \in \mathbb{R}^d$, and that
\[
\omega_{\mathcal P}(x) =
\begin{cases}
1 & \text{if } x \in \interior({\mathcal P}) \\
0 & \text{if } x \notin {\mathcal P}.
\end{cases}
\]
But when $x \in \partial {\mathcal P}$, we have $\omega_{\mathcal P}(x) >0$. For example, if $x$ lies on a
codimension-two face of ${\mathcal P}$, then $\omega_{\mathcal P}(x)$ is the fractional dihedral angle
subtended by ${\mathcal P}$ at $x$.
Returning to discrete volumes, Ehrhart and Macdonald analyzed a different discrete volume for
any polytope ${\mathcal P}$. Namely, for each positive integer $t$, define the finite sum
\begin{equation} \label{anglesum1}
A_{\mathcal P}(t) := \sum_{n\in \mathbb{Z}^d} \omega_{t{\mathcal P}}(n),
\end{equation}
where $t{\mathcal P}$ is the $t$'th dilation of the polytope ${\mathcal P}$.
for ${\mathcal P}$.
In other words, $A_{\mathcal P}(1)$ is
a new discrete volume for ${\mathcal P}$, obtained by placing at each integer point $n\in \mathbb{Z}^d$ the weight
$\omega_{t{\mathcal P}}(x)$, and summing all of the weights.
\begin{example}
\rm{
In Figure \ref{triangle solid angle sum}, the solid angle sum of the polygon ${\mathcal P}$ is
\[
A_{\mathcal P}(1) = \theta_1 + \theta_2 + \theta_3 + 3\left( \tfrac{1}{2} \right)+ 4 = 6.
\]
Here the $\theta_j$'s are the three angles at the vertices of
${\mathcal P}$.
}
\hfill $\square$
\end{example}
Using purely combinatorial methods, Macdonald showed that for any integer polytope ${\mathcal P}$,
and for {\bf positive integer values} of $t$,
\begin{equation} \label{solidanglesum2}
A_{\mathcal P}(t) = (\vol {\mathcal P}) t^d + a_{d-2} t^{d-2} + a_{d-4} t^{d-4} + \cdots +
\begin{cases}
a_1 t & \text{if } d \text{ is odd},\\
a_2 t^2 & \text{if } d \text{ is even}.
\end{cases}
\end{equation}
We will call $A_{\mathcal P}(t)$ the {\bf angle-polynomial} of ${\mathcal P}$, \index{angle polynomial}
for integer polytopes ${\mathcal P}$ and positive
integer dilations $t$. However, when these restrictions are lifted, the sum still captures crucial
geometric information of ${\mathcal P}$, and we will simply call it the (solid) angle-sum of ${\mathcal P}$.
We define the {\bf heat kernel}, \index{heat kernel}
for each fixed positive $\varepsilon$, by
\begin{equation}
G_{\varepsilon}(x) := \varepsilon^{-\frac{d}{2}} e^{-\frac{\pi}{\varepsilon} \| x \|^2},
\end{equation}
for all $x \in \mathbb{R}^d$. By Exercises \ref{Gaussian1} and \ref{Gaussian2}, we know that
$\int_{\mathbb{R}^d} G_{\varepsilon}(x)dx = 1$ for each fixed $\varepsilon$, and that
\begin{equation} \label{Fourier transform of the Gaussian, formal}
\hat G_{\varepsilon}(\xi) = e^{-\varepsilon \pi \| \xi \|^2}.
\end{equation}
The convolution of the indicator function $1_{\mathcal P}$ by the heat kernel $G_{\varepsilon}$ will be called the
{\bf Gaussian smoothing} \index{Gaussian smoothing}
of $1_{\mathcal P}$:
\begin{align} \label{Gaussian smoothing}
(1_{\mathcal P} * G_{\varepsilon})(x) &:= \int_{\mathbb{R}^d} 1_{\mathcal P}(y) G_{\varepsilon} (x-y) dy
= \int_{{\mathcal P}} G_{\varepsilon} (y-x) dy \\
&= \varepsilon^{-\frac{d}{2}} \int_{{\mathcal P}} e^{-\frac{\pi }{\varepsilon} \| y-x \|^2} dy,
\end{align}
a $C^{\infty}$ function of $x\in \mathbb{R}^d$, and in fact a Schwartz function
(Exercise \ref{convolution of the indicator function with a Gaussian}). The following Lemma provides a first
crucial link between the discrete geometry of a local solid angle and the convolution of $1_{\mathcal P}$ with a
Gaussian-based approximate identity.
\bigskip
\begin{lem} \label{basic connection for solid angle}
Let ${\mathcal P}$ be a full-dimensional polytope in $\mathbb{R}^d$. Then for each point $x\in\mathbb{R}^d$, we have
\begin{equation}
\lim_{\varepsilon \rightarrow 0} (1_{\mathcal P} * G_{\varepsilon})(x) = \omega_P(x).
\end{equation}
\end{lem}
\begin{proof}
We have
\begin{align*}
(1_{\mathcal P} * G_{\varepsilon})(x) &= \int_{{\mathcal P}} G_{\varepsilon} (y-x) dy \\
&= \int_{u \in P- x} G_{\varepsilon} (u) du = \int_{\frac{1}{\sqrt{\varepsilon}}(P- x)} G_1(v) dv.
\end{align*}
In the calculation above, we make use of the evenness of $G_{\varepsilon}$ in the second equality. The substitutions $u = y-x$ and $v = u/\sqrt{\varepsilon}$ are also used. Following those substitutions, we change the domain of integration from $P$ to the translation $P- x$,
and to the dilation of $P-x$ by the factor $\frac{1}{\sqrt{\varepsilon}}$.
Now, when $\varepsilon$ approaches $0$,
$\frac{1}{\sqrt{\varepsilon}}(P- x)$ tends to a cone $K$ at the origin, subtended by $P- x$.
The cone $K$ is in fact a translation of the tangent cone \index{tangent cone}
of $P$ at $x$. Thus, we arrive at
\[
\lim_{\varepsilon \to 0} (1_{\mathcal P} * G_{\varepsilon}) (x) = \int_{K} G_1(v)dv = \omega_K(0) = \omega_P(x).
\]
\end{proof}
Putting things together, the definition \ref{anglesum1} and
Lemma \ref{basic connection for solid angle}
above tell us that
\begin{equation}
A_{\mathcal P}(t) = \sum_{n\in\mathbb{Z}^d} \omega_{tP}(x) =
\sum_{n\in\mathbb{Z}^d} \lim_{\varepsilon \rightarrow 0} (1_{t{\mathcal P}} * G_{\varepsilon})(n).
\end{equation}
We would like to interchange a limit with an infinite sum over a lattice, so that we may use Poisson summation, and although this is subtle in general, it's possible to carry out here, because the summands are rapidly decreasing.
\begin{lem} \label{interchanging limit and sum, solid angle sum}
Let ${\mathcal P}$ be a full-dimensional polytope in $\mathbb{R}^d$.
Then
\begin{equation}
A_{\mathcal P}(t) = \lim_{\varepsilon \rightarrow 0} \sum_{n\in\mathbb{Z}^d} (1_{t{\mathcal P}} * G_{\varepsilon})(n).
\end{equation}
\hfill $\square$
\end{lem}
(For a proof of Lemma \ref{interchanging limit and sum, solid angle sum} see \cite{RicardoNhatSinai}).
Next, we apply the Poisson summation formula to the Schwartz function \\
$f(x) := (1_{\mathcal P} * G_{\varepsilon})(x)$:
\begin{align} \label{Gaussian smoothing for Angle sum}
A_P(t) &= \lim_{\varepsilon \rightarrow 0} \sum_{n\in\mathbb{Z}^d} (1_{t{\mathcal P}} * G_{\varepsilon})(n) \\
&= \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}^d} \hat 1_{t{\mathcal P}}(\xi) \hat G_{\varepsilon}(\xi) \\
&= \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}^d} \hat 1_{t{\mathcal P}}(\xi) \ e^{-\varepsilon \pi \| \xi \|^2} \\
&= t^d \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}^d} \hat 1_{{\mathcal P}}( t\xi)
\ e^{-\varepsilon \pi \| \xi \|^2} \\
&= t^d \ \hat 1_{{\mathcal P}}( 0 ) + \lim_{\varepsilon \rightarrow 0} t^d \sum_{\xi \in\mathbb{Z}^d-\{0\}} \hat 1_{{\mathcal P}}( t\xi)
\ e^{-\varepsilon \pi \| \xi \|^2} \\
&= t^d (\vol {\mathcal P}) + \lim_{\varepsilon \rightarrow 0}t^d \sum_{\xi \in\mathbb{Z}^d-\{0\}} \hat 1_{{\mathcal P}}( t\xi)
\ e^{-\varepsilon \pi \| \xi \|^2},
\label{last line of Poisson summation}
\end{align}
where we used the fact that Fourier transforms interact nicely with dilations of the domain:
\[
\hat 1_{t{\mathcal P}}(\xi) = \int_{t{\mathcal P}} e^{-2\pi i \langle \xi, x \rangle} dx =
t^d \int_{{\mathcal P}} e^{-2\pi i \langle \xi, ty \rangle} dy = t^d \int_{{\mathcal P}} e^{-2\pi i \langle t \xi, y \rangle} dy=
t^d \hat 1_{{\mathcal P}}(t\xi).
\]
We also used the simple change of variable $x =t y$, with $y \in {\mathcal P}$, implying that $dx = t^d dy$,
as well as the Fourier transform formula for the heat kernel \eqref{Fourier transform of the Gaussian, formal}.
Altogether, we now have:
\begin{equation} \label{phase 2 of angle polynomial}
A_{\mathcal P}(t) =
t^d (\vol {\mathcal P}) + t^d \lim_{\varepsilon \rightarrow 0}
\sum_{n\in\mathbb{Z}^d-\{0\}} ( \hat 1_{{\mathcal P}}(t\xi) * G_{\varepsilon})(n),
\end{equation}
suggesting a polynomial-like behavior for the angle polynomial $A_{\mathcal P}(t)$.
\medskip
The next step will be to use our knowledge of the Fourier transform of the polytope ${\mathcal P}$, in
the right-hand-side of \eqref{phase 2 of angle polynomial}, for which even
a $1$-dimensional example is interesting.
\medskip
\begin{example} \label{rigorous example of P_1}
\rm{
Let's compute the angle polynomial of the $1$-dimensional polytope ${\mathcal P}:= [a,b]$, with
$a,b \in \mathbb{R}$. We will use our knowledge of the $1$-dimensional Fourier transform of an interval,
from Exercise \ref{transform.of.interval.a.to.b}, to compute:
\begin{align}
A_P(t) &= (b-a) t + \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}} \hat 1_{{\mathcal P}}( t\xi) \ e^{-\varepsilon \pi \xi^2} \\
&= (b-a) t + \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\left( \frac{e^{-2\pi i t\xi b} - e^{-2\pi i t\xi a} }{-2\pi i \xi} \right)
e^{-\varepsilon \pi \xi^2} \\ \label{strange limit1}
&= (b-a) t + \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\frac{e^{-2\pi i tb\xi -\varepsilon \pi \xi^2}}{-2\pi i \xi}
- \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\frac{e^{-2\pi i ta\xi -\varepsilon \pi \xi^2}}{-2\pi i \xi} \\
\end{align}
Throughout this example, all series converge absolutely (and quite rapidly) due to the existence of the Gaussian damping factor~$e^{-\varepsilon \pi \xi^2}$.
Let's see what happens when we specialize the vertices $a$ or $b$ - perhaps we can solve for these
new limits?
case 1. \ $a, b \in \mathbb{Z}$. This is the case of an integer polytope, which in this case is an interval in~$\mathbb{R}^1$. Because we are restricting attention to integer dilates $t$, and since $a, b, \xi \in \mathbb{Z}$, we have $e^{-2\pi i t\xi b} = e^{-2\pi i t\xi a} =1$.
Therefore
\[
A_P(t) = (b-a) t + \lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\left( \frac{e^{-2\pi i t\xi b} - e^{-2\pi i t\xi a} }{-2\pi i \xi} \right)
e^{-\varepsilon \pi \xi^2} =(b-a) t + 0.
\]
We arrive at
\[
A_P(t) = (b-a)t,
\]
so that the solid angle sum $A_P(1)$ is exactly the length of the interval we considered. We may compare this discrete volume with the other discrete volume, namely the Ehrhart polynomial of this interval: $L_{[a, b]}(t) = (b-a)t + 1$.
\medskip
case 2. \ $a= 0, b \notin \mathbb{Z}$. Here one of the two series in \eqref{strange limit1} is:
\[
\sum_{\xi \in\mathbb{Z}-\{0\}}
\frac{e^{-2\pi i ta\xi -\varepsilon \pi \xi^2}}{-2\pi i \xi} =
\sum_{\xi \in\mathbb{Z}-\{0\}}
\frac{e^{ -\varepsilon \pi \xi^2}}{-2\pi i \xi} = 0,
\]
because the summand is an odd function of $\xi$. But we already know by direct computation
that in this case
$A_{[0, b]}(t) = \frac{1}{2} + \lfloor bt \rfloor$, we can solve for the other limit:
\[
\frac{1}{2} + \lfloor bt \rfloor= b t +
\lim_{\varepsilon \rightarrow 0} \sum_{\xi \in\mathbb{Z}-\{0\}}
\left( \frac{e^{-2\pi i t\xi b} }{-2\pi i \xi} \right)
e^{-\varepsilon \pi \xi^2}
\]
So this simple example has given us a nice theoretical result. We record this rigorous proof above as Lemma \ref{rigorous approach for P_1(x)} below, after relabelling $bt := x\in \mathbb{R}$.
}
\hfill $\square$
\end{example}
\medskip
\begin{lem} \label{rigorous approach for P_1(x)}
For any $x\in \mathbb{R}$, we have
\[
\frac{1}{2\pi i} \lim_{\varepsilon \rightarrow0} \sum_{\xi \in \mathbb{Z} - \{0\}}
\frac{e^{-2\pi i x\xi -\varepsilon \pi \xi^2}}{\xi} = x - \lfloor x \rfloor - \frac{1}{2}.
\]
\end{lem}
\bigskip
\begin{thm}
Let ${\mathcal P}$ be an integer polygon. Then the angle polynomial of ${\mathcal P}$ is:
\[
A_{\mathcal P}(t) = (\rm{area}{\mathcal P})t^2,
\]
for all positive integer dilations $t$.
\end{thm}
It turns out that this result, for $A_{\mathcal P}(1)$, is easily
equivalent to the well-known Pick's formula \index{Pick's formula}
for an integer polygon.
\begin{thm}[Pick's formula, 1899]
Let ${\mathcal P}$ be an integer polygon. Then
\[
\rm{Area} {\mathcal P} = I + \frac{1}{2} B -1,
\]
where $I$ is the number of interior integer points in ${\mathcal P}$, and B is the number of boundary integer points in ${\mathcal P}$.
\end{thm}
\begin{figure}[!h]
\centering
\begin{tikzpicture}[scale=0.45]
\draw (0,0) node[below left] {$0$};
\draw[loosely dotted] (-1,-1) grid (7,5);
\draw[->] (-1.25,0) -- (7.25,0) node[right] {$x$};
\draw[->] (0,-1.25) -- (0,5.25) node[above] {$y$};
\draw[fill = green] (3,4) circle (.1cm);
\draw[fill = green] (4,3) circle (.1cm);
\draw[fill = green] (3,3) circle (.1cm);
\draw[fill = green] (3,2) circle (.25cm);
\draw[fill = green] (2,2) circle (.1cm);
\draw[fill = green] (2,3) circle (.25cm);
\draw[fill = green] (4,4) circle (.25cm);
\draw[fill = green] (5,3) circle (.5cm);
\draw[fill = green] (3,5) circle (.5cm);
\draw[fill = green] (1,1) circle (.5cm);
\draw[thick] (1,1) -- (5,3) -- (3,5) -- cycle;
\filldraw[nearly transparent, blue] (1,1) -- (5,3) -- (3,5) -- cycle;
\draw (3,-2) node {$P_1$};
\draw (9,2) node[scale = 2] {$\cup$};
\draw (0+12,0) node[below left] {$0$};
\draw[loosely dotted] (-1+12,-1) grid (7+12,5);
\draw[->] (-1.25+12,0) -- (7.25+12,0) node[right] {$x$};
\draw[->] (0+12,-1.25) -- (0+12,5.25) node[above] {$y$};
\draw[fill = green] (5+12,2) circle (.1cm);
\draw[fill = green] (4+12,2) circle (.1cm);
\draw[fill = green] (3+12,2) circle (.25cm);
\draw[fill = green] (5+12,3) circle (.5cm);
\draw[fill = green] (6+12,2) circle (.5cm);
\draw[fill = green] (1+12,1) circle (.5cm);
\draw[thick] (1+12,1) -- (6+12,2) -- (5+12,3) -- cycle;
\filldraw[semitransparent, blue] (1+12,1) -- (5+12,3) -- (6+12,2) -- cycle;
\draw (3+12,-2) node {$P_2$};
\draw (9+12,2) node[scale = 2] {$=$};
\draw (0+24,0) node[below left] {$0$};
\draw[loosely dotted] (-1+24,-1) grid (7+24,5);
\draw[->] (-1.25+24,0) -- (7.25+24,0) node[right] {$x$};
\draw[->] (0+24,-1.25) -- (0+24,5.25) node[above] {$y$};
\draw[fill = green] (3+24,4) circle (.1cm);
\draw[fill = green] (4+24,3) circle (.1cm);
\draw[fill = green] (3+24,3) circle (.1cm);
\draw[fill = green] (5+24,2) circle (.1cm);
\draw[fill = green] (4+24,2) circle (.1cm);
\draw[fill = green] (3+24,2) circle (.25cm);
\draw[fill = green] (2+24,2) circle (.1cm);
\draw[fill = green] (2+24,3) circle (.25cm);
\draw[fill = green] (4+24,4) circle (.25cm);
\draw[fill = green] (5+24,3) circle (.5cm);
\draw[fill = green] (6+24,2) circle (.5cm);
\draw[fill = green] (3+24,5) circle (.5cm);
\draw[fill = green] (1+24,1) circle (.5cm);
\draw[thick] (1+24,1) -- (6+24,2) -- (3+24,5) -- cycle;
\draw[thick] (1+24,1) -- (5+24,3);
\filldraw[nearly transparent, blue] (1+24,1) -- (5+24,3) -- (3+24,5) -- cycle;
\filldraw[semitransparent, blue] (1+24,1) -- (5+24,3) -- (6+24,2) -- cycle;
\draw (3+24,-2) node {$P_1 \cup P_2$};
\end{tikzpicture}
\caption{Additive property of the angle polynomial}
\end{figure}
There is also a way to characterize the polytopes that $k$-tile $\mathbb{R}^d$ by translations, using
solid angle sums. Gravin, Robins, and Shiryaev~\cite[Theorem 6.1]{GravinShiryaevRobins} gave the following characterization.
\begin{thm}
A polytope $P$ $k$-tiles $\mathbb{R}^d$ by integer translations if and only if
\[
\sum_{\lambda \in \mathbb{Z}^d} \omega_{P + v}(\lambda) = k,
\]
for every $v \in \mathbb{R}^d$.
\end{thm}
\bigskip
\section{The Gram relations for solid angles}
\index{Gram relations}
How does our elementary school identity, giving us the sum of the angles of a triangle, extend to higher dimensions? We describe the extension here, mainly due to Gram. First, for each face
$F$ of a polytope ${\mathcal P} \subset \mathbb{R}^d$, we define the {\bf solid angle of} $F$,
\index{solid angle of a face}
as follows. Fix any
$x_0 \in \interior F$, and let
\[
\omega_F := \omega_P(x_0).
\]
We notice that this definition is independent of $x_0$, as long as we restrict $x_0$ to the relative interior of $F$.
\begin{example}
If ${\mathcal P}$ is the $d$-dimensional cube $[0, 1]^d$, then each of its facets $F$ has
$\omega_F = \frac{1}{2}$. However, it is a fact that for the cube, a face of dimension $k$ has a solid angle of
$\frac{1}{2^{d-k}}$ (Exercise \ref{solid angle of a face of the cube}). In particular
a vertex $v$ of this cube, having dimension $0$, has solid angle $\omega_v = \frac{1}{2^d}$.
\hfill $\square$
\end{example}
\begin{thm}[Gram relations] \label{Gram relations} \index{Gram relations}
Given any $d$-dimensional polytope $P\subset \mathbb{R}^d$, we have
\[
\sum_{F\subset {\mathcal P}} (-1)^{\dim F} \omega_F = 0.
\]
\end{thm}
\hfill $\square$
(For a proof of Lemma \ref{Gram relations} see \cite{BeckRobins}).
\begin{example}
\rm{
Let's see what the Gram relations tell us in the case of a triangle $\Delta$.
For each edge $E$ of $\Delta$, placing a small sphere at a point in the interior of $E$ means half of it is inside $\Delta$ and half of it is outside of $\Delta$, so that $\omega_E = \frac{1}{2}$. Next, each vertex of $\Delta$ has a solid angle equal to the usual (normalized) angle $\theta(v)$ at that vertex.
Finally $\Delta$ itself has a solid angle of $1$, because picking a point $p$ in the interior of $\Delta$, and placing a small sphere centered at $p$, the whole sphere will be contained in $\Delta$. Putting it all together, the Gram relations read:
\begin{align*}
0 &= \sum_{F\subset \Delta} (-1)^{\dim F} \omega_F \\
&= (-1)^0 (\theta(v_1) + \theta(v_2) +\theta(v_3) )
+ (-1)^1 \left(\frac{1}{2} + \frac{1}{2} +\frac{1}{2}\right)
+ (-1)^2 \cdot 1 \\
&= \theta(v_1) + \theta(v_2) +\theta(v_3) -\frac{1}{2},
\end{align*}
which looks familiar! We've retrieved our elementary-school knowledge, namely that the three
angles of a triangle sum to $\pi$ radians. So the Gram relations really are an extension of this fact.
}
\hfill $\square$
\end{example}
What about $\mathbb{R}^3$?
\begin{example}
\rm{
Let's see what hidden secrets lie behind the Gram relations for the standard simplex
$\Delta \subset \mathbb{R}^3$.
\index{standard simplex}
At the origin $v_0 = 0$, the tangent cone is the positive orthant,
so that $\omega(v_0) = \frac{1}{8}$. The other $3$ vertices all ``look alike'', in the sense that
their tangent cones are all isometric, and hence have the same solid angle $\omega_v$.
What about the edges?
In general, it's a fact that the solid angle of an edge equals the dihedral angle between the planes of
its two bounding facets (Exercise \ref{dihedral angle=solid angle}). There are two types of edges here, as in the figure.
For an edge $E$ which lies on the boundary of the skew facet, we have the dihedral angle
$\cos \phi = \left\langle \frac{1}{\sqrt 3}\icol{ 1\{\bf 1}\{\bf 1}}, \icol{ 0\{\bf 1}\{\bf 0}} \right\rangle = \frac{1}{\sqrt 3}$, so that
$\omega_E = \phi = \cos^{-1}\frac{1}{\sqrt 3}$. It's straightforward that for the other type of edge,
each of those $3$ edges has a solid angle of $\frac{1}{4}$. Putting it all together, we see that
\begin{align*}
0 &= \sum_{F\subset \Delta} (-1)^{\dim F} \omega_F \\
&= (-1)^0 \left( \frac{1}{8} + 3\omega_v \right)
+ (-1)^1 \left( 3 \frac{1}{4}+ 3 \cos^{-1}\frac{1}{\sqrt 3} \right)
+ (-1)^2 \frac{1}{2} \cdot 4
+ (-1)^3 \cdot 1.
\end{align*}
Solving for $\omega_v$, we get $\omega_v = \cos^{-1}\frac{1}{\sqrt 3} -\frac{1}{8}$.
So we were able to compute the solid angle of at a vertex of $\Delta$ in $\mathbb{R}^3$, using the
Gram relations, together with a bit of symmetry.
}
\hfill $\square$
\end{example}
Related to the topics above is the fact that the angle polynomial possesses the following fascinating
functional equation
(For a proof of Theorem \ref{Angle polynomial functional equation}, and an extension of it, see \cite{DesarioRobins}).
\begin{thm}[Functional equation for the angle polynomial] \label{Angle polynomial functional equation}
\index{angle polynomial: functional equation}
Given a $d$-dimensional \\
rational polytope ${\mathcal P}\subset \mathbb{R}^d$, we have
\[
A_{{\mathcal P}}(-t) = A_{{\mathcal P}}(t),
\]
for all $t \in \mathbb{Z}$.
\hfill $\square$
\end{thm}
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.angle.polynomial}
\begin{enumerate}[(a)]
\item Let's compare and contrast the two notions of discrete volumes that we have encountered so far.
For a given rational polytope ${\mathcal P}$, we notice that the Ehrhart quasi-polynomial $L_{\mathcal P}(t)$ is invariant when
we map ${\mathcal P}$ to any of its unimodular images. That is, any rational polytope in the whole orbit of the unimodular group
$\rm{SL}_d(\mathbb{Z})({\mathcal P})$ has the same discrete volume $L_{\mathcal P}(t)$. This is false for the second discrete volume $A_{\mathcal P}(t)$ - it is not invariant under the modular group
(Exercise \ref{counterexamples for angle polynomials}). But $A_{\mathcal P}(t)$ is invariant under the
large finite group of the isometries of $\mathbb{R}^d$ that preserve the integer lattice (known as the hyperoctahedral group).
So we see that $A_{\mathcal P}(t)$ is more sensitive to the particular embedding of ${\mathcal P}$ in space, because it is dependent upon a metric. It is reasonable to expect that it can distinguish between ``more'' rational polytopes, but such a question remains to be formalized.
The angle polynomial also has the advantage of being a much more symmetric polynomial, with half as many coefficients that occur
in the Ehrhart polynomial of integer polytopes.
However, $L_{\mathcal P}(t)$ has its advantages as well - to compute a \emph{local summand} for $A_{\mathcal P}(t):= \sum_{n\in\mathbb{Z}^d} \omega_{tP}(x) $ requires finding the volume of a local spherical polytope, while to compute a \emph{local summand} for $L_{\mathcal P}(t):= \sum_{n\in\mathbb{Z}^d} 1$ is quite easy: it is equal to $1$.
But as we have seen, computing the full global sum for $A_{\mathcal P}(t)$ turns out to have its simplifications.
\item The interesting undergraduate dissertation of Nhat Le Quang \cite{Nhat}, from $2010$, gives a thorough analysis of solid angle sums in $\mathbb{R}^2$, for rational polygons.
\item The recent work of Gerv\'asio \cite{GervasioSantos} gives an online implementation for the calculation of solid angles in any dimension, with open source code.
\item In \cite{RicardoNhatSinai}, there is an explicit description for some of the coefficients of the solid angle polynomial
$A_{\mathcal P}(t)$ of a $d$-dimensional polytope, for all positive real dilations $t>0$. Indeed, the approach in \cite{RicardoNhatSinai} uses the Fourier analytic landscape.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\medskip
\begin{prob}
Let $
{\mathcal K} = \{ \lambda_1
\left( \begin{smallmatrix}
1 \\ 0 \\ 0
\end{smallmatrix}
\right) +
\lambda_2
\left( \begin{smallmatrix}
1 \\ 1 \\ 0
\end{smallmatrix}
\right)
+
\lambda_3
\left( \begin{smallmatrix}
1 \\ 1 \\ 1
\end{smallmatrix}
\right) \mid \lambda_1, \lambda_2, \lambda_3 \geq 0 \},
$
a simplicial cone. Show that the solid angle of ${\mathcal K}$ is $\omega_{\mathcal K} = \frac{1}{48}$.
\end{prob}
\medskip
\begin{prob}
We recall the $2$-dimensional cross-polytope
$
\Diamond:=\left\{ \left( x_1, x_2 \right) \in \mathbb{R}^2 \mid
\, \left| x_1 \right| + \left| x_2 \right| \leq 1 \right\}.
$
Find, from first principles, the angle quasi-polynomial for the rational polygon
${\mathcal P}:= \frac{1}{3}\Diamond$,
for all integer dilations of ${\mathcal P}$.
\end{prob}
\medskip
\begin{prob}
We recall that the $3$-dimensional cross-polytope was defined by
\[
\Diamond:=\left\{ \left( x_1, x_2, x_3 \right) \in \mathbb{R}^3 \mid
\, \left| x_1 \right| + \left| x_2 \right| + \left| x_3 \right| \leq 1 \right\}.
\]
Compute the angle polynomial of $A_{\Diamond}(t)$.
\end{prob}
\medskip
\begin{prob}
We recall that the $d$-dimensional cross-polytope \index{cross-polytope}
was defined by
\[
\Diamond:=\left\{ \left( x_1, x_2, \dots, x_d \right) \in \mathbb{R}^d \mid
\, \left| x_1 \right| + \left| x_2 \right| + \cdots + \left| x_d \right| \leq 1 \right\}.
\]
Compute the angle polynomial of $A_{\Diamond}(t)$.
\end{prob}
\medskip
\begin{prob}
Let ${\mathcal P}$ be an integer zonotope. Prove that the angle polynomial of ${\mathcal P}$ is
\[
A_{{\mathcal P}}(t) = (\vol {\mathcal P})t^d,
\]
valid for all positive integers $t$.
\end{prob}
\medskip
\begin{prob}
Let ${\mathcal P}$ be a rational interval $[\frac{a}{c}, \frac{b}{d}]$. Compute the angle quasi-polynomial
$A_{{\mathcal P}}(t)$ here.
\end{prob}
\medskip
\begin{prob}
Define the rational triangle $\Delta$ whose vertices are $(0, 0), (1, \frac{N-1}{N}), (N, 0)$, where $N \geq 2$ is a fixed integer.
Find the angle quasi-polynomial $A_\Delta(t)$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{counterexamples for angle polynomials}
For each dimension $d$, find an example of an integer polytope ${\mathcal P} \subset \mathbb{R}^d$ and a unimodular matrix
$U \in \rm{SL}_d(\mathbb{Z})$, such that the angle quasi-polynomials
$A_{{\mathcal P}}(t)$ and $A_{U({\mathcal P})}(t)$ are not equal to each other for all $t \in \mathbb{Z}_{>0}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{solid angle of a face of the cube}
For the cube $\square:= [0, 1]^d$, show that any face $F\subset \square$ that has dimension $k$ has the solid angle
$\omega_F = \frac{1}{2^{d-k}}$.
\end{prob}
\medskip
\begin{prob} $\clubsuit$
\label{dihedral angle=solid angle}
Show that the solid angle $\omega_E$ of an edge E ($1$-dimensional face) of a polytope
equals the dihedral angle between the hyperplanes defined by
its two bounding facets. (Hint: use the unit normal vectors for both facets)
\end{prob}
\medskip
\begin{prob}
Using the Gram relations, namely Theorem \ref{Gram relations}, compute the solid angle at any vertex of the following regular tetrahedron:
\[
T:= \conv\Big\{ \icol{1\{\bf 0}\{\bf 0}} \icol{0\{\bf 1}\{\bf 0}}, \icol{0\{\bf 0}\{\bf 1}}, \icol{1\{\bf 1}\{\bf 1}} \Big\}.
\]
\end{prob}
\chapter{Sphere packings} \label{Sphere packings}
\index{sphere packings}
\begin{quote}
The problem of packing, as densely as possible, an unlimited number of equal nonoverlapping circles in a plane
was solved millions of years ago by the bees, who found that the best arrangement consists of circles inscribed
in the hexagons of the regular tessellation. \ -- \ H. S. M. Coxeter \index{Coxeter}
\end{quote}
\begin{quote}
There is geometry in the humming of the strings. There is music in the spacing of the spheres.
\ -- \ Pythagoras \index{Pythagoras}
\end{quote}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.6in]{HexagonalPacking}
\end{center}
\caption{A lattice sphere packing, using the hexagonal lattice, which gives the densest packing in 2 dimensions.} \label{periodic packing, Eisenstein}
\end{figure}
\section{Intuition}
The sphere packing problem traces its roots back to Kepler, and it asks for a packing
of solid spheres in Euclidean space that achieves the maximum possible density. In all of the known cases,
such optimal configurations - for the centers of the spheres - form a lattice. It's natural, therefore, that Fourier analysis comes into the picture. We prove here a result of Cohn and Elkies, from $2003$, which is a beautiful application of Poisson summation, and gives upper bounds for the maximum densities of sphere packings
in $\mathbb{R}^d$.
At this point it may be wise to define carefully all of the terms - what is a packing? what is density?
Who was Kepler?
\begin{wrapfigure}{R}{0.4\textwidth}
\centering
\includegraphics[width=0.27\textwidth]{JohannesKepler}
\caption{Johannes Kepler} \label{Kepler.pic}
\end{wrapfigure}
\section{Definitions}
A {\bf sphere packing} in $\mathbb{R}^d$ is any arrangement
of spheres of fixed radius $r>0$ such that no two interiors overlap, so we do not preclude the possibility that the
spheres may touch one another at some points on their boundary.
A {\bf lattice packing} is a sphere packing with the property that the centers of the spheres form a lattice
${\mathcal L} \subset \mathbb{R}^d$, as in Figure \ref{Sphere Packing 1}.
Relaxing this restriction - in order to allow more general packings - we define
a {\bf periodic packing} by a sphere packing with a lattice ${\mathcal L}$, together with a finite collection of its translates, say ${\mathcal L} + v_1, \dots, {\mathcal L} + v_N$, such that the differences $v_i-v_j \notin L$. This means that the centers of the spheres may be placed at any points belonging to the disjoint
union of ${\mathcal L}$, together with its $N$ translates, as in Figure \ref{periodic packing}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=1.5in]{SpherePacking1}
\end{center}
\caption{A lattice packing, with small packing density.} \label{Sphere Packing 1}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=2in]{PeriodicPacking}
\end{center}
\caption{A periodic packing with two translates of the same lattice. This packing is not a lattice packing.} \label{periodic packing}
\end{figure}
The density of any sphere packing is intuitively the proportion
of Euclidean space covered by the spheres, in an asymptotic sense, but rather than go into these technical
asymptotic details,
we will simply define a density function for lattice packings and for general periodic packings, as follows.
Given a lattice packing, with the lattice ${\mathcal L} \subset \mathbb{R}^d$, and with spheres of radius $r$,
we define its {\bf lattice packing density} by
\begin{equation}
\Delta({\mathcal L}) := \frac{ \vol B^d(r) }{\det {\mathcal L}},
\end{equation}
where $B^d(r) $ is a ball of radius $r$. This lattice packing corresponds to placing a sphere of
radius $r$ at each lattice point of ${\mathcal L}$, guaranteeing that the spheres do not overlap.
\begin{wrapfigure}{R}{0.36\textwidth}
\centering
\includegraphics[width=0.22\textwidth]{IntegerLatticePacking}
\caption{The densest sphere packing for the lattice $\mathbb{Z}^2$, with a packing density of
$\frac{\pi}{4} \approx .785$, which means that about $78.5 \%$ of the plane is covered by this configuration of balls.}
\label{densest integer lattice packing}
\end{wrapfigure}
\begin{example}
\rm{
Consider the integer lattice ${\mathcal L}:= \mathbb{Z}^2$. It is clear that we can place a sphere of radius $r = \frac{1}{2}$ at each integer point, so that we have packing, and it is also clear that any larger radius for our spheres will not work with this lattice (see Figure \ref{densest integer lattice packing}). So this particular packing gives us a sphere packing density of
\[
\frac{ \vol B^2(r) }{\det {\mathcal L}}:= \frac{ \frac{\pi}{4} }{ \det \mathbb{Z}^2 } = \frac{\pi}{4}.
\]
}
\hfill $\square$
\end{example}
More generally, given a period packing with a lattice ${\mathcal L}$ and a set of translates $v_1, \dots, v_N$,
we define its {\bf periodic packing density} by
\begin{equation} \label{periodic packing density}
\Delta_{periodic}({\mathcal L}) := \frac{ N \vol B^d(r) }{\det {\mathcal L}},
\end{equation}
corresponding to placing a sphere of radius $r$ at each point of ${\mathcal L}$, and also at each point
of its translates ${\mathcal L} + v_1, \dots, {\mathcal L} + v_N$. It's not hard to prove that the latter definition
\ref{periodic packing density} matches our intuition that any fixed fundamental parallelepiped of ${\mathcal L}$
intersects this configuration of spheres in a set whose measure is exactly
$N \vol B^d(r) $ (Exercise \ref{volume of periodic packing}).
Henceforth, we use the words `packing density' to mean `periodic packing density', and we always restrict attention to periodic packings - see the Notes for technical remarks involving any sphere arrangement, and why periodic packings are sufficient.
We define the {\bf sphere packing problem} as follows:
\begin{question}
What is the maximum possible packing density, in any periodic packing of spheres?
\end{question}
In other words, the problem asks us to find the maximum density
$\Delta_{periodic}{{\mathcal L}}$, among all lattices ${\mathcal L}$, allowing also any finite collection of translates of ${\mathcal L}$. The sphere packing
problem also asks us to find, if possible, the lattice ${\mathcal L}$ that achieves this optimal density.
Many other questions naturally arise:
\begin{question} \label{lattice or a few lattices?}
Is the densest sphere packing always achieved by using just one lattice, in each dimension $d$?
\end{question}
In other words, are there dimensions $d$ for which we in fact need to use some translates of a lattice?
\begin{question} \label{unique lattice?}
If the answer to Question \ref{lattice or a few lattices?} is affirmative, then is such an optimal lattice unique in each dimension?
\end{question}
The only dimensions $d$ for which we know the answers to Question \ref{lattice or a few lattices?} and
Question \ref{unique lattice?} are $d=1, 2, 3, 8, 24$, and in these known cases the answer is affirmative.
The sphere packing problem is a very important
problem in Geometry, Number theory, Coding theory, and information theory.
\bigskip
\section{The volume of the ball, and of the sphere}
\label{Volume of the ball, the Gamma function}
To warm up, we compute volumes of $d$-dimensional balls and spheres. For these very classical computations, we need the Gamma function:
\begin{equation}
\Gamma(x):= \int_0^\infty e^{-t} t^{x-1} dt,
\end{equation}
valid for all $x>0$. The Gamma function $\Gamma(x)$ interpolates smoothly between the integer values of the factorial function $n!$, in the following sense.
\begin{lem} \label{Gamma properties}
Fix $x>0$. Then
\begin{enumerate}[(a)]
\item $\Gamma(x+1) = x \Gamma(x)$.
\item $\Gamma(n+1) = n!$, for all nonnegative integers $n$.
\item $\Gamma\Big( \frac{1}{2} \Big) = \sqrt \pi$.
\item $\Gamma$ extends to an infinitely smooth function on the complex plane, except at $0$ and at the
negative integers, where it has simple poles.
\end{enumerate}
\end{lem}
The verifications of parts (a), (b), and (c) are good exercises (Exercise \ref{prove Gamma properties}), and we don't want to deprive the reader of that pleasure. Part (d) requires some knowledge of complex analysis, but we include the statement here for general knowledge.
What is the volume of the unit ball $B:= \left\{ x \in \mathbb{R}^d \mid \|x\| \leq 1 \right\}$? And what about the volume of the unit sphere $S^{d-1} := \left\{ x\in \mathbb{R}^d \mid \| x \| \right\}= 1 \|$?
\begin{lem} \label{lem:volume of ball and sphere}
For the unit ball $B$, and unit sphere $S^{d-1}$, we have:
\begin{equation} \label{volume of ball and sphere}
\vol B = \frac{ \pi^{\frac{d}{2}} }{\Gamma\left(\frac{d}{2} +1\right)}, \text{ and }
\vol \left(S^{d-1}\right) = \frac{2 \pi^{\frac{d}{2}} }{ \Gamma\left(\frac{d}{2}\right)}.
\end{equation}
\end{lem}
\begin{proof}
We let $\kappa_{d-1}:= \vol(S^{d-1})$ denote the surface area of the unit sphere
$S^{d-1} \subset \mathbb{R}^d$. We use polar coordinates in $\mathbb{R}^d$, meaning that we may write each $x \in \mathbb{R}^d$ in the form $x = (r, \theta)$, where $r>0$ and $\theta \in S^{d-1}$. Thus $\|x\| = r$, and we also have the calculus fact that $dx = r^{d-1} dr d\theta$.
Returning to our Gaussians $e^{-\pi \|x\|^2}$, we recompute their integrals using polar coordinates in $\mathbb{R}^d$:
\begin{align*}
1=\int_{\mathbb{R}^d} e^{-\pi \|x\|^2} dx &= \int_{S^{d-1} }
\int_0^{\infty} e^{-\pi r^2} r^{d-1} dr \, d\theta \\
&=\kappa_{d-1} \int_0^{\infty} e^{-\pi r^2} r^{d-1} dr \\
&= \kappa_{d-1} \frac{1}{ 2\pi^{\frac{d}{2}} }
\int_0^{\infty} e^{- t} t^{\frac{d}{2} -1} dt \\
\end{align*}
where we've used $t:= \pi r^2$, implying that
$r^{d-1} dr = r^{d-2} r dr =
\Big(\frac{t}{\pi} \Big)^{\frac{d-2}{2}} \frac{dt}{2\pi} $. Recognizing the latter integral as
$\Gamma\left(\frac{d}{2}\right)$, we find that
$1= \frac{ \kappa_{d-1} } { 2\pi^{\frac{d}{2}} } \Gamma\left(\frac{d}{2}\right)$, as desired.
For the volume of the unit ball $B$, we have:
\[
\vol B = \int_0^1 \kappa_{d-1} r^{d-1} dr = \frac{\kappa_{d-1}}{d}
= \frac{\pi^{\frac{d}{2}}}{\frac{d}{2} \Gamma\left(\frac{d}{2}\right) }
= \frac{\pi^{\frac{d}{2}}}{ \Gamma\left(\frac{d}{2} +1\right) }.
\]
\end{proof}
It is easy, but worth mentioning (Exercise \ref{explicit volume for ball and sphere}), that we may also
rewrite the formulas \eqref{volume of ball and sphere} in terms of ratios of factorials, by using the recursive properties of the $\Gamma$ function.
While we are at it, let's dilate the unit ball by $r>0$, and recall our definition of the ball of radius $r$:
\[
B^d(r):= \left\{ x \in \mathbb{R}^d \mid \|x\| \leq r \right\}.
\]
We know that for any $d$-dimensional body $K$, we have $\vol(rK) = r^d \vol K$, so we also get the volumes of the ball of radius $r$, and the sphere of radius $r$:
\begin{equation} \label{dilated volumes of balls and spheres}
\vol B^d(r) = \frac{ \pi^{\frac{d}{2}} }{\Gamma\left(\frac{d}{2} +1\right)} r^d , \text{ and }
\vol \left(r S^{d-1}\right) = \frac{2 \pi^{\frac{d}{2}} }{ \Gamma\left(\frac{d}{2}\right)} r^{d-1}.
\end{equation}
Intuitively, the derivative of the volume is the surface area, and now we can confirm this intuition:
\[
\frac{d}{dr} \vol B^d(r) = \frac{ d \pi^{\frac{d}{2}} }{\Gamma\left(\frac{d}{2} +1\right)} r^{d-1}
=\frac{ 2\frac{d}{2} \pi^{\frac{d}{2}} }{\frac{d}{2} \Gamma\left(\frac{d}{2} \right)} r^{d-1}
=\frac{ 2 \pi^{\frac{d}{2}} }{\Gamma\left(\frac{d}{2}\right)} r^{d-1} =\vol \left(r S^{d-1}\right).
\]
\bigskip
\section{The Fourier transform of the ball}
Whenever considering packing or tiling by a convex body $B$, we have repeatedly seen that taking the Fourier transform of the body, namely
$\hat 1_B$, is very natural, especially from the perspective of Poisson summation.
It's also very natural to consider the FT of a ball in $\mathbb{R}^d$.
To compute the Fourier transform of $1_{B(r)}$, a very classical computation, we first define
the \textbf{Bessel function} \index{Bessel function}
$J_p$ of order $p$ (\cite{EpsteinBook}, page 147), which comes up naturally here:
\begin{equation}\label{Bessel definition}
J_p(x) :=\left(\frac{x}{2} \right)^p \frac{1}{\Gamma\left(p + \frac12\right)\sqrt{\pi}}\int_0^\pi e^{ix\cos\varphi} \sin^{2p}(\varphi) \, d\varphi,
\end{equation}
valid for $p>-\frac{1}{2}$, and all $x \in \mathbb{R}$. We call a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ {\bf radial} \index{radial function}
if it is invariant under all rotations of $\mathbb{R}^d$. In other words, we have the definition
\[
f \text{ is radial } \iff f \circ M = f,
\]
for all $M \in SO_d(\mathbb{R})$, the orthogonal group. Another way of describing a radial function is to say that the function $f$ is constant on each sphere that is centered at the origin, so that a radial function only depends on the norm of its input: $f(x) = f(\|x\|)$, for all $x\in \mathbb{R}^d$.
A very useful fact in various applications of
Fourier analysis (in particular medical imaging) is that the Fourier transform of a radial function is again a radial function (Exercise \ref{radial function transform}).
\bigskip
\begin{lem} \label{FT of the ball}
The Fourier transform of $B^d(r)$, the ball of radius $r$ in $\mathbb{R}^d$ centered at the origin, is
\[\hat{1}_{B^d(r)}(\xi) := \int_{B^d(r)} e^{-2\pi i \langle \xi, x \rangle } dx = \left(\frac{r}{\| \xi\|}\right)^{d/2}J_{d/2}\big(2\pi r\|\xi\|\big).\]
\end{lem}
\begin{proof}
Taking advantage of the inherent rotational symmetry of the ball, and also using the fact that the Fourier transform
of a radial function is again radial (Exercise \ref{radial function transform}), we have:
\[
\hat{1}_{B^d(r)}(\xi) = \hat{1}_{B^d(r)}(0, \dots, 0, \|\xi\|),
\]
for all $\xi \in \mathbb{R}^d$. With $r=1$ for the moment, we therefore have:
\[
\hat{1}_{B}(\xi) =
\int_{\| x\| \leq 1} e^{-2\pi i x_d\|\xi\|}\, dx_1\, \dotsc\, dx_d,
\]
Now we note that for each fixed $x_d$, the function being integrated is constant and the integration domain for the
variables $x_1, \dots, x_{d-1}$ is a $(d-1)$-dimensional ball of radius $(1-x_d^2)^{1/2}$.
By equation \eqref{volume of ball and sphere}, the
volume of this ball is $(1-x_d^2)^{\frac{d-1}{2}} \frac{ \pi^{\frac{d-1}{2}} }{ \Gamma\left(\frac{d+1}{2} \right) }$, we have
\[
\hat{1}_{B}(\xi)
= \frac{ \pi^{\frac{d-1}{2}} }{ \Gamma(\frac{d+1}{2}) }
\int_{-1}^1
e^{-2\pi i x_d\|\xi\|}
(1-x_d^2)^{\frac{d-1}{2}} \,dx_d
= \frac{ \pi^{ \frac{d}{2} }}{\sqrt{\pi}\Gamma\left(\frac{d+1}{2}\right)}
\int_0^\pi
\, e^{2\pi i\|\xi\|\cos\varphi}
\sin^d\varphi \,d\varphi.
\]
Using the definition~\eqref{Bessel definition} of the $J$-Bessel function, we get
\[\hat{1}_{B}(\xi) = \| \xi\|^{-\frac{d}{2} }J_{\frac{d}{2} }\big(2\pi\|\xi\|\big),\]
and consequently
\[\hat{1}_{B^d(r)}(\xi) = \left(\frac{r}{\| \xi\|}\right)^{\frac{d}{2} }J_{\frac{d}{2} }\big(2\pi r\|\xi\|\big).
\qedhere\]
\end{proof}
\bigskip
\begin{example}\label{ex:integral using Bessel functions}
\rm{
Using the $J$-Bessel functions, let's work out the following explicit evaluation of the following interesting
integrals, for all $p>0$:
\begin{equation} \label{identity of J-Bessel example}
\int_0^\pi \sin^{2p}(\varphi) \, d\varphi =
\sqrt \pi \frac{ \Gamma\left(p + \frac12\right) }{ \Gamma\left(p + 1 \right)}.
\end{equation}
Whenever we raise a negative real number to an arbitrary real exponent, some care has to be taken to avoid `branch problems' with the definition of exponentiation. Here we can easily avoid such problems by defining $\sin^{2p}(x) := \left( \sin^2(x)\right)^p$, so that we are always exponentiating a nonnegative real number, and everything is copacetic. We will use the following equivalent formulation for the $J_p$ Bessel function in terms of a hypergeometric series:
\begin{equation} \label{Bessel infinite series}
J_p(x) = \frac{x^p}{2^p} \sum_{k=0}^\infty (-1)^k \frac{x^{2k} }{2^{2k} k! \, \Gamma(p+k+1) }.
\end{equation}
(\cite{EpsteinBook}, p. 684).
Using the definition of the Bessel function \eqref{Bessel definition}, we can rewrite it slightly:
\begin{equation}\label{fancy integral 1}
\frac{J_p(x)}{x^p} 2^p \sqrt{\pi} \, \Gamma\left(p + \frac12\right)
=
\int_0^\pi e^{ix\cos\varphi} \sin^{2p}(\varphi) \, d\varphi.
\end{equation}
Taking the limit as $x\rightarrow 0$, we can safely move this limit inside the integral in \eqref{fancy integral 1}
because we are integrating a differentiable function over a compact interval:
\begin{align*}
\int_0^\pi \sin^{2p}(\varphi) \, d\varphi
&= \lim_{x\rightarrow 0}
\frac{J_p(x)}{x^p} 2^p \sqrt{\pi} \, \Gamma\left(p + \frac12\right).
\end{align*}
So if we knew the asymptotic limit $ \lim_{x\rightarrow 0} \frac{J_p(x)}{x^p}$, we'd be in business.
From \eqref{Bessel infinite series}, we may divide both sides by $x^p$, and then take the limit as $x\rightarrow 0$ to obtain the constant term of the remaining series, giving us
\[
\lim_{x\rightarrow 0} \frac{J_p(x)}{x^p} = \frac{1}{2^p \Gamma(p+1)}.
\]
Altogether, we have
\begin{align*}
\int_0^\pi \sin^{2p}(\varphi) \, d\varphi
&= \lim_{x\rightarrow 0}
\frac{J_p(x)}{x^p} 2^p \sqrt{\pi} \, \Gamma\left(p + \frac12\right) \\
&= \frac{1}{2^p \Gamma(p+1)} 2^p \sqrt{\pi} \, \Gamma\left(p + \frac12\right) \\
&= \sqrt \pi \frac{ \Gamma\left(p + \frac12\right) }{ \Gamma\left(p + 1 \right)},
\end{align*}
valid for all $p>0$.
}
In the special case that $p$ is a positive integer, the latter identity can of course be written in terms of a ratio of factorials (Exercise \ref{the integral of the example in terms of factorials}).
\hfill $\square$
\end{example}
\bigskip
\bigskip
\section{Upper bounds for sphere packings via Poisson summation}
Here we give an exposition of the ground-breaking result of Henry Cohn and Noam Elkies on the sphere packing problem.
This result sets up the machinery for finding certain {\bf magical functions} $f$, as defined
in Theorem \ref{Cohn-Elkies} below,
that allow us to give precise upper bounds on
$\Delta_{periodic}{{\mathcal L}}$. The main tool is Poisson summation again, for arbitrary lattices. We recall that we defined a function $f$ to be \emph{nice} if $f$ satisfies the Poisson summation formula
\[
\sum_{n \in {\mathcal L}} f(n+v) = \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) e^{2\pi i \langle v, \xi \rangle},
\]
pointwise for all $v\in \mathbb{R}^d$.
\begin{thm}[Cohn-Elkies] \label{Cohn-Elkies}
Let $f:\mathbb{R}^d \rightarrow \mathbb{R}$ be a nice function, not identically zero, which enjoys the following
three conditions:
\begin{enumerate}
\item $f(x) \leq 0$, for all $\|x\| \geq r$. \label{condition 1}
\item $\hat f(\xi) \geq 0$, for all $\xi \in \mathbb{R}^d$. \label{condition 2}
\item $f(0) >0$, and $\hat f(0) > 0$. \label{condition 3}
\end{enumerate}
Then the periodic packing density of any $d$-dimensional sphere packing has the upper bound
\[
\Delta_{periodic}({\mathcal L}) \leq \frac{ f(0) }{ \hat f(0) } \vol B^d(r) .
\]
\end{thm}
\begin{proof}
Suppose we have a periodic packing with spheres of radius $r$, a lattice ${\mathcal L}$, and translation vectors
$v_1, \dots, v_N$, so that by definition the packing density is
$\Delta_{periodic}({\mathcal L}) := \frac{ N \vol B^d(r) }{\det {\mathcal L}}$.
By Poisson summation, \index{Poisson summation formula}
we have
\begin{equation}
\sum_{n \in {\mathcal L}} f(n+v) = \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) e^{2\pi i \langle v, \xi \rangle},
\end{equation}
converging absolutely for all $v \in \mathbb{R}^d$. Now we form the following finite sum and rearrange
the right-hand-side of Poisson summation:
\begin{align}\label{fancy Poisson}
\sum_{1\leq i \leq j \leq N}
\sum_{n \in {\mathcal L}} f(n+v_i - v_j) &= \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) \sum_{1\leq i \leq j \leq N} e^{2\pi i \langle v_i - v_j, \xi \rangle} \\
&= \frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) \Big| \sum_{1\leq k \leq N} e^{2\pi i \langle v_k, \xi \rangle} \Big|^2.
\label{RHS of Poisson}
\end{align}
Now, every summand on the right-hand-side of \eqref{RHS of Poisson} is nonnegative, because by the second
assumption of the Theorem, we have
$\hat f (\xi) \geq 0$, so that the whole series can be bounded from below by its constant term, which for $\xi = 0$
gives us the bound $ \frac{ \hat f(0) N^2}{\det {\mathcal L}}$.
On the other hand, let's ask what the positive contributions are, from the left-hand-side of \eqref{fancy Poisson}.
Considering the vectors $n+v_i - v_j$ on the left-hand-side of \eqref{fancy Poisson},
suppose we have $\| n+v_i - v_j \| \geq r$. Then the first hypothesis of the Theorem guarantees
that $f(n+v_i - v_j ) \leq 0 $. So we may restrict attention to those vectors that satisfy $\| n+v_i - v_j \| < r$. Here
the vector $n+v_i - v_j$ is contained in the sphere of radius $r$, centered at the origin, but this means (by the packing assumption) that it must be the zero vector: $n + v_i - v_j = 0$. By assumption, the difference between any two translations $v_i-v_j$ is never a nonzero element of ${\mathcal L}$, so we have $i=j$, and now $v_i = v_j \implies n=0$. We conclude that the only positive contribution from the left-hand-side of \eqref{fancy Poisson} is the $n=0$ term, and so the
left-hand-side of \eqref{fancy Poisson} has an upper bound of $N f(0) >0$.
Altogether, Poisson summation gave us the bound:
\[
N f(0)
\geq
| \sum_{1\leq i \leq j \leq N} \sum_{n \in {\mathcal L}} f(n+v_i - v_j) |
=
\frac{1}{\det {\mathcal L}} \sum_{\xi \in {\mathcal L}^*}
\hat f(\xi) \Big| \sum_{1\leq k \leq N} e^{2\pi i \langle v_k, \xi \rangle} \Big|^2
\geq
\frac{ \hat f(0) N^2}{\det {\mathcal L}}.
\]
Simplifying, we have
\[
\frac{ f(0) }{ \hat f(0) } \geq \frac{N}{\det {\mathcal L}} :=
\frac{ \Delta_{periodic}({\mathcal L})}{ \vol B^d(r) }.
\]
\end{proof}
\bigskip
\begin{example} [The trivial bound]
\rm{
Let ${\mathcal L}$ be a full-rank lattice in $\mathbb{R}^d$, whose shortest nonzero vector has length $r>0$.
We define the function
\[
f(x):= 1_{K}(x) * 1_{K}(x),
\]
where $K$ is the ball
of radius $r$, centered at the origin.
We claim that $f$ satisfies all of the conditions of
Theorem \ref{Cohn-Elkies}. Indeed, by the convolution Theorem,
\[
\hat f(\xi) = \widehat{\left(1_{K} * 1_{K}\right)}(\xi)
= \Big( \hat 1_{K}(\xi) \Big)^2 \geq 0,
\]
for all $\xi \in \mathbb{R}^d$, verifying condition \ref{condition 2}. Condition \ref{condition 1} is also easy to verify, because the support of $f$ is equal to the Minkowski sum (by Exercise \ref{support of convolution})
\index{Minkowski sum}
$K + K = 2K$, a sphere of radius $2r$.
It follows that $f$ is identically zero outside a sphere of radius $2r$. For condition \ref{condition 3}, by the definition of convolution we have $f(0) =
\int_{\mathbb{R}^d} 1_{K}(0-x) 1_{K}(x)dx = \int_{\mathbb{R}^d} 1_{K}(x)dx =
\vol K >0$. Finally, $\hat f(0) = \Big( \hat 1_{K}(0) \Big)^2 =
\vol^2( K) > 0$.
By the Cohn-Elkies Theorem \ref{Cohn-Elkies}, we know that the packing density of such a lattice is therefore bounded above by
\[
\frac{ f(0) }{ \hat f(0)} \vol B^d(r)
= \frac{\vol K }{ \vol^2( K)} \vol K = 1,
\]
the trivial bound. So we don't get anything interesting, but all this tells us is that our particular choice of function $f$ above was a poor choice, as far as density bounds are concerned. We need to be more clever in picking our magical $f$.
}
\hfill $\square$
\end{example}
Although it is far from trivial to find magical functions $f$ that satisfy the hypothesis of the
Cohn-Elkies Theorem, and simultaneously give a strong upper bound,
there has been huge success recently in finding exactly such functions - in dimensions $8$ and $24$. These recent magical functions gave the densest sphere packings in these dimensions, knocking off the whole sphere packing problem in dimensions $8$ and $24$.
It also turns out that if we have a magical function $f$ that enjoys all three
hypotheses of the Cohn-Elkies
Theorem \ref{Cohn-Elkies}, then $f\circ \sigma$ also satisfies the same hypotheses,
for any $\sigma \in SO_d(\mathbb{R})$ (Exercise \ref{invariance of magical functions under orthogonal transformations}).
We may therefore take certain radial functions as candidates for magical functions.
This exciting story continues today, and we mention some of the recent spectacular applications of the
Cohn-Elkies Theorem, initiated recently by
Maryna Viazovska for $\mathbb{R}^8$, and then extended by a large joint effort from
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska,
for $\mathbb{R}^{24}$ \cite{Cohn.etal}.
Here is a synopsis of some of their results.
\begin{thm}
The lattice $E_8$ is the densest periodic packing in $\mathbb{R}^8$. The Leech lattice is the densest
periodic packing in $\mathbb{R}^{24}$. In addition, these lattices are unique,
in the sense that there do not exist any
other periodic packings that achieve the same density.
\end{thm}
At the moment, the provably densest packings are known only in dimensions
$1, 2, 3, 8$, and $24$. Each dimension seems to require slightly different methods, and sometimes wildly different methods,
such as $\mathbb{R}^3$.
For $\mathbb{R}^3$, the sphere packing problem was solved by Hales,
and before Hales' proof, it was an open problem since the time of Kepler.
Somewhat surprisingly, the sphere packing problem is still open in all other dimensions.
In $\mathbb{R}^4$, it is very tempting and natural to think of the lattice $D_4$
as a possible candidate for the densest lattice sphere packing in $\mathbb{R}^4$, but this is still unknown.
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.SpherePackings}
\begin{enumerate}[(a)]
\item Each dimension $d$ appears to have a separate theory for sphere packings. This intuition is sometimes tricky to conceptualize, but there are facts that help us do so. For example, it is a fact that the Gram matrix (see \ref{Gram matrix positive semidefinite}) of a lattice ${\mathcal L} \subset \mathbb{R}^d$ consists entirely of integers, with even diagonal elements $\iff \ d$ is divisible by $8$.
For this reason, it turns out that the theta series of a lattice possesses certain functional equations (making it a modular form) if and only if $8 \mid d$, which in turn allows us to build some very nice related `magical' functions $f$ that are sought-after in Theorem \ref{Cohn-Elkies}, at least for $d=8$ and $d=24$ so far.
In dimension $2$, it is an open problem to find such magical functions, even though we have an independent proof that the hexagonal lattice is the optimal sphere packing lattice.
\item Johannes Kepler (1571 --1630) \index{Kepler, Johannes}
was a German astronomer and mathematician.
Kepler's laws of planetary motion motivated Sir Isaac Newton to develop further the theory of gravitational attraction and planetary motion. Kepler conjectured that the densest packing of sphere is given by the
``face-centered cubic'' packing. It was Gauss (1831) \index{Gauss}
who first proved that, if we assume the packing to be a lattice packing, then Kepler's conjecture is true.
In $1998$ Thomas Hales (using an approach initiated by L. Fejes T\'oth (1953)), gave an unconditional proof of the Kepler conjecture.
\item It is also possible, of course, to pack other convex bodies. One such variation is to pack regular tetrahedra in $\mathbb{R}^3$.
The interesting article by Jeffrey Lagarias and Chuanming Zong \cite{LagariasZong} gives a nice account of this story.
\item Regarding lower bounds for the optimal density of sphere packings, Keith Ball \cite{KeithBall.1} discovered the following lower bound in all dimensions:
\[
\Delta_{periodic}({\mathcal L}) \geq \frac{(n-1)}{2^{n-1}} \zeta(n),
\]
where $\zeta(s)$ is the Riemann zeta function.
Akshay Venkatesh \cite{Venkatesh} has given an improvement over the known lower bounds by a multiplicative constant. For all sufficiently large dimensions, this improvement is by a factor of at least $10, 000$.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
``It is better to do the right problem the wrong way, than the wrong problem the right way.''
-- Richard Hamming \index{Hamming, Richard}
\end{quote}
\medskip
\begin{prob} \label{explicit volume for ball and sphere}
Using Lemma \ref{lem:volume of ball and sphere},
show that for the unit ball $B$ and unit sphere $S^{d-1}$ in $\mathbb{R}^d$, we have:
\begin{enumerate}[(a)]
\item
\[
\vol S^{d-1} =
\begin{cases}
\frac{ \left(2 \pi\right)^{\frac{d}{2}} }{ 2\cdot 4 \cdot 6 \cdots (d-2) }, & \text{if } d \text{ is even}, \\
\frac{ 2 \left(2 \pi\right)^{\frac{d-1}{2}} }{ 1\cdot 3 \cdot 5 \cdots (d-2) }, & \text{if } d \text{ is odd}.
\end{cases}
\]
\item
\[
\vol B =
\begin{cases}
\frac{ \left(2 \pi\right)^{\frac{d}{2}} }{ 2\cdot 4 \cdot 6 \cdots d }, & \text{if } d \text{ is even}, \\
\frac{ 2 \left(2 \pi\right)^{\frac{d-1}{2}} }{ 1\cdot 3 \cdot 5 \cdots d }, & \text{if } d \text{ is odd}.
\end{cases}
\]
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{volume of periodic packing}
Given a periodic lattice packing, by $N$ translates of a lattice ${\mathcal L} \subset \mathbb{R}^d$, show that any fixed fundamental
parallelepiped of ${\mathcal L}$
intersects the union of all the spheres in a set of measure $N \vol B^d(r) $, where $r:= \frac{1}{2}\lambda_1({\mathcal L})$.
Thus, we may compute the density of a periodic sphere packing by just considering the
portions of the spheres that lie in one fundamental parallelepiped.
\end{prob}
\medskip
\begin{prob} Here we show that the integer lattice is a very poor choice for sphere packing.
\begin{enumerate}[(a)]
\item Compute the packing density of the integer lattice $\mathbb{Z}^d$.
\item Compute the packing density of the lattices $D_3$ and $D_4$.
\item Compute the packing density of the lattices $D_n$, for $n\geq 5$.
\end{enumerate}
\end{prob}
\medskip
\begin{prob} \label{radial function transform}
If $f\in L^1(\mathbb{R}^d)$ is a radial function, prove that its Fourier transform $\hat f$
is also a radial function.
\end{prob}
\medskip
\begin{prob}
Suppose we pack equilateral triangles in the plane, by using only translations of a fixed equilateral triangle.
What is the maximum packing density of such a packing? Do you think it may be the worst possible density among translational packings of any convex body in $\mathbb{R}^2$?
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{prove Gamma properties}
Prove the elementary properties of the $\Gamma$ function, in Lemma \ref{Gamma properties}.
\end{prob}
\medskip
\begin{prob} \label{invariance of magical functions under orthogonal transformations}
Show that if we have a magical function $f$ that enjoys all $3$ hypotheses of Theorem~\ref{Cohn-Elkies},
then $f\circ \sigma$ also satisfies the same hypotheses, for any orthogonal transformation $\sigma \in SO_d(\mathbb{R})$.
\end{prob}
\medskip
\begin{prob}
We define a rigid motion of a compact set $K$ to be any orthogonal transformation of $K$, composed with any translation of $K$.
\begin{enumerate}[(a)]
\item When $d=1$, find a (nontrivial) continuous function $f:\mathbb{R} \rightarrow \mathbb{C}$ such that:
\[
\int_c^{c+R} f(x) dx = 0,
\]
for all constants $c, R>0$.
\item \label{ex:part b, for Pompeiu}
More generally, in any dimension $d$, find a (nontrivial) continuous function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ that allows the following integrals (taken over any ball of radius $r$) to vanish:
\[
\int_{B^d(r)+c} f(x) dx = 0,
\]
for all constants $c, R>0$.
\end{enumerate}
Notes. For part \ref{ex:part b, for Pompeiu},
it's advisable to think about the Fourier transform of the ball.
It is conjectured that for any bounded set $K$ with nonempty interior, the balls in this example are the only examples of objects that allow such nonzero continuous functions $f$ to exist. This is known as the {\bf Pompeiu problem} - see also Question \ref{Pompeiu conjecture}.
\end{prob}
\begin{prob} \label{the integral of the example in terms of factorials}
Show that when $p$ is a positive integer, the identity
\eqref{identity of J-Bessel example} of Example \ref{ex:integral using Bessel functions} simplifies to a ratio of factorials.
\end{prob}
\medskip
\begin{prob}
\rm{ [hard-ish]
\label{positive FT over R^d}
Using the idea of Exercise \ref{positive FT over R} in Chapter \ref{Fourier analysis basics},
and using the sum of two
indicator functions of balls (with incommensurable radii) in $\mathbb{R}^d$,
show that there exists a compactly supported function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ such that
\[
\hat f(\xi) >0,
\]
for all $\xi \in \mathbb{R}^d$.
}
\end{prob}
\bigskip
\chapter{Shannon sampling, in one and several dimensions}
\label{Chapter:Shannon sampling}
\begin{quote}
``It is easy to argue that real signals must be band-limited. \\
\quad It is also easy to argue that they cannot be so.''
-- David Slepian \index{Slepian, David}
\end{quote}
Sampling theory consists in the reconstruction of a continuous function with only a discrete or finite amount of data and has many applications in signal processing and other engineering applications. At a first glance this task sounds impossible, however it can be done well in practice. One of the reasons for this success comes from the Fourier analysis, which deals with the representation of a function in terms of its ``frequencies", and functions without high frequencies (bandlimited) represents very well the real-world signals.
In one dimension, the classical example is a sound signal, and since typical humans can only hear sounds with frequencies smaller than $20$ kHz, the bandlimited assumption is appropriate.
Examples in higher dimensions include images or MRI exams where higher frequencies are associated with random noises and measurement errors, more connected to the physical apparatus than the object being measured \cite{EpsteinBook}. In this sense it is even desirable to remove the high
frequency information.
More recently, the interest in bandlimited functions increased in the machine learning community, because
it was observed that neural networks learns low frequencies faster and this might explain why they often generalize quickly from the training sets.
On the other hand, by a basic uncertainty principle of Theorem \ref{basic uncertainty principle}, we know that a function with compact support can never be bandlimited, so representing an arbitrary function using
this class of functions is in general not exact. It is therefore desirable to also give
some theoretical results concerning the error of such approximations.
Here we introduce the classical sampling theorem by Shannon and Whittaker
for one dimensional sampling, and then we study some of its generalizations to higher dimensions,
where much less is known.
An excellent introduction to Sampling Theory, from an expository as well as a rigorous perspective,
is the book of J. R. Higgins \cite{Higgins1996}.
\bigskip
\section{The Shannon-Whittaker sampling Theorem}
\label{sec:one-dimensional-stuff}
Claude Shannon~\cite{Shannon1} showed how to reconstruct a complete signal $f$ by sampling it only discretely,
in a classical paper that gave rise to the field of information theory.
To accomplish this, Shannon used an interesting assumption, namely that the Fourier transform of $f$
vanishes outside of some interval.
One of the main characters of this story is our old friend, the ${\rm{sinc}}$ function:
\begin{equation}\label{eq:sinc-def}
{\rm{sinc}}(x):= \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{2\pi i \xi x} d\xi = \begin{cases}
\frac{\sin(\pi x)}{\pi x}, &\mbox{if } x \not= 0 \\
1 & \mbox{if } x= 0,
\end{cases}
\end{equation}
which plays a central role in the sampling theory for functions in $\mathbb{R}$, because it turns out to be a building block for a basis of the Paley-Wiener space $PW_c$, as the Shannon-Whittaker sampling theorem shows.
Reviewing some of the Fourier facts that we learned in Chapter \ref{Fourier analysis basics},
we recall that if $f \in L^1(\mathbb{R})$, then $\hat f$ is uniformly continuous and $\hat f(\xi) \to 0$ as $|\xi| \to \infty$.
So not every function can be the Fourier transform of some other function in $L^1(\mathbb{R})$.
In practice, we are often interested in functions that are not absolutely integrable, and yet possess a (conditionally convergent) Fourier transform, such as the important ${\rm{sinc}}$ function. To resolve this issue, the theory progresses by first defining the transform in the space $L^1(\mathbb{R})$, and then extending the definition of $\hat f$ to all of $L^2(\mathbb{R})$, by taking the limit $\lim_{n \to \infty} \int_{|x| < n}f(x)e^{-2\pi i x \xi}dx$. This unique extension of the Fourier transform,
from the $L^1(\mathbb{R})$ space to the $L^2(\mathbb{R})$ space, is sometimes called the Plancherel-Fourier transform. From now on, we'll follow the usual Fourier convention and simply call both transforms ``the Fourier transform''.
For a given number $c>0$, a function $f\in L^2(\mathbb{R})$ is called {\bf c-bandlimited} if
\[
\hat{f}(x) = 0 \text{ for all } x \not\in [-c, c].
\]
We will sometimes just say `bandlimited' if the $c$ is not contextually important. A bandlimited function $f$ has a Fourier transform that decays at the `best possible rate', in the sense that its Fourier transform is identically zero outside the interval $[-c, c]$.
It's easy to notice that any $c$-bandlimited function $f$ must be equal (almost everywhere)
to an infinitely smooth
function, because by Fourier inversion, we have:
\begin{equation}\label{eq:bandlimit-inversion}
f(x) = \int_{\mathbb{R}} \hat f(\xi) e^{2\pi i \xi x} d\xi
=\int_{-c}^c \hat f(\xi) e^{2\pi i \xi x} d\xi.
\end{equation}
This identity implies that we can differentiate the last expression with respect to $x$ as many times as we like under the integral sign, because the integrand is a smooth function of $x$, and we are integrating over a compact domain. Therefore $f$ is infinitely smooth.
For simplicity, when considering a bandlimited function $f$,
we will always assume that $f$ is also continuous, which is
consistent with the equality in~\eqref{eq:bandlimit-inversion}.
Given any $c>0$, we define the space of all $c$-bandlimited functions in
$L^2(\mathbb{R})$ by
\[
PW_c:= \{ f \in L^2(\mathbb{R}) \mid \, \hat f(\xi) = 0 \text{ for } \xi \notin [-c, c], \text{ and } f
\text{ is continuous} \},
\]
called {\bf the Paley-Wiener space} \cite{Higgins1996}.
\begin{figure}[htb]
\begin{center}
\includegraphics[totalheight=4.6in]{Shannon1}
\end{center}
\caption{We can visualize the isometries of the various spaces of functions, given by the Fourier transform.
First, the Fourier transform $\mathcal{F}$ gives us an isometry from $L^2(\mathbb{R})$ onto itself.
Second, restricting attention to the subspace $PW_c \subset L^2(\mathbb{R})$ of bandlimited functions, $\mathcal{F}$ also gives us an isometry from $PW_c$ to $L^2([-c, c])$, carrying the basis of translated ${\rm{sinc}}$ functions to the basis of exponentials. }
\label{Shannon1}
\end{figure}
\begin{thm}[Shannon-Whittaker] \label{Shannon}
Suppose that $f \in PW_c$. Then we have
\begin{equation}
\label{shannon_sampling}
f(x) = \sum_{n \in \mathbb{Z}} f\left( \frac{n}{2c} \right) {\rm sinc } \left( 2c x - n \right),
\end{equation}
and the series converges absolutely and uniformly over $\mathbb{R}$.
\hfill $\square$
\end{thm}
In other words, if we sample a $c$-bandlimited function $f$ at only the discrete set of points $\{ \frac{n}{2c} \mid n \in \mathbb{Z}\}$, we may reconstruct the whole function $f(x)$ for all $x\in \mathbb{R}$! In the next two sections, we give two different proofs of Theorem \ref{Shannon}.
The quantity $\frac{1}{2c}$ is called the {\bf sample spacing} and its reciprocal $2c$ is the {\bf sampling rate}.
So what the theorem says is that to reconstruct a function with bandlimit $c$, one has to sample at a rate~$2c$. Offhand, it seems rather incredible that some (non-periodic) functions $f:\mathbb{R} \rightarrow \mathbb{C}$ may be completely recovered by knowing only their values $f(n)$ at a discrete set of points.
This phenomenon shows, in a sense, how the Paley-Wiener space is a very special subspace of $L^2(\mathbb{R})$.
\bigskip
\section{The approach of G. H. Hardy}\label{sec:Hardy}
G. H. Hardy's proof \cite{Hardy41} of Theorem \ref{Shannon} is particularly interesting because it also answers the following informal question:
\begin{question}\rm{[Rhetorical]}
How large is the space of bandlimited functions?
\end{question}
Hardy's approach also clarifies some of the underlying structure of bandlimited functions. It relies on the following isometry.
\begin{lem}\label{isometry}
The Fourier transform ${\mathcal F}$ gives a bijection between the following two Hilbert spaces:
\begin{equation*}
{\mathcal F}: PW_{c} \rightarrow
L^2(\left[-c, c\right]).
\end{equation*}
Moreover, this bijection is an isometry.
\end{lem}
\begin{proof}
First, given any $f \in PW_{c} \subset L^2(\mathbb{R})$, we need to show that $\hat f \in L^2(\left[-c, c\right])$.
By definition $f$ is $c$-bandlimited, hence its Fourier transform can be naturally viewed as a function with domain $[-c,c]$.
We need to show that $\hat f$ has a finite $L^2(\left[-c, c\right])$ norm.
The following computation uses Parseval's identity, namely that
$\| f \|^2_{L^2(\mathbb{R})} = \|\hat f \|^2_{L^2(\mathbb{R})}$:
\begin{align}
{\| \hat f \|}^2_{L^2(\left[-c, c\right])}
&:=\int_{-c}^{c}
|\hat f(\xi)|^2 d\xi
=\int_{\mathbb{R}}
|\hat f(\xi)|^2 d\xi
=\int_{\mathbb{R}}
|f(x)|^2 dx
=: \| f \|^2_{L^2(\mathbb{R})}
< \infty,
\end{align}
proving that
$\hat f \in L^2(\left[-c, c\right])$.
Conversely, given any $g \in L^2(\left[-c, c\right])$, we need to show that ${\mathcal F}^{-1}(g) \in PW_c$.
We may extend $g$ to be equal to $0$ outside the interval $\left[-c, c\right]$, so that now $g \in L^2(\mathbb{R})$. By construction of $g$, we also
have $g \in L^1(\mathbb{R})$, so that now Lemma \ref{uniform continuity} guarantees that ${\mathcal F}^{-1}(g)$ is uniformly continuous on $\mathbb{R}$.
Because the Fourier transform is an isometry of $L^2(\mathbb{R})$, and $g \in L^2(\mathbb{R})$, we also have ${\mathcal F}^{-1}(g) \in L^2(\mathbb{R})$. So now we have
${\mathcal F}^{-1}(g) \in PW_c$.
Finally, the Fourier transform is invertible, and since we have
$\|f-g\|_{L^2(\mathbb{R})} = \|\hat f-\hat g\|_{L^2(\mathbb{R})}$ by Parseval again, we have an isometry between the two Hilbert spaces $PW_c$ and $L^2([-c, c])$.
\end{proof}
\medskip
Hardy's insight is to consider an orthonormal basis for $L^2([-c,c])$ and then pull it back to an orthonormal basis for $PW_c$. We recall the classical fact (Theorem \ref{Fourier series for periodic functions}) that the set of exponentials
\[ \Big\{ \frac{1}{\sqrt{2c}}e^{\frac{2\pi i n x}{2c}} \mid n \in \mathbb{Z}\Big\}\]
form a complete orthonormal basis for the Hilbert space $L^2(\left[-c, c\right])$.
Moreover, any $g \in L^2(\left[-c, c\right])$ has a unique representation in this basis (which we called its Fourier series), that converges in the $L^2$-norm on $[-c, c]$:
\begin{equation}\label{good series for now}
g(\xi)
\underset{L^2([-c,c])}{=}
\sum_{m\in \mathbb{Z}} \hat g_m e^{\frac{2\pi i m \xi}{2c}},
\end{equation}
with coefficients equal to
\begin{equation} \label{good series for now: coefficients}
\hat g_m = \frac{1}{2c}\big\langle g(\xi), e^{\frac{2\pi i m \xi}{2c}} \big\rangle
:= \frac{1}{2c}\int_{-c}^c g(\xi) e^{-\frac{2\pi i m \xi}{2c}} d\xi.
\end{equation}
Using the Fourier series~\eqref{good series for now}, we may expand $\|g\|_{L^2([-c,c])} = \langle g, g\rangle^{1/2}$, obtaining
\begin{equation*}
\| g \|_{L^2([-c,c])} = \Big( 2c \sum_{m \in \mathbb{Z}} |\hat g_m|^2 \Big)^{1/2}.
\end{equation*}
Although we've only scratched the surface, we've already scratched it enough in order to prove
Theorem \ref{Shannon}.
\begin{comment}
\begin{figure}
\centering
\fbox{
\includegraphics[width=1.0\textwidth]{images/L^2.png}
}
\caption{Visualizing the isometries of various spaces of functions of one variable, given by the Fourier transform. First, the Fourier transform $\mathcal{F}$ gives us an isometry from $L^2(\mathbb{R})$ onto itself. Next, restricting attention to the subspace $PW_c \subset L^2(\mathbb{R})$ of bandlimited functions, $\mathcal{F}$ also gives us an isometry from $PW_c$ to $L^2([-c, c])$, carrying the basis of translated ${\rm{sinc}}$ functions to the basis of exponentials. \red{[Fabricio: the Fourier transform of $2c{\rm{sinc}}(2cx)$ is $1_{[-c,c]}(\xi)$ and the basis element should be $2c{\rm{sinc}}(2cx-n)$.]}}
\label{fig:L^2 spaces}
\end{figure}
\end{comment}
\bigskip
\begin{proof}[Proof of Theorem~\ref{Shannon}]
For any $f \in PW_{c}$, we know by Lemma~\ref{isometry} that $\hat f \in L^2(\left[-c, c\right])$, so that $\hat f$ has a Fourier series that converges in the $L^2$-norm on $[-c, c]$:
\begin{equation*}
\hat f(\xi)
\underset{L^2([-c,c])}{=}
\sum_{m\in \mathbb{Z}} c_m e^{\frac{2\pi i m \xi}{2c}},
\end{equation*}
with coefficients equal to
$c_m = \frac{1}{2c}\int_{-c}^c \hat f(\xi) e^{-\frac{2\pi i m \xi}{2c}} d\xi = \frac{1}{2c}f(-\frac{m}{2c})$ by the Fourier inversion formula \eqref{eq:bandlimit-inversion}. It follows that
\begin{equation}
\label{eq:fhat-L2series}
\hat f(\xi) \underset{L^2([-c,c])}{=} \frac{1}{2c} \sum_{m\in \mathbb{Z}} f\left(\frac{m}{2c}\right) e^{-\frac{2\pi i m \xi}{2c}}.
\end{equation}
From the orthonormality of the exponentials and Parseval's identity (Lemma~\ref{isometry}) we have
\begin{equation}\label{eq:series-fnorm}
\| f \|_{L^2(\mathbb{R})} = \| \hat f \|_{L^2([-c,c])} = \Big( \frac{1}{2c} \sum_{m \in \mathbb{Z}} \Big|f\left(\frac{m}{2c}\right)\Big|^2 \Big)^{1/2}.
\end{equation}
We recall that the equality in the norm in~\eqref{eq:fhat-L2series} means that
\[
\lim_{N \to \infty} \bigg\| \hat f(\xi) - \frac{1}{2c}\sum_{|m| < N} f\left(\frac{m}{2c}\right) e^{-\frac{2\pi i m \xi}{2c}} \bigg \|_{L^2([-c,c])} = 0.
\]
Using the Fourier inversion and the isometry stated in Lemma~\ref{isometry},
\begin{align*}
0 &= \lim_{N \to \infty} \bigg\| \int_{-c}^{c} \Big( \hat f(\xi) - \frac{1}{2c}\sum_{|m| < N} f\left(\frac{m}{2c}\right) e^{-\frac{2\pi i m \xi}{2c}}\Big) e^{2\pi i \xi x}d\xi \bigg \|_{L^2(\mathbb{R})}\\
&= \lim_{N \to \infty} \bigg\| \int_{-c}^{c} \hat f(\xi)e^{2\pi i \xi x}d\xi - \frac{1}{2c}\sum_{|m| < N} f\left(\frac{m}{2c}\right) \int_{-c}^{c} e^{-\frac{2\pi i m \xi}{2c}} e^{2\pi i \xi x}d\xi \bigg \|_{L^2(\mathbb{R})}\\
&= \lim_{N \to \infty} \bigg\| f(x) - \sum_{|m| < N} f\left(\frac{m}{2c}\right) \frac{\sin(\pi(2cx-m))}{\pi(2cx-m)} \bigg \|_{L^2(\mathbb{R})}\\
&= \lim_{N \to \infty} \bigg\| f(x) - \sum_{|m| < N} f\left(\frac{m}{2c}\right) {\rm{sinc}}(2cx-m) \bigg \|_{L^2(\mathbb{R})},
\end{align*}
and therefore
\begin{equation}\label{eq:formulal2norm}
f(x) \underset{L^2(\mathbb{R})}{=} \sum_{m \in \mathbb{Z}} f\left(\frac{m}{2c}\right) {\rm{sinc}}(2cx-m).
\end{equation}
To pass from the convergence in the norm to pointwise convergence, we need to show that the latter
series converges uniformly, so that we can conclude that it represents a continuous function and hence by Lemma \ref{norm convergence plus absolute convergence implies equality} it is equal to $f$ everywhere.
To prove the uniform convergence, we make use of the Cauchy-Schwartz inequality for infinite series, namely
\begin{equation}\label{C-S for infinite series}
\sum_{m=N}^\infty \Big|f\left(\frac{m}{2c}\right) {\rm{sinc}}(2cx-m)\Big| \leq
\Big(\sum_{m=N}^\infty \Big|f\left(\frac{m}{2c}\right)\Big|^2\Big)^{1/2}
\Big(\sum_{m=N}^\infty {\rm{sinc}}^2(2cx-m)\Big)^{1/2}.
\end{equation}
The rest of the proof consists in showing that the right-hand side of \eqref{C-S for infinite series}
goes to zero as $N \rightarrow \infty$, uniformly for $x \in \mathbb{R}$.
The same proof will also work for the series defined from $-N$ to $-\infty$. Together these results show that the expression in~\eqref{eq:formulal2norm} converges absolutely and uniformly over~$\mathbb{R}$, giving the result stated in the theorem.
From~\eqref{eq:series-fnorm}, we see that
\begin{equation}\label{eq:fgoestozero}
\Big(\sum_{m = N}^\infty \Big|f\left(\frac{m}{2c}\right)\Big|^2\Big)^{1/2} \to 0,\quad \text{as } N \to \infty.
\end{equation}
Clearly
\[
\sum_{m = N}^\infty {\rm{sinc}}^2(2cx - m) \leq \sum_{m \in \mathbb{Z}} {\rm{sinc}}^2(2cx - m),
\]
and since the latter series are periodic function of $x$, with period $\frac{1}{2c}$,
we may assume that $0 \leq x < \frac{1}{2c}$.
For $m = 0$ and $1$, we note that ${\rm{sinc}}^2(2 c x - m) \leq 1$. For $m \geq 2$, we use the estimate
\[
{\rm{sinc}}^2(2cx -m) = \frac{\sin^2(2\pi cx - \pi m)}{(2\pi cx - \pi m)^2} \leq \frac{1}{\pi^2(m-1)^2},
\]
so that
\[
\sum_{m = 2}^\infty {\rm{sinc}}^2(2cx - m) \leq \sum_{m=2}^\infty \frac{1}{\pi^2(m-1)^2}
= \frac{1}{6}.
\]
Similarly, for $m \leq -1$,
\[
\sum_{m = -\infty}^{-1} {\rm{sinc}}^2(2cx - m) \leq \sum_{m=1}^\infty \frac{1}{\pi^2 m^2} =\frac{1}{6}.
\]
We conclude that for all $x \in \mathbb{R}$,
$\sum_{m = N}^\infty {\rm{sinc}}^2(2cx-m) \leq \sum_{m \in \mathbb{Z}} {\rm{sinc}}^2(2cx-m) \leq 2 + \frac{1}{6}+ \frac{1}{6} = \frac{7}{3}$,
and therefore the series in~\eqref{eq:holder} converges uniformly to $0$ as $N\rightarrow \infty$.
\end{proof}
\bigskip
It follows from this approach of G.H. Hardy, that despite being a subspace of $L^2(\mathbb{R})$, the Paley-Wiener space $PW_{c}$ has
a concrete, countable basis, which we record as follows.
\begin{cor}\label{cor:sinc-orthonormal}
The set of translated ${\rm{sinc}}$ functions
\begin{equation}
\{(\sqrt{2c}) \,{\rm sinc}(2cx-n) \mid n \in \mathbb{Z}\}
\end{equation}
is a complete orthonormal basis for the Hilbert space
$PW_{c}$ of $c$-bandlimited functions.
\hfill $\square$
\end{cor}
\noindent
It is also worthwhile recording here the orthonormality of the ${\rm{sinc}}$ functions explicitly. For each
$n, m \in \mathbb{Z}$, we have:
\begin{equation}\label{eq:sinc-orthonormal}
2c\int_{\mathbb{R}} {\rm{sinc}}(2cx-n) {\rm{sinc}}(2cx-m) dx =
\begin{cases}
1 &\mbox{if } n = m,\\
0 &\mbox{otherwise.}
\end{cases}
\end{equation}
\bigskip
\section{An alternative proof, using Poisson summation}\label{sec:proof-Poisson}
Here we give Shannon's proof of the classical Shannon-Whittaker sampling theorem (Theorem~\ref{Shannon}), with some added details.
This proof uses the Poisson summation formula. As we've seen several times before, Poisson summation often simplifies proofs in surprising ways.
To state the formula more precisely, we use $\underset{L^1(\mathbb{R})}{=}$ and $\underset{L^1([-c,c])}{=}$ to denote convergence in the $L^1$-norm, so that equality between functions holds almost everywhere but cannot be assumed at a specific point, unless we have an additional assumption like continuity.
Assuming only that $f \in L^1(\mathbb{R})$, the Poisson summation formula (See \cite{SteinWeiss})
states that the periodized function defined by the series
$
\sum_{n \in \mathbb{Z}} f(x+2cn)
$
converges in the norm of $L^1([-c,c])$ to a function whose Fourier expansion is
\begin{equation}\label{first Poisson summation-one dimension}
\sum_{n \in \mathbb{Z}} f(x+2cn)
\underset{L^1([-c,c])}{=}
\frac{1}{2c} \sum_{m \in \mathbb{Z}} \hat f\left(\frac{m}{2c}\right) e^{\frac{2\pi i m x}{2c} }.
\end{equation}
\begin{proof}[Proof of Theorem~\ref{Shannon}]
We begin with the Fourier series \eqref{first Poisson summation-one dimension}, which converges in the $L^1([-c, c])$ norm.
Step $1$. \ Our first goal will be to
exchange the roles of $f$ and $\hat f$. To justify this,
we begin by noting that our assumption that $f \in PW_c$ implies
$\hat f \in L^2([-c,c])$ by Lemma~\ref{isometry}, and $ L^2([-c,c]) \subset L^1([-c,c])$ by
Lemma \ref{proper containment of L^2 in L^1 for torus}. So we have $\hat f \in L^1([-c,c] \subset L^1(\mathbb{R})$,
allowing us to apply the same Poisson summation formula as above, together with Fourier inversion:
\begin{equation}\label{Poisson strikes again-one dimension}
\sum_{n \in \mathbb{Z}} \hat f(\xi + 2cn)
\underset{L^1([-c,c])}{=}
\frac{1}{2c} \sum_{m \in \mathbb{Z}} f\left( \frac{m}{2c} \right) e^{-\frac{2\pi i m \xi }{2c}}.
\end{equation}
Step $2$. \ Since we are assuming that $f \in PW_c$ and thus $\hat f$ is supported on $[-c, c]$, we may use the indicator function $1_{[-c, c]}$, defined as $1_{[-c, c]}(\xi) = 1$ if $\xi \in [-c,c]$ and $1_{[-c, c]}(\xi) = 0$ otherwise, and write the trivial identity
\begin{equation} \label{step 2}
\hat f( \xi ) = 1_{[-c, c]}(\xi) \sum_{n\in \mathbb{Z}} \hat f(\xi + 2c n ),
\end{equation}
for all real $\xi \not= c, \xi \not= -c$. The reason is that the series on the right-hand-side contains only one term, namely
the $n=0$ term $\hat f(\xi)$.
Step $3$. \ Using the Poisson summation formula \eqref{Poisson strikes again-one dimension} above, together with \eqref{step 2}, we see that
\begin{equation}\label{step 3}
\hat f( \xi ) = 1_{[-c, c]}(\xi) \sum_{n\in \mathbb{Z}} \hat f(\xi + 2c n)
\underset{L^1(\mathbb{R})}{=} \sum_{n \in \mathbb{Z}} f\left( \frac{n}{2c} \right) \Big(
\frac{1}{2c} 1_{[-c, c]}(\xi) e^{ - \frac{2\pi i n \xi }{2c}} \Big).
\end{equation}
We recall that the inverse Fourier transform of the interval $[-c, c]$ is
\[
\int_\mathbb{R} 1_{[-c, c]}(\xi) e^{-2\pi i \xi x} d\xi =2c \, {\rm{sinc}}(2c x),
\]
so that after composing the ${\rm{sinc}}$ function with a translation, we know that the Fourier transform of
${\rm{sinc}} \left( 2c \left(x - \frac{n}{2c} \right)\right)$
is $\frac{1}{2c} 1_{[-c, c]}(\xi)
e^{ - \frac{2\pi i n \xi }{2c}}$.
Multiplying both sides of \eqref{step 3} by $e^{-2\pi i \xi x}$ and integrating term-by-term over $x \in [-c, c]$, we get:
\[
f(x) \underset{L^1(\mathbb{R})}{=} \sum_{n \in \mathbb{Z}} f\left( \frac{n}{2c} \right) {\rm{sinc}} \left( 2c x - n \right),
\]
applying Fourier inversion again on the left-hand side.
Finally, we recall that we are assuming $f$ is continuous, since $f\in PW_c$. So to pass from the convergence in the norm to the pointwise convergence we may apply the same procedure from the first proof to conclude that the series on the right converges uniformly in $\mathbb{R}$ and hence also represents a continuous function.
\end{proof}
\bigskip
There are many different possible extensions of the Shannon-Whittaker sampling theorem to higher dimensions, and below we glimpse some of them below.
\section{Special properties of bandlimited and sinc functions}
\label{sec:special}
Here we focus on some special properties of bandlimited functions. We've already seen in Section~\ref{sec:Hardy} that the space $PW_c$ is isometric to $L^2([-c,c])$,
so many of its special properties comes from $L^2([-c,c])$, and they are then pulled back via inverse Fourier transform.
The special case $c=\frac{1}{2}$ of Theorem \ref{Shannon} is worth pointing out:
\begin{equation}\label{Shannon simplified}
f(x) = \sum_{n \in \mathbb{Z}} f(n) \ {\rm sinc } \left( x - n \right),
\end{equation}
a classical version of the Shannon-Whittaker formula.
The choice of $c = \frac{1}{2}$ means that we begin with the interval $[-\frac{1}{2}, \frac{1}{2}]$ in the frequency space;
this interval is a Voronoi cell of the integer lattice $\mathbb{Z}$.
\begin{example} \label{ex-1}
\rm{
What happens if we apply the Shannon-Whittaker formula
\eqref{Shannon simplified} to the ${\rm{sinc}}$ function itself? Let's try it!
With $f(x):= {\rm{sinc}}(y - x)$, and any fixed $y\in \mathbb{R}$, we have:
\begin{equation}
{\rm{sinc}}(y - x) = \sum_{n \in \mathbb{Z}} {\rm{sinc}}(y - n) \, {\rm{sinc}}( x - n ).
\end{equation}
As a special case, if we let $x=y$, we get:
\begin{equation}
1= {\rm{sinc}}(0) = \sum_{n \in \mathbb{Z}} {\rm{sinc}}^2(x - n).
\end{equation}
\hfill $\square$
}
\end{example}
In Corollary \ref{cor:sinc-orthonormal} we showed that the functions ${\rm{sinc}}(x-n)$ with $n \in \mathbb{Z}$
form an orthonormal basis for the space of bandlimited functions. This has some nice consequences.
\begin{thm} \label{thm:discrete-inner-product}
If $f$ and $g$ are $\tfrac{1}{2}$-bandlimited,
then
\[
\int_\mathbb{R} f(x) \overline{g(x)} dx
= \sum_{n \in \mathbb{Z}} f(n) \overline{g(n)}.
\]
\end{thm}
[{\bf Intuitive proof}] \ If we work formally, then we can use the orthonormality of the sinc functions \eqref{eq:sinc-orthonormal},
together with \eqref{Shannon simplified} to quickly see that:
\begin{align*}
\langle f(x), g(x) \rangle &=
\Big\langle \sum_{m \in \mathbb{Z}} f(m) {\rm{sinc}}(x-m), \sum_{n \in \mathbb{Z}} g(n){\rm{sinc}}(x-n) \Big\rangle \\
&= \sum_{m, n \in \mathbb{Z}}
f(m)\overline{g(n)} \Big\langle {\rm{sinc}}(x-m), {\rm{sinc}}(x-n) \Big\rangle \\
&=\sum_{n \in \mathbb{Z}}f(n) \overline{g(n)},
\end{align*}
and we're done. Although this intuitive proof may seem `fast and loose', these steps can
be made rigorous if we would prove just a bit more about Hilbert spaces,
because the Paley-Wiener space $PW_c$ is a Hilbert space, and the translated
functions ${\rm{sinc}}(x-n)$ are a basis
for this Hilbert space.
\begin{comment}
[{\bf Rigorous proof}] \ Let's redo the steps above, adding rigor and using the theorems that we've already
proved and/or seen. Using the Shannon sampling identity \eqref{Shannon simplified} for both $f$ and $g$,
we have:
\begin{align*}
\int_\mathbb{R} f(x) \overline{g(x)} dx &= \int_\mathbb{R} \sum_{m \in \mathbb{Z}} f(m) {\rm{sinc}}(x-m)
\sum_{n \in \mathbb{Z}} \overline{g(n)} {\rm{sinc}}(x-n) dx \\
&= \int_\mathbb{R} \sum_{(m, n) \in \mathbb{Z}^2} f(m) \overline{g(n)} \, {\rm{sinc}}(x-m) \, {\rm{sinc}}(x-n) dx.
\end{align*}
If we can move the integral past the sum, then we're done, because we would have:
\begin{align*}
\int_\mathbb{R} f(x) \overline{g(x)} dx
&= \sum_{(m, n) \in \mathbb{Z}^2} f(m) \overline{g(n)} \, \int_\mathbb{R} {\rm{sinc}}(x-m) \, {\rm{sinc}}(x-n) dx \\
&= \sum_{n \in \mathbb{Z}} f(n) \overline{g(n)},
\end{align*}
by \eqref{eq:sinc-orthonormal}, the orthogonality relations of the ${\rm{sinc}}$ functions.
To justify moving the integral past the sum, we may use \eqref{Application of dominated convergence} (in the Appendix), once we show that $\sum_{n=1}^\infty \int_{\mathbb{R}^d} | f_n(x) | dx < \infty$, where
$ f_n(x) := $
\hfill $\square$
\end{comment}
\begin{comment}
Let $F$ be a class of functions defined in a Hilbert space $E$.
A function $f:E \times E$ is a Reproducing Kernel of $F$ if
\begin{enumerate}
\item For every $x$, $f(y,x)$ as function of $y$ belongs to $F$.
\item For every $x \in E$, for every $f \in F$, $f(x) = \langle f(y),K(y,x) \rangle$
\end{enumerate}
\end{comment}
One important case of the previous theorem is when $g(x) := {\rm{sinc}}(x - y)$, which combined again with the Shannon-Whittaker formula
\eqref{Shannon simplified} results in the next theorem.
\begin{thm} \label{thm:ReproducingSinc}
The space $PW_{\frac{1}{2}}$ is a space with a reproducing kernel ${\rm{sinc}}(x-y)$, which means by definition
that any $f \in PW_{\frac{1}{2}}$ can be written as
\[
f(x) = \int_\mathbb{R} f(y) {\rm{sinc}}(x-y) d y.
\]
\end{thm}
\begin{proof}
For $x \in \mathbb{R}$, take $g(y) := {\rm{sinc}}(x - y)$ in Theorem~\ref{thm:discrete-inner-product}:
\[
\int_R f(y) {\rm{sinc}}(x-y) d y = \sum_{n \in \mathbb{Z}} f(n) {\rm{sinc}}(x-n) = f(x),
\]
where the second equality follows from the Shannon-Whittaker formula.
\end{proof}
\bigskip
We also have the following somewhat surprising properties of bandlimited functions on $\mathbb{R}$. First we recall
that we called a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ nice if $f, \hat f \in L^1(\mathbb{R}^d)$, and $f$ satisfies the Poisson summation formula:
\begin{equation} \label{Poisson summation again}
\sum_{n \in \mathbb{Z}^d} f(n+ x) = \sum_{\xi \in \mathbb{Z}^d} \hat f(\xi) e^{2\pi i \langle \xi, x \rangle},
\end{equation}
valid pointwise for each $x \in \mathbb{R}^d$.
\begin{thm}
\label{bandlimited, Poisson summation}
Let $f:\mathbb{R}\rightarrow \mathbb{C}$ be a nice function, such that $f$ is $c$-bandlimited. Then we have:
\begin{enumerate}[(a)]
\item \label{firstpart}
We have, for each $k>c$,
\begin{equation}\label{exact Riemann approximation}
\frac{1}{k}\sum_{n\in\mathbb{Z}}
f\left(\frac{n}{k}\right)
= \int_{\mathbb{R}} f(x) dx,
\end{equation}
\blue{We note that the identity \eqref{exact Riemann approximation}
can be interpreted to mean that the Riemann approximation to the integral is always exact for such an $f$, provided that the step size is $\Delta x:=\frac{1}{k}<\frac{1}{c}$.}
\item \label{secondpart}
For all $a, k$ with $a>c$ and $k > a+c$, we have
\begin{equation}
\sum_{n\in\mathbb{Z}} f\left(\frac{n}{k}\right)
e^{\frac{2\pi i n a}{k}}=0.
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}
To prove \ref{firstpart}, we use Poisson summation \eqref{Poisson summation again}, with ${\mathcal L}:= \frac{1}{k} \mathbb{Z}$:
\begin{equation} \label{equality of sum and integral}
\sum_{n\in\mathbb{Z}}
f\left(\frac{n}{k}\right)=
\sum_{\xi \in {\mathcal L}} f(\xi) =
\frac{1}{\det {\mathcal L}}
\sum_{m \in {\mathcal L}^*} \hat f(m) =
k \sum_{m \in \mathbb{Z}} \hat f(mk) = k\int_\mathbb{R} f(x) dx,
\end{equation}
which is the desired identity. In the last equality we used the assumption that the indices of summation satisfy $ |mk| > c$,
for $m\not=0$,
so that $\hat f(mk) = 0$ because $f$ is $c$-bandlimited by assumption.
We also used the fact that
$\hat f(0) = \int_\mathbb{R} f(x) dx$.
\noindent
To prove \ref{secondpart}, we apply the following small variation of Poisson summation:
\begin{equation}\label{magic2}
\frac{1}{k}
\sum_{n \in \mathbb{Z}} f\left( \frac{n}{k} \right) e^{\frac{2\pi i n a }{k}}
=\sum_{n\in \mathbb{Z}} \hat f(-a+kn),
\end{equation}
which follows quickly from the Poisson summation formula given above in \eqref{Poisson summation again} (Exercise \ref{Exercise:PoissonSummation1}).
But by the assumption that $f$ is $c$-bandlimited, we also have
\begin{equation}
\sum_{n\in \mathbb{Z}} \hat f(-a+kn)=0,
\end{equation}
provided that
\begin{equation}\label{constraint}
|-a+kn|>c, \ \text{ for all } n\in \mathbb{Z}.
\end{equation}
For $n=0$, we see that a necessary condition for \eqref{constraint} is $|a|>c$. Geometrically, \eqref{constraint} tells us that the arithmetic progression $\{ kn -a\}_{n \in \mathbb{Z}}$ does not intersect the interval $[-c, c]$. It is easily checked that the additional constraint $k>c+a$ gives us a sufficient condition for \eqref{constraint} to hold.
\end{proof}
As is easily observed, sums and products of bandlimited functions are again bandlimited. In particular, more precise statements such as the following are possible.
\begin{lem}\label{lemma:product}
Suppose that $f$ is $c$-bandlimited, and $g$ is $d$-bandlimited. \\
Then $fg$ is $(c+d)$-bandlimited.
\end{lem}
\begin{proof} By assumption, $\hat f(\xi) = 0$ outside of $[-c, c]$, and
$\hat g(\xi) = 0$ outside of $[-d, d]$.
We know that the support of the convolution $\hat f(\xi)*\hat g(\xi)$ is contained in the closure of the Minkowski sum \index{Minkowski sum}
of the individual supports of $f$ and $g$ (by Exercise \ref{support of convolution}), which is $[-c, c]+[-d, d] = [-c-d, c+d]$.
This argument justifies the second equality in the chain of equalities below.
Also, the assumption that $f, g \in L^2(\mathbb{R})$ implies $fg \in L^1(\mathbb{R})$, by the Cauchy-Schwartz argument we gave in
\eqref{product of two L^2 functions is L^1}.
We'd like to say that $\widehat{(fg)}(\xi)=(\hat f*\hat g)(\xi)$, so let's brute-force
the computation, using our knowledge of Lemma \ref{isometry}:
\begin{align*}
(\hat f*\hat g)(\xi) &:= \int_{\mathbb{R}} \hat f(x-\xi)\hat g(x) dx\\
&= \int_{-c-d}^{c+d} \hat f(x-\xi)\hat g(x) dx\\
&:= \int_{-c-d}^{c+d} \int_{-c}^c f(u) e^{-2\pi i (x-\xi)u} du \int_{-d}^d g(v) e^{-2\pi i v x} dv \, dx\\
&:= \int_{-c}^c \int_{-d}^d f(u) g(v) e^{2\pi i \xi u}
\left(
\int_{-c-d}^{c+d} e^{-2\pi i x(u + v)}
\right) \, dx \\
& \text{ ... and now using the orthogonality relations for the exponentials ....} \\
& \text{ In other words, ` a miracle happens here ' .... } \\
&= \int_{\mathbb{R}} f(u) g(u) e^{-2\pi i u \xi }du = \widehat{(fg)}(\xi).
\end{align*}
So we conclude that $fg$ is $(c+d)$-bandlimited. Another argument is also possible, using
$L^2[-c, c] \subset L^1[-c, c]$, which we proved in Lemma \ref{proper containment of spaces over the torus}.
\end{proof}
\bigskip
\begin{example}
\rm{
Here are some fun consequences of Theorem \ref{bandlimited, Poisson summation}. Let's fix any $\epsilon >0$.
By Theorem \ref{bandlimited, Poisson summation}, part \rm{\ref{secondpart}} , we can pick $k=1, c=\frac{1}{2}-\epsilon$, and $a=\frac{1}{2}$, all of
which satisfy the hypothesis, so that $f$ is $(\frac{1}{2}-\epsilon)$-bandlimited by definition. We then have
\begin{equation}
0= \sum_{n\in\mathbb{Z}} f(n)
e^{\pi in}
= \sum_{n\in\mathbb{Z}} f(n)
(-1)^n.
\end{equation}
Seperating the lattice sum into $n$ even and $n$ odd, we have
\begin{equation}
\sum_{m\in \mathbb{Z}} f(2m)
=\sum_{m \in \mathbb{Z}} f(2m+1).
\end{equation}
Generalizing the latter identity, we fix any positive integer $N$, and we let $k=\frac{2}{N}, c=\frac{1}{N}-\epsilon$, and $a=\frac{1}{N}$, so that $f$ is $(\frac{1}{N}-\epsilon)$-bandlimited by definition.
By Theorem
\ref{bandlimited, Poisson summation}, part \rm{\ref{secondpart}}:
\begin{equation}
0= \sum_{n\in\mathbb{Z}} f\left(\frac{n}{k}\right)
e^{\frac{2\pi i n a}{k}}
=\sum_{n\in\mathbb{Z}} f\left(\frac{Nn}{2}\right)
(-1)^n,
\end{equation}
so that we get the identity
\begin{equation}
\sum_{m\equiv 0\text{ mod N }} f(m)
=\sum_{m\equiv 0\text{ mod N }}
f\Big(m+ \frac{N}{2}\Big).
\end{equation}
}
\hfill $\square$
\end{example}
\bigskip
\begin{example}
With $k=1, c=\frac{1}{3}-\epsilon$, and $a=\frac{1}{3}$, part \rm{\ref{secondpart}} of Theorem
\ref{bandlimited, Poisson summation}
gives:
\begin{align*}
0&=\sum_{n\in\mathbb{Z}} f(n) e^{\frac{2\pi i n}{3}}
=\sum_{n\equiv 0\text{ mod 3 }} f(n)
+ \omega\sum_{n\equiv 1\text{ mod 3 }} f(n)
+ \omega^2\sum_{n\equiv 2\text{ mod 3 }} f(n),
\end{align*}
where $\omega:= e^{2\pi i / 3}$.
\hfill $\square$
\end{example}
\bigskip
\begin{example}
\rm{
Consider ${\rm{sinc}}^N(x / \pi)$, which has the bandlimit $c:= \frac{N}{2\pi}$.
By Theorem~\ref{bandlimited, Poisson summation}, the strange relation
\begin{equation} \label{example:strange 1}
\sum_{n \in \mathbb{Z}} {\rm{sinc}}^N\left(\frac{n}{\pi}\right) = \int_{\mathbb{R}} {\rm{sinc}}^N\left(\frac{x}{\pi}\right) dx,
\end{equation}
holds for $N = 2, \dots, 6$, because in this range we have
$c=\frac{N}{2\pi} \leq \frac{6}{2\pi} < 1=: k$. It's also true for $N=1$, with some care:
\[
\lim_{M\rightarrow \infty} \sum_{|n| < M \atop n \in \mathbb{Z}}
{\rm{sinc}} \left(\frac{n}{\pi}\right) = \lim_{M\rightarrow \infty} \int_{-M}^M {\rm{sinc}} \left(\frac{x}{\pi}\right) dx.
\]
It turns out that this identity fails, however, for $N \geq 7$. Indeed, by
Poisson summation~\eqref{Poisson summation again}, for a nice function $f$ we have:
\[
\sum_{n \in \mathbb{Z}}f(n) = \sum_{m \in \mathbb{Z}}\hat f(m) = \int_\mathbb{R} f(x) dx + \sum_{m \in \mathbb{Z} \setminus \{0\}} \hat f(m).
\]
Taking $f(x) := {\rm{sinc}}^N\left( \frac{x}{\pi}\right)$, we see that
that the last sum is zero when $N \leq 6$ and positive when $N \geq 7$, since $\hat f$ has support $[-\frac{N}{2\pi}, \frac{N}{2\pi}]$ and is positive inside this interval.
In a similar manner to eq. \eqref{example:strange 1}, we have:
\begin{equation} \label{example:strange 2}
\sum_{n \in \mathbb{Z}}\, \prod_{k = 0}^N {\rm{sinc}}\left(\frac{n}{(2k+1)\pi}\right) = \int_{\mathbb{R}}\, \prod_{k = 0}^N {\rm{sinc}}\left(\frac{x}{(2k+1)\pi}\right) dx,
\end{equation}
holds for $N = 0, \dots, 40248$, since for these $N$ we have $1 + \frac{1}{3} + \dots + \frac{1}{2N+1} < 2\pi$.
It can be also checked that the equality above fails for $N = 40249$. These facts are easy corollaries of Theorem~\ref{bandlimited, Poisson summation}, but may seem surprising when taken out of this context. The identities \eqref{example:strange 1}
and \eqref{example:strange 2} appeared in \cite{Baillie}.
}
\hfill $\square$
\end{example}
\bigskip
\section{Shannon sampling in higher dimensions}
The first research into higher-dimensional Shannon-type sampling theorems, as far as we know,
was
the work of Petersen and Middleton~\cite{PetersenMiddleton62}. We'll also follow a bit of
Chapter $14$ from Higgins~\cite{Higgins1996}.
For a convex body ${\mathcal P}$, we say that a function $f$ is {\bf ${\mathcal P}$-bandlimited} if $\hat f$ vanishes outside of ${\mathcal P}$.
We note that this does not preclude the possibility that $\hat f$ may only be nonzero on some proper subset of ${\mathcal P}$.
Assuming that $f$ is real-valued, we know that the image of $\hat f$ is symmetric about the origin
(Lemma \ref{symmetric iff FT is real});
so the assumption that ${\mathcal P}$ is symmetric is natural.
By analogy with the $1$-dimensional Paley-Wiener space $PW_c$, we define the
{\bf Paley-Wiener space} of ${\mathcal P}$-bandlimited functions in $L^2(\mathbb{R}^d)$:
\begin{equation}
PW_{\mathcal P}:= \{ f \in L^2(\mathbb{R}^d) \mid
f \text{ is continuous and ${\mathcal P}$-bandlimited} \}.
\end{equation}
A new twist in higher dimensions is the strong distinction between packing and tiling,
so the following question motivates some of these research directions.
\begin{question}
Given a convex $d$-dimensional body ${\mathcal P} \subset \mathbb{R}^d$, suppose we want to have a sampling theorem for functions that are
${\mathcal P}$-bandlimited. Does ${\mathcal P}$ have to tile $\mathbb{R}^d$ by translations, or is it sufficient to consider a packing of ${\mathcal P}$ by some lattice ${\mathcal L}$?
\end{question}
Interestingly, we don't observe this distinction in dimension $1$, because optimal packing and tiling are equivalent. But they are quite different in dimensions $d \geq 2$.
Luckily, our elementary $1$-dimensional Lemma~\ref{isometry} does extend directly to our new $d$-dimensional setting.
\begin{lem}
\label{isometry lemma for R^d}
Let ${\mathcal P}$ be a bounded convex body in $\mathbb{R}^d$. The Fourier transform ${\mathcal F}$ is an isometry between the two Hilbert spaces:
\begin{equation*}
{\mathcal F}: PW_{{\mathcal P}} \rightarrow
L^2({\mathcal P}).
\end{equation*}
\end{lem}
\begin{proof}
Given $f \in PW_{{\mathcal P}} \subset L^2(\mathbb{R}^d)$, by definition $\mathrm{supp}(\hat f) \subseteq {\mathcal P}$,
so using Parseval's identity we have:
\begin{align}
{\| \hat f \|}^2_{L^2({\mathcal P})}
&:=\int_{\mathcal P}
|\hat f(\xi)|^2 d\xi =
\int_{\mathbb{R}^d}
|\hat f(\xi)|^2 d\xi
=\int_{\mathbb{R}^d}
|f(\xi)|^2 d\xi =: \| f \|^2_{L^2(\mathbb{R}^d)}
< \infty,
\end{align}
which shows that $\hat f \in L^2({\mathcal P})$.
Using the fact that the Fourier transform is an isometry of $L^2(\mathbb{R}^d)$, and is invertible, we are done.
\end{proof}
\bigskip
\begin{thm}
\label{Higher dimensional sampling}
Suppose we have a lattice packing for a symmetric convex body ${\mathcal P}$, with a lattice~${\mathcal L}^*\subset \mathbb{R}^d$.
We let $\phi(x):= \hat 1_{\mathcal P}(x)$, our old friend.
If $f \in PW_{\mathcal P}$, then
then we can reconstruct the function $f$ completely by sampling it only at the lattice points of ${\mathcal L}$:
\[
f(x) = \det {\mathcal L} \sum_{n \in {\mathcal L}} f(n) \phi(x-n),
\]
and the series converges absolutely and uniformly over $\mathbb{R}^d$.
\end{thm}
Before diving into the proof, let's see why Theorem \ref{Higher dimensional sampling} indeed generalizes the classical $1$-dimensional
Theorem~\ref{Shannon}. In the one dimensional case, the lattice ${\mathcal L}$ is just the sampling domain $\{\frac{n}{2c} \mid n \in \mathbb{Z}\}$ and
hence $\det {\mathcal L} = \frac{1}{2c}$, while ${\mathcal P}:= [-c,c]$ is simply an interval.
Therefore:
\[
\phi(x) = \int_{-c}^c e^{2\pi i \xi x}d\xi
= \frac{1}{2\pi i x}e^{2\pi i xc} - \frac{1}{2\pi i x}e^{-2\pi i xc}
= \frac{1}{\pi x} \sin(2 \pi c x)
= 2c \, {\rm{sinc}}(2cx).
\]
The formula from Theorem~\ref{Higher dimensional sampling} reduces to the formula from Theorem~\ref{Shannon}:
\begin{align*}
f(x) = \det {\mathcal L} \sum_{n \in {\mathcal L}}f(n)\phi(x-n)
&= \frac{1}{2c} \sum_{n \in \mathbb{Z}} f\Big(\frac{n}{2c}\Big) 2c \, {\rm{sinc}}\Big(2c\big(x-\frac{n}{2c}\big)\Big) \\
&= \sum_{n \in \mathbb{Z}} f\Big(\frac{n}{2c}\Big) {\rm{sinc}}(2cx-n),
\end{align*}
which is the Shannon-Whittaker sampling formula.
\bigskip
\begin{proof}
[Proof of Theorem~\ref{Higher dimensional sampling}]
The assumption that $f \in PW_{\mathcal P}$, together with ${\mathcal P}$ being compact, implies that
$\hat f \in L^2({\mathcal P}) \subseteq L^1({\mathcal P}) \subseteq L^1(\mathbb{R}^d)$. Now we may use the Poisson summation formula
\eqref{Poisson summation again}, but with $f$ replaced by $\hat f$, and with ${\mathcal L}$ replaced by ${\mathcal L}^*$:
\begin{equation}\label{Poisson strikes again}
\sum_{m \in {\mathcal L}^*} \hat f(\xi + m)
\underset{L^1(\mathbb{R}^d/{\mathcal L}^*)}{=}
\det {\mathcal L} \sum_{n \in {\mathcal L}} f\left( n \right) e^{-2\pi i \langle \xi, n \rangle},
\end{equation}
where we also have used that $\hat{\hat{f}}(m) = f(-m)$.
Because $\hat f$ is supported on ${\mathcal P}$, we have by definition $\sum_{m \in {\mathcal L}^*} \hat f(\xi + m) = \hat f(\xi)$,
so that we may write
\begin{equation} \label{step 2.1}
\hat f( \xi ) = 1_{{\mathcal P}}(\xi) \sum_{m \in {\mathcal L}^*} \hat f(\xi + m),
\end{equation}
for all $\xi\in \mathbb{R}^d$ that do not lie on the boundary of ${\mathcal P}$.
Because of our packing assumption, all of the translated supports of $ \hat f(\xi + m)$ are disjoint,
as $m$ varies over the lattice ${\mathcal L}^*$. In other words, these supports are
\begin{equation*}
\{ \supp( \hat f ) + m \mid \, m \in {\mathcal L}^* \} \subseteq \{{\mathcal P}+ m \mid \, m \in {\mathcal L}^*\},
\end{equation*}
a disjoint collection of translates of ${\mathcal P}$.
This means that the latter identity \eqref{step 2.1} holds
because the series on the right-hand-side contains only one term, namely
the $m=0$ term $\hat f(\xi)$.
Next, we combine~\eqref{Poisson strikes again} with~\eqref{step 2.1} to get
\begin{equation}\label{step 3.1}
\hat f( \xi ) \underset{L^1(\mathbb{R}^d)}{=} \det {\mathcal L} \sum_{n \in {\mathcal L}} f\left( n \right) 1_{{\mathcal P}}(\xi) e^{-2\pi i \langle \xi, n \rangle}.
\end{equation}
Now we'd like to take the inverse Fourier transform of both sides of \eqref{step 3.1}.
We'll use the following elementary identity, for a fixed $n$:
\[
{\mathcal F}^{-1} \big( 1_{{\mathcal P}}(\xi) e^{-2\pi i \langle \xi, n \rangle} \big)(x) = {\mathcal F}^{-1}(1_{{\mathcal P}})(x - n)
= \int_{\mathcal P} e^{2\pi i \langle \xi, x-n \rangle}d\xi = \phi(x-n),
\]
Arriving finally at
\begin{align*}
f(x) \underset{L^1(\mathbb{R}^d)}{=} \det {\mathcal L} \sum_{n \in {\mathcal L}} f\left( n \right) \phi(x-n),
\end{align*}
it seems like we're done. We just to pass from the $L^1$-convergence of the latter identity, to its pointwise and uniform convergence.
There is just a slippery issue with uniform convergence that we need to
justify. To finish the argument, we would like to show that the series
$\sum_{n \in {\mathcal L}} f\left( n \right) \phi(x-n)$ converges
uniformly on $\mathbb{R}^d$, which is analogous to the proof in the previous Section, and for which we call on Lemma \ref{nasty uniform convergence}.
\end{proof}
\bigskip
\begin{lem} \label{nasty uniform convergence}
Given $f \in PW_{\mathcal P}$, $\phi:= \hat 1_{\mathcal P}$, and any full-rank lattice ${\mathcal L}$, the series
\[
\sum_{n \in {\mathcal L}} f\left( n \right) \phi(x-n)
\]
converges uniformly on $\mathbb{R}^d$.
\end{lem}
(See \cite{Higgins1996})
\hfill $\square$
Intuitively, if we pick a larger set ${\mathcal P}$, then the vectors from ${\mathcal L}^*$ will have to be more widely spaced in order to satisfy the packing condition
for ${\mathcal P}$.
Therefore our samples, which we always take from the lattice ${\mathcal L}$, will have to be denser due to the reciprocal relation $(\det {\mathcal L} )(\det {\mathcal L}^*) = 1$.
On the other hand, for a given sampling lattice ${\mathcal L}$, in this multidimensional case we can consider infinitely many different
bodies ${\mathcal P}$ that form a packing of $\mathbb{R}^d$ with the lattice ${\mathcal L}$.
One of the most natural choices for such a convex set ${\mathcal P}$ is the Voronoi cell of ${\mathcal L}^*$.
In closing, we note that it is impossible for both $f$ and $\hat f$ to be simultaneously bandlimited, by the basic uncertainty principle,
Theorem \ref{basic uncertainty principle}. However, in practice, if we are given a function $f\in L^2(\mathbb{R}^d)$ that is not bandlimited,
we can form a sequence of bandlimited functions that approach $f$ as $n\rightarrow \infty$, as follows. To make $\hat f$ compactly supported, we'll multiply $\hat f$ by $1_{[-n, n]^d}$, the indicator function of the cube.
Pulling things back to the space domain, we have:
\begin{align}
{\mathcal F}^{-1} \left(
1_{[-n, n]^d} \, \hat f
\right)
&= {\mathcal F}^{-1} \left( 1_{[-n, n]^d} \right) * {\mathcal F}^{-1} (\hat f) \\ \label{inverting sinc^d}
&= {\rm{sinc}}^d* f.
\end{align}
So if we define $f_n:= {\rm{sinc}}^d* f$, then ${\mathcal F}(f_n) = 1_{[-n, n]^d} \, \hat f$, a compactly supported function that is bandlimited to the cube $ [-n, n]^d$.
The careful reader might notice that in \eqref{inverting sinc^d}, we are applying the Fourier inversion formula on $L^2(\mathbb{R}^d)$, and not on
$L^1(\mathbb{R}^d)$. We do this because although ${\rm{sinc}}^d(x) \notin L^1(\mathbb{R}^d)$, we do have ${\rm{sinc}}^d(x) \in L^2(\mathbb{R}^d)$.
\bigskip \bigskip
\section*{Notes} \label{Notes.chapter.Shannon}
\begin{enumerate}[(a)]
\item Higgins' book \cite{Higgins1996}, Chapter $6$, has an excellent account of the Paley-Wiener space, and its connections to the Paley-Wiener theorem, which also answers the question: ``how may we analytically continue bandlimited functions of a real variable, to $\mathbb{C}$?'' Moreover, Higgin's book has more mathematical rigor than many other books.
\item For further reading, the following articles are of interest: \cite{Entezari09}, \cite{Unser00}, \cite{Ye12}.
\item Interesting relations between rates of convergence of bandlimited functions,
Nikol'skij type functions spaces, and Plancherel-Polya type inequalities is given in \cite{SchmeisserSickel2000}.
\end{enumerate}
\bigskip
\section*{Exercises}
\addcontentsline{toc}{section}{Exercises}
\markright{Exercises}
\begin{quote}
\end{quote}
\begin{prob}
By using an example, show that a bandlimited function $f\in L^2(\mathbb{R})$ may not be in $L^1(\mathbb{R})$.
\end{prob}
\medskip
\begin{prob}
Let $f \in PW_c$, and fix any $x_0 \in \mathbb{R}$. Prove that $f$ is completely determined
by the samples
\[
\left\{ f \left(x_0 +\frac{\pi n}{c} \right) \mid n \in \mathbb{Z} \right\}.
\]
\end{prob}
\medskip
\begin{prob} $\clubsuit$ \label{Exercise:PoissonSummation1}
Here we give another small variation on Poisson summation, namely that for any $a, k \in \mathbb{R}$, we have
\begin{equation}
\frac{1}{k}
\sum_{n \in \mathbb{Z}} f\left( \frac{n}{k} \right) e^{\frac{2\pi i n a }{k}}
=\sum_{n\in \mathbb{Z}} \hat f(-a+kn),
\end{equation}
where $f:\mathbb{R}\rightarrow \mathbb{C}$ is a nice function.
\end{prob}
Hint: you can use the usual Poisson summation formula given in \eqref{Poisson summation again}.
\bigskip
\chapter{Appendix A: The dominated convergence theorem, and other goodies}
\label{Appendix A}
A frequent question that comes up in proofs is ``when may we take the limit inside the integral''?
A general tool that allows us to do so is the Dominated convergence theorem. Here we remind the reader of
some of the basic results from real analysis, but we skip the proofs and give references for them. For our purposes, we only need these results in Euclidean spaces, although all of these theorems have extensions to arbitrary measure spaces. All functions here are assumed to be measurable.
\begin{thm}[Fatou's lemma] \label{Fatou}
\index{Fatou's lemma}
Fixing any subset $E\subset \mathbb{R}^d$, let $f_n:E \rightarrow [0, \infty)$ be a sequence of nonnegative functions. Then we have:
\begin{equation}
\int_{E} \lim \inf \, f_n(x) dx \leq \lim \inf \int_{E} \, f_n(x) dx.
\end{equation}
\hfill $\square$
\end{thm}
The inherent flexibility in {\bf Fatou's lemma} allows it to be useful in many different contexts, because the lim inf $f_n$ always exists, and are even allowed to be equal to $\pm$ infinity. In fact, Fatou's lemma is the main tool in proving Lebesgue's dominated convergence theorem, below.
Another essential fact for us is {\bf Fubini's theorem}, which allows us to interchange integrals with integrals, and series with integrals, for product spaces.
If we write $\mathbb{R}^d = \mathbb{R}^m \times \mathbb{R}^n$, and we denote a point $z\in \mathbb{R}^d$ by $z:= (x,y)$, then we may also write
$f(z):= f(x, y)$.
\begin{thm}[Fubini] \label{Fubini}
\index{Fubini's theorem}
Let $f \in L^1(\mathbb{R}^d)$. Then:
\begin{equation}
\int_{\mathbb{R}^d} f(z) dz = \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^m} f(x, y) dx \right) dy,
\end{equation}
and
\begin{equation}
\int_{\mathbb{R}^d} f(z) dz = \int_{\mathbb{R}^m} \left( \int_{\mathbb{R}^n} f(x, y) dy \right) dx.
\end{equation}
\hfill $\square$
\end{thm}
There is also a version of Fubini's theorem that uses the counting measure in one of the
factors of $\mathbb{R}^m \times \mathbb{R}^n$, giving us:
\begin{equation}\label{Fubini for sums and integrals}
\sum_{\xi \in \mathbb{Z}^n} \left( \int_{\mathbb{R}^m} f(x, \xi) dx \right) = \int_{\mathbb{R}^m} \left( \sum_{\xi \in \mathbb{Z}^n} f(x, \xi) \right) dx.
\end{equation}
(See \cite{PaulSally1}, p. 220, for a proof of Theorem \ref{Fubini})
\medskip
\section{The Dominated Convergence Theorem}
\begin{thm}[Dominated convergence theorem] \label{Dominated convergence theorem}
\index{Lebesgue dominated convergence theorem}
\ Suppose that we have a sequence of functions $ f_n(x):\mathbb{R}^d \rightarrow \mathbb{C}$, for $n = 1, 2, 3, \dots $, and suppose
there exists a limit function $f(x) =\lim_{n\rightarrow \infty} f_n(x)$, valid for all $x\in \mathbb{R}^d$.
If there exists a function $g \in L^1(\mathbb{R}^d)$ such that for all $x \in \mathbb{R}^d$, we have:
\[
\left| f_n(x) \right| \leq g(x), \quad n = 1, 2, 3, \dots
\]
then:
\begin{enumerate}[(a)]
\item $f \in L^1(\mathbb{R}^d)$.
\item $ \lim_{n\rightarrow \infty} \int_{\mathbb{R}^d} | f_n(x) - f(x) | dx = 0$.
\item And finally, we may interchange limits and integrals:
\[
\lim_{n \rightarrow \infty} \int_{\mathbb{R}^d} f_n(x) dx = \int_{\mathbb{R}^d} f(x) dx.
\]
\end{enumerate}
\hfill $\square$
\end{thm}
Theorem \ref{Dominated convergence theorem} is sometimes called the \emph{Lebesgue dominated convergence theorem}, honoring the work of Lebesgue.
There is a useful application of Lebesgue's dominated convergence theorem, which allows us to interchange summations with integrals as follows.
\medskip
\begin{thm} \label{Application of dominated convergence}
\ Suppose that we have a sequence of functions $ f_n(x):\mathbb{R}^d \rightarrow \mathbb{C}$, such that
\[
\sum_{n=1}^\infty \int_{\mathbb{R}^d} | f_n(x) | dx < \infty.
\]
Then the series $ \sum_{n=1}^\infty f_n(x)$
converges for all $x\in \mathbb{R}^d$, and we have:
\[
\sum_{n=1}^\infty \int_{\mathbb{R}^d} f_n(x) dx = \int_{\mathbb{R}^d} \sum_{n=1}^\infty f_n(x) dx.
\]
\hfill $\square$
\end{thm}
(See \cite{RudinGreenBook}, p. 26 for a proof of Theorem \ref{Dominated convergence theorem}, and
\cite{RudinGreenBook}, p. 29 for a proof of Theorem \ref{Application of dominated convergence})
\section{Big-O}
Very often we'd like to compare, in a quick-and-dirty way that avoids uncountably many details,
how fast two functions grow. We review here two of the most common ways to do this.
Suppose we are given two functions $f, g:\mathbb{R}^d \rightarrow \mathbb{C}$.
We say that $f(x) = O( g(x) )$ (pronounced ``Big o''), as $x \rightarrow x_0$,
if {\bf there exists a positive constant} $C$ such that
\begin{equation}
|f(x)| \leq C |g(x)|,
\end{equation}
for all $x$ that are sufficiently close to $x_0$. Here we allow $x_0$ to be any real vector,
and we also allow the very common case $x_0 = \pm \infty$.
Equivalently, we may say
\[
\left| \frac{f(x)}{g(x)} \right| \text{ is eventually bounded above}.
\]
\begin{example}
\rm{
We write
$e^x = 1 + x + \frac{1}{2} x^2 + O(x^3)$,
as $x \rightarrow 0$. We could, of course, also write $e^x -(1 + x + \frac{1}{2} x^2 ) = O(x^3)$, though the
former way of writing it is much more common.
In this case, we can give a `better' Big-O estimate by adding more terms of the Taylor series:
$e^x = 1 + x + \frac{1}{2} x^2 + \frac{1}{6} x^3 + O(x^4)$,
as $x \rightarrow 0$.
\hfill $\square$
}
\end{example}
\begin{example}
\rm{
Given $f(x) := x \sin \left(\frac{1}{x}
\right)$, and $g(x) := x^2 - 12$, we have
\[
f(x) = O( g(x) ), \text{ as } x \rightarrow \infty.
\]
In other words, for all sufficiently large $x$, $|f(x)| \leq C g(x)$, despite the fact that
this statement is false for these particular functions, for some small positive values of $x$.
\hfill $\square$
}
\end{example}
One of the useful properties of the Big-O notation is transitivity:
if $f = O(g)$, and $g = O(h)$, then $f= O(h)$.
\begin{proof}
For all $x$ sufficiently close to $x_0$, there exists positive constants $C_1, C_2$ such that
$|f(x)| \leq C_1 |g(x)|$ and $|g(x)| \leq C_2 |h(x)|$, implying that
\[
|f(x)| \leq C_1 |g(x)| \leq C_1 C_2 |h(x)|.
\]
\end{proof}
\bigskip
\section{little-o}
We are again given two functions $f, g:\mathbb{R}^d \rightarrow \mathbb{C}$.
We say that $f(x) = o( g(x) )$ (pronounced ``little o''), as $x \rightarrow x_0$,
if {\bf for all positive constants} $C$, we have:
\begin{equation}
|f(x)| \leq C | g(x) |,
\end{equation}
for all $x$ that are sufficiently close to $x_0$. Again
we allow $x_0$ to be any real vector, and we also allow the very common case $x_0 = \pm \infty$.
Equivalently, we may also write
\[
\lim_{x \rightarrow x_0} \left| \frac{ f(x) }{ g(x) } \right| = 0,
\]
which intuitively means that $g$ approaches $x_0$ faster than $f$ does.
\begin{comment}
\begin{example}
\rm{
Given $f(x) := 3^x e^{-x \log x}$, and $g(x) := 2^{-x} $, we have
\[
f(x) = o( g(x) ), \text{ as } x \rightarrow \infty.
\]
Proof. \ We have
\[
\lim_{x \rightarrow \infty} \left| \frac{ f(x) }{ g(x) } \right| =
\lim_{x \rightarrow \infty} \left| \frac{ 3^x }{ x^x } \right| 2^x = 0.
\]
So for large values of $x$, both functions are decreasing, but $g$ is decreasing much faster than $f$.
\hfill $\square$
}
\end{example}
\end{comment}
\begin{example}
\rm{
Given $f(x) := \sqrt x$, and $g(x) :=x$, where we restrict the domain of
both functions to be $(0, +\infty)$.
We claim
$f(x) = o( g(x) ), \text{ as } x \rightarrow 0$.
\begin{proof}
\[
\lim_{x \rightarrow 0} \left| \frac{ f(x) }{ g(x) } \right| =
\lim_{x \rightarrow 0} \left| \frac{ \sqrt x }{ x } \right| =
\lim_{x \rightarrow 0} \left| \frac{ 1}{ \sqrt x } \right|
= 0.
\]
So $g$ approaches $0$ much faster than $f$.
\end{proof}
}
\end{example}
\medskip
Here we have transitivity as well: if $f = o(g)$ and $g=o(h)$, then $f=o(h)$.
\begin{proof}
The two given limits
$\lim_{x \rightarrow x_0} \left| \frac{ f(x) }{ g(x) } \right| = 0$ and
$\lim_{x \rightarrow x_0} \left| \frac{ g(x) }{ h(x) } \right| = 0$ together imply that
\[
\lim_{x \rightarrow x_0} \left| \frac{ f(x) }{ h(x) } \right|
=\lim_{x \rightarrow x_0} \left| \frac{ f(x) }{ g(x) } \right| \left| \frac{ g(x) }{ h(x) } \right|
= 0.
\]
\end{proof}
\bigskip
\chapter{Appendix B: Various forms of convergence}
\label{Appendix B}
\section{Weierstrass M-test}
How can we quickly conclude that certain series converge uniformly? The following criterion, discovered by Karl Weierstrass, comes to the rescue.
\begin{thm} \rm{[Weierstrass M-test]}
Suppose that $f_n(x)$ is a sequence of complex-valued functions defined on a set $E\subset \mathbb{R}$,
such that there exists a sequence of numbers $M_n\geq 0$ satisfying the following conditions:
\begin{enumerate}
\item $ |f_{n}(x)| \leq M_{n}, \ \forall n \in \mathbb{Z} \text{ and all } x \in E$.
\item $\sum_{n \in \mathbb{Z}} M_n < \infty$.
\end{enumerate}
Then the series
$\sum _{n\in \mathbb{Z}} f_n(x) $
converges absolutely and uniformly on $E$.
\hfill $\square$
\end{thm}
In practice, the Weierstrass $M$-test gets used together with the following test, which allows us to partially answer the question:
\begin{question}
When does a series $\sum_{n\in \mathbb{Z}} f_n(x)$ converge to a continuous function of $x$?
\end{question}
\begin{thm} \label{uniform limit test} \rm{[Uniform limit]}
Suppose that $s_n(x):E \rightarrow \mathbb{C}$ is a sequence of continuous functions defined on a set $E\subset \mathbb{R}$,
and that $s_n$ converges uniformly to $s(x)$, on $E$. Then $s(x)$ is continuous on $E$.
\hfill $\square$
\end{thm}
\bigskip
\section{Some things you wanted to know about convergence but were afraid to ask}
It's often useful to pass from $L^2$ convergence to pointwise convergence, under some additional hypothesis on $f$. Throughout, we fix a real number $1\leq p < \infty$. Given a subset $E\subset \mathbb{R}^d$, and a sequence of functions $f_n:E \rightarrow \mathbb{C}$, we say that
$f_n(x) \rightarrow f(x)$ in the $p$-norm} if
\begin{equation} \label{convergence in L^p norm, take 2}
\lim_{n \rightarrow \infty}
\int_{E} \left| f_n(x) - f(x) \right |^p dx =0,
\end{equation}
for which we will also use here the notation $\lim_{n\rightarrow \infty} \| f_n - f \|_{L^p(E)} = 0$.
Sometimes, if the constant $p$ is not specified, it is common to simply call \eqref{convergence in L^p norm, take 2}
{\bf convergence in norm}.
The two most common subsets are $E:= \mathbb{R}^d$, and $E:= [0, 1]^d$.
A natural question arises:
\begin{question}
When can we pass from convergence in norm to pointwise convergence?
\end{question}
Given a series $\sum_{n \in \mathbb{Z}} f_n(x)$, we consider the sequence of partial sums
$S_N(x):= \sum_{|n| < N} f_n(x)$.
By definition, we say the series converges
\begin{enumerate}
\item {\bf pointwise on $E$} if the sequence $\{S_N(x)\}_{N=1}^\infty$ converges, for each $x \in E$.
\item {\bf absolutely on $E$} if the series $\sum_{n \in \mathbb{Z}} |f_n(x)|$ converges pointwise,
for each $x \in E$.
\item {\bf uniformly on $E$} if the sequence of partial sums $S_N(x)$ converge uniformly on $E$.
\item {\bf in the $p$-norm} on $E$ if $\lim_{n\rightarrow \infty} \| f_n - f \|_{L^p(E)} = 0$.\end{enumerate}
\medskip
\begin{lem} \label{technical equality a.e.}
Consider the partial sums
\[
S_N(x):= \sum_{|n| < N \atop n \in \mathbb{Z}^d} f_n(x),
\]
for all $x$ in a given subset $E\subset \mathbb{R}^d$. Suppose we have the following two properties:
\begin{enumerate}
\item There exists a function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ such that $S_N(x) \rightarrow f(x)$ in the $p$-norm, on $E$.
\item $S_N(x)$ converges uniformly to the series $S(x):= \sum_{n\in \mathbb{Z}^d} f_n(x)$ on $E$.
\end{enumerate}
Then:
\[
S(x) = f(x) \text{ a.e. on } E.
\]
\hfill $\square$
\end{lem}
\begin{lem} \label{absolute convergence of Fourier series implies continuity}
Let $f\in L^1([-c,c])$, and suppose we already know that its Fourier series converges pointwise:
\begin{equation} \label{Fourier series converging pointwise}
f(x) = \sum_{n\in \mathbb{Z}} \hat f_n e^{\frac{2\pi i n x}{c}},
\end{equation}
If the series \eqref{Fourier series converging pointwise}
converges absolutely, then $f$ is a continuous function on $[-c,c]$, and $f(-c) = f(c)$.
\end{lem}
\begin{proof}
The idea is to use the uniform limit Theorem \ref{uniform limit test}, together with the fact that the summands $\hat f_n e^{\frac{2\pi i n x}{c}}$ are continuous functions of $x$. So it remains to show that the convergence of the series is uniform.
\[
|S_N(x)| :=
\left| \sum_{|n|<N} \hat f_n e^{\frac{2\pi i n x}{c}} \right|
\leq \sum_{|n|<N} \left| \hat f_n e^{\frac{2\pi i n x}{c}} \right|
= \sum_{|n|<N} | \hat f_n | < \infty,
\]
where the penultimate equality holds because $| e^{\frac{2\pi i n x}{c}} |=1$, and the last inequality holds by assumption.
So by the $M$-test, with $M_n := | \hat f_n |$, we have uniform convergence of the series.
Finally, the claim $f(-c) = f(c)$ is trivial, because $f(\pm c) := \sum_{n\in \mathbb{Z}} \hat f_n e^{\pm 2\pi i n} = \sum_{n\in \mathbb{Z}} \hat f_n$.
\end{proof}
In the previous lemma, we could have also used the alternate notation of the circle $ \mathbb{R}/c\mathbb{Z}$, and rewrite everything
in terms of it, which automatically incorporates periodicity.
The following passage from convergence in the $ L^2([-c,c])$ norm, to pointwise convergence, is often useful.
\begin{lem} \label{norm convergence plus absolute convergence implies equality}
Let $f\in L^2([-c,c])$ be a continuous function, and write its Fourier series as
\begin{equation} \label{L^2 convergent series}
f(x) \underset{L^2([-c,c])}{=}
\sum_{n\in \mathbb{Z}} \hat f_n e^{\frac{2\pi i n x}{c}},
\end{equation}
which by definition means that this series converges in the $L^2([-c,c])$-norm. If the series \eqref{L^2 convergent series}
converges absolutely, then it also converges pointwise and uniformly to $f(x)$, for all $x \in [-c,c]$.
\end{lem}
\begin{proof}
Repeating the computation of the previous proof, we have:
\[
|S_N(x)| :=
\left| \sum_{|n|<N} \hat f_n e^{\frac{2\pi i n x}{c}} \right|
\leq \sum_{|n|<N} \left| \hat f_n e^{\frac{2\pi i n x}{c}} \right|
\leq \sum_{|n|<N} | \hat f_n | < \infty,
\]
Therefore by the $M$-test again, the sequence $S_N(x)$ converges
uniformly to the series $S(x):= \sum_{n\in \mathbb{Z}} \hat f_n e^{\frac{2\pi i n x}{c}}$, for all $x \in [-c, c]$.
We also know, by Lemma \ref{absolute convergence of Fourier series implies continuity},
that $S(x)$ is continuous on $[-c, c]$.
We still need to prove that the series converges to $f$, but now we at least know that both hypotheses of Lemma \ref{technical equality a.e.} are satisfied (with $p=2$ and $E:= [-c, c]$),
and therefore $S(x) = f(x) \text{ a.e. on } E$.
To prove that $S(x) = f(x)$ for all $x \in [-c, c]$, we observe that the summands $\hat f_n e^{\frac{2\pi i n x}{c}}$ are continuous functions of $x$, and hence by the uniform limit theorem (Theorem \ref{uniform limit test}), the series $S(x)$ is itself a continuous function of $x$. Since $f$ is also continuous on $[-c, c]$, and $S(x) = f(x)$ almost everywhere, they must agree everywhere.
\end{proof}
\begin{comment}
\begin{lem}
Suppose a continuous function $F:\mathbb{R}^d\rightarrow \mathbb{C}$ is represented by a series
\begin{equation}
F(x) \underset{L^2(\mathbb{R}^d)}{=}
\sum_{n\in \mathbb{Z}} a_n(x),
\end{equation}
where the convergence is in $L^2(\mathbb{R}^d)$-norm, and where each $a_n(x)$ is continuous on $\mathbb{R}^d$. If the series
converges absolutely, then it also converges pointwise and uniformly to $F(x)$, for all $x \in \mathbb{R}^d$.
\end{lem}
\end{comment}
\section{Bump functions}
Perhaps the easiest bump function to define is the function (see \cite{SteinShakarchi}, page 209):
\begin{equation*}
\varphi(x):=
\begin{cases}
ce^{-\frac{1}{1-\|x\|^2}} & \text{ if } \|x\|<1, \\
0 & \text{ if } \|x\| \geq1. \\
\end{cases}
\end{equation*}
By definition, $\varphi$ is compactly supported, on the unit ball. Here the constant $c$ is chosen so that
$\int_{\mathbb{R}^d} \varphi(x) dx = 1$. It turns out that $\varphi$ is infinitely smooth.
As usual, using $\varphi$ we can build a family of integrable functions:
\begin{equation*}
\varphi_{\varepsilon}(x):=\varepsilon^{-d}\varphi(x\varepsilon^{-1}), \text{ for all } 0<\varepsilon\leq1.
\end{equation*}
Thus, the family $\{\varphi_\varepsilon\}$ is an approximate identity.
More generally, a {\bf bump function} is defined to be any infinitely smooth function $\varphi:\mathbb{R}^d\rightarrow \mathbb{C}$ that is compactly supported.
By Lemma \ref{useful Schwartz fact}, we know that any such bump function
$\varphi$ lies in the Schwartz class $S(\mathbb{R}^d)$. Clearly finite linear combinations of bump functions are again bump functions, making the space of bump functions a vector subspace of the space
of Schwartz functions.
\chapter{Solutions and hints}
\begin{quote}
``There are no problems, just pauses between ideas.''
-- David Morrell, Brotherhood of the Rose
\end{quote}
{\bf \Large Chapter \ref{Chapter.Tiling.A.Rectangle}}
\bigskip
\index{tiling}
\bigskip
Exercise \ref{TrivialExponential}
\quad
By Euler, we have $1 = e^{i \theta} = \cos\theta + i\sin\theta$, which holds
if and only if
$\cos\theta = 1$, and $\sin\theta = 0$. The latter two conditions hold simultaneously if and only if
$\theta \in 2 \pi k$, with
$k \in \mathbb{Z}$.
\medskip
Exercise \ref{bound of the exponential function}
\quad Let $z := a + bi$, so that $|e^z| = |e^{a+bi}| =|e^a| | e^{bi} | = e^a \cdot 1
\leq e^{\sqrt{ a^2 + b^2}} = e^{|z|}$.
\bigskip
Exercise \ref{orthogonality for exponentials}
\quad
In case $a \not= b$, we have
\begin{equation*}
\int_0^1 e_a(x) \overline{e_b(x)} dx = \int_0^1 e^{2\pi i (a-b) x} dx
= \frac{e^{2\pi i (a-b)}}{2\pi i(a-b)} - 1=0,
\end{equation*}
because we know that $a-b \in \mathbb{Z}$. In case $a = b$, we have
\begin{equation*}
\int_0^1 e_a(x) \overline{e_a(x)} dx = \int_0^1 dx = 1.
\end{equation*}
\bigskip
Exercise \ref{definition of complex integral} \
\quad By definition,
\begin{align*}
\int_{[0,1]} e^{-2\pi i \xi x} dx &:= \int_{[0,1]} \cos(2\pi \xi x) dx + i \int_{[0,1]} \sin(2\pi \xi x) dx\\
&= \frac{\sin(2\pi \xi)}{2\pi\xi} +i \frac{-\cos(2\pi \xi)+1}{2\pi\xi}\\
&= \frac{i\sin(2\pi \xi)}{2\pi i \xi} + \frac{\cos(2\pi \xi)-1}{2\pi i \xi}\\
&= \frac{e^{2\pi i \xi}-1}{2\pi i \xi}.
\end{align*}
\bigskip
Exercise \ref{SumOfRootsOfUnity} \
\quad Let $S:= \sum_{k = 0}^{N-1} e^{\frac{2\pi i k}{N}}$, and note that we may write
\[
S = \sum_{k\text{ mod } N} e^{\frac{2\pi i k}{N}}.
\]
Now, pick any $m$ such that $e^{\frac{2\pi i m}{N}} \not=1$. Consider
\begin{align*}
e^{\frac{2\pi i m}{N}} S &= \sum_{k\text{ mod } N} e^{\frac{2\pi i (k + m)}{N}} \\
&= \sum_{n\text{ mod } N} e^{\frac{2\pi i n}{N}} = S,
\end{align*}
so that $0=( e^{\frac{2\pi i m}{N}} -1)S$, and since
by assumption $e^{\frac{2\pi i m}{N}} \not=1$, we have $S = 0$.
\bigskip
Exercise \ref{DivisibilityUsingExponentials}
\quad
We use the finite geometric series:
$1+x+ x^2 + \cdots + x^{N-1} = \frac{x^{N} - 1}{x-1}$. Now, if $N \not{\mid} M$, then $x:= e^{\frac{2\pi i M}{N}} \not=1$, so we may substitute this value of $x$ into the finite geometric series to get:
\begin{align*}
\frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i kM}{N}} &=
\frac{ e^{\frac{2\pi i MN}{N}}- 1}{ e^{\frac{2\pi i M}{N}}-1} \\
&= \frac{0}{e^{\frac{2\pi i M}{N}}-1}=0.
\end{align*}
On the other hand, if $N \mid M$, then $ \frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i kM}{N}} =
\frac{1}{N} \sum_{k = 0}^{N-1} 1 = 1$.
\bigskip
Exercise \ref{Orthogonality.for.roots.of.unity} \
\begin{align*}
\frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i ka}{N}} e^{-\frac{2\pi i kb}{N}}
&= \frac{1}{N} \sum_{k = 0}^{N-1} e^{\frac{2\pi i k(a-b)}{N}}. \\
\end{align*}
Therefore, using Exercise \ref{DivisibilityUsingExponentials}, we see that the latter sum equals
$1$ exactly when $N \mid a-b$, and vanishes otherwise.
\bigskip
Exercise \ref{trick-write an integer as a product with roots of unity} \
\quad We begin with the factorization of the polynomial
$x^n-1= \prod_{k=1}^n (x - \zeta^k)$, with $\zeta:= e^{2\pi i /n}$. Dividing both sides by
$x-1$, we obtain $1+ x + x^2 + \cdots + x^{n-1} = \prod_{k=1}^{n-1} (x - \zeta^k)$. Now
substituting $x=1$, we have $n = \prod_{k=1}^{n-1} (1 - \zeta^k)$.
\bigskip
Exercise \ref{PrimitiveRootsOfUnity} \
\quad Suppose to the contrary, that a primitive $N$'th root of unity is of the form $e^{2\pi i m/N}$, where
$\gcd(m,N) > 1$. Let $m_1 := \frac{m}{\gcd(m, N)}$, and $k:=\frac{N}{\gcd(m, N)}$, so that by assumption both
$m_1$ and $k$ are integers. Thus $e^{2\pi i m/N} = e^{2\pi i m_1/k}$, a $k$'th root of unity, with $k < N$, a contradiction.
\bigskip
Exercise \ref{zeros of the sin function} \
\quad We recall Euler's identity:
\[
e^{iw} = \cos w + i \sin w,
\]
which is valid for all $w \in \mathbb C$. Using Euler's identity first with $w:= \pi z$, and then with $w := -\pi z$, we have the two identities $e^{\pi i z} = \cos \pi z + i \sin \pi z$, and $e^{-\pi i z} = \cos \pi z - i \sin \pi z$. Subtracting the second identity from the first, we have
\[
\sin(\pi z) = \frac{1}{2i}\left( e^{\pi i z} - e^{-\pi i z} \right).
\]
Now it's clear that $\sin(\pi z) = 0 \iff e^{\pi i z} = e^{-\pi i z} \iff e^{2\pi i z} = 1 \iff z \in \mathbb{Z}$, by Exercise~\ref{TrivialExponential}.
\bigskip \medskip
Exercise \ref{Erdos lattice partition problem} \
\quad We will assume, to the contrary,
that we only have one arithmetic progression with a common difference of $a_N$, the largest of the common differences.
We hope to obtain a contradiction. To each arithmetic progression $\{ a_k n + b_k \mid n \in \mathbb{Z}\}$, we associate the generating function
\[
f_k(q):= \sum_{a_k n + b_k \geq 0, \ n\in \mathbb{Z}} q^{a_k n + b_k },
\]
where $|q| < 1$, in order to make the series converge. The hypothesis that we have a tiling
\index{tiling}
of the integers by these $N$
arithmetic progressions translates directly into an identity among these generating functions:
\[
\sum_{a_1 n + b_1 \geq 0, \ n\in \mathbb{Z}} q^{a_1 n + b_1 } + \cdots +
\sum_{a_N n + b_N \geq 0, \ n\in \mathbb{Z}} q^{a_N n + b_N } = \sum_{n=0}^\infty q^n.
\]
Next, we use the fact that we may rewrite each generating function in a `closed form' of the following kind, because they are geometric series:
$f_k(q):= \sum_{a_k n + b_k \geq 0, \ n\in \mathbb{Z}} q^{a_k n + b_k } = \frac{q^{b_k}}{1-q^{a_k}}$. Thus, we have:
\[
\frac{q^{b_1}}{1-q^{a_1}} + \cdots + \frac{q^{b_N}}{1-q^{a_N}} = \frac{1}{1-q}.
\]
Now we make a `pole-analysis' by observing that each rational function $f_k(q)$ has poles at precisely all of the $k$'th roots of unity.
The final idea is that the `deepest' pole, namely $e^{ \frac{2\pi i}{N} }$, cannot cancel with any of the other poles. To make this idea precise,
we isolate the only rational function that has this pole (by assumption):
\[
\frac{q^{b_N}}{1-q^{a_N}} = \frac{1}{1-q} - \left( \frac{q^{b_1}}{1-q^{a_1}} + \cdots + \frac{q^{b_{N-1}}}{1-q^{a_{N-1}}} \right).
\]
Finally, we let $q\rightarrow e^{ \frac{2\pi i}{N} }$, to get a finite number on the right-hand-side,
and infinity on the left-hand-side of the latter identity, a contradiction.
\bigskip \bigskip
{\bf \Large Chapter \ref{Chapter.Examples}}
\medskip
Exercise \ref{transform.of.interval.a.to.b} \
\quad If $\xi = 0$, we have $\hat 1_{[a,b]}(0) := \int_a^b e^0 dx = b-a$.
If $\xi \not=0$, we can compute the integral:
\begin{align*}
\hat 1_{[a,b]}(\xi) &:= \int_a^b e^{-2\pi i \xi x} dx \\
&=\frac{e^{-2\pi i \xi b} - e^{-2\pi i \xi a} }{-2\pi i \xi}.
\end{align*}
\medskip
Exercise \ref{transform.of.unit.cube} \
\quad Beginning with the definition of the Fourier transform of the unit cube $[0,1]^d$, we have:
\begin{align*}
\hat 1_{\square}(\xi) &= \int_{\square} e^{2\pi i \langle x, \xi \rangle}dx \\
&= \int_0^1 e^{2\pi i \xi_1 x_1} dx_1 \int_0^1 e^{2\pi i \xi_2 x_2} dx_2 \cdots
\int_0^1 e^{2\pi i \xi_d x_d} dx_d \\
&= \frac{1}{(-2\pi i)^d} \prod_{k=1}^d \frac{ e^{-2\pi i \xi_k} -1 }{ \xi_k },
\end{align*}
valid for all $\xi \in \mathbb{R}^d$, except for the finite union of hyperplanes defined by \\
$H := \{ x \in \mathbb{R}^d \mid \xi_1 = 0 \text{ or } \xi_2 = 0 \dots \text{ or } \xi_d = 0 \}$.
\medskip
Exercise \ref{brute force Bernoulli polys} \
\quad To see that the generating-function definition of the Bernoulli polynomials in fact gives polynomials, we first write the Taylor series of the following two analytic functions:
\[
\frac{t}{e^t - 1} = \sum_{k=0}^\infty \frac{B_k}{k!} t^k
\]
\[
e^{xt} = \sum_{j=0}^\infty \frac{ x^j t^j}{j!}.
\]
Multiplying these series together by brute-force gives us:
\begin{align}
\frac{t}{e^t - 1} e^{xt} &= \left( \sum_{k=0}^\infty \frac{B_k}{k!} t^k \right)
\left( \sum_{j=0}^\infty \frac{ x^j}{j!} t^j \right) \\
&= \sum_{n=0}^\infty \left( \sum_{j+k = n} \frac{B_k}{k!} \frac{ x^j}{j!} \right) t^n \\
&= \sum_{n=0}^\infty \left( \sum_{k = 0}^n \frac{B_k}{k!} \frac{ x^{n-k}}{(n-k)!} \right) t^n.
\end{align}
The coefficient of $t^n$ on the LHS is by definition $\frac{1}{n!} B_n(x)$, and by uniqueness of Taylor series, this must also be the coefficient on the RHS, which is seen here to be a polynomial in $x$. In fact, we see more, namely that
\[
\frac{1}{n!} B_n(x) = \sum_{k = 0}^n \frac{B_k}{k!} \frac{ x^{n-k}}{(n-k)!},
\]
which can be written more cleanly as
$ B_n(x) = \sum_{k = 0}^n {n\choose k} B_k x^{n-k}$.
\medskip
Exercise \ref{Reflection property for B_n(x)} \
\quad Commencing with the generating-function definition of the Bernoulli polynomials,
equation
\ref{generating function for Bernoulli polynomials},
we replace $x$ with $1-x$ in order to observe the coefficients $B_k(1-x)$:
\begin{align*}
\sum_{k=0}^\infty \frac{B_k(1-x)}{k!} t^k &= \frac{te^{t(1-x)}}{e^t - 1} \\
&= \frac{te^t e^{-tx}}{e^t - 1} \\
&= \frac{t e^{-tx}}{1- e^{-t} } \\
&= \frac{-t e^{-tx}}{e^{-t}-1 } \\
&= \sum_{k=0}^\infty \frac{B_k(x)}{k!} (-t)^k,
\end{align*}
where the last equality follows from the definition of the same generating function, namely equation
\ref{generating function for Bernoulli polynomials}, but with the variable $t$ replaced by $-t$.
Comparing the coefficient of $t^k$ on both sides, we have $B_k(1-x) = (-1)^k B_k(x)$.
\medskip
Exercise \ref{difference of Bernoulli polys} \
To show that
$B_n(x+1) - B_n(x) = n x^{n-1}$, we play with:
\begin{align*}
\sum_{k=0}^\infty \left(\frac{B_k(x+1)}{k!} t^k - \frac{B_k(x)}{k!} t^k \right) &=
\frac{te^{t(x+1)}}{e^t - 1} - \frac{te^{t(x)}}{e^t - 1} \\
&= e^t \frac{te^{tx}}{e^t - 1} - \frac{te^{t(x)}}{e^t - 1} \\
&= (e^t - 1) \frac{te^{tx}}{e^t - 1} \\
&= te^{tx} \\
&= \sum_{k=0}^\infty \frac{x^k}{k!} t^{k+1} \\
&= \sum_{k=1}^\infty \frac{x^{k-1}}{(k-1)!} t^{k} \\
&= \sum_{k=1}^\infty \frac{k x^{k-1}}{k!} t^{k}.
\end{align*}
Therefore, again comparing the coefficients of $t^k$ on both sides, we arrive at the required identity.
\medskip
Exercise \ref{derivative of Bernoulli polys} \
\quad We need to show that $\frac{d}{dx} B_n(x) = n B_{n-1}(x)$. Well,
\begin{align*}
\sum_{k=0}^\infty \frac{d}{dx} \frac{B_k(x)}{k!} t^k
&= \frac{d}{dx} \frac{te^{tx}}{e^t - 1} \\
&= t \sum_{k=0}^\infty \frac{B_k(x)}{k!} t^k \\
&= \sum_{k=0}^\infty \frac{B_k(x)}{k!} t^{k+1} \\
&= \sum_{k=1}^\infty \frac{B_{k-1}(x)}{(k-1)!} t^{k} \\
&= \sum_{k=1}^\infty k \frac{B_{k-1}(x)}{k!} t^{k},
\end{align*}
so that comparing the coefficient of $t^n$ on both sides, the proof is complete.
\medskip
Exercise \ref{Dirichlet's convergence test}
\quad Considering the partial sum $S_n:= \sum_{k=1}^n a_k b_k$, we know by Abel summation that
\[
S_n = a_n B_n + \sum_{k=1}^{n-1} B_k(a_k - a_{k+1}),
\]
for each $n \geq 2$, where $B_n := \sum_{k=1}^n b_k$.
By assumption, $|B_n|:= | \sum_{k=1}^n b_k | \leq M$, and the $a_k$'s are going to $0$,
so we see that the first part of the right-hand-side approaches
zero, namely:
$|a_n B_n| := |a_n| | \sum_{k+1}^n b_k| \rightarrow 0$, as $n \rightarrow \infty$.
Next, we have
\[
| \sum_{k=1}^{n-1} B_k(a_k - a_{k+1}) | \leq \sum_{k=1}^{n-1} | B_k| | a_k - a_{k+1} |
\leq M \sum_{k=1}^{n-1} | a_k - a_{k+1} | = M \sum_{k=1}^{n-1} (a_k - a_{k+1}),
\]
where the last equality holds because by assumption the $a_k$'s are decreasing. But the last finite
sum equals $ -M a_{n} + M a_1$, and we have $ \lim_{n\rightarrow \infty} (-M a_{n} +M a_1)= M a_1$, a finite limit.
Therefore $\sum_{k=1}^{n-1} B_k(a_k - a_{k+1})$ converges absolutely, and so $S_n$ converges, as desired.
\medskip
Exercise \ref{exponential sum bound}
\quad We fix $x \in \mathbb{R}-\mathbb{Z}$, and let $z:= e^{2\pi i x}$, which lies on the unit circle, and by assumption $z \not= 1$.
Then
\begin{equation}
\left | \sum_{k= 1}^n e^{2\pi i k x} \right |
= \left | \sum_{k= 1}^n z^k \right | = \left | \frac{z^{n+1} -1}{z-1} \right | \leq \frac{2}{z-1},
\end{equation}
because $|z^{n+1} -1| \leq |z^{n+1}| +1 = 2$. We also have
\begin{align*}
|z-1|^2 &= |e^{2\pi ix}-1||e^{-2\pi ix}-1| = |2-2\cos(2\pi x)| = 4\sin^2(\pi x),
\end{align*}
so that we have the equality
$\left | \frac{2}{z-1} \right | =\left | \frac{1}{\sin(\pi x)} \right |$.
Altogether, we see that
\begin{equation}
\left | \sum_{k= 1}^n e^{2\pi i k x} \right | \leq \frac{1}{ | \sin(\pi x) | }.
\end{equation}
\medskip
Exercise \ref{rigorous convergence of P_1(x)}
\quad We fix $a \in \mathbb{R} - \mathbb{Z}$ and need to prove that $\sum_{m = 1}^\infty \frac{e^{2\pi i m a}}{m}$ converges.
Abel's summation formula \eqref{actual Abel summation} gives us
\[
\sum_{k = 1}^n \frac{e^{2\pi i k a}}{k} = \frac{1}{n}\sum_{r=1}^n e^{2\pi i r a}
+ \sum_{k=1}^{n-1} \Big( \sum_{r=1}^k e^{2\pi i r a} \Big) \frac{1}{k(k+1)},
\]
so that
\[
\sum_{k = 1}^\infty \frac{e^{2\pi i k a}}{k} =
\sum_{k=1}^{\infty} \Big( \sum_{r=1}^k e^{2\pi i r a} \Big) \frac{1}{k(k+1)}.
\]
and the latter series in fact converges absolutely.
\begin{comment}
\medskip
Exercise \ref{rigorous limit formula for sinc}
\quad
The main idea here is to transform everything into exponentials, which we can then explicitly integrate.
For added simplicity, we initially omit the $\pi$'s everywhere. To prove the first part, we compute the integral:
\[
\int_0^\infty e^{-xt} dt = -\frac{1}{x} e^{-xt}\Big|_0^\infty = \frac{1}{x},
\]
valid for all $x >0$.
For the second part, it suffices to prove $\int_0^\infty \frac{\sin x}{ x} dx = \frac{\pi}{2}$.
We will substitute for $\frac{1}{x}$ using part (a), and
then we use Fubini's theorem:
\begin{align*}
\int_{0}^\infty \frac{\sin x}{ x} dx
&= \frac{1}{\pi} \int_{0}^\infty \sin x
\Big(\int_0^\infty e^{-xt} dt \Big) dx \\
&=
\int_0^\infty \int_{0}^\infty \sin x \, e^{-xt} \, dx \, dt.
\end{align*}
To justify the use of Fubini's theorem, we need to check that $ \sin x \, e^{-xt}\in L^1(\mathbb{R}^2)$, which
doesn't seem correct?!?!?
Now we can give an explicit formula for the inner integral, using exponentials. The computation simplifies
if we now let $N\in 2\pi\mathbb{Z}$, and we integrate from $0$ to $N$:
\begin{align*}
\int_0^N \sin x \, e^{-xt} \, dx
&= \frac{1}{2i} \int_0^N \left( e^{ i x} - e^{- i x} \right) e^{-xt} dx \\
&= \frac{1}{2i} \int_0^N \left( e^{-x( t - i ) } - e^{-x( t + i ) } \right) dx \\
&= \frac{1}{2i} \left( \frac{ e^{-x( t - i ) } }{ i -t }
+ \frac{ e^{-x( t + i ) } }{ i + t }
\right) \Big|_{x=0}^{x=N} \\
&= \frac{1}{2i} \left( \frac{ e^{-Nt } }{ i -t }
+ \frac{ e^{-Nt } }{ i + t }
\right)
-
\frac{1}{2i} \left( \frac{ 1 }{ i -t }
+ \frac{ 1 }{ i + t }
\right) \\
&= \frac{ 1- e^{-Nt } }{1+t^2}.
\end{align*}
Letting $N\rightarrow \infty$, we get $\int_0^\infty \sin x \, e^{-xt} \, dx = \frac{1}{1+t^2}.$
Altogether, we have
\[
\int_0^\infty \frac{\sin x}{ x} dx = \int_0^\infty \int_{0}^\infty \sin x \, e^{-xt} \, dx \, dt =
\int_0^\infty \frac{ dt }{1+t^2} = \tan^{-1} (t) \Big|_0^\infty = \frac{\pi}{2}.
\]
\end{comment}
\bigskip \bigskip
{\bf \Large Chapter \ref{Fourier analysis basics}}
\medskip
Exercise \ref{elementary norm relations}
\quad For all four inequalities, we will use an arbitrary vector $a \in \mathbb{R}^d$.
For the first inequality, $a_1^2 + \cdots + a_d^2 \geq \max\{ |a_1|, \dots, |a_d| \}^2 := \|a\|_\infty^2$.
The second inequality $\|a\|_2 \leq \|a\|_1$ means that $\sqrt{a_1^2 + \cdots + a_d^2} \leq |a_1| + \cdots + |a_d|$, which is clear by squaring both sides.
To prove the third and most interesting inequality here, we use the Cauchy-Schwarz inequality, with the two vectors
$x:= (a_1, \dots, a_d)$ and $(1, 1, \dots, 1)$:
\[
\| a \|_1:= |a_1| \cdot 1 + \cdots + |a_d| \cdot 1 \leq \sqrt{ a_1^2 + \cdots + a_d^2} \sqrt{ 1 + \cdots + 1 }
= \sqrt{d} \ \| a \|_2,
\]
which also shows that we obtain equality if and only if $(a_1, \dots, a_d)$ is a
scalar multiple of $(1, 1, \dots, 1)$.
For the fourth inequality, we have:
\[
\sqrt{a_1^2 + \cdots + a_d^2} \leq
\sqrt{ d \max\{ |a_1|, \dots, |a_d| \}^2 }:=
\sqrt{d} \|a\|_\infty.
\]
\medskip
Exercise \ref{exercise:hyperbolic cosine and sine}
\quad
To prove part (a), we compute:
\begin{align*}
\left( \frac{e^{t} + e^{-t}}{2} \right)^2 - \left( \frac{e^{t} - e^{-t}}{2} \right)^2
&= \frac{e^{2t} + 2 + e^{-2t} - \left( e^{2t} - 2 + e^{-2t} \right) }{4} = 1.
\end{align*}
To prove part (b), we begin with the definition of the hyperbolic cotangent:
\begin{align*}
t \coth t &= t\frac{ e^t + e^{-t}}{e^t - e^{-t}} = t \frac{ e^t }{e^t - e^{-t}} + t \frac{ e^{-t}}{e^t - e^{-t}} \\
&= \frac{ t }{1 - e^{-2t}} + \frac{ t }{ e^{2t}-1}.
\end{align*}
Recalling the definition of the Bernoulli numbers, namely
$
\frac{t}{e^t-1} = \sum_{k =0}^\infty B_k \frac{t^k}{k!},
$
we see that
\begin{align*}
t \coth t &= \frac{1}{2} \left( \frac{ -2t }{ e^{-2t}-1} \right) + \frac{1}{2} \left( \frac{ 2t }{ e^{2t}-1} \right) \\
&= \frac{1}{2} \sum_{k =0}^\infty B_k \frac{(-2t)^k}{k!} + \frac{1}{2} \sum_{k =0}^\infty B_k \frac{(2t)^k}{k!} \\
&= \sum_{k =0}^\infty \tfrac{1}{2} \left( (-1)^k + 1 \right) B_k \frac{(2t)^k}{k!},
\end{align*}
so the only surviving terms in the latter series are the terms whose index $k$ is
an even integer. This yields
$
t \coth t = \sum_{n=0}^\infty \frac{2^{2n}}{(2n)!} B_{2n} t^{2n}.
$
\medskip
Exercise \ref{compute FT for exponential of abs value}
\quad
We know, by equation \eqref{FT of the abs value exponential}, that the Fourier transform of
$f(x):=e^{-2\pi t |x|}$ is equal to $\hat f(\xi) = \frac{ t }{\pi (t^2 + \xi^2)}$.
So using Poisson summation, we have:
\index{Poisson summation formula}
\begin{align*}
\sum_{n \in \mathbb{Z}} e^{-2\pi t |n|} = \sum_{n \in \mathbb{Z}} f(n)
= \sum_{\xi \in \mathbb{Z}}\hat f(\xi)
= \frac{t}{\pi} \sum_{\xi \in \mathbb{Z}} \frac{1}{\xi^2 + t^2}.
\end{align*}
\medskip
Exercise \ref{Elementary bounds for sin(x), sinc(x)}
We'll prove part \ref{Elementary trig bounds, part b}. To begin, we have:
\begin{align*}
\big| e^{i\theta} - 1 \big|^2 &= \big| \cos \theta -1 + i \sin \theta \big|^2 = (\cos \theta -1)^2 + \sin^2\theta \\
&= 2 - 2 \cos \theta = 4 \sin^2\left(\frac{\theta}{2}\right).
\end{align*}
So it suffices to show that
$4 \sin^2\left(\frac{\theta}{2}\right) \leq \theta^2$, for all $0 \leq \theta \leq 2\pi$.
In other words, the problem is reduced to the Calculus I problem of showing that
$ \sin\left(\frac{\theta}{2}\right) \leq \frac{\theta}{2}$, for $\theta \in [0, 2\pi]$.
To prove this, we let $y(x)=x- \sin x $, so that it suffices to prove that $y \geq 0$ on $[0, \pi]$.
Computing its derivative,
$y'(x) = 1-\cos x \geq 0$ on $[0, \pi]$, and since $y(0) = 0$, we conclude that $y$ is an increasing function. This proves $y \geq 0$ on $[0, \pi]$.
\medskip
Exercise \ref{positive FT over R}
We need to show that there exist two real numbers $r, s$ such that
\[
f:= 1_{[-r, r]}*1_{[-r, r]} + 1_{[-s, s]}*1_{[-s, s]}
\]
enjoys the property:
\[
\hat f(\xi) >0,
\]
for all $\xi \in \mathbb{R}$. Let's pick any two real numbers $r, s$ that are incommensurable, meaning that
$\frac{r}{s} \notin \mathbb{Q}$. Using \eqref{Stretch lemma for the sinc function}, we compute $\hat f$:
\[
\hat f(\xi):= \Big( \hat 1_{[-r, r]}(\xi) \Big)^2 + \Big( \hat 1_{[-s, s]}(\xi) \Big)^2 =
\left( \frac{ \sin(2r\pi \xi) }{ \pi \xi } \right)^2 + \left( \frac{ \sin(2s\pi \xi) }{ \pi \xi } \right)^2 \geq 0.
\]
To prove positivity, suppose to the contrary that there exists a nonzero $\xi\in \mathbb{R}$ such that $\hat f(\xi)=0$. Then
$\left( \sin(2r\pi \xi) \right)^2 + \left( \sin(2s\pi \xi) \right)^2 = 0$, but the vanishing of a sum of two squares implies that
they must both equal $0$:
\[
\sin(2r\pi \xi) =0, \text{ and } \sin(2s\pi \xi) = 0.
\]
Therefore $2r \pi \xi = m \pi$ and $2s \pi \xi = n \pi$, for some integers $m, n$. We conclude that $\xi = \frac{m}{2r} = \frac{n}{2s}$,
so $\frac{r}{s}=\frac{m}{n} \in \mathbb{Q} $, a contradiction that proves $\hat f(\xi) >0$ for all nonzero real $\xi$.
\bigskip
Exercise \ref{tricky application of Poisson summation}
By assumption, $g:\mathbb{R}^d\rightarrow \mathbb{C}$ is infinitely smooth, and compactly supported.
By Corollary \ref{cor: f smoother implies FT of F decays faster}, $\hat g$ is a rapidly decreasing function. Because $g$ has compact support, we also know that $\hat g$ is infinitely smooth. So $\hat g$ is a Schwartz function (and
$g$ is also a Schwartz function - in fact $g$ is a `bump function', by definition). Therefore we may apply the Poisson summation formula for Schwartz functions (Theorem \ref{Poisson.Summation}) to $\hat g$:
\[
\sum_{\xi \in \mathbb{Z}^d} \hat g(\xi) = \sum_{n \in \mathbb{Z}^d} g(n),
\]
which is a finite sum due to the compact support of $g$.
\bigskip \bigskip
{\bf \Large Chapter \ref{Geometry of numbers}}
\medskip
Exercise \ref{c.s. C equals its symmetrized body}
\quad
For part (a), we suppose that
\begin{equation} \label{symmetrized toy}
\frac{1}{2}C - \frac{1}{2}C = C.
\end{equation}
Let's pick any $x \in C$; we need to
show that $-x \in C$. Since $x \in \frac{1}{2}C - \frac{1}{2}C$,
we know that there must exist $y, z\in C$ such that $x=\frac{1}{2}y - \frac{1}{2}z$.
This implies that $-x = \frac{1}{2}z -\frac{1}{2}y \in \frac{1}{2}C - \frac{1}{2}C\subseteq C$. Therefore $C$ is centrally symmetric.
To show part (b), we note that the convexity of $C$ gets used in the step
$\frac{1}{2}C + \frac{1}{2}C = C$.
First, we suppose that $C$ is centrally symmetric, so that $C = -C$. This implies that
$\frac{1}{2}C - \frac{1}{2}C
= \frac{1}{2}C + \frac{1}{2}(-C) = \frac{1}{2}C + \frac{1}{2}(C) = C$.
Conversely, suppose that $C = \frac{1}{2}C - \frac{1}{2}C$. Then by part (a), we already know that
$C$ is centrally symmetric.
For part (c), consider the counter-example $C:= [-2, -1] \cup [1, 2]$, a nonconvex set in $\mathbb{R}$. Here $C$ is centrally symmetric, yet
$C - C = [-3, 3] \not= [-4, -2] \cup [2, 4] = 2C$.
\bigskip
Exercise \ref{support of convolution}
\quad To prove part (a), we are given two convex bodies $A, B \subset \mathbb{R}^d$, so by definition we have
\[
\supp( 1_A * 1_B) := \closure \left\{ y \in \mathbb{R}^d \bigm | \int_{\mathbb{R}^d} 1_A(x) 1_B(y-x) dx \not= 0 \right\},
\]
and we must prove that $\supp( 1_A * 1_B) = A + B$, their Minkowski sum. \index{Minkowski sum}
In general, we have:
\begin{align} \label{equivalences for Minkowski sums}
1_A(x) 1_B(y-x)>0
& \iff 1_A(x) =1 \text{ and } 1_B(y-x) = 1 \\
& \iff x\in A \text{ and } y-x \in B \\
& \iff y \in A+B.
\end{align}
If we fix any $y \notin \supp( 1_A * 1_B)$,
then $\int_{\mathbb{R}^d} 1_A(x) 1_B(y-x) dx = 0$, which implies that $1_A(x) 1_B(y-x)=0$ for all $x\in \mathbb{R}^d$.
But by the equivalences \eqref{equivalences for Minkowski sums}
above, we see that
$1_A(x) 1_B(y-x)=0 \iff y\notin A+B$, proving that $ A + B \subset \supp( 1_A * 1_B)$.
Conversely, suppose that $y \in \supp( 1_A * 1_B)$, meaning that there exists a sequence $y_n\in \mathbb{R}^d$ with
$ \int_{\mathbb{R}^d} 1_A(x) 1_B(y_n-x) dx \not= 0$. This implies that for each such $y_n$, there exists at least
one $x \in \mathbb{R}^d$ with $1_A(x) 1_B(y_n-x) >0$. This last inequality, using our
our equivalences \eqref{equivalences for Minkowski sums}, implies that the sequence $y_n \in A+B$.
Because $A+B$ is a closed set, we finally have $y := \lim_{n\rightarrow \infty} y_n \in A+B$.
\bigskip
To prove part (b), we must show that
$ \supp(f*g)\subseteq C$, where
\[
C:= \closure\left( \supp(f) + \supp(g) \right).
\]
We'll prove the contrapositive: if $x \notin C$, then $x \notin \supp(f*g)$ . So we suppose $x \notin C$, and we have to prove that $(f*g)(x)=0$.
By our assumption on $x$, for each $y \in \supp(g)$, we have that $x-y \notin \supp(f)$. The last assertion means that $f(x-y) = 0$, so we now know that $f(x-y) g(y)=0$ for all $y\in \mathbb{R}^d$.
Finally, we have $(f*g)(x) := \int_{\mathbb{R}^d} f(x-y) g(y) dy = 0$.
\bigskip
Exercise \ref{non-unimodular but empty simplex}
\quad
Define $\Delta:= \conv\{ (0, 0, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1)\}$, an integer $3$-simplex. It's clear that
$\Delta$ is subset of the unit cube $[0, 1]^3$, and therefore $\Delta$ has no integer points in its interior.
To see that $\Delta$ is not a unimodular simplex, we can consider its tangent $K_0$ cone at the origin, which has primitive integer vectors $(1, 1, 0), (1, 0, 1), (0, 1, 1)$, so that the determinant of $K_0$ is
equal to $\left| \det
\begin{pmatrix}
1 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 1
\end{pmatrix}
\right| = 2 > 1.
$
\medskip
Exercise \ref{FT of a polytope is not Schwartz}
\quad
Suppose to the contrary, that for some polytope ${\mathcal P}$ we have $\hat 1_{{\mathcal P}}(\xi)= g(\xi)$, a Schwartz function. Taking the Fourier transform of both sides of the latter equality, and using the fact that the Fourier transform takes Schwartz functions to Schwartz functions, we conclude that $1_{{\mathcal P}}(-x) = \hat g(-x)$ is a Schwartz function. But this is a contradiction, because the indicator function of a polytope is not even continuous.
\medskip
Exercise \ref{an application of Cauchy-Schwartz 1}
\quad We use the Cauchy-Schwartz inequality:
\[
{\Big \langle \icol{a \\ b} , \icol{ \sin x \\ \cos x} \Big \rangle}^2
:= ( a \sin x+ b \cos x )^2
\leq \big( a^2 + b^2 \big) \big( \sin^2 x+ \cos^2 x \big) = a^2 + b^2.
\]
By the equality condition of Cauchy-Schwartz, we see that the maximum is obtained when
the two vectors are linearly dependent, which gives
$\tan x = \frac{a}{b}$.
\bigskip \bigskip
{\bf \Large Chapter \ref{chapter.lattices}}
\medskip
Exercise \ref{distance between hyperplanes}
\quad We are given the hyperplanes $H_1:= \{ x\in \mathbb{R}^d \mid c_1 x_1 + \cdots + c_d x_d = k_1\}$,
and $H_2:= \{ x\in \mathbb{R}^d \mid c_1 x_1 + \cdots + c_d x_d = k_2\}$. First we'll pick a point $x \in H_1$,
and then we'll `walk along its normal vector', until we get to $H_2$. With this `walk' in mind, we may assume WLOG
that $k_2 > k_1$, and that the normal vector is pointing from $H_1$ towards $H_2$.
For simplicity, we'll let $L:= \sqrt{ c_1^2 + \cdots + c_d^2}$, and with this definition
the unit normal vector to $H_1$ is
$n:= \frac{1}{ L }(c_1, \dots, c_d)^T$, and we want to find $\delta>0$ such that
$x + \delta n \in H_2$.
Unraveling the definition of the latter statement, we must have
\begin{align*}
& c_1 ( x_1 + \delta \tfrac{1}{L} c_1) + \cdots + c_d (x_d + \delta \tfrac{1}{L} c_d) = k_2 \\
\iff & (c_1 x_1 + \cdots + c_d x_d) + \frac{\delta}{L}( c_1^2 + \cdots + c_d^2) = k_2 \\
\iff & k_1 + \delta\sqrt{c_1^2 + \cdots + c_d^2} = k_2 \\
\iff & \delta = \frac{k_2 - k_1}{\sqrt{c_1^2 + \cdots + c_d^2}}.
\end{align*}
\medskip
Exercise \ref{Hadamard's inequality, exercise}
\quad We consider each $k$'th row of $M$ as a vector, call it $v_k$. By assumption,
the norm of $v_k$ is bounded by $\|v\| \leq \sqrt{ B^2 + \cdots B^2 } = B\sqrt d$.
Using Hadamard's inequality \ref{Hadamard inequality}, we have:
\begin{align*}
|\det M| \leq \|v_1\| \cdots \|v_d\| \leq \left(B\sqrt d \right)^d.
\end{align*}
\medskip
Exercise \ref{Ellipsoid problem}
\quad It's easy to see that the inverse matrix for $M$ is
\[
M^{-1} := \begin{pmatrix} | & | & ... & | \\
\frac{1}{c_1} b_1 & \frac{1}{c_2} b_2 & ... & \frac{1}{c_d} b_d \\
| & | & ... & | \\
\end{pmatrix}^T.
\]
The image of the unit sphere under the matrix $M$ is, by definition:
\begin{align*}
M(S^{d-1}) &:= \{ u \in \mathbb{R}^d \mid u = Mx, x \in S^{d-1} \} \\
&= \{ u \in \mathbb{R}^d \mid M^{-1}u \in S^{d-1} \} \\
&= \{ u \in \mathbb{R}^d \mid \frac{1}{c_1^2} \langle b_1, u \rangle^2 + \cdots +
\frac{1}{c_d^2} \langle b_d, u \rangle^2 = 1 \},
\end{align*}
using our description of $M^{-1}$ above.
For part (b), we begin with the definition of volume, and we want to compute the
volume of the region $M(B):= \{ u \in \mathbb{R}^d \mid u = My, \text{ with } \| y \| \leq 1 \}$,
where $B$ is the unit ball in $\mathbb{R}^d$.
\begin{align*}
\vol(Ellipsoid_M) &:= \int_{M(B)} du \\
&= | \det M | \int_{B} dy \\
&= | \det M | \vol(B).
\end{align*}
using the change of variable $u = My$, with $y \in B$. We also used the Jacobian,
which gives $du = | \det M | dy$.
Finally, we note that the matrix $M^T M$ is a diagonal matrix, with diagonal entries $c_k^2$, due
to the fact that the $b_k$'s form an orthonormal basis. Thus we use: $| \det M |^2 = | \det M^T M |
= \prod_{k=1}^d c_k^2$, so taking the positive square root, we arrive at $| \det M | = \prod_{k=1}^d c_k$,
because all of the $c_k$'s are positive by assumption.
\medskip
Exercise \ref{exercise:2by2 positive definite matrix}
\quad Let $A:=
\left(\begin{smallmatrix} a & b \\ b & d \end{smallmatrix}\right) $ be an invertible, symmetric matrix.
Because $A$ is symmetric, we know both of its eigenvalues $\lambda_1, \lambda_2$ are real.
The characteristic polynomial of $A$, namely
$(a-\lambda)(d-\lambda) - b^2 $, may also be factored and rewritten as
\[
\lambda^2 - (a+ d) \lambda + (ad-b^2) = (\lambda - \lambda_1)(\lambda - \lambda_2) = \lambda^2 - (\lambda_1 + \lambda_2)\lambda + \lambda_1 \lambda_2.
\]
Equating coefficients of the latter identity between polynomials, we therefore have $\lambda_1 + \lambda_2 = {\rm Trace } A$, and $\lambda_1 \lambda_2= \det A$.
From these last two relations, we see that if both eigenvalues are positive, then ${\rm Trace } A>0$ and $\det A>0$.
Conversely, suppose that ${\rm Trace } A>0$ and $\det A>0$.
Then $\lambda_1 \lambda_2 >0$, so either both eigenvalues are positive, or both eigenvalues are negative.
But the eigenvalues cannot both be negative, for this would contradict our assumption that $\lambda_1 + \lambda_2>0$.
\medskip
Exercise \ref{translating the Voronoi cell around}
\quad
Given a full rank lattice ${\mathcal L}\subset \mathbb{R}^d$, and any $m \in {\mathcal L}$, we have:
\begin{align}
{\rm Vor}_0({\mathcal L}) + m
&:=
\left\{ x +m \in \mathbb{R}^d \bigm | \|x\| \leq \|x - v\|, \ \text{ for all } v \in {\mathcal L} \right\} \\ \label{last expression for Voronoi}
&= \left\{ y \in \mathbb{R}^d \bigm | \|y-m\| \leq \|y-m - v\|, \ \text{ for all } v \in {\mathcal L} \right\}.
\end{align}
But as $v$ varies over ${\mathcal L}$, so does $m + v$, because $m \in {\mathcal L}$. Hence the expression
\eqref{last expression for Voronoi} above is ${\rm Vor}_m({\mathcal L})$.
\bigskip \bigskip
{\bf \Large Chapter \ref{chapter.Brion}}
\medskip
Exercise \ref{independent of edge vectors}
We are given $\alpha >0$, and a simplicial cone ${\mathcal K}_v$, with edge vectors $w_1, \dots, w_d \in \mathbb{R}^d$.
By definition, $\det {\mathcal K}_v$ is the determinant of the matrix whose columns are the $w_k$'s.
Replacing each $w_k(v)$ by $\alpha_k w_k(v)$, we see that the determinant $|\det {\mathcal K}_v|$
gets multiplied by $\alpha^d$, and so
\[
\frac{ \alpha^d |\det {\mathcal K}_v| }{\prod_{k=1}^d \langle \alpha w_k(v), z \rangle}
= \frac{ |\det {\mathcal K}_v| }{\prod_{k=1}^d \langle w_k(v), z \rangle}.
\]
\medskip
Exercise \ref{polytope from pentagons}
\quad
Euler's formula gives us
\[
V-E+F =2,
\]
and the hypotheses also imply that:
\begin{align}
5F&=2E \\
5F &\geq 3V.
\end{align}
Altogether, we get
\begin{equation*}
2=V-E+F \leq \frac{5}{3} F - \frac{5}{2} F + F = \frac{1}{6} F,
\end{equation*}
so that $F \geq 12$.
\bigskip \bigskip
{\bf \Large Chapter \ref{chapter:Discrete Brion}}
Exercise \ref{unimodular cone, integer point transform}
\quad
The main point here is that at each vertex $v\in V$, the edge vectors form a basis for $\mathbb{Z}^d$, and therefore
the only integer point in the (half-open) fundamental parallelepiped $\Pi_v$ is $v$ itself. So we see that
its integer point transform is $\sigma_{\Pi_v}(x) = e^{\langle v, z \rangle}$.
Now we use Theorem \ref{brion, discrete form}, followed by Theorem \ref{closed form for integer point transform of a cone}:
\begin{equation*}
\sigma_{\mathcal P}(z) = \sum_{v \in V} \sigma_{{\mathcal K}_v}(z) =
\frac{e^{\langle v, z \rangle} }{\prod_{k=1}^d \left( 1 - e^{\langle w_k, z\rangle} \right) }.
\end{equation*}
Exercise \ref{bound for integer point transform}
\quad
Because $|e^{2\pi i \langle x, n \rangle}|=1$ for all $x\in \mathbb{R}^d$, we have:
\[
\left | \sigma_{\mathcal P}(2\pi i x) \right |
\leq \sum_{n\in {\mathcal P} \cap \mathbb{Z}^d } \left| e^{2\pi i \langle x, n \rangle} \right|
= \sum_{n\in {\mathcal P} \cap \mathbb{Z}^d } 1 = \left |\mathbb{Z}^d \cap {\mathcal P} \right |.
\]
\bigskip \bigskip
{\bf \Large Chapter \ref{Ehrhart theory}}
\medskip
Exercise \ref{Bernoulli polynomial as an Ehrhart polynomial}
\quad
Here ${\mathcal P}:= \conv\{ C, {\bf e_d} \}$, where $C$ is the $(d-1)$-dimensional unit cube $[0, 1]^{d-1}$. To compute the Ehrhart
polynomial ${\mathcal L}_{{\mathcal P}}(t)$ here, we use the fact that a `horizontal' slice of ${\mathcal P}$, meaning a slice parallel to $C$, and orthogonal
to $e_d$, is a dilation of $C$. Thus, each of these slices counts the number of points in a $k$-dilate of $C$, as $k$ varies from $0$ to $t+1$. Summing over these integer dilations of $C$, we have
\[
{\mathcal L}_{{\mathcal P}}(t) = \sum_{k=0}^{t+1} (t+1 - k)^{d-1} = \sum_{k=0}^{t+1} k^{d-1} =
\frac{1}{d}(B_d(t+2) - B_d),
\]
where the last step holds thanks to Exercise \ref{historical origin of Bernoulli poly}.
\medskip
Exercise \ref{unimodular triangle}
Using Pick's formula, the unimodular triangle ${\mathcal P}$ has area:
\[
\rm{Area } {\mathcal P} = I + \frac{1}{2} B -1 = 0 + \frac{1}{2} 3 -1 = \frac{1}{2}.
\]
\medskip
Exercise \ref{properties of floor, ceiling, fractional part}
Throughout, we first write $x = n + \alpha$, with $\lfloor x \rfloor := n \in \mathbb{Z}$ and $0 \leq \alpha < 1$.
We prove part \ref{ex:part 1 of fractional parts}, namely that $- \floor{-x} = \left\lceil x \right\rceil$.
Case $1$: $x \in \mathbb{Z}$. Here $\alpha = 0$ and $x=n$, so that $- \floor{-x} = - (-n)= n = \left\lceil x \right\rceil$.
Case $2$: $x \notin \mathbb{Z}$. In this case
$\left\lceil x \right\rceil = n+1$. We have $-x = -n - \alpha = - n-1 + (1-\alpha)$,
from which we see that $- \floor{-x} = - (-n-1) = n + 1 = \left\lceil x \right\rceil$.
To prove part \ref{ex:part 2 of fractional parts}, we need to show that $\floor{x} - \left\lceil x \right\rceil +1=1_{\mathbb{Z}}(x)$.
Case $1$: $x \in \mathbb{Z} \implies \floor{x} - \left\lceil x \right\rceil +1 = n - n + 1 = 1 = 1_{\mathbb{Z}}(x)$.
Case $2$: $x \notin \mathbb{Z} \implies \floor{x} - \left\lceil x \right\rceil +1
= n - (n + 1) + 1 = 0 = 1_{\mathbb{Z}}(x)$.
To prove part \ref{ex:part 3 of fractional parts}, we need to show that
$ \{ x \} + \{-x\} = 1- 1_{\mathbb{Z}}(x)$.
This follows from part \ref{ex:part 2 of fractional parts} if we use the definitions
$\floor{x} := x - \{x\}, \lceil x \rceil:= x + \{x\}$. Using the identity of part \ref{ex:part 2 of fractional parts}, we have
\[
1- 1_{\mathbb{Z}}(x) = \left\lceil x \right\rceil - \floor{x} = x + \{x\} - ( x - \{x\}) = \{x\} + \{x\}.
\]
Finally, for part \ref{ex:part 4 of fractional parts}, we have to prove that
if $m \in \mathbb{Z}_{>0}, n \in \mathbb{Z}$, then $\floor{ \frac{n-1}{m} } + 1 = \left\lceil \frac{n}{m} \right\rceil$.
We begin by using the division algorithm, which gives us $n=qm+r$, with integers $q$ and $0\leq r < m$.
Case $1$: $r=0$. Here $n = qm$, and we have $\floor{ \frac{n-1}{m} } + 1 = \floor{ q- \frac{1}{m} } + 1=q
= \frac{n}{m} = \left\lceil \frac{n}{m} \right\rceil$.
Case $2$: $0 < r < m$. Here
$\floor{ \frac{n-1}{m} } + 1 = \floor{ \frac{qm + r - 1}{m} } + 1
= \floor{ q + \frac{ r - 1}{m} } + 1 = \floor{ \frac{ r - 1}{m} } + 1 = 1$.
On the other hand, $ \left\lceil \frac{n}{m} \right\rceil = \left\lceil \frac{qm + r}{m} \right\rceil
= \left\lceil q + \frac{ r}{m} \right\rceil = \left\lceil \frac{ r}{m} \right\rceil =1 $.
\bigskip
|
1506.03926
|
\section{Introduction}
Our understanding of the physical cosmology strongly depends on the data that we observe and the model assumptions
that we make. Assumptions in the standard model sometime pose problems in understanding the true nature of the
Universe. Statistical isotropy is one of such assumptions which we assume in almost all the data analysis. In this assumption
we claim that the statistical properties of the observables in the Universe are same in all direction. While in parameter
estimations from the datasets, isotropy is assumed, there have been a number of tests to confirm this assumption, using
data from different cosmological probes, such as Cosmic Microwave Background (CMB)~\cite{Hinshaw:1996ut,Spergel:2003cb,
Copi:2010na,Ade:2013nlj,Akrami:2014eta,de OliveiraCosta:2003pu,Abramo:2006gw,Land:2005ad,Land:2006bn,Rakic:2007ve,
Samal:2007nw,Samal:2008nv,Eriksen:2007pc,Hoftuft:2009rq,Copi:2006tu,Copi:2005ff,Schwarz:2004gk,Souradeep:2006dz} and Large Scale
Structure (LSS)~\cite{Fernandez-Cobos:2013fda,Cai:2013lja,Keenan:2009jh,Keenan:2012gr,Keenan:2013mfa,Whitbourn:2013mwa,
Frith:2003tb,Busswell:2003ta,Frith:2005az,Frith:2004wd,Frith:2004tw,Appleby:2014lra}.
In this paper we investigate isotropy of the matter dominated Universe using the
Lyman-$\alpha$ forest datasets in the redshift range $z \sim 2 -3$. Spectra of high redshift
quasars contain absorption lines that trace the components of the IGM along the line of sight. In the case of Lyman-$\alpha$
forest, we find absorption lines from the first ionization state of the Hydrogen atoms. The wavelength of the absorption
identifies the redshift of the neutral hydrogen cloud. Analyzing these absorption lines, collectively from a number of
quasars, we can constrain properties of the IGM. In particular, the Lyman-$\alpha$ transmitted
flux ($F$) (absorption {\it w.r.t} the estimated continuum spectrum) can be related to the dark matter overdensity
($\delta$) as $F=\bar{F}\exp[-A(1+\delta)^{2-0.7(\gamma-1)}]$~\cite{fdelta}, where, $\bar{F}$ is the mean transmitted
flux, $\gamma-1$ dictates the temperature density relation and $A$ is a redshift dependent constant. We shall address the statistics
of the transmitted flux in the Lyman-$\alpha$ region at different redshifts. To have a theoretical model independent analysis, in this
work we only consider the statistical properties of the observational data. In the last decade, detection of
large number of quasar spectrum with high SNR enables us to perform this type of analysis. We use
Baryon Oscillation Spectroscopic Survey (BOSS) DR9 from Sloan Digital Sky Survey (SDSS)-III~\cite{sdss,boss}. We perform our test
in three different redshifts in the matter dominated epoch that enables us to track the isotropy along time. In each
redshift, the isotropy is tested in medium, high and highest SNR of detection. We divide the sky
in three patches using two different patch selection criteria. Since SDSS is a ground base survey which only has partial sky
coverage, our test of isotropy will be limited by the survey area. Throughout the analysis we closely follow the
survey parameters for the selection of data pixels from the quasar spectrum in order to obtain the
PDF of the transmitted flux.
The paper is organized as follows : In section~\ref{sec:formalism} we discuss the data from BOSS-DR9 and the selection
criteria for the data pixels for our analysis. We also outline the error estimation procedure for the flux PDF.
Afterwards we discuss and tabulate the properties of the sky patches selected. In the results section~\ref{sec:results}
we discuss the main outcome of our analysis. Finally in section~\ref{sec:conclusions} we conclude along with
highlighting future prospects.
\section{Data analysis}~\label{sec:formalism}
We use the latest available BOSS DR9 quasar Lyman-$\alpha$ forest data~\cite{Lyman-sample}. The ninth data release
contains 54,468 spectra of Lyman-$\alpha$ quasars with redshifts more than 2.15. They provide Lyman-$\alpha$ forest data in the redshift
range $z\sim2-5.7$. However due to a smaller number of quasars with redshift higher than $z\sim3$ we restrict our analysis
in the redshift $z\sim2-3$. BOSS-DR9 has a survey area of 3275 square degrees and hence our test of isotropy will be
limited by this area.
As has been discussed in~\cite{Lyman-sample}, out of 87,822 total spectra, the set of 54,468 spectra
was selected after removing low redshift quasars, quasars having broad absorption lines, too low SNR and negative continuum.
Note that the continuum to each quasar spectrum is estimated using mean-flux regulated principal component analysis (MF-PCA)
techniques~\cite{Leeetal} and they are provided along with the data. We would like to mention that even at this stage not all the
data are appropriate for the isotropy test due to low SNR, damped Lyman-$\alpha$ (DLA), limited exposures and few other
characteristics. Below we point out the data cuts that we have used to select a spectrum.
\subsection{Quasar selection, data cuts}
Our selection of data closely follows the BOSS criteria used to obtain constraints on the IGM~\cite{bossigm}. As we mentioned earlier,
we test the isotropy of the sky in three redshift bins with the central redshifts being 2.3, 2.6 and 2.9. The redshift bins have a bin width of
$\Delta z=0.3$. In each of the redshift bins we impose the following data cuts. Firstly since large number of low SNR quasars can have a systematic bias
in our results, we reject all the quasars with SNR $<6$. The rest of the quasars are binned in {\it good} ($6\le{\rm SNR}<8$),
{\it better} ($8\le{\rm SNR}<10$) and {\it best} (${\rm SNR}\ge10$) categories. In the quasar rest frame we use
$1041-1185\mathring{A}$ as the Lyman-$\alpha$ domain.
Lyman-$\alpha$ forest typically refers to neutral hydrogen atoms with column density of $10^{14}$ atoms per ${\rm cm}^2$ in the
line of sight. The presence of DLA in a spectrum indicates extremely high column density (${\cal O} (10^{20})$ atoms per
${\rm cm}^2$). Following~\cite{bossigm} we discard spectra with identified DLA in the sightlines. The provided data uses
a DLA concordance catalogue for the identification of DLA's. The detection efficiency of the catalogue decreases below
neutral hydrogen column densities less than $10^{20.3}/{\rm cm}^2$. Hence, below this criteria, in our analysis, for each of the forest we
correct the flux for the damping wings of DLA in the sightlines, provided with the data~\cite{Lyman-sample}.
We leave the spectra with neutral hydrogen column density of more than $10^{20.3}/{\rm cm}^2$ out of our analysis.
As mentioned earlier, the survey provides the continuum of each spectrum estimated using MF-PCA technique. Note that
the transmitted flux is the observed flux {\it w.r.t.} the estimated continuum and hence a bad continuum estimation
shall bias the calculated flux~\footnote{The effects of different continuum estimations are discussed in~\cite{Busca:2012bu}}.
Here too, following the survey criteria we reject the quasars which do not provide
good fit to the spectrum redwards to the Lyman-$\alpha$ forest. We also reject spectra which had
less than three individual exposures.
Our final selection criteria depends on the resolution of the BOSS spectographs. As has been mentioned in~\cite{bossigm},
BOSS spectographs do not resolve the Lyman-$\alpha$ forest. To evade this systematic effect that can reflect in the
transmitted flux, we use similar procedure used in~\cite{bossigm}. We stack the spectra according to the intrinsic
wavelength dispersion ($\sigma_{\rm disp}$) at the Lyman-$\alpha$ wavelength of each central redshift bin. Note that in each pixel
the $\sigma_{\rm disp}$ is provided along with the spectrum. From each redshift bin we then discard the 5$\%$ spectra
from below and 10$\%$ spectra from above.
Each quasar spectrum contain certain number of data pixels in the Lyman-$\alpha$ region. In a redshift bin (say, $2.15<z<2.45$)
some quasars may have all the Lyman-$\alpha$ pixels, some will have partial pixels from either blue end or red end of the
Lyman-$\alpha$ domain. We here consider all the quasars satisfying the above criteria and also contributing more than
30 pixels in the redshift window.
\subsection{Selection of sky patches}
Since BOSS-DR9 surveys in 3275 ${\rm deg}^2$ area, we need to make patches in the sky in particular ways. First we
convert the coordinate of the quasars from J2000 equatorial system to galactic coordinate ($l,b$) system. Below, in
Fig.~\ref{fig:skycut} we provide our selection of patches with different colors. In the first system we divide the sky
using quadrant convention. Here we divide the sky in three patches where one patch remains in the southern
hemisphere (red, $b<0^{\circ}$) and the other two patches are in northern
hemisphere ($b>0^{\circ}$). The two patches in the northern hemisphere are divided in $l>180^{\circ}$ (blue) and
$l<180^{\circ}$ (green) parts. In principle, the quadrant convention divides the southern hemisphere into two parts
($l>180^{\circ}$ and $l<180^{\circ}$) too, but since there are barely any data in $b<0^{\circ},l>180^{\circ}$, we take only one patch
from southern hemisphere. In our other selection, we divide the sky according
to galactic latitude. The red patch remains same as it stays in the southern hemisphere. The patches in the northern
hemisphere are divided in one patch with $b>50^{\circ}$ (magenta) and another with $0^{\circ}<b\le50^{\circ}$ (black). Our first selection
of patches is provided in the left of Fig.~\ref{fig:skycut} and the other selection is shown to the right.
\begin{figure*}[!htb]
\begin{center}
\resizebox{230pt}{120pt}{\includegraphics{plots/patch-selection1.png}}\hskip -22 pt
\resizebox{230pt}{120pt}{\includegraphics{plots/patch-selection2.png}}
\end{center}
\caption{\footnotesize\label{fig:skycut} The BOSS-DR9 survey area in galactic coordinates and our
selection of sky patches for the test of isotropy. To accommodate comparable number of quasars in each
patch we have divided the sky in two manner, namely patch selection 1 (left) and patch selection 2 (right).}
\end{figure*}
In the following table~\ref{tab:patch-info} we tabulate the essential information about the selected patches and the
overall sample in different redshift and different SNR. For different patches we provide the number of quasars and
the number of pixels that pass all our data cuts discussed in the paragraph before. Note that apart from the red patch
that stays in the southern hemisphere, other patches contain quasar number and the pixels that are comparable to each
other. As we go towards higher redshifts we find lesser number of data pixels which is expected as we know there are not
many high redshift quasars. This decrease in number of high redshift quasars limit our analysis to $z<3$. However, in
each SNR bin we have enough pixels to perform a relatively robust statistical analysis. Also note that compared to $6\le {\rm SNR}<8$
bin, in the two higher SNR bins the numbers of quasars or pixels are approximately half. We should mention that this
is not a concern since we assume the data from different SNR are independent and we compare the statistical properties
of the transmitted flux in each SNR separately. Moreover the lack of large number of quasars in higher SNR bins are
compensated by the less dispersion in the data from higher SNR quasar samples.
\bgroup
\renewcommand{\arraystretch}{1.1}
\begin{table*}[!t]
\begin{scriptsize}
\begin{center}
\begin{tabular}{c | c | c | c | c}
\hline\hline
Redshift range($z$) & SNR & Patch location & Number of quasars & Number of Lyman-$\alpha$ pixels\\
\cline{1-5}
& & Complete sky& 1091&285878 \\
& & $b<0^\circ$ (Red)& 226& 60631\\
& & $b>0^\circ,~l<180^\circ$ (Green)&390 &100289 \\
& $6-8$ & $b>0^\circ,~l>180^\circ$ (Blue)&475 &124958 \\
& & $0^\circ<b<50^\circ$ (Black)&401 &106772 \\
& & $b>50^\circ$ (Magenta)&464 &118475 \\
\cline{2-5}
& & Complete sky&498 &130242 \\
& & $b<0^\circ$ (Red)&106 &28686 \\
& & $b>0^\circ,~l<180^\circ$ (Green)&185 &48158 \\
$2.15-2.45$ ($\bar{z}=2.3$) & $8-10$ &$b>0^\circ,~l>180^\circ$ (Blue) &207 &53398\\
& & $0^\circ<b<50^\circ$ (Black)&168 &45290 \\
& & $b>50^\circ$ (Magenta)&224 &56266 \\
\cline{2-5}
& & Complete sky&567 &147679 \\
& & $b<0^\circ$ (Red)&122 &30899 \\
& & $b>0^\circ,~l<180^\circ$ (Green)& 187&47806 \\
& $>10$ &$b>0^\circ,~l>180^\circ$ (Blue)&258 &68974\\
& & $0^\circ<b<50^\circ$ (Black)& 228& 62655\\
& & $b>50^\circ$ (Magenta)&217 & 54125\\
\cline{1-5}
& & Complete sky&975 &230165 \\
& & $b<0^\circ$ (Red)& 221&53062 \\
& & $b>0^\circ,~l<180^\circ$ (Green)&373 &86995 \\
& $6-8$ & $b>0^\circ,~l>180^\circ$ (Blue)&381 &90108 \\
& & $0^\circ<b<50^\circ$ (Black)&348 & 80712\\
& & $b>50^\circ$ (Magenta)& 406&96391 \\
\cline{2-5}
& & Complete sky&475 &107539 \\
& & $b<0^\circ$ (Red)&110 &25099 \\
& & $b>0^\circ,~l<180^\circ$ (Green)& 181&41564 \\
$2.45-2.75$($\bar{z}=2.6$) & $8-10$ &$b>0^\circ,~l>180^\circ$ (Blue) &184 &40876\\
& & $0^\circ<b<50^\circ$ (Black)& 141& 32220\\
& & $b>50^\circ$ (Magenta)&224 &50220 \\
\cline{2-5}
& & Complete sky&607 &144797 \\
& & $b<0^\circ$ (Red)&139 &33690 \\
& & $b>0^\circ,~l<180^\circ$ (Green)&224 &54156 \\
& $>10$ &$b>0^\circ,~l>180^\circ$ (Blue)&244 &56951\\
& & $0^\circ<b<50^\circ$ (Black)&248 & 56434\\
& & $b>50^\circ$ (Magenta)&220 &54673 \\
\cline{1-5}
& & Complete sky&628 &139373 \\
& & $b<0^\circ$ (Red)&135 &30291 \\
& & $b>0^\circ,~l<180^\circ$ (Green)&250 &57698 \\
& $6-8$ & $b>0^\circ,~l>180^\circ$ (Blue)&243 & 51384\\
& & $0^\circ<b<50^\circ$ (Black)&225 &50412 \\
& & $b>50^\circ$ (Magenta)&268 &58670 \\
\cline{2-5}
& & Complete sky&372 &87257 \\
& & $b<0^\circ$ (Red)& 75&17879 \\
& & $b>0^\circ,~l<180^\circ$ (Green)& 150&35068 \\
$2.75-3.05$ ($\bar{z}=2.9$)& $8-10$ &$b>0^\circ,~l>180^\circ$ (Blue) &147 &34310\\
& & $0^\circ<b<50^\circ$ (Black)&112 &25351 \\
& & $b>50^\circ$ (Magenta)& 185& 44027\\
\cline{2-5}
& & Complete sky&432 &97826 \\
& & $b<0^\circ$ (Red)&111 & 25645\\
& & $b>0^\circ,~l<180^\circ$ (Green)&174 & 37713\\
& $>10$ &$b>0^\circ,~l>180^\circ$ (Blue)&147 &34468\\
& & $0^\circ<b<50^\circ$ (Black)&164 & 39660\\
& & $b>50^\circ$ (Magenta)&157 &32521 \\
\cline{1-5}
\end{tabular}
\end{center}
\caption{~\label{tab:patch-info}The number of quasars and the number of Lyman-$\alpha$ pixels that contribute
in different bins and patches. We bin our samples in three different redshift bins with mean redshift being
2.3, 2.6 and 2.9 and also in three signal-to-noise-ratio, namely $6\le{\rm SNR}<8$, $8\le{\rm SNR}<10$ and ${\rm SNR}\ge10$. The whole
sky is divided in two different patch types (see, Fig.~\ref{fig:skycut}). Each type contains three different patches with different color
codes provided in this table.}
\end{scriptsize}
\end{table*}
\egroup
\clearpage
\subsection{Error estimation}
In this paper, as we have mentioned earlier, we compare the statistical properties of the observed data in
different direction of the sky without comparing with any theoretical model. To obtain the uncertainties/errors in the estimated PDF of the
transmitted flux, we need to generate the covariance matrix associated with the flux PDF. Here too, we follow similar
procedure as adopted in the BOSS analysis, in order to obtain the covariance matrix. We follow bootstrap resampling of
the data in each bin. We have a total of 36 bins in both type of patch selection: 3 SNR bins, 3 redshift bins and 4 bins corresponding to
each of the 3 patches and a complete sample. In each bin, we gather all the pixels that pass through our data cut and
provided in Table~\ref{tab:patch-info}. We perform bootstrap resampling of chunks upto $100\mathring{A}$ wavelengths
obtained from each quasar spectrum and generate 1000 realizations of the data. From these samples covariance matrix
of the flux PDF can be easily calculated. Using different realizations we did check the convergence of
the diagonal terms of the covariance matrix and we could conclude that the choice of 1000 realizations has been conservative.
We concentrate on the statistical properties of the PDF or the data. Hence we calculate different statistical moments
such as mean, median, variance, skewness and kurtosis of the PDF. We report the moments of the distribution that we obtain
from the original data as well as the upper and lower error bounds on each of the moments that we calculate from the bootstrap
simulations. For distribution of each moment, we generate the empirical cumulative distribution function (ECDF) and report the
(34.15\%) upper and lower error bounds, which we hereafter shall refer as 1$\sigma$ uncertainties in the distribution of the
particular moment.
Before discussing the isotropy test on patches in next section, in Table~\ref{tab:flux} we report the mean transmitted flux
and 1$\sigma$ uncertainties of the flux PDF that we obtain from the complete sample in the three redshifts and in three SNR bins.
The number of quasars and the number of pixels that are used in our analysis are provided in Table~\ref{tab:patch-info} for
the complete sample size. Note that as we go back in time (higher in redshift) the increase of neutral hydrogen
is evident from the decrease of the transmitted flux in all SNR. In each SNR, the error on the
flux are comparable and that indicates, though at high SNR the signal is better, the less number of
quasars in high SNR keeps the uncertainties comparable to the uncertainties at lower SNR which contain larger number of quasars
(for example, compare the {\it good} and the {\it best} SNR cases).
\bgroup
\renewcommand{\arraystretch}{1.2}
\begin{table*}[!htb]
\begin{center}
\vskip -15 pt
\begin{tabular}{c | c | c}
\hline\hline
Redshift range($z$) & SNR & $\bar{F}\pm\Delta F$\\
\cline{1-3}
& $6-8$&$0.826^{+0.154}_{-0.375}$\\
$2.15-2.45$ ($\bar{z}=2.3$)&$8-10$&$0.822^{+0.138}_{-0.405}$\\
& $>10$&$0.819^{+0.129}_{-0.487}$\\
\cline{1-3}
& $6-8$&$0.762^{+0.172}_{-0.39}$\\
$2.45-2.75$ ($\bar{z}=2.6$)&$8-10$&$0.758^{+0.159}_{-0.427}$\\
& $>10$&$0.756^{+0.152}_{-0.454}$\\
\cline{1-3}
& $6-8$&$0.69^{+0.191}_{-0.377}$\\
$2.75-3.05$ ($\bar{z}=2.9$)&$8-10$&$0.687^{+0.181}_{-0.396}$\\
& $>10$&$0.686^{+0.176}_{-0.413}$\\
\cline{1-3}
\end{tabular}
\end{center}
\caption{~\label{tab:flux}The flux statistics of the Lyman-$\alpha$ forest transmitted flux in different redshift
bins and different signal-to-noise bins, provided for the samples across the complete sky. The mean flux and the $1\sigma$
error bars on the PDF on both sides of the mean are provided.}
\end{table*}
\egroup
\section{Analysis, results and discussions}~\label{sec:results}
In the previous section we mentioned that we trisect the survey area in two ways.
One patch, that is in southern galactic hemisphere remains common
to both the selection. For the 5 patches, that are color coded (red, green, blue, magenta and black)
we obtain the PDF of transmitted flux. However, we compare the statistical properties of the PDF
within each selection type.
\begin{figure*}[!htb]
\begin{center}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch1-snr-low-23-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch1-snr-med-23-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch1-snr-high-23-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch1-snr-low-26-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch1-snr-med-26-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch1-snr-high-26-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch1-snr-low-29-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch1-snr-med-29-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch1-snr-high-29-eps-converted-to.pdf}}
\end{center}
\caption{\footnotesize\label{fig:comparison-PDF1}Comparison of PDF of the Lyman-$\alpha$ transmitted flux in different
patches for patch selection 1 (left of Fig.~\ref{fig:skycut}). The color codes represent the PDF's from the corresponding
patches. Along the rows we plot PDF's for different redshifts and along the columns we plot for different SNR. The error
on the PDF is estimated through bootstrap resampling over $100\mathring{A}$ data chunks.}
\end{figure*}
To start with, we provide the flux PDF with the errors in Fig.~\ref{fig:comparison-PDF1} for patch selection 1 (corresponding
to the selection in the left of Fig.~\ref{fig:skycut}) and in Fig.~\ref{fig:comparison-PDF2} for patch selection 2 (the right
selection in the Fig.~\ref{fig:skycut}). The colors of the PDF's correspond to different patches (as shown in the Fig.~\ref{fig:skycut}).
The PDF's are calculated in the flux range [-0.2-1.5] in 34 bins. Theoretically
the transmitted flux should be within 0 and 1 (for the complete and no absorption respectively) but due to the noise in
observations and the bias in the continuum estimations, the transmitted flux might be obtained outside the theoretical
boundary. The PDF's are normalized such that the total area under each PDF is 1. In both the figures, we provide the PDF for different redshifts and
different SNR. As we mentioned in the previous section, due to less number of quasars in the higher
SNR bins, the errors on the flux PDFs are comparable to that of the lower SNR bins. In the plots of the PDF, the
difference between the PDF's in {\it good, better} and {\it best} SNR bins are evident. For the higher SNR bins
the flux PDF are more sharp around the peak. It is interesting to note that at the same SNR and at the same redshift,
the PDF of transmitted flux from different patches are very similar. In some of the bins we find the flux from one patch is different
from other ones. To quantify the difference, we next look for the different moments of the PDFs.
\begin{figure*}[!htb]
\begin{center}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch2-snr-low-23-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch2-snr-med-23-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch2-snr-high-23-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch2-snr-low-26-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch2-snr-med-26-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch2-snr-high-26-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch2-snr-low-29-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch2-snr-med-29-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/pdf-patch2-snr-high-29-eps-converted-to.pdf}}
\end{center}
\caption{\footnotesize\label{fig:comparison-PDF2} Comparison of PDF of the Lyman-$\alpha$ transmitted flux in different
patches for patch selection 2 (right of Fig.~\ref{fig:skycut}). The color codes represent the PDF's from the corresponding
patches. Along the rows we plot PDF's for different redshifts and along the columns we plot for different SNR. The error
on the PDF is estimated through bootstrap resampling over $100\mathring{A}$ data chunks.}
\end{figure*}
In this paper, we restrict ourselves to calculating up-to the fourth moment, {\it i.e.} till kurtosis of the PDF.
We calculate the mean ($\bar{F}$), the median ($F_{1/2}$), the variance ($\sigma^2$), the skewness ($s$) and
the kurtosis ($\kappa$) of the flux PDFs in each bin. Since we have noticed in Fig.~\ref{fig:comparison-PDF1} and
~\ref{fig:comparison-PDF2} that the PDFs are very different in different redshifts, we report the residual of such
quantities obtained in a patch {\it w.r.t.} the complete sample. In Fig.~\ref{fig:comparison-1} we plot the
5 statistical quantities in residual space for the patch selection 1 and in Fig.~\ref{fig:comparison-2} we plot the
same quantities for patch selection 2. As we are plotting in residual space, we find the moments are distributed about
the zero. We should mention that here the errors represent the uncertainties of the corresponding statistical
moments obtained from the ECDFs of the moments. The plotted errors represent 1$\sigma$ uncertainties.
\begin{figure*}[!b]
\begin{center}
\resizebox{140pt}{110pt}{\includegraphics{plots/mean-low-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/mean-med-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/mean-high-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/median-low-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/median-med-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/median-high-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/variance-low-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/variance-med-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/variance-high-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/skewness-low-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/skewness-med-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/skewness-high-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/kurtosis-low-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/kurtosis-med-snr_patch_type1-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/kurtosis-high-snr_patch_type1-eps-converted-to.pdf}}
\end{center}
\caption{\footnotesize\label{fig:comparison-1}First few statistical moments of the Lyman-$\alpha$ transmitted flux
PDF for different parts of the sky. For relative comparison we plot the residual moments from the total sample. The
error bars correspond to the 1$\sigma$ bounds on that particular moment. This
figure represents the comparison between the properties of the patches for patch selection 1 (left plot of Fig.~\ref{fig:skycut}).}
\end{figure*}
Note that almost all the statistical moments from different patches are comparable to each other within their
associated uncertainties. We would like to stress that
we find this consistency throughout redshift $2-3$ and at all SNRs. Hence our results are {\it consistent with
the isotropic distribution of neutral hydrogen} and also {\it consistent with isotropic absorption of photons by
the hydrogen clouds in the IGM at different redshifts}.
A closer examination shows that almost none of the patches contain any moment which is systematically
higher/lower than the total sample during the time evolution. Hence the properties
in each patch is preserved during the matter dominated expansion of the Universe.
We should mention that we have obtained some deviations from isotropy in our analysis. We find the residual moments
for the blue patch and the red patch (for patch selection 1) do not agree at 2$\sigma$ level in cases.
For example, we refer to the median mismatch (at $z=2.3$, $6\le{\rm SNR}<8$), the variance mismatches (at $z=2.9$, $6\le{\rm SNR}<8$ and
at $z=2.6$, ${\rm SNR}\ge10$) and skewness mismatches (at $z=2.9$ at $8\le{\rm SNR}<10$). Similarly, for patch selection 2
we find such deviations at 2$\sigma$ level within the red and the black patches. Since, the number of such deviations are only a few
and all the moments agree within 3$\sigma$, we do not report any statistically significant deviation. Moreover,
we should note that due to less number of quasars, the red patch at the southern hemisphere contains the largest
dispersion in the distribution of the statistical moments which in turn can lead to such fluctuations. Hence, in order to rule out or confirm
any deviations between the blue and the red patches (or between black and red patches) we need more detection of
Lyman-$\alpha$ forest in the southern galactic hemisphere. Future surveys will be able to provide
significantly more quasar spectra with higher sky coverage, which are essential to extend this initiative beyond the
flux PDF level.
\begin{figure*}[!b]
\begin{center}
\resizebox{140pt}{110pt}{\includegraphics{plots/mean-low-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/mean-med-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/mean-high-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/median-low-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/median-med-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/median-high-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/variance-low-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/variance-med-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/variance-high-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/skewness-low-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/skewness-med-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/skewness-high-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/kurtosis-low-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/kurtosis-med-snr_patch_type2-eps-converted-to.pdf}}
\resizebox{140pt}{110pt}{\includegraphics{plots/kurtosis-high-snr_patch_type2-eps-converted-to.pdf}}
\end{center}
\caption{\footnotesize\label{fig:comparison-2}First few statistical moments of the Lyman-$\alpha$ transmitted flux
PDF for different parts of the sky. For relative comparison we plot the residual moments from the total sample. The
error bars correspond to the 1$\sigma$ bounds on that particular moment.This figure represents the comparison
between the properties of the patches for patch selection 2 (right plot of Fig.~\ref{fig:skycut}).}
\end{figure*}
\clearpage
\section{Conclusions}~\label{sec:conclusions}
As a first approach to test the isotropy in the matter dominated epoch we have used the Lyman-$\alpha$ forest
data in this paper. In order to remain independent from theoretical assumptions of the IGM, we have used only the
observational data and compared the statistical properties of the PDF of the observed flux in different directions
of the sky. Detection of large number high redshift quasar spectra by BOSS has enabled us to perform this test. Though we report
the distribution of neutral hydrogen is consistent with isotropic Universe during $z\sim 2-3$, making any general claim about the isotropy
of the Universe requires much more sky coverage.
As we have mentioned before, our test of anisotropy is {\it partial} due to only
3275 ${\rm deg}^2$ sky coverage. With a larger sky coverage we can address the precise direction and significance of
the possible anisotropy, if present. With upcoming observations like e-BOSS~\cite{eboss} and DESI~\cite{desi} we expect to detect
higher quality of Lyman-$\alpha$ forest data, significantly higher number of quasar spectra with larger sky coverage. Hence,
with the upcoming data the assumption of isotropy can be falsified with higher precision. Cross-correlating the
data from different surveys will be also important to rule out any systematic effect.
Any presence of anisotropy in the Lyman-$\alpha$ forest is interesting and it points towards the distribution of neutral
hydrogen, temperature-density relation and few other properties in the IGM. A straightforward extension of this topic
would be to model the IGM using some semi-analytical modeling or simulations. With the modeling we can address if
the significance of such anisotropy being statistical or physical. Moreover, if any physical anisotropy is found,
we need to examine the change in the properties of the IGM it refers to and search for the probable cause.
We would like to conclude by mentioning that the major finding of our analysis is {\it consistency of the data with the isotropic Universe in the
final stage of matter dominated epoch ($z\sim2-3$), and unfortunately we found no preferred direction in the Lyman-$\alpha$ forest to guide the travelers}. Though
our result is a partial test, if similar tests on larger survey area
also confirms the isotropy, that might hint towards the possibility that any late time anisotropy is probably caused by bulk flow and not an
intrinsic anisotropy in the Universe.
\section{Acknowledgments}
We would like to thank Tapomoy Guha Sarkar for important discussions, suggestions and comments on the manuscripts. We would also
like to thank Amir Aghamousa, Stephen Appleby, Eric Linder, Pat McDonald, Graziano Rossi and Tirthankar Roy Choudhury for their comments and suggestions.
We thank Khee-Gan Lee for various clarifications regarding the BOSS analysis of the Lyman-$\alpha$ forest data.
D.K.H. wish to acknowledge support from the Korea Ministry of Education, Science and Technology, Gyeongsangbuk-Do and Pohang
City for Independent Junior Research Groups at the Asia Pacific Center for Theoretical Physics. A.S. would like to acknowledge
the support of the National Research Foundation of Korea (NRF-2013R1A1A2013795). We acknowledge
the use of data from SDSS III. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions,
the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of
Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group,
the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group,
Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics,
New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University,
the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and
Yale University.
|
1903.04908
|
\section{Introduction}
The Gauss-Green divergence theorem
\eqn{gg}
$$
\int_{A}\div \mathbf u(x)\,dx= \int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu_A\,d\Hnmj
$$
holds whenever $A\subset\rn$ is a bounded $BV$ set (or, in another terminology, a
bounded set of finite perimeter) and $\mathbf u\in C^1(\rn,\rn)$.
Here, $\partial_*A$ is the essential boundary and $\boldsymbol\nu_A$ is
the measure-theoretic unit exterior normal. This setting
and its history can be found e.g.\ in \cite{AFP}.
If we want to allow discontinuous derivatives, routine approximation
arguments give \eqref{gg} if $\mathbf u\in C(\rn,\rn)$ and
$\div\mathbf u(x)\in L^1(\rn)$.
Beyond Lebesgue integrability of $\div\mathbf u(x)$,
a natural idea is to consider the divergence in the sense of distributions.
Particularly deep results have been obtained for divergence measure vector
fields,
see e.g.\ Chen, Torres and Ziemer \cite{ZiemGG}, Ziemer \cite{ZiemC} or
\v{S}ilhav\'{y} \cite{Silh1, Silh2, Silh3}.
We pursue another direction. If $\mathbf u$ is differentiable,
the divergence formula still holds even if the divergence
is not Lebesgue integrable. This phenomenon indicates that the $L^1$
setting is not the ultimate generality if we want to consider the divergence
as
\textit{a pointwise function}.
Such
a
divergence
still plays the
role of divergence in the sense of distributions, but the task is to what
extent non-absolutely integrable pointwise functions can be
represented as distributions.
The problem exists already in the one-dimensional case where it has been
solved by the Denjoy-Perron integral. The multidimensional case
has been treated by many authors, among the most important
contribution we mention \cite{JKS,Ma1,He}.
The most important progress in this direction
has been done by Pfeffer \cite{Pf}, who developed a theory which
can be used for the divergence theorem on $BV$ sets. In his setting,
indefinite integral is a function on $BV$ sets, so that the definite
integral on the left of \eqref{gg} is the evaluation of the indefinite
integral at $A$.
An interesting extension has been introduced by Pfeffer and Mal\'{y} in \cite{mpf}.
Their effort leads to the $\R^*$ integral, which is stable under reasonable operations and
has a rich family of integrable functions. In particular, the $\R^*$ integral includes Pfeffer's $\R$ integral \cite{Pf}
and the $1$-dimensional Henstock-Kurzweil integral.
In a series of papers \cite{KM,malydistr,ph},
a new non-absolutely convergent
integral with respect to distributions, called \textit{packing
integral}, has been introduced. Since main
motivation comes from the divergence theorem and related results again,
it is natural to ask on comparison of this integral with Pfeffer's
approach. In its original setting, the indefinite packing integral is a
functional on smooth (or Lipschitz) test functions and its evaluation
at $BV$ sets does not make sense. Therefore, the definite integral
on the left of \eqref{gg} is the evaluation of the indefinite integral
of $\chi_A \div \mathbf u$ at a test function which is $1$ on a neighborhood of
$\partial A$.
The Pfeffer integral (one of the equivalent versions) is based on
Riemann-type sums
$$
\sum_{i=1}^m\big|\F(E_i)-f(x_i)\L(E_i)\big|
$$
where $E_i\subset\rn$ are disjointed
$BV$ sets, $x_i\in\rn$ are tags, $\L$ is Lebesgue measure
and $\F$ is the candidate for the
indefinite integral.
In our setting, we also use sums
\eqn{oursums}
$$
\sum_{i=1}^m q_{x_i,r_i}(\F-f(x_i)\L)
$$
where $(q_{x,r})_{x,r}$ is a system of suitable seminorms.
Let $A\subset\rn$ be a bounded $BV$ set.
Suppose that $\mathbf u\in C(\rn,\rn)$ and the indefinite packing integral of a function $f$ is
the flux of $\mathbf u$, so that $f=\Div \mathbf u$ in a general sense.
We would be happy to see that
$$
\int_{A}\Div \mathbf u(x)\,dx= \int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu\,d\Hnmj,
$$
where the integral on the left means the integration of $f\chi_A$.
(In other words, the characteristic function of $A$ acts as a multiplier
for the integration of $f$.)
However, in the setting of \cite{KM} it is not clear
how to estimate the sums \eqref{oursums}
(and it is probably impossible without additional hypotheses).
It helps if we can omit $x_i$ belonging to a small set, say of
$\sigma$-finite $\Hnmj$ Hausdorff measure, namely, just $\partial_*A$.
This change of definition requires the indefinite integral to be
a \textit{charge}, a functional on $BV\cap L^{\infty}$ functions continuous
with respect to a convergence specified below. Charges can be
represented as functions on $BV$ sets, and by this series of thoughts
we recover most ingredients of Pfeffer's setting.
In this paper we present modifications of the packing integral
which contains Pfeffer's $\R$ integral and Pfeffer's and Mal\'{y}'s $\R^*$ integral. We
apply the new integrals to obtain more general versions of the divergence theorem.
In the end we discuss the relationships between particular integrals
including the one-dimensional Henstock-Kurzweil-Stieltjes integral and
$MC_\alpha$ integral.
\iffalse
PRIDAT DO UVODU
to o tom prostoru mer
\begin{definition}
Let $E$ be a measurable subset of $\rn$.
Then the \emph{perimeter} of $E$ in an open set $A$ is the number
$$
P(E,A)=\Hnmj(\partial_*E\cap A),
$$
which has the meaning of $\|D\chi_E\|$ in $(C_0(A))*$ - the space of measures on A.
The \emph{relative perimeter} of $E$ in a measurable set $A$ is the number
$$
P(E,\inn A)=\Hnmj(\partial_*E\cap \inter_* A).
$$
The \emph{perimeter} of $E$, denoted by $P(E)$ or $\|E\|$, is the number
$$
P(E)=P(E,\rn)=\Hnmj(\partial_*E).
$$
We say that a measurable set $E$ is a \emph{locally $\BV$ set}, if $P(E,A)<\infty$ for each bounded open set $A$.
A measurable set $E$ is called a \emph{$\BV$ set}, if $|E|+\|E\|<\infty$.
The family of all $\BV$ sets, or all locally $\BV$ sets, is denoted by $\cBV$, or $\cBV_{\loc}$ respectively.
The family of all bounded $\BV$ sets, or all bounded locally $\BV$ sets, is denoted by $\tbv$, or $\tbv_{\loc}$ respectively.
\end{definition}
\begin{theorem}[Gauss-Green divergence theorem]
Let $A\subset\rn$ be a bounded $BV$ set,
let $\mathbf u\in C(\rn,\rn)$. Let us suppose that there exists a generalized divergence $\Div \mathbf u$
on a set $\Omega$ containing $\cl_* A$.
Then
$$
\int_{A}\Div \mathbf u(x)\,dx= \int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu_A\,d\Hnmj.
$$
\end{theorem}
\fi
\section{Notation and Preliminaries}
\iffalse
\begin{definition}
Let $\I$ denote the system of all nondegenerate compact intervals in $\rn$.
We say that a finite subset $\D\subset \I$ is a \emph{partition of an interval $I\in\I$} if the intervals from $\D$ are
nonoverlapping (i.e. have disjoint interiors) and $\bigcup_{Q\in\D} Q=I$.
Any subset of a partition is called a \emph{subpartition}.
A function $F:\I\to\er$ is said to be
\emph{additive}, if for each interval $I\in\I$ and each partition
$P=\{A_1,\ldots,A_k\}$ of $I$ we have
$$
\sum_{i=1}^k F(A_i)=F(I).
$$
\end{definition}
\fi
\begin{notation}
Let $E$ be a subset of $\rn$. Then $d(E)$ denotes the diameter of $E$, i.e.
$$
d(E)=\sup\{|y-x|; x,y\in E \}.
$$
Let $x\in\rn$ and $r>0$. Then $B(x,r)$ denotes the open ball
$$
B(x,r)=\{y\in\rn; |y-x|<r\}
$$
and $\bar{B}(x,r)$ denotes the closed ball
$$
\bar{B}(x,r)=\{y\in\rn; |y-x|\leq r\}.
$$
The Lebesgue measure of $E$ is denoted by $|E|$ or $\L(E)$.
\end{notation}
\begin{definition}
We say, that measurable sets $A$ and $B$ are \emph{equivalent} (or $A$ and $B$ belong to the same \emph{equivalence class})
if $|A\triangle B|=0$, where $A\triangle B$ denotes the symmetric difference of the sets $A$ and $B$.
\end{definition}
\begin{definition}
Let $s\geq 0$.
The \emph{$s$-dimensional outer Hausdorff measure} of a set $E\subset\rn$
is defined as $\Hs(E)=\lim_{\delta\to0+} \H^s_\delta(E)$, where
$$\H^s_\delta(E)=\inf\left\{
\sum_{i=1}^\infty \alpha_s \left(
\frac{\diam(C_i)}{2}
\right)^s; C_i \subset\rn,
E\subset \bigcup_{i=1}^\infty C_i, \diam(C_i)<\delta
\right\}
$$
and $\alpha_s=\frac{\pi^{\frac{s}{2}}}{\Gamma\left(\frac{s}{2}+1\right)}$.
\end{definition}
\iffalse
\begin{notation}
Let $s\geq 0$. Then
$$
\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}\dt
$$
denotes the Gamma function. The volume of $n$-dimensional unit ball is then denoted by
$$
\alpha_n=\frac{\Gamma\left(\frac{1}{2}\right)^n}{\Gamma\left( \frac{n}{2}+1\right)}.
$$
\end{notation}
\begin{definition}
Let $\delta\in[0,\infty]$, $s\geq0$ and
$A\subset \rn$.
Then
$$
\Hsdelta (A):=\inf\left\{
\sum_{j=1}^\infty
\alpha_s 2^{-s}\diam(A_j)^s
:
\bigcup_{j=1}^\infty A_j \supset A, \diam A_j\leq\delta
\right\}
$$
denotes the \emph{approximating outer measure $\Hsdelta$ }
and
$$
\Hs(A):=\sup_{\delta>0}\Hsdelta(A)=\lim_{\delta\to 0_+}\Hsdelta(A)
$$
denotes the \emph{$s$-dimensional outer Hausdorff measure $\Hs$}.
\end{definition}
\begin{definition}
We say that a set $A\subset \rn$ is of \emph{$\sigma$-finite Hausdorff measure $\Hs$} if
$A=\bigcup_{i=1}^\infty A_i$ and $\Hs(A_i)<\infty$ for $i=1,2,\ldots$.
\end{definition}
\begin{remark}
Especially, the Hausdorff measure $\H^0$ is counting measure in $\rn$ and the Hausdorff measure $\Hn=\lambda_n$, the
$n$-dimensional Lebesgue measure in $\rn$.
\end{remark}
\fi
\iffalse
\begin{definition}
We say that a measure $\mu$ on $\rn$ is \emph{doubling}, if $\mu$ is Radon and
there exists a constant $c_D$ such that
$$
\mu(B(x,2r))\le c_D\mu(B(x,r))
$$
for each $x\in \rn$ and $r>0$.
\end{definition}
\fi
\begin{proposition}\label{l:Pflipeo}
Let $A\subset \rn$ be a set and let $\varphi:A\to\rn$ be a Lipschitz mapping. Then
$\Hnmj(\varphi(A))\leq (\Lip \varphi)^{n-1}\Hnmj(A)$.
\end{proposition}
\begin{proof}
For the proof and further details see \cite[Section 2.4.1]{EG}.
\end{proof}
\begin{definition}
Let $A\subset \rn$ be a measurable set and let $x\in\rn$. Then we define the \emph{lower density} of $A$ at $x$ as
$$
\underline{\Theta}
(A,x)
:=\liminf_{r\to0+} \frac{|A\cap {B}(x,r)|}{|{B}(x,r)|}
$$
and the \emph{upper density} of $A$ at $x$ as
$$
\overline{\Theta}
(A,x)
:=\limsup_{r\to0+} \frac{|A\cap {B}(x,r)|}{|{B}(x,r)|}
.
$$
The \emph{essential closure $\cl_*A$}, \emph{essential interior $\inter_*A$} and \emph{essential boundary $\partial_* A$}
are then defined as
$$
\cl_*A=\{x\in\rn; \overline{\Theta}(A,x)>0\},
$$
$$
\inter_*A=\{x\in\rn; \underline{\Theta}(A,x)=1\}
$$
and
$$
\partial_*A=\cl_*A\setminus \inter_*A.
$$
\end{definition}
\begin{definition}
We say that a measurable set $A\subset\rn$ is \emph{admissible} if
$\inter_* A\subset A\subset \cl_* A$
\end{definition}
\begin{remark}
Our definition of admissible set differs from that used by Mal\'{y} and Pfeffer
in \cite{mpf}, according to which $\partial A$ is required to be compact.
\end{remark}
\begin{remark}
Let $A$, $A'$ be measurable sets such that $|A\triangle A'|=0$.
Then $\cl_*A=\cl_*A'$, $\intt_* A=\intt_* A'$ and $\partial_* A=\partial_* A'$.
Hence, for every bounded measurable set $A$ we can find an admissible set $A'$ such that $|A\triangle A'|=0$.
\end{remark}
\iffalse
\begin{definition}
Let $\Omega$ be a subset of $\rn$.
Then $C^1(\Omega,\rn)$ denotes the space of all functions $v:\Omega\to\rn$ with continuous partial derivatives of the
first order such that the norm
$$
\|v\|_{C^1(\Omega)}=\sum_{|\alpha|\leq1}\sup_\Omega |\partial^\alpha v|
$$
is finite.
The
set of all functions in $C^1(\Omega, \rn)$ with compact support is denoted by
$C^1_c(\Omega,\rn)$.
\end{definition}
\fi
\section{$\BV$ sets and charges}
In this section we will present some basic facts about spaces of sets of bounded variation ($\BV$ sets) and about charges
which will be essential in further definitions.
For details see \cite{Pf}, \cite{npf} and \cite{dpw}.
\begin{definition}
Let $U\subset\rn$ be an open set. For a measurable set $E\subset\rn$
we define the \emph{perimeter of $E$ in $U$} as
$$
P(E,U)=\sup
\Big\{\int_{U\cap E}\div \ff\colon \ff\in C_c^1(U),\;\|\ff\|_{\infty}\le 1\Big\}.
$$
If $P(E,U)<\infty$, then the distributional gradient $D\chi_E$ of $\chi_E$
in $U$ is a vector-valued Radon measure and
$P(E,U)$ is exactly its total variation. By the De Giorgi--Federer theorem,
we can compute $P(E,U)$ as
$$
P(E,U)=\H^{n-1}(\partial_*E\cap U).
$$
The particular choice $U=\rn$ gives the
\emph{perimeter} of $E$
$$
P(E)=\|E\|=\H^{n-1}(\partial_*E).
$$
If $A\subset \rn$ is just measurable, we define
also the \emph{relative perimeter of $E$ in $A$} as
$$
P(E,\inn A)=\H^{n-1}(\partial_*E\cap\inter_* A).
$$
There is a distinction between $P(E,\inn U)$ and $P(E,U)$ if $U$
is open, see Example \ref{ex:perim} below.
We say that a measurable set $E$ is a \emph{locally $\BV$ set}, if $P(E,A)<\infty$ for each bounded open set $A$.
A measurable set $E$ is called a \emph{$\BV$ set}, if $|E|+\|E\|<\infty$.
The family of all $\BV$ sets and all locally $\BV$ sets is denoted by $\cBV$ and $\cBV_{\loc}$, respectively.
The family of all bounded $\BV$ sets is denoted by $\tbv$.
\end{definition}
\begin{example}\label{ex:perim}
Let $E=B(0,1)$ and $A=B(0,2)\setminus\{x\in\er^2:|x|=1\}$ be subsets of $\er^2$.
Then $P(E,A)=\Hnmj(\emptyset)=0$, whereas $P(E,\inn A)=\Hnmj\big(\partial\left(B\left(0,1\right)\right)\big)=2\pi$.
\end{example}
\iffalse
\begin{definition}
Let $E$ be a subset of $\rn$.
The number
$$
\|E\|:=\sup_v \left\{\int_E \div v(x)\dx; \quad v\in C^1_c(\rn,\rn),\, \|v\|_{C^1(\rn)}\leq 1\right\}
$$
is called the \emph{perimeter of $E$}.
The set $E$ is called a \emph{$\BV$ set}, if $\|E\|+|E|<\infty$.
Let $\Omega$ be an open subset of $\rn$. Then the \emph{variation of $E$ in $\Omega$} is defined as
$$
V(E,\Omega):=\sup_v \left\{\int_\Omega \chi_E\div v(x)\dx; \quad v\in C^1_c(\rn,\rn),\, \|v\|_{C^1(\rn)}\leq 1\right\}.
$$
We say that the set $E$ is \emph{locally $\BV$ set}, if $V(E,\Omega)<\infty$ for every $\Omega\subset\rn$ open.
\end{definition}
\fi
\begin{remark}
If $n=1$,
each $BV$ set $E$ is equivalent to a set $\bigcup_{i=1}^k (a_i,b_i)$, where $a_1<b_1<\cdots<a_k<b_k$
are real numbers.
In this case, $\|E\|=2k$.
\end{remark}
\begin{definition}
Let $A$ be a locally $\BV$ set. Then we define the \emph{critical boundary} of $A$ as
$$
\partial_cA=\left\{
x\in\rn; \limsup_{r\to 0+} \frac{P(A,B(x,r))}{r^{n-1}}>0
\right\}.
$$
The \emph{critical interior} $\intt_c A$ and \emph{critical exterior} $\ext_c A$ are then defined as
$$
\intt_c A=\intt_* A\setminus \partial_c A,\qquad \ext_c A=\ext_*A \setminus \partial_c A.
$$
\end{definition}
In the following, we will define the regularity of a $BV$ set.
This concept has been first introduced by Kurzweil, Mawhin and Pfeffer in \cite{mawhin}.
In this article, we use the modification established by Pfeffer in \cite{npf}.
\begin{definition}
Let $E\subset\rn$ be a bounded $\BV$ set and let $x\in\rn$.
The \emph{regularity} of the set $E$ is the number
$$
r(E) =
\begin{cases}
\frac{|E|}{d(E)\|E\|} & \text{if } |E| >0,\\
0& \text{if } |E| = 0.\\
\end{cases}
$$
The \emph{regularity} of the pair $(E,x)$ is the number
$$
r(E,x) =
r(E\cup\{x\})=
\begin{cases}
\frac{|E|}{d(E\cup\{x\})\|E\|} & \text{if } |E| >0,\\
0& \text{if } |E| = 0.\\
\end{cases}
$$
Let $\varepsilon>0$. We say that the set $E$ and the pair $(E,x)$ are \emph{$\varepsilon$-regular} if $r(E)>\varepsilon$
and $r(E,x)>\varepsilon$, respectively.
A system $P=\{(A_1,x_1),\ldots, (A_m,x_m)\}$, $A_i\subset \rn$ and $x_i\in \rn$, is called
$\varepsilon$-regular if $r(A_i,x_i)>\varepsilon$ for $i=1,\ldots,m$.
Let us note that every $\varepsilon$-regular $BV$ set is bounded.
\end{definition}
\begin{remark}
For every bounded $\BV$ set $E$ we have the estimate
$r(E)\leq 1/(2n)$.
Especially, the regularity of a ball is equal to $1/(2n)$
(see \cite[Chapter 2.3]{Pf}).
\end{remark}
\iffalse
\begin{example}
The regularity of a ball $B$ in $\rn$ is $\|B\|=1/(2n)$.
\end{example}
\fi
\begin{definition}
A \emph{dyadic cube} is an interval
$$
\prod_{i=1}^n \left[ \frac{k_i}{2^m},\frac{k_i+1}{2^m}\right],
$$
where $m,k_1,\ldots,k_n$ are integers.
A dyadic cube $C'$ is called the \emph{mother} of a dyadic cube $C$ if $C'$ is the smallest (with respect to inclusion)
dyadic cube properly containing $C$.
A finite (possibly empty) union of nondegenerate compact intervals in $\rn$
is called a \emph{figure}.
A \emph{dyadic figure} is a figure that is a union of finitely many dyadic cubes.
\end{definition}
\begin{definition}
Let $B$ be a bounded $\BV$ set.
We say that a sequence $\{B_i\}\subset\tbv$
\emph{converges} to $B$ in $\tbv$ if
\begin{enumerate}
\item $\bigcup_{i=1}^\infty B_i$ is a bounded set,
\item $\lim_{i\to\infty} |B_i\triangle B|=0$ and $\sup_i \|B_i\|<\infty$.
\end{enumerate}
\end{definition}
\iffalse
Let us endow $\tbv$ with the topology $\TT$ for which the injection
$$
\chi:B\mapsto \chi_B: (\tbv(\rn,\TT)\to
(\BV_c^\infty(\rn),\T)
$$
is a homeomorphism.
\fi
\begin{lemma}\label{l:aproxkrychle}
Let $A$ be a bounded $\BV$ set. Then there exists a sequence $\{A_i\}$ of dyadic figures which
converges to $ A$ in $\tbv$.
\end{lemma}
\begin{proof}
See \cite[Proposition 1.10.3]{Pf}.
\end{proof}
\begin{definition}
We say that a function $\F:\tbv\to\er$ is a \emph{charge} if $\F$ satisfies the following conditions:
\begin{enumerate}
\item $\F(A\cup B)=\F(A)+\F(B)$ for each disjoint bounded $\BV$ sets $A$ and $B$.
\item Given $\varepsilon>0$ there exists an $\eta>0$ such that
$|\F(C)|<\varepsilon$ for each $\BV$ set $C\subset B(0,1/\varepsilon)$ with $\|C\|<1/\varepsilon$ and $|C|<\eta$.
\end{enumerate}
\end{definition}
\begin{remark}
Let $E$ be a bounded $\BV$ set and $\F$ be a charge. Since $\F$ is additive and vanishes on bounded negligible sets,
$\F(E)$ depends only on the equivalence class of the set $E$.
\end{remark}
\begin{notation}
Let $E$ be a locally $BV$ set and $\F$ be a charge. Then $\F\lfloor_E$ denotes the charge
$\F\lfloor_E(A):=\F(A\cap E)$, $A\in\tbv$.
\end{notation}
\begin{definition}
Let $E$ be a locally $\BV$ set and let $\F$ be a charge.
We say that $\F$ is a \emph{charge in $E$}
if $\F=\F\lfloor_E$.
\end{definition}
\begin{proposition}\label{l:charchar}
An additive function $\F$ on $\tbv$ is a charge if and only if
either of the following conditions is satisfied.
\begin{enumerate}
\item
For given $\varepsilon$ there is a $\theta>0$ such that for every $\BV$ set $B\subset B(1/\varepsilon)$ we have
$$
|\F(B)|<\theta|B|+\varepsilon(\|B\|+1).
$$
\item
$\lim \F(A_i)=0$ for each sequence $\{A_i\}$ with $A_i\to\emptyset$ in $\tbv$ .
\end{enumerate}
\end{proposition}
\begin{proof}
See \cite[Proposition 2.2.6, Proposition 2.1.2]{Pf}.
\end{proof}
\begin{definition}
Let $A$ be a locally $\BV$ set.
We say that an additive function $\F:\tbv\to\er$ is a \emph{flux in $A$} of a vector field
$\mathbf u\in C(\bar{A},\rn)$, if for each $E\in\tbv$ we have
$$
\F(E)=
\int_{\partial_* (E\cap A)} \mathbf u\cdot\boldsymbol\nu_{E\cap A}\d\Hnmj,
$$
where $\boldsymbol\nu_{E\cap A}$ denotes the unit exterior normal of $E\cap A$.
In the case $A=\rn$ we say that $\F$ is just a \emph{flux} of $\mathbf u$.
\end{definition}
\begin{examples}\label{r:ident}
\begin{enumerate}
\item
Let $n=1$.
Since every bounded set $E\subset \er$ is equivalent to a finite disjoint union of compact intervals $\bigcup_{i=1}^k [a_i,b_i]$,
each additive function $\F$ on $\tbv$ can be written as
$$\F(E)
=\sum_{i=1}^k \left(u(b_i)-u(a_i)\right),$$
where $u:\er\to\er$.
The additive function $\F$ is a charge if and only if $u$ is continuous (see \cite[Remark 2.1.5]{Pf}).
In other words, $\F$ can be represented as the distributional derivative of a continuous function $u$.
\item
Let $\F$ be a flux in $A$ of a continuous vector field
$\mathbf u\in C(\bar{A},\rn)$, where $A$ is a locally $\BV$ set.
Then $\F$ is a charge (see \cite[Example 2.1.4]{Pf}).
On the other hand, a charge needs not to be of this form.
For an example see \cite[Example 2.1.10]{Pf}.
\item
Let $f\in\L_{\loc}^1 (\rn)$ be a function.
Then the function $\F:\tbv\to\er$ defined as
$$
\F(A)=\int_{A} f\d\L
$$
is a charge. (See \cite[Example 2.1.3]{Pf}.)
\item Let $A$ be a measurable set with $\Hnmj(A)>0$.
Then the function $\F:\tbv\to\er$ defined as $\F(E)=\Hnmj(E\cap A)$ is not a charge.
Without loss of generality we may assume $A$ to be bounded.
At first let us suppose that $\Hnmj(A)<\infty$. Then there is a constant $c$ such that
for every $k\in\en$ we can find a sequence of balls $\{B_i\}$
with $A\subset \bigcup_{i=1}^\infty B_i$, $\diam B_i<1/k$ and
$\sum_{i=1}^\infty \|B_i\|<c$.
Then for $E_k:=\bigcup_i B_i$ we have $E_k\subset \bigcup_{x\in A} B(x,1)$, $\|E_k\|<c$
and $|E_k|<\frac ck$.
It follows that $E_k\to\emptyset$ in $\tbv$, whereas $\F(E_k)=\Hnmj(E_k\cap A)=
\Hnmj(A)>0$.
By Proposition \ref{l:charchar} $\F$ cannot be a charge.
It is easy to check that $\F$ is not a charge
if $\Hnmj(A)=\infty$.
\end{enumerate}
\end{examples}
\section{Packing $\R$ integral
In this section we set up concept of the packing $\R$ integral, which will be further developed in the next section.
\begin{definition}
A pairwise disjoint finite system of balls
$(B(x_i,r_i))_{i=1}^k$
in $\rn$ is called a \textit{packing}.
A function $\delta:E\to[0,\infty)$, where $E\subset \rn$, is called a \emph{gage}
if the set $N=\{x;\delta(x)=0\}$ is of $\sigma$-finite $\Hnmj$ Hausdorff measure.
We say that a system $P=\{(A_1,x_1),\ldots, (A_k,x_k)\}$, $A_i\subset \rn$ and $x_i\in \rn$,
is \emph{$\delta$-fine} if
$d(A_i\cup {x_i})<\delta(x_i)$.
Let us remark that we do not require $x_i\in A_i$.
Especially, a packing
$(B(x_i,r_i))_{i=1}^k$ is $\delta$-fine if and only if $2r_i<\delta(x_i)$ for $i=1,\ldots, k$.
\end{definition}
\begin{notation
Let $x\in \er^n$, $r,\varepsilon>0$ and $\F$ be a charge. Then we will use the seminorms
$$
\bar{p}_{x,r}^\varepsilon(\F)=
\sup\left\{
|\F(E)|; E\subset\subset B(x,r), E\in \tbv, (E,x) \mbox{ is }\varepsilon\text{-regular}
\right\}.
$$
\end{notation}
\iffalse
\begin{definition
We say that a charge $\F$ is an \textit{indefinite packing $\R$
integral}
of a function $f\colon \rn\to\er$ with respect to a charge $\G$
if there exists
$\tau\in (0,1]$
such that for every $\varepsilon>0$ there exists a gage $\delta\colon \rn\to[0,\infty)$
such that
for every $\delta$-fine packing
$\sys{B(x_i,r_i)}_{i=1}^k
we have
$$
\sum_{i=1}^k
\bar{p}^\varepsilon_{x_i,\tau r_i}(\F-f(x_i)\G)<\varepsilon.
$$
The family of all functions which are packing $\R$ integrable with respect to a charge $\G$ is denoted by $\P\R(\G)$.
\end{definition}
\begin{theorem}[Uniqueness of the integral]
Let $f:\rn\to\er$ be a function and $\G$ be a charge. Then there exists at most one indefinite packing $\R$ integral of $f$ with
respect to $\G$.
\end{theorem}
\begin{proof}
Proof is analogous to the proof of Theorem \ref{thm:uniq} below.
\end{proof}
\fi
\begin{definition}
Let $A\subset\rn$ be a locally $\BV$ set.
We say that a charge $\F$ in $A$ is an \textit{indefinite packing $\R$
integral} of a function $f:\cl_*A\to\er$ in $A$ with respect to a charge $\G$
if there exists
$\tau\in (0,1]$
such that for every $\varepsilon>0$ there exists a gage $\delta\colon \cl_* A\to[0,\infty)$
such that for every $\delta$-fine packing
$\sys{B(x_i,r_i)}_{i=1}^k$, $x_i\in \cl_* A$, we have
$$
\sum_{i=1}^k
\bar{p}^\varepsilon_{x_i,\tau r_i}(\F-f(x_i)\G)<\varepsilon.
$$
\end{definition}
\begin{remark}
In the previous definition, as well as in forthcoming Definitions
\ref{d:pih}, \ref{d:dpih}, \ref{d:gdiv}, \ref{def:pff}, \ref{def:pffj} and \ref{def:mp}
it is possible to consider a function $f$ defined only on $\cl_*A\setminus T$,
where $T$ is of $\sigma$-finite $\Hnmj$ Hausdorff measure.
The integral is well defined since we can consider gages $\delta$ with
$\delta=0$ on $T$. For the same reason, the
indefinite packing $\R$ integral with respect to any charge $\G$
does not depend on values of $f$ on a set of
$\sigma$-finite $\Hnmj$ Hausdorff measure.
\end{remark}
\begin{remark}
The uniqueness of the indefinite packing integral of $f$ in $A$ will be discussed later.
\end{remark}
\begin{remark}
The indefinite packing $\R$ integral is linear with respect to a function $f$.
\end{remark}
\section{Packing $\R^*$ integral
Let us continue with so called packing $\R^*$ integral. We will prove its uniqueness, basic properties and finally we will
formulate and prove the Gauss-Green theorem.
Its definition relies on the concept of an $\varepsilon$-isoperimetric set, which was introduced by Mal\'{y} and Pfeffer in
\cite{mpf}.
We will be inspired by their work also further in this section.
\begin{definition}
Let $\varepsilon>0$ and $E\subset\rn$ be a bounded $BV$ set. We say that $E$ is \emph{$\varepsilon$-isoperimetric} if for each
$T\in\tbv$
$$
\min\{
P(E\cap T),P(E\setminus T)\}\leq\frac{1}{\varepsilon}P(T,\inn E).
$$
Since $P(T, \inn E)=P(E\cap T,\inn E)$, it is enough to consider only $T\subset E$. (See \cite[Lemma 2.1]{mpf}.)
\end{definition}
\begin{notation
Let $x\in \er^n$, $r,\varepsilon>0$ and $\F$ be a charge. Then we will use the seminorms
$$
\aligned
\bar{q}_{x,r}^\varepsilon(\F)&=
\sup\{
|\F(E)|; E\subset\subset B(x,r), E\in \tbv, x\in\cl_*E,\\
& (E,x) \mbox{ is }\varepsilon\text{-regular and }E\text{ is }\varepsilon
\text{-isoperimetric}
\}.
\endaligned
$$
\end{notation}
\begin{definition}\label{d:pih}
Let $A\subset \rn$ be a locally $\BV$ set.
We say that a charge $\F$ in $A$ is an \textit{indefinite packing $\R^*$
integral}
of a function $f\colon \cl_* A\to\er$ in $A$ with respect to a charge $\G$
if there exists
$\tau\in (0,1]$
such that for every $\varepsilon>0$ there exists a gage $\delta\colon \cl_* A\to[0,\infty)$
such that
for every $\delta$-fine packing
$\sys{B(x_i,r_i)}_{i=1}^k$, $x_i\in \cl_* A$,
we have
$$
\sum_{i=1}^k
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-f(x_i)\G)<\varepsilon.
$$
In the case $A=\rn$ we say that $\F$ is just an \textit{indefinite packing $\R^*$ integral} of $f$ with respect to $\G$.
The family of all functions packing $\R^*$ integrable with respect to a charge $\G$ is denoted by $\P\R^*(\G)$.
\end{definition}
\begin{lemma}\label{l:predkey}
Let $\tau\in(0,1]$ and $\varepsilon>0$.
Then there exists a constant $c_T$ (depending only on $\tau$ and $n$)
with the following property:
for each function $\Phi:\er\to (0,\infty)$, $x\in \rn$ and $R>0$ there exists
$0<r<R$ such that
$$
\Phi(10r)+\varepsilon|B(x,10r)|\leq c_T\bigl(\Phi(\tau r)+\varepsilon|B(x,\tau r)|\bigr).
$$
\end{lemma}
\begin{proof}
See \cite[Lemma 3.7]{KM}.
\end{proof}
\begin{lemma}\label{l:pomery}
Let $0<\varepsilon\leq1/(2n)$ and $Q=[0,a_1]\times[0,a_2]\times\cdots\times[0,a_{n}]$
be an $\varepsilon$-regular interval.
Then
$$
\max\{a_1,\ldots,a_n\}\leq \frac{1}{\varepsilon} \min\{a_1,\ldots,a_n\}.
$$
\end{lemma}
\begin{proof}
For simplicity, let us suppose that $a_1\leq a_2\leq \cdots \leq a_n$.
Since $Q$ is $\varepsilon$-regular, we can estimate
$$
a_n(a_2\cdots a_n)\leq d(Q)\|Q\|<\frac{1}{\varepsilon}|Q|=\frac{1}{\varepsilon}a_1a_2\cdots a_n.
$$
Dividing by $a_2\cdots a_n$ we obtain
$a_n<\frac{1}{\varepsilon}a_1$, which establishes the formula.
\end{proof}
\begin{lemma}\label{l:regiso}
Let $\varepsilon>0$, $Q$ be an $\varepsilon$-regular interval and $T\in\tbv$, $T\subset Q$ satisfying $|T|\leq |Q|/2$.
Then there exists a constant $\gamma=\gamma(\varepsilon,n)$ such that
\begin{equation}\label{e:regiso}
\Hnmj(\partial Q\cap \partial_*T)\leq
\gamma\Hnmj(\intt Q\cap \partial_*T).
\end{equation}
\end{lemma}
\begin{proof}
At first let $Q$ be a cube. By \cite[Lemma 6.7.2]{Pffin} there exists a constant $\eta=\eta(n)$ such that
\begin{equation}\label{e:cube}
\Hnmj(\partial Q\cap \partial_*T)\leq
\eta\Hnmj(\intt Q\cap \partial_*T).
\end{equation}
Further, let $Q$ be an $\varepsilon$-regular interval.
We can suppose $Q=[0,a_1]\times[0,a_2]\times\cdots\times[0,a_{n}]$, $a_1\leq a_2\leq\cdots\leq a_n$.
Let $L:\rn\to\rn$ be a linear mapping represented by the diagonal matrix
$$
\begin{pmatrix}
a_n/a_1 &0 &\cdots &0\\
0&a_n/a_2 & \cdots &0\\
\vdots&\vdots & \ddots & \vdots \\
0&0 & \cdots &1\\
\end{pmatrix}.
$$
Then $L(Q)$ is a cube and $|L(T)|\leq|L(Q)|/2$.
Moreover,
$\intt L(Q) \cap \partial_* L(T)=L(\intt Q \cap \partial_* T)$.
Further, we can estimate the Lipschitz constant of $L$ as $\Lip(L)=\max_i\{a_i/a_n\}\leq \frac{1}{\varepsilon}$,
which follows from Lemma \ref{l:pomery}.
Since $L^{-1}$ can be represented by the matrix
$$
\begin{pmatrix}
a_1/a_n &0 &\cdots &0\\
0&a_2/a_n & \cdots &0\\
\vdots&\vdots & \ddots & \vdots \\
0&0 & \cdots &1\\
\end{pmatrix},
$$
we have $\Lip(L^{-1})=1$.
Applying Lemma \ref{l:Pflipeo}, inequality \eqref{e:cube} and properties of $L$
we obtain
$$
\aligned
\Hnmj(\partial Q\cap \partial_*T)&=
\Hnmj(L^{-1}(L(\partial Q\cap \partial_* T)))\leq
\Hnmj(\partial L(Q) \cap \partial_* L(T))\\
&\leq
\eta \Hnmj(\intt L(Q) \cap \partial_* L(T))
=
\eta \Hnmj(L(\intt Q \cap \partial_* T))\\
&\leq
\frac{\eta}{\varepsilon^{n-1}} \Hnmj(\intt Q \cap \partial_* T).
\endaligned
$$
Hence \eqref{e:regiso} holds with $\gamma(\varepsilon,n):=\frac{\eta}{\varepsilon^{n-1}}$.
\end{proof}
\begin{lemma}\label{l:regular}
For every $n\in\en$ there exists an increasing function $\beta:(0,\infty)\to\er$ such that
every $\varepsilon$-regular interval $Q\subset\rn$ is $\beta(\varepsilon)$-isoperimetric.
\end{lemma}
\begin{proof}
We set $\beta(\varepsilon)=1/(1+\gamma(\varepsilon,n))$, the constant $\gamma(\varepsilon,n)$ being as in Lemma \ref{l:regiso}.
Now let us fix an $\varepsilon$-regular interval $Q$ and a set $T\in\tbv$, $T\subset Q$. We need to show that
$$
\min\{
P(Q\cap T),P(Q\setminus T)\}\leq\frac{1}{\beta(\varepsilon)}P(T,\inn Q),
$$
Let us assume $|T|\leq |Q|/2$. Since $Q$ is an interval, we have $\intt Q=\intt_*Q$.
Then by Lemma \ref{l:regiso}
there exists a $\gamma=\gamma(\varepsilon,n)$ such that
$$\aligned
P(T)&\leq \Hnmj(\intt Q\cap\partial_*T)+\Hnmj(\partial Q\cap\partial_*T)
\leq(1+\gamma)\Hnmj(\intt Q\cap\partial_*T)\\
&\leq(1+\gamma)P(T,\inn Q)=\frac{1}{\beta(\varepsilon)}P(T,\inn Q).
\endaligned$$
In the case $|T|>|Q|/2$ we have $|Q\setminus T|<|Q|/2$ and then we obtain
$$
P(Q\setminus T)=P(Q\cap(Q\setminus T))\leq(1+\gamma)P(Q\setminus T,\inn Q)
=\frac{1}{\beta(\varepsilon)}P(T,\inn Q).
$$
\end{proof}
\begin{lemma}\label{l:predkvadry}
Let $r>0$, $x\in\rn$ and $Q=[a_1,b_1]\times[a_2,b_2]\times\cdots\times[a_n,b_n]$ be an
interval such that
$Q\subset B(x,2r)$ and $\frac{r}{2\sqrt{n}}\leq\min_l\{|b_l-a_l|\}$.
Then $(Q,x)$ is $\rho$-regular,
where
$\rho=\rho(n)=\frac{1}{n^{\frac{n+1}{2}}2^{3n-2}}$.
\end{lemma}
\begin{proof}
Let us denote
$s:=\min_l\{|b_l-a_l|\}$ and $w:=\max_l\{|b_l-a_l|\}$.
Since $\frac{r}{2\sqrt{n}}\leq s$, $w\leq 4r$ and $\diam(Q\cup\{x\})\leq 4r$,
we can estimate the regularity of $Q$ as
$$
\aligned
r(Q,x)&=
\frac{|Q|}{\diam(Q\cup\{x\})\cdot\|Q\|}\\
&\geq\frac{s^{n-1}w}{4r\cdot 2nw^{n-1}}\\
&\geq\frac{\left(\frac{r}{2\sqrt{n}}\right)^{n-1}}{8rn\left(4r\right)^{n-2}}\\
&=\frac{1}{n^{\frac{n+1}{2}}2^{3n-2}}=\rho(n).\\
\endaligned
$$
\end{proof}
\begin{lemma}\label{l:kvadry}
Let $\F$ be a charge and $B(x,r)\subset\rn$, $x=(x_1,x_2,\ldots,x_n)$,
be a ball.
Further, let
$Q=[a_1,b_1]\times[a_2,b_2]\times\cdots\times[a_n,b_n]$ be an
interval such that
$Q\subset B(x,2r)$ and $\frac{r}{2\sqrt{n}}\leq\min_l\{|b_l-a_l|\}$.
Then
\begin{equation}\label{e:kvadry}
|\F(Q)|\leq
2^m\bar{q}^{\varepsilon}_{x,2r}(\F),
\end{equation}
where
$m=\#\{l; x_l\not\in[a_l,b_l]\}$, $\varepsilon=\min\{\beta(\rho),\rho\}$ and $\beta$ and $\rho$ are as in Lemma
\ref{l:regular} and Lemma \ref{l:predkvadry}.
\end{lemma}
\begin{proof}
The proof proceeds by induction on $m$.
First, for $m=0$ we have $x\in Q$.
Since $Q\subset B(x,2r)$ and $\frac{r}{2\sqrt{n}}\leq\min_l\{|b_l-a_l|\}$,
$Q$ is $\rho(n)$-regular. Furthermore, by Lemma \ref{l:regular} we obtain $Q$ is also $\beta(\rho)$-isoperimetric.
Then we can estimate
$$
|\F(Q)|\leq
\bar{q}^{\varepsilon}_{x,2r}(\F).
$$
Now let us fix $m\geq 1$ and suppose that \eqref{e:kvadry} holds for $m-1$.
Without loss of generality we can assume that
$x_l\not\in[a_l,b_l]$ for $l=1,\ldots,m$.
Our next purpose is to define an auxiliary interval
$$\widetilde{Q}=
[{a}_1,{b}_1]\times\cdots\times
[a_{m-1},b_{m-1}]
\times
[\tilde{a}_{m},\tilde{b}_{m}]\times
[a_{m+1},b_{m+1}]\times
\cdots\times [a_n,b_n],$$
where $[\tilde{a}_{m},\tilde{b}_{m}]$ is defined as follows:
In the case $x_m<a_m$ let us set $\widetilde{a}_m=x_m-(b_m-x_m)$ and $\widetilde{b}_m=b_m$.
If $x_m>b_m$, let us set $\widetilde{a}_m=a_m$, $\widetilde{b}_m=x_m+(x_m-a_m)$.
We see that $Q\subset \widetilde{Q}\subset B(x,2r)$ and $x\in\widetilde{Q}$.
For simplicity, let us assume $x_m<a_m$. Then
$$
{Q}=\widetilde{Q} \setminus
\widetilde{Q}',
$$
where
$$
\widetilde{Q}'=
[{a}_1,b_1]\times\cdots\times
[{a}_{m-1},b_{m-1}]\times
[\tilde{a}_{m},a_{m}]\times
[a_{m+1},b_{m+1}]\times\cdots\times [a_n,b_n].
$$
In the following, we need to estimate the regularity of subintervals $\widetilde{Q}$ and $\widetilde{Q'}$.
Since $\min_l\{|b_l-a_l|\}\geq \frac{r}{2\sqrt{n}}$
and $\widetilde{Q}\subset B(x,2r)$,
by Lemma \ref{l:predkvadry} we obtain
$\widetilde{Q}$ is $\rho(n)$-regular.
Analogously we obtain the regularity of $\widetilde{Q}'$.
By Lemma \ref{l:regular} we have $\widetilde{Q}$ and $\widetilde{Q}'$ are $\beta(\rho)$-isoperimetric.
Using the additivity of $\F$ and the inductive assumption we obtain
$$
|\F(Q)|\leq |\F(\widetilde{Q})|+|\F(\widetilde{Q}')|
\leq 2\cdot 2^{m-1}\bar{q}^{\varepsilon}_{x,2r}(\F)=
2^{m}\bar{q}^{\varepsilon}_{x,2r}(\F),
$$
which completes the proof.
\end{proof}
\begin{theorem}[Uniqueness of the integral]\label{thm:uniq}
Let $f$ be a function and $\G$ be \linebreak a charge. Then there exists at most one indefinite packing $\R^*$ integral of $f$ with
respect to $\G$.
\end{theorem}
\begin{proof}
Let $\F_1$, $\F_2$ be indefinite packing $\R^*$ integrals of $f$ with respect to $\G$. Then
$\F_1-\F_2$ is the integral of $0$ with respect to $\G$.
So it is sufficient to show that if
$\F$ is an indefinite packing $\R^*$ integral of $f\equiv 0$,
then $\F\equiv 0$.
By Lemma \ref{l:aproxkrychle} it is enough to prove that $\F(K)=0$ for each dyadic cube $K$.
Let $\tau$ be as in Definition \ref{d:pih}.
Now, let us fix a dyadic cube $K$
of side $a_0$ and choose $\varepsilon>0$ such that
$\varepsilon<\min\{\beta(\rho),\rho\}$,
where $\beta$ and $\rho$ are as in Lemma \ref{l:regular} and Lemma \ref{l:predkvadry}.
Finally, denote $K_0:=\bigcup_{x\in K}B(x,1)$.
STEP 1.
Since $\F$ is an indefinite packing $\R^*$ integral of $f\equiv0$, there exists a
gage $\delta:\rn\to [0,\infty)$
such that for every $\delta$-fine packing
$\sys{B(x_i,r_i)}_{i=1}^h
we have
\begin{equation}\label{e:defint}
\sum_{i=1}^h
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F)<\varepsilon.
\end{equation}
STEP 2.
In this step we construct the covering of the set $K\setminus N$, where $N=\{x;\delta(x)=0\}$.
By Lemma \ref{l:predkey}, applied to $\Phi(r):=\bar{q}^\varepsilon_{x,r}(\F)$, we can
find
a constant $c_T$ such that
for every $x$ there exists $r(x)<\delta(x)$, $10 r(x)<1$,
with the following properties:
\begin{equation}\label{e:mkrd}
20 r(x)<a_0
\end{equation}
and
\begin{equation}\label{e:predkey}
\bar{q}^\varepsilon_{x,10r(x)} (\F) +\varepsilon|B(x,10r(x))|\leq
c_T (\bar{q}^\varepsilon_{x,\tau r(x)} (\F) +\varepsilon|B(x,\tau r(x))|).
\end{equation}
Now, let us consider the covering $\C=\{\bar{B}(x,r(x)); x\in K\setminus N\}$.
By the Vitali theorem we can construct a pairwise disjoint subsystem
$\C'\subset \C$, such that
$\bigcup_{{B}(x,R)\in\C''} {B}(x,R)\supset K\setminus N$,
where
$\C''=\{B(x,5r);\bar{B}(x,r)\in\C' \}$.
STEP 3.
Now we will cover the set $N$.
Since $N$ is of $\sigma$-finite $\Hnmj$ measure, we can write out
$N=\bigcup_{s=1}^\infty N^s$, where $\H^{n-1}(N^s)=c_s<\infty$
for every $s=1,2,\ldots$
Let us fix $s\in\en$
and $\varepsilon_s\in(0,\varepsilon)$ such that
\begin{equation}\label{e:epsilon}
\varepsilon_s (c_1c_c2^{n-1}(c_s+\varepsilon)+1)<2^{-s}\varepsilon,
\end{equation}
where
$c_1= 2^n n^{(3-n)/2}$ and $c_c=\alpha_n2^{2n}n^{n/2}$.
By Lemma \ref{l:charchar}, with $\varepsilon_s$ we can associate $\theta_s$ such that
for every $\BV$ set $E\subset B(1/\varepsilon)$ we have
\begin{equation}\label{e:charchar}
|\F(E)|<\theta_s|E|+\varepsilon_s(\|E\|+1).
\end{equation}
Furthermore, there exist $\zeta_s<1/2$ and a system of balls $\N^s=\{B(x_i^s,R^s_i)\}$
covering $N^s$ such that
$R^s_i\leq\zeta_s$,
\begin{equation}\label{e:mkr}
4R^s_i<a_0,
\end{equation}
$$
c_2\zeta_s\theta_s(c_s+\varepsilon)<2^{-s}\varepsilon\alpha_{n-1}
$$
and
\begin{equation}\label{e:nmna}
\sum_{B(x_i^s,R_i^s)\in\N^s}
\alpha_{n-1}\left(\frac{\diam B(x_i^s,R^s_i)}{2}\right)^{n-1}\leq
(\alpha_{n-1}+1)\sum_{B(x_i^s,R_i^s)\in\N^s}
(R^s_i)^{n-1}
<c_s+\varepsilon.
\end{equation}
Note that
\begin{equation}\label{e:nmnadva}
c_2\theta_s \sum_{B(x_i^s,R_i^s)\in\N^s} (R_i^s)^{n}
\leq c_2\zeta_s\theta_s \sum_{B(x_i^s,R_i^s)\in\N^s} (R_i^s)^{n-1}<2^{-s}\varepsilon,
\end{equation}
where $c_2=\alpha_n c_c 2^{n}$.
Let us denote $\N:=\bigcup_s \N^s$.
Now, let us consider the covering $\V:=\C''\cup\N$. Since $\V$ covers the compact set $K$,
we can choose a finite system of balls
$B(x_i,R_i)\in\V$, $i=1,\ldots, k$, covering $K$.
Without loss of generality we can assume that $B(x_1,R_1),\ldots, B(x_h,R_h)\in\C''$
and $B(x_{h+1},R_{h+1}),\ldots,B(x_k,R_k)\in \N$.
STEP 4.
In this step we construct a partition of the cube $K$ in the sense that
we look for a finite system of nonoverlapping cubes whose union is $K$.
Recall that $Q'$ denotes the mother cube of a cube $Q$.
Let $\K$ denote the family of all dyadic subcubes of $K$.
For fixed $i\in\{1,\ldots,k\}$ set
$$\aligned
\tilde\Q_i=\{Q\in\K;
Q\cap B(x_i,R_i)\neq \emptyset,\,
Q\subset B(x_{i},2R_{i})
\text{ and } Q'\not\subset B(x_{i},2R_{i})
\}.
\endaligned
$$
\begin{comment}
For instance, cubes of the first generation arise from bisecting the sides of $K$.
This yields $2^n$ closed cubes, each of side length $a/2$.
The cubes of the second generation are obtained by bisecting each cube of the first generation and so on.
The partitioning of a cube $Q$ stops if we find a
ball $B(x_{i},R_{i})$, ($i\in\{1,\ldots, k\}$), such that
$Q\cap B(x_i,R_i)\neq \emptyset$,
$Q\subset B(x_{i},2R_{i})$ and $Q'\not\subset B(x_{i},2R_{i})$, where
$Q'$ denotes the mother cube of $Q$.
Especially, let us construct systems $\K_p$ and $\Q_p$
of cubes with side length $a/2^p$,
$p=0,1,2,\ldots$, by induction.
Let us set $\K_0:=\{K\}$ and $\Q_0:=\emptyset$.
Further, let us denote
$$\K_p:=\{Q; Q'\in \K_{p-1}\setminus Q_{p-1}\}$$
and
$$\aligned
\Q_p&:=\{Q\in\K_p;
\exists i\in\{1,\ldots, k\}:
Q\cap B(x_i,R_i)\neq \emptyset,\,
Q\subset B(x_{i},2R_{i}),\\
&\qquad\qquad Q'\not\subset B(x_{i},2R_{i})
\},
\endaligned$$
where $Q$ are cubes arisen in the $p$-th step and
$Q'$ denotes the mother cube of $Q$.
Finally we obtain a finite partition $\{Q_j, j=1,\ldots, m\}$, $Q_j\in\Q_p$ for some $p$, of a cube $K$
and a finite system $\qak$ of pairs $(Q_j;B(x_i,R_i))$, where $i:=\min\{i;
Q_j\cap B(x_{i},R_{i})\neq\emptyset;\,
Q_j\subset B(x_{i},2R_{i}),
Q_j'\not\subset B(x_{i},2R_{i})\}$.
Assumptions \eqref{e:mkrd} and \eqref{e:mkr} ensure that there exists a system $\Q_p$, which is nonempty.
\end{comment}
We show that the union
$\tilde\Q=\bigcup^k_{i=1}\tilde\Q_i$
is all of $K$. Choose $y\in K$.
Consider a sequence $P_l$ of dyadic cubes such that $P_0=K$,
$P_{l-1}=P_l'$ for $l=1,2,\ldots$ and $\{y\}=\bigcap_{l=0}^\infty
P_l$. There exists $i\in\{1,\ldots,k\}$ such that $y\in B(x_i,R_i)$.
Since $\diam P_l \searrow 0$,
there exists $l$ such that $P_{l}\subset B(x_{i},2R_{i})$.
We find the smallest $l$ such that $P_l\subset B(x_{i},2R_{i})$.
By \eqref{e:mkrd} and \eqref{e:mkr}, $l\ge 1$.
We easily verify that $y\in P_l\in\tilde\Q$.
Next we show that the system $\tilde \Q$ is finite.
Let us fix $Q\in\tilde\Q_i$ and
denote the side length of $Q$ by $a$.
The length of the diagonal can be expressed as $\sqrt{n}a$.
Since
$Q'$ intersects both $B(x_i,R_i)$ and $B(x_i,2R_i)^c$, we obtain
\begin{equation}\label{dolni}
R_i/2<\sqrt{n}a.
\end{equation}
Hence the side length of all cubes in $\tilde\Q_i$ is bounded from below.
Therefore,
the systems $\tilde\Q_i$ and hence the system $\tilde\Q$ are finite.
Now we can define
the system of cubes
$$
\Q=\tilde\Q\setminus\{
Q\in\tilde\Q;\, \exists P\in\tilde\Q\text{ such that }\, P\supsetneq Q
\}.
$$
Since two dyadic cubes are either in inclusion or nonoverlapping,
$\Q$ is a finite partition of $K$; we enumerate it as
$\Q=\{Q_j, j=1,\ldots, m\}$.
Finally, let us define the systems
$$
\Q_i=\Big\{Q\in \Q\cap\tilde \Q_i\colon Q\notin \bigcup_{l<i} \tilde \Q_l\Big\}.
$$
Let us fix $Q_j\in \Q_i$ and denote it side length by $a_j$.
Recall that the length of the diagonal can be expressed as $\sqrt{n}a_j$ and
since $Q_j$ is included in $B(x_i,2R_i)$, we have
$\sqrt{n}a_j<4R_{i}$. Hence
we can estimate the perimeter of $Q_j$:
\begin{equation}\label{e:perQ}
\|Q_j\|=2n a_j^{n-1}
\leq 2n\left(\frac{4R_i}{\sqrt{n}}\right)^{n-1}
= 2^nn^{(3-n)/2}2^{n-1}R_i^{n-1}=
c_12^{n-1}R_i^{n-1}.
\end{equation}
Let us estimate the number of the cubes $Q_j\in\Q_i$.
Applying \eqref{dolni}, we obtain
$$
\alpha_n(2R_i)^n=|B(x_i,2R_i)|\geq \left| \bigcup_{Q_j\in\Q_i} Q_j\right|\geq \#\Q_i \left(\frac{R_i}{2\sqrt{n}}\right)^n.
$$
Hence $\# \Q_j\leq c_c$.
(Let us remind that $c_c= \alpha_n 2^{2n}n^{n/2}$.)
STEP 5.
In this step we estimate $\F(K)$.
By the additivity of $\F$ we obtain
$$
\F(K)=
\F\left(\bigcup_{i=1}^k \bigcup_{Q_j\in \Q_i} Q_j \right)=
\F\left(\bigcup_{i=1}^h \bigcup_{Q_j\in \Q_i} Q_j \right)+
\F\left(\bigcup_{i=h+1}^k \bigcup_{Q_j\in \Q_i} Q_j \right).
$$
Firstly let us suppose that $i\in\{1,\ldots, h\}$.
Then $B(x_{i},R_{i})\in\C''$.
Let us fix a pair $(Q_j,B(x_i,R_i))$. Since $q_j\geq\frac{R_i}{2\sqrt{n}}$,
we can apply Lemma \ref{l:kvadry} and obtain
\begin{equation}\label{e:konkrkvadr}
|\F(Q_j)|\leq 2^n\bar{q}^\varepsilon_{x_i,2R_i}(\F).
\end{equation}
Using the fact that $\#Q_j\leq c_c$, the system $\{B(x_1,r_1),\ldots, B(x_h,r_h)\}$ is a $\delta$-fine packing
and applying \eqref{e:konkrkvadr}, \eqref{e:predkey} and \eqref{e:defint} we can estimate
$$
\aligned
\left|\F\left(\bigcup_{i=1}^h \bigcup_{Q_j\in \Q_i} Q_j \right)\right|&=
\sum_{i=1}^h \sum_{Q_j\in \Q_i} |\F(Q_j)|\\
&\leq
\sum_{i=1}^h c_c2^n\bar{q}^\varepsilon_{x_i,2R_i}(\F)\\
&\leq 2^nc_c\sum_{i=1}^h
c_T(\bar{q}^\varepsilon_{x_i,\tau r_i}(\F)+\varepsilon|B(x_i,\tau r_i)|)\\
&<c_c2^n(c_T\varepsilon +c_T\varepsilon|K_0|)
=c_c2^nc_T\varepsilon (1+|K_0|).
\endaligned
$$
Secondly, let us fix $s\in\en$ and set
$\A^s:=\{i\in\{h+1,\ldots,k\}; B(x_{i},R_{i})\in\N^s\}$.
Then, applying the fact that $\# \Q_j\leq c_c$ and inequalities \eqref{e:charchar}, \eqref{e:nmna},
\eqref{e:epsilon}, \eqref{e:perQ} and \eqref{e:nmnadva} we obtain
$$
\aligned
\left|\F\left(\bigcup_{{i\in\A^s }} \bigcup_{Q_j\in\Q_i}
Q_j\right)\right|
&\leq
\theta_s\left|\bigcup_{{i\in\A^s }} \bigcup_{Q_j\in\Q_i}
Q_j\right|
+\varepsilon_s\left(\left\|\bigcup_{{i\in\A^s }} \bigcup_{Q_j\in\Q_i}
Q_j
\right\|+1\right)\\
&<
\varepsilon_s + \theta_s \alpha_n c_c2^{n}\sum_{{i\in\A^s }}
R_i^n +\varepsilon_s
\sum_{{i\in\A^s }}
\sum_{{Q_j\in\Q_i }}
\|Q_j\| \\
&<
\varepsilon_s + \theta_s c_2\sum_{{i\in\A^s }}
R_i^n +\varepsilon_s c_c c_1
2^{n-1}\sum_{{i\in\A^s }} R_i^{n-1}\\
&\leq
\varepsilon_s +
\varepsilon 2^{-s} +\varepsilon_s c_c c_1 2^{n-1}(c_s+\varepsilon)
<
2\cdot2^{-s}\varepsilon.
\endaligned
$$
Since the union
$
\bigcup_{s=1}^\infty\bigcup_{{i\in\A^s }}
\bigcup_{Q_j\in\Q_i} Q_j
$
has only finite number of nonempty elements, we can use the additivity of $\F$ and we obtain
$$
|\F(K)|<
c_c2^nc_T\varepsilon(1+|K_0|)+\varepsilon\sum_{s=1}^\infty 2^{-s+1}
=
\varepsilon(c_Tc_c2^n(1+|K_0|)+2),
$$
which completes the proof.
\end{proof}
\begin{remark}
The indefinite packing $\R^*$ integral of a function $f$ with respect to a charge $\G$ depends linearly on $f$.
\end{remark}
In the preceding, we were concerned with an indefinite packing $\R^*$ integral of a function $f:\rn\to\er$.
Now we will concentrate on a packing $\R^*$ integral in $A$, where $A$ is a locally $\BV$ set.
\iffalse
\begin{proposition}[Linearity of packing $\R^*$ integral]
Let $\F_1$, $\F_2$ and $\G$ be charges and $f_1,f_2:\rn\to\er$ be functions.
Further, let $\F_1$ (resp., $\F_2$) be the indefinite packing $\R^*$ integral of $f_1$ (resp., $f_2$)
with respect $\G$.
Then
\begin{enumerate}
\item $\F_1+\F_2$ is the indefinite packing $\R^*$ integral of the function $f_1+f_2$ with respect to $\G$,
\item $\alpha\F_1$ is the indefinite packing $\R^*$ integral of the function $\alpha f_1$ with respect to $\G$.
\end{enumerate}
\end{proposition}
\begin{proof}
The proof is analogous to the proof of Theorem \ref{thm:add}
\end{proof}
\fi
\begin{theorem}\label{thm:rhr}
Let $A\subset\rn$ be a locally $\BV$ set and let a
charge $\F$ be an indefinite packing $\R$ integral of a function $f:\cl_*A\to\er$ in $A$
with respect to a charge $\G$.
Then $\F$ is also an indefinite packing $\R^*$ integral of $f$ in $A$ with respect to $\G$.
\end{theorem}
\begin{proof}
The proof follows from the fact that $\bar{q}^\varepsilon_{x,r}\leq \bar{p}^\varepsilon_{x,r}$.
\end{proof}
\begin{notation}
Let $A\subset\rn$ and $f:A\to\er$ be a function. Then $\bar{f}_A$ denotes the
zero extension of $f$:
$$
\bar{f}_A= \begin{cases}
f(x) & \text{if } x\in A,\\
0 & \text{if } x\not\in A.\\
\end{cases}
$$
\end{notation}
\iffalse
\begin{definition}\label{d:urcin}
Let $A\subset\rn$ be a $\BV$ set.
We say that a charge $\F$ in $A$ is an \textit{indefinite packing $\R^*$
integral of a function $f:A\to\er$ in $A$ with respect to a charge $\G$}
if there exists
$\tau\in (0,1]$
such that for every $\varepsilon>0$ there exists a gage $\delta\colon \rn\to[0,\infty)$
such that for every $\delta$-fine packing
$\sys{B(x_i,r_i)}_{i=1}^k$, $x_i\in A$, we have
$$
\sum_{i=1}^k
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-f(x_i)\G)<\varepsilon.
$$
\end{definition}
\fi
The two following lemmas with proofs can be found in \cite[Lemma 2.5 and 3.7]{mpf}.
\begin{lemma}\label{l:meas}
Let $\F$ be a charge. Then for $\varepsilon>0$ there is an absolutely continuous Radon measure $\mu$ in $\rn$ such that
for each $BV$ set $E\subset B(0,1/\varepsilon)$,
$$
|\F(E)|\leq \mu(E)+\varepsilon P(E).
$$
\end{lemma}
\begin{lemma}\label{l:admis}
Let $A\in\cBV_{\loc}$ and $\varepsilon>0$. For each $x\in \ext_c A$, there is $\delta>0$ such that
every strongly $\varepsilon$-regular set $E$ with $x\in\cl_* E$ and $d(E)<\delta$ satisfies
$$
P(E\cap A)\leq P(E\setminus A).
$$
\end{lemma}
The proof of the next theorem follows the lines of the proof in \cite[Lemma 3.8]{mpf}.
\begin{theorem}\label{thm:admis}
Let $\F$ be a charge and $A$ be an admissible locally $\BV$ set.
For given $\tau\in(0,1]$ and $\varepsilon>0$ there is a
gage $\delta:\rn\to [0,\infty)$ such that
$$
\sum_{x_i\in A}\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_{A^c})<\varepsilon
\quad \mbox{and }\quad
\sum_{x_i\not \in A}\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_A)<\varepsilon
$$
for each $\delta$-fine packing $\sys{B(x_i,r_i)}_{i=1}^k$.
\end{theorem}
\begin{proof}
At first let us suppose that $A$ is bounded.
Let us fix $\varepsilon>0$ such that $\bar A\subset B:=B(0,1/\varepsilon')$,
where
\eqn{ep'}
$$\ep'=\frac{\ep^2}{P(A)}.
$$
By Lemma \ref{l:meas}, there is an absolutely continuous Radon measure $\mu$ in $\rn$ such that
$$
|F(E)|\leq \mu(E)+\ep'\,P(E)
$$
for each $E\in\BV$, $E\subset B$.
Then there exists a compact $K$ such that $K\subset B\setminus A$ and
\begin{equation}\label{e:mi}
\mu((B\setminus A)\setminus K)<\frac12\,\varepsilon.
\end{equation}
Applying Lemma \ref{l:admis} to $A^c$, for each $x\in B\cap \ext_c A^c=B\cap \intt_c A$ we can find $\delta_x>0$ such that $B(x,\delta_x)\subset B$,
and
\begin{equation}\label{e:p}
P(E\setminus A)\leq P(E\cap A)
\end{equation}
for each strongly $\varepsilon$-regular set $E$ with $x\in\cl_*E$ and $d(E)<\delta_x$.
\begin{comment}
Further, applying Lemma \ref{l:admis} to $A$, for each $x\in B\cap \ext_c A$ we can find $\delta_x>0$ such that $B(x,\delta_x)\subset B$,
and
\begin{equation}\label{e:p}
P(E\cap A)\leq P(E\setminus A)
\end{equation}
for each strongly $\varepsilon$-regular set $E$ with $x\in\cl_*E$ and $d(E)<\delta_x$.
\end{comment}
Making $\delta_x$ smaller, we may assume that $K\cap B(x,\delta_x)=\emptyset$ for $x\in \intt_c A$.
Since $A$ and is an admissible set, it follows that also $A^c$ is admissible and hence $\intt_c A^c\subset A^c$ and $A^c\cap \ext_cA^c=\emptyset$.
Let us set $N:=\partial_c A^c=\partial_c A$. Then $N$ is of $\sigma$-finite Hausdorff measure $\Hnmj$, which follows from the criterion for
finite perimeter \cite[p. 222]{EG}.
Now we can define a gage $\tilde\delta$ on $\rn$ in the following way:
$$
\tilde\delta(x) = \begin{cases}
0 & \text{if } x\in N,\\
1 & \text{if } x\in \ext_c A,\\
\delta_x & \text{if } x\in \intt_c A.
\end{cases}
$$
Let us fix a $\tilde\delta$-fine packing $\sys{B(x_i,r_i)}_{i=1}^k$
and sets $E_i$, where
$E_i\subset\subset B(x_i,\tau r_i)$, $E_i\in \BV$, $(E_i,x_i)$ is $\varepsilon$-regular and $E_i$ is $\varepsilon$-isoperimetric
for each $i=1,\ldots, k$.
By the $\varepsilon$-regularity of $E_i$, inequality \eqref{e:p} and definition of $\tilde\delta$, we obtain
\begin{enumerate}
\item $x_i\not\in N$ for $i=1,\ldots, k$;
\item $E_i\setminus A\subset (B\cap A^c)\setminus K$ when $x_i\in \intt_cA$;
\item $P(E_i\setminus A)\leq (1/\varepsilon)P(A^c,\inn E_i)$ when $x_i\in \intt_c A$.
\end{enumerate}
Hence, using the inequality \eqref{e:mi} and
the fact that packing is pairwise disjoint, we can estimate
$$
\aligned
\sum_{x_i\in A} |\F(E_i\setminus A)|&=
\sum_{x_i\in \intt_c A} |\F(E_i\setminus A)|
\\
&\leq
\sum_{x_i\in \intt_c A} \mu(E_i\setminus A)+\frac{\varepsilon^2}{2P(A)}\;P(E_i\setminus A)\\
&\leq
\mu(B\cap A^c\setminus K)+\frac{\varepsilon}{2P(A)}
\sum_{x_i\in \intt_c A} P(A^c, \inn E_i)\leq
\frac\varepsilon2+\frac\varepsilon2.
\endaligned
$$
Passing to the supremum we obtain
$$
\sum_{x_i\in A}\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_{A^c})<\varepsilon,
$$
which we needed.
We now turn to the case $A$ is unbounded.
Let us consider a sequence of balls $\{B_m\}$ which forms a
locally finite
covering of $\rn$.
Choose $\ep>0$.
Let us fix $m\in\en$ and set $A_m=A\cap B_m$. Then $A_m$ is a bounded admissible locally $\BV$ set
and we can use the previous step to find $\ep_m\le 2^{-m}\ep$ and
a gage $\delta_m:\rn\to[0,\infty)$ such that
$$
\sum_{x_i\in A_m}\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_{A_m^c})<
\varepsilon_m
$$
for every $\delta_m$-fine packing $((B(x_i,r_i))_{i=1}^k$.
Further, let us set
$$
\tilde\delta(x):=\min\{\delta_m(x)\colon x\in B_m\}.
$$
It is easily seen that $\tilde\delta$ is a gage.
Let us fix a $\tilde\delta$-fine packing $((B(x_i,r_i))_{i=1}^k$. Then
$$
\aligned
\sum_{x_i\in A}\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_{A^c})&\leq
\sum_{m=1}^\infty
\sum_{x_i\in A_m}\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_{A_m^c})\\
&<
\sum_{m=1}^\infty 2^{-m}\varepsilon=\ep,
\endaligned
$$
which establishes the formula.
Finally, we proceed similarly to find
$\tilde\delta^c$ which yields the second inequality and set
$$
\delta=\min\{\tilde\delta,\tilde\delta^c\},
$$
which gives both inequalities at the same time.
\end{proof}
In the proof of the next theorem we are inspired by \cite[Proposition 3.9]{mpf}.
\begin{theorem}\label{thm:urc}
Let $\G$, $\F$ be charges, $f\in\P\R^*(\G)$ and let $\F$ be an indefinite packing $\R^*$ integral of $f$ with respect to $\G$.
If $A$ is an admissible locally $\BV$ set, then $\chi_Af\in\P\R^*(\G)$ and $\F\lfloor_A$ is an indefinite packing $\R^*$ integral
of $\chi_A f$ with respect to $\G$.
\end{theorem}
\begin{proof}
Let us fix $\tau\in(0,1]$ as in Definition \ref{d:pih} and $\varepsilon>0$.
By the definition of packing $\R^*$ integral and Theorem \ref{thm:admis}
there exists a gage $\delta:\rn\to[0,\infty)$ such that
for every $\delta$-fine packing $(B(x_i,r_i))_{i=1}^k$
we have
$$
\sum_{i=1}^k \bar{q}^\varepsilon_{x_i,\tau r_i} (\F-f(x_i)\G)<\varepsilon,
$$
$$
\sum_{x_i\in A}\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_{A^c})<\varepsilon
\quad \mbox{and }\quad
\sum_{x_i\not \in A}\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_A)<\varepsilon.
$$
Hence
$$
\aligned
&\sum_{i=1}^k \bar{q}^\varepsilon_{x_i,\tau r_i} (\F\lfloor_A-f(x_i)\chi_A(x_i)\G)\\
&\qquad=
\sum_{x_i\in A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F\lfloor_A-f(x_i)\G)+
\sum_{x_i\not\in A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F\lfloor_A)\\
&\qquad<
\sum_{x_i\in A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F\lfloor_A-\F)+
\sum_{x_i\in A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F-f(x_i)\G)+
\sum_{x_i\not\in A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F\lfloor_A)\\
&\qquad=
\sum_{x_i\in A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F\lfloor_{A^c})+
\sum_{x_i\in A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F-f(x_i)\G)+
\sum_{x_i\not\in A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F\lfloor_A)\\
&\qquad<3\varepsilon,
\endaligned
$$
which completes the proof.
\end{proof}
\begin{theorem}\label{thm:admisn}
Let $A$ be an admissible locally $\BV$ set.
Then the charge $\F$ in $A$ is an indefinite packing $\R^*$
integral of a function $f:\cl_*A\to\er$ with respect to a charge $\G$ in $A$ if and only if
$\F$ is an indefinite packing $\R^*$ integral of $\bar{f}_A$ with respect to $\G$ in $\rn$.
\end{theorem}
\begin{proof}
Let us suppose that $\F$ in $A$ is the indefinite packing $\R^*$ integral of $f$ with respect to $\G$ in $A$.
Let us fix $\varepsilon>0$.
Now let $\tau\in (0,1]$ and a gage $\delta_1$ on $\cl_*A$ be as in Definition \ref{d:pih}
and let $\delta_2$ on $\rn$ be as in Theorem \ref{thm:admis}.
Then let us fix a $\delta$-fine packing $\sys{B(x_i,r_i)}_{i=1}^k$ and
set $$\delta=
\begin{cases}
\min\{\delta_1(x),\delta_2(x)\} & \text{if } x\in \cl_*A,\\
\delta_2(x) & \text{if } x\in \rn\setminus \cl_*A.
\end{cases}
$$
At first, let us consider the sum over $x_i\in A$.
Since $\F$ in $A$ is the indefinite packing $\R^*$ integral of $f$ in $A$, we have
$$
\sum_{x_i\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-\bar{f}_A(x_i)\G)=
\sum_{x_i\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-f(x_i)\G)<\varepsilon.
$$
\iffalse
Furthermore, Theorem \ref{thm:admis} gives
$$
\sum_{x_i\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_{A^c})<\varepsilon.
$$
Since $\F\lfloor_A=\F-\F\lfloor_{A^c}$, we obtain
\begin{equation}\label{e:intrj}
\left|
\sum_{x_i\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_A-\bar{f}_A(x_i)\G)
\right|=
\left|
\sum_{x_i\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-\bar{f}_A(x_i)\G)-
\sum_{x_i\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F\lfloor_{A^c})
\right|
<\varepsilon+\varepsilon.
\end{equation}
\fi
Further, for the case $x_i\not\in A$,
we have by Theorem \ref{thm:admis} the estimate
$$
\sum_{x_i\not\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-\bar{f}_A(x_i)\G)=
\sum_{x_i\not\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F)<\varepsilon.
$$
Therefore we obtain
$$
\aligned
\sum_{i=1}^k
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-\bar{f}_A(x_i)\G)&=
\sum_{x_i\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-\bar{f}_A(x_i)\G)+
\sum_{x_i\not\in A}
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-\bar{f}_A(x_i)\G)\\
&<
2\varepsilon.
\endaligned
$$
Hence $\F$ is the indefinite packing $\R^*$ integral of $\bar{f}_A$ with respect to $\G$.
Conversely, let $\F$ be the indefinite packing $\R^*$ integral of $\bar{f}_A$ with respect to $\G$.
By Theorem \ref{thm:urc} it follows that $\F\lfloor_A$ is the indefinite packing $\R^*$ integral of $\bar{f}_A$ with respect to
$\G$ in $\rn$.
In other words, for fixed $\varepsilon>0$
there exists a gage $\delta:\rn\to[0,\infty)$ such that
$\delta=0$ on $\cl_*A\setminus A$ and
for every $\delta$-fine packing $(B(x_i,r_i))_{i=1}^k$
we have
$$
\sum_{i=1}^k \bar{q}^\varepsilon_{x_i,\tau r_i} (\F\lfloor_A-\bar{f}_A(x_i)\G)<\varepsilon.
$$
By the uniqueness of packing $\R^*$ integral we have $\F\lfloor_A=\F$ and hence
$$
\sum_{x_i\in \cl_*A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F-f(x_i)\G)
=
\sum_{x_i\in A} \bar{q}^\varepsilon_{x_i,\tau r_i} (\F-\bar{f}_A(x_i)\G)
<\varepsilon,
$$
which we needed.
\end{proof}
\iffalse
\begin{corollary}
Let $A\in\tbv$ be an admissible set, $f:A\to\er$ be a function and let charges $\F_1$ and $\F_2$ be an indefinite
integral with respect to a charge $\G$ in $A$.
Then $\F_1\lfloor_A=\F_2\lfloor_A$.
\end{corollary}
\fi
\begin{corollary}
Let $A$ be an admissible locally $BV$ set and let a charge $\F$ be an indefinite packing $\R^*$ integral of a
function $f:\cl_*A\to\er$ in $A$
with respect to a charge $\G$.
Then, by Theorem \ref{thm:admisn}, $\F$ is the indefinite packing $\R^*$ integral of $\bar{f}_A$ with respect to $\G$,
which is unique by Theorem \ref{thm:uniq}.
Therefore the indefinite packing $\R^*$ integral in $A$ is unique as well.
Further, let $A$ be an admissible locally $BV$ set
and let a charge $\F$ be an indefinite packing $\R$ integral of a function $f:\cl_*A\to\er$ in $A$
with respect to a charge $\G$.
Then $\F$ is also the packing $\R^*$ integral of in $A$ with respect to $\G$ by Theorem \ref{thm:rhr}.
Hence the uniqueness holds also for the indefinite packing $\R$ integral in $A$.
\end{corollary}
\begin{remark}\label{r:wrtL}
Since the function $f$ is defined on $\cl_* A$, the requirement that $A$ be admissible might seem to be unnecessary.
This is really the case with $\G=\L$, because
sets of measure zero (such as $A\triangle \cl_*A$) does not play a role in integration with respect to Lebesgue measure.
On the other hand, Lebesgue null sets cannot be neglected in general.
For example, the classical Cantor set cannot be neglected
for integration with respect to the Cantor measure in $\er$, which is a charge
by Example \ref{r:ident}(1).
\end{remark}
\begin{definition}\label{d:dpih}
Let $A\in\tbv$ be an admissible set, $f:\cl_*A\to\er$ be a function and $\F$, $\G$ be charges.
We say that the number $\F(A)$ is a \textit{definite packing $\R^*$
integral} of $f$ over $A$ with respect to $\G$
if $\F$ is an indefinite packing $\R^*$ integral of $\bar{f}_A$ with respect to $\G$.
More generally: if $A\subset \rn$ is a bounded measurable
set and $\F$ is the indefinite packing $\R^*$ integral of $\bar{f}_A$ with respect to $\G$,
then the definite packing $\R^*$ integral of $f$ over $A$ with respect to $\G$ is the number
$\F(A')$, where $A'\in\tbv$, $A'\supset A$ is a bounded admissible set.
The family of all functions packing $\R^*$ integrable with respect to $\G$ over $A$ is denoted by $\P\R^*(A,\G)$.
\end{definition}
\begin{remark}
The integral does not depend on the choice of $A'$.
Indeed, let $A'$ and $A''$ be bounded admissible $\BV$ sets.
Since $\bar{f}_{A}\cdot\chi_{A'}=\bar{f}_{A}\cdot\chi_{A''}$,
by Theorem \ref{thm:urc} and by the uniqueness of the packing $\R^*$
integral we obtain $\F\lfloor_{A'}=\F\lfloor_{A''}$.
Then
$
\F(A')=\F\lfloor_{A'}(A'\cup A'')=\F\lfloor_{A''}(A'\cup A'')=\F(A'').
$
\end{remark}
\begin{remark}
Let $A\in \tbv$ be an admissible set, $\G$ be a charge and $f\in\P\R^*(A,\G)$.
Let $\F$ be the indefinite packing $\R^*$ integral of $f$
in $A$ with respect to $\G$.
Then the definite $\P\R^*$ integral of $f$ over $A$ wich respect to $\G$ is just
$\F(A)$.
This fact follows from Theorem \ref{thm:admisn}.
\end{remark}
\begin{remark}
If $f$ is a merely an indefinite packing $\R^*$ integrable function,
it does not make sense to define the definite integral over unbounded sets in general.
If we want to set up the definite integral over an unbounded set, we must suppose some additional limiting behaviour
of the indefinite integral at infinity.
There are several nonequivalent ways how to do it and we do not pursue this direction.
\end{remark}
\begin{remark}
Let $A\subset\rn$ be a bounded measurable set and let $f:A\to\er$ be a Lebesgue integrable function.
Then $\bar{f}_A$ is also a Lebesgue integrable function.
Then there exists an indefinite packing $\R^*$ integral of $\bar{f}_A$ with respect to Lebesgue measure.
Hence the definite packing $\R^*$ integral
of $f$ over $A$
is well defined.
\end{remark}
In the following theorem, we will focus on the convergence of a sequence of sets.
The importance of this property will be demonstrated in Section \ref{s:sedm}.
The proof uses ideas from Pfeffer and Mal\'y in \cite[Theorem 3.20]{mpf}.
\iffalse
VERZE 2
\begin{theorem}\label{thm:conmn}
Let $\G$ and $\F$ be charges and let $f$ be a function defined on a $\BV$ set $A$.
Let $\{A_j\}_{j=1}^\infty$ be a sequence of $BV$ sets such $A_j\subset A$ for $j=1,2,\ldots$ and $A_j\to A$.
Further, let $\F\lfloor_{A_j}$ be an indefinite packing $\R^*$ integral of $f$ on $A_j$ with respect to $\G$.
Then there exists an indefinite packing $\R^*$ integral of $f$ on $A$ with respect to $\G$ and is equal to $\F\lfloor_{A}$.
\end{theorem}
\fi
\begin{theorem}\label{thm:conmn}
Let $A$ be a bounded admissible $\BV$ set, $\G$ and $\F$ be charges and let $f:\rn\to\er$ be a function.
Let $\{A_j\}_{j=1}^\infty$ be a sequence of bounded admissible $BV$ sets such that
$A_j\subset A$ for $j=1,2,\ldots$ and $A_j\to A$ in $\tbv$.
Further, let $f\chi_{A_j}\in \P\R^*$ and
$\F\lfloor_{A_j}$ be an indefinite packing $\R^*$ integral of $f\chi_{A_j}$ with respect to $\G$
with constants $\tau_j$ as in Definition \ref{d:pih}.
Let $\inf_j \tau_j>0$.
Then there exists an indefinite packing $\R^*$ integral of $f\chi_{A}$ with respect to $\G$ and is equal to $\F\lfloor_{A}$.
\end{theorem}
\begin{proof}
Let us fix $\tau=\inf_j \tau_j$ and let us denote $N:=A\setminus \bigcup_{j=1}^\infty A_j$.
Then
$N$ is of $\sigma$-finite $\Hnmj$ measure (see \cite[Cor. 6.2.7]{Pf}).
Let us choose $\varepsilon>0$.
Since $\bar{q}^\varepsilon_{x,\tau r}\leq
\bar{q}^\varepsilon_{x,\tau' r}$ for $\tau\leq \tau'$,
we can by the definition of packing $\R^*$ integral and by Theorem \ref{thm:admis}
for $j\in\en$ find a gage $\delta_j$ such that
for each $\delta_j$-fine packing $\sys{B(x_i,r_i)}_{i=1}^k$
we obtain
\begin{equation}\label{e:conmnj}
\sum_{x_i\in A_j}
\bar{q}^{\varepsilon}_{x_i,\tau r_i}(\F\lfloor_{A_j}-f(x_i)\G)<\varepsilon 2^{-j
\end{equation}
and
\begin{equation}\label{e:conmnd}
\sum_{x_i\in A_j}
\bar{q}^\varepsilon_{x_i, \tau r_i}(\F\lfloor_{A_j^c})
<\varepsilon 2^{-j}.
\end{equation}
Further, for $x\in\bigcup_{j=1}^\infty A_j$ let us set $j_x:=\min\{j\in\en; x\in A_j\}$.
Now we can define a gage
$$
\delta(x) = \begin{cases}
\delta_{j_x}(x) & \text{if } x\in\bigcup_{j=1}^\infty A_j,\\
0& \text{if } x\in N.\\
\end{cases}
$$
By Theorem \ref{thm:admisn} it is enough to show that $\F\lfloor_A$ is the indefinite packing $\R^*$ integral of $f$ in $A$
with respect to $\G$.
Let us choose $\delta$-fine packing $(B(x_i,r_i))_{i=1}^k$, $x_i\in A$, and denote $j_i:=j_{x_i}$.
\iffalse
Now, let us consider a set $A_p$ with the following property: there exists $i\in\{1,\ldots,k\}$ such that $j_i=p$.
\fi
Using the additivity of $\F$ and estimates \eqref{e:conmnj} and \eqref{e:conmnd} we can for fixed $p\in\en$ estimate
$$
\aligned
\sum_{x_i:j_i=p}
\bar{q}^{\varepsilon}_{x_i,\tau r_i}(\F\lfloor_{A}-f(x_i)\G)
&\leq
\sum_{x_i:j_i=p}
\bar{q}^{\varepsilon}_{x_i,\tau r_i}(\F\lfloor_{A_j}-f(x_i)\G)
+
\bar{q}^\varepsilon_{x_i, \tau r_i}(\F\lfloor_{A\setminus A_j})\\
&<
\varepsilon 2^{-p+1}.\\
\endaligned
$$
Summing over $p$ we obtain
$$
\aligned
\sum_{p=1}^\infty
\sum_{x_i:j_i=p}
\bar{q}^{\varepsilon}_{x_i,\tau r_i}(\F\lfloor_{A}-f(x_i)\G)
&<\sum_{p=1}^\infty
\varepsilon 2^{-p+1}
=2\varepsilon,
\endaligned
$$
which completes the proof.
\end{proof}
\iffalse
\begin{remark}
In the remainder of the article we will denote by $\div \mathbf u$ the usual divergence of a function $\mathbf u:\rn\to\rn$.
\end{remark}
\fi
\begin{definition}\label{d:gdiv}
Let $A\subset \rn$ be a locally $\BV$ set and let $f:\cl_*A\to\er$ and $\mathbf u\in C(\bar{A},\rn)$ be functions.
Further, let a charge $\F$ be the flux of $\mathbf u$ in $A$.
We say that $f$ is a \emph{generalized divergence} of $\mathbf u$ in $A$
if $\F$ is an indefinite packing $\R^*$ integral of $f$ in $A$.
The generalized divergence of $\mathbf u$ will be denoted by $\Div \mathbf u$.
\end{definition}
The following three definitions was mentioned by Pfeffer in Chapters 2.3 and 2.5 of \cite{Pf}.
\begin{definition}
Let $A\subset\rm$ be a measurable set and let $x\in A\cap \intt_* A$.
A map $\mathbf u:A\to\rn$ is called \emph{differentiable at $x$ relative to $A$}
if there is a linear map $L:\rm\to\rn$ such that for given $\varepsilon>0$ there exists a $\delta>0$ so that
$$
|\mathbf u(y)-\mathbf u(x)-L(y-x)|<\varepsilon |y-x|
$$
for each $y\in A\cap B(x,\delta)$.
The linear map $L$ is called the
\emph{differential of $\mathbf u$ at $x$ relative to $A$} and is denoted by $D_A\mathbf u(x)$.
Let $x\in\intt_* A$ and $\mathbf u:\cl_*A\to\er^m$ be a vector field. Let $\mathbf u$ be differentiable at $x$ relative to
$\cl_* A$.
The \emph{divergence of $\mathbf u$ at $x$ relative to $\cl_* A$} is the number
$\div_* \mathbf u(x):=\tr D_{\cl_*A}\mathbf u(x)$,
where $\tr D_{\cl_*A}\mathbf u(x)$ denotes the trace of the matrix representation of the linear transformation
$D_{\cl_*A}\mathbf u(x):\rm\to\rm$.
By $\div \mathbf u$ we will denote the pointwise divergence defined on interior points of $A$ at
which $\mathbf u$ is differentiable.
Especially, $\div_*\mathbf u=\div \mathbf u$ whenever $\div \mathbf u$ is defined.
\end{definition}
\begin{definition}
Let $\F$ be a charge and let $x\in\rn$. Then for $\eta\geq0$ we define
$$
\underline{D}_\eta\F(x):=\sup_{\delta>0}\inf_E \frac{\F(E)}{|E|}
\quad\mbox{ and }\quad
\overline{D}_\eta\F(x):=\inf_{\delta>0}\sup_E \frac{\F(E)}{|E|},
$$
where $E\in\BV$ such that $d(E\cup\{x\})<\delta$ and $r(E,x)>\eta$.
The \emph{lower} and \emph{upper derivative} of $\F$ at $x$ are defined as
$$
\underline{D}\F(x):=\inf_{\eta>0}\underline{D}_\eta\F(x)
\quad\mbox{ and }\quad
\overline{D}\F(x):=\sup_{\eta>0}\overline{D}_\eta\F(x).
$$
We say that $\F$ is \emph{derivable} at $x$, if
$$
\underline{D}\F(x)=
\overline{D}\F(x)\neq \pm\infty.
$$
The \emph{derivative} of $\F$ at $x$ is then defined as $D\F(x):=\underline{D}\F(x)=\overline{D}\F(x)$.
\end{definition}
\begin{definition}
Let $E$ be a locally $\BV$ set, $\mathbf u:\cl_*E\to\rn$ be a bounded Borel measurable vector field
and $\F$ be the flux of $\mathbf u$.
If $\F$ is derivable at $x\in\intt_c E$, we call the number
$\ndiv \mathbf u(x):=D\F(x)$
the \emph{mean divergence} of $\mathbf u$ at $x$.
\end{definition}
Applying the inclusion between $\R$ integral and packing $\R^*$ integral we can state
sufficient conditions for existence of generalized divergence. For further details see
\cite[Example 2.3.2, Remark 2.5.9, Theorem 5.1.12, Proposition 2.5.7 and Corollary 5.1.13]{Pf}.
\begin{proposition}
Let $A$ be a locally $BV$ set and $\mathbf u\in C(\bar{A},\rn)$.
\begin{enumerate}
\item
If $A=\rn$ and $\mathbf u$ is differentiable in $\rn$,
then $\div \mathbf u$ is a generalized divergence of $\mathbf u$.
\item
If $\mathbf u$ is differentiable relatively to $\cl_* A$ on $\intt_c A$,
then $\div_* \mathbf u$ is a generalized divergence of $\mathbf u$.
\item
If $\mathbf u$ is differentiable relatively to $\cl_* A$ on $\intt_c A$,
then $\ndiv \mathbf u$ is a generalized divergence of $\mathbf u$.
\item
If $\mathbf u$ is Lipschitz on $\cl_* A\setminus T$, where $T$ is of $\sigma$-finite Hausdorff measure $\Hnmj$,
then $\div_* \mathbf u$ is a generalized divergence of $\mathbf u$.
\iffalse
\item SPATNE
If the flux of $\mathbf u$ is almost derivable on $\cl_* A\setminus T$,
where $T$ is of $\sigma$-finite Hausdorff measure $\Hnmj$,
then $\ndiv \mathbf u$ is a generalized divergence of $\mathbf u$.
\fi
\end{enumerate}
\end{proposition}
\begin{theorem}[Gauss-Green divergence theorem]
Let $A\subset\rn$ be a bounded $BV$ set,
let $\mathbf u\in C(\bar{A},\rn)$. Let us suppose that there exists a generalized divergence $\Div \mathbf u$
in $A$.
Then
$$
\int_{A}\Div \mathbf u(x)\,dx= \int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu_A\,d\Hnmj,
$$
where the integral on the left is the definite packing $\R^*$ integral.
\end{theorem}
\begin{proof}
Since
$|A\triangle \cl_*A|=0$,
it is enough to show that
$\int_{\cl_* A}\Div \mathbf u(x)\,dx= \int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu_A\,d\Hnmj$ (see Remark \ref{r:wrtL}).
Let $\F$ denote the indefinite packing $\R^*$ integral of $\Div \mathbf u$ in $A$.
Since $\F$ is the flux of $\mathbf u$ in $A$, we have $\F(A)=\int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu_A\,d\Hnmj$.
By Theorem \ref{thm:admisn} we obtain
$\int_{\cl_*A}\Div \mathbf u(x)\,dx=\F(A)$, which completes the proof.
\end{proof}
\iffalse
\begin{theorem}[Gauss-Green divergence theorem]
Let $A\subset\rn$ be a bounded $BV$ set,
let $\mathbf u\in C(\bar{A},\rn)$. Let us suppose that there exists a generalized divergence $\Div \mathbf u$
on a set $\Omega$ containing $\cl_* A$.
Then
$$
\int_{A}\Div \mathbf u(x)\,dx= \int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu_A\,d\Hnmj.
$$
\end{theorem}
\begin{proof}
Let $\F$ denote the indefinite packing $\R^*$ integral of $\Div \mathbf u$.
Since $\F$ is the flux of $\mathbf u$ we have $\F(A)=\int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu_A\,d\Hnmj$.
By Theorem \ref{thm:urc} we obtain
$\int_{A}\Div \mathbf u(x)\,dx=\F(A)$, which completes the proof.
\end{proof}
\fi
\iffalse
\begin{theorem}[Gauss-Green divergence theorem]
Let $A\subset\rn$ be a bounded $BV$ set,
let $\mathbf u\in C(\rn,\rn)$. Let us suppose that the generalized divergence $\Div \mathbf u$ is an
indefinite packing integral of a function $f:\rn\to\er$ in an open set $\Omega$
containing $\cl_* A$. Then
$$
\int_{A}f(x)\,dx= \int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu_A\,d\Hnmj.
$$
\end{theorem}
\begin{proof}
Let us denote by $\F$ the indefinite packing $\R^*$ integral of $f$.
By definition of the flux we obtain $\F(A)=\int_{\partial_*A}\mathbf u\cdot\boldsymbol\nu_A\,d\Hnmj$.
Now we need to prove the equality
$\int_{A}f(x)\,dx=\F(A)$, which follows by Theorem \ref{thm:urc}, and the proof is complete.
\end{proof}
\fi
\section{$\R$ integral
In this section we will introduce Pfeffer's $\R$ integral described in \cite{Pf}.
For easier comparison of integrals we use
the characterization of $\R$ integral \cite[Proposition 5.5.6]{Pf} rather than original definition.
\begin{definition}
A \emph{$\BV$ partition} is a system of couples $\{(A_1,x_1),\ldots,(A_k,x_k)\}$
of pairwise disjoint bounded $\BV$ sets $A_i$ and points $x_i\in\rn$ for $i=1,\ldots,k$.
It is not required $x_i\in A_i$.
\end{definition}
\begin{definition}\label{def:pff}
Let $A$ be a locally $\BV$ set and $\F$, $\G$ be charges in $A$.
Let $f$ be a function defined on $\cl_* A$.
We say that $\F$
is an \emph{intrinsic indefinite $\R$ integral} of $f$ in $A$ with respect to $\G$ if
for given $\varepsilon>0$ we can find a gage $\delta:\cl_* A\to[0,\infty)$
so that
$$
\sum_{i=1}^k \big|
\F(A_i)-f(x_i)\G(A_i)
\big|<\varepsilon
$$
for each $\varepsilon$-regular $\delta$-fine $\BV$ partition
$\{(A_1,x_1),\ldots,(A_k,x_k)\}$ with
$\bigcup_{i=1}^k A_i\subset A$ and $x_i\in \cl_* A$ for $i=1,\ldots, k$.
The family of all $\R$ integrable functions in $A$ with respect to $\G$ is denoted by $\R(A,\G)$.
The family of all $\R$ integrable functions in $A$ with respect to Lebesgue measure is denoted just by $\R(A)$.
\end{definition}
\begin{remark}
The intrinsic indefinite $\R$ integral is well defined, unique and linear. For the proof and other properties see
\cite[p. 211-213]{Pf}.
\end{remark}
\begin{definition}\label{def:pffj}
Let $A$ be a locally $\BV$ set and $\F$, $\G$ be charges in $A$.
Let $f$ be a function defined on $\cl_* A$.
We say that $\F$
is an \emph{indefinite $\R$ integral} of $f$ in $A$ with respect to $\G$ if
for given $\varepsilon>0$ we can find a gage $\delta:\cl_* A\to[0,\infty)$
so that
$$
\sum_{i=1}^k \big|
\F(A_i)-f(x_i)\G(A_i)
\big|<\varepsilon
$$
for each $\varepsilon$-regular $\delta$-fine $\BV$ partition
$\{(A_1,x_1),\ldots,(A_k,x_k)\}$ with
$x_i\in \cl_* A$ for $i=1,\ldots, k$.
(We do not require that $A_i\subset A$.)
The family of all $\R$ integrable functions in $A$ with respect to $\G$ is denoted by $\I\R(A,\G)$.
\end{definition}
\begin{remark}
Let us remark that our terminology slightly differs from that used in \cite{Pf}.
Namely, what we call ``intrinsic indefinite $\R$ integral in $A$'' is termed simply ``indefinite $\R$
integral'' in \cite{Pf}.
Furthermore, in \cite{Pf} it is distinguished between the $\R$ integral (with respect to Lebesgue measure)
and $\S$ integral (Stieltjes version; with respect to an arbitrary charge).
\end{remark}
\begin{lemma}\label{l:krit}
Let $\varepsilon>0$ and $A\subset \rn$ be an $\varepsilon$-regular bounded $\BV$ set.
Then $[\diam (A)]^n\leq\frac{1}{\varepsilon^n}c |A|$, where $c=c(n)$ is a constant depending only on $n$.
\end{lemma}
\begin{proof}
Since $A$ is $\varepsilon$-regular, we have $\diam (A)P(A)\leq \frac{1}{\varepsilon}|A|$.
Further, by the isoperimetric inequality (see \cite[Theorem 1.8.7]{Pf}) we have
$|A|^{\frac{n-1}{n}}\leq p(n)P(A)$, where $p(n)$ is a constant depending on $n$.
Thus $\diam (A)P(A)\leq \frac{1}{\varepsilon}|A|\leq \frac{1}{\varepsilon}p(n)^{\frac{n}{n-1}}P(A)^{\frac{n}{n-1}}$.
Hence
$$
\aligned
\diam (A)^{n-1}P(A)^{n-1}&\leq \frac{1}{\varepsilon^{n-1}}p(n)^nP(A)^{{n}},\\
\diam (A)^{n-1}&\leq \frac{1}{\varepsilon^{n-1}}p(n)^nP(A),\\
\diam (A)^{n}&\leq \frac{1}{\varepsilon^{n-1}}p(n)^nP(A)\diam (A)\\
&\leq \frac{1}{\varepsilon^{n}}c|A|,\\
\endaligned
$$
where $c=p(n)^n$.
\end{proof}
\begin{theorem}\label{thm:rinnein}
Let $A$ be a locally $\BV$ set, $\F$, $\G$ be charges in $A$. Let $f$ be a function defined on $\cl_* A$.
If $\F$ is an intrinsic indefinite $\R$ integral of $f$ in $A$ with respect to $\G$,
then $\F$ is also an indefinite $\R$ integral of $f$ in $A$ with respect to $\G$.
\end{theorem}
\begin{proof}
Let us choose $\varepsilon\in(0,1/(c\alpha_n))$
and set $\varepsilon'=\varepsilon(1-c\alpha_n\varepsilon)/(1+c)$,
where $c=c(n)$ is as in Lemma \ref{l:krit}.
Let us find a gage $\delta_1:\cl_* A\to[0,\infty)$
so that
$$
\sum_{i=1}^k \big|
\F(A_i)-f(x_i)\G(A_i)
\big|<\varepsilon'
$$
for each $\varepsilon'$-regular $\delta_1$-fine $\BV$ partition
$\{(A_1,x_1),\ldots,(A_k,x_k)\}$ with
$\bigcup_{i=1}^k A_i\subset A$ and $x_i\in \cl_* A$ for $i=1,\ldots, k$.
Further, for each $x\in\intt_c A$ let us find $R=R(x)>0$ such that
for every $r<R$ we have
\begin{equation}\label{e:krit}
P(A,B(x,r))\leq \varepsilon^{n-1} r^{n-1}.
\end{equation}
Since $\intt_c A\subset \intt_*A$, for every $x\in\intt_cA$ we can find $R'=R'(x)>0$ such that
\begin{equation}\label{e:hust}
|B\setminus A|<\varepsilon^{n+1}|B|
\end{equation}
for every $B=B(x,r)$, $r<R'$.
Now let us define
$$
\delta(x)=\begin{cases}
0 &\text{if } x\in \cl_*A\setminus \intt_cA,\\
\min\{\delta_1(x),R(x),R'(x)\} &\text{if } x\in \intt_cA.\\
\end{cases}
$$
Since the set $\cl_*A\setminus \intt_c A$ is of $\sigma$-finite $\Hnmj$ Hausdorff measure (see \cite[Theorem 1.8.2]{Pf} and \cite[Proposition 7.3.1]{Pffin}),
$\delta$ defines a gage.
Let us fix an
$\varepsilon$-regular $\delta$-fine $\BV$ partition
$\{(A_1,x_1),\ldots,(A_k,x_k)\}$ such that
$x_i\in \intt_c A$ for $i=1,\ldots, k$.
We need to show that
$$
\sum_{i=1}^k \big|
\F(A_i)-f(x_i)\G(A_i)
\big|<\varepsilon.
$$
Let us set $A'_i=A_i\cap A$, $i=1,\ldots, k$.
Then $\{(A'_1,x_1),\ldots,(A'_k,x_k)\}$ is obviously a $\delta$-fine $\BV$ partition.
We need to show that the system $\{(A'_1,x_1),\ldots,(A'_k,x_k)\}$ is $\varepsilon'$-regular.
Let us fix $i\in\{1,\ldots,k\}$.
Since $(A_i,x_i)$ is $\varepsilon$-regular, we have
$$\diam(A_i\cup\{x_i\})P(A_i)\leq\frac{1}{\varepsilon}|A_i|.$$
Further, let us find a minimal ball ${B}={B}(x_i,r)$ with the property that $A_i\subset \bar{B}$.
Then $r<\delta(x)$.
By Lemma \ref{l:krit} there exists a constant $c$ such that
$$|B|\leq \alpha_n \left(\diam (A_i\cup\{x_i\})\right)^n \leq \frac{c\alpha_n}{\varepsilon^n}|A_i|.$$
Then applying \eqref{e:hust} we can estimate
$$
\aligned
|A_i|&\leq |A_i\cap A|+|A_i\setminus A|\\
&\leq |A_i\cap A|+|B\setminus A|\\
&\leq |A_i\cap A|+\varepsilon^{n+1}|B|\\
&\leq |A_i\cap A|+c\alpha_n\varepsilon |A_i|.
\endaligned
$$
Hence
\begin{equation}\label{e:deleni}
(1-c\alpha_n\varepsilon)|A_i|\leq |A_i\cap A|.
\end{equation}
Applying \eqref{e:krit}, \eqref{e:deleni} and Lemma \ref{l:krit} then gives
$$
\aligned
\diam(A_i\cap A\cup\{x_i\})P(A_i\cap A)&\leq
\diam(A_i\cup\{x_i\})\left[P(A_i)+ P(A, {B})\right]\\
&\leq
\frac{1}{\varepsilon}|A_i|+\diam (A_i\cup\{x_i\})\varepsilon^{n-1} r^{n-1}\\
&\leq
\frac{1}{\varepsilon}|A_i|+\varepsilon^{n-1}[\diam (A_i\cup\{x_i\})]^{n}\\
&\leq
\left(\frac{1}{\varepsilon}+\frac{c}{\varepsilon}\right)
|A_i|\\
&\leq
\frac{1+c}{\varepsilon(1-c\alpha_n\varepsilon)}
|A_i\cap A|\\
&=
\frac{1}{\varepsilon'}
|A_i\cap A|.
\endaligned
$$
Thus the system $\{(A'_1,x_1),\ldots,(A'_k,x_k)\}$ is $\delta$-fine $\varepsilon'$-regular $\BV$ partition.
Since $\F$ and $\G$ are charges in $A$, we have $\F(A_i)=\F(A'_i)$ and $\G(A_i)=\G(A'_i)$ for $i=1,\ldots,k$.
Further, since $\F$ is the intrinsic indefinite $\R$ integral of $f$ with respect to $\G$, we can estimate
$$
\aligned
&\sum_{i=1}^k \big|
\F(A_i)-f(x_i)\G(A_i)
\big|
=
\sum_{i=1}^k \big|
\F(A'_i)-f(x_i)\G(A'_i)
\big|
<\varepsilon'<\varepsilon,
\endaligned
$$
which completes the proof.
\end{proof}
\begin{corollary}\label{c:rir}
Let $A$ be a locally $\BV$ set, $\F$, $\G$ be charges in $A$. Let $f$ be a function defined on $\cl_* A$.
Then $\F$ is an intrinsic indefinite $\R$ integral of $f$ in $A$ with respect to $\G$
if and only if $\F$ is an indefinite $\R$ integral of $f$ in $A$ with respect to $\G$.
\end{corollary}
\begin{theorem}\label{thm:pr}
Let $A$ be an admissible locally $\BV$ set, $\F$, $\G$ be charges in $A$. Let $f$ be a function defined on $\cl_* A$.
Let $\F$ be an (intrinsic) indefinite $\R$ integral of $f$ in $A$ with respect to $\G$.
Then $\F$ is also an indefinite packing $\R$ integral of $f$ in $A$ with respect to $\G$.
\end{theorem}
\begin{proof}
Let us set $\tau:=1$. Now let us choose $\varepsilon>0$ and
find a gage $\delta$ as in Definition \ref{def:pffj}.
Let us fix a $\delta$-fine packing
$\sys{B(x_i,r_i)}_{i=1}^k$, $x_i\in \cl_* A$.
We need to show that
$$
\sum_{i=1}^k
\bar{p}^\varepsilon_{x_i,\tau r_i}(\F-f(x_i)\G)<\varepsilon,
$$
where
$\bar{p}_{x,r}^\varepsilon(\F)=
\sup\left\{
|\F(E)|; E\subset\subset B(x,r), E\in \tbv, (E,x) \mbox{ is $\varepsilon$-regular}
\right\}.
$
Now, let us fix test sets $E_i$
such that
$E_i\subset\subset B(x_i,r_i)$,
$E_i$ are $\BV$ sets
and $(E_i,x_i)$ are $\varepsilon$-regular for $i=1,\ldots,k$.
Obviously, the system $\{ (E_i,x_i)\}$ is $\varepsilon$-regular $\delta$-fine $BV$ partition and hence by Definition
\ref{def:pffj} and Theorem \ref{thm:rinnein}
we have
$$
\sum_{i=1}^k
\left|\F(E_i)-f(x_i)\G(E_i)\right|
<\varepsilon.
$$
Passing to the supremum we obtain
$$
\sum_{i=1}^k
\bar{p}^\varepsilon_{x_i,\tau r_i}(\F-f(x_i)\G)\leq\varepsilon.
$$
\end{proof}
\section{$\GR$ integral}\label{s:sedm}
It can happen that a function which is $\R$ integrable in sets $A_1$ and
$A_2$ is not $\R$ integrable in their union. Also, $\R$ integrability
is not closed with respect to $\BV$ convergence of sets.
To correct this deficiency, Pfeffer \cite{Pf} extended the definition
of the $\R$ integral.
Fortunately, the construction based on the closure with respect to
$\BV$ convergence of sets solves automatically the problem of
additivity. The result of this construction is called $\GR$ integral
(the generalized Riemann integral).
Using our Theorem \ref{thm:sonmn} we show that also this $\GR$ integral
is contained in our packing $\R^*$ integral.
\begin{notation}
Let $f$ be a function whose domain contains a locally $\BV$ set $E$ and let $\F$ be a charge.
Then we denote by $\tR(f,\F,E)$ the family of all bounded $\BV$ sets $A\subset E$ such that
$f\chi_A$ belongs to $\R(A)$ and the charge $\F\lfloor_A$ is the indefinite $\R$ integral of $f\chi_A$.
Further, let us denote
$\tpR(f,\F,E)$ the minimal system of bounded $BV$ sets containing $\tR(f,\F,E)$
and closed with respect to convergence in $\tbv$.
\end{notation}
\begin{definition}
Let $f$ be a function defined on a locally $\BV$ set $E$.
We say that a charge $\F$ is an
\emph{indefinite $\GR$ integral} of $f$ in $E$ if
$\tpR(f,\F,E)=\tbv(E)$,
where $\tbv(E)=\{A\in\tbv;A\subset E\}$ .
The family of all $\GR$ integrable functions in $E$ is denoted by $\GR(E)$.
\end{definition}
\begin{remark}
The indefinite $\GR$ integral is well defined, unique and linear. For further details see \cite[Sec. 6.3]{Pf}.
\end{remark}
The next theorem with proof can be found in \cite[Proposition 6.3.12]{Pf}.
\begin{theorem}\label{thm:RGR}
Let $E$ be a locally $\BV$ set. Then
\begin{enumerate}
\item\label{thm:RGRj} If $n=1$, then $\R(E)=\GR(E)$.
\item\label{thm:RGRd} If $n\geq 2$ and $\intt E\neq \emptyset$, then $\R(E)\subsetneq \GR(E)$.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{thm:sonmn}
Let $E$ be a bounded admissible $BV$ set. Then $\GR(E)\subset\P\R^*(E)$.
\end{theorem}
\begin{proof}
The proof follows from Theorems \ref{thm:rhr}, \ref{thm:pr} and \ref{thm:conmn}.
\end{proof}
\section{$\R^*$ integral}
The $\R^*$ integral was introduced by Mal\'{y} and Pfeffer in \cite{mpf}.
It is an alternative approach to overcome drawbacks of the $\R$ integral.
Moreover, in $\er^1$ this integral coincides with
the Henstock-Kurzweil integral.
\begin{definition}
Let $\varepsilon>0$.
We say that an $\varepsilon$-regular $\BV$ partition $\{(A_1,x_1),\ldots,\penalty 0
(A_k,x_k)\}$ is \emph{strongly $\varepsilon$-regular} if
$A_i$ is $\varepsilon$-isoperimetric and $x_i\in\cl_* A_i$ for $i=1,\ldots, k$.
\end{definition}
\begin{definition}\label{def:mp}
Let $A\subset \rn$ be a locally $\BV$ set.
We say that a charge $\F$ in $A$ is an \emph{indefinite $\R^*$ integral} of a function $f:\cl_* A\to\er$ in $A$
with respect to a charge $\G$ if
for given $\varepsilon>0$ we can find a gage $\delta:\cl_* A\to[0,\infty)$
so that
$$
\sum_{i=1}^k \big|
\F(A_i)-f(x_i)\G(A_i)
\big|<\varepsilon
$$
for each strongly $\varepsilon$-regular $\delta$-fine $\BV$ partition
$\{(A_1,x_1),\ldots,(A_k,x_k)\}$.
The family of all $\R^*$ integrable functions in $A$ is denoted by $\R^*(A,\G)$.
The family of all $\R^*$ integrable functions in $A$ with respect to Lebesgue measure is denoted just by $\R^*(A)$.
\end{definition}
\begin{remark}
It is easily seen that for an admissible $\BV$ set $E$ we have $\R(E)\subset \R^*(E)$.
\end{remark}
\begin{theorem}\label{thm:mp}
Let $A\subset\rn$ be a locally $\BV$ set.
Let a charge $\F$ be an indefinite $\R^*$ integral of a function $f:\cl_* A\to\er$ in $A$ with respect to a charge $\G$.
Then $\F$ is also an indefinite packing $\R^*$ integral of $f$ in $A$ with respect to $\G$.
\end{theorem}
\begin{proof}
Let us set $\tau:=1$. Then let us choose $\varepsilon>0$ and
find a gage $\delta$ as in Definition \ref{def:mp}.
Let us fix a $\delta$-fine packing
$\sys{B(x_i,r_i)}_{i=1}^k$, $x_i\in \cl_*A$.
We need to show that
$$
\sum_{i=1}^k
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-f(x_i)\G)<\varepsilon,
$$
where
$$
\aligned
\bar{q}_{x,r}^\varepsilon(\F)&=
\sup\{
|\F(E)|; E\subset\subset B(x,r), E\in \tbv, x\in\cl_*E, (E,x)
\mbox{ is }\varepsilon\text{-regular} \\
&\qquad \mbox{ and }\varepsilon\text{-isoperimetric}
\}.
\endaligned
$$
Now let us fix test sets $E_i$, $E_i\subset\subset B(x_i,r_i)$, $x_i\in\cl_*E_i$,
$E_i$ is $\BV$ and $(E,x)$ is $\varepsilon$-regular and $\varepsilon$-isoperimetric for $i=1,\ldots, k$.
Obviously, the system $\{ (E_i,x_i)\}$ is strongly $\varepsilon$-regular $\delta$-fine $\BV$ partition
and hence by Definition \ref{def:mp}
we obtain
$$
\sum_{i=1}^k
\left|\F(E_i)-f(x_i)\G(E_i)\right|
<\varepsilon.
$$
Passing to the supremum we obtain
$$
\sum_{i=1}^k
\bar{q}^\varepsilon_{x_i,\tau r_i}(\F-f(x_i)\G)\leq\varepsilon.
$$
\end{proof}
\begin{remark}\label{r:rhgr}
Let $E$ be an admissible locally
$BV$ set. Then $\GR(E)\subsetneq \R^*(E)$.
The inclusion follows from \cite[Corollary 3.18]{mpf} and
\cite[Theorem 3.20]{mpf}.
An example of function which is $\R^*$ integrable but not $\GR$ integrable can be found in
\cite[Example 6.9]{Pf-cl} and \cite[Proposition 10.8]{Pf-cl}.
\end{remark}
\section{Henstock-Kurzweil-Stieltjes integral}
In the next two sections we will investigate packing $\R$ and packing $\R^*$ integral on the real line.
For this purpose let us note that a charge $\F$ in $\er^1$ can be identified
with an ``ordinary'' function $F$ through the relation
$\F\left((u,v)\right)=F(v)-F(u)$.
Since for those integral $\F$ is supposed to be a charge, in the view of Example \ref{r:ident}, $F$ is continuous.
\begin{definition}
Let $[a,b]\subset\er^1$ be a compact interval.
A finite collection\linebreak $\left( [a_i,b_i], \xi_i\right)_{i=1}^k$ of tagged intervals is called a \emph{subpartition} of $[a,b]$
if intervals $[a_i,b_i]$ are nonoverlapping and $\xi_i\in[a_i,b_i]$ for every $i=1,\ldots,k$.
A function $\delta:[a,b]\to(0,\infty)$ is called a \emph{positive gage}.
We say that a subpartition is $\delta$-fine if $|b_i-a_i|<\delta(\xi_i)$.
\end{definition}
\begin{definition}
Let $f,G,F:[a,b]\to\er$ be functions.
We say that $F$ is the \emph{strong Henstock-Kurzweil-Stieltjes integral} of $f$ with respect to $G$ if
for every $\varepsilon>0$, there exists a positive gage $\delta:[a,b]\to(0,\infty)$, so that
for every $\delta$-fine subpartition $\left( [a_i,b_i], \xi_i\right)_{i=1}^k$ we have
$$
\sum_{i=1}^k \left|F(b_i)-F(a_i)-f(\xi_i)(G(b_{i})-G(a_{i}))\right|<\varepsilon.
$$
In the case $G$ is the identity function we say that $F$ is just the \emph{strong Henstock-Kurzweil integral} of $f$.
The families of all strongly Henstock-Kurzweil-Stieltjes integrable functions on $[a,b]$ with respect to $G$
and all strongly Henstock-Kurzweil integrable functions on $[a,b]$
are denoted by $HKS([a,b],G)$ and $HK([a,b])$, respectively.
\end{definition}
\begin{definition}
Let $f,G,F:\er\to\er$ be functions.
We say that $F$ is the \emph{indefinite Henstock-Kurzweil-Stieltjes integral} of $f$ with respect to $G$ if
$F$ is the strong Henstock-Kurzweil-Stieltjes integral of $f$ with respect to $G$ on every compact interval $[a,b]\subset\er$.
In the case $G$ is the identity function we say that $F$ is just the \emph{indefinite Henstock-Kurzweil integral} of $f$.
The families of all Henstock-Kurzweil-Stieltjes integrable functions with respect to $G$
and all Henstock-Kurzweil integrable functions
are denoted by $HKS(\er,G)$ and $HK(\er)$, respectively.
\end{definition}
\begin{comment}
\begin{remark}
\red
Although Henstock-Kurzweil-Stieltjes integral usually appears as definite integral,
indefinite integrals are in center of our interest
and then our definition slightly differs.
\endred
\end{remark}
\end{comment}
The proof of the following proposition can be found in \cite[Proposition 3.6]{mpf}.
\begin{proposition}\label{p:RhHK}
Let $f:\er\to\er$ be a function. Then $f$ is $\R^*$ integrable with respect to Lebesgue measure
if and only if $f$ is strongly Henstock-Kurzweil integrable on every compact interval $[a,b]\subset\er$.
\end{proposition}
Applying Theorem \ref{thm:RGR}, Remark \ref{r:rhgr} and Proposition \ref{p:RhHK} we obtain the following theorem.
\begin{theorem}\label{t:RHK}
$\R(\er)\varsubsetneq HK(\er)$.
\end{theorem}
\iffalse
\section{Henstock-Kurzweil-Stieltjes integral}
\begin{definition}
Let $[a,b]\subset\er^1$ be a compact interval.
A finite collection $\left( [u_i,v_i], \xi_i\right)_{i=1}^k$ of tagged intervals is called a \emph{subpartition of $[a,b]$}
if intervals $[u_i,v_i]$ are nonoverlapping and $\xi_i\in[u_i,v_i]$ for every $i=1,\ldots,k$.
A subpartition is called a \emph{complete partition of $[a,b]$} if $\bigcup_{i=1}^k [u_i,v_i]=[a,b]$.
A function $\delta:[a,b]\to(0,\infty)$ is called a \emph{positive gage}.
We say that a subpartition is $\delta$-fine, if $|v_i-u_i|<\delta(\xi_i)$.
\end{definition}
\begin{definition}[Henstock-Kurzweil-Stieltjes integral]
Let $f,g$ be functions defined on a compact interval $[a,b]$. We say that $f$ is \emph{Henstock-Kurzweil-Stieltjes integrable on $[a,b]$ with
respect to $g$}
if there is a number $I\in\er$ so that for every $\varepsilon>0$ exists a positive gage $\delta:[a,b]\to(0,\infty)$ such that
for every $\delta$-fine partition $([x_{i-1},x_i],\xi_i)_{i=1}^k$ of $[a,b]$ we have
$$
\left|
I-\sum_{i=1}^k f(\xi_i)(g(x_i)-g(x_{i-1}))\right|<\varepsilon.
$$
The family of all $HKS$ integrable functions on $[a,b]$ with respect to $g$ is denoted by $HKS([a,b],g)$.
Especially, the family of all $HKS$ integrable functions on $(a,b)$ with respect to the identity function is denoted by
$HK([a,b])$.
\end{definition}
\begin{lemma}[Henstock-Saks
Let $f,g:[a,b]\to\er$ be functions.
Let $f$ be Henstock-Kurzweil-Stieltjes integrable with respect to $g$.
Then $f$ is Henstock-Kurzweil-Stieltjes integrable with respect to $g$ on every closed subinterval of $[a,b]$.
Furthermore, for every $\varepsilon>0$, there exists a positive gage $\delta:[a,b]\to(0,\infty)$, so that
for every $\delta$-fine subpartition $\left( [u_i,v_i], \xi_i\right)_{i=1}^k$ we have
$$
\sum_{i=1}^k \left|F(v_i)-F(u_i)-f(\xi_i)(g(x_{i})-g(x_{i-1}))\right|<\varepsilon,
$$
where $F(x)$ is the Henstock-Kurzweil-Stieltjes integral of $f$ with respect to $g$ on $[a,x]$, $x\leq b$.
\end{lemma}
\begin{proof}
For the proof see \cite[Lemma 1.13]{SchwabikODE}.
\end{proof}
\begin{corollary}
Let $f,g,F:[a,b]\to\er$ be functions.
Then $F$ is the Henstock-Kurzweil-Stieltjes integral of $f$ with respect to $g$ if and only if
for every $\varepsilon>0$, there exists a positive gage $\delta:[a,b]\to(0,\infty)$, so that
for every $\delta$-fine subpartition $\left( [u_i,v_i], \xi_i\right)_{i=1}^k$ we have
$$
\sum_{i=1}^k \left|F(v_i)-F(u_i)-f(\xi_i)(g(x_{i})-g(x_{i-1}))\right|<\varepsilon.
$$
\end{corollary}
\fi
\section{$MC$ and $MC_\alpha$ integrals}
In this section we will introduce $MC$ and $MC_\alpha$ integrals.
The monotonically controlled Stieltjes ($MC$) integral was defined by Bendov\'{a} and Mal\'{y} in \cite{BM}.
The theory of the $MC_\alpha$ integral with respect to Lebesgue measure was further developed by
Ball and Preiss in \cite{dball}. Their ideas will be used in the proofs of
Propositions \ref{p:mccont} and \ref{p:albe}.
\begin{definition}\label{d:amc}
Let $\alpha>0$ be a real number and $f,F,G:\er\to\er$ be functions, let $G$ be continuous.
We say that $F$ is an \emph{indefinite $MC_\alpha$ integral} of $f$ with respect to $G$ if there exists a strictly increasing control function
$\varphi:\er\to\er$ such that for each $x\in \er$ we have
$$
\lim_{h\to 0}\frac{F(x+h)-F(x)-f(x)(G(x+h)-G(x))}{\varphi(x+\alpha h)-\varphi(x)}=0.
$$
The families of all $MC_\alpha$ integrable functions with respect to $G$
and all $MC_\alpha$ integrable functions with respect to identity function
are denoted by
$MC_\alpha(G)$ and $MC_\alpha$, respectively.
Especially, if $\alpha=1$, we say that $F$ is an \emph{indefinite $MC$ integral of $f$ with respect to $G$}.
We write $MC(G)=MC_1(G)$ and $MC=MC_1$.
\end{definition}
\begin{remark}\label{r:bfi}
In Definition \ref{d:amc} the control function $\varphi$ can be chosen to be bounded. (See \cite[Lemma 1]{BM}.)
\end{remark}
\begin{proposition}\label{p:mccont}
Let $\alpha>0$ and let $f,F,G:\er\to\er$ be functions, let $G$ be continuous.
If $F$ is an indefinite $MC_\alpha$ integral of $f$ with respect to $G$, then $F$ is continuous.
\end{proposition}
\begin{proof}
Let us fix $\varepsilon>0$ and $x\in \er$.
We need to find $\delta$ such that for every $|h|<\delta$ we have
$$
|F(x+h)-F(x)|<\varepsilon.
$$
Since $G$ is continuous at $x$, we can find $\delta_1$ such that
for every $|h|<\delta_1$ we have $|G(x+h)-G(x)|<\varepsilon$.
Further, since $F$ is the indefinite $MC_\alpha$ integral of $f$ with respect to $G$, there exists a strictly increasing control function
$\varphi:\er\to\er$ and a $\delta<\delta_1$ such that for every $|h|<\delta$ we have
$$
\left|
\frac{F(x+h)-F(x)-f(x)(G(x+h)-G(x))}{\varphi(x+\alpha h)-\varphi(x)}
\right|<\varepsilon.
$$
Applying Remark \ref{r:bfi} we can assume that there exists a constant $M$ such that $|\varphi(x)|<M$ for every $x\in \er$.
Hence
$$\aligned
&|F(x+h)-F(x)|\\
&\qquad
\leq
\left|\frac{F(x+h)-F(x)-f(x)(G(x+h)-G(x))}{\varphi(x+\alpha h)-\varphi(x)}(\varphi(x+\alpha
h)-\varphi(x))\right|\\
&\qquad\quad+\left|f(x)(G(x+h)-G(x))
\right|\\
&\qquad<\varepsilon (2M+f(x)).
\endaligned
$$
\end{proof}
\begin{proposition}\label{p:albe}
Let $0<\alpha<\beta$ be real numbers,
$f,F,G:\er\to\er$ be functions and let $G$ be continuous.
If $F$ is an indefinite $MC_\alpha$ integral of $f$ with respect to $G$, then
$F$ is also an indefinite $MC_\beta$ integral of $f$ with respect to $G$.
\end{proposition}
\begin{proof}
The proof follows from the fact that for $0<\alpha<\beta$ we have
$|\varphi(x+\alpha h)-\varphi(x)|\leq
|\varphi(x+\beta h)-\varphi(x)|$ for $h\in\er$.
\end{proof}
The two following theorems can be found in \cite[Theorem 3]{dball}.
\begin{theorem}\label{thm:pmca}
For every $\alpha\geq2$ there exists a function which is not $MC_\alpha$ integrable but
is $MC_\beta$ integrable for every $\beta>\alpha$.
\end{theorem}
\begin{theorem}\label{thm:mcmca}
Let $\alpha>2$. Then $MC$ is a proper subspace of $MC_\alpha$.
\end{theorem}
For the proof of the next theorem see
\cite[Theorem 3]{dball}.
\begin{theorem}\label{thm:mcmci}
Let $\alpha\in[1,2]$. Then $MC=MC_\alpha$.
\end{theorem}
\begin{theorem}\label{thm:mchks}
Let $G,F,f:\er\to\er$ be functions.
Suppose that $G$ is continuous.
Then $F$ is an indefinite $MC$ integral of $f$ with respect to $G$ if and only if
$F$ is an indefinite Henstock-Kurzweil-Stieltjes integral of $f$ with respect to $G$.
\end{theorem}
\begin{proof}
For the proof and further details see \cite[Theorem 3]{BM} and \cite[Theorem 17]{dball}.
\end{proof}
\begin{theorem}\label{thm:pmc}
Let $\alpha\geq 1$, $G,F,f:\er\to\er$ be functions.
Suppose that $G$ is continuous.
Let $F$ be an indefinite $MC_\alpha$ integral of $f$ with respect to $G$.
Further, let $\F$ and $\G$ be charges induced by $F$ and $G$ in the sense of Example \ref{r:ident}.
Then $\F$ is also an indefinite packing $\R$ integral of $f$ with respect to $\G$.
\end{theorem}
\begin{proof}
First, let us note that $F$ is continuous by Proposition \ref{p:mccont}. Hence it is legitimate to
use the term charges for the set functions $\F$ and $\G$ constructed as in Example \ref{r:ident}.
Let us set $\tau:=1/\alpha$.
Further, let us fix $\varepsilon>0$ and write $\varepsilon':=\varepsilon^2$.
Since $f$ is $MC_\alpha$ integrable, there exists a strictly increasing function $\varphi:\er\to\er$ with the following
property:
for each $x\in \er$ there exists $\delta(x)>0$ such that for every $|h|<\delta(x)$ we have
\begin{equation}\label{e:fi}
|F(x+h)-F(x)-f(x)(G(x+h)-G(x))|<\varepsilon'|\varphi(x+\alpha h)-\varphi(x)|.
\end{equation}
Moreover, by Remark \ref{r:bfi} we can suppose that there exists $M>0$ such that $|\varphi|\leq M$.
We need to show that for fixed $\delta$-fine packing
$\sys{B(x_i,r_i)}_{i=1}^k$, we have
$$
\sum_{i=1}^k
\bar{p}^{\varepsilon}_{x_i,\tau r_i}(\F-f(x_i)\G)<\varepsilon,
$$
where
$
\bar{p}_{x_i,\tau r_i}^{\varepsilon}(\F)=
\sup\left\{
|\F(E)|; E\subset\subset B(x_i,\tau r_i), E\in \tbv, (E,x_i) \mbox{ is $\varepsilon$-regular}
\right\}.
$
Let us fix $i\in\{1,\ldots, k\}$ and a test set
$E_i\in\tbv$ such that $E_i\subset\subset B(x_i,\tau r_i)$ and $(E_i,x_i)$ is $\varepsilon$-regular.
In other words, $E_i=\bigcup_{j=1}^{l_i} (a^i_j,b^i_j)$ is a finite union of disjoint nondegenerate intervals in
$B(x_i,\tau r_i)$
(up to a Lebesgue null set).
Moreover, since
$E_i$ is $\varepsilon$-regular
and $\H^0$ is the counting measure, we estimate
$$\frac{1}{\|E_i\|}\geq
\frac{|E_i|}{d(E_i\cup \{x_i\})\|E_i\|}>\varepsilon$$
and
$$
\frac{1}{\varepsilon}\geq \|E_i\|
=\H^0(\partial_*E_i)=2l_i.
$$
Let us set $m$ to be the greatest natural number such that $m\leq 1/(2\varepsilon)$. Then $l_i\leq m\leq 1/(2\varepsilon)$.
Further, since for each $a_j^i$ and $b_j^i$, $i=1,\ldots,k$ and $j=1,\ldots,l_i$, we have $|a_j^i-x_i|<\delta(x_i)$ and
$|b_j^i-x_i|<\delta(x_i)$, by \eqref{e:fi} and the fact that $\varphi$ is increasing we have the estimates
$$
\left|F\left(b_j^i\right)-\F(x_i)-f(x_i)(G(b_j^i)-G(x_i))\right|
<\varepsilon'\left|\varphi\left(x_i+{r_i}\right)-\varphi(x_i-r_i)\right|
$$
and
$$
\left|F(a_j^i)-\F(x_i)-f(x_i)(G(a_j^i)-G(x_i))\right|
<\varepsilon'\left|\varphi\left(x_i+{r_i}\right)-\varphi(x_i-r_i)\right|.
$$
Moreover, since the system $(B(x_i,r_i))_{i=1}^k$ is pairwise disjoint and $\varphi$ is strictly increasing and bounded, we have
\begin{equation}\label{e:fik}
\aligned
&\sum_{i=1}^k
\left| F(b^i_j)-F(a^i_j)-f(x_i)(G(b^i_j)-G(a^i_j))\right|\\
&\qquad \leq \sum_{i=1}^k
\left|F(b_j^i)-\F(x_i)-f(x_i)(G(b_j^i)-G(x_i))\right|\\
&\quad\qquad +\left|F(a_j^i)-\F(x_i)-f(x_i)(G(a_j^i)-G(x_i))\right|\\
&\qquad\leq \sum_{i=1}^k
2\varepsilon'\left|\varphi\left(x_i+r_i\right)-\varphi(x_i-r_i)\right|\\
&\qquad<2\varepsilon'(\varphi(x_k+r_k)-\varphi(x_1-r_1))\\
&\qquad<
4\varepsilon'M.
\endaligned
\end{equation}
Let us denote $L:=\max_i l_i$.
For $j\in\{1,\ldots,L\}$ let $I_j$ be the set of indices $i\in\{1,\ldots,k\}$ for which
$l_i\geq j$.
Then applying estimates in \eqref{e:fik} we obtain
$$
\aligned
\sum_{i=1}^k
\left|\F(E_i)-f(x_i)\G(E_i)\right|&\leq
\sum_{i=1}^k \sum_{j=1}^{l_i}
\left| F(b^i_j)-F(a^i_j)-f(x_i)(G(b^i_j)-G(a^i_j))\right|\\
&\leq \sum_{i=1}^k \sum_{j=1}^{l_i}
\left|F(b_j^i)-\F(x_i)-f(x_i)(G(b_j^i)-G(x_i))\right|\\
&\quad+
\left|F(a_j^i)-\F(x_i)-f(x_i)(G(a_j^i)-G(x_i))\right|\\
&\leq \sum_{j=1}^L \sum_{i\in I_j}
\left|F(b_j^i)-\F(x_i)-f(x_i)(G(b_j^i)-G(x_i))\right|\\
&\quad +
\left|F(a_j^i)-\F(x_i)-f(x_i)(G(a_j^i)-G(x_i))\right|\\
&<\sum_{j=1}^{L}4\varepsilon'M=4L\varepsilon^2M
\leq \frac{4\varepsilon^2M}{2\varepsilon}
=2M\varepsilon.
\endaligned
$$
Finally, passing to the supremum we obtain
$$
\sum_{i=1}^k
\bar{p}^\varepsilon_{x_i, r_i}(\F-f(x_i)\G)\leq 2M\varepsilon,
$$
which we needed.
\end{proof}
\section{Summary of relations
Let $A\subset \rn$ be an admissible locally $\BV$ set.
The relation between classes of integrable functions in $A$ is shown in the following diagram.
$$
\begin{tikzcd}
\I\R \arrow[draw=none]{r}[sloped,auto=false]{\subsetneq} \arrow[draw=none]{d}[sloped,auto=false]{=}& \G\R \arrow[draw=none]{r}[sloped,auto=false]{\subsetneq}& \R^*\arrow[draw=none]{d}[sloped,auto=false]{\subsetneq}\\
\R \arrow[draw=none]{r}[sloped,auto=false]{\subsetneq} & \P\R \arrow[draw=none]{r}[sloped,auto=false]{\subset}& \P\R^*\\
\end{tikzcd}
$$
The strictness of the inclusion $\I\R \subset \G\R$ holds for $n\ge 2$ and can be found in Theorem \ref{thm:RGR}(\ref{thm:RGRd}) and
Corollary \ref{c:rir}; the case $n=1$ is discussed below.
The fact that $\G\R \subsetneq \R^*$ is mentioned in Remark \ref{r:rhgr}.
Corollary \ref{c:rir} shows the equality of $\I\R$ and $\R$.
The relationship $\R \subsetneq\P\R$ is described in Theorem \ref{thm:pr};
Theorems \ref{t:RHK}, \ref{thm:pmc} and \ref{thm:mchks} show that this inclusion is strict.
The inclusion $\P\R \subset \P\R^*$ is proved in Theorem \ref{thm:rhr}.
Theorem \ref{thm:mp} proves the inclusion $\R^*\subsetneq \P\R^*$,
the fact, that this inclusions is proper follows from Theorems \ref{thm:pmc}, \ref{thm:mchks}, \ref{thm:mcmca} and Proposition \ref{p:RhHK}.
In the case $A=\er$, we can compare integrable functions in the following way.
$$
\begin{tikzcd}
\G\R \arrow[draw=none]{r}[sloped,auto=false]{=}&
\R \arrow[draw=none]{r}[sloped,auto=false]{\subsetneq}&
HK \arrow[draw=none]{r}[sloped,auto=false]{=}
\arrow[draw=none]{d}[sloped,auto=false]{=}&
MC\arrow[draw=none]{d}[sloped,auto=false]{=}
\arrow[draw=none]{r}[sloped,auto=false]{\subsetneq}&
MC_\beta \arrow[draw=none]{r}[sloped,auto=false]{\subsetneq}&
\P\R \arrow[draw=none]{r}[sloped,auto=false]{\subset} &
\P\R^*
\\
&&\R^* & MC_\alpha \\
\end{tikzcd}
$$
The equality $\GR=\R$ is described in Theorem \ref{thm:RGR}(\ref{thm:RGRj})
and the inclusion $\R\subsetneq HK$ in Theorem \ref{t:RHK}.
The fact that $HK$ integral coincides with $\R^*$ integral can be found in \ref{p:RhHK}.
Theorem \ref{thm:mchks} shows the equality $HK=MC$.
Theorem \ref{thm:mcmci} proves the equality $MC=MC_\alpha$ for $\alpha\in[1,2]$.
The inclusion $MC\subsetneq MC_\beta$ for $\beta>2$ is proved in Proposition \ref{p:albe},
the fact, that this inclusion is proper is shown in Theorem \ref{thm:mcmca}.
The relationship $MC_\beta\subsetneq\P\R$ (not only) for $\beta\geq2$ is proved in Theorem \ref{thm:pmc}
and \ref{thm:pmca}.
Finally, the inclusion $\P\R \subset \P\R^*$ is shown in Theorem \ref{thm:rhr}.
\section*{Acknowledgements}
The research was supported by the grants GA\,\v{C}R P201/15-08218S and P201/18-07996S.
The author would also like to express deep gratitude to
Jan Mal\'{y} for many valuable suggestions and helpful comments.
\bibliographystyle{abbrv}
\def\cprime{$'$} \def\cprime{$'$}
|
1903.04934
|
\section{Method}
A selected single crystalline sample of volborthite \cite{IshikawaActa} with the dimension of $2 \times0.4 \times0.2$ mm$^{3}$ is glued to a FBG strain sensor along the $b$ axis.
Magnetostriction is measured using the FBG based strain measurement system, with a resolution of $\Delta L/ L \sim1\times10^{-6}$ where the optical filter method is employed as a detection scheme \cite{IkedaFBGHR}.
High magnetic fields are generated using a non-destructive pulsed magnet in IMGSL, ISSP, UTokyo, Japan.
All results reported here are for the longitudinal magnetostriction along the $b$ axis.
The results of the longitudinal magnetostriction measurements of volborthite in the $b$ axis at 4.2 and 2.2 K are shown in Fig. \ref{result}(a).
The magnetostriction is negative and the data for the up sweep and down sweep of the pulsed magnetic fields coincide with each other without hysteresis within the resolution, indicating that the magnetic phase transitions are continuous.
The magnitude of the magnetostriction shows a qualitatively similar behavior as the magnetization curve reported in Ref. \cite{IshikawaPRL} as shown in Fig. \ref{result}(b).
In Fig. \ref{result}(c), the field derivatives of the magnetostriction, $d(\Delta L/L)/dB$, at 4.2 and 2.2 K are shown, where the temperature dependence is apparent.
Compared with the data at 4.2 K, the magnitude of the peak of $d(\Delta L/L)/dB$ becomes larger and the peak position shifts slightly to the lower field at 2.2 K.
This suggests that the plateau phase is more stable at 2.2 K, resulting in the lower entrance field to the plateau phase.
Similar enlargement of the peak is also observed in $dM/dB$ as shown in Fig. \ref{result}(d).
It should be noted that the peak of $dM/dB$ at 1.4 K has some shoulder structure, which is not seen in that of $d(\Delta L/L)/dB$ at 2.2 K.
This may be due to the possible spin nematic phase that appears below 2.2 K \cite{IshikawaPRL, YoshidaPRB}.
In Figs. \ref{result}(e) and \ref{result}(f), the data of $\Delta L/L$ is compared with the magnetization curves with the power of $p$.
It is apparently seen that the trend of $\Delta L/L$ data agrees best with $\Delta L/L \propto M^{p}$ with $p=1.3$, whereas $M^{p}$ curves with $p=1.0$ and $2.0$ clearly cannot be fitted to the data of $\Delta L/L$.
We discuss the origins of the negative magnetostriction and the relation of $\Delta L/L\propto M^{1.3}$ observed in volborthite in terms of the exchange striction model.
As a possible origin of the negative magnetostriction, we propose a pantograph-like change of the Cu-O-Cu chain in the $b$ axis.
As for the dependence $\Delta L/L\propto M^{1.3}$, we argue that the local spin correlator is responsible.
Magnetoelastic couplings arising from on-site spin-orbit couplings, crystal field effects and Jahn-Teller effects are considered to be insignificant in the present system.
We note that, based on the coupled-trimer model \cite{JansonPRL}, it is reasonable to focus on the second strongest $J_{1}$ bond along the $b$ axis and neglect the magnetostriction in the strongest $J$ bond of in Fig. \ref{lattice}(b) up to the 1/3 magnetization plateau.
This is because the total spin of each trimer is fixed to the spin doublet $S_{\rm{tri}}=1/2$ up to the 1/3 magnetization plateau.
In the exchange striction model, magnetostriction is dependent on the spin correlation $\braket{\bm{S}_{i}\cdot \bm{S}_{j}}$ as discussed later, which is fixed at -1/2 in the above doublet sector.
Thus, one does not need to worry about the intra-trimer magnetostriction up to the 1/3 magnetization plateau.
\begin{figure}[h]
\includegraphics[scale=0.4]{fig04.pdf}
\caption{Schematic drawing of a pantograph-like motion of the Cu-O-Cu chain in the $b$ direction of volborthite.
\label{model}}
\end{figure}
We consider the pantograph-like lattice change in the Cu chains in the $b$ axis to discuss the magnetoelastic coupling on the $J_1$ bonds.
As shown in Fig. \ref{model}, the 3$d_{x^{2}-y^{2}}$ orbital of each Cu2 site is connected to the 3$d_{x^{2}-y^{2}}$ orbital of the adjacent Cu2 site through two paths of Cu-O-Cu bonds \cite{YoshidaNC}.
On the left in Fig. \ref{model}, the lattice model at $M=0$ is shown, which is drawn from the lattice parameter at 55 K \cite{IshikawaPRL}, where the Cu-O-Cu bond angles are 93.4$^{\circ}$ and 98.9$^{\circ}$.
On the right, the lattice at $M>0$ is shown, where the Cu-O-Cu bond angles are reduced from the original values, approaching 90$^{\circ}$.
In Fig. \ref{model}, the way the crystal lattice changes mimics the motion of a pantograph, which was originally discussed for Cu dimers in the magnetostriction of SrCu(BO$_{2}$)$_{3}$ \cite{NarumiJPSJ, JaimePNAS, RadtkePNAS}.
The proposed model of the pantograph-like lattice modification is tested with density functional theory calculations with the on-site Coulomb term (DFT$+U$).
To calculate the evolution of the exchange integrals $J_{1}$, $J$, $J'$, and $J_{2}$ in Fig. \ref{lattice}(b), we vary the lattice parameter $b$ and calculate the DFT+$U$ total energies using the full-potential code FPLO \cite{Koepernik}.
For further details of the computational procedure, we refer the reader to Ref. \cite{JansonPRL}.
Note that the extremely low energy scale of the experimentally measured magnetostriction is beyond the reach of DFT total energy calculations.
To overcome this difficulty, we perform calculations for a largely enhanced lattice contraction.
The main outcome of the DFT+$U$ calculations is the enhancement of the ferromagnetic exchange constant $J_{1}$ upon a striction along the $b$ axis as shown in Fig. \ref{DFT}.
The lattice constant $b$ is the only parameter varied in the calculations.
The changes in the exchange constants $\Delta J$ and their ratios $\Delta J/J_{\Delta L=0}$ to the original values for different bonds are shown in Figs. \ref{DFT}(a) and \ref{DFT}(b).
The values of exchange constants at $\Delta b=0$ are, $J_{1}=-84.6$ $(-82.7)$ K, $J=156.3$ $(167.4)$ K, $J'=-30.0$ $(-24.0)$ K, $J_{2}=26.4$ $(25.1)$ K \cite{JansonPRL}, where the two values for each constant are for the two structurally inequivalent layers.
The initial value of $b$ is 5.8415 \AA \ \cite{YoshidaNC}.
The DFT$+U$ calculations qualitatively support our model of the pantograph-like lattice modification.
As can be seen in Fig. \ref{DFT}(b), both the ferromagnetic (blue-colored symbols) and antiferromagnetic (red-colored symbols) exchange constants are enhanced with decreasing $b$, which results from the enhanced overlap of the relevant orbitals responsible for the exchange bonds.
As seen in the absolute values [Fig. \ref{DFT}(a)] and the ratios of changes [Fig. \ref{DFT}(b)], enhancement of the exchange constant with decreasing $b$ is the most prominent for $J_{1}$ [Fig. \ref{lattice}(b)].
On the other hand, the changes in $J'$ and $J_2$ are relatively small, and the change in $J$ is expected to be irrelevant to the magnetostriction as discussed above.
These results support the idea that the change in $b$ originates primarily from the change in $J_{1}$ within the exchange striction model.
\begin{figure}
\includegraphics[scale=0.52]{fig05.pdf}
\caption{(a) Changes and (b) relative changes in the exchange constants as a function of the decreasing lattice constant $b$ obtained with the DFT+$U$ calculations.
Blue- and red-colored symbols correspond to ferromagnetic (negative) and antiferromagnetic (positive) exchange interactions, respectively.
Two sets of data for each exchange constant are for the two structurally inequivalent layers.
\label{DFT}}
\end{figure}
We, here, introduce a Hamiltonian based on the spin model in Fig. \ref{lattice}(b) as $\mathcal{H}_{\rm{es}}=J\sum_{i,j}\bm{S}_{i}\cdot\bm{S}_{j}+(J_{1}-p\epsilon)\sum_{i,j}\bm{S}_{i}\cdot\bm{S}_{j} + k\epsilon^{2}/2+J_{2}\sum_{i,j}\bm{S}_{i}\cdot\bm{S}_{j}+J'\sum_{i,j}\bm{S}_{i}\cdot\bm{S}_{j}+h_{z}\sum_{i}S^{z}_{i}$, where each summation is taken to satisfy the configuration of the exchange bonds in Fig. \ref{lattice}(b).
We have introduced the elastic energy term with $\epsilon=\Delta L/L$ in the $b$ axis and a strain dependence to the $J_{1}$ bond.
This is a reasonable simplification considering the result of the DFT+$U$ calculation in Fig. \ref{DFT}(a).
By taking $dE/d\epsilon=0$ with $E=\braket{\mathcal{H}_{\rm{es}}}$,
one obtains a relation of $\epsilon=(p/k')\braket{\bm{S}_{i}\cdot\bm{S}_{j}}$ with $k'=k/N$ where $N$ is the number of Cu2 sites.
The result of the DFT$+U$ calculations is quantitatively analyzed, showing a reasonable agreement with the experimental result in an order of magnitude.
The total energy increase at $\Delta L/L (=\epsilon) = - 1.7\times10^{-3}$ $(- 3.4\times10^{-3})$ is $\Delta E=240$ (539) K per unit cell in the DFT$+U$ calculation.
This sizable increase is dominated by the elastic energy, because the changes in the magnetic exchange energy are of the order of several K as seen in Fig. \ref{DFT}(a).
Considering the increase in the elastic energy per bond as $\Delta E/8=c\epsilon+k'\epsilon^{2}/2$, we obtain the value of the elastic constant $k'$ to be $2.46\times10^{6}$ K per bond, where there are 8 bonds for $J_{1}$ in a unit cell.
The obtained value of $k'$ corresponds to a Young's modulus $\lambda$ of 303 GPa with the relation $\lambda=zk'/V_{\rm{unit}}$ where $z=8$ is the number of $J_{1}$ bonds in a unit cell and $V_{\rm{unit}}=892.125$ \AA$^{3}$ \cite{IshikawaPRL}.
We note that in the DFT+$U$ result, the minimum point of the total energy is shifted slightly from $\epsilon=0$ to $\epsilon=-c/k'=1.25\times 10^{-2}$, which is irrelevant to the magnitude of the magnetostriction.
The change in the exchange constant $J_{1}$ at $\epsilon = - 3.4\times10^{-3}$ is -5.17 K in the DFT$+U$ calculation.
Considering the linear relation $\Delta J_{1}=-p\epsilon$ as seen in Fig. \ref{lanchoz}, we obtain the value of $p$ to be -1520 K.
Using the obtained values of $k'$ and $p$ and the relation $\epsilon=(p/k')\braket{\bm{S}_{i}\cdot\bm{S}_{j}}$, one obtains a value of $\epsilon_{\rm{DFT}} =- 6.9\times10^{-5}$, where we assumed $\braket{{\bm{S}}_{i}\cdot{\bm{S}}_{j}}=1/9$.
The value of 1/9 for the spin correlation is based on the product of the $S_{\rm{tri}}^z=1/2$ states at the 1/3 magnetization plateau in the spin-1/2 system \cite{JansonPRL}.
The obtained value of $\epsilon_{\rm{DFT}}$ is of the same order of magnitude as
$\epsilon_{\rm{EXP}} =- 1.7\times10^{-5}$ at the plateau, indicating that the pantograph-like lattice modification is a plausible mechanism.
We finally discuss the observed dependence of $\Delta L/L\propto M^{1.3}$.
First, we compare the present system with other spin systems.
Then, we consider the behavior of the local spin correlations in the present system.
Relation of $\Delta L/L\propto M^{1.0}$ is observed in the dimer-spin systems, SrCu$_{2}$(BO$_{3}$)$_{2}$\cite{NarumiJPSJ, JaimePNAS, RadtkePNAS} and KCuCl$_{3}$ \cite{sawai}, where the pantograph-like lattice change is discussed for Cu dimers.
In these systems, the dominant interaction is the intra-dimer antiferromagnetic coupling, which results in the formation of the singlet ground state below a spin gap \cite{Kageyama}.
Magnetostriction is proportional to the effective number of spin dimers in the excited $S_{\rm{dim}}=1$ state that are transformed from the low-lying $S_{\rm{dim}}=0$ state under magnetic fields, which also is proportional to the magnetization.
In other cases, a dependence in the form of an even function of the magnetization, $\Delta L/L=c_{2}M^{2}+c_{4}M^{4} +\dots$ with $c_{2}$, $c_{4}$, $\dots$ being constants, is usually expected for a variety of magnetic systems based on the symmetry considerations on the magnetic point groups \cite{JaimeNC, Romanov}.
\begin{figure}[t]
\includegraphics[scale=0.43]{fig06.pdf}
\caption{Magnetization $m=\braket{T_{i}^{z}}$, $m^{2}$ and the nearest-neighbor spin correlation $\braket{{\bm{T}}_{i}\cdot {\bm{T}}_{j}}$ on the ${\cal J}_1$ bond calculated with the exact diagonalization of the effective model in Fig.\ref{lattice}(c) as a function of the external magnetic field.
The saturation $m=1/2$ of pseudospins corresponds to the 1/3 magnetization plateau in the original model in Fig.\ref{lattice}(b).
The number of pseudospins is (a) $N=28$ and (b) $30$.
We set ${\cal J}_1=-34.9$ K and ${\cal J}_2=36.5$ K, and also include small further-neighbor couplings \cite{JansonPRL}.
Each curve is normalized so that it becomes unity at the saturation.
\label{lanchoz}}
\end{figure}
We argue that the local spin correlator $\braket{{\bm{S}}_{i} \cdot {\bm{S}}_{j}}$ on the $J_1$ bond should be responsible as an origin of the present observation of $\Delta L/L\propto M^{1.3}$,
To calculate it, we resort to the effective model shown in Fig. \ref{lattice}(c) where the low-lying spin-1/2 sector of each trimer is described by the pseudospin-1/2 operator $\bm{T}_{i}$.
Note that $\langle \bm{S}_i\cdot\bm{S}_j\rangle$ on the $J_1$ bond is equal to $(4/9)\langle \bm{T}_i\cdot\bm{T}_j\rangle$ on the ${\cal J}_1$ bond in the effective model.
Calculated results of $\braket{\bm{T}_{i} \cdot \bm{T}_{j}}$ and $m=\braket{T^{z}_{i}}$ are compared in Figs. \ref{lanchoz}(a) and \ref{lanchoz}(b), which are obtained with exact diagonalization (ED) for finite clusters.
Notice that the curves of the local spin correlation tend to be located in between those of $m^{1.0}$ and $m^{2.0}$ both for $N=28$ and 30 with some exception at low fields where finite-size effects are still found to be large.
This result is consistent with the experimental observation of $\Delta L/L\propto M^{1.3}$ and the relation $\Delta L/L=(p/k')\braket{\bm{S}_{i}\cdot\bm{S}_{j}}$ derived from the exchange striction mechanism.
The observed deviation from the quadratic behavior $\Delta L/L\propto M^2$ can also be interpreted in terms of a spin density wave (SDW) order, whose indications have been observed over a wide range of magnetic field up to 22 T below 3 K in NMR experiment \cite{IshikawaPRL, YoshidaPRB}.
Such an order is also predicted to appear in the effective model in Fig. \ref{lattice}(b) with ${\cal J}_2>0$ (irrespective of the sign of ${\cal J}_1$) \cite{Starykh}.
This order leads to the non-zero covariance $\Delta=\braket{\bm{T}_{i}\cdot\bm{T}_{j}}-m^{2}=\braket{(\bm{T}_{i}-m\hat{z})\cdot(\bm{T}_{j}-m\hat{z})}\neq 0$ as we explain in the following.
In the effective model, a SDW can be described as $\braket{{T}^{z}_{i}}=m+A\cos(qx/b)$ \cite{Starykh}, where $A$ is the amplitude of the SDW order, $q$ is the wave number with $q=\pi - 2\pi m$, and $x$ is the position of the site $i$ in the $b$ axis of Fig. \ref{lattice}(b).
Note that, on the nearest-neighbor site $j$, $\braket{{T}^{z}_{j}}=m-{\rm sgn}({\cal J}_1)A\cos(qx/b+q/2)$.
Assuming $\braket{(T^{z}_{i}-m)\cdot(T^{z}_{j}-m)}\simeq \braket{T^{z}_{i}-m}\braket{T^{z}_{j}-m}$, one obtains $\Delta \simeq -{\rm sgn}({\cal J}_1)(A^2/2)\cos(q/2)$ after taking the spatial average and neglecting the terms of $\braket{T^{x}_{i}T^{x}_{j}}+\braket{T^{y}_{i}T^{y}_{j}}$.
The experimental data in Figs. \ref{result}(e) and \ref{result}(f) indicate $\Delta>0$, which corresponds to ferromagnetic ${\cal J}_1<0$.
Even above $T_{\rm{SDW}}$, $\Delta$ is still expected to be non-zero because the short-range correlation develops prior to the long-range one
The present discussion explains the deviation from $\Delta L/L\propto M^2$ both for 4.2 and 2.2 K.
Another possible microscopic origin of $\Delta \neq 0$ is the lateral spin correlation, $\braket{{T}^{x}_{i}{T}^{x}_{j}}+\braket{{T}^{y}_{i}{T}^{y}_{j}}$, which is neglected in the above discussion.
In the magnon Bose-Einstein condensates (BEC), the lateral spin moment is fixed globally, resulting in the infinite correlation length in the lateral spin correlation which appears in the magnetostriction \cite{zapf2008}.
Even without a long-range order, the pair correlation function is expected to be non-zero at short separations \cite{Hikihara}.
The spin nematic phase whose indications have been observed in volborthite below 2 K \cite{IshikawaPRL, YoshidaPRB}, is a BEC of bi-magnons.
Though its order parameter, $\braket{T_{i}^{+}T_{j}^{+}}$ \cite{Shannon}, differs from that of the single-magnon BEC, $\braket{T_{i}^{+}}$ \cite{ZapfRMP}, the magnetostriction may still show some anomaly at phase transitions as it is related to the energy on the $J_1$ bonds.
In summary, the single crystalline volborthite is found to show a negative longitudinal magnetostriction in the $b$ axis up to 45 T with a relation of $\Delta L/L \propto M^{1.3}$ by means of the FBG-based magnetostriction measurement.
The results are discussed in terms of the exchange striction model.
It is argued that the negative magnetostriction arises from the pantograph-like lattice change in the Cu-O-Cu chain in the $b$ axis, strengthening the ferromagnetic exchange coupling in $J_{1}$ bond, which is supported by the DFT+$U$ calculations.
The relation of $\Delta L/L \propto M^{1.3}$ is discussed in terms of the local spin correlator, which is reproduced in the ED of the effective model.
The scope of the future studies includes a possible observation of the signature of the spin nematic phase below 2.0 K and a search for a possible structural transition at higher fields using the high-speed 100 MHz magnetostriction monitor \cite{IkedaFBGHS1, IkedaFBGHS2}.
This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) Grant No. 16K17738, Grant-in-Aid for Scientific Research (B) Grant No. 16H04009 and the internal research grant from ISSP, UTokyo.
OJ was supported by the Austrian Science Fund (FWF) through the Lise Meitner programme, project no. M2050. DFT calculations have been done on the Vienna Scientific Cluster (VSC).
|
1601.03904
|
\section{Introduction}
The dark energy and dark matter problems are the most important unsolved issues in cosmology.
Although many models of dark energy and dark matter have been proposed, none of them are conclusive at the present time.
Phenomenologically, the cold dark matter with a cosmological constant ($\Lambda$CDM model) is currently the most successful cosmological model.
The most promising candidate of the cold dark matter (CDM) is supersymmetric particles, the so-called neutralino.
While CDM works quite well especially on large scales, it is known that there exists a problem on small scales.
In fact, this model predicts a cusp of dark matter halo profile and overabundance of dwarf galaxies, which are not consistent with observations.
Moreover, the LHC has not reported any signature of supersymmetry.
Given this situation, it is worth seeking another possibility, namely axion dark matter.
The axion, the pseudo-Nambu-Goldstone boson, is originally introduced by Peccei and Quinn to resolve the strong CP problem of QCD~\cite{770620}.
Nowadays, it is known that the string theory also predicts such scalar fields with a wide range of mass scales~\cite{060626}.
Note that we use the term ``axion" in more general meaning, e.g., axionlike particles and other ultralight scalar particles.
The axion with the mass around $10^{-23}~\text{eV}$ behaves as nonrelativistic matter on cosmological scales, and hence it can be regarded as a candidate of dark matter.
Furthermore, it is known that such an ultralight axion can resolve the cusp problem on subgalactic scales because of its wave nature~\cite{000807, 14:Schive}.
A peculiarity of the axion is the oscillating pressure in time with frequency at the twice of the axion mass, $2m$.
Therefore, in order to identify the axion dark matter, we should detect the effect of the oscillating pressure of the axion.
The period of the oscillation corresponds to about one year, and this time scale is much shorter than the cosmological time scale, i.e. $H_{0}^{-1} \sim 10^{10}~\text{years}$.
Hence, after averaging the oscillating pressure over the cosmological time scale, the axion behaves as pressureless dust on cosmological scales.
Thus, it might be difficult to distinguish the axion from other dark matter candidates by cosmological scale observations.
For this reason, we should pay attention to smaller scales to prove the existence of the axion dark matter.
From this point of view, it is pointed out by Khmelnitsky and Rubakov that the effect of oscillating pressure of the axion can be detected by pulsar timing array experiments~\cite{140212}.
The oscillating pressure induces the oscillation of the gravitational potential with frequency in the nano-Hz range.
This effect can be observed as a shift of the arrival time of the signal from the pulsar.
In the previous paper, the axion oscillation was studied in Einstein's theory.
However, since the main energy component of the universe is the dark energy, it might be necessary to consider this issue in the context of modified gravity.
The reason is as follows:
The simplest candidate of dark energy, i.e., the cosmological constant, has several problems, e.g. the fine-tuning problem and coincidence problem.
One possibility to resolve these issues is to consider unknown matter such as the quintessence.
However, there is no natural candidate of quintessence in particle physics.
Therefore, it is natural to assume that theory of gravity is different from Einstein's theory on cosmological scales.
In this paper, as a first step to this direction, we focus on the $f(R)$ theory, which is the simplest modified gravity.
We discuss the detectability of the oscillation of the gravitational potential induced by the time-oscillating pressure of the axion in this context.
This paper is organized as follows:
In Sec. II, we review the results obtained by Khmelnitsky and Rubakov in Einstein's theory.
In Sec. III, we formulate the procedure to determine the amplitude of the gravitational potential in the framework of $f(R)$ theory and discuss two specific models: $R^{2}$ model which can be solved exactly, and the Hu-Sawicki model which is known as a viable cosmological model.
The final section is devoted to conclusion.
\section{Pulsar Timing and Ultralight Axions in Einstein's Theory}
In this section, we review the results obtained by Khmelnitsky and Rubakov in Einstein's theory~\cite{140212}.
We consider the situation that the dark matter halo is composed out of a free ultralight axion field.
The trace of the Einstein equation gives
\begin{equation}
R = -T \ ,
\label{eq1}
\end{equation}
where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor of the axion field.
We will use this equation to determine the gravitational potentials with given $T$.
Now let us consider both sides of this equation in turn.
On the scale of the dark matter halo, the expansion of the universe is completely negligible and the gravitational potentials can still be treated as perturbation.
Thus, we use the Newtonian gauge for the metric:
\begin{equation}
g_{\mu\nu} = \left( \begin{array}{cc}
-1 - 2\Psi & 0 \\ 0 & (1 - 2\Phi)\delta_{ij}
\end{array} \right).
\label{eq2}
\end{equation}
Note that this convention is different from that in \cite{140212}: it is $\Phi$ that affects the signal from the pulsar in this paper.
The Ricci scalar $R$ can be calculated from the metric in the usual manner: at the first order of potentials, it is given by
\begin{equation}
R = -6\ddot{\Phi} + 2\nabla^{2}(2\Phi - \Psi) \ ,
\label{eq3}
\end{equation}
where a dot denotes the derivative with respect to time and $\nabla$ represents the spatial gradient.
This gives the left-hand side of Eq.~(\ref{eq1}).
Next we consider the right-hand side of Eq.~(\ref{eq1}).
Since the occupation number of the axion in the dark matter halo is huge, we can treat it as a classical scalar field.
The axion field satisfies the Klein-Gordon equation in the flat space-time at the leading order, and the solution is given by the superposition of waves of different frequencies.
Since the typical scale of the dark matter halo, $(10~\text{kpc})^{-1} \sim 10^{5}H_{0}$, is much smaller than the mass of the axion, $m \sim 10^{-23}~\text{eV} \sim 10^{10}H_{0}$, we can assume that the axion field oscillates monochromatically with frequency of its mass.
Under these assumptions, we can write the energy density $\rho$ and pressure $p$ of the axion in the following form:
\begin{equation}
\rho \simeq \rho_{\text{DM}}, \quad p \simeq -\rho_{\text{DM}}\cos(2mt) \ ,
\label{eq4}
\end{equation}
where $\rho_{\text{DM}}$ is a constant.
The typical energy density of the dark matter halo is about $0.3~\text{GeV} / \text{cm}^{3}$.
The negative sign of the pressure is just a convention of choosing a phase.
Therefore, the trace of the energy-momentum tensor of the axion can be written as
\begin{equation}
T = -\rho + 3p \simeq -\rho_{\text{DM}}[1 + 3\cos(2mt)] \ .
\label{eq5}
\end{equation}
From the above results, we can rewrite Eq.~(\ref{eq1}) as follows:
\begin{equation}
-6\ddot{\Phi} + 2\nabla^{2}(2\Phi - \Psi) = \rho_{\text{DM}}[1 + 3\cos(2mt)] \ .
\label{eq6}
\end{equation}
Now let us separate the gravitational potential $\Phi~(\Psi)$ into the time-independent part $\Phi_{0}~(\Psi_{0})$ and the time-dependent part $\delta\Phi~(\delta\Psi)$.
To this aim, we should recall the Poisson equation derived from the time-time component of the Einstein equation
\begin{equation}
2\nabla^{2} \Psi_0 = \rho_{\text{DM}} \ .
\label{eq7}
\end{equation}
We also have the equation $\Psi_0 = \Phi_0$ from the traceless part of the space-space component of the Einstein equation.
Thus, we obtain the equation determining the time dependence of the gravitational potential $\delta\Phi$,
\begin{equation}
-6\delta\ddot{\Phi} = 3\rho_{\text{DM}}\cos(2mt) \ .
\label{eq8}
\end{equation}
The above equation can be easily solved as
\begin{equation}
\delta\Phi = \frac{\pi G\rho_{\text{DM}}}{m^{2}}\cos(2mt) \ ,
\label{eq9}
\end{equation}
where we wrote $8\pi G$ explicitly.
Note that $\delta\Phi \ll \Phi_0$ because $k^2 \ll m^2$ in the present situation~\cite{140212}.
They calculated the timing residuals of the signal from the pulsar and showed that the axion dark matter has the same effect on the pulsar timing measurements as gravitational wave background with characteristic strain
\begin{align}
h_{\text{c}} &= 2\sqrt{3}|\delta\Phi| \nonumber\\
&= 2 \times 10^{-15} \left( \frac{\rho_{\text{DM}}}{0.3~\text{GeV} / \text{cm}^{3}} \right) \left( \frac{10^{-23}~\text{eV}}{m} \right)^{2},
\label{eq10}
\end{align}
at frequency
\begin{equation}
f \equiv \frac{\omega}{2\pi} = 5 \times 10^{-9}~\text{Hz} \left( \frac{m}{10^{-23}~\text{eV}} \right).
\label{eq11}
\end{equation}
This signature is detectable in the planned SKA pulsar timing array experiments.
\section{Axions in $f(R)$ Theory}
In the previous section, we explained how the axion dark matter produces the oscillating gravitational potential in Einstein's theory and the oscillation can be detected through the observation of pulsar timing residuals.
The aim in this paper is to extend the analysis to $f(R)$ theory.
In this section, we will show how to obtain the gravitational potential from axion oscillations in $f(R)$ theory and discuss two specific models.
The action of $f(R)$ theory is given by
\begin{equation}
S = \frac{1}{2} \int d^{4}x\sqrt{-g}[R + f(R)] + S_{\text{m}} \ ,
\label{eq12}
\end{equation}
where $f(R)$ is a function of the Ricci scalar $R$, and $S_{\text{m}}$ is the action for matter fields.
Hereafter, we consider the axion field as the matter.
We assume $f(R) \ll R$ and $f_{R} \equiv f'(R) \ll 1$ so that the deviation from Einstein's theory is small.
The variation of the action with respect to the metric gives the field equation:
\begin{equation}
G_{\mu\nu} - \frac{1}{2}g_{\mu\nu}f + (R_{\mu\nu} + g_{\mu\nu}\Box - \nabla_{\mu}\nabla_{\nu})f_{R} = T_{\mu\nu} \ ,
\label{eq13}
\end{equation}
where $G_{\mu\nu} \equiv R_{\mu\nu} - (1 / 2)g_{\mu\nu}R$ is the Einstein tensor.
The trace of this equation gives
\begin{equation}
-R - 2f + (R + 3\Box)f_{R} = T \ .
\label{eq14}
\end{equation}
We assume that the spatial derivative of $f_{R}$ is much smaller than the time derivative of it, i.e. $\Box f_{R} \simeq -\ddot{f}_{R}$.
Then, we obtain
\begin{equation}
3\ddot{f}_{R} + R = -T \ ,
\label{eq15}
\end{equation}
or equivalently,
\begin{equation}
3f''(R)\ddot{R} + 3f'''(R)\dot{R}^{2} + R = -T \ ,
\label{eq16}
\end{equation}
where we used the approximations $f \ll R$ and $f_{R}\ll 1$.
Since the axion field minimally couples to gravity, we can use the same form for the trace of the energy-momentum tensor of the axion, $T$, as the previous one (\ref{eq5}).
We will use Eq.~(\ref{eq16}) to determine the time-dependent part of the Ricci scalar.
Now, let us consider two specific models.
First, we discuss the $f(R) \propto R^{2}$ model, which can be solved exactly.
Second, we consider the more realistic model known as the Hu-Sawicki model~\cite{070910}.
While it is known that this model can pass both cosmological and solar system tests, we will see that there is a tension between the Hu-Sawicki model and the axion dark matter for some parameters.
\subsection{$f(R) = R^{2} / 6M^{2}$ model}
Let us consider a simple model given by
\begin{equation}
f(R) = \frac{R^{2}}{6M^{2}} \ ,
\label{eq17}
\end{equation}
where $M$ is a constant mass scale.
When we discuss the model in terms of the scalar-tensor theory, $M$ is indeed the mass of the scalar field.
This type of model was introduced by Starobinsky to explain the inflationary universe~\cite{800524}.
Now, however, we use this model in the context of the dark energy.
Now, the field equation (\ref{eq16}) becomes
\begin{equation}
\frac{1}{M^{2}}\ddot{R} + R = \rho_{\text{DM}}[1 + 3\cos(2mt)] \ ,
\label{eq18}
\end{equation}
and the solution is given by
\begin{equation}
R = \rho_{\text{DM}} + \frac{3\rho_{\text{DM}}}{1 - (2m / M)^{2}}\cos(2mt) \ .
\label{eq19}
\end{equation}
Here, we ignored the homogeneous solutions which oscillate freely with frequency $M$, and we will discuss this point at the end of this subsection.
Following the same procedure done in the case of Einstein's theory, we obtain the time-dependent part of the gravitational potential as follows:
\begin{equation}
\delta\Phi = \frac{1}{1 - (2m / M)^{2}}\frac{\pi G\rho_{\text{DM}}}{m^{2}}\cos(2mt) \ .
\label{eq20}
\end{equation}
Therefore, in the $R^{2}$ model, we obtained the amplitude of the gravitational potential relative to that in Einstein's theory:
\begin{equation}
\frac{\delta\Phi}{\delta\Phi_{\text{E}}} = \frac{1}{1 - (2m / M)^{2}} \ ,
\label{eq21}
\end{equation}
where $\delta\Phi_{\text{E}}$ is the amplitude of the gravitational potential predicted in Einstein's theory.
This result is illustrated in Fig.~\ref{Fig1}.
In the large $M$ limit, $M \gg 2m$, the prediction in $f(R)$ theory is the same as in Einstein's theory.
This can be understood from the form of $f(R) = R^{2} / 6M^{2}$.
In fact, when the mass of the scalar field $M$ becomes large, $f(R)$ can be neglected compared to the Ricci scalar $R$.
Thus, Einstein's theory is reproduced in this limit.
In the opposite limit, $M \ll 2m$, the amplitude of the gravitational potential goes to zero.
Hence, in this case, it would be difficult to detect the oscillation of the gravitational potential.
When the mass scale $M$ gets close to the frequency of the pressure, $2m$, resonance would occur and the amplitude of the gravitational potential would be dramatically amplified.
Of course, the amplitude cannot reach to infinity: the approximation becomes worse when the oscillating part of $f(R)$ cannot be ignored compared to the Ricci scalar, $R$.
\begin{figure}
\includegraphics[width = 150pt]{Figures/Fig1.pdf}
\caption{The amplitude of the gravitational potential in $R^{2}$ model normalized by the value in Einstein's theory.
Note that we plotted the absolute value because the sign is not important.}
\label{Fig1}
\end{figure}
Now, we make a comment on homogeneous solutions ignored before.
It is pointed out by Starobinsky that the homogeneous solutions decay in the expansion universe and can be completely ignored at the present time~\cite{071000}.
In addition, it is supposed that such scalar degrees of freedom should be highly suppressed by some mechanisms in the solar system scale in order not to mediate the so-called fifth force.
For example, taking into account the interactions, the stabilization mechanism called chameleon mechanism~\cite{040227} or Vainshtein mechanism~\cite{720501} would work and such degrees of freedom might be killed in the solar system scale.
However, if such modes were alive in the dark matter halo scale for some reasons and the mass scale $M$ were sufficiently close to the frequency of the pressure, $2m$, a beat would occur with a frequency $|M - 2m|$.
In this situation, after averaging over the time scale corresponding to the high-frequency $M \sim 2m$, we would observe the beat frequency, $|M - 2m|$.
If such a thing happened, the detectable mass range of the axion by the pulsar timing observation would shift to more heavy mass regions.
\subsection{Hu-Sawicki model}
In the previous subsection, we discussed the simplest $f(R)$ model which can be solved exactly.
Now, in this subsection, we consider a more realistic model.
While there are several $f(R)$ models that explain the late time acceleration of the universe and also pass the solar system tests, now let us focus on the specific model known as the Hu-Sawicki model:
\begin{equation}
f(R) = -\mu R_{\text{c}}\frac{(R / R_{\text{c}})^{2n}}{(R / R_{\text{c}})^{2n} + 1} \ ,
\label{eq22}
\end{equation}
where $n, \mu, R_{\text{c}} > 0$.
For this model to mimic the $\Lambda$CDM model, $\mu R_{\text{c}} \simeq 2\Lambda$ is needed, where $\Lambda$ is the cosmological constant.
Since the energy density of the dark matter halo is much larger than the cosmological critical density, we can assume $R \gg R_{\text{c}}$.
In this limit, Eq.~(\ref{eq22}) takes the following form:
\begin{equation}
f(R) \simeq -\mu R_{\text{c}} \left[ 1 - (R / R_{\text{c}})^{-2n} \right].
\label{eq23}
\end{equation}
Note that the Starobinsky model~\cite{071000} has the same form as Eq.~(\ref{eq23}) in the high curvature limit.
In order to pass the local gravity tests, the Ricci scalar should oscillate around its average value $R_{0} = \rho_{\text{DM}}$.
The mass scale of this model is given by
\begin{equation}
M^{2} \equiv \frac{1}{3f''(R_{0})} \simeq \frac{R_{\text{c}}}{6n(2n + 1)\mu} \left( \frac{\rho_{\text{DM}}}{R_{\text{c}}} \right) ^{2n + 2}.
\label{eq24}
\end{equation}
Using $R_{\text{c}} \simeq 2\Lambda / \mu$ and plausible cosmological parameters~\cite{150209}, the mass is roughly evaluated as
\begin{equation}
M \sim 1.5\mu \times 10^{-23}~\text{eV} \ ,
\label{eq25}
\end{equation}
for $n = 1$.
This rough estimate tells us that $M$ has a value around the critical mass, $2m$, for $\mu = \mathcal{O}(1)$.
Since $M$ is strongly dependent on $n$ [see Eq.~(\ref{eq24})], $M$ can be larger or smaller compared to $2m$.
When $M \ll 2m$, completely the same situation as $R^{2}$ model is realized and the amplitude of the gravitational potential is given by Eq.~(\ref{eq20}).
This is because the amplitude of the Ricci scalar is much smaller than its average value in this limit and the field equation (\ref{eq16}) is reduced to Eq.~(\ref{eq18}).
Hence, this behavior should be universal for more general models which pass the local gravity tests.
When $M \gtrsim 2m$, however, a problem arises.
Since $f'''(R) / f''(R) \sim 1 / R$, we can evaluate
\begin{equation}
\frac{f'''(R)\dot{R}^{2}}{f''(R)\ddot{R}} \sim \frac{\delta\dot{R}^{2}}{R\delta\ddot{R}} \sim \frac{\delta R}{R} \ ,
\label{eq26}
\end{equation}
where $\delta R$ is the oscillating part of $R$.
Therefore, once $\delta R$ becomes of the order of $R_{0}$, the second term of the field equation (\ref{eq16}) prohibits $\delta R$ from oscillating stably.
With numerical calculations, we verified the Ricci scalar diverges for these parameters.
This is also true for the Einstein limit, $M \gg 2m$.
Thus, the Hu-Sawicki model is not compatible with the axion dark matter for these parameters.
Note that other viable $f(R)$ models also suffer from the same problem.
In order to avoid the instability of oscillations, the condition $M \lesssim 2m$ is needed.
This constraint is shown in Fig.~\ref{Fig2} with other constraints from cosmological and local gravity tests~\cite{100623}:
The three downward-sloping curves are the upper bounds on $\mu$ for three different axion masses.
The almost horizontal line denotes the lower bound on $\mu$ from cosmological tests.
The vertical line corresponding to $n = 0.9$ represents the lower bound on $n$ from the local tests.
From Fig.~\ref{Fig2}, the ultralight axion dark matter and the Hu-Sawicki model are compatible only in the certain parameter regions~(shaded in Fig.~\ref{Fig2}).
Note that the upper bounds on $\mu$ are somewhat underestimated:
From numerical calculations, we found that the upper bounds on $\mu$ are about 5 times smaller than the roughly estimated values illustrated in Fig.~\ref{Fig2}.
Of course, since the Hu-Sawicki model works well on large scales, it might be natural to modify the Hu-Sawicki model on small scales to circumvent this instability problem.
\begin{figure}
\includegraphics[width = 200pt]{Figures/Fig2.pdf}
\caption{
The constraints on the Hu-Sawicki model.
The axion dark matter model and the Hu-Sawicki model are compatible in the shaded regions.
}
\label{Fig2}
\end{figure}
If the axion dark matter were detected by pulsar timing experiments, we can determine the axion mass $m$ from the oscillation frequency
and the mass scale $M$ of $f(R)$ model from the amplitude of oscillation Eq.~(\ref{eq20}).
In the Hu-Sawicki model , since $M$ monotonically increases as $\mu$ and $n$ increase, it has the minimum value, $M_{\text{min}}$,
corresponding to the lower bounds for $\mu$ and $n$.
Numerically, we obtain the minimum value as
\begin{equation}
M_{\text{min}} \sim 0.76 \times 10^{-23}~\text{eV} \ .
\label{eq27}
\end{equation}
Hence, if the observed mass $M$ were lower than Eq.~(\ref{eq27}), the Hu-Sawicki model would be excluded.
\section{Conclusion}
We studied the pulsar timing signal from the ultralight axion field in $f(R)$ theory.
First, we discussed the simplest $f(R) = R^{2} / 6M^{2}$ model.
Then, it turned out that the amplitude of the gravitational potential in this model is enhanced or suppressed depending on the mass parameter $M$ compared to the case in Einstein's theory.
If $M$ is larger than the frequency of the pressure, $2m$, the results in Einstein's theory are reproduced.
On the other hand, if $M$ is smaller than $2m$, the amplitude is suppressed and difficult to be detected.
Furthermore, when $M$ approaches $2m$, the amplitude is dramatically amplified due to the resonance.
Next we discussed the Hu-Sawicki model.
Although the Hu-Sawicki model is known to pass both cosmological and solar system tests, we showed that this model is not compatible with the ultralight axion dark matter for some parameters.
When the mass scale $M$ given by Eq.~(\ref{eq24}) is much smaller than $2m$, completely the same situation as $R^{2}$ model is realized.
In this case, unfortunately, the amplitude is too small to be detected by near-future experiments.
Meanwhile, when $M$ reaches $2m$, the oscillation cannot be stable owing to the ``nonlinear" term of the field equation.
Remarkably, the model does not work even in the Einstein limit, $M \gg 2m$, for the same reason.
This gives rise to the new constraint on the Hu-Sawicki model.
In order to circumvent this instability problem, a modification on small scales would be needed.
In fact, if the detected mass scale $M$ were lower than $M_{\text{min}} \sim 0.76 \times 10^{-23}~\text{eV}$, the Hu-Sawicki model would be excluded.
\acknowledgements
This work was supported by JSPS KAKENHI Grant No. 25400251, MEXT KAKENHI Grants No. 26104708 and No. 15H05895.
\bibliographystyle{apsrev4-1}
|
2110.14004
|
\section*{Acknowledgement}
This work was performed at the Center for Integrated Nanotechnologies at Los Alamos National Laboratory (LANL), a U.S. Department of Energy, Office of Basic Energy Sciences user facility, under user proposal 2018BU0083. It was primarily supported through the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering via FWP No. 2018LANLBES16 (M.-C.L. and R.P.P.). Work at MIT (C.O., J.L. Z.Z., and R.C.) was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-SC0019126.
\section*{Author contributions}
M.-C.L., R.C., and R.P.P. conceived and designed the project. $\alpha$-Sr$_2$CrO$_4$ single crystalline thin films were grown and characterized by J.L., Z.Z., C.O., and R.C. M.-C.L. performed the experiments with help from N.S.S and L.T.M. The data was analyzed by M.-C.L., C.O., J.L., R.C., and R.P.P. The manuscript was written by M.-C.L. and R.P.P. with significant contributions from C.O., J.L., Z.Z., R.C., and D.A.Y.
\section*{Competing Interests}
The authors declare no competing interests.
|
0907.3801
|
\section{Introduction}
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$.
An {\em incidence} in $G$ is a pair
$(v,e)$ with $v \in V(G)$ and $e \in E(G)$, such that $v$ and $e$
are incident. We denote by $I(G)$ the set of all incidences in
$G$. Two incidences $(v,e)$ and $(w,f)$ are {\em adjacent} if one
of the following holds: (i) $v=w$, (ii) $e=f$ or (iii) the edge
$vw$ equals $e$ or $f$.
An {\em incidence $k$-coloring} of $G$ is a mapping from $I(G)$
to a set of $k$ colors such that adjacent incidences are assigned
distinct colors. The {\em incidence chromatic number} $\chi_i(G)$ of $G$ is the
smallest $k$ such that $G$ admits an incidence $k$-coloring.
Incidence colorings have been introduced by Brualdi and Massey in~\cite{BM}.
In this paper, the authors also conjectured that
the relation $\chi_i(G)\le\Delta(G)+2$ holds for every graph $G$, where $\Delta(G)$ denotes
the maximum degree of $G$. In~\cite{G}, Guiduli disproved this
{\em Incidence Coloring Conjecture} (ICC for short).
However, the ICC conjecture has been proved for several
graph classes~\cite{G,HS,HSZ,HWC,LL,M,SS,W,WCP}.
Let $G$ and $H$ be graphs. The {\em Cartesian product} $G \Box H$ of $G$ and $H$
is the graph with vertex set $V(G) \times V(H)$ where two vertices
$(u_1, v_1)$ and $(u_2,v_2)$ are adjacent if and only if either
$u_1=u_2$ and $v_1v_2 \in E(H)$, or $v_1=v_2$ and $u_1u_2 \in E(G)$.
Let $P_n$ and $C_n$ denote respectively the path and the cycle on $n$ vertices.
We will denote by $G_{m,n}=P_m\Box P_n$ the {\em grid} with $m$ rows and $n$ columns
and by $T_{m,n}=C_m\Box C_n$ the {\em toroidal grid} with $m$ rows and $n$ columns.
In this paper, we determine the incidence chromatic number of toroidal
grids and prove that this class of graphs satisfies the ICC:
\begin{theorem}
For every $m,n\geq 3$,
$\chi_i(T_{m,n})=5$ if $m,n \equiv 0 \pmod 5$ and
$\chi_i(T_{m,n})=6$ otherwise.
\label{SW-Theorem}
\end{theorem}
In~\cite{HWC}, Huang {\em et al.} proved that
$\chi_i(G_{m,n})=5$ for every $m$, $n$. Since every toroidal graph $T_{m,n}$ contains
the grid $G_{m,n}$ as a subgraph, we get that
$\chi_i(T_{m,n})\ge 5$ for every $m$, $n$.
The paper is organized as follows.
In Section~\ref{sec:2} we give basic properties and illustrate
the techniques we shall use in the proof of our main result, which
is given in Section~\ref{sec:3}.
\section{Preliminaries}
\label{sec:2}
Let $G$ be a graph, $u$ a vertex of $G$ with maximum
degree and $v$ a neighbour of $u$.
Since in any incidence coloring of $G$
all the incidences of the form $(u,e)$ have to get
distinct colors and since all of them have to get
a color distinct from the color of $(v,vu)$, we
have:
\begin{proposition}
\label{prop-delta}
For every graph $G$, $\chi_i(G)\ge\Delta(G)+1$.
\end{proposition}
The {\em square} $G^2$ of a graph $G$ is given by
$V(G^2)=V(G)$ and $uv\in E(G^2)$ if and only if $uv\in E(G)$ or there exists $w\in V(G)$
such that $uw$, $vw\in E(G)$. In other words, any two vertices within distance at most
two in $G$ are linked by an edge in $G^2$.
Let now $c$ be a proper vertex coloring of $G^2$ and $\mu$
be the mapping defined by $\mu(u,uv)=c(v)$ for every incidence
$(u,uv)$ in $I(G)$. It is not difficult to check that $\mu$
is indeed an incidence coloring of $G$
(see Example~\ref{ex1} below). Therefore we have:
\begin{proposition}
\label{prop-square}
For every graph $G$, $\chi_i(G)\le\chi(G^2)$.
\end{proposition}
In~\cite{SW}, we studied the chromatic number of the squares
of toroidal grids and proved the following:
\begin{theorem}
Let $T_{m,n}=C_m \Box C_n$. Then $\chi(T_{m,n}^2) \leq 7$ except $\chi(T_{3,3}^2)=9$ and
$\chi(T_{3,5}^2)=\chi(T_{4,4}^2)=8$.
\label{SW-square-Theorem}
\end{theorem}
By Proposition~\ref{prop-square}, this result provides upper bounds on
the incidence chromatic number of toroidal grids.
In~\cite{SW}, we also proved the following:
\begin{theorem}
For every $m,n\ge 3$, $\chi(T_{m,n}^2) \ge 5$.
Moreover, $\chi(T_{m,n}^2)=5$ if and only if
$m,n\equiv 0 \pmod 5$.
\label{SW-square5-Theorem}
\end{theorem}
In~\cite{W}, the second author proved the following:
\begin{theorem}
If $G$ is regular, then $\chi_i(G)=\Delta(G)+1$
if and only if $\chi(G^2)=\Delta(G)+1$.
\label{W-theorem}
\end{theorem}
\begin{figure}
\begin{center}
\begin{tabular}{ccc}
$A=$
&
\begin{tabular}{|cccc|}
\hline
1& 2& 3& 4 \\
3& 4& 5& 6 \\
5& 6& 7& 8 \\
7& 8& 1& 2 \\
\hline
\end{tabular}
\ \ \
&
\ \ \
\begin{tabular}{@{}r@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}l}
& \vl & & \vl & & \vl & & \vl & \\
& 7 & & 8 & & 1 & & 2 \\
\hl 4 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 1 \hl \\
& 3 & & 4 & & 5 & & 6 \\
& \vl & & \vl & & \vl & & \vl & \\
& 1 & & 2 & & 3 & & 4 \\
\hl 6 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl 5 & & 3 \hl \\
& 5 & & 6 & & 7 & & 8 \\
& \vl & & \vl & & \vl & & \vl & \\
& 3 & & 4 & & 5 & & 6 \\
\hl 8 & & 6 \hl 5 & & 7 \hl 6 & & 8 \hl 7 & & 5 \hl \\
& 7 & & 8 & & 1 & & 2 \\
& \vl & & \vl & & \vl & & \vl & \\
& 5 & & 6 & & 7 & & 8 \\
\hl 2 & & 8 \hl 7 & & 1 \hl 8 & & 2 \hl 1 & & 7 \hl \\
& 1 & & 2 & & 3 & & 4 \\
& \vl & & \vl & & \vl & & \vl & \\
\end{tabular}
\end{tabular}
\caption{A pattern $A$ and the
corresponding incidence coloring of $T_{4,4}$}
\label{fig-pattern}
\end{center}
\end{figure}
Since toroidal graphs are 4-regular, by
combining Proposition~\ref{prop-delta},
Theorem~\ref{SW-square5-Theorem} and Proposition~\ref{W-theorem}
we get the following:
\begin{corollary}
For every $m,n\ge 3$, $\chi_i(T_{m,n})\ge 5$.
Moreover, $\chi_i(T_{m,n})=5$ if and only if
$m,n\equiv 0 \pmod 5$.
\label{square5-corollary}
\end{corollary}
Note here that this corollary is part of our main result.
Any vertex coloring of the square of a toroidal grid $T_{m,n}$ can be
given as an $m\times n$ matrix whose entries correspond in an obvious way
to the colors of the vertices.
Such a matrix will be called a {\em pattern} in the following.
\begin{example}
\label{ex1}
{\rm Figure~\ref{fig-pattern} shows a $4\times 4$ pattern $A$, which defines a vertex
coloring of $T_{4,4}^2$, and the incidence coloring of $T_{4,4}$
induced by this pattern, according to the discussion
before Proposition~\ref{prop-square}.
Note for instance that the four incidences of the form $(u,uv)$, for $v$
being the second vertex in the third row, have color 6, which corresponds
to the entry in row 3, column 2, of pattern $A$.
}
\end{example}
If $A$ and $B$ are patterns of size
$m\times n$ and $m\times n'$ respectively, we shall denote by
$A+B$ the pattern of size $m\times (n+n')$ obtained by
``gluing'' together the patterns $A$ and $B$.
Moreover, we shall denote by $\ell A$, $\ell\ge 2$, the pattern
of size $m\times\ell n$ obtained by gluing
together $\ell$ copies of the pattern $A$.
We now shortly describe the technique we shall use
in the next section.
The main idea is to use a pattern for coloring the square
of a toroidal grid in order to get an incidence coloring
of this toroidal grid.
However, as shown in~\cite{SW}, the squares of toroidal grids
are not all 6-colorable.
Therefore, we shall use the notion of a {\em quasi-pattern} which corresponds
to a vertex 6-coloring of the square of a {\em subgraph} of a toroidal grid
obtained by deleting some edges.
We can then use such a quasi-pattern in the same way as before to obtain
a {\em partial} incidence coloring of the toroidal grid.
Finally, we shall prove that such a partial incidence coloring can
be extended to the whole toroidal grid without using any additional color.
We shall also use the following:
\begin{observation}
For every $m,n\ge 3$, $p,q\ge 1$, if $\chi_i(T_{m,n})\le k$
then $\chi_i(T_{pm,qn})\le k$.
\label{obs}
\end{observation}
To see that, it is enough to observe that every incidence $k$-coloring
$c$ of $T_{m,n}$ can be extended to an incidence $k$-coloring $c_{p,q}$ of
$T_{pm,qn}$ by ``repeating'' the pattern given by $c$, $p$ times ``vertically'' and $q$ times
``horizontally''.
\section{Proof of Theorem~1}
\label{sec:3}
According to Corollary~\ref{square5-corollary} above, we only
need to prove that $\chi_i(T_{m,n})\le 6$ for every
$m,n\ge 3$. The proof is based on a series of Lemmas, according
to different values of $m$ and $n$.
We first consider the case when $m\equiv 0\pmod 3$.
We have proved in~\cite{SW} the following:
\begin{proposition}
If $k\ge 1$, $n\ge 3$ and $n$ even, then $\chi(T_{3k,n}^2)\le 6$.
\label{prop-3k}
\end{proposition}
Here we prove:
\begin{lemma}
If $k\ge 1$ and $n\ge 3$, then $\chi_i(T_{3k,n})\le 6$.
\label{lemma-3k}
\end{lemma}
\begin{figure}
$$\begin{array}{cccc}
B=
\begin{array}{|c|}
\hline
3\\
1\\
2\\
\hline
\end{array}
&
C=
\begin{array}{|cccc|}
\hline
1& 4 & 2 & 5\\
2& 5 & 3 & 6\\
3& 6 & 1 & 4\\
\hline
\end{array}
&
D=
\begin{array}{|cccc|}
\hline
1& 4 & 3 & 6\\
2& 5 & 1 & 4\\
3& 6 & 2 & 5\\
\hline
\end{array}
&
E=
\begin{array}{|cc|}
\hline
2 & 5 \\
3 & 6 \\
1 & 4 \\
\hline
\end{array}
\\
\end{array}$$
$$\begin{array}{cc}
B+C=
\begin{array}{|c|cccc|}
\hline
3& 1& 4 & 2 & 5\\
1& 2& 5 & 3 & 6\\
2& 3& 6 & 1 & 4\\
\hline
\end{array}
&
B+D+E=
\begin{array}{|c|cccc|cc|}
\hline
3& 1& 4 & 3 & 6& 2 & 5 \\
1& 2& 5 & 1 & 4& 3 & 6 \\
2& 3& 6 & 2 & 5& 1 & 4 \\
\hline
\end{array}
\\
\end{array}$$
\caption{Patterns and quasi-patterns for Lemma~\ref{lemma-3k}}
\label{fig-pattern-1}
\end{figure}
\begin{figure}
{\small
$$
\begin{tabular}{@{}r@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}l}
& \vl & & \vl & & \vl & & \vl & & \vl & \\
& 2 & & \fbox{5} & & 6 & & 1 & & 4\\
\hl 5 & & \fbox{6} \hl 3 & & 4 \hl 1 & & 2 \hl 4 & & 5 \hl 2 & & 3 \hl \\
& 1 & & 2 & & 5 & & 3 & & 6 \\
& \vl & & \vl & & \vl & & \vl & & \vl & \\
& 3 & & \fbox{6} & & 4 & & 2 & & 5 \\
\hl 6 & & \fbox{4} \hl 1 & & 5 \hl 2 & & 3 \hl 5 & & 6 \hl 3 & & 1 \hl \\
& 2 & & 3 & & 6 & & 1 & & 4\\
& \vl & & \vl & & \vl & & \vl & & \vl & \\
& 1 & & \fbox{4} & & 5 & & 3 & & 6 \\
\hl 4 & & \fbox{5} \hl 2 & & 6 \hl 3 & & 1 \hl 6 & & 4 \hl 1 & & 2 \hl \\
& 3 & & 1 & & 4 & & 2 & & 5 \\
& \vl & & \vl & & \vl & & \vl & & \vl & \\
\end{tabular}$$
}
\caption{Incidence coloring for Lemma~\ref{lemma-3k}\label{fig-lemma-1}}
\end{figure}
\begin{proof}
If $n$ is even, the result follows from
Propositions~\ref{prop-square} and~\ref{prop-3k}.
We thus assume that $n$ is odd,
and we let first $k=1$. We consider three cases.
\begin{enumerate}
\item $n=3$.\\
We can easily get an incidence 6-coloring by coloring the incidences of one dimension with $\{1,2,3\}$ and the incidences of
the other dimension with $\{4,5,6\}$.
\item $n=4\ell+1$.\\
Let $B$ and $C$ be the patterns depicted in Figure~\ref{fig-pattern-1}
and consider the quasi-pattern $B+\ell C$ (the quasi-pattern $B+C$
is depicted in Figure~\ref{fig-pattern-1}).
This quasi-pattern provides a $6$-coloring of $T_{m,n}^2$ if we delete all
the edges linking vertices in the first column to
vertices in the second column.
We can use this quasi-pattern to obtain an incidence $6$-coloring of $T_{m,n}$
by modifying six incidence colors, as shown in Figure~\ref{fig-lemma-1}
(modified colors are in boxes).
\item $n=4\ell+3$.\\
Let $B$, $D$ and $E$ be the patterns depicted in Figure~\ref{fig-pattern-1}
and consider the quasi-pattern $B+\ell D+E$ (the quasi-pattern $B+D+E$
is depicted in Figure~\ref{fig-pattern-1}).
As in the previous case,
we can use this quasi-pattern to obtain an incidence $6$-coloring of $T_{m,n}$
by modifying the same six incidence colors.
\end{enumerate}
For $k\ge 2$, the result now directly follows from Observation~\ref{obs}.
\end{proof}
We now consider the case when $m\equiv 0\pmod 4$.
For $m\equiv 0\pmod 5$, we have proved in~\cite{SW} the following:
\begin{proposition}
If $k\ge 1$, $n\ge 5$ and $n\neq 7$, then $\chi(T_{5k,n}^2)\le 6$.
\label{prop-5k}
\end{proposition}
Here we prove:
\begin{lemma}
If $k\ge 1$, $n\ge 3$ and $(k,n)\neq (1,5)$, then $\chi_i(T_{4k,n})\le 6$.
\label{lemma-4k}
\end{lemma}
\begin{figure}
$$ \begin{array}{cc}
F=\begin{array}{|ccc|}
\hline
1& 2& 4 \\
1& 2& 4 \\
3& 5& 6 \\
3& 5& 6 \\
\hline
\end{array}
\
&
\
G=\begin{array}{|cccc|}
\hline
1& 2& 3& 4 \\
1& 2& 3& 4 \\
3& 4& 5& 6 \\
3& 4& 5& 6 \\
\hline
\end{array}
\\
\end{array}
$$
$$H=2F+2G=\begin{array}{|ccc|ccc|cccc|cccc|}
\hline
1& 2& 4& 1& 2& 4& 1& 2& 3& 4& 1& 2& 3& 4 \\
1& 2& 4& 1& 2& 4& 1& 2& 3& 4& 1& 2& 3& 4 \\
3& 5& 6& 3& 5& 6& 3& 4& 5& 6& 3& 4& 5& 6 \\
3& 5& 6& 3& 5& 6& 3& 4& 5& 6& 3& 4& 5& 6 \\
\hline
\end{array}$$\caption{Quasi-patterns for Lemma~\ref{lemma-4k}}
\label{fig-pattern-2}
\end{figure}
\begin{figure}
$$\begin{tabular}{cc}
{\small
\begin{tabular}{@{}r@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}l}
& \vl & & \vl & & \vl & \\
& 3 & & 5 & & 6 & \\
\hl 4 & & 2 \hl 1 & & 4 \hl 2 & & 1 \hl \\
& \fbox{x} & & \fbox{x} & & \fbox{x} & \\
& \vl & & \vl & & \vl & \\
& \fbox{x} & & \fbox{x} & & \fbox{x} & \\
\hl 4 & & 2 \hl 1 & & 4 \hl 2 & & 1 \hl \\
& 3 & & 4 & & 5 & \\
& \vl & & \vl & & \vl & \\
& 1 & & 2 & & 4 & \\
\hl 6 & & 5 \hl 3 & & 6 \hl 5 & & 3 \hl \\
& \fbox{x} & & \fbox{x} & & \fbox{x} & \\
& \vl & & \vl & & \vl & \\
& \fbox{x} & & \fbox{x} & & \fbox{x} & \\
\hl 6 & & 5 \hl 3 & & 6 \hl 5 & & 3 \hl \\
& 1 & & 2 & & 4 & \\
& \vl & & \vl & & \vl & \\
\end{tabular}
}
\
&
\
{\small
\begin{tabular}{@{}r@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}l}
& \vl & & \vl & & \vl & & \vl & \\
& 3 & & 4 & & 5 & & 6 & \\
\hl 4 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 1 \hl \\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & \\
& \vl & & \vl & & \vl & & \vl & \\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & \\
\hl 4 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 1 \hl \\
& 3 & & 4 & & 5 & & 6 & \\
& \vl & & \vl & & \vl & & \vl & \\
& 1 & & 2 & & 3 & & 4 & \\
\hl 6 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl 5 & & 3 \hl \\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & \\
& \vl & & \vl & & \vl & & \vl & \\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & \\
\hl 6 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl 5 & & 3 \hl \\
& 1 & & 2 & & 3 & & 4 & \\
& \vl & & \vl & & \vl & & \vl & \\
\end{tabular}
}
\\
\end{tabular}$$
\caption{Partial incidence colorings for Lemma~\ref{lemma-4k}\label{fig-lemma-2}}
\end{figure}
\begin{proof}
For $n=5$, the result holds by Proposition~\ref{prop-5k}, except for $k=1$.
Assume now $k=1$ and $n \neq 5$ and consider
the quasi-patterns $F$ and $G$ depicted
in Figure~\ref{fig-pattern-2}.
From these patterns, we can derive a partial incidence
6-coloring of $T_{4,3}$ and $T_{4,4}$, respectively,
as shown in Figure~\ref{fig-lemma-2}, where the
uncolored incidences are denoted by $x$.
It is easy to check that every such incidence has
only four forbidden colors and that only incidences
belonging to a same edge have to be distinct.
Therefore, these partial incidence colorings can be
extended to incidence $6$-colorings of $T_{4,3}$ and $T_{4,4}$.
For $n\ge 6$, we shall use the quasi-pattern $H=pF+qG$
where $p$ and $q$ are such that $n=3p+4q$ (recall that
every integer except 1,2 and 5 can be written in this form).
The quasi-pattern $H=2F+2G$ is depicted in Figure~\ref{fig-pattern-2}.
As in the previous case, this quasi-pattern provides a partial
incidence $6$-coloring of $T_{4,n}$ that can be extended
to an incidence $6$-coloring of $T_{4,n}$.
For $k\ge 2$, the result now directly follows from Observation~\ref{obs}.
\end{proof}
\begin{figure}
$$ \begin{array}{cc}
I=\begin{array}{|cccccc|}
\hline
6& 1& 2& 3& 4& 5 \\
3& 4& 5& 6& 1& 2 \\
5& 6& 1& 2& 3& 4 \\
2& 3& 4& 5& 6& 1 \\
4& 5& 6& 1& 2& 3 \\
\hline
\end{array}
\
&
\
I'=\begin{array}{|cccccc|}
\hline
6& 1& 2& 3& 4& 5 \\
6& 1& 2& 3& 4& 5 \\
6& 1& 2& 3& 4& 5 \\
3& 4& 5& 6& 1& 2 \\
5& 6& 1& 2& 3& 4 \\
2& 3& 4& 5& 6& 1 \\
4& 5& 6& 1& 2& 3 \\
\hline
\end{array}
\end{array}$$
\caption{Patterns for Lemma~\ref{lemma-mn}}
\label{fig-pattern-3}
\end{figure}
\begin{figure}
{\small
$$
\begin{tabular}{@{}r@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}l}
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& 4 & & 5 & & 6 & & 1 & & 2 & & 3\\
\hl 5 & & 1 \hl 6 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl\\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x}\\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& \fbox{y} & & \fbox{y} & & \fbox{y} & & \fbox{y} & & \fbox{y} & & \fbox{y}\\
\hl 5 & & 1 \hl 6 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl\\
& \fbox{z} & & \fbox{z} & & \fbox{z} & & \fbox{z} & & \fbox{z} & & \fbox{z}\\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x}\\
\hl 5 & & 1 \hl 6 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl\\
& 3 & & 4 & & 5 & & 6 & & 1 & & 2\\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& 6 & & 1 & & 2 & & 3 & & 4 & & 5\\
\hl 2 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl 5 & & 1 \hl 6 & & 2 \hl 1 & & 3 \hl \\
& 5 & & 6 & & 1 & & 2 & & 3 & & 4\\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& 3 & & 4 & & 5 & & 6 & & 1 & & 2\\
\hl 4 & & 6 \hl 5 & & 1 \hl 6 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 5 \hl \\
& 2 & & 3 & & 4 & & 5 & & 6 & & 1\\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& 5 & & 6 & & 1 & & 2 & & 3 & & 4\\
\hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl 5 & & 1 \hl 6 & & 2 \hl \\
& 4 & & 5 & & 6 & & 1 & & 2 & & 3\\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& 2 & & 3 & & 4 & & 5 & & 6 & & 1\\
\hl 3 & & 5 \hl 4 & & 6 \hl 5 & & 1 \hl 6 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl \\
& 6 & & 1 & & 2 & & 3 & & 4 & & 5\\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
\end{tabular}$$
}
\caption{A partial incidence coloring of $T_{7,6}$}
\label{fig-lemma-3}
\end{figure}
We now consider the remaining cases.
\begin{lemma}
If $m, n \geq 5$, $m \neq 6, 8$ and $n \neq 7$, then $\chi_i(T_{m,n}) \leq 6$.
\label{lemma-mn}
\end{lemma}
\begin{proof}
Assume $m, n \geq 5$, $m \neq 6, 8$ and $n \neq 7$.
By Proposition~\ref{prop-5k}, we have
$\chi(T^2_{5k,n})\le 6$ for $n \neq 7$. Hence, there exists a vertex $6$-coloring
of $T_{5k,n}^2$ which corresponds to some pattern $M$ of size $5k\times n$.
We claim that each row of pattern $M$ can be repeated one or three times
to get quasi-patterns that can be extended to incidence 6-colorings of
the corresponding toroidal grids.
Let for instance $M'$ be the quasi-pattern obtained from $M$ by repeating the first row
of $M$ three times. The quasi-pattern $M'$ has thus size $(5k+2)\times n$.
The quasi-pattern $M'$ induces a partial incidence coloring of $T_{5k+2,n}$
in which the only uncolored incidences are those lying on the edges
linking vertices in the first row to vertices in the second row and on the edges
linking vertices in the second row to vertices in the third row.
We illustrate this in Figure~\ref{fig-pattern-3} with a pattern $I$
of size $5\times 6$ (this pattern induces a vertex 6-coloring
of $T_{5,6}^2$) and its associated pattern $I'$ of size $7\times 6$.
The partial incidence coloring of $T_{7,6}$ obtained from $I'$ is then
given in Figure~\ref{fig-lemma-3}, where uncolored incidences are
denoted by $x$, $y$ and $z$.
Observe now that in each column, the two incidences denoted by $x$
have three forbidden colors in common and each of them has four forbidden colors in total.
Therefore, we can assign them the same color. Now, in each column,
the incidences denoted by $y$ and $z$ have four forbidden colors in common (the color
assigned to $x$ is one of them) and each of them has five forbidden colors
in total. They can be thus colored with distinct colors.
Doing that, we extend the partial incidence
coloring of $T_{7,6}$ to an incidence $6$-coloring of $T_{7,6}$.
The same technique can be used for obtaining an incidence 6-coloring
of $T_{5k+2,n}$ since all the columns are ``independent'' in the
quasi-pattern $M'$, with respect to uncolored incidences.
If we repeat three times several distinct rows of pattern $M$, each
repeated row will produce a chain of four uncolored incidences, as before,
and any two such chains in the same column will be ``independent'', since they will be
separated by an edge whose incidences
are both colored.
Hence, we will be able to extend to corresponding quasi-pattern to an
incidence 6-coloring of the toroidal grid, by assigning available colors
to each chain as we did above.
Starting from a pattern $M$ of size $5k\times n$, we can thus obtain
quasi-patterns of size $(5k+2)\times n$,
$(5k+4)\times n$,
$(5k+6)\times n$ and
$(5k+8)\times n$, by repeating respectively one, two, three or four lines from $M$.
Using these quasi-patterns, we can produce incidence 6-colorings of the
toroidal grid $T_{m,n}$, $m,n\ge 5$, $n\neq 7$, for every $m$
except $m=6$, $8$.
\end{proof}
\begin{figure}
{\small
$$
\begin{tabular}{@{}r@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}l}
& \vl & & \vl & & \vl & & \vl & & \vl & \\
& 1 & & 1 & & 1 & & 2 & & 3 & \\
\hl 4 & & \fbox{3} \hl \fbox{6} & & \fbox{4} \hl \fbox{3} & & 6 \hl 5 & & 4 \hl 6 & & 5 \hl\\
& 2 & & 2 & & 2 & & 3 & & 1 & \\
& \vl & & \vl & & \vl & & \vl & & \vl & \\
& 5 & & 5 & & 5 & & 6 & & 4 & \\
\hl 1 & & \fbox{6} \hl \fbox{3} & & \fbox{1} \hl \fbox{6} & & 3 \hl 2 & & 1 \hl 3 & & 2 \hl\\
& 4 & & 4 & & 4 & & 5 & & 6 & \\
& \vl & & \vl & & \vl & & \vl & & \vl & \\
& 2 & & 2 & & 2 & & 3 & & 1 & \\
\hl 6 & & \fbox{3} \hl \fbox{5} & & \fbox{6} \hl \fbox{3} & & 5 \hl 4 & & 6 \hl 5 & & 4 \hl\\
& 1 & & 1 & & 1 & & 2 & & 3 & \\
& \vl & & \vl & & \vl & & \vl & & \vl & \\
& 4 & & 4 & & 4 & & 5 & & 6 & \\
\hl 3 & & \fbox{6} \hl \fbox{2} & & \fbox{3} \hl \fbox{6} & & 2 \hl 1 & & 3 \hl 2 & & 1 \hl\\
& 5 & & 5 & & 5 & & 6 & & 4 & \\
& \vl & & \vl & & \vl & & \vl & & \vl & \\
\end{tabular}$$
}
\caption{An incidence 6-coloring of $T_{4,5}$}
\label{fig-45}
\end{figure}
\begin{figure}
$$\begin{array}{cc}
J=
&
\begin{array}{|ccccccc|}
\hline
3& 5& 6 & 3 & 4& 5 & 6 \\
1& 2& 4 & 1 & 2& 3 & 4 \\
1& 2& 4 & 1 & 2& 3 & 4 \\
3& 5& 6 & 3 & 4& 5 & 6 \\
3& 5& 6 & 3 & 4& 5 & 6 \\
1& 2& 4 & 1 & 2& 3 & 4 \\
1& 2& 4 & 1 & 2& 3 & 4 \\
\hline
\end{array}
\end{array}$$
\caption{A quasi-pattern for Lemma~\ref{lemma-77}}
\label{fig-pattern-77}
\end{figure}
\begin{figure}
{\small
$$
\begin{tabular}{@{}r@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}l}
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& 1 & & 2 & & 4 & & 1 & & 2 & & 3 & & 4 & \\
\hl 6 & & \fbox{4} \hl 3 & & \fbox{1} \hl 5 & & \fbox{2} \hl 6 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl 5 & & 3 \hl\\
& \fbox{5} & & \fbox{6} & & \fbox{3} & & \fbox{5} & & \fbox{6} & & \fbox{1} & & \fbox{2} & \\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& 3 & & 5 & & 6 & & 3 & & 4 & & 5 & & 6 & \\
\hl 4 & & 2 \hl 1 & & 4 \hl 2 & & 1 \hl 4 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 1 \hl \\
& \fbox{y} & & \fbox{y} & & \fbox{y} & & \fbox{y} & & \fbox{y} & & \fbox{y} & & \fbox{y} & \\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& \fbox{5} & & \fbox{6} & & \fbox{3} & & \fbox{5} & & \fbox{6} & & \fbox{1} & & \fbox{2} & \\
\hl 4 & & 2 \hl 1 & & 4 \hl 2 & & 1 \hl 4 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 1 \hl \\
& 3 & & 5 & & 6 & & 3 & & 4 & & 5 & & 6 & \\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& 1 & & 2 & & 4 & & 1 & & 2 & & 3 & & 4 & \\
\hl 6 & & 5 \hl 3 & & 6 \hl 5 & & 3 \hl 6 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl 5 & & 3 \hl\\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & \\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & \\
\hl 6 & & 5 \hl 3 & & 6 \hl 5 & & 3 \hl 6 & & 4 \hl 3 & & 5 \hl 4 & & 6 \hl 5 & & 3 \hl\\
& 1 & & 2 & & 4 & & 1 & & 2 & & 3 & & 4 & \\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& 3 & & 5 & & 6 & & 3 & & 4 & & 5 & & 6 & \\
\hl 4 & & 2 \hl 1 & & 4 \hl 2 & & 1 \hl 4 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 1 \hl \\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & \\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
& \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & & \fbox{x} & \\
\hl 4 & & 2 \hl 1 & & 4 \hl 2 & & 1 \hl 4 & & 2 \hl 1 & & 3 \hl 2 & & 4 \hl 3 & & 1 \hl \\
& 3 & & 5 & & 6 & & 3 & & 4 & & 5 & & 6 & \\
& \vl & & \vl & & \vl & & \vl & & \vl & & \vl & & \vl &\\
\end{tabular}$$
}
\caption{A partial incidence 6-coloring of $T_{7,7}$}
\label{fig-77}
\end{figure}
The only remaining cases are $m=4$, $n=5$ and $m=n=7$.
Then we have:
\begin{lemma}\label{lemma-77}
$\chi_i(T_{4,5}) \leq 6$ and $\chi_i(T_{7,7}) \leq 6$.
\end{lemma}
\begin{proof}
Let $m=4$ and $n=5$. Consider the pattern $C$ of size $3\times 4$ depicted
in Figure~\ref{fig-pattern-1}.
As in the proof of Lemma~\ref{lemma-mn}, we can repeat the first row of $C$
three times to get a quasi-pattern $C'$ that can be extended to an incidence
6-coloring of $T_{5,4}$. We then exchange $m$ and $n$ to get an incidence
6-coloring of $T_{4,5}$, depicted in Figure~\ref{fig-45} (the colors
assigned to uncolored incidences are drawn in boxes).
Let now $m=n=7$ and consider the quasi-pattern $J$ depicted in Figure~\ref{fig-pattern-77}.
This quasi-pattern provides the partial incidence coloring of $T_{7,7}$
given in Figure~\ref{fig-77}, where incidences with modified colors are
in boxes and uncolored incidences are denoted
by $x$ and $y$.
Observe now that the incidences denoted by $y$ have five forbidden
colors while the incidences denoted by $x$ have four forbidden
colors.
Therefore, this partial coloring can be extended to an incidence
6-coloring of $T_{7,7}$.
\end{proof}
We are now able to prove our main result:
\noindent {\bf Proof of Theorem~\ref{SW-Theorem}.}
From Corollary~\ref{square5-corollary}, we get that $\chi_i(T_{m,n})\ge 5$
for every $m$, $n$, and that equality holds if and only if
$m,n\equiv 0\pmod 5$.
For $m=3$ or $6$, the result follows from Lemma~\ref{lemma-3k}
and, for $m=4$ or $8$, the result follows from Lemma~\ref{lemma-4k},
except the case when $m=4$ and $n=5$ which follows from Lemma~\ref{lemma-77}.
Assume now that $m>6$ and $m\neq 7$.
If $n=7$, the result follows from Lemma~\ref{lemma-77} when $m=7$
and from Lemma~\ref{lemma-mn} when $m\neq 7$, by exchanging $m$ and $n$.
Finally, if $n\neq 7$, the result follows from Lemma~\ref{lemma-mn}.
\mbox{}\hfill\rule{0.5em}{0.809em}
|
2111.08804
|
\section{Introduction}
Given a Markov process $\{\eta_t:t\geq 0\}$ with state space $\Omega$, and a proper function $f:\Omega\to\bb R$, one would like to understand the long-time behavior of
$$\Sigma(t)\,:=\,\int_0^tf(\eta_s)ds$$
under a suitable space-time rescaling. Since the seminal work of Kipnis and Varadhan \cite{kv}, plenty of results has been obtained on the additive function of Markov processes, especially a particular class of Markov processes, the interacting particle systems. For the interacting particle systems on the one-dimensional lattice, we refer to \cite{s00}\cite{s03}\cite{b04}\cite{fgn14} on the scaling limits of the additive functionals of exclusion processes of various cases, and refer to \cite{qjs02} for those of zero range processes. We also point out a landmark work \cite{gj13} of Gon\c calves and Jara, where the authors obtained the Central Limit Theorem of a quite general class of additive functionals of one-dimensional, conservative, stationary interacting particle systems.
To the best of our knowledge, all the previous results in this topic are obtained under the assumption that the interacting particle system starts from its invariant measure. The purpose of this work is to present the first result on the Central Limit Theorem of the additive functionals of non-equilibrium systems.
The interacting particle systems that we work on are a sequence of spatially inhomogeneous, weakly asymmetric simple exclusion processes on the one-dimensional discrete torus $\bb T_n$ with $n$ sites. The dynamics of these processes can be informally described as follows. Fix a smooth function $F$ defined on the one-dimensional continuous torus. To each site $x$ of the torus $\bb T_n$ it is associated a Poisson clock with parameter $2$. When the clock at site $x$ rings, with probability $\frac{1}{2}(1-\frac{1}{n}F(\frac{x}{n}))$, the particle at this site attempts to jump to the left neighbour site $x+1$, and with probability $\frac{1}{2}(1+\frac{1}{n}F(\frac{x}{n}))$ it attempts to jump to the right neighbour site. The system obeys the exclusion rule, namely, at most one particle is allowed at each site. When the destination site of the attempted jump is occupied, the jump is denied.
Let us denote the WASEP on $\bb T_n$ by $\{\eta_t^n: t\geq 0\}$. The system is taken to start from a non-equilibrium state, which consists of a Bernoulli product measure with a smooth non-constant profile. We focus on a particular type of additive functional, namely, the occupation time of the origin. More precisely, we show that the limit of
\begin{equation}\label{1time}
\Gamma_n(t)\,:=\,\frac{1}{\sqrt{n}}\int_0^{t}\frac{\eta_{sn^2}(0)-\rho_s(0)}{q_s}ds
\end{equation}
is a Gaussian as $n\to\infty$,
where $q_{\cdot}:[0,T]\to\bb R$ is a smooth function that is bounded away from zero, and $\rho_s$ is the solution of the corresponding hydrodynamic limit equation obtained under the diffusive time scale:
$$\partial_s\rho=\Delta \rho\,-\,2\nabla\big\{\rho(1-\rho)F\big\}.$$
Our proof follows the approach adopted in \cite{gj13}, which can be roughly described as follows. Once the limit of the denstiy fluctuation field is derived, one then needs to choose a proper test function which can well approximate the Dirac delta function. For the non-equilibrium fluctuations of the density field, in general this is very difficult and has been one of the main open problems in statistical mechanics. It was just recently that Jara and Menezes \cite{jm} made a major breakthrough by introducing a robust refined relative entropy method to derive the non-equilibrium fluctuations of the interacting particle systems. In the present work we choose to work on the same model that Jara and Menezes studied in \cite{jm}.
One of the main novelties of the present work is the local replacement lemma (Lemma \ref{ZG}) out of equilibrium. The choice of the test function to approximate the Dirac delta function is elaborate. The usual choice for the equilibrium system, namely $\varepsilon^{-1}\mathds{1}_{(0,\varepsilon)}$, no longer works. With the help of the integration by parts formula introduced in \cite{jm}, we find that the correct approximation function is
\begin{equation}\label{app}
\varepsilon^{-1}\mathds{1}_{(0,\varepsilon)}\frac{\mc X(\rho(0))}{\mc X(\rho(\cdot))},
\end{equation}
where $\mc X(\rho)=\rho(1-\rho)$. The proof of the local replacement lemma usually relies on the Kipnis-Varadhan inequality, which is known only in the stationary case, unfortunately. Our proof is inspired by the techniques developed in \cite{jm}\cite{jm1}. We show that the quantity \eqref{1time} can be well approximated by the density field acting with the test function given in \eqref{app}(or its smooth version) in the $L^{\lambda}$ norm, with $\lambda\in(1,2)$, integrated up to time $t$.
The paper is organized in the following way. In Section \ref{sec2} we define our model, recall the hydrodynamic limit and fluctuations of the density obtained in \cite{jm}, and state our main results about scaling limits of occupation times. In Section \ref{sec3} we first give sketch of the proof to the relative entropy estimate from \cite{jm}, then apply some estimates in that sketch to prove the local replacement lemma. Using the local replacement lemma, we prove in Section \ref{sec4} the tightness of two sequences, one is used to characterize the limiting process, the other is the sequence $\{\Gamma_n(t): t\in[0,T]\}_{n\in\bb N}$ with $\Gamma_n(t)$ defined in \eqref{1time}. In Section \ref{sec5} we prove the theorems stated in Section \ref{sec2}.
\section{The Model and Main results}\label{sec2}
\subsection{WASEP}
For each $n\in\bb N$, denote by $\bb T_n=\bb Z/n\bb Z$ the one-dimensional discrete torus. Consider the exclusion process on $\bb T_n$, denoted by $\{\eta^n_t: t\geq 0\}$, whose generator $L_n$ acting on function $f:\{0,1\}^{\bb T_n}\to\bb R$ is given by
$$L_n f(\eta)\,=\,\sum_{x,y\in\bb T_n}r_n(x,y)\eta(x)(1-\eta(y)) [f(\eta^{x,y})\,-\,f(\eta)]$$
In this formula and below, the configurations are represented by the Greek letters $\eta$, $\xi$, so that $\eta(x)=1$ if site $x\in \bb T_n$ is occupied for the configuration $\eta$ and $\eta(x)=0$ otherwise. The symbol $\eta^{x,y}$ represents the configuration obtained from $\eta$ by switching the occupation of sites $x$ and $y$:
\begin{equation*}
(\eta^{x,y})(z)=
\begin{cases}
\eta(y) & \mbox{ if } z=x\\
\eta(x) & \mbox{ if } z=y\\
\eta(z) & \mbox{ if } z\neq x,y\;.
\end{cases}
\end{equation*}
Let us denote the state space $\{0,1\}^{\bb T_n}$ by $\Omega_n$. Let $\bb T=\bb R/\bb Z$ and fix a smooth function $F:\bb T\to\bb R$. We choose the jump rate function $r_n:\bb T_n\times \bb T_n\to\bb R_{\geq 0}$ such that $r_n(x,x\pm1)=1\pm \frac{1}{n}F(\frac{x}{n})$, and $r_n(x,y)=0$ if $\lvert x-y\rvert>1$. In order to guarantee that $r_n$ is non-negative, since $F$ is a bounded function, we assume that $n$ is sufficiently large. This process is called the weakly asymmetric exclusion process(WASEP). In particular if $F\equiv 0$, the process $\{\eta_t^n: t\geq 0\}$ becomes the simple symmetric exclusion process(SSEP).
\subsection{Scaling limit}
Consider an integrable function with respect to the Lebesgue measure $\rho(\cdot):\bb T\to\bb R_+$ . Denote by $\nu_\rho^n$ the Bernoulli product measure with slowly varying parameter associated to the profile $\rho(\cdot)$:
\begin{equation}\label{defnu}
\nu_{\rho(\cdot)}^n\{\eta:\eta(x)=1\}\,=\,\rho(x/n), \,\,\,\,x\in \bb T_n.
\end{equation} Throughout this article, we shall fix a smooth function $\rho_0$ such that there exists $\varepsilon_0\in(0,1/2)$ such that $\rho_0(u)\in (\varepsilon_0,1-\varepsilon_0)$ for every $u\in\bb T$. This function $\rho_0$ will be used as the initial profile of our model.
Given two probability measures $\mu$ and $\pi$ on the same probability space $E$ such that: $\mu$ is absolutely continuous with respect to $\pi$, the relative entropy of $\mu$ with respect to $\pi$, denoted by $H(\mu|\pi)$, is defined by
$$H(\mu|\pi)\,=\,\int_E\frac{d\mu}{d\pi}\log\frac{d\mu}{d\pi}d\pi,$$
where $\frac{d\mu}{d\pi}$ represents the Radon-Nikodym derivative of $\mu$ with respect to $\pi$.
Let $\mc M$ be the space of positive measures on $\bb T$ with total mass bounded by one, endowed with the weak topology. Let
$\pi^{n}_{t} \in \mc M$ be the empirical measure on $\bb T$ obtained by rescaling time by $n^2$, rescaling space by $n^{-1}$, and assigning mass $n^{-1}$ to each particle, i.e.,
\begin{equation}\label{f01}
\pi^{n}_{t}(\eta,du) \;=\; \frac{1}{n} \sum _{x\in \bb T_n} \eta_{tn^2} (x)\,
\delta_{x/n}(du)\,,
\end{equation}
where $\delta_u$ is the Dirac measure concentrated on $u$. Let $\mc D(\bb R_+, \Omega_n)$ be the path space of c\`adl\`ag trajectories with values in $\Omega_n$, endowed with the Skorokhod topology. For a measure $\mu_n$ on $\Omega_n$, denote by $\bb P_{\mu_n}$( resp. $\bb E_{\mu_n}$) the probability measure(resp. expectation) on $\mc D(\bb R_+, \Omega_n)$ induced by the
initial state $\mu_n$ and the Markov process $\{\eta_t : t\ge 0\}$.
Suppose that the process $\{\eta_t^n: t\geq 0 \}$ starts from an initial distribution $\mu^n$. We shall assume $\mu^n$ is close to the Bernoulli product measure $\nu^n_{\rho_0(\cdot)}$ in the sense of relative entropy. To simplify the notation, we write
$$H_n(0)\,=\,H(\mu^n\mid \nu^n_{\rho_0(\cdot)}).$$ In order to observe the evolution of the density in the macroscopic level, the process has to be accelerated by $n^2$. The next theorem on the hydrodynamic limit of WASEP is a well known result. See Proposition 2.1 of \cite{jm} for instance.
\begin{theorem}\label{hdl}
Assume that $H_n(0)=o(n)$.
Then for every $t\in [0,T]$, for any continuous function $f:\bb T\to\bb R$ and any $\delta>0$,
$$\lim_{n\to\infty}\bb P_{\mu^n}\Big\{\eta\,:\,\Big|\frac{1}{n}\sum_{x\in\bb T_n}f\big(\frac{x}{n}\big)\eta_{tn^2}(x)\,-\,\int_{\bb T}f(u)\rho_t(u)du\Big|>\delta\Big\}\,=\,0,$$
where $\rho_t(u)$ is the unique solution of the heat equation
\begin{equation}\label{pde}
\begin{cases}
\partial_t\rho_t(u)=\partial_{uu}^2 \rho_t(u)\,-\,2\partial_u\big\{\rho_t(u)(1-\rho_t(u))F(u)\big\}, \quad u\in \bb T \\
\rho(0,\cdot)=\rho_0(\cdot) \,
\end{cases}
\end{equation}
\end{theorem}
The theorem above can be interpreted as the law of large numbers of the empirical measure $\pi^n$.
The next result is the associated central limit theorem, which is the non-equilibrium fluctuation of the density field. Let us start by introducing the density fluctuation field. The density fluctuation field $X^n$ is defined by
\begin{equation}\label{dff}
X_t^n(f)\,:=\,\frac{1}{\sqrt{n}}\sum_{x\in\bb T_n}\big[\eta_{tn^2}(x)\,-\,\rho_t\big(\frac{x}{n}\big)\big]f(\frac{x}{n})
\end{equation}
for every test function $f\in C^{\infty}(\bb T)$ and $t\geq 0$.
Given an integer $k\in \bb Z$, denote by $H_{k}(\bb T)$ the Sobolev space:
$$H_k(\bb T)\,:=\,\Big\{f\in L^2(\bb T): \,\sum_{n\in \bb Z}(1+n^2)^k\lvert \widehat{f}(n)\rvert^2<\infty\Big\}$$
where $\widehat{f}$ is the Fourier series of $f$. The norm $\|\cdot\|_k$ of $H_k(\bb T)$ is naturally given by
$$\|f\|_k\,=\, \sum_{n\in \bb Z}(1+n^2)^k\lvert \widehat{f}(n)\rvert^2.$$
It is shown in \cite{jm} that the process $\{X_t^n, t\geq 0\}$ takes value in $H_{-k}(\bb T)$ for any $k>1/2$. Throughout this article except in Theorem \ref{hdl}, we assume that $H_n(0)=O(1)$. The next theorem gives the limit of the density fluctuation field $X^n$ as $n\to\infty$.
\begin{theorem}\label{CLT}[Theorem 2.4 and 7.1 in \cite{jm}]
Let $k>3$. Assume that the random variable $X_0$ taking value in $H_{-k}(\bb T)$ and $X_0^n$ converges to $X_0$ in law with respect to the topology of $H_{-k}(\bb T)$. Then $\{X_t^n, 0\leq t\leq T\}$ converges in law to the process $\{X_t, 0\leq t\leq T\}$, which is the solution of the stochastic heat equation
\begin{equation}\label{spde}
\partial_t X_t\,=\, \nabla\Big( \nabla X_t\,-\,2X_t(1-\rho_t)F\,+\,\sqrt{2\rho_t(1-\rho_t)}\dot{W}_t\Big)
\end{equation}
with initial condition $X_0$, where $\dot{W}_t$ is a space-time one-dimensional white noise.
\end{theorem}
In the last theorem, a stochastic process $\{X_t: t\in [0,T]\}$ is said to be the solution of \eqref{spde} if for any $f\in C^\infty\big([0,T]\times \bb T\big)$, the process $\{M_t(f): t\in[0,T]\}$ defined as
\begin{equation}\label{mardef}
M_t(f)\,=\,X_t(f_t)\,-\,X_0(f_0)\,-\,\int_0^t\,X_s\big((\partial_s+\bb L_s)f_s\big)ds
\end{equation}
is a continuous martingale with respect to the filtration $\mc F_t=\sigma\{X_s(f_s): s\leq t, f\in C^\infty\big([0,T]\times \bb T\big)\}$,
whose quadratic variation is
$$\<M_t(f)\>=\int_0^t\int2\rho_s(u)(1-\rho_s(u))|\partial_u f_s(u)|^2duds.$$
The generator $\bb L_t$ in \eqref{mardef} is defined as
\begin{equation}\label{Lt}
\bb L_t f(u)\,:=\,\Delta f(u)\,+\,2\big(1-2\rho_t(u)\big)F(u) f'(u)
\end{equation}
for any $f\in C^\infty(\bb T)$ and $u\in\bb T$.
\subsection{Occupation time at the origin}
The idea to study the occupation time at the origin is to take a test function in \eqref{dff}, which is an approximation of the Dirac delta function.
Let $\phi:\bb R\to\bb R$ be a nonnegative smooth function with support $(0,1)$ and has integral $1$. Define $\phi_\varepsilon(u):=\varepsilon^{-1}\phi(u/\varepsilon)$ for each $\varepsilon\in(0,1)$. Let $\mathds{1}_{A}$ be the indicator function of the set $A\subset \bb T$. When performing the replacement lemma, the smooth function $\phi_\varepsilon(u)$ plays the same role as $\varepsilon^{-1}\mathds{1}_{(0,\varepsilon)}$, the the normalized indicator function on the set $(0,\varepsilon)$. In the case where the exclusion process starts from the invariant measure $\nu_\rho^n$(see \cite{gj13}), the approximation of the Dirac delta function can be chosen as $\varepsilon^{-1}\mathds{1}_{(0,\varepsilon)}$. The idea behind this is that, roughly speaking, after taking the time integral, $\eta(0)$ can be replaced by $\frac{1}{\varepsilon n}\sum_{x=1}^{\varepsilon n}\eta(x)$. However, this idea no longer works if the exclusion process starts from a non-invariant measure, mainly because $\rho_t$ is not a constant function. The correct object to perform the replacement lemma turns out to be $w(0)$, which can be replaced by $\frac{1}{\varepsilon n}\sum_{x=1}^{\varepsilon n}w(x)$ with the help of the integration by parts formula given in Lemma E.3 in \cite{jm}, after taking the time integral, where
$$w(x)\,=\,w_{tn^2}(x)\,=\,\frac{\eta_{tn^2}(x)\,-\,\rho_t\big(\frac{x}{n}\big)}{\rho_t\big(\frac{x}{n}\big)\big(1-\rho_t\big(\frac{x}{n}\big)\big)} \quad \text{for all }\,\, x\in\bb T_n.$$
Based on the intuition above, we can define the following sequence of stochastic processes. Given $T>0$, fix a smooth function $q:[0,T]\to\bb R$. In addition, we assume that there exists a constant $M\geq1$ such that $\frac{1}{M}\leq q_s\leq M$ for all $s\in[0,T]$. For each $\varepsilon\in(0,1)$, let us define the real-valued process $\{Z_t^\varepsilon:t\in[0,T]\}$ as
\begin{equation}\label{Zt}
Z_t^\varepsilon\,=\,\int_0^tX_s\Big(\frac{\phi_\varepsilon(\cdot)\mc X(\rho_s(0)) }{q_s\mc X(\rho_s(\cdot))}\Big)ds,
\end{equation}
where $\mc X(\rho):= \rho(1-\rho)$ is the compressibility of the system.
We also define the discrete version of $Z_t^\varepsilon$ to be used later:
\begin{equation}\label{Ztn}
Z_{t,n}^\varepsilon\,=\,\int_0^tX^n_s\Big(\frac{\phi_\varepsilon(\cdot)\mc X(\rho_s(0)) }{q_s\mc X(\rho_s(\cdot))}\Big)ds.
\end{equation}
Our first main result is the characterization of the limit of $Z_t^\varepsilon$ as $\varepsilon\to 0$.
\begin{theorem}\label{limit}
Assume that $X_0$ is a Gaussian random field. The process $\{Z_t^\varepsilon:t\in[0,T]\}$ converges in distribution with respect to the uniform topology of $C([0,T],\bb R)$, as $\varepsilon\to 0$, to a Gaussian process $\{Z_t:t\in[0,T]\}$.
\end{theorem}
We next consider an object which is a generalization of the occupation time at the origin. Define the generalized rescaled occupation time at the origin
$$\Gamma_n(t)\,:=\,\frac{1}{\sqrt{n}}\int_0^{t}\frac{\eta_{sn^2}(0)-\rho_s(0)}{q_s}ds.$$
Now it becomes more clear why the test function in \eqref{Zt} is chosen in that way. Notice that $q^{-1}\big(\eta(0)\,-\rho(0)\big)$ is equal to $q^{-1}w(0)\mc X(\rho(0))$ by definition of $w(\cdot)$ and $\mc X(\cdot)$. To approximate it, the correct quantity should be
$$\frac{1}{\varepsilon n q}\sum_{x=1}^{\varepsilon n}w(x) \mc X(\rho(0)).$$
The test function appearing in \eqref{Zt} is just the smooth version of it.
We now state our second main result.
\begin{theorem}\label{time}
Assume that $X_0$ is a Gaussian random field. As $n\to\infty$, the sequence of process $\{\Gamma_n(t): t\in [0,T]\}_{n\in\bb N}$ converges in law, to the same Gaussian process $\{Z_t:t\in[0,T]\}$ as in Theorem \ref{limit}, with respect to the uniform topology of $C([0,T], \bb R)$.
\end{theorem}
\section{Entropy estimate and local replacement lemma}\label{sec3}
\subsection{Entropy estimate}
In this subsection we collect some results on the relative entropy estimate obtained in \cite{jm}, which will be used later.
For any $f:\bb T_n\to\bb R$, define the Dirichlet form with respect to a probability measure $\mu$ on $\Omega_n$ by
$$D(f;\mu)\,=\,\sum_{\eta\in\Omega_n}\sum_{x,y\in\bb T_n}[f(\eta^{x,y})\,-\,f(\eta)]^2 \mu(\eta).$$
Note that here $\frac{1}{2}D(f;\mu)$ is not equal to $-\<f, L_nf \>_{\mu}$. However they satisfy the following relation: for every probability measure $\mu$,
\begin{equation}\label{relD}
-\<f,L_nf\>_{\mu}\,\geq\, \frac{1}{2}D(f;\mu).
\end{equation}
Recall the definition \eqref{defnu}. For every $t\in(0,T]$, let $\mu_t^n$ be the Bernoulli product measure with slowing varying parameter $\rho_t(\cdot)$ and $\nu^n_{1/2}$ be the Bernoulli product measure with slowing varying parameter $1/2$.
Let $f_t^n=\frac{d\eta^n_{tn^2}}{d\mu_t^n}$ and let $H_n(t)$ be the relative entropy of the law of $\eta_{tn^2}$ with respect to the reference measure $\mu_t^n$:
$$H_n(t)\,=\,\int_{\Omega_n} f_t^n\log f_t^n d\mu_t^n.$$
The following theorem gives a sharp upper bound on the relative entropy $H_n(t)$, which depends on $H_n(0)$ and $\rho(\cdot,\cdot)$. Recall that we assume that $\rho_0\in (\varepsilon_0,1-\varepsilon_0)$ for some small $\varepsilon_0>0$. It is shown in Lemma B.3 in \cite{jm} that $\rho(t,\cdot)\in (\varepsilon_1,1-\varepsilon_1)$ for some $\varepsilon_1\in (0,1/2)$. For the simplicity of notation, we just assume that $\rho_t(u)\in (\varepsilon_0,1-\varepsilon_0)$ for all $t\in[0,T]$ and all $u\in\bb T$. It is also proved in Proposition B.1 in \cite{jm} that $\rho_t$ is smooth. Therefore we can assume that there exists $\kappa>0$ such that
$$n\Big\lvert \rho_t\big(\frac{x+1}{n}\big)\,-\,\rho_t\big(\frac{x}{n}\big)\Big\rvert \leq \kappa, \quad \text{for any \,\,} x\in T_n, \,\, \text{any}\,\,n\in\bb N, \,\, \text{and any\,\,} t\in\bb [0,T].$$
\begin{theorem}\label{entropy}
For every $t\in [0,T]$, there exists a constant $C=C(\varepsilon_0, T,\kappa)$ such that
\begin{equation*}
H_n(t)\,\leq\, C(H_n(0)\,+\,8).
\end{equation*}
\end{theorem}
One of the main steps to prove the above theorem is the following lemma, which is a particular case that $A=\{0\}$ of Lemma 3.1 of \cite{jm}.
To keep notation simple, whenever $t$ is fixed in the context, we write $\rho_t(x/n)$ as $\rho_x$. Throughout this article, the constant $C$ may change from line to line, and never depends on $n$.
\begin{lemma}\label{main}
Given a function $G:\bb T_n\to\bb R$, there exists a finite constant $C=C(\varepsilon_0)$ such that, for any density $f$ with respect to $\mu_t^n$ and any $\delta>0$,
\begin{equation*}
\begin{split}
&\int \sum_{x\in\bb T_n}w(x) w(x+1) G(x)f d\mu_t^n\\
\leq\,&\delta n^2 D(\sqrt{f};\mu_t^n)\,+\, \frac{C(1+\kappa^2)}{\delta}\big(\|G\|_\infty\,+\,\|G\|_\infty^2\big)\big(H(f;\mu_t^n)\,+\, 8\big),
\end{split}
\end{equation*}
where $H(f;\mu_t^n):=\int f\log f d\mu_t^n$.
\end{lemma}
To introduce notation and list some estimates that will be used later, we give the sketch of the proof. For more details, please check the original proof of Lemma 3.1 in \cite{jm}.
\begin{proof}[Sketch of the proof]
Given $\l\in\bb N$, let $\Lambda_l\,=\,\{0,1\cdots,\l\}$ and let $p_{\l}$ be the uniform measure on $\Lambda_{\l}$. Let $q_{\l}$ be the measure on $\bb Z$ given by
$$q_{\l}(z):=\sum_{y\in\bb Z}p_{\l}(y)p_{\l}(z-y).$$
For any $\l<n/2$, define $$w^{\l}(x):=\sum_{y\in\bb Z}w(x+y)q_{\l}(y).$$
Write
$$w(x)\,-\,w^{\l}(x)\,=\, \sum_{z\in\bb Z} \phi_{\l}(z) \big(w(x+z+1)\,-\,w(x+z)\big),$$
then $\sum_{z\in\bb Z} \phi_{\l}(z)^2\,\leq\, C\l$ by Lemma 3.2 in \cite{jm}. From the previous identity we can get
$$\sum_{x\in\bb T_n}\big( w(x+1)-w^{\l}(x+1)\big) w(x) G(x)\,=\, \sum_{x\in\bb T_n}h_{x-1}^{\l}\big(w(x+1)\,-\,w(x)\big),$$
where each $x\in\bb T_n$, we define
$$h_x^{\l}\,:=\,\sum_{z\in\bb Z} \phi_{\l}(z)w(x-z)G(x-z).$$
Moreover, we define $$V(G)\,:=\,\sum_{x\in\bb T_n}w(x) w(x+1) G(x)$$
$$V^{\l}(G)\,:=\,\sum_{x\in\bb T_n}w(x) w^{\l}(x+1) G(x)$$
$$W^{\l}(G)\,:=\, \sum_{x\in\bb T_n}(h_x^{\l})^2$$
$$\widetilde{V}^{\l}(G)\,:=\,\sum_{x\in\bb T_n}n(\rho_{x+1}-\rho_x)h_{x-1}^{\l}w(x)w^{\l}(x+1)$$
$$\widetilde{W}^{\l}(G)\,:=\,\sum_{x\in\bb T_n}\Big\{\sum_{z\in\bb Z}\phi_{\l}(z)n(\rho_{x-z+1}-\rho_{x-z}) h_{x-z-1}^{\l} w(x-z)\Big\}^2$$
and
$$\widetilde{Z}^{\l}(G)\,:=\,\sum_{x\in\bb T_n}h_{x-1}^{\l}n(\rho_{x+1}-\rho_x)w(x)w(x+1).$$
With the help of Lemma 3.3 in \cite{jm}, for any $\delta>0$,
\begin{equation}\label{v2}
\begin{split}
\int V(G) fd\mu_t^n\,\leq\,\delta n^2D(\sqrt{f};\mu_t^n)\,&+\,\int\Big( V^{\l}(G)\,+\,\frac{C(\varepsilon_0)}{\delta n^2}W^{\l}(G)\,+\,\frac{1}{n}\widetilde{V}^{\l}(G)\\
&+\,\frac{C(\varepsilon_0)}{\delta n^4}\widetilde{W}^{\l}(G)\,+\,\frac{1}{n^2}\widetilde{Z}^{\l}(G)\Big)fd\mu_t^n.
\end{split}
\end{equation}
Choosing $\l=\frac{n}{8}$, it is then proved that
$$\int V^{\l}(G) f d\mu_t^n\,\leq\,C(\varepsilon_0) \|G\|_{\infty}\big(H(f;\mu_t^n)\,+\,8\big),$$
$$\frac{C(\varepsilon_0)}{\delta n^2}\int W^{\l}(G)f d\mu_t^n\,\leq\,\frac{C(\varepsilon_0)\|G\|_\infty^2}{\delta}\big(H(f;\mu_t^n)\,+\,8\big),$$
$$\int \frac{1}{n}\widetilde{V}^{\l}(G)f d\mu_t^n\,\leq\, C(\varepsilon_0)\kappa\|G\|_\infty \big(H(f;\mu_t^n)\,+\,1\big),$$
$$\int \frac{C(\varepsilon_0)}{\delta n^4}\widetilde{W}^{\l}(G) fd\mu\,\leq\, \frac{C(\varepsilon_0)\kappa^2\|G\|_\infty^2}{\delta}\big(H(f;\mu_t^n)\,+\,8\big),$$
and
$$\int \frac{1}{n^2}\widetilde{Z}^{\l}(G)fd\mu_t^n\,\leq\, C(\varepsilon_0)\kappa^2\|G\|_\infty\big(H(f;\mu_t^n)\,+\,8\big).$$
With these estimates, the proof of the lemma finishes.
\end{proof}
For the constant $C(\varepsilon_0)$ and $\delta$ appearing in \eqref{v2}, choosing $\l=n/8$ ,
we define
\begin{equation}\label{defUG}
U_{\delta}(G)\,:=\,V^{\l}(G)\,+\,\frac{C(\varepsilon_0)}{\delta n^2}W^{\l}(G)\,+\,\frac{1}{n}\widetilde{V}^{\l}(G)\,+\,\frac{C(\varepsilon_0)}{\delta n^4}\widetilde{W}^{\l}(G)\,+\,\frac{1}{n^2}\widetilde{Z}^{\l}(G).
\end{equation}
In the proof of Theorem \ref{entropy}, actually the following estimate from Lemma \ref{main} is used:
\begin{equation}\label{sum}
\int U_\delta(G)f_t^nd\mu_t^n\,\leq\, C(\varepsilon_0,\kappa, G)\delta^{-1} \big(H_n(t)\,+8\big).
\end{equation}
Moreover, a stronger statement is also proved in \cite{jm}:
\begin{equation}\label{sum2}
\int |U_\delta(G)|f_t^nd\mu_t^n\,\leq\, C(\varepsilon_0,\kappa, G)\delta^{-1} \big(H_n(t)\,+8\big).
\end{equation}
\subsection{Replacement lemma}
We first state some results that will be used in this subsection.
\begin{lemma}\label{pe}[Lemma 4.3 of \cite{jm}]
Let $X$ be a nonnegative random variable. Assume $P(X>\delta)\,\leq\,C/\delta^2$ for any $\delta>0$. Then for any $\lambda\in(1,2)$, there exists an universal constant $C(\lambda)$ such that $E[X^\lambda]\,\leq\, C(\lambda) C^{\lambda/2}$.
\end{lemma}
\begin{proposition}\label{entpro}[Proposition 8.2 in Appendix 1 of \cite{kl}]
Let $\mu$ be a measure on a finite space $\Omega $ and let $f$ be a density with respect to $\mu$. Then for any $A\subset \Omega$,
$$\int_A fd\mu\,\leq\,\frac{H(f;\mu)\,+\,\log 2}{\log \big(1+\mu(A)^{-1}\big)}.$$
\end{proposition}
Recall the definition \eqref{Ztn} of $Z_{t,n}^\varepsilon$. The next replacement lemma shows that $Z_{t,n}^\varepsilon$ and $\Gamma_n(t)$ are close under the $L^{\lambda}$ norm if $1<\lambda<2$.
\begin{lemma}\label{ZG}
Fix $0\leq s < t \leq T$. Then for every $\lambda\in (1,2)$, there exists a positive constant $C=C(\lambda,M,\varepsilon_0,\kappa,T)$ independent of $n$ and $\varepsilon$ such that
$$\bb E_{\mu^n}\Big[\big|Z_{t,n}^\varepsilon\,-\,Z_{s,n}^\varepsilon\,-\,\big(\Gamma_n(t)-\Gamma_n(s)\big)\big|^\lambda\Big]\,\leq\, C|t-s|^{\frac{\lambda}{2}}\varepsilon^{\frac{\lambda}{2}} $$
for all $n\in \bb N$.
\end{lemma}
\begin{proof}
It is enough to prove the theorem only for $s<t$ such that $t-s\leq 1$. In view of Lemma \ref{pe}, to prove the theorem, it is enough to show that for any $\Delta>0$, there exists a constant $C>0$ such that
$$\bb P_{\mu^n}\Big[\big\lvert Z_{t,n}^\varepsilon\,-\,Z_{s,n}^\varepsilon\,-\,\big(\Gamma_n(t)-\Gamma_n(s)\big)\big\rvert>\Delta\Big]\,\leq\, \frac{C(t-s)\varepsilon }{\Delta^2}.$$
Define
$$\overline{\eta}_{rn^2}(x)=\eta_{rn^2}(x)-\rho_r(\frac{x}{n}) \quad \text{for every}\, x\in\bb T_n,$$
and
$$V^\varepsilon_r:=\sqrt{n}\Big\{\frac{\overline{\eta}_{rn^2}(0)}{q_r}\,-\,\frac{1}{n}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w_{rn^2}(x)\frac{\mc X(\rho_r(0))}{q_r}\Big\}.$$
Note that
$$Z_{t,n}^\varepsilon\,-\,Z_{s,n}^\varepsilon\,-\,\big(\Gamma_n(t)-\Gamma_n(s)\big)\,=\,\int_s^t V_r^\varepsilon ds.$$
It remains to show that
\begin{equation}\label{eqtone}
\bb P_{\mu^n}\Big[\int_s^t \pm V_r^\epsilon \,-\,\frac{\gamma}{2}\, U_\delta(G) dr >\Delta\Big]\,\leq\, \frac{C(t-s)\varepsilon }{\Delta^2}
\end{equation}
\begin{equation}\label{eqttwo}
\bb P_{\mu^n}\Big[\Big\lvert\int_s^t \frac{\gamma}{2}\, U_\delta(G)dr\Big\rvert>\Delta\Big]\,\leq\, \frac{C(t-s)\varepsilon }{ \Delta^2}
\end{equation}
for some constants $\gamma>0$ and $\delta>0$ that will be decided later.
The second inequality is easy to deal with. Choose $\delta=1/4$ and write $U_{1/4}(G)$ simply as $U(G)$ in the proof of this lemma. By Markov inequality, the left hand side of \eqref{eqttwo} is bounded by
$$\Delta^{-1}\bb E_{\mu^n}\Big[\Big\lvert\int_s^t\frac{\gamma}{2}\, U(G)dr\Big\rvert\Big]\,\leq\, \frac{(t-s)\gamma}{2\Delta}\bb E_{\mu^n}\big[\big\lvert U(G)\big\rvert\big].$$
Since $H_n(r)$ is of order $O(1)$ for every $0\leq r\leq T$, by \eqref{sum2}, $\bb E_{\mu^n}\big[\big\lvert U(G)\big\rvert\big]$ is bounded by some constant $C$. Choosing $\gamma=(t-s)\varepsilon\Delta^{-1}$ with $B=B(\varepsilon_0, \phi, M)$ that will be determined later, since $t-s\leq1$, inequality \eqref{eqttwo} is proved.
Now we turn to prove the first inequality \eqref{eqtone}. By Proposition \ref{entpro},
$$\bb P_{\mu^n}\Big[\int_s^t \pm V_r^\varepsilon\,-\,\frac{\gamma}{2} U(G) dr >\Delta\Big]\,\leq\,\frac{\log 2}{\log\Big(1\,+\, \bb P_{\mu^n}\Big[\int_s^t \pm V_r^\varepsilon\,-\,\frac{\gamma}{2}\, U(G) dr>\Delta\Big]^{-1} \Big)}.$$
Thus it is enough to show that
\begin{equation}\label{midexp}
\bb P_{\mu^n}\Big[\int_s^t \pm V_r^\varepsilon\,-\,\frac{\gamma}{2}\, U(G) dr>\Delta\Big]\,\leq\, \exp\Big\{\frac{-\Delta^2}{C(t-s)\varepsilon}\Big\},
\end{equation}
for sufficiently large $n$.
By Markov inequality,
$$\bb P_{\mu^n}\Big[\int_s^t \pm V_r^\varepsilon\,-\,\frac{\gamma}{2}\, U(G) dr>\Delta\Big]\,\leq\, e^{-\frac{\Delta}{\gamma}}\bb E_{\mu_n}\Big[\exp\Big\{\int_s^t \pm \frac{V_r^\varepsilon}{\gamma}\,-\,\frac{U(G)}{2} dr\Big\}\Big].$$
If we can show that
\begin{equation}\label{expbound}
\bb E_{\mu^n}\Big[\exp\Big\{\int_s^t \pm \frac{V_r^\varepsilon}{\gamma}\,-\,\frac{U(G)}{2} dr\Big\}\Big]\,\leq\, \exp\Big\{\frac{2\|\phi\|_\infty^2 M^2}{\varepsilon_0} \frac{\varepsilon(t-s)}{\gamma^2} \Big\},
\end{equation}
then we claim that \eqref{midexp} holds by choosing $B=4\|\phi\|_\infty^2 M^2\varepsilon_0^{-1}$. Indeed, since we have chosen $\gamma=B(t-s)\varepsilon\Delta^{-1}$, a direct computation gives
$$\frac{2\|\phi\|_\infty^2 M^2}{\varepsilon_0}\frac{(t-s)\varepsilon}{\gamma^2}= \frac{\Delta}{2\gamma}.$$
By Lemma A.2 of \cite{jm} and inequality \eqref{relD}, the logarithm of the expectation at the left hand side of \eqref{expbound} is bounded by
\begin{equation}\label{stbound}
\begin{split}
\int_s^t \Big\{\sup_{f} &\frac{-n^2}{2}D(\sqrt{f};d\mu_r^n)+\int\frac{\pm V^\varepsilon_r}{\gamma}fd\mu_r^n\\
&+\,\frac{1}{2} \int\big(L_{n,r}^\star 1\,-\,\frac{d}{dr}\log\Psi_r^n-U(G)\big)f d\mu_r^n \Big\}dr,
\end{split}
\end{equation}
where the supremum is taken over all the density $f$ with respect to the reference measure $\mu_r^n$, $L_{n,r}^\star$ is the adjoint operator of $L_n$ with respect to $\mu_r^n$ and $\Psi_r^n$ is the Radon-Nikydim derivative of $\mu_r^n$ with respect to $\nu^n_{1/2}$.
In section A.3 of \cite{jm} it is shown that for every $0<t\leq T$,
\begin{equation*}
L_{n,t}^\star 1\,-\,\frac{d}{dt}\log\Psi_t^n\,=\, \frac{1}{n^2}\sum_{x\in\bb T_n} w(x) R_x^n(t)\,+\, \sum_{x\in\bb T_n}w(x) w(x+1) G^n_{x}
\end{equation*}
where $|R_x^n(t)|\,\leq\,\sup_{u\in\bb T} \frac{d}{du^4}\rho_t(u)$, and
$$G_{x}^n\,=\, n(\rho_{x+1}-\rho_x) F\big(\frac{x}{n}\big)\big(\rho_{x+1}+\rho_x-2\rho_{x+1}\rho_x\big) -n^2(\rho_{x+1}\,-\,\rho_x)^2.$$
The discrete function given above is exactly our choice of the function $G$ in $U(G)$.
For every density $f$ with respect to $\mu_r^n$, since $|w(x)|\,\leq C(\varepsilon_0)$ and $\sup_{u\in\bb T} \frac{d}{du^4}\rho_r(u)$ is bounded,
$$\frac{1}{n^2}\int \sum_{x\in\bb T_n} w(x) R_x^n(r) fd\mu_r^n$$
is of order $O(\frac{1}{n})$. On the other hand, by inequality \eqref{v2}, since $\delta$ was chosen to be $1/4$,
$$\int \sum_{x\in\bb T_n}w(x) w(x+1)G^n_{x} fd\mu_r^n\,\leq\,\int U(G)fd\mu_r^n\,+\,\frac{n^2}{4} D(\sqrt{f};\mu_r^n)$$
for any density $f$ with respect to the measure $\mu_t^n$.
Up to now we have proved that
\begin{equation*}
\int_s^t \sup_{f}\Big\{-\frac{n^2}{4}D(\sqrt{f};d\mu_r^n)+\,\frac{1}{2} \int\big(L_{n,r}^\star 1\,-\,\frac{d}{dr}\log\Psi_r^n-U(G)\big)f d\mu_r^n \Big\}dr
\end{equation*}
vanishes as $n\to\infty$.
Applying Lemma \ref{dif} with $A=\pm\,\gamma^{-1}$, we get
$$\int_s^t \sup_{f} \Big\{ \frac{-n^2}{4}D(\sqrt{f};d\mu_r^n)\,+\, \int\frac{\pm V^\varepsilon_r}{\gamma}fd\mu_r^n \,\Big\}dr\,\leq\, \frac{2\|\phi\|_\infty^2 M^2}{\varepsilon_0} \frac{\varepsilon(t-s)}{\gamma^2}.$$
This proves \eqref{expbound} and hence the lemma.
\end{proof}
\begin{remark}
We applied Lemma A.2 of \cite{jm} to obtain the upper bound stated in \eqref{stbound}. The original statement of that Lemma is limited to the setting $s=0$. It is easy to extend the lemma to $s>0$, which is actually needed in the proof of the previous Theorem. We just need to consider a time-shifted version of the original Markov chain:
$$\xi_{tn^2}(\cdot)=\eta_{(t+s)n^2}(\cdot), \quad \text{for all}\,\, t\geq 0,$$
and then apply Lemma A.2 of \cite{jm} to the new process \{$\xi_{tn^2}: 0\leq t\leq T-s\}$.
\end{remark}
\begin{lemma}\label{dif}
Fix $A\in\bb R$, $q>0$ and $\varepsilon\in(0,1)$. Assume that $f$ is density with respect to $\mu$ and $H(f;\mu)=o(\sqrt{n})$. Then for any $\delta>0$,
\begin{equation}
\begin{split}
\limsup_{n\to\infty}\Big\{&\int A\sqrt{n}\Big\{\frac{\overline{\eta}(0)}{q}-\frac{1}{n}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w(x)\frac{\mc X(\rho(0))}{q}\Big\}f d\mu\,-\, \frac{n^2}{4}D(\sqrt{f},\mu)\Big\}\\
&\leq\,\frac{2A^2\varepsilon \|\phi\|_\infty^2}{\varepsilon_0 q^2}.
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
Note that by the smoothness and compact support of $\phi$, one can show that
$$\Big|1-\frac{1}{n}\sum_{x=1}^n\phi_\varepsilon\big(\frac{x}{n}\big)\Big|\,\leq\,\frac{\|\phi'\|_\infty}{\varepsilon n}.$$
Therefore
$$\int A\sqrt{n}\Big\{\frac{\overline{\eta}(0)}{q}-\frac{1}{n}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w(x)\frac{\mc X(\rho(0))}{q}\Big\}f d\mu$$ is less than or equal to
$$\int \frac{A\mc X(\rho(0))\sqrt{n}}{q n}\sum_{x=0}^{n}\sum_{y=x+1}^{n-1}\phi_\varepsilon\big(\frac{y}{n}\big)\big(w(x)-w(x+1)\big) f d\mu$$
added with an error term $\frac{A\sqrt{n}\|\phi'\|_\infty^2}{\varepsilon nq}$ which vanishes as $n\to\infty$. By Lemma E.3 in \cite{jm}, for any $\delta_x>0$, $x\in\bb T_n$, it is bounded by
\begin{equation}\label{sumA}
\begin{split}
\frac{\lvert A\rvert\mc X(\rho(0))\sqrt{n}}{q n}\sum_{x=0}^{ n}\sum_{y=x+1}^{n-1}\phi_\varepsilon\big(\frac{y}{n}\big)\Big\{ &\delta_x n^2D_{x,x+1}(\sqrt{f};\mu)\,+\, \frac{4}{\delta_x \varepsilon_0 n^2}\\
\,-\, &sgn(A)(\rho_{x+1}-\rho_x)\int w(x)w(x+1)fd\mu\Big\}
\end{split}
\end{equation}
where for any function $g:\Omega_n\to\bb R$, $D_{x,x+1}(g;\mu)$ is defined as
$$D_{x,x+1}(g;\mu)\,:=\, \sum_{\eta\in\Omega_n}[g(\eta^{x,x+1})\,-\,g(\eta)]^2 \mu(\eta),$$
and
$$sgn(A):= \frac{A}{\lvert A\rvert} \quad \text{if}\,\, A\neq 0, \quad \text{and}\,\, sgn(A):=0\quad \text{if}\,\, A=0.$$
Taking
$$\delta_x=\frac{qn}{8\lvert A\rvert\mc X(\rho(0))\sqrt{n}\sum_{y=x+1}^{n-1}\phi_\varepsilon\big(\frac{y}{n}\big)}\,,$$
the sum of the first two terms in \eqref{sumA} is less than or equal to
$$\frac{n^2}{8}D(\sqrt{f};\mu)\,+\,\frac{32\varepsilon A^2\big(\mc X(\rho(0))\big)^2\|\phi\|_\infty^2}{q^2\varepsilon_0}.$$
On the other hand, choosing
$$G(x)\,=\,-\mathds{1}_{\{1\leq x\leq \varepsilon n\}}\frac{\big(\rho_{x+1}\,-\,\rho_x\big)}{|\rho_{x+1}\,-\,\rho_x|}sgn(A)$$
and $\delta=\frac{1}{8}$ in Lemma \ref{main}, we have that
$$\sum_{x=1}^{n} G(x)\int w(x)w(x+1)fd\mu\,\leq\, \frac{n^2}{8} D(\sqrt{f};\mu)\,+\, C(\kappa)\big(H(f;\mu)\,+\, 8\big).$$
Therefore the third term in \eqref{sumA}
\begin{equation*}
\begin{split}
-&\frac{A\mc X(\rho(0))\sqrt{n}}{q n}\sum_{x=0}^{ n}\sum_{y=x+1}^{n-1}\phi_\varepsilon\big(\frac{y}{n}\big)(\rho_{x+1}-\rho_x)\int w(x)w(x+1)fd\mu\\
\leq\, &\frac{\lvert A\rvert\mc X(\rho(0))\sqrt{n}}{q n}\frac{\kappa n\|\phi\|_\infty}{n}\Big\{ \frac{1}{8} n^2 D(\sqrt{f};\mu)\,+\, C(\kappa)\big(H(f;\mu)\,+\, 8\big)\Big\}\\
\leq\, &\frac{A\mc X(\rho(0)) C(\kappa)\|\phi\|_\infty}{q\sqrt{n}} \big\{n^2D(\sqrt{f};\mu)\,+\, H(f;\mu)\,+\,8\}.
\end{split}
\end{equation*}
In summary, we have proved that
\begin{equation*}
\begin{split}
&A\sqrt{n}\Big\{\frac{\overline{\eta}(0)}{q}-\frac{1}{n}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w(x)\frac{\mc X(\rho(0))}{q}\Big\}f d\mu\,-\, \frac{n^2}{4}D(\sqrt{f},\mu)\\
\leq\,&\Big(\frac{A\mc X(\rho(0)) C(\kappa)\|\phi\|_\infty}{q\sqrt{n}}\,-\,\frac{1}{8}\Big) n^2D(\sqrt{f};\mu)\,+\,\frac{32\varepsilon A^2\big(\mc X(\rho(0))\big)^2\|\phi\|_\infty^2}{q^2\varepsilon_0}\\
&\,+\,\frac{A\mc X(\rho(0)) C(\kappa)\|\phi\|_\infty}{q\sqrt{n}} \big\{H(f;\mu)\,+\,8\}\,+\,\frac{A\sqrt{n}\|\phi'\|_\infty^2}{\varepsilon nq}.
\end{split}
\end{equation*}
Under the assumption $H(f;\mu)=o(\sqrt{n})$ and by the fact $\mc X(\rho(0))\leq \frac{1}{4}$, taking the limit $n\to\infty$, the first, third and fourth terms at the right hand side of the above inequality either vanish or become negative, so we finish the proof.
\end{proof}
\section{Tightness }\label{sec4}
\subsection{Tightness of $\{Z_t^\varepsilon: t\in [0,T]\}_{\varepsilon}$}
We start by introducing the concept of subgaussian variables. We say that a real-valued random variable $X$ is subgaussian of order $\sigma^2$, if for every $\theta\in \bb R$,
$$\log E[e^{\theta X}]\,\leq\,\frac{1}{2}\sigma^2\theta^2.$$
As a simple consequence of Hoeffding's Lemma, if $X$ is a Bernoulli random variable with mean $\rho$, then $X-\rho$ is subgaussian of order $1/4$. Then we can easily deduce that under the product measure $\mu_r^n$, there exists a constant $C=C(\varepsilon_0,M)$ such that
\begin{equation}\label{subw}
\frac{1}{\sqrt{n}}\phi_\varepsilon\big(\frac{x}{n}\big) w(x)\frac{\mc X(\rho_r(0))}{q_r} \quad \text{is subgaussian of order} \quad \frac{C}{n\varepsilon^2},
\end{equation}
for every $1\leq x\leq \varepsilon n$. We refer to Appendix F.3 of \cite{jm} for a more detailed discussion on subgaussian variables.
\begin{lemma}\label{Z}
Fix $0\leq s < t \leq T$. Then for every $\lambda\in(1,2)$, there exists a positive constant $C=C(\lambda,M,\varepsilon_0)$ such that
$$\bb E_{\mu^n}\Big[\big|Z_{t,n}^\varepsilon\,-\,Z_{s,n}^\varepsilon\big|^\lambda\Big]\,\leq\, C|t-s|^\lambda\varepsilon^{-\lambda/2}$$
for all $n\in\bb N$.
\end{lemma}
\begin{proof}
By Jensen's inequality,
\begin{equation*}
\begin{split}
&\bb E_{\mu^n}\Big[\Big\lvert\int _s^t \frac{1}{\sqrt{n}}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w_{rn^2}(x)\frac{\mc X(\rho_r(0))}{q_r}dr\Big\rvert^\lambda \Big]\\
\leq\,&(t-s)^{\lambda-1}\int_s^t\bb E_{\mu^n}\Big[\Big\lvert \frac{1}{\sqrt{n}}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w_{rn^2}(x)\frac{\mc X(\rho_r(0))}{q_r}\Big\rvert^\lambda \Big]dr.
\end{split}
\end{equation*}
Therefore it is enough to prove that there exists a constant $C>0$ independent of $n$ and $r$ such that
$$\bb E_{\mu^n}\Big[\Big\lvert \frac{1}{\sqrt{n}}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w_{rn^2}(x)\frac{\mc X(\rho_r(0))}{q_r}\Big\rvert^\lambda \Big]\,\leq\, C\varepsilon^{-\lambda/2}.$$
In view of Lemma \ref{pe}, since $\lambda>1$, we just need to show
\begin{equation}\label{prob1}
\bb P_{\mu^n}\Big[\Big|\frac{1}{\sqrt{n}}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w_{rn^2}(x)\frac{\mc X(\rho_r(0))}{q_r}\Big|>\Delta\Big]\,\leq\, \frac{C}{\varepsilon \Delta^2}
\end{equation}
for every $\Delta>0$.
In view of Proposition \ref{entpro}, the probability at the left hand side of \eqref{prob1} is less than or equal to
\begin{equation}\label{stepent}
\frac{H_n(r)\,+\,\log 2}{\log \Big(1\,+\,\mu_r^n\Big[\Big|\frac{1}{\sqrt{n}}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w_{rn^2}(x)\frac{\mc X(\rho_r(0))}{q_r}\Big|>\Delta\Big]^{-1}\Big)}.
\end{equation}
By Hoeffding's inequality and \eqref{subw}, since $\mu_r^n$ is a product measure, we have
\begin{equation}\label{steph}
\mu_r^n\Big[\Big|\frac{1}{\sqrt{n}}\sum_{x=1}^{n}\phi_\varepsilon\big(\frac{x}{n}\big) w_{rn^2}(x)\frac{\mc X(\rho_r(0))}{q_r}\Big|>\Delta\Big]\,\leq\,2\exp\Big( -\frac{\Delta^2}{C\sum_{x=1}^{\varepsilon n} n^{-1}\varepsilon^{-2}}\Big).
\end{equation}
By \eqref{stepent} and \eqref{steph}, the probability at the left hand side of \eqref{prob1} is bounded by
$$\frac{H_n(r)\,+\,\log 2}{\log\big(1+\frac{1}{2}\exp\{C\varepsilon \Delta^2\}\big)}.$$
Recall that it is proved in Theorem \ref{entropy} that $H_n(r)$ is of order $O(1)$ for every $0\leq r\leq T$. Using the elementary inequality
$$\log(1+\frac{e^x}{2})\geq \frac{x}{3}, \quad \forall \,\, x\geq 0,$$
we conclude the proof of \eqref{prob1}.
\end{proof}
\begin{lemma}\label{Zdif}
For any $0<\delta<\varepsilon<1$, $\lambda\in (1,2)$, there exists a positive constant $C=C(\varepsilon_0, \kappa, T, M, \lambda )$ such that
$$\bb E_{\mu^n}\Big[\big(Z_t^\delta-Z_t^\varepsilon\big)^\lambda\Big]\,\leq\,Ct^{\frac{\lambda}{2}}\varepsilon^{\frac{\lambda}{2}}$$
\end{lemma}
\begin{proof}
Notice that the expectation in the lemma is bounded by a constant $C(\lambda)$ times the sum of following four terms:
\begin{equation*}
\bb E_{\mu^n}\Big[\big|Z_t^\varepsilon\,-\,Z_{t,n}^\varepsilon\big|^\lambda\Big]\,+\,\bb E_{\mu^n}\Big[\big|Z_{t,n}^\delta\,-\,Z_t^\delta\big|^\lambda\Big]
\end{equation*}
\begin{equation*}
\bb E_{\mu^n}\Big[\big|\Gamma_n(t)\,-\,Z_{t,n}^\varepsilon\big|^\lambda\Big]\,+\,\bb E_{\mu^n}\Big[\big|Z_{t,n}^\delta\,-\,\Gamma_n(t)\big|^\lambda\Big]
\end{equation*}
The sum of two terms in the first line vanishes as $n\to\infty$ due to Theorem \ref{CLT}. The terms in the second line is bounded by $Ct^{\frac{\lambda}{2}}\varepsilon^{\frac{\lambda}{2}}$ by Lemma \ref{ZG}, since $\delta<\epsilon$ and $\lambda\geq 1$.
\end{proof}
We now show that the sequence $\{Z_t^\varepsilon: t\in[0,T]\}_{\varepsilon\in(0,1)}$ is tight with respect to the uniform topology in $C([0,T];\bb R)$ . By Kolmogorov-Centov criterion(see Problem 2.4.11 in \cite{ks}), we only need to show the following theorem by choosing $\lambda\in(\frac{4}{3},2)$.
\begin{theorem}\label{Ztbound}
For every $0\leq s<t\leq T$, there exists a constant $C=C(\varepsilon_0, \kappa, T, M, \lambda )$ such that
$$\sup_{\varepsilon\in(0,1)}\bb E_{\mu^n}\Big[\big|Z_t^\varepsilon\,-\,Z_s^\varepsilon\big|^\lambda\Big]\,\leq\,C (t-s)^{\frac{3\lambda}{4}}$$
for any $\lambda\in (1,2)$.
\end{theorem}
\begin{proof}
By Lemma \ref{Z} and Theorem \ref{CLT}, there exists a constant $C=C(\lambda)>0$ such that
\begin{equation}\label{Zlambda}
\bb E_{\mu^n}\Big[\big\lvert Z_t^\varepsilon\big\rvert^\lambda\Big]\,\leq\,C\,\bb E_{\mu^n}\Big[\big\lvert Z_t^\varepsilon\,-\,Z_{t,n}^\varepsilon\big\rvert^\lambda\Big] \,+\,C\,\bb E_{\mu^n}\Big[\big\lvert Z_{t,n}^\varepsilon\big\rvert^\lambda\Big]\,\leq\, Ct^\lambda\varepsilon^{-\lambda/2}.
\end{equation}
Fix $0<\varepsilon<1$. For $\delta<\varepsilon$, by \eqref{Zlambda} and Lemma \ref{Zdif},
\begin{equation*}
\begin{split}
\bb E_{\mu^n}\Big[\big\lvert Z_t^\delta\big\rvert^\lambda\Big]\,\leq\, &C(\lambda)\bb E_{\mu^n}\Big[\big\lvert Z_t^\varepsilon\big\rvert^\lambda\Big]\,+\, C(\lambda)\bb E_{\mu^n}\Big[\big\lvert Z_t^\varepsilon-Z_t^\delta\big\rvert^\lambda\Big]\\
\,\leq\, &Ct^\lambda\varepsilon^{-\lambda/2}\,+\, Ct^{\frac{\lambda}{2}}\varepsilon^{\frac{\lambda}{2}}.
\end{split}
\end{equation*}
If $t\geq \delta^2$, let $\varepsilon=\sqrt{t/T}$, then $\bb E_{\mu^n}\Big[\big\lvert Z_t^\delta\big\rvert^\lambda\Big]\,\leq\, C(\lambda,T)t^{\frac{3\lambda}{4}}$. If $t<\delta^2$, using \ref{Zlambda}, we still get the same upper bound of $\bb E_{\mu^n}\Big[\big\lvert Z_t^\delta\big\rvert^\lambda\Big]$.
Since we have proved a uniform bound on $H_n(t)\leq C$ for all $t\in[0,T]$ and this entropy bound is the only property of the initial measure we use to obtain the bound of $\bb E\Big[\big\lvert Z_t^\delta\big\rvert^\lambda\Big]$, shifting the time,
\begin{equation*}
\begin{split}
\bb E_{\mu^n}\Big[\big|Z_t^\varepsilon\,-\,Z_s^\varepsilon\big|^\lambda\Big]\,=\,&\bb E_{\eta_{sn^2}}\Big[\big|Z_{t-s}^\varepsilon\,-\,Z_0^\varepsilon\big|^\lambda\Big]\\
\,=\,&\bb E_{\eta_{sn^2}}\Big[\big|Z_{t-s}^\varepsilon\big|^\lambda\Big]\,\leq\,C(t-s)^{\frac{3\lambda}{4}}.
\end{split}
\end{equation*}
\end{proof}
\subsection{Tightness of $\{\Gamma_n(t):t\in[0,T]\}_n$}
The tightness of $\{\Gamma_n(t):t\in[0,T]\}_n$ with respect to the uniform topology in space $D([0,T]; \bb R)$ is much easier based on our previous estimates. Again We use Kolmogorov-Centov criterion and the following theorem to prove.
\begin{theorem}\label{tightTau}
Fix $0\leq s < t \leq T$. For any $\lambda\in(1,2)$, there exists a constant $C = C(\varepsilon_0, \kappa, T, M, \lambda )$ such that
$$\bb E_{\mu^n}\Big[\big|\Gamma_n(t)-\Gamma_n(s)\big|^\lambda\Big]\,\leq\, C\lvert t-s\rvert^{\frac{3\lambda}{4}} $$
holds for every $n\in\bb N$.
\end{theorem}
\begin{proof}
It is enough to prove the theorem only for the case $t-s<1$. The expectation in the theorem is bounded by a constant $C(\lambda)$ times the sum
\begin{equation*}
\bb E_{\mu^n}\Big[\big|Z_{t,n}^\varepsilon\,-\,Z_{s,n}^\varepsilon\,-\,\big(\Gamma_n(t)-\Gamma_n(s)\big)\big|^\lambda\Big]\,+\,\bb E_{\mu^n}\Big[\big|Z_{t,n}^\varepsilon\,-\,Z_{s,n}^\varepsilon\big|^\lambda\Big].
\end{equation*}
Choosing $\varepsilon=\sqrt{t-s}$ in Lemma \ref{ZG}, the first expectation is bounded by $C(t-s)^{\frac{3\lambda}{4}}$. The second expectation is also bounded by $C(t-s)^{\frac{3\lambda}{4}}$ by Theorem \ref{Ztbound}.
\end{proof}
\section{The limit}\label{sec5}
In this section we are going to prove Theorem \ref{limit} and \ref{time}.We start by showing that $X_t(f)$ is Gaussian assuming $X_0$ is a Gaussian random field.
Recall the definition of $\bb L_t$ given in \eqref{Lt}. Given $f\in C^\infty(\bb T)$ and $t\in[0,T]$, let $\{P_{s,t}f: 0\leq s\leq t\}$ be the solution of the backwards Fokker-Planck equation
\begin{equation*}
\begin{cases}
\partial_s v_s\,+\,\bb L_sv_s\,=\,0 \quad \text{for} \,\, s\leq t\\
v_t\,=\,f
\end{cases}
\end{equation*}
It follows from Theorem 5.1 of Chapter IV in \cite{lsu} that, if $f$ is smooth, then $P_{s,t}f(\cdot)$ is a smooth function on $\bb T$ and $s\mapsto P_{\cdot, t}f(u)$ is smooth on $[0,t]$ for any $u\in \bb T$.
\begin{theorem}\label{gaussian}
For every $f\in C^\infty( \bb T)$, conditional to $\mc F_s$ with $s<t$, the distribution of $X_t(f)$ is normal with mean $X_s(P_{s,t}f)$ and variance $\int_s^t\int\rho_r(u)(1-\rho_r(u))|\partial_u (P_{r,t}f)(u)|^2dudr$. In particular, assume that $X_0$ is a Gaussian random field, then $\{X_t(f): t\geq0\}$ is a Gaussian process.
\end{theorem}
The proof of this theorem is very similar to the proof presented in Section 4 of Chapter 11 in \cite{kl}. In the proof we need the following standard result on the local martingale.
\begin{proposition}\label{localm}[Proposition 5.5 in \cite{efgnt}]
Suppose $M_t$ is a local martingale with respect to a filtration $\mc F_t$ and
$$\bb E\big[ \sup_{0\leq s\leq t} \lvert M_s\rvert \big]\,<\,+\infty $$
for any $t\geq 0$, then $M_t$ is a martingale.
\end{proposition}
Now we prove Theorem \ref{gaussian}.
\begin{proof}[Proof of Theorem \ref{gaussian}]
Recall definition \eqref{mardef} of the martingale $M_t$. Fix $0\leq s\leq T$ and $f\in C^\infty(\bb T)$. We first claim that the process $\{Y_t^s(f): t\geq s\}$ defined by
\begin{equation}\label{Yt}
\begin{split}
Y_t^s(f)\,=\,\exp\Big\{&\int_s^t\int\rho_r(u)(1-\rho_r(u))|\partial_u f(u)|^2dudr\\
&+\,i \Big(X_t(f)\,-\,X_s(f)\,-\,\int_s^t\,X_r\big(\bb L_r f\big)dr\Big)\Big\}
\end{split}
\end{equation}
is a martingale. By Proposition 3.4 of Chapter IV in \cite{ry}, $Y_t^s(f)$ is a local martingale.
Since $\rho\in(\varepsilon_0,1-\varepsilon_0)$ and $f$ is smooth,
$$\sup_{s\leq r\leq t}\lvert M_r\rvert\,=\,\sup_{s\leq r\leq t}\int_s^t\int\rho_r(u)(1-\rho_r(u))|\partial_u f(u)|^2dudr\,<\,+\infty.$$
From this estimate and Proposition \ref{localm} we deduce that $Y_t^s(f)$ is a martingale.
Fix $T'\leq T$. We claim that the process $\{U_t: 0\leq t\leq T'\}$ defined by
$$U_t(f)\,:=\,\exp\Big\{\int_0^t\int \rho_r(u)(1-\rho_r(u))|\partial_u (P_{r,T'}f)(u)|^2dudr\,+\,i X_t(P_{t,T'}f)\Big\}$$
is also a martingale. To prove the claim, for any $0\leq t_1\leq t_2\leq T'$ and $n\in\bb N$, let
$$s_j\,=\,t_1\,+\,\frac{t_2-t_1}{n}j \quad \text{for every} \,\, 0\leq j\leq n.$$
Observe that
\begin{equation*}
\begin{split}
&\prod_{j=0}^{n-1} Y_{s_{j+1}}^{s_j}(P_{s_j,T'}f)\,=\,\exp\Big\{\sum_{j=0}^{n-1}\int_{s_j}^{s_{j+1}}\int \rho_r(u)(1-\rho_r(u))|\partial_u (P_{s_j,T'}f)(u)|^2dudr \\
&\,+\,i \sum_{j=0}^{n-1}\Big(X_{s_j+1}(P_{s_j,T'}f)\,-\,X_{s_j}(P_{s_j,T'}f)\,-\,\int_{s_j}^{s_j+1}X_r(\bb L_rP_{s_j,T'}f) dr \Big)\Big\}
\end{split}
\end{equation*}
Since $s\mapsto P_{\cdot,t}f$ is smooth, the first sum inside the exponential in the above equation converges to
$$\int_{t_1}^{t_2}\int \rho_r(u)(1-\rho_r(u))|\partial_u (P_{r,T'}f)(u)|^2dudr$$
as $n\to\infty$. On the other hand, the second sum can be written as
\begin{equation*}
\begin{split}
&X_{t_2}(P_{t_2-1/n,T'}f)\,-\,X_{t_1}(P_{t_1,T'}f)\\
\,+\,&\sum_{j=1}^{n+1}\Big(X_{s_j}(P_{s_{j-1},T'}f\,-\,P_{s_j,T'}f)\,-\,\int_{s_j}^{s_{j+1}}X_r(\bb L_rP_{s_j,T'}f) dr\Big)
\end{split}
\end{equation*}
Since $s\mapsto P_{\cdot,t}f$ is smooth, by definition of $P_{s,t}$,
$$P_{s_{j-1},T'}f\,-\,P_{s_j,T'}f\,=\,\frac{1}{n}\bb L_{s_j}P_{s_j,T'}f\,+\,o\big(\frac{1}{n}\big).$$
From that it is not hard to derive that the second sum converges to
$$X_{t_2}(P_{t_2,T'})-X_{t_1}(P_{t_1,T'}f)$$
almost surely as $n\to\infty$, and hence
\begin{equation*}
\begin{split}
&\lim_{n\to\infty}\prod_{j=0}^{n-1} Y_{s_{j+1}}^{s_j}(P_{s_j,T'}f)\\
=\,&\exp\Big\{\int_{t_1}^{t_2}\int \rho_r(u)(1-\rho_r(u))|\partial_u (P_{s_j,T'}f)(u)|^2dudr\\
&\quad\quad+\, i\Big( X_{t_2}(P_{t_2,T'})-X_{t_1}(P_{t_1,T'}f)\Big)\Big\}
\end{split}
\end{equation*}
which is exactly $\frac{U_{t_2}}{U_{t_1}}$. By the Dominated Convergence Theorem, since the complex exponential is bounded,
$$\lim_{n\to\infty}\bb E\Big[g\prod_{j=0}^{n-1} Y_{s_{j+1}}^{s_j}(P_{s_j,T'}f)\Big]\,=\,\bb E\Big[g\,\frac{U_{t_2}}{U_{t_1}}\Big]$$
for any bounded function $g:[0,T]\times\bb T\to\bb R$. Given a bounded and $\mc F_{t_1}$-measurable function $g$, taking conditional expectation with respect to $\mc F_{s_{n-1}}$, since $\{Y_t^s: t\geq s\}$ is a mean one martingale, we have
$$\bb E\Big[g\prod_{j=0}^{n-1} Y_{s_{j+1}}^{s_j}(P_{s_j,T'}f)\Big]\,=\, \bb E\Big[g\prod_{j=0}^{n-2} Y_{s_{j+1}}^{s_j}(P_{s_j,T'}f)\Big].$$
By induction, we can finally get
$$\bb E\Big[g\,\frac{U_{t_2}}{U_{t_1}}\Big]\,=\,\bb E[\,g\,]$$
for any bounded and $\mc F_{t_1}$-measurable function $g$. This implies that $\{U_t:t\geq 0\}$ is a martingale.
Since $\bb E[U_t\mid \mc F_s]=U_s$ for $s\leq t$, we have
\begin{equation*}
\begin{split}
&\bb E\Big[\exp\Big\{i X_t(P_{t,T'}f)\Big\}\mid \mc F_s\Big]\\
=\,&\exp\Big\{-\int_s^t\int\rho_r(u)(1-\rho_r(u))|\partial_u (P_{r,T'}f)(u)|^2dudr\,+\,i X_s(P_{s,T'}f)\Big\}
\end{split}
\end{equation*}
Choosing $T'=t$ and replacing $f$ by $\lambda f$, the identity above becomes
\begin{equation*}
\begin{split}
&\bb E\Big[\exp\Big\{i \lambda X_t(f)\Big\}\mid \mc F_s\Big]\\
=\,&\exp\Big\{-\lambda^2\int_s^t\int\rho_r(u)(1-\rho_r(u))|\partial_u (P_{r,t}f)(u)|^2dudr\,+\,i \lambda X_s(P_{s,t}f)\Big\}.
\end{split}
\end{equation*}
This means that, conditional to $\mc F_s$, $X_t(f)$ has normal distribution with mean $X_s(P_{s,t}f)$ and variance $\int_s^t\int\rho_r(u)(1-\rho_r(u))|\partial_u (P_{r,t}f)(u)|^2dudr$.
\end{proof}
A sufficient condition to guarantee that $X_0$ is a Gaussian random field is that the initial distribution $\mu_n$ is taken to be the Bernoulli product measure $\nu^n_{\rho_0(\cdot)}$. Under this condition, with the same proof of Proposition B.1 in \cite{efgnt}, one can show that $X_0^n$ converges in distribution to $X_0$, where $X_0$ is a mean zero Gaussian field of covariance given by
$$\bb E\big[X_0(f)X_0(g)\big]\,=\,\int_{\bb T}\mc X(\rho_0(u))f(u)g(u)du$$
for any $f,g\in C^{\infty}(\bb T)$. Fix $f\in C^\infty(\bb T)$ and choose $T'=t$ in the definition of $U_t$. Since $U_t(f)$ is a martingale, $\bb E[U_t(f)]=\bb E[U_0(f)]$ gives
\begin{equation*}
\begin{split}
\bb E\Big[\exp\{iX_t(f)\}\Big]\,=\,\exp\Big\{-&\frac{1}{2}\int_{\bb T}\mc X(\rho_0(u))\big( (P_{0,t}f)(u)\big)^2du\\
-\,&\int_0^t\int_{\bb T}\mc X(\rho_r(u))\big(\partial_u(P_{r,t}f)(u)\big)^2dudr\big)\Big\}.
\end{split}
\end{equation*}
Replacing $f$ by a linear combination of functions in the above identity and then applying the
Cr\'amer-Wold device , we can conclude that $X_t(f)$ is mean zero Gaussian with variance given by
$$\int_{\bb T}\mc X(\rho_0(u))\big( (P_{0,t}f)(u)\big)^2du\,+\,\int_0^t\int_{\bb T}2\mc X(\rho_r(u))\big(\partial_u(P_{r,t}f)(u)\big)^2dudr.$$
Now we give the proof of Theorem \ref{limit} and \ref{time}.
\begin{proof}[Proof of Theorem \ref{limit}]
We have shown in Theorem \ref{Ztbound} that the sequence the process $\{Z_t^\varepsilon: t\in [0,T]\}_{\varepsilon>0}$ is tight with respect to the uniform topology of $C([0,T]; \bb R)$. Suppose that $\{Z_t: t\in [0,T]\}$ is one of the limits. Fix any $t\in[0,T]$. By Lemma \ref{Zdif}, $\{Z_t^\varepsilon\}_{\varepsilon>0}$ is a Cauchy sequence in $L^\lambda(\bb P)$ for any $\lambda\in(1,2)$. Therefore $Z_t^\varepsilon$ converges to some limit as $\varepsilon\to0$ and this limit has to be $Z_t$. Since the law of the a continuous process is determined by its finite dimensional distributions, this proves the uniqueness of the limit $\{Z_t: t\in [0,T]\}$ in $C([0,T];\bb R)$.
As a consequence of Theorem \ref{gaussian}, $Z_t^\varepsilon$ is the integral of some Gaussian random variables and hence it is Gaussian. Since $Z_t^\varepsilon$ converges to $Z_t$ in $L^\lambda(\bb P)$ with $\lambda\in(1,2)$, $Z_t $ is Gaussian as well.
\end{proof}
\begin{proof}[Proof of Theorem \ref{time}]
The tightness of $\{\Gamma_n(t):t\in[0,T]\}_{n\in\bb N}$ with respect to the uniform topology in space $D([0,T]; \bb R)$ was proved in Theorem \ref{tightTau}. Let $\{\Gamma_t:t\in[0,T]\}$ be a limit point of $\{\Gamma_n(t):t\in[0,T]\}_n$. Without loss of generality, we can assume that $\{\Gamma(t):t\in[0,T]\}$ is defined in the same probability space on which the process $\{X_t: t\in[0,T]\}$ is defined. For any $\lambda\in (1,2)$, since $L^\lambda$ upper bounds are preserved by convergence in distribution, by Lemma \ref{ZG} and triangle inequality,
$$\bb E\Big[\big(\Gamma(t)\,-\,Z_t^\varepsilon\big)^\lambda\Big]\,\leq\, C t^{\frac{\lambda}{2}}\varepsilon^{\frac{\varepsilon}{2}}.$$
Taking $\varepsilon\to 0$, it follows that $\{\Gamma(t): t\in [0,T]\}$ has the same finite dimensional distributions as those of $\{Z_t: t\in [0,T]\}$ and we finish the proof.
\end{proof}
\smallskip\noindent{\bf Acknowledgments.} L.R.F. was partially supported by CNPq grant 307884/2019-8, and FAPESP grant 2017/10555-0. T. X. would like to thank the financial support of FAPESP Grant No.2019/02226-2.
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\section{Introduction}
One of the main challenges of modern science is the understanding of our Universe, and the nature of the dark sector which dominates its energy content in the concordance model of cosmology. Since the discovery of the accelerated expansion of our Universe \citep{1998Natur.391...51P,1998AJ....116.1009R}, cosmologists have for decades searched for new probes to understand its properties and successfully confronted the $\Lambda$CDM model to them \citep{planck2018cosmological, Abbott:2018wog, 2018ApJ...859..101S, riess2019large, wong2019holicow, freedman2019CCHP,alam2021completed}. All of these probes have in common that they consist in information coming from light, which acts as messengers for us to observe the Universe. Only since very recently the use of other messengers such as gravitational waves are starting to be used in order to get complementary information \citep{2005ApJ...629...15H,2016JCAP...10..006C}. However, photons still remain today the most used messenger to get cosmological information, and one can ask how the properties of light modify our perception of the Universe.
Because photons propagate on null geodesics, they are sensitive to inhomogeneities in the matter density field. This leads to mainly two effects which have received a lot of attention over the years and are accounted for to interpret observational data. First, gravitational lensing \citep{schneider1992gravitational, bartelmann2001weak} that modifies the apparent position of sources but also alter their observed properties (shape, luminosity) with respect to the case where photons would propagate in a homogeneous Friedmann-Lemaître-Robertson-Walker (FLRW) universe. Second, the position of the emission lines of sources are shifted due to their own motion, which in return lead to an error when estimating distances with a fiducial cosmology: this is called Redshift-Space Distortions \citep[RSD,][]{kaiser1987clustering,hamilton1992measuring}. This means that the image we have of our Universe is
altered due to gravitational lensing and RSD. However, these effects also leave distinct imprints on various cosmological observables, which in turn can help us infer the properties of the Universe. Various current or future missions such as Euclid \citep{laureijs2011euclid} or DESI \citep{desi2016desi} aim at studying these effects through the shape of distant galaxies or their observed spatial distribution.
Due to the increasing quality of data, it is becoming necessary to model the mapping from a statistically homogeneous and isotropic universe to the observed one more accurately. This was done for example regarding the galaxy number counts, accounting for all the effects at first order in metric perturbations within linear theory \citep{yoo2009new,challinor2011linear,bonvin2011what}. However, theoretical prescriptions are limited since they usually cannot properly model the non-linear regime of structure formation. Instead, the use of numerical simulations becomes mandatory to fully understand the clustering of matter, from linear to non-linear scales.
The use of numerical simulations, and more precisely dark-matter (DM) $N$-body simulations \citep{hockney1981computer} have been widely used in cosmology to study large-scale structures beyond analytical methods. A lot of work has been done to develop optimised $N$-body codes \citep{kravtsov1997ART,couchman1991mesh, knebe2001MLAPM, teyssier2002cosmological, o'shea2004introducing,springel2005cosmological,bryan2014enzo,aubert2015emma,garrison2021abacus} which allow us to have access to good spatial resolution (small scales) and beat cosmic variance (large scales), while limiting shot noise. Cosmological $N$-body simulations run on super-computers and use HPC techniques, so that today the largest numerical simulations reach trillions of DM particles \citep{potter2017pkdgrav3,heitman2019outerrim,ishiyama2021uchuu}.
$N$-body simulations compute the evolution of the matter density field from initial perturbations at high redshift up to now. However, it is not sufficient to accurately model what we actually observe as one still needs to construct past light-cones for some given observers and compute the trajectory of light. This was usually done through the multiple lens formalism \citep{blandford1986fermat}, and ray-tracing post-processing tools have been developed mostly to model the effect of gravitational lensing
\citep{fluke1999raybundle,jain2000raytracing, fosalba2008onion, hilbert2009raytracing, giocoli2015disentangling, fabbian2018cmb,gouin2019weak}. However, these methods often compute the lens equation on 2D planes and rely on multiple approximations. Alternatively, several authors have developed ray-tracing algorithms which instead directly compute the geodesic equations in 3D \citep{killedar2012gravitational,barreira2016ray}, however the applications are still restricted to gravitational lensing.
Ideally, ray-tracing should be general enough to accurately reconstruct the past light-cone of an observer and produce a wide range of cosmological observables. This approach has been gaining more and more attention recently \citep{reverdy2014propagation,borzyszkowski2017liger, breton2019imprints,adamek2019bias,lepori2020weak,breton2020theoretical} as it allows for an in-depth study of subtle effects which are currently neglected but could play an important role in future surveys. As evidence of this, \cite{breton2019imprints} have for the first time used ray-tracing to find the null geodesic connecting an observer to various sources, in order to estimate the impact of relativistic effects on the clustering of haloes. They found that the dipole of the correlation function could be used to probe their gravitational potential in next-generation surveys \citep{saga2021detectability} and therefore test the nature of gravity \citep{bonvin2018testing}.
The goal of this paper is to present the \textsc{Magrathea-Pathfinder}{} framework, initially developed in \cite{reverdy2014propagation} and further developed in several directions since then. It aims at modelling the observed Universe as accurately as possible. To do so, we have developed ray-tracing techniques which allow us to construct various cosmological observables beyond standard assumptions. In this article we will briefly present the basics of \textsc{Magrathea-Pathfinder}{} \citep{reverdy2014propagation} and focus on recent developments and code validation.
In \Cref{sec:methods} we review the main features of our ray-tracing code. In \Cref{sec:tests} we perform convergence tests and conclude in \Cref{sec:conclusion}.
\section{Numerical methods}
\label{sec:methods}
\textsc{Magrathea-Pathfinder}{} is a post-processing numerical framework\footnote{Available at \\ \url{https://github.com/vreverdy/magrathea-pathfinder}} to propagate photons throughout light-cones produced by $N$-body astrophysics simulations. The framework is built on top of \textsc{Magrathea}{} (Multi-processor Adaptive Grid Refinement Analysis for THEoretical Astrophysics), a high-performance library developed to provide highly-optimized building blocks for the construction of AMR-based astrophysics applications \citep{reverdy2014propagation}. More specifically, the ray-tracing framework leverage \textsc{Magrathea}{}'s generic $N$-dimensional hyperoctree abstraction and the associated numerical methods to simplify the handling of the adaptive mesh refinement while ensuring the highest level of performance. Internally the AMR structure is flattened, each cell being associated with a unique binary index that encodes information about its exact location in the tree. All canonical hyperoctree operations such as finding parents, children or neighbour cells as well as calculating inter-cell distances are implemented in terms of bit manipulation operations. Each cell is also associated with a data tuple to store the physics quantities attached to a particular location. One of the key features of \textsc{Magrathea}{} is the heavy use of C\texttt{++} template metaprogramming approaches to guide the compiler throughout the optimization process and ensure the highest level of performance regarding the manipulation of indices and physics data \citep{reverdy2015edsl}. In that sense, \textsc{Magrathea}{} acts as an active library running at compile-time to pre-process \textsc{Magrathea-Pathfinder}{}'s code. The exact implementation of the hyperoctree and low-level numerical algorithms can be found in \cite{reverdy2014propagation} and will be presented in more details in an upcoming paper focusing on the \textsc{Magrathea}{} library. For the present paper, we focus on the ray-tracing algorithms in \textsc{Magrathea-Pathfinder}{} made possible by the above-mentioned library.
\subsection{From simulations to \textsc{Magrathea}{} octree}
\label{subsec:sim_to_magrathea}
As ray-tracing simulations are performed as an independent and subsequent phase of dynamic cosmological simulations, it is possible to leverage the particular geometry of the problem to optimize parallelization schemes and minimize inter-node communications. In practice, the very first step consists in generating a 3D light-cone containing all the information relative to the gravitational field and converting it into \textsc{Magrathea}{}'s 3D-octree data structure. Usually, the light-cone is built from a $N$-body simulation. If the simulation uses a Particle-Mesh (PM) method, then the identification between the cells of the simulations and that of \textsc{Magrathea}{} can be trivially achieved through the conversion of cartesian positions into \textsc{Magrathea}{} indices. The same applies to AMR-based simulation. However, when cosmological simulations rely on another method to compute the gravitational field one has to first interpolate the gravitational information available at particle locations onto a fixed or adaptive grid that can then be processed by \textsc{Magrathea}{}'s hyperoctree engine.
Once the light-cone is converted into \textsc{Magrathea}{}'s octree format, geometrical inconsistencies are checked to ensure that the ray-tracer can work on clean data. One typical problem that may arise during concentric shell extraction happening at the cosmological simulation level is the production of sparse AMR cells at the boundary of two shells. In this case, the geometrical checker adds cells interpolated from a coarser level to build a full tree with either zero or eight children per cell that preserves the original gravitational information. The operation is repeated at every level, from the coarser to the most refined one.
Because the light-cone of high-resolution simulations can reach several terabytes of data, it is often not possible to load it on a single node. In this case, a conic domain decomposition of the light-cone with overlaps is performed which allows to propagate photons almost without the need of cross-node communications for the most part of the ray-tracing phase. On top of the conic domain, every computational node gets a copy of a small spherical region centered on the observer to ensure proper independent propagation at very low redshift. The hybrid MPI/C++ thread parallelization approach of \textsc{Magrathea}{} allows to maximize the size of geometrical subdomains, minimize the total size of overlaps, while still ensuring a maximal exploitation of computational resources with many threads working at the same time on the same shared memory, each one of them taking care of a particular photon.
Once the octree containing all the relevant gravitational information is built, checked, and distributed across computational nodes, the main ray-tracing phase can begin.
\subsection{Geodesic integration and light propagation}
\label{subsec:geodesic_integration}
\textsc{Magrathea-Pathfinder}{} propagates photons on the null geodesics of the weakly perturbed FLRW metric in Newtonian gauge
\begin{equation}
\mathrm{d} s^2 = a^2(\eta)\left[-\left(1+2\frac{\Phi}{c^2}\right)c^2\mathrm{d}\eta^2 + \left(1-2\frac{\Phi}{c^2}\right)\mathrm{d}\vect{x}^2\right] ,
\label{eq:perturbed_FLRW_metric}
\end{equation}
where $\eta$ and $\bm{x}$ are respectively the conformal time and comoving coordinates, $a(\eta)$ is the scale factor, $\Phi$ the gravitational potential and $c$ the speed of light. The null geodesic equations are
\begin{eqnarray}
\label{eq:geodesic_equation1}
\frac{\textrm{d}^2 \eta}{\textrm{d}\lambda^{2}} &=& -\frac{2a'}{a}\frac{\textrm{d}\eta}{\textrm{d}\lambda}\frac{\textrm{d}\eta}{\textrm{d}\lambda} - \frac{2}{c^2}\frac{\textrm{d}\Phi}{\textrm{d}\lambda}\frac{\textrm{d}\eta}{\textrm{d}\lambda} + 2\frac{\partial\Phi}{\partial\eta}\left(\frac{\textrm{d}\eta}{\textrm{d}\lambda}\right)^2, \\
\label{eq:geodesic_equation2}
\frac{\textrm{d}^2 x^i}{\textrm{d}\lambda^2} &=& -\frac{2a'}{a}\frac{\textrm{d}\eta}{\textrm{d}\lambda}\frac{\textrm{d}x^i}{\textrm{d}\lambda} + \frac{2}{c^2}\frac{\textrm{d}\Phi}{\textrm{d}\lambda}\frac{\textrm{d}x^i}{d\lambda} - 2\frac{\partial\Phi}{\partial x^i}\left(\frac{\textrm{d}\eta}{\textrm{d}\lambda}\right)^2,
\end{eqnarray}
where a prime denotes a derivative with respect to conformal time, and $\lambda$ is an affine parameter.
We perform backward ray-tracing, meaning that we start the propagation of the photons at the observer today towards the past. The geodesic equations are solved using a 4th-order Runge-Kutta integrator (RK4) with $\mathcal{N}$ steps per AMR cell ($\mathcal{N} = 4$ in common settings), where the photons are initialised using $k^\nu k_\nu = 0$ (with $k^\nu = \mathrm{d} x^\nu/\mathrm{d}\lambda$) given the initial direction of the photon $k^i$, and $k^0 = 1$, meaning that at the observer conformal time and affine parameter coincides.
To compute \cref{eq:geodesic_equation1,eq:geodesic_equation2}, we have access to $\eta$, $x^i$, $k^0$ and $k^i$ that are given by the integrator, $a(\eta)$ and $a'(\eta)$ by external pre-computed tables in the fiducial cosmology of the simulation given $\eta$, and $\partial\Phi/\partial x^i$ by the simulation itself. The last subtlety lies in the use of $\mathrm{d}\Phi/\mathrm{d}\lambda$ or $\partial\Phi/\partial\eta$, knowing that these terms are simply related by
\begin{equation}
\label{eq:dphidl_1}
\frac{\textrm{d}\Phi}{\textrm{d}\lambda} = \frac{\partial\Phi}{\partial x}\frac{\textrm{d}x}{\textrm{d}\lambda}+\frac{\partial\Phi}{\partial y}\frac{\textrm{d}y}{\textrm{d}\lambda}+\frac{\partial\Phi}{\partial z}\frac{\textrm{d}z}{\textrm{d}\lambda}+\frac{\partial\Phi}{\partial \eta}\frac{\textrm{d}\eta}{\textrm{d}\lambda}.
\end{equation}
Usually, $N$-body simulations do not provide $\partial\Phi/\partial\eta$ and hence it is easier to rewrite the geodesic equations in terms of $\mathrm{d}\Phi/\mathrm{d}\lambda$ only since it can be estimated by differentiating between two steps of the integration as
\begin{equation}
\label{eq:dphidl_2}
\frac{\textrm{d}\Phi^i}{\textrm{d}\lambda} = \frac{\Phi^{i}-\Phi^{i-1}}{\lambda^{i} - \lambda^{i-1}},
\end{equation}
where the superscripts refer to the integration step. The main drawback of this method is that it strongly depends on the procedure used to build the light-cone. For example, if one uses the onion-shell method \citep{fosalba2008onion}, then some artefacts will appear on $\mathrm{d}\Phi/\mathrm{d}\lambda$ at the crossing between shells. It is important to note that these subtleties are only important when studying very small effects (such as ISW or time delay).
Ideally, $\partial\Phi/\partial\eta$ should be provided by the simulation. It is the case for example with the RayGal simulations (Rasera et al. in prep) where the authors use a double-layer strategy, meaning that two light-cones slightly shifted in time are stored. In this case, the time derivative of the potential can be straightforwardly computed for every cell using
\begin{equation}
\frac{\partial\Phi}{\partial \eta} = \frac{\Phi_2 - \Phi_1}{a_2 - a_1}\frac{\textrm{d}a}{\textrm{d}\eta}.
\end{equation}
where the subscripts denote which light-cone is used. The main advantage of this method is that it is non-perturbative and therefore accurate even at non-linear scales. Last, we note that in practice $\partial\Phi/\partial\eta$ could also be computed from the density and velocity fields \citep{cai2010fullsky}.
All of the components needed to compute the geodesic equations are available at the location of cells. This means that we need to interpolate them at the photon position at each step (and each sub-step in the RK4 integrator), while accounting for the AMR structure of the light-cone. We have implemented three different interpolation schemes, namely the Nearest-Grid Point (NGP), Cloud-in-Cell (CIC) and Triangular-Shaped Cloud (TSC) interpolations, where the first two were already present in \cite{reverdy2014propagation}. For consistency, the interpolation scheme must be the same as the one used to produce the gravity grid (see \cref{subsec:sim_to_magrathea}). The weighting functions associated to NGP, CIC and TSC are
\begin{equation}
W\e{NGP}(r_i) =
\left\lbrace
\begin{array}{lcl}
1 & \rm{for} & |r_i| < 0.5, \\
0& \rm{otherwise,} &
\end{array}\right.
\end{equation}
\begin{equation}
W\e{CIC}(r_i) =
\left\lbrace
\begin{array}{lcl}
1 - |r_i| & \rm{for} & |r_i| < 0.5, \\
0& \rm{otherwise,} &
\end{array}\right.
\end{equation}
\begin{equation}
W\e{TSC}(r_i) =
\left\lbrace
\begin{array}{lcc}
0.75 - r_i^2 & \rm{for} & |r_i| < 0.5, \\
\left(1.5-|r_i|\right)^2/2 & \rm{for} & 0.5 < |r_i| < 1.5, \\
0& \rm{otherwise,} &
\end{array}\right.
\end{equation}
where $\bm{r}$ is the separation between a cell and the location at which we interpolate, normalised by the cell size.
For NGP, the interpolation is trivial as we take the gravity information from the most refined cell which contain the photon. For CIC and TSC, the interpolation procedures go as follow:
\begin{enumerate}
\item Estimate the refinement level of the most refined cell containing the photon
\item Check if there are 8 (27) neighbouring cells to perform the CIC (TSC) interpolation in three dimensions.
\item If all the neighbouring cells exist in the octree, compute the interpolation.
\item If at least one of the neighbouring cells does not exist in the octree, repeat the full procedure at a coarser level.
\item If there are not enough neighbours even at coarse level (meaning that we reach the edges of the numerical light-cone), the integration is stopped.
\end{enumerate}
We illustrate this procedure in \cref{fig:tsc_scheme}.
\begin{figure}
\includegraphics[width=0.496\columnwidth]{Figures/ray_amr_tsc_1.pdf}
\includegraphics[width=0.496\columnwidth]{Figures/ray_amr_tsc_2.pdf}
\includegraphics[width=0.496\columnwidth]{Figures/ray_amr_tsc_3.pdf}
\includegraphics[width=0.496\columnwidth]{Figures/ray_amr_tsc_4.pdf}
\includegraphics[width=0.496\columnwidth]{Figures/ray_amr_tsc_5.pdf}
\includegraphics[width=0.496\columnwidth]{Figures/ray_amr_tsc_6.pdf}
\caption{Illustration of the TSC interpolation scheme at a given location in an AMR grid in 2D. Red points refer to some locations on the grid (which can be for example the position of a photon during its propagation), the cells used to perform the interpolation are highlighted in blue. The interpolation is done at fixed level, meaning that when the interpolation level is set, we do not use the information from coarser or finer cells.}
\label{fig:tsc_scheme}
\end{figure}
On the top left panel, we see that there are the 9 neighbouring cells to perform the 2D TSC interpolation. In this case the gravity information is easily interpolated at the photon location to compute the geodesic equations. The photon location is updated in the top right panel, where there are also enough neighbours at coarse level. Here, even if the top-right cell also contain more refined cells, they do not contribute to the interpolation and we rather use the coarser cell. In the middle left panel, the photon is in a more refined cell than previously. The code tries to interpolate at this finer level, but cannot find the associated neighbouring cells. It therefore performs the interpolation at a coarser level, where all the neighbours exist in the grid. In the middle right panel, the photon is in a new cell at the same level as previously, the only difference being that now there are enough neighbouring cells to perform the interpolation at this finer level. The bottom panels shows a similar behaviour as previously but with more refinement. The CIC procedure is similar to that of TSC, except for the fact that in this case we need 4 neighbouring cells in two dimensions. A nice feature about TSC compared to CIC is that for the latter, the neighbours depend on the position of the photon within a cell, while for the former it is not the case. We take advantage of this by keeping in memory all the neighbours of a given photon at each step, and if the next step of the photon is in the same cell and the TSC interpolation performed at the same level, then we do not have to search again for the neighbouring cells in the octree and therefore gain in performance.
It shall be noted that this interpolation procedure does not prevent discontinuities at the crossing between AMR levels, however we expect this effect to be very faint, especially for the commonly used CIC and TSC interpolation schemes as we will see in \cref{subsec:test_trajectory}.
At each step of the propagation, we keep in memory the step number, scale factor, conformal time, comoving position, affine parameter, redshift, wavevector $k^\nu$, level of the most refined cell, density, gravitational potential and its derivatives with respect to conformal time, affine parameter and comoving coordinates (that is, the force). It is also possible to estimate these quantities along the propagation of an unperturbed light ray (this is the so-called `Born approximation') by setting the gravity to zero in the geodesic equations.
Now that we have seen the methodology to propagate photons on null geodesics within the 3D AMR structure of the light-cone, we shall turn to the implementation of gravitational lensing and simulated catalogues.
\subsection{Distortion matrix and gravitational lensing}
One direct consequence of light propagation in an inhomogeneous universe is the modification of the apparent position of a source, as well as the modification of its properties (shape and observed luminosity). To derive these properties, we start from the lens equation
\begin{equation}
\bm{\beta} = \bm{\theta} - \bm{\alpha},
\end{equation}
where $\bm{\beta}$ and $\bm{\theta}$ are the comoving and observed angular position on the sky of a source, and $\bm{\alpha}$ the deflection angle. We note that $\bm{\theta}$ is the photon direction at the observer which is directly given by $k^i$ at this location.
The relevant information for weak gravitational lensing is encoded in the Jacobian matrix $\bm{\mathcal{A}}$ which describes the mapping from $\mathrm{d}^2\bm{\beta}$ to $\mathrm{d}^2\bm{\theta}$. This gives
\begin{equation}
\label{eq:jacobian_matrix}
\bm{\mathcal{A}} = \frac{\mathrm{d}^2\bm{\beta}}{\mathrm{d}^2\bm{\theta}} =
\begin{pmatrix}
1 - \kappa - \gamma_1 & - \gamma_2 + \omega \\
- \gamma_2 - \omega & 1 - \kappa + \gamma_1
\end{pmatrix},
\end{equation}
where $\kappa$, $\gamma = \gamma_1 + i\gamma_2$ and $\omega$ are respectively the convergence, shear and rotation of an image. The magnification $\mu$ is the change in observed flux and size of an image (due to the conservation of surface brightness), and is defined as the inverse determinant of $\bm{\mathcal{A}}$.
Usually, ray-tracing codes aim at computing $\kappa$ and $\gamma$ (and sometimes $\mu$). However, these often make use of several approximations such as Born, plane-parallel and multiple lens. We will now see how to implement the computation of weak-lensing quantities using the 3D propagation on null geodesics described in \cref{subsec:geodesic_integration}. To do so, we use two methods which we refer to as `infinitesimal' and `finite' beams.
\subsubsection{Infinitesimal light beams}
\label{subsubsec:infinitesimal_beams}
First, we consider the usual definition of the distortion matrix, which describes the behaviour of infinitesimal light beams. Formally, \cref{eq:jacobian_matrix} is given by
\begin{equation}
\label{eq:infinitesimal_matrix}
\mathcal{A}_{ab} = \delta_{ab} - \frac{2}{c^2}\int^{\chi\e{s}}_0 \mathrm{d} \chi \; \frac{(\chi\e{s}-\chi)\chi}{\chi\e{s}}\nabla_a\nabla_b\Phi[\eta(\chi),\vect{x}(\chi)]
\ ,
\end{equation}
where the subscripts refer to the angular coordinates, $\delta_{ab}$ is the Kronecker delta, $\chi$ is the comoving distance of the photon during its propagation and $\chi\e{s}$ is the distance at the source. Note that here the derivatives are performed along angular spherical coordinates. Because spherical derivatives are not straightforward to compute, we find it easier to first perform derivatives along the 3D Cartesian coordinates, and then rewrite it in term of spherical ones. At any location on the light-cone, we have access to the gravitational field $-\nabla_i\Phi(\bm{x}) \equiv F_i(\bm{x}) = \{F_x(\bm{x}), F_y(\bm{x}), F_z(\bm{x})\}$. It is possible to compute its Cartesian derivatives by differentiating as
\begin{equation}
\label{eq:laplacian_finite_difference}
-\nabla_i\nabla_j \Phi(\bm{x}) = \frac{F_j(\bm{x} + h \bm{e}_i)-F_j(\bm{x} - h \bm{e}_i)}{2h},
\end{equation}
where $\bm{e}_i = \{\bm{e}_x, \bm{e}_y, \bm{e}_z\}$ is a unit vector, and $h$ is the derivation step. We found that choosing a derivation step equal to the size of the most refined AMR cell (and therefore highest AMR level) the photon is in, that is $h = 2^{-\rm{level}}$ is the optimal choice to compute these derivatives (see also \cref{subsec:test_derivation_infinitesimal}). From a numerical perspective, $\nabla_i\nabla_j \neq \nabla_j\nabla_i$ when $i \neq j$. However, the difference is so small that we consider them equal so as to avoid the costly computation of all the possible permutations. The last step is to go from Cartesian to spherical derivatives. Our method is similar to that of \cite{barreira2016ray}, except that we do not resort to the Born approximation and we compute ray-tracing as post-processing. The components of the Laplacian in \cref{eq:infinitesimal_matrix} are given by
\begin{align}
\nabla_1\nabla_1\Phi &= \sin^2\varphi~ \nabla_x\nabla_x\Phi + \cos^2\varphi~\nabla_y\nabla_y\Phi - \sin 2\varphi ~\nabla_x\nabla_y\Phi, \\
\nabla_2\nabla_2\Phi &= \cos^2\varphi \cos^2\vartheta~\nabla_x\nabla_x\Phi + \sin^2\varphi\cos^2\vartheta~\nabla_y\nabla_y\Phi
\nonumber\\&\quad
+ \sin^2\vartheta~\nabla_z\nabla_z\Phi + \sin 2\varphi\cos^2\vartheta~\nabla_x\nabla_y\Phi
\nonumber\\&\quad
-\sin\varphi\sin 2\vartheta~\nabla_y\nabla_z\Phi - \cos\varphi\sin 2\vartheta ~\nabla_x\nabla_z\Phi, \\
\nabla_1\nabla_2\Phi &= \cos\vartheta\cos\varphi\sin\varphi~(\nabla_y\nabla_y\Phi-\nabla_x\nabla_x\Phi)
\nonumber\\&\quad
+ (\cos^2\varphi-\sin^2\varphi)\cos\vartheta~\nabla_x\nabla_y\Phi
\nonumber\\&\quad
+ \sin\varphi\sin\vartheta~\nabla_x\nabla_z\Phi-\cos\varphi\sin\vartheta~\nabla_y\nabla_z\Phi.
\end{align}
where $(\varphi, \vartheta)$ is the angular position of the photon in spherical coordinates at each step. Last, we use $\nabla_1\nabla_2\Phi = \nabla_2\nabla_1\Phi$, meaning that we consider that there is no rotation of the image.
We note that the option to use the Born approximation has been implemented by using this method along an FLRW trajectory.
\subsubsection{Finite beams}
\label{subsubsec:finite_beams}
In fact sources are not infinitesimal, but they are rather extended. The usual weak-lensing formalism is, therefore, an approximation of the more accurate finite-beam formalism \citep{fleury2017weak,fleury2019cosmic,fleury2019weak}. In this case we can compute the lensing distortion matrix by launching a beam composed of several close-by light rays which are all integrated on null geodesics independently. The idea to consider several rays to characterise a light beam is similar to the `ray-bundle method' proposed by \cite{fluke1999raybundle, fluke2011shape}. First, we launch a reference photon in the direction in which we want to compute the lensing matrix. This photon is used as a reference to know where to stop the beam. For example, we might want to know the lensing quantities at some parameter $p_0$ where $p = \{a, \eta, \chi, z, \lambda\}$ (see \cref{subsec:geodesic_integration}), but we still have the choice to stop the beam light rays at some other parameter $\tilde{p}_0 = \tilde{p}(p_0)$ for the reference photon. Note that all of the parameters are equivalent in an FLRW universe, but this is no longer true when accounting for inhomogeneities.
To compute the distortion matrix, we therefore need to know $p_0$, $\varepsilon$ the beam semi-aperture, and $\bm{\theta} = (\theta_1, \theta_2)$ the photon direction at the observer. In Cartesian coordinates, the direction of the target is $\hat{\bm{r}} = (\cos\theta_1\sin\theta_2, \sin\theta_1\sin\theta_2, \cos\theta_2)$ where a hat denotes a unit vector. We can define a screen perpendicular to this direction with two orthogonal vectors $\bm{e}_1 = (-\sin\theta_1, \cos\theta_1, 0)$ and $\bm{e}_2 = (-\cos\theta_1\cos\theta_2, -\sin\theta_1\cos\theta_2, \sin\theta_2)$. Now we can launch four rays denoted A, B, C and D (see also Fig.5 in \citealt{breton2020theoretical}), with initial directions
\begin{eqnarray}
\hat{\bm{r}}\e{A} &=& \hat{\bm{r}} + \tan(\varepsilon)~\bm{e}_1\cdot\bm{u}, \\
\hat{\bm{r}}\e{B} &=& \hat{\bm{r}} - \tan(\varepsilon)~\bm{e}_1\cdot\bm{u}, \\
\hat{\bm{r}}\e{C} &=& \hat{\bm{r}} - \tan(\varepsilon)~\bm{e}_2\cdot\bm{u}, \\
\hat{\bm{r}}\e{D} &=& \hat{\bm{r}} + \tan(\varepsilon)~\bm{e}_2\cdot\bm{u},
\end{eqnarray}
where $\bm{u} = (\bm{e}_x, \bm{e}_y, \bm{e}_z)$. Each ray is propagated on the light-cone until $\tilde{p}_0$ so that their final position is given by $\vect{\xi}\e{A}$, $\vect{\xi}\e{B}$, $\vect{\xi}\e{C}$ and $\vect{\xi}\e{D}$. To compute the lensing distortion matrix, we differentiate between the positions of the light rays of the beam. Taking advantage of the fact that the beam is supposed to be small, we can write $\Delta\vect{\beta}=\Delta\vect{\xi}/\chi_0=\hat{\vect{\mathcal{A}}}\,\Delta\vect{\theta}$, where $\chi_0$ is the comoving distance of the reference ray at $p_0$ and $\hat{\vect{\mathcal{A}}}$ the finite-beam distortion matrix.
A last subtlety is the choice of screen onto which we compute the finite differences. From \cref{eq:jacobian_matrix}, a natural choice for this screen is the one orthogonal to $\bm{\beta} = (\beta_1, \beta_2)$, defined as $\tilde{\bm{e}}_1 = (-\sin\beta_1, \cos\beta_1, 0)$ and $\tilde{\bm{e}}_2 = (-\cos\beta_1\cos\beta_2, -\sin\beta_1\cos\beta_2, \sin\beta_2)$. Alternatively, it is possible to use the more physically motivated Sachs screen, which is orthogonal to the central (reference) photon direction. In this case the screen is defined with $\tilde{\bm{e}}_1 = (-\sin\zeta_1, \cos\zeta_1, 0)$ and $\tilde{\bm{e}}_2 = (-\cos\zeta_1\cos\zeta_2, -\sin\zeta_1\cos\zeta_2, \sin\zeta_2)$, with $\zeta_1 = \arctan(\hat{k}_y/\hat{k}_x)$ and $\zeta_2 = \arccos(\hat{k}_z)$, where $\hat{\bm{k}} \equiv \hat{\bm{k}}(p_0) = (k_x, k_y, k_z)$ is the direction of the central photon at $p_0$. Finally, the lensing distortion matrix is computed as
\begin{equation}
\hat{\vect{\mathcal{A}}}
\equiv
\frac{1}{2\chi_0\tan(\varepsilon)}
\begin{bmatrix}
(\vect{\xi}\e{A}-\vect{\xi}\e{B})\cdot\tilde{\vect{e}}_1 & (\vect{\xi}\e{C}-\vect{\xi}\e{D})\cdot\tilde{\vect{e}}_1 \\
(\vect{\xi}\e{A}-\vect{\xi}\e{B})\cdot\tilde{\vect{e}}_2 &
(\vect{\xi}\e{C}-\vect{\xi}\e{D})\cdot\tilde{\vect{e}}_2
\end{bmatrix} \ .
\end{equation}
Moreover, instead of stopping the light rays of the beam at $\tilde{p}_0$, we also implemented the possibility to stop them directly on the screen of interest.
Using finite beams to compute the Jacobian matrix allow us to make an accurate treatment of extended sources, which smooths the effect of gravitational lensing on the scale of the beam \citep{fleury2017weak, fleury2019cosmic, fleury2019weak}. This impacts the convergence and shear angular power spectra with respect to the infinitesimal case. A nice agreement between theoretical prediction and numerical estimation was found in \cite{breton2020theoretical} (see also the Appendix B. therein for a visualisation of the finite-beam effect on convergence maps).
Since this method is purely geometrical and depends on the differential deflection of photons within a beam, we cannot adapt it with the Born approximation. Also, this method does not assume that the off-diagonal terms of the Jacobian matrix are equal, meaning that we have access to the image rotation.
Having implemented the tools to propagate photons and compute the weak gravitational lensing quantities, we now turn to the production of cosmological observables: maps and simulated catalogues.
\subsection{Producing Healpix maps}
\label{subsec:healpix_maps}
First, we consider `direction-averaged' observables \citep{kibble2005average}: these relate to observations in random directions of the sky that are especially relevant for the Cosmic Microwave Background.
Usually, ray-tracing is performed in a pencil beam where all the photons propagate almost in parallel towards the pixels of a plane. In \textsc{Magrathea-Pathfinder}{}, all the photons start from the observer and in this case we find that the most natural frame to homogeneously sample the sky is to use Healpix \citep{gorski2005healpix}. Given a resolution level $N\e{side}$,\footnote{The total number of pixels on the full sky is $N\e{pix} = 12\times N\e{side}^2$} Healpix gives the position of the centre of the pixels. These positions are then used to initialise the photon direction $k^i$ at the observer.
In practice, we first assign the pixels to the different MPI subdomains (see \cref{subsec:sim_to_magrathea}) to ray-trace the different parts of the sky in parallel. Within one MPI task, we use C\texttt{++}11 \texttt{std::thread} multithreading to propagate light rays towards the pixels simultaneously. Then, the user must specify \texttt{z\_stop\_min} (minimum redshift), \texttt{z\_stop\_max} (maximum redshift) and \texttt{nb\_z\_maps} (number of redshifts at which we compute the maps), which sets (roughly) the redshifts $z_n$ of the output maps. More precisely, the maps are computed at some iso-parameter $p$ surfaces (using the keyword \texttt{stop\_ray}, see also \cref{subsubsec:finite_beams}), and evaluated at $p_n = p(z_n)$, where $p_n$ is the stop criterion computed by launching an FLRW light ray in a very refined homogeneous grid.
Now, we need to specify which quantities we want to estimate. In \texttt{map\_components}, the user can write a list of keywords to output several maps containing these informations:
\begin{itemize}
\item \texttt{lensing}: The code computes the weak-lensing quantities $\kappa$, $\gamma_1$, $\gamma_2$ and $1/\mu$ using either the infinitesimal method (\texttt{jacobiantype=infinitesimal}, see \cref{subsubsec:infinitesimal_beams}) or the finite-beam method (\texttt{jacobiantype=bundle}, see \cref{subsubsec:finite_beams}). For the latter, an additional map is written which contains the image rotation, and one must specify the stop criterion of bundle (that is, $\tilde{p}$, see \cref{subsubsec:finite_beams}) using \texttt{stop\_bundle}.
\item \texttt{lensing\_born}: Here we propagate an FLRW light ray in the pixel directions. Then we use the infinitesimal method along these trajectories to estimate the weak-lensing quantities.
\item \texttt{deflection}: The deflection angle, computed as $\bm{\alpha} = \bm{\theta}-\bm{\beta}$, where $\bm{\theta}$ is given by Healpix and $\bm{\beta}$ by the photon position at the map.
\item \texttt{dens}: The density (computed in the $N$-body solver) interpolated at the iso-$p$ surfaces.
\item \texttt{dens\_max}: The maximum density probed by the photon during its trajectory until the maps.
\item \texttt{phi}: The gravitational potential.
\item \texttt{steps}: The number of integration steps for the photon.
\item Relative differences with respect to their FLRW counterpart for various quantities: $\chi$, $\lambda$, $\eta$, $a$, $z$, with the keywords \texttt{dr}, \texttt{dl}, \texttt{dt}, \texttt{da}, \texttt{dz} respectively.
\end{itemize}
These are the map types currently implemented in \textsc{Magrathea-Pathfinder}{}, and in the future this number will be easily expanded to add more functionalities. The only limitation regarding the number of map components and redshifts at some Healpix resolution is the memory available.
Last, there is one subtlety regarding the computation of the redshift: since we estimate the redshift of the photon at each step of integration, we only perturb the redshift with gravity informations (local and integrated terms). Indeed, we do not have access to the velocity which is only available at the position of DM particles (or haloes). However, by adding the compile flag \texttt{-DVELOCITYFIELD}, \textsc{Magrathea-Pathfinder}{} computes the velocity field at the position of the AMR grid by interpolating (using either CIC or TSC) the velocity from all the particles available in the light-cone shells around $z_n$ in the subdomain of interest. This means that we need to add data slots on the octree to store $\{v_x, v_y, v_z\}$ at each cell, which produces a heavier octree. This is interesting in particular when we want to compute a map at some constant-redshift surface, where the redshift is notably impacted by the Doppler contributions.
\subsection{Geodesics finder and relativistic catalogues}
\label{subsec:catalogues}
Alternatively, we can produce `source-averaged' observables: these are simulated catalogues which relate to observations at the direction of sources on the sky (such as galaxy or supernovae surveys). Since we observe sources thanks to photons that propagated between their emission location and us, we must reproduce the same procedure numerically to construct realistic simulated catalogues. \textsc{Magrathea-Pathfinder}{} already integrates the trajectory of light rays on null geodesic, the only remaining element is to find the appropriate initial condition to link the observer to sources on the light-cone.
As described in \cite{breton2019imprints}, we start from the comoving position of a source $\bm{r}$, with comoving angular position $\bm{\beta}$. The goal is to find $\bm{\theta}$ so that the photon angular position at the comoving distance of the source is very close to $\bm{\beta}$. To do so, we iterate over $\bm{\theta}$ and use a Newton-like method to find the null geodesic which connects the observer to the source. This reads
\begin{equation}
\label{eq:newton_method}
\bm{\theta}_{i+1} = \bm{\theta}_i - \bm{\mathcal{A}}^{-1} (\bm{\beta}_i - \bm{\beta}),
\end{equation}
where the subscripts refer to the iteration of the root-finding method, and $\bm{\beta}_i$ is the photon angular position at $\chi$ with initial direction at the observer $\bm{\theta}_i$. To avoid any problems due to the system of coordinates, we make all the calculations on a screen orthogonal to $\bm{\theta}_i$. We consider that our method converged when $|\bm{\beta}_i-\bm{\beta}|< \epsilon$, with $\epsilon$ the convergence criterion set with the keyword \texttt{cat\_accuracy}. To speed-up the calculation, for the first three iterations we impose $\bm{\mathcal{A}} = \mathcal{I}$, with $\mathcal{I}$ the identity matrix. This should be a good enough approximation when there are no large gravitational fields along the photon trajectory. If the iterations did not converge, we then estimate $\bm{\mathcal{A}}$ with the infinitesimal method (which is faster than the finite-beam one) for two iterations. If convergence is still not achieved, we use a finite-beam method to compute the Jacobian matrix. If after ten iterations convergence is still not achieved (which represents in our tests about one part per million), the sources are saved in some separate files, which we can decide to use if $|\bm{\beta}_i-\bm{\beta}|$ is small enough, or we can run an alternative root finder where we re-run the ray-tracing on sources with higher resolution. In this case, we launch photons in the direction of a regular grid centred on $\bm{\theta}_{10}$, and compute $\bm{\theta}_{11}$ from the grid pixel which gives the best agreement. We repeat this process until we achieve the desired accuracy. The size of the grid decreases at every iteration of this procedure. While this last method is slow, it should in the end converge for all the remaining sources.
Finally, for each source we therefore have $\bm{\beta}$, $\bm{\theta}$, $\mathcal{A}$ (computed when $\bm{\theta}$ is known) using either the infinitesimal (with or without the Born approximation) or finite-beam method, as well as various redshifts containing local and integrated terms, depending on the contributions we are interested in. These read
\begin{eqnarray}
\label{eq:firstredshift}
z_0 &=& \frac{a_0}{a} - 1, \\
z_1 &=& z_0 + \frac{a_0}{a} \frac{\left[\Phi_o - \Phi_s\right]}{c^2}, \\
z_2 &=& z_1 + \frac{a_0}{a} \frac{\left[(\bm{v}_s - \bm{v}_o)\cdot\bm{n}\right]}{c}, \\
z_3 &=& z_2 + \frac{1}{2} \frac{a_0}{a} \frac{\left[|\bm{v_s}|^2-|\bm{v_o}|^2\right]}{c^2}, \\
z_4 &=& z_3 - \frac{2 a_0}{c^2a} \int^{\eta_o}_{\eta_s} \frac{\partial\Phi}{\partial\eta}\textrm{d}\eta, \\
\label{eq:lastredshift}
z_5 &=& \frac{(g_{\mu\nu}k^{\mu}u^{\nu})_s}{(g_{\mu\nu}k^{\mu}u^{\nu})_o} - 1,
\end{eqnarray}
where $z_0$ is the FLRW redshift, and $z_1$ to $z_4$ contain the added contribution of the gravitational potential, Doppler effect, Transverse Doppler effect and ISW. The scale factor today is given by $a_0$, the subscripts `$o$' and `$s$' refer to evaluations at the observer and at the source respectively, and $g_{\mu\nu}k^{\mu}u^{\nu} = -ak^0 \left[ 1 + \Phi/c^2 + \bm{v}\cdot\bm{n}/c + \frac{1}{2}|\bm{v}|^2/c^2 \right]$. Note that $z_4$ and $z_5$ contain all the terms at first order in metric perturbations (plus the Transverse Doppler term which is second order). This redshift decomposition is particularly interesting to study these effects either in isolation or in combination (see also \citealt{breton2019imprints} for an analysis of these effects on the dipole of the correlation function).
\subsection{Light-ray statistics}
\label{subsec:light_statsistics}
Last, we implemented the possibility to propagate light rays and bundles in random directions on the sky, and save several relevant statistics along their trajectories. In each subdomain, the user sets the number of trajectories (that is, the number of lines of sight). For each trajectory, \textsc{Magrathea-Pathfinder}{} ray-traces a light bundle which contains one central ray, and $N$ photons in a circular beam around. The photons of the beam are evenly spaced on the circle, with semi-aperture set by the user, and each photon propagates on null geodesics independently. From this, \textsc{Magrathea-Pathfinder}{} can either output the full trajectory for all the photons (see \cref{subsec:geodesic_integration} to see which kind of information is saved), or summary statistics for each subdomain or for the full light-cone. Saving the full trajectories of all the bundles is interesting to study in detail what happens during light propagation at each step of integration, and the bundle method proposed here is more general than that of \cref{subsubsec:finite_beams} which consists in only four surrounding rays. Furthermore, using a spherical bundle with an arbitrary number of photons in principle enables us to study higher-order effects of gravitational lensing such as flexion with high accuracy.
\section{Tests and convergence study}
\label{sec:tests}
In this section we perform several tests to check the convergence of the numerical methods described in \cref{sec:methods}. To do so, we use \textsc{Magrathea-Pathfinder}{} on the RayGal simulation\footnote{\url{https://cosmo.obspm.fr/public-datasets/}} (Rasera et al. in prep) which is based on the PM-AMR RAMSES code \citep{teyssier2002cosmological}. This simulation has evolved $4096^3$ particles (and as many coarse cells) in a (2.625~$h^{-1}$Gpc)$^3$ volume, with the $\Lambda$CDM, WMAP-7 year data best-fit parameters \citep{komatsu2011seven}. The RayGal simulation outputs three light-cones using the onion-shell method: a full-sky light-cone and two narrow cones with 2500 and 400 deg$^2$ aperture which reach a maximum redshift of $z = 0.5, 2$ and 10 respectively.
\subsection{Performance tests}
\label{subsec:performance_tests}
First, we estimate the run-time performance of \textsc{Magrathea-Pathfinder}{} when propagating photons within the AMR structure of the RayGal light-cone. Since the MPI and multithreading parallelizations are almost `embarrassingly parallel', that is there are little to no communication between tasks/thread, we expect \textsc{Magrathea-Pathfinder}{} to scale almost linearly with the number of CPUs. Furthermore, the number of steps per photon depends on the stop criterion, size of the light-cone and the number of integration steps per AMR cell chosen by the user. Therefore, the relevant quantity to estimate the performances of \textsc{Magrathea-Pathfinder}{} is the time needed to perform a single integration step. This run-time depends on the integrator (we implemented the Euler integration as well as RK4, however only the latter is used) but also on the type of integration. We can identify four types of trajectories which will give different run-times:
\begin{itemize}
\item FLRW: At any photon location, we assign the gravitational field of an FLRW Universe to compute the geodesic equations.
\item NGP, CIC, TSC: We use the NGP, CIC and TSC interpolation schemes to estimate the gravitational field from the AMR grid.
\end{itemize}
To perform our tests, we run \textsc{Magrathea-Pathfinder}{} on the French supercomputer Irene, on the Skylake partition composed of Intel Xeon Platinum 8168 processors. We propagate a photon in a given direction of the narrow light-cone of RayGal (with 400 deg$^2$ aperture and maximum redshift $z = 10$) using a single CPU (and single thread). We show the results in \cref{tab:performances}.
\begin{table}
\centering
\begin{tabular}{lc}
\hline\hline
Integration & time per step ($\mu$s)\\
\hline\hline
FLRW & 0.55\\
\hline
NGP & 1.10\\
\hline
CIC & 6.0\\
\hline
TSC & 14.6\\
\hline
\end{tabular}
\caption{Ray-tracing run-time for a single integration step, averaged over 100 realisations of the same trajectory and depending on the type of interpolation. We consider a photon trajectory with 4 steps per AMR cell on the narrow light-cone of the RayGal simulation until $z = 10$. The total number of integration steps is roughly $5\times 10^4$.}
\label{tab:performances}
\end{table}
For an FLRW trajectory, \textsc{Magrathea-Pathfinder}{} takes roughly $0.55~\mu $s. Since we do not need to estimate the gravitational field, the run-time is mainly that needed to compute the geodesic equations in \cref{eq:geodesic_equation1,eq:geodesic_equation2} with the RK4 integrator. For NGP, we see that it takes 1.10~$\mu$s, which is twice the time of FLRW. The additional time is that needed to get the index of the most refined cell the photon is in, find it in the octree, and get the associated data. Note that we search for an index in a sorted vector, meaning that in principle, the larger the octree vector, the longer it takes to find the index (in our case the octree of the subdomain we consider contains roughly $3\times10^8$ elements). When using CIC, \textsc{Magrathea-Pathfinder}{} takes 6~$\mu$s to perform one integration step. It is expected that the CIC interpolation schemes takes more time than NGP, and at first one could have expected $9\times 0.55 = 5~\mu$s to compute the geodesic equations and find the 8 neighbouring cells. The slight discrepancy can be explained by the fact that for CIC (and TSC) we need to perform the interpolation at some fixed level, and we loop over the coarser levels if not enough neighbours are found (see \cref{subsec:geodesic_integration}), which add some additional run-time. Finally, we see that TSC takes 14.6~$\mu$s, which is less than an optimistic expected run-time $28\times 0.55 + 1 = 16.4~\mu$s from the 27 neighbouring cells (instead of 8 for CIC). This comes from the fact that we keep in memory the neighbouring cells, so that we do not need to find them again when the photon is in the same cell and we perform the interpolation at the same level as previously.
Last, we need to estimate how many steps $N\e{tot}$ are needed to reach a given comoving distance $\chi$. Using the $\Lambda$CDM light-cones of RayGal, with coarse cell size $x\e{coarse}$ and $n\e{steps}$ the number of steps per AMR cell (set by the user), we found that the total number of integration steps as a function of the distance is well fitted by
\begin{equation}
N\e{tot}(\chi) \approx \left[1.33 -2.37\times 10^{-5} ~\raisebox{0.4ex}{$\chi$}\right] \frac{n\e{steps}}{x\e{coarse}}\chi,
\label{eq:ntot_steps_distance}
\end{equation}
between $z = 0.1$ and $z = 10$, where $\chi$ and $x\e{coarse}$ are in units of $h^{-1}$Mpc. If there was no AMR, we would expect $N\e{tot}(\chi) = n\e{steps}~\raisebox{0.3ex}{$\chi$}/x\e{coarse}$, which is different from \cref{eq:ntot_steps_distance}. This difference comes from grid refinement (AMR), especially at low redshift where the late-time small-scale clustering is more important.
\subsection{Accuracy of the interpolation schemes}
\label{subsec:test_trajectory}
One of the main features of AMR is the fact that we have access to very non-linear scales of structure formation in high-density regions. Since our ray-tracing procedure adapts to the AMR level of the simulation light-cone, we can wonder how well we recover the gravitational potential in these regions, and how it depends on the interpolation schemes described in \cref{subsec:geodesic_integration}. We use the methods in \cref{subsec:light_statsistics} to save the full trajectory of a single light ray for a given line of sight, with the three interpolations schemes previously described by setting the compile option \texttt{-DORDER} equal to 0, 1 or 2 for NGP, CIC or TSC respectively. We note that RayGal uses a TSC version of \textsc{Ramses} to compute the potential, which means that for consistency we should use the same interpolation scheme. However, it can be interesting to visualise the differences between these different types of interpolation and how it behaves with AMR.
In \cref{fig:potential_amr}, we show the gravitational potential at the photon position for each integration step, as a function of the comoving distance to the observer.
\begin{figure*}
\includegraphics[width=\columnwidth]{Figures/potential_level_interpolation.png}
\includegraphics[width=\columnwidth]{Figures/potential_level_interpolation_zoom.png}
\caption{Gravitational potential and AMR level along a photon trajectory, as a function of the comoving distance for different interpolation schemes. The right panel is a zoom on the second potential well seen in the left panel around 2910$~h^{-1}$Mpc. The potential minima coincide with a higher clustering of matter, which refines the AMR grid (as seen in the subplots).}
\label{fig:potential_amr}
\end{figure*}
Additionally, we also show the AMR level of the most-refined cell at the photon location. In the left panel, we see two potential wells around 2887 and 2910 $h^{-1}$Mpc, which is roughly $z\approx 1.3$ in the fiducial cosmology of the simulation. These potential wells reach roughly $-2.5\times 10^{-5}$, which is slightly less than 10~km/s (the order of the gravitational redshift for galaxies and cluster). We can clearly see the difference between NGP and the other two interpolations schemes, as the former shows some sharp `jumps' when the photon goes in a different cell while the others seem to exhibit a much smoother behaviour. At the same time, we see that the AMR level follows a similar pattern as the gravitational potential, starting from level 12 (which is the coarse level), and increase when the photon enters the potential wells. This makes sense since these potential wells come from small-scale clustering of matter, which also causes the $N$-body solver to refine the grid in these high-density regions. In particular, we see that for the second, deeper and sharper potential well, the AMR level goes to 16 (while for the first well it only reaches level 14). The right panel shows a zoom of the second potential well, in order to better see the differences between the interpolation schemes in these high-density regions. We indeed see the `stair' behaviour of the NGP interpolation, while the CIC interpolation is very smooth. However, we can see that CIC seems to give a potential which linearly evolves by parts along the trajectory. This is expected since CIC is just a tri-linear interpolation and we use 4 steps per AMR cells. This means that the value of a field estimated by the same 8 neighbours of the CIC scheme necessarily evolve linearly along any direction. Last, we see that the TSC scheme agrees very well with CIC far from the trough, while at the minimum of the potential it seems much smoother and would give better results when estimating its derivatives (which is important because weak lensing is sensitive to the second derivative of the gravitational potential) compared to CIC. This is also expected since TSC is higher order.
As a final note, we can see that although our method does not necessarily prevent discontinuities at the crossing between AMR level, we do not see sharp cuts in the gravitational potential when using either CIC or TSC. This shows that our methodology should give good results even for very small scales.
\subsection{On the importance of AMR}
Because most of the gravitational lensing power lies in small scales, we already expect AMR to be an important aspect of the ray-tracing procedure. In \cref{fig:convergencePS_AMR}, we quantify its impact through the estimation of the angular power spectrum of the convergence computed with \textsc{anafast} on the full-sky light-cone at $z = 0.45$, between $\ell\e{min} = 10$ and $\ell\e{max} = 2000$ (because in that case $\ell\e{max} \sim N\e{side}$, which should give a good enough accuracy).
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/convergence_powerspectra_withorwithout_AMR.png}
\caption{Convergence power spectra with AMR (red) or without (blue) at $z = 0.45$, using Healpix with $N\e{side} = 2048$. The black line is the theoretical prediction computed using \textsc{Nicaea} \citep{kilbinger2017precision} with \textsc{Halofit} \citep{smith2003stable} parameters fitted on our simulation. When AMR is accounted for, we have a nice agreement between the numerical results and theoretical prediction. When it is not, the power spectrum experiences a damping at small angular scales (high $\ell$), where the discrepancy becomes large already at $\ell \approx 300$.}
\label{fig:convergencePS_AMR}
\end{figure}
First, we remark that the angular power spectrum computed with AMR is in excellent agreement with the theoretical prediction. This validates our methodology (in the present case, that of the finite-beam method) to compute the lensing distortion matrix. Second, we see that the angular power spectrum without AMR (that is, we only consider the coarse level the propagation of light) departs early (around $\ell \sim 200-300$) from the AMR case and exhibits a strong damping at small scales. A similar trend was noticed in \cite{lepori2020weak}. This shows that AMR is extremely important for light propagation, in order to recover the correct statistical properties of gravitational lensing.
\subsection{Propagation and number of steps per AMR cell}
\label{subsec:nsteps_per_amr_cell}
We now verify that our choice to make 4 integration steps per AMR cells is well motivated. It was first shown in \cite{reverdy2014propagation} that using 4 steps was ideal to correctly recover the redshift up to double precision with respect to an analytical calculation when propagating an FLRW light ray with an RK4 integrator until $z = 25$ in the DEUS-Full Universe Run simulations \citep{alimi2012deus, rasera2014cosmic,bouillot2015probing}. However, one might wonder if this choice is still relevant when accounting for an inhomogeneous universe.
In \cref{fig:reldiff_steps_per_cell}, we show the impact of taking 1, 2 or 4 steps on the convergence angular power spectrum with respect to the very conservative case where we use 8 steps per AMR cell.
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/compare_nsteps_nside4096_born_z1p5}
\caption{Relative difference on the angular power spectrum of the convergence, depending on the number of steps per AMR cell during the propagation. The reference for the relative difference is taken as the angular power spectrum with 8 steps per AMR cell. The convergence is computed using an infinitesimal method on a Healpix map with $N\e{side} = 4096$ at $z = 1.5$. We see that we achieve numerical convergence when using 4 steps per AMR cell.}
\label{fig:reldiff_steps_per_cell}
\end{figure}
To do so, we used a narrow light-cone with 2500 deg$^2$ aperture, and produced Healpix maps (see \cref{subsec:healpix_maps}) of the convergence at $z = 1.5$ with different values of \texttt{nsteps}. To correctly estimate the power spectrum with a mask we use PolSpice \citep{szapudi2001fast,chon2004fast}.
We see that the effect here is very small (at most a few percents). When taking 1 step per AMR cell, we see a roughly $0.5\%$ bias at large angular scales, which increases up to $2\%$ at $\ell = 5000$. For 2 steps, the angular power spectrum departs from the reference one around $\ell \approx 1000$ to reach $0.5\%$ at most. Finally, we find that the convergence angular power spectrum with 4 steps is indistinguishable from its 8 steps counterparts. This is further evidence that there is no need to use more that 4 steps per AMR cell to propagate light rays.
We note that if sub-percent precision is not needed, one can decide to use 2 steps per AMR cell. This should decrease the run-time by a factor two and hence be very interesting for HPC. In any case, the user can decide to use an arbitrary number of steps by using the keyword \texttt{nsteps}.
\subsection{Infinitesimal case, choice of the derivation step}
\label{subsec:test_derivation_infinitesimal}
Now we turn to the estimation of the lensing distortion matrix with the infinitesimal method as described in \cref{subsubsec:infinitesimal_beams}. The reason we only study the convergence of the infinitesimal method and not that of the finite-beam one is that the former depends on an arbitrarily chosen derivation step, while the latter does not depend on any arbitrary choice (except for the beam aperture at the observer, which is a parameter set by the user and is physically motivated, with its impact clearly understood from a theoretical perspective, see \citealt{fleury2019cosmic, breton2020theoretical}).
The Laplacian of the gravitational potential along the line of sight in \cref{eq:infinitesimal_matrix} is computed using finite differences of the force. To compute these differences, we need a derivation step $h$ in \cref{eq:laplacian_finite_difference}, which is set to the size of the most refined cell the photon is in. In \cref{fig:reldiff_derivation_step_infinitesimal} we show the impact of the derivation step on the estimation of the convergence angular power spectrum.
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/compare_infinitesimal_clkk_nside4096_vs_cellsize_z1p9.png}
\caption{Relative difference on the convergence angular power spectrum depending on the derivation step used in the infinitesimal method (see \cref{subsubsec:infinitesimal_beams}) to compute the lensing distortion matrix at $z = 1.9$, using Healpix with $N\e{side} = 4096$. Using a larger derivation step considerably biases the estimation of the power spectrum, while smaller steps mostly affect the very small scales. Our choice for $h$ seems ideal as it is stable for $\ell < 10^3$ and should not lead to any large bias at smaller scales.}
\label{fig:reldiff_derivation_step_infinitesimal}
\end{figure}
We multiplied the size of our step choice by a factor 8, 4, 2, 1/2, 1/4 and 1/8 to clearly test our method. For a derivation step smaller than our reference choice, we see that there is a large bias (several order of magnitudes larger that in \cref{subsec:nsteps_per_amr_cell}) on the angular power spectrum even at linear scales (small $\ell$), and then at damping at small scales (high $\ell$). For derivation steps larger than the reference one, there is no noticeable difference at $\ell < 10^3$, while at $\ell > 10^3$ the angular power spectrum seem to be over-estimated with increasing $h'$. This overall behaviour shows that we found a `sweet spot' when setting $h = 2^{\rm level}$. This choice was also shown to give a very good agreement with theoretical predictions from CLASS \citep{lesgourgues2011cosmic} in Rasera et al. (in prep).
\subsection{Mean convergence and the Born approximation}
Here, we estimate the impact of the commonly used Born approximation on weak-lensing quantities, and more precisely on the mean convergence. A standard test of the Born approximation with respect to real ray-tracing is to compute the convergence angular power spectrum and estimate the differences in both cases. In this particular configuration, the Born approximation is known to give results that are extremely close to real ray-tracing \citep{hilbert2020accuracy}, however this should be taken with caution as it can be deceiving when considering galaxy surveys. Indeed, to compute the angular power spectra, it is common to propagate light rays in the direction of homogeneously distributed pixels on a map (or plane). However, this procedure gives the same weight to each pixel, which is not what really happens in galaxy surveys where solid angles on the sky are magnified. This is the difference between direction-averaging and source-averaging \citep{kibble2005average,bonvin2015cosmological,kaiser2016bias,breton2020theoretical}. When propagating light rays in random (or statistically isotropic) directions, it is expected that the Born approximation should give overall similar results as real ray-tracing \citep{breton2020theoretical}. In particular, in both cases, we expect $\ev{\kappa} = 0$ (which also implies $\ev{\mu} > 1$ at second order), since photons evenly sample the Universe. However, when we compute the null geodesics between the observer and sources, it is known that $\ev{\mu} = 1$ \citep{weinberg1976apparent} and $\kappa < 0$. Indeed, photons tend to propagate in more under-dense regions between two fixed points for several realisations of the matter density field.
In \cref{fig:meankappa}, we compute the distortion matrix for a given simulated halo catalogue and estimate the averaged convergence in tomographic redshift bins.
\begin{figure}
\includegraphics[width=\columnwidth]{Figures/comparison_meankappa_born_vs_geodesic.png}
\caption{Mean convergence on the deep narrow cone of RayGal (400 deg$^2$ aperture) as function of redshift, when evaluating the distortion matrix along null geodesics (black) or using the Born approximation in the direction of sources ($\bm{\beta}$, red) or images ($\bm{\theta}$, blue). The black solid line shows the expected value of the mean convergence when using null geodesics, that is $\ev{\kappa} = \frac{1}{2}(\ev{\mu}-1) - 2\ev{\kappa}^2$, where the averaged quantities are numerically evaluated. The error bars contain Poisson and super-sample variance \citep{breton2020theoretical}. The red and black points are consistent with the expectation values for direction and source averaging procedures respectively. The blue points, which come from a hybrid prescription, seem to lie between the former two.}
\label{fig:meankappa}
\end{figure}
In any case, we use the infinitesimal method in \cref{subsubsec:infinitesimal_beams}, and distinguish three different types of trajectories: first we consider the null geodesics between the observer and sources, then we use the Born approximation where photons propagate in straight lines between the observer and the sources or the source images. We note that the last case is strictly theoretical since, to our knowledge, it is not necessarily used nor discussed, however it is interesting from a pedagogical point of view to cover all the possible scenarios. We see that for the standard Born approximation, that is when photons propagate in straight lines in the source direction ($\bm{\beta}$), the averaged convergence is consistent with zero within error bars, which is expected. For null geodesics, we find that the mean convergence is indeed negative and follows well the theoretical prediction. Interestingly, when we use the Born approximation in the direction of images, the result seems to lie in between the first two cases.
There are two main implications of this result: first, a precise theoretical modelling of weak-lensing observables might need to account for the fact that $\ev{\kappa} \neq 0$ for galaxy surveys. Secondly (and more importantly), one should take with caution the distortion matrix evaluated using the Born approximation when constructing realistic simulated catalogues. Because in this case, the mean magnification is superior to unity (while in reality it should be strictly equal before the flux cut), and furthermore the magnification probability density function is that of the null-geodesic case multiplied by $\mu$ (this was also discussed in \citealt{takahashi2011probability}). When emulating a flux-limited survey, one has to magnify the fluxes (or magnitude) with the magnification, however if one uses the Born approximation, the final number count on the flux-limited sample will be larger than what it should realistically be in comparison to observations.
\subsection{Relativistic effects and galaxy clustering analyses}
Last, we discuss the importance of the global relativistic treatment for light propagation. While ray-tracing codes usually only allow for gravitational lensing studies, \textsc{Magrathea-Pathfinder}{} offers a unified framework which accounts for both gravitational lensing and redshift perturbations. For the latter, we implemented all the corrections at first order in metric perturbations (as seen in \cref{subsec:catalogues}). This allows in particular to study the impact of relativistic effects for galaxy clustering analysis. The full relativistic number counts has been analytically estimated within linear theory in \cite{yoo2009new,bonvin2011what,challinor2011linear}, and it has been shown in \cite{bonvin2014asymmetric} that the dipole of the correlation function could be very sensitive to the gravitational potential and could therefore be a useful probe to test the nature of gravity \citep{bonvin2018testing}. In \cite{breton2019imprints}, the authors found a nice agreement at large scale with linear-theory based predictions, which shows the accuracy of our method. However, it was noted that the small scales are completely dominated by the gravitational potential, way beyond the expectation from linear theory. This was later analytically modelled using non-linear prescriptions \citep{didio2019relativistic,saga2020modelling,beutler2020modeling,saga2021detectability}. The dipole of the correlation function of galaxies is expected to be detected with a high signal-to-noise ratio for next-generation galaxy surveys \citep{saga2021detectability}, for which it will be mandatory to perform a full relativistic treatment of light propagation.
\section{Conclusion}
\label{sec:conclusion}
In this paper we presented \textsc{Magrathea-Pathfinder}{}, a ray-tracing framework which post-processes numerical simulations to accurately rebuild the past light-cone of an observer from linear to non-linear scales. This framework propagates light rays in the AMR structure of a simulation light-cone by solving the null geodesic equations of a perturbed FLRW metric in the Newtonian gauge and does not resort to any approximation, except that we consider the weak-field regime for the gravitational potential. Moreover, \textsc{Magrathea-Pathfinder}{} is optimised for HPC and have already been run on very large $N$-body simulations such as DEUS- Full Universe Run \citep{alimi2012deus} and RayGal (Rasera et al. in prep).
Our code produces relativistic simulated catalogues (where the null geodesic between the observer and sources are identified), Healpix maps, as well as various light-ray statistics along their propagation. By accounting all the effects at first order in metric perturbations, \textsc{Magrathea-Pathfinder}{} opens up a wide range of possible applications, some of which have already been studied in the literature:
\begin{itemize}
\item \emph{Weak gravitational lensing}: In Rasera et al. (in prep), the authors studied the impact of relativistic effects on various lensing-matter angular power spectra. In particular, they emphasised the importance of peculiar velocities and magnification bias beyond linear order. Furthermore, the weak gravitational lensing of light beams with finite extension give different two-point statistic than infinitesimal ones. This effect was successfully confronted to theoretical predictions for the convergence and shear angular power spectra in \cite{breton2020theoretical}.
\item \emph{Galaxy clustering and relativistic effects}: The simulated catalogues produced by \textsc{Magrathea-Pathfinder}{} contain both the comoving and apparent angular position, as well as the full redshift decomposition for a given source. From this, one can perform galaxy clustering analysis beyond standard assumptions (distant observer, redshift only perturbed by peculiar velocities\ldots). In particular, in \cite{breton2021impact} the authors studied the impact of gravitational lensing (also known as magnification bias) on the estimation of the growth rate of structure when performing a standard RSD analysis and found that if lensing is not accounted for in the modelling, this leads to an underestimation of $f\sigma_8$. These catalogues also enabled for the first time the study of relativistic effects on the dipole of the correlation function at all scales \cite{breton2019imprints, taruya2020wide, saga2020modelling}, which will be detectable in next-generation surveys with high signal-to-noise ratio \citep{saga2021detectability}. Using the same data, \cite{beutler2020modeling} performed a similar analysis on the power spectrum dipole.
\item \emph{Distance measures}: Distance measures are crucial to interpret observational data and cosmological inference. When it comes to the Hubble diagram, the average distance is often modelled by that of an FLRW universe. In \cite{breton2020theoretical}, the authors studied in details the perturbations on the distance-redshift relation for a wide range of redshifts. They numerically tested Weinberg's conjecture \citep{weinberg1976apparent}, which states that the area of constant redshift is unaffected by inhomogeneities in the matter density field and showed that even a full non-linear treatment gave similar results to the theoretical predictions of \cite{kaiser2016bias} regarding possible biases.
\item \emph{Integrated Sachs-Wolfe effect}: \cite{adamek2019raytracing} used \textsc{Magrathea-Pathfinder}{} on the DEUS- Full Universe Run simulations up to $z = 25$ to study the ISW angular power spectra for various scenarios such as $\Lambda$CDM, phantom dark energy and modified gravity. In any case, the authors found a good agreement with theoretical predictions.
\end{itemize}
These works show the importance of an exhaustive modelling of the light-cone in order to correctly interpret future surveys, which will probe the Universe at the largest scales with unprecedented precision. In particular, with the increasing data accuracy, omissions in the model might lead to biases in the inference of cosmological parameters.
The methods developed in \textsc{Magrathea-Pathfinder}{} will be useful, in coordination with theoretical predictions and observations to refine existing probes and construct new ones, in order to shed light on the true nature of our Universe.
\begin{acknowledgements}
We thank Yann Rasera for helpful comments on the draft, and the Laboratoire Univers et Théories for its support and for providing a stimulating scientific environment from the early development of \textsc{Magrathea}{} to the last raytracing analyses with \textsc{Magrathea-Pathfinder}{}. The design, development, and implementation of the foundational \textsc{Magrathea}{} library as well as the initial version of \textsc{Magrathea-Pathfinder}{} were carried out at LUTH during VR's PhD under the direction of Jean-Michel Alimi and Yann Rasera. The further developments of \textsc{Magrathea-Pathfinder}{} were carried out during MAB's thesis under the direction of Yann Rasera. We thank Jean-Michel Alimi and Yann Rasera for their scientific contributions and LUTH for its hospitality. MAB also thanks Julian Adamek for an early implementation of Healpix in the \textsc{Magrathea-Pathfinder}{} code. VR also thanks the countless dedicated developers of the software stack without which none of this would have been possible. This work was granted access to HPC resources of TGCC/CINES through allocations made by GENCI (Grand Équipement National de Calcul Intensif) under the allocation 2020-A0070402287.
\end{acknowledgements}
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\section{Introduction}
\label{S-1}
Let $f:X\la Y$ be a morphism of noetherian schemes. At the
level of derived categories
there exist
natural functors $\LL f^*:\Dqc(Y)\la\Dqc(X)$, its right adjoint
$\R f_*:\Dqc(X)\la
\Dqc(Y)$,
as well as a right adjoint for $\R f_*$, nowadays (following Lipman)
denoted
$f^\times:\Dqc(Y)\la\Dqc(X)$.
For general $f$ the functor
$f^\times$ can be dreadful---it can take
a bounded complex of coherent sheaves, that is an
object in $\dcoh(Y)\subset\Dqc(Y)$,
to a truly enormous object in $\Dqc(X)$.
This functor $f^\times$ only behaves well under
strong restrictions, the usual being that $f$ be
proper.
To remedy this one introduces a better-behaved functor $f^!$. If $f$ is
proper then $f^!=f^\times$, but for general $f$ one traditionally
does some finicky
manipulations to arrive at $f^!$. And, until very recently, the recipe
worked only for cohomologically bounded-below complexes. That is
$f^!$ has always been viewed as a functor $f^!:\Dqcpl(Y)\la\Dqcpl(X)$.
Against this background came the striking work of
Avramov, Iyengar, Lipman
and Nayak, see~\cite{Avramov-Iyengar08,Avramov-Iyengar-Lipman-Nayak10},
relating Grothendieck duality with Hochschild homology
and cohomology. To give the flavor of
the results let me present just one formula, and for simplicity
let me give only the affine version.
Suppose therefore that $X=\spec S$, $Y=\spec R$, assume that $R$ and
$S$ are noetherian, and that $f:X\la Y$ is a flat, finite-type
map. In an abuse of notation we will write $f:R\la S$ for the induced
ring homomorphism, and also identify $\D(R)\cong\Dqc(Y)$ and $\D(S)=\Dqc(X)$.
Let $\se=S\oo_R^{}S$ be the enveloping algebra. Then, for any
object $N\in\Dqcpl(Y)=\D^+(R)$, we have a canonical isomorphism
\[
f^!N\cong S\oo_{\se}^{}\Hom_R^{}(S,S\oo_R^{}N)\ .
\]
In this formula the tensor products and the Hom are all derived.
The reader might find it interesting to note that, in the special case
where $f:R\la S$ is finite and \'etale, we recover the classical
formula
\[
f^!N\cong \Hom_R^{}(S,N)\cong S\oo_{\se}^{}\Hom_R^{}(S,S\oo_R^{}N)\ .
\]
Of course for finite, \'etale maps the Homs and tensors are underived.
We will revisit \'etale maps (not necessarily finite) in Remark~\ref{R1.13}.
Perhaps one needs some familiarity with the classical literature to
appreciate how striking this is---assuming only that $f$ is
flat we have produced a formula for $f^!$, which took
a mere paragraph to state, and is clearly free of
auxiliary choices and
functorial. And although the left-hand-side was defined on the
assumption that $N$ is bounded
below---after all we only knew $f^!$ on the
bounded-below derived category---the right-hand-side makes sense for any $N$.
In fact the formula tells us the surprising fact that if $N$ is an
object in $\D^+(R)$ then $S\oo_{\se}^{}\Hom_R^{}(S,S\oo_R^{}N)$ must belong
to $\D^+(S)$. We know, from the complicated
classical construction, that $f^!$ takes
$\D^+(R)$ to $\D^+(S)$, but $S$ is not of finite Tor-dimension
over $\se$ and we have
no reason to expect an expression of the form $S\oo_{\se}^{}M$ to be
bounded below. The derived tensor product tends to introduce lots of
negative cohomology.
In joint work with Iyengar and Lipman we revisited these results,
and along the way developed a useful
new natural transformation $\psi(f):f^\times\la f^!$,
see~\cite{Iyengar-Lipman-Neeman13}. Hints of $\psi$ may be found in
Lipman~\cite[Exercise~4.2.3(d)]{Lipman09}, but without the naturality
properties that make it so valuable.
With all these unexpected new tools
it was becoming clear that the time may have come to revisit
the foundations of Grothendieck duality.
In this article we sketch what has come out of this.
Finally we should tell the reader the structure of this survey. The
early sections, \S\ref{S1} and \S\ref{S2}, survey recent
results that can be found elsewhere in the literature. The results
are new, meaning new in this generality---there are older avatars,
what's unusual here
is that the theory is developed in the unbounded derived category. The
results might be innovative but we still omit the proofs.
With the
exception of Proposition~\ref{P2.202}, where the argument
is included, the proofs are all to be found
in recent preprints available electronically.
In \S\ref{S99} and \S\ref{S95} this changes. Special cases of the
results are known, with what turn out to be artificial boundedness
restrictions.
We give a general treatment---both to show that the
results are true more generally, and to illustrate the power of the
new techniques. Because of this our treatment is complete, with
proofs.
The reader interested in the highlights is advised to read the
statement (not proof) of Lemma~\ref{L99.5}, as well as
Corollary~\ref{C95.13} and Example~\ref{E95.15}.
The final sections, \S\ref{S3} and \S\ref{S4}, are again ``soft'',
with no proofs presented. They review the history and suggest open problems.
\section{Conventions}
\label{S-2}
In this article we consider schemes $X$ and the corresponding
derived categories $\Dqc(X)$, whose objects are complexes
of sheaves of $\co_X^{}$--modules
with quasicoherent cohomology. Since abelian categories never come
up, whenever there is a possible ambiguity our functors should be
assumed derived---thus we will write $f^*$ for $\LL f^*$, $f_*$
for $\R f_*$, $\Hom$ for $\RHom$ and $\oo$ for the derived tensor
product $\oo^\LL_{}$.
For simplicity, in \S\ref{S1}, \S\ref{S2}, \S\ref{S99}, \S\ref{S95}
and \S\ref{S3} we
will assume that our schemes are noetherian
and morphisms of schemes are separated and of finite
type---occasionally, but not always, we will explicitly remind the reader of
these standing assumptions.
Unless we specifically say otherwise
all derived categories will be unbounded. For a
morphism of schemes $f:X\la Y$ we let
$f^*\dashv f_*\dashv f^\times$ be the adjoint functors
which, back in \S\ref{S-1}, we referred to as
$\LL f^*\dashv \R f_*\dashv f^\times$.
\section{The formal theory}
\label{S1}
In this section and the next
we sketch the current state of
the formal
theory, without worrying about who proved
what and when.
Let $f:X\la Y$ be a morphism of schemes. The functor $f^*:\Dqc(Y)\la\Dqc(X)$
is a strict monoidal functor, meaning it respects the tensor product.
Therefore for any pair of objects $E\in\Dqc(Y)$ and $F\in\Dqc(X)$ we have a
natural map
\[
\CD
f^*[E\oo f_*F] @>\sim>> f^*E\oo f^*f_*F @>\id\oo\e>> f^*E\oo F
\endCD
\]
where the first map is the natural isomorphism, and $\e:f^*f_*\la\id$ is
the counit of the adjunction $f^*\dashv f_*$. By adjunction we obtain a
natural map $p(E,F):E\oo f_*F\la f_*(f^*E\oo F)$. The map $p(E,F)$
is known to be an
isomorphism, usually called the
\emph{projection formula.} This leads us to
\dfn{D1.1}
Let $f:X\la Y$ be a morphism of schemes and let $E,F$ be objects in
$\Dqc(Y)$. The map $\chi(f,E,F):f^*E\oo f^\times F\la f^\times (E\oo F)$
is defined by applying the adjunction $f_*\dashv f^\times$ to the
composite
\[
\xymatrix@C+30pt{
f_*(f^*E\oo f^\times F) \ar[r]^-{p(E,f^\times F)^{-1}} & E\oo f_*f^\times F
\ar[r]^-{\id\oo\e'} & E\oo F\ ,
}
\]
where the first map is the inverse of the isomorphism in the projection
formula, while $\e':f_*f^\times\la \id$ is the counit of the
adjunction $f_*\dashv f^\times$.
\edfn
The first result in the theory is
\thm{T1.3}
The map $\chi(f,E,F)$ is an isomorphism whenever
\be
\item
$f$ is arbitrary, but $E$ is a perfect complex.
\item
$E$ and $F$ are arbitrary, but $f$ is proper and of finite Tor-dimension.
\ee
\ethm
Next recall the base-change maps. Given a commutative square
of morphisms of schemes
\[
\CD
W @>u>> X \\
@VfVV @VVgV \\
Y @>v>> Z
\endCD
\]
there is a canonical isomorphism of functors $\alpha:f^*v^*\la u^*g^*$. Consider
the composite
\[
\CD
f^*v^*g_* @>\alpha g_*>> u^*g^*g_* @>u^*\e>> u^*\ ,
\endCD
\]
where $\e:g^*g_*\la\id$ is the counit of the adjunction $g^*\dashv g_*$.
Adjunction gives us a base-change map $\beta:v^*g_*\la f_*u^*$; this map
is not always an isomorphism, but there are important situations
in which it is. This leads us to
\dfn{D1.5}
Assume we are in a situation where the base-change map $\beta:v^*g_*\la f_*u^*$
is an isomorphism; for this article the important case
where this happens is when the square
\[
\CD
W @>u>> X \\
@VfVV @VVgV \\
Y @>v>> Z
\endCD
\]
is cartesian and the map $v$ is flat. In this scenario consider the
composite
\[
\xymatrix@C+20pt{
f_*u^*g^\times \ar[r]^-{\beta^{-1}g^\times} & v^*g_*g^\times \ar[r]^-{v^*\e'} &
v^*
}
\]
where the first map is the inverse of the isomorphism $\beta$ while
$\e':g_*g^\times\la\id$ is the counit of the adjunction
$g_*\dashv g^\times$.
The (second) base change map $\Phi:u^*g^\times\la f^\times v^*$ corresponds
to this composite under the adjunction $f_*\dashv f^\times$.
\edfn
One can wonder when the base-change map $\Phi$ is an isomorphism.
The best result to date says
\thm{T1.7}
Let the notation be as in the case of
Definition~\ref{D1.5} which interests us in this article---that
is we assume the square cartesian and $v$ flat.
Let $E$ be an object
in $\Dqc(Z)$. Then the
base-change map $\Phi(E):u^*g^\times(E)\la f^\times v^*(E)$
is an isomorphism provided $g$ is proper
and one of the conditions below holds:
\be
\item
$E$ belongs to $\Dqcpl(Z)\subset\Dqc(Z)$.
\item
$E\in\Dqc(Z)$ is arbitrary, but the map $f:W\la Y$ is of finite Tor-dimension.
\ee
\ethm
Now one proceeds as follows:
given any morphism $f:X\la Y$ we factor it as
$X\stackrel u\la\ov X\stackrel p\la Y$ with $u$ an open immersion
and $p$ proper, and then define $f^!:\Dqc(Y)\la\Dqc(X)$ by
the formula $f^!=u^*p^\times$. One of the consequences of
Theorem~\ref{T1.7} is that $f^!$
is well-defined, meaning that it is canonically independent of
the choice of factorization. And we have the following
theorem.
\thm{T1.107}
The assignment, taking a morphism of schemes $f:X\la Y$
to the functor $f^!:\Dqc(Y)\la\Dqc(X)$, satisfies a long list
of compatibility properties. We list some highlights.
\sthm{ST1.107.1}
Let
$X\stackrel f\la Y\stackrel g\la Z$
be composable morphisms of schemes.
There is a map $\rho(f,g):{(gf)}^!\la f^!g^!$, which has the property
that the two ways of using $\rho$ to go
from $(hgf)^!$ to $f^!g^!h^!$ are equal.
\esthm
\sthm{ST1.107.3}
The two functors $f^\times, f^!:\Dqc(Y)\la\Dqc(X)$ are related
by a natural transformation $\psi(f):f^\times\la f^!$. The $\psi$ is compatible
with composition, in the obvious sense that the square below commutes
\[\xymatrix@C+15pt{
{(gf)}^\times \ar[r]^{\delta(f,g)}\ar[d]_{\psi(gf)} &f^\times g^\times \ar[d]^{\psi(f)\psi(g)}\\
{(gf)}^! \ar[r]_{\rho(f,g)} & f^!g^!
}\]
where $\rho(f,g)$ is the map of \ref{ST1.107.1}, while $\delta(f,g):(gf)^\times
\la f^\times g^\times$ is the canonical isomorphism.
\esthm
\sthm{ST1.107.4}
The map $\rho(f,g)$ is an
isomorphism if $f$ is of finite Tor-dimension or if either
$gf$ or $g$ is proper. The map $\psi(f)$ is an isomorphism
whenever $f$ is proper.
\esthm
\sthm{ST1.107.5}
Given a pair of object $E,F\in\Dqc(Y)$ then there is a way to mimick
the construction in Definition~\ref{D1.1} with $f^!$ in place of $f^\times$.
More precisely: there is a map $\s(f,E,F):f^*E\oo f^!F\la f^!(E\oo F)$ so that
the natural square commutes
\[
\CD
f^*E\oo f^\times F @>\chi(f,E,F)>> f^\times(E\oo F)\\
@V\id\oo\psi(f)VV @VV\psi(f)V \\
f^*E\oo f^!F @>\sigma(f,E,F)>> f^!(E\oo F)
\endCD
\]
Furthermore we have the analog of Theorem~\ref{T1.3}, that is
$\s(f,E,F)$ is an isomorphism if one of the conditions below
holds
\be
\item
$f$ is arbitrary, but $E$ is a
perfect complex.
\item
$E$ and $F$ are arbitrary, but
$f$ is of finite Tor-dimension.
\ee
\esthm
\sthm{ST1.107.7}
The base-change map $\Phi$ of Definition~\ref{D1.5} also has an $(-)^!$
analog. Precisely: given a cartesian square as in Definition~\ref{D1.5},
there is a base-change map $\theta:u^*g^!\la f^!v^*$.
\esthm
\sthm{ST1.107.9}
There is an analog of Theorem~\ref{T1.7} for $(-)^!$ in place of $(-)^\times$.
Precisely: the map $\theta(E):u^*g^!(E)\la f^!v^*(E)$ is an isomorphism
as long as one of the following holds
\be
\item
$E$ belongs to $\Dqcpl(Z)\subset\Dqc(Z)$.
\item
$E\in\Dqc(Z)$ is arbitrary, but the map $f:W\la Y$ is of finite Tor-dimension.
\ee
\esthm
\ethm
The full list of compatibility properties is quite long, and in any case it
is clearer and more compact to present it in a 2-category formulation.
For this paper we content ourselves with what's in Theorem~\ref{T1.107}.
\rmk{R1.109}
In the introduction we mentioned that people have traditionally preferred
$f^!$ to $f^\times$ because it is ``better behaved''. Theorem~\ref{T1.107}
allows us to make this more precise. If we compare \ref{ST1.107.5} with
Theorem~\ref{T1.3} we see that
\be
\item
If $f$ is proper then $\s(f,E,F)$ and $\chi(f,E,F)$ agree up to canonical
isomorphism.
To see this observe that, when $f$ is proper, then the vertical
maps in the commutative square of \ref{ST1.107.5} are isomorphisms
by \ref{ST1.107.4}.
\item
The maps $\s(f,E,F)$ and $\chi(f,E,F)$ are defined for every
triple $f,E,F$, but
$\s(f,E,F)$ is an isomorphism more often. If $E$ is perfect
then both are isomorphisms. But for non-perfect $E$ the result
\ref{ST1.107.5}(ii)
says that $\s(f,E,F)$ is
an isomorphism whenever $f$ is of finite
Tor-dimension, whereas Theorem~\ref{T1.3}(ii)
guarantees that $\chi(f,E,F)$ is an isomorphism only if $f$ is
proper as well as of finite Tor dimension.
\setcounter{enumiii}{\value{enumi}}
\ee
The same pattern repeats itself for the base-change maps $\Phi$ and $\theta$.
They are defined for every cartesian square with flat horizontal morphisms,
and coincide if the vertical
maps are proper---if our Theorem~\ref{T1.107} were less pared down this could
be shown to follow from the general structure, the reader can see the
introduction to \cite{Neeman13} for the fullblown formalism.
But if we ask ourselves when $\Phi$ and $\theta$ induce isomorphisms,
the conditions on
$\theta$ are less restrictive than on $\Phi$. Precisely: when we compare
Theorem~\ref{T1.7} with \ref{ST1.107.9} we discover
\be
\setcounter{enumi}{\value{enumiii}}
\item
Assume $E$ is bounded below. Then
$\Phi(E):u^*g^\times E\la f^\times v^*E$ is an isomorphism if $g$ is proper,
while $\theta(E):u^*g^! E\la f^! v^*E$ is an isomorphism unconditionally.
\item
Let $E$ be arbitrary. Then
$\Phi(E)$ is an isomorphism as long as
$g$ is proper and $f$ is of
finite Tor-dimension, while $\theta(E)$ is an isomorphism whenever
$f$ is of
finite Tor-dimension (no need for any properness).
\setcounter{enumiii}{\value{enumi}}
\ee
\ermk
\rmk{R1.11}
If $f:X\la Y$ is an open immersion then the square
\[
\xymatrix@C+2pt@R+2pt{
X \ar[r]^{\id}\ar[d]_{\id} & X\ar[d]^{f}\\
X \ar[r]^{f} & Y
}\]
is cartesian. By \ref{ST1.107.9}(ii) we have that
$f^!=\id^*f^!\stackrel{\theta}\la \id^! f^*=f^*$
is an isomorphism.
Given any morphism $g:Y\la Z$, the map
$\rho(f,g):(gf)^!\la f^!g^!$ is an isomorphism by
\ref{ST1.107.4}---after all the open immersion
$f$ is of finite Tor-dimension.
Combining these isomorphisms gives
\be
\item
If $X\stackrel f\la Y\stackrel g\la Z$ are composable
morphisms of schemes, with $f$ an open immersion, then
we have a canonical isomorphism $(gf)^!\stackrel{\rho}\la
f^!g^!\stackrel\theta\la f^*g^!$.
\ee
If $g:Y\la Z$ happens to be a proper morphism
then $\psi(g):g^\times\la g^!$ is an isomorphism by \ref{ST1.107.4},
which we may combine with the isomorphism of (i) to deduce
a canonical isomorphism
$(gf)^!\cong f^*g^!\cong f^*g^\times$. The compatibilities
of Theorem~\ref{T1.107} force upon us the formula for $(gf)^!$. That is:
any time we can factor a map $X\la Z$ as a composite
$X\stackrel f\la Y\stackrel g\la Z$, with $f$ an open immersion and $g$
proper, then $(gf)^!$ must be given by the formula $(gf)^!\cong f^*g^\times$.
\ermk
\rmk{R1.13}
In passing we observe that the
formula of Remark~\ref{R1.11}(i) generalizes, we need only
assume $f$ \'etale.
Suppose $f:X\la Y$ is an \'etale morphism of noetherian schemes. Consider the
following diagram
\[
\xymatrix@C+15pt@R+2pt{
X\ar[r]^-{\Delta} & X \times_Y^{}X\ar[r]^-{\pi_1^{}}\ar[d]_-{\pi_2^{}} & X\ar[d]^{f}\\
&X \ar[r]^{f} & Y
}\]
where the square is cartesian and $\Delta$ is the diagonal map. We are
given that $f$ is \'etale, meaning that the diagonal map $\Delta$ is
an open immersion. This gives a series of isomorphisms
\[
f^! \,\,\cong\,\, \Delta^*\pi_1^*f^! \,\,\cong\,\, \Delta^*\pi_2^!f^* \,\,\cong\,\,
(\pi_2^{}\Delta)^! f^* \,\,\cong\,\, f^*.
\]
The first isomorphism is $\Delta^*\pi_1^*\cong\id^*=\id$, the second
isomorphism is the map $\pi_1^*f^!\la
\pi_2^!f^*$ of \ref{ST1.107.7}, which is an isomorphism by \ref{ST1.107.9}(ii), the third
isomorphism is Remark~\ref{R1.11}(i) applied to the composable maps
$X\stackrel\Delta\la X\times_Y^{} X\stackrel{\pi_2^{}}\la X$ with
$\Delta$ an open immersion, and the last isomorphism is because
$\pi_2^{}\Delta$ is the identity. Remark~\ref{R1.11}(i) therefore also
generalizes, we have
\be
\item
If $X\stackrel f\la Y\stackrel g\la Z$ are composable morphisms with
$f$ \'etale, then $(gf)^!\stackrel\rho\la f^!g^!\cong f^*g^!$
combines to an isomorphism. The map $\rho(f,g)$ is an isomorphism
because $f$ is of finite Tor-dimension, while
$f^!\cong f^*$ is the isomorphism above.
\ee
\ermk
\section{Still formal, but less familiar---the way the abstract theory is related to explicit computations}
\label{S2}
Our first aim is to obtain a fomula for $f^!$ free
of auxiliary choices---that is, one that does not involve
factoring $f$ as $X\stackrel u\la\ov X\stackrel p\la Y$
with $u$ an open immersion and $p$ proper. We begin with a little Lemma.
\lem{L2.2}
Let $U\stackrel\alpha\la V\stackrel\beta\la W$ be finite-type morphisms of
noetherian schemes
so that $\alpha$ is a closed immersion and $\beta\alpha$ is
proper. Then the maps $\alpha^\times\psi(\beta):\alpha^\times\beta^\times
\la\alpha^\times\beta^!$ and
$\alpha^*\psi(\beta):\alpha^*\beta^\times
\la\alpha^*\beta^!$
are both isomorphisms.
\elem
\prf
The proof is an easy consequence of Theorem~\ref{T1.107}, coupled
with standard facts from support theory.
First~\ref{ST1.107.3} gives us the commutative square
\[\xymatrix@C+15pt{
{(\beta\alpha)}^\times \ar[r]^{\delta(\alpha,\beta)}\ar[d]_{\psi(\beta\alpha)} &\alpha^\times \beta^\times \ar[d]^{\psi(\alpha)\psi(\beta)}\\
{(\beta\alpha)}^! \ar[r]_{\rho(\alpha,\beta)} & \alpha^!\beta^!
}\]
Recalling
that $\alpha$ and $\beta\alpha$ are proper maps,
we conclude from \ref{ST1.107.4}
that $\rho(\alpha,\beta):(\beta\alpha)^!\la\alpha^!\beta^!$ is an isomorphism,
as are $\psi(\alpha):\alpha^\times\la\alpha^!$ and
$\psi(\beta\alpha):(\beta\alpha)^\times\la(\beta\alpha)^!$.
Now $\delta(\alpha,\beta)$
is trivially an isomorphism, hence in the square above
the indicated maps are all isomorphisms
\[\xymatrix@C+15pt{
{(\beta\alpha)}^\times \ar[r]^{\sim}\ar[d]_{\wr} &\alpha^\times \beta^\times \ar[d]^{\psi(\alpha)\psi(\beta)}\\
{(\beta\alpha)}^! \ar[r]_{\sim} & \alpha^!\beta^!
}\]
The commutativity implies that the vertical map
$\psi(\alpha)\psi(\beta):\alpha^\times\beta^\times
\la
\alpha^!\beta^!$
must be an isomorphism. This isomorphism can be written as the
composite
\[
\CD
\alpha^\times\beta^\times
@>{\alpha^\times\psi(\beta)}>>
\alpha^\times\beta^!
@>{\psi(\alpha)\beta^!}>>
\alpha^!\beta^!\ .
\endCD
\]
but, as $\psi(\alpha)$ is an isomorphism,
we conclude that $\alpha^\times\psi(\beta)$
is also an isomorphism. Support theory, more precisely
\cite[Proposition~A.3(ii)]{Iyengar-Lipman-Neeman13}, tells us that
$\alpha^*\psi(\beta)$ is also an isomorphism.
\eprf
We will apply this little lemma in the following situation.
\con{C2.2002}
Let $f:X\la Y$ be a finite-type, flat morphism of noetherian schemes.
We may form the diagram
\[\xymatrix@C+10pt@R+10pt{
X \ar[dr]^-{\Delta} & & \\
& X\times_Y^{}X \ar[r]^-{\pi_1^{}}\ar[d]_-{\pi_2^{}} & X\ar[d]^f \\
& X \ar[r]^-{f} & Y\ar@{}[ul]|{(\diamondsuit)}
}\]
where the square is cartesian, $\pi_1^{}$ and $\pi_2^{}$ are the first
and second projections, and $\Delta:X\la X\times_Y^{}X$ is the
diagonal inclusion. We assert
\econ
\pro{P2.202}
With the notation as in Construction~\ref{C2.2002},
there is a canonical isomorphism $\Delta^*\pi_2^\times f^*\la f^!$.
\epro
\prf
We apply Lemma~\ref{L2.2} to the
composable maps $X\stackrel\Delta\la X\times_Y^{}X\stackrel {\pi_2^{}}\la X$;
the map $\Delta$ is a closed immersion and the composite $\id=\pi_2^{}\Delta$
is proper, and Lemma~\ref{L2.2} tells us that
$\Delta^*\psi(\pi_2^{}):\Delta^*\pi_2^\times\la \Delta^*\pi_2^!$ is an isomorphism.
On the other hand all the maps in the cartesian square $(\diamondsuit)$
are flat, and
\ref{ST1.107.9}(ii) gives that
$\theta:\pi_1^*f^!\la\pi_2^!f^*$ is an isomorphism.
Consider therefore the composite
\[
\xymatrix@C+25pt@R+5pt{
\Delta^*\pi_2^\times f^* \ar[r]^-{\Delta^*\psi(\pi_2^{})f^*} &
\Delta^*\pi_2^! f^* \ar[r]^-{\Delta^*\theta^{-1}_{}} &
\Delta^*\pi_1^* f^! \ar[r]^-{\sim} & \id^* f^!\ar@{=}[r] & f^!
}
\]
The first and second maps are isomorphisms by the discussion above,
and the
third map comes from applying the functor $(-)^*$ to the equality
$\id=\pi_1^{}\Delta$. The composite is therefore
an isomorphism $\Delta^*\pi_2^\times f^*\la f^!$.
\eprf
\con{C2.3}
Assume $f:X\la Y$ is flat and let the notation be as above.
If $\e:\Delta_*\Delta^\times\la\id$
is the counit of adjunction $\Delta_*\dashv\Delta^\times$, let
$\ph:\Delta_*\la\pi_2^\times$ be the composite
\[
\xymatrix@C+20pt{
\Delta_* \ar@{=}[r] & \Delta_*\Delta^\times\pi_2^\times
\ar[r]^-{\e\pi_2^\times} & \pi_2^\times
}
\]
where the equality is the observation that
$\id=\id^\times=(\pi_2^{}\Delta)^\times
\cong\Delta^\times\pi_2^\times$.
Define $c_f^{}:\Delta^*\Delta_*f^*\la\Delta^*\pi_2^\times f^*$
to be $\Delta^*\ph f^*$; combining with the isomorphism
$\Delta^*\pi_2^\times f^*\la f^!$ of Proposition~\ref{P2.202} we
arrive at a map which, in an abuse
of notation, we will also denote $c_f^{}:\Delta^*\Delta_*f^*\la f^!$.
If we apply this map to the object $\co_Y^{}\in\Dqc(Y)$ and note
that $f^*\co_Y^{}=\co_X^{}$, we obtain a map
$c_f^{}(\co_Y^{}):\Delta^*\Delta_*\co_X^{}\la f^!\co_Y^{}$.
For any integer $d\in\zz$ let $\gamma_f^{}(d)$ be the composite
\[
\xymatrix@C+20pt{
(\Delta^*\Delta_*\co_X^{})^{\leq -d}\ar[r] &
\Delta^*\Delta_*\co_X^{}\ar[r]^-{c_f^{}(\co_Y^{})} & f^!\co_Y^{}
}
\]
where the first map is given by the standard \tstr\ truncation.
\econ
Note that we have defined the maps in great generality, globally and
without auxiliary choices of coordinates. What is known so far is
\thm{T2.5}
Suppose the map $f:X\la Y$ is smooth and of relative dimension $d$.
Then the map $\gamma_f^{}(d)$ is an isomorphism.
\ethm
\rmk{R2.7}
The object $f^!\co_Y^{}$ might appear mysterious but $\Delta^*\Delta_*\co_X^{}$
isn't, it is just the obvious object in the derived category whose cohomology
is the Hochschild homology of $\co_X^{}$. If $f:X\la Y$ is smooth and of
relative dimension $d$ then $(\Delta^*\Delta_*\co_X^{})^{\leq -d}$ is
nothing other than $HH^d(\co_X^{})$, which is the relative canonical bundle
in degree $-d$. In symbols
$\Omega^d_{X/Y}[-d]=(\Delta^*\Delta_*\co_X^{})^{\leq -d}$.
Note that the maps $c_f^{}$ and $\gamma_f^{}(d)$ are defined for any flat
$f$ and any integer $d$, and might contain interesting information for
$f$ which aren't smooth. I don't believe anyone has computed examples yet.
\ermk
\rmk{R2.9}
The reader can find (different) proofs of Theorem~\ref{T2.5} in either
Alonso, Jerem{\'{\i}}as and
Lipman~\cite[Proposition~2.4.2]{Alonso-Jeremias-Lipman14} or else in
\cite[\S1]{Neeman13A}.
The point we want to make here is that the proof can't be hard:
the map is defined globally, but proving it an isomorphism
is local in $Y$ in the flat topology---hence we may assume $Y$ an
affine scheme---and
local in $X$ in the \'etale topology\footnote{To see that it suffices
to prove the map an isomorphism flat-locally in $Y$ and \'etale-locally
in $X$ one needs the
full strength of Theorem~\protect{\ref{T1.107}}, the extract we presented
in this survey doesn't suffice. The reader can find the complete statement
in~\protect{\cite{Neeman13}}}.
And \'etale-locally
any smooth map of degree $d$ is of the
form $\spec S\la \spec R$, where
$f:R\la S$ identifies $S$ as the polynomial ring
$S=R[x_1^{},x_2^{},\ldots,x_d^{}]$.
Let $\se=S\oo_R S$; since $S$ is flat over $R$ it makes
no difference whether we view this particular
tensor product as ordinary or derived.
The expression $\Delta^*\Delta_*\co_X^{}$ is nothing other than
the derived tensor product $S\oo_{\se}^{}S$, while
$f^!\co_Y^{}\cong\Delta^*\pi_2^\times f^*\co_Y^{}$ comes down to
$S\oo_{\se}^{}\Hom_R^{}(S,S)$, where the tensor and the Hom are
both derived. And the map $c_f^{}:S\oo_{\se}^{}S\la
S\oo_{\se}^{}\Hom_R^{}(S,S)$
is the tensor product over $\se$ of the identity
map $\id:S\la S$ and the obvious inclusion
$S\la \Hom_R^{}(S,S)$.
OK: a little computation is necessary to finish off the proof,
the details may be found
in \cite[\S1]{Neeman13A}.
\ermk
\con{C2.11}
Now let $W\subset X$ be the union of closed subsets $W_i\subset X$,
such that
the restriction of $f$ to each $W_i$ is proper.
For every point $x\in X$ write $k(x)$ for its residue field;
then the full subcategory $\D_{\mathrm{qc},W}^{}(X)\subset\Dqc(X)$
has objects given by the formula
\[
\D_{\mathrm{qc},W}^{}(X)=\{E\in\Dqc(X)\mid E\oo k(x)=0\text{ for all }x\notin W\}.
\]
The inclusion
$I:\D_{\mathrm{qc},W}^{}(X)\la\Dqc(X)$ is well-known to admit a Bousfield
localization, meaning it has a right adjoint
$R:\Dqc(X)\la\D_{\mathrm{qc},W}^{}(X)$. The composite functor
$\Dqc(X)\stackrel R\la\D_{\mathrm{qc},W}^{}(X)\stackrel I\la\Dqc(X)$
is nowadays denoted
$\Gamma_W^{}:\Dqc(X)\la \Dqc(X)$,\footnote{
Classically it was denoted $\R\Gamma_W^{}$---it is the right derived functor
of some functor on abelian categories.
But we have been suppressing all the notation
that usually reminds us of the functors of abelian categories that
we are deriving, and in this case it brings our notation into
concert with that of Benson, Iyengar and
Krause~\cite{Benson-Iyengar-Krause08,Benson-Iyengar-Krause11,Benson-Iyengar-Krause11A,Benson-Iyengar-Krause12,Benson-Iyengar-Krause13}. Their
choice of the letter $\Gamma$ was quite unrelated to Grothendieck's,
see the comment at the top of
\cite[page~582]{Benson-Iyengar-Krause08}.
By a fortuitous accident the notations coincide
(once the $\R$ is eliminated in $\R\Gamma$).}
and this Bousfield localization is
even smashing, meaning there is a natural isomorphism
$E\oo\Gamma_W^{}F\la\Gamma_W^{}(E\oo F)$. We assumed that, for each
$W_i$, the composite $W_i\stackrel{\alpha_i}\la X\stackrel f\la Y$
is proper, and
a slight refinement of Lemma~\ref{L2.2} tells us that $\alpha_i^*\psi(f)$ is
an isomorphism for each $\alpha_i$. Support theory, more precisely
\cite[Proposition~A.3(ii)]{Iyengar-Lipman-Neeman13}, allows us to
deduce that
the map $\Gamma_W^{}\psi(f)$ is also an isomorphism.
Let $\e:\Gamma_W^{}=IR\la\id$ be the counit of the adjunction $I\dashv R$
and let $\e':f_*f^\times\la\id$ be the counit of the adjunction
$f_*\dashv f^\times$. We define the map $\int_W^{}:f_*\Gamma_W^{}f^!\la\id$ to
be the composite
\[
\xymatrix@C+40pt{
f_*\Gamma_W^{}f^!\ar[r]_-{f_*\big[\Gamma_W^{}\psi(f)\big]^{-1}}
\ar@/^2pc/[rrr]^-{\ds\int_W^{}}
& :f_*\Gamma_W^{}f^\times \ar[r]_-{f_*\e f^\times} & f_*f^\times
\ar[r]_-{\e'} &\id
}\]
If we apply this natural transformation to the
object $\co_Y^{}\in\Dqc(Y)$, and combine with the map
$\gamma_f^{}(d)$, we obtain a composite
\[
\xymatrix@C+40pt{
f_*\Gamma_W^{}\big[\Delta^*\Delta_*\co_X^{}\big]^{\leq -d}
\ar[r]^-{f_*\Gamma_W^{}\gamma_f^{}(d)} &
f_*\Gamma_W^{}f^!\co_Y^{}
\ar[r]^-{\ds\int_W^{}}
&\co_Y^{}
}\]
Now assume $f:X\la Y$ is smooth of relative dimension $d$. Theorem~\ref{T2.5}
tells us that $\gamma_f^{}(d)$ is an isomorphism,
and in Remark~\ref{R2.7} we observed that
$\big[\Delta^*\Delta_*\co_X^{}\big]^{\leq -d}$ is nothing more
than the relative canonical bundle in degree $-d$, that
is $\Omega^d_{X/Y}[-d]=\big[\Delta^*\Delta_*\co_X^{}\big]^{\leq -d}$.
We are assuming $W=\cup W_i$ is the union
of closed subschemes $W_i\subset X$ so that, for each $i$, the composite
$W_i\stackrel{\alpha_i}\la X\stackrel f\la Y$ is proper. Let us make the
stronger assumption that $f\alpha_i$ is a finite morphism for each $i$. Then
the functor $f_*\Gamma_W^{}$ takes an object $E\in\Dqc(X)$ to its
local cohomology at $W$ and, in the particular case of $E=\Omega^d_{X/Y}[-d]$,
this comes down (locally and modulo boundaries)
to the relative meromorphic $d$--forms
$\omega/s_1^{}s_2^{}\cdots s_d^{}$ on $X$ with $(s_1^{},s_2^{},\ldots, s_d^{})$
a relative system of parameters. And now we are ready for the next computation.
\econ
\thm{T2.13}
Assume $f:X\la Y$ is smooth of relative dimension $d$.
Then the composite
\[
\xymatrix@C+0pt{
f_*\Gamma_W^{}\Omega^d_{X/Y}[-d]\ar@{=}[r] &
f_*\Gamma_W^{}\big[\Delta^*\Delta_*\co_X^{}\big]^{\leq -d}
\ar[rr]^-{f_*\Gamma_W^{}\gamma_f^{}(d)} & &
f_*\Gamma_W^{}f^!\co_Y^{}
\ar[rr]^-{\ds\int_W^{}}&
&\co_Y^{}
}\]
is just the map taking a relative meromorphic $d$--form to its residue.
\ethm
\rmk{R2.15}
Early incarnations of Theorem~\ref{T2.13} may be found in
Verdier~\cite[pp.~398-400]{Verdier68} and H\"ubl and
Sastry~\cite[Residue Theorem, p.~752]{Hubl-Sastry93}.
The proof of Theorem~\ref{T2.13}, as stated here, may be found in \cite[\S2]{Neeman13A}. Once
again we note that while the global definition might be slightly subtle,
the local computation can easily be reduced to the simple, affine case of
Remark~\ref{R2.9}.
\ermk
We have mentioned, several times, that the functor $f^!$ is ``better behaved''
than its cousin $f^\times$. One facet is that it is amenable
to local computations. To illustrate this we end the section with
a couple of little lemmas. The first of the lemmas just records, for a
morphism $f:X\la Y$, the open subsets $U\subset X$ which we will find
useful for these local computations.
\lem{L2.2055}
Suppose $f:X\la Y$ is a finite-type morphism of
noetherian schemes.
Suppose $U\subset X$ is an open affine subset so that the composite
$U\stackrel u\la X\stackrel f\la Y$ can be factored through an open affine subset $V\subset Y$.
Then
\be
\item
If $f$ is of finite Tor-dimension then $fu$ admits a factorization
$fu=hg$, where $g$ is a closed immersion of finite Tor-dimension and
$h$ is smooth.
\item
For arbitrary $f$, the map $fu$ may be factored as $fu=khg$, where $g$
is an open immersion, $h$ is a closed immersion and $k$ is smooth and
proper.
\ee
\elem
\prf
By hypothesis the composite $U\stackrel u\la X\stackrel f\la Y$ is
equal to the composite
$U\stackrel \alpha\la V\stackrel\beta\la Y$ with $V$ affine and $\beta$
an open immersion.
Because $\alpha:U\la V$ is a finite-type morphism of affine schemes we may
factor it
as $U\stackrel {j'}\la \pp^n_V\stackrel{\pi'}\la V$ where
$j'$ is a locally closed
immersion, and this allows us to factor $fu$ as
$U\stackrel {j}\la \pp^n_Y\stackrel{k}\la Y$
where $j$ is also a locally closed immersion.
Now we separate the treatment into two cases. If $f$ is of finite
Tor-dimension we factor $j$ as
$U\stackrel g\la W\stackrel \delta\la \pp_Y^{n}$ with $g$ a
closed immersion and $\delta$ an open immersion. This gives
us a factorization of $fu$ as $U\stackrel g\la W\stackrel{k\delta}\la Y$.
Since $k$ and $\delta$ are both smooth so is $h=k\delta$. And
the fact that $fu=hg$ is of finite Tor-dimension and $h$ is smooth means
that $g$ must be of finite Tor-dimension. We have found a factorization
satisfying (i) of the Lemma.
It remains to treat the case where $f$ is arbitrary. In
this case
we factor
$j:U\la\pp_Y^n$ as $U\stackrel g\la W\stackrel h\la \pp_Y^{n}$, with
$g$ an open immersion and $h$ a closed immersion.
In total this factors $fu$ as $khg$, with $k:\pp_Y^n\la Y$
smooth and proper, $h:W\la\pp_Y^n$ a closed immersion and
$g:U\la W$ an open immersion.
\eprf
\lem{L1.13}
Let $f:X\la Y$ be a finite-type, separated morphism of noetherian
schemes. We record the following boundedness and coherence statements:
\be
\item
There exists an integer $n$ so that $f^!\Dqc(Y)^{\geq 0}\subset\Dqc(X)^{\geq n}$.
\item
If $E\in\Dqc(Y)$ is bounded below and has coherent cohomology, then the
same is true for $f^!E$.
\setcounter{enumiii}{\value{enumi}}
\ee
For the next few assertions assume furthermore that $f$ is
of finite Tor-dimension.
\be
\setcounter{enumi}{\value{enumiii}}
\item
For any object $E\in\Dqc(Y)$, the support of $f^!E$ is equal to
the support of $f^*E$.
\item
If $E\in\Dqc(Y)$ has coherent cohomology then so has
$f^!E$ [no need to assume $E$
bounded below.]
\item
There exists an integer $n$ so that $f^!\Dqc(Y)^{\leq 0}\subset\Dqc(X)^{\leq n}$.
\item
There exists an integer $n$ so that, if the Tor-amplitude of $E\in\Dqc(Y)$
is contained in the interval $[0,\infty)$, then the
Tor-amplitude of $f_*f^!E$ is contained in the interval
$[n,\infty)$.
\setcounter{enumiii}{\value{enumi}}
\ee
\elem
\prf
First we show that (vi) follows from (i) and Theorem~\ref{T1.107}.
Suppose we know (i); we may choose an integer $n$ with
$f^!\Dqc(Y)^{\geq0}\subset\Dqc(X)^{\geq n}$. Let $E\in\Dqc(Y)$
be an object with Tor-amplitude contained in
$[0,\infty)$. Then $\Dqc(Y)^{\geq0}\oo E\subset\Dqc(Y)^{\geq0}$,
and
\[
f^*\Dqc(Y)^{\geq0}\oo f^!E\eq f^!\big[\Dqc(Y)^{\geq0}\oo E\big]\quad\subset\quad
\Dqc(X)^{\geq n}
\]
where the equality is by \ref{ST1.107.5}(ii).
Applying $f_*$ we deduce
\[
\Dqc(Y)^{\geq0}\oo f_*f^!E\eq
f_*\big[f^*\Dqc(Y)^{\geq0}\oo f^!E\big] \quad\subset\quad
f_*\Dqc(X)^{\geq n}\quad\subset\quad \Dqc(Y)^{\geq n}
\]
where the equality is by the projection formula. We conclude that
$f_*f^!E$ has Tor-amplitude in the interval $[n,\infty)$.
It remains to prove (i)--(v), which are all local in $X$.
This means the following: cover $X$ by open immersions $u_i:U_i\subset X$.
Remark~\ref{R1.11}(i) tells us that $(fu_i)^!\cong u_i^*f^!$.
If we had a cover so that we could prove (i)--(v) for all
the $(fu_i)^!\cong u_i^*f^!$, then the statement would follow for $f$.
Cover $Y$ by finitely many open affine subsets $V_i$, then cover each
$f^{-1}V_i$ by finitely many open affine subsets $U_i$, and we have covered $X$ by open
subsets as in Lemma~\ref{L2.2055}. We are therefore
reduced to proving the Lemma under the assumption that
$f=fu$ has a factorization as in Lemma~\ref{L2.2055}~(i) or (ii).
We first prove (i), (ii), (iv) and (v), all of which
respect composition: this means
\be
\setcounter{enumi}{\value{enumiii}}
\item
If $X\stackrel f\la Y\stackrel g\la Z$
are morphisms so that
the map $\rho(f,g):(gf)^!\la f^!g^!$ is an isomorphism
and
(i) and (ii) hold for each of $f$ and $g$,
then it formally follows that (i) and (ii) hold for $gf$.
\item
If we furthermore assume that $f$ and $g$ are of finite Tor-dimension
and (iv) and (v) hold for each of $f$ and $g$, then (iv) and (v)
hold for $gf$.
\setcounter{enumiii}{\value{enumi}}
\ee
Next we observe that the factorizations of Lemma~\ref{L2.2055}
behave well with respect to the functor $(-)^!$, meaning
\be
\setcounter{enumi}{\value{enumiii}}
\item
If $f=hg$ is a factorization as in Lemma~\ref{L2.2055}(i)
then
$\rho(g,h):(hg)^!\la g^!h^!$ is an isomorphism. This is because
$g$ is of finite Tor-dimension, see \ref{ST1.107.4}.
\item
If $f=khg$ is a factorization as in Lemma~\ref{L2.2055}(ii)
then
the map $(khg)^!\la g^!h^!k^!$ is an isomorphism.
The map $\rho(hg, k):(khg)^!\la(hg)^!k^!$ is
an isomorphism because $k$ is proper,
while the map $\rho(g,h):(hg)^!\la g^!h^!$ is an isomorphism
because $g$ is of finite Tor-dimension. See \ref{ST1.107.4}.
\setcounter{enumiii}{\value{enumi}}
\ee
This means that it suffices to prove (i), (ii), (iv) and (v)
in the special cases where $f$ is either smooth or a closed immersion,
and for the proof of (iv) and (v) we may assume that the closed
immersion is of finite Tor-dimension.
If $f:X\la Y$ is a smooth map
then \ref{ST1.107.5}(ii) tells us that $f^!(-)\cong f^*(-)\oo f^!\co_Y^{}$,
while from Theorem~\ref{T2.5} we learn that $f^!\co_Y^{}$ is just
a shift of the relative
canonical bundle. Hence (i), (ii), (iv) and (v) are all true
for smooth maps $f$.
Next we prove (i), (ii), (iv) and (v) for closed immersions $f$. By
\ref{ST1.107.4} we know that $\psi(f):f^\times\la f^!$
is an isomorphism. We need to prove the coherence and/or vanishing
of cohomology sheaves of $f^!E\cong f^\times E$, and
as $f$ is a closed immersion these are equivalent to the
coherence and/or vanishing of the cohomology sheaves
of $f_*f^\times E$.
Now recall the isomorphisms
$f_*f^\times(-)\cong f_*\HHom\big[\co_X^{},f^\times(-)\big]
\cong\HHom(f_*\co_X^{},-)$.
If $f$ is of finite Tor-dimension then $f_*\co_X^{}\in\Dqc(Y)$ is a perfect
complex, hence (iv) and (v) are clear. It remains to prove
(i) and (ii), which become assertions about the vanishing and coherence
of $\HHom(f_*\co_X^{},E)$ where $E$ is bounded below.
Illusie~\cite[Proposition~3.7]{Illusie71A} tells us that this
may be computed locally in $Y$, and if $Y$ is affine
the assertions are obvious.
It remains to prove (iii), the assertion about the supports.
The map $f$ is assumed of finite Tor-dimension and \ref{ST1.107.5}(ii)
gives an isomorphism $f^!(E)\cong f^*(E)\oo f^!\co_Y^{}$.
Support theory tells us that the support of the tensor
product $f^*(E)\oo f^!\co_Y^{}$ is the intersection of the support
of $f^*E$ and the support of $f^!\co_Y^{}$, hence it suffices
to show that the support of $f^!\co_Y^{}$ is all of $X$.
But we have reduced to the case where
$f$ has
a factorization $X\stackrel g\la W\stackrel h\la Y$ as
in Lemma~\ref{L2.2055}(i), and in (ix) we noted that
$\rho(g,h):(hg)^!\la g^!h^!$ is an
isomorphism. Thus
\[
(hg)^!\co_Y^{}\,\cong\, g^![h^!\co_Y^{}]\,\cong\, \big[g^*h^!\co_Y^{}\big]\oo g^!\co_W^{}
\]
where the last isomorphism is by \ref{ST1.107.5}(ii). We wish to
show that the support of $(hg)^!\co_Y^{}$ is all of $X$,
and it suffices to prove that the support of
$h^!\co_Y^{}$ is all of $W$ and the support of $g^!\co_W^{}$ is
all of $X$.
Theorem~\ref{T2.5} tells us that $h^!\co_Y^{}$ is just
a shift of the relative canonical bundle---its support is all of $W$.
Since $g$ is a closed immersion the support of
$g^!\co_W^{}\cong g^\times\co_W^{}$ is equal
to the support of $g_*g^\times\co_W^{}\cong\HHom(g_*\co_X^{},\co_W^{})$. But
as $g$ has finite Tor-dimension the object $g_*\co_X^{}$ is a perfect
complex on $W$, and its support (all of $X$) is equal to the support of the
dual $\big[g_*\co_X^{}\big]^\vee=\HHom(g_*\co_X^{},\co_W^{})$. This completes
the proof of (iii).
\eprf
\section{Some basic isomorphisms}
\label{S99}
In this section we prove some formal corollaries of the theory
presented so far, establishing that certain natural maps are
isomorphisms. None of the results is hard, but they
are a little technical---their value will only become
apparent when we see the applications later on. The one result we
will need, in \S\ref{S95}, is Lemma~\ref{L99.5}. At a first
reading we recommend that the reader study the statement of
Lemma~\ref{L99.5} and skip the rest of this section.
\lem{L99.-1}
Let $f:X\la Y$ be a finite-type morphism of noetherian schemes, of
finite Tor-dimension. Let
$E\in\Dqc(Y)$ be a perfect complex, and let $\{A_\lambda,\,\lambda\in\Lambda\}$
be a set of objects of $\Dqc(Y)$. Then
the natural map
$\Big[\prod_{\lambda\in\Lambda}^{}f^*A_\lambda\Big]\oo f^!E
\la \prod_{\lambda\in\Lambda}^{}\big[f^*A_\lambda\oo f^!E\big]$
is an isomorphism.
\elem
\prf
We begin by proving a special case: we show that the Lemma is true
$f$ is proper and of finite Tor-dimension.
Assume therefore that $f$ is proper and of finite Tor-dimension. Then $f_*$
takes perfect complexes to perfect complexes,
and \cite[Theorem~1.7(GN2)]{Balmer-Dellambrogio-Sanders16} tells us that $f^*$
has a left adjoint and respects products.
If we contemplate the commutative diagram
\[\xymatrix@C+20pt{
\ds f^*\left[\prod_{\lambda\in\Lambda}A_\lambda\right]\oo f^! E
\ar[r]^-{(1)}\ar[d]_{\s}&
\ds\left[\prod_{\lambda\in\Lambda}f^*A_\lambda\right]\oo f^! E
\ar[r]^-{(2)}&
\ds\prod_{\lambda\in\Lambda}\big[f^*A_\lambda\oo f^! E\big]\ar[d]^\s\\
\ds f^!\left(\left[\prod_{\lambda\in\Lambda}A_\lambda\right]\oo E\right)
\ar[r]^-{(3)} &
\ds f^!\left[\prod_{\lambda\in\Lambda}\big(A_\lambda\oo E\big)\right]
\ar[r]^-{(4)} &
\ds\prod_{\lambda\in\Lambda}f^!\big[A_\lambda\oo E\big]
}\]
the vertical maps $\s$ are isomorphisms by \ref{ST1.107.5}(ii), the map
$(1)$ is an isomorphism because $f^*$
respects products, the map $(3)$ is an isomorphism because $E$
is perfect and hence $(-)\oo E\cong\HHom(E^\vee,-)$ respects products,
and the map $(4)$ is an isomorphism because, for the proper morphism
$f$, the functor $f^!\cong f^\times$ has a left adjoint and respects
products. Hence the map $(2)$ must be an isomorphism, as we asserted.
We have proved the Lemma if $f$ is proper
and of finite Tor-dimension. Now suppose $f$ may be factored
as $X\stackrel g\la W\stackrel h\la Y$
with $h$ smooth and $g$ proper and of finite Tor-dimension.
Because $h$ is of finite Tor-dimension \ref{ST1.107.5} gives an
isomorphism $h^*E\oo h^!\co_Y^{}\la h^!E$, while
Theorem~\ref{T2.5} tells us that $h^!\co_Y^{}$ is a shift
of the relative
canonical bundle. Since $E$ is assumed perfect
it follows that $h^!E$, being the
tensor product of two perfect complexes $h^*E$ and
$h^!\co_Y^{}$, must be perfect. But then we may apply
the
special case
of the Lemma to the map $g:X\la W$, the perfect complex $h^!E\in\Dqc(W)$
and the set of objects $\{h^*A_\lambda,\,\lambda\in\Lambda\}$
of $\Dqc(W)$. We deduce that the map $(2)$ in the
commutative square below is an isomorphism
\[\xymatrix@C+30pt{
\ds\left(\prod_{\lambda\in\Lambda}\big[(hg)^*A_\lambda\big]\right)\oo (hg)^!E
\ar[d]_{\tau(g,h)\oo\rho(g,h)}
\ar[r]^-{(1)} &\ds\prod_{\lambda\in\Lambda}\big[(hg)^*A_\lambda\oo (hg)^!E\big]
\ar[d]^{\tau(g,h)\oo\rho(g,h)}\\
\ds\left(\prod_{\lambda\in\Lambda}\big[g^*h^*A_\lambda\big]\right)\oo g^!h^!E
\ar[r]^-{(2)} &\ds\prod_{\lambda\in\Lambda}\big[g^*h^*A_\lambda\oo g^!h^!E\big]
}\]
The vertical maps are induced by the isomorphism $\tau(g,h):(hg)^*\la g^*h^*$
and the map $\rho(g,h):(hg)^!\la g^!h^!$ of \ref{ST1.107.1}.
Because $g$ is of finite Tor-dimension \ref{ST1.107.4}
informs us that $\rho(g,h)$ is an isomorphism, hence
both vertical maps are isomorphisms.
From the commutativity we deduce that $(1)$ is an isomorphism.
In other words: the Lemma is true for any $f$ which admits a factorization
as $X\stackrel g\la W\stackrel h\la Y$ with $h$ smooth and
$g$ proper and of finite Tor-dimension.
Now let $f:X\la Y$ be arbitrary, fix a perfect
complex $E\in\Dqc(Y)$, as well as
a set of objects $\{A_\lambda,\,\lambda\in\Lambda\}$
of $\Dqc(Y)$. If $u:U\la X$ is an open immersion
with $U$ affine, and assuming that $fu:U\la Y$ can be factored through
an open affine subset $V\subset Y$, then Lemma~\ref{L2.2055}(i)
guarantees that $fu$ may be factored as $hg$ with $h$ smooth and
$g$ proper and of finite Tor-dimension. By the above the natural
map
$\big[\prod_{\lambda\in\Lambda}(fu)^*A_\lambda\big]\oo(fu)^!E
\la\prod_{\lambda\in\Lambda}\big[(fu)^*A_\lambda\oo(fu)^!E\big]$
is an
isomorphism. Remark~\ref{R1.11}(i) gives an
isomorphism $(fu)^!\cong u^*f^!$, allowing us to rewrite the
isomorphism above as
$\big[\prod_{\lambda\in\Lambda}(fu)^*A_\lambda\big]\oo u^*f^!E
\la\prod_{\lambda\in\Lambda}\big[(fu)^*A_\lambda\oo u^*f^!E\big]$.
If we apply the functor $u_*$, and use the projection formula
and the fact that $u_*$ has a left adjoint and hence
respects products, we obtain that the natural map
is an isomorphism
$\big[\prod_{\lambda\in\Lambda}u_*u^*f^*A_\lambda\big]\oo f^!E
\la\prod_{\lambda\in\Lambda}\big[u_*u^*f^*A_\lambda\oo f^!E\big]$.
Now glue these isomorphisms: any time we have open
immersions $u:U\la X$, $v:V\la X$, $j:U\cap V\la X$ and $w:U\cup V\la X$,
then for each $\lambda$ we obtain a triangle
\[\xymatrix{
w_*w^*f^*A_\lambda \ar[r] &
u_*u^*f^*A_\lambda \oplus
v_*v^*f^*A_\lambda \ar[r] &
j_*j^*f^*A_\lambda \ar[r] &
}\]
Taking the product over $\lambda$ and tensoring with $f^!E$ gives
a triangle, as does tensoring with $f^!E$ and then forming the product
over $\lambda$. There is a map between the triangles: if two
of the morphisms are isomorphisms then so is the third. Starting with
the fact that we know the map to be an isomorphism if $U\subset X$
is a sufficiently
small open affine, we glue to discover that it is an isomorphism for
every open immersion $u:U\la X$. The case of the identity map $\id:X\la X$
gives the Lemma.
\eprf
\lem{L99.3}
As in Lemma~\ref{L2.2}
let $U\stackrel\alpha\la V\stackrel\beta\la W$ be
finite-type morphisms of
noetherian schemes
so that $\alpha$ is a closed immersion and $\beta\alpha$
proper. Given a set of objects $\{A_\lambda,\,\lambda\in\Lambda\}$ in
the category $\Dqc(W)$, then the functors $\alpha^*$ and
$\alpha^\times$ take the natural morphism
\[\xymatrix@C+20pt{
\ds\beta^!\left[\prod_{\lambda\in\Lambda}A_\lambda\right]\ar[r]^-{J} &
\ds\prod_{\lambda\in\Lambda}\beta^! A_\lambda
}\]
to isomorphisms.
If $\beta$ is of finite Tor-dimension the functors
$\alpha^*$ and
$\alpha^\times$ also take the natural morphism
\[\xymatrix@C+20pt{
\ds\beta^*\left[\prod_{\lambda\in\Lambda}A_\lambda\right]\ar[r]^-{K} &
\ds\prod_{\lambda\in\Lambda}\beta^* A_\lambda
}\]
to isomorphisms.
\elem
\prf
Consider the commutative square
\[\xymatrix@C+20pt{
\ds\beta^\times
\left[\prod_{\lambda\in\Lambda}A_\lambda\right]\ar[r]^-{I}\ar[d]_-{\psi(\beta)} &
\ds\prod_{\lambda\in\Lambda}\beta^\times A_\lambda
\ar[d]^-{\prod_\Lambda \psi(\beta)}\\
\ds\beta^!\left[\prod_{\lambda\in\Lambda}A_\lambda\right]\ar[r]^-{J} &
\ds\prod_{\lambda\in\Lambda}\beta^! A_\lambda
}\]
The map $I$ is an isomorphism, after all $\beta^\times$ is a right
adjoint and respects products. If we apply the functor $\alpha^\times$
and recall (i) that it respects products and (ii) that
$\alpha^\times\psi(\beta)$ is an isomorphism by Lemma~\ref{L2.2}, then
we deduce the commutative diagram where the indicated maps are
isomorphisms
\[\xymatrix@C+20pt{
\ds \alpha^\times\beta^\times
\left[\prod_{\lambda\in\Lambda}A_\lambda\right]\ar[r]^-{\sim}\ar[d]_-{\wr} &
\ds \alpha^\times\left[\prod_{\lambda\in\Lambda}\beta^\times A_\lambda\right]
\ar[d]^-{\wr}\\
\ds \alpha^\times\beta^!\left[\prod_{\lambda\in\Lambda}A_\lambda\right]\ar[r]^-{\alpha^\times J} &
\ds \alpha^\times\left[\prod_{\lambda\in\Lambda}\beta^! A_\lambda\right]
}\]
Hence $\alpha^\times J$ must be an isomorphism. Support theory, more
concretely \cite[Proposition~A.3(ii)]{Iyengar-Lipman-Neeman13}, tells us that
$\alpha^*J$ is also an isomorphism.
Now assume that $\beta$ is of finite Tor-dimension.
We wish to show that the maps $\alpha^*K$ and $\alpha^\times K$ are
isomorphisms, with $K$ as in the Lemma, and
by \cite[Proposition~A.3(ii)]{Iyengar-Lipman-Neeman13} it
suffices to consider $\alpha^*K$.
From
\ref{ST1.107.5}(ii) we have a natural isomorphism
$\beta^*(-)\oo\beta^!\co_W^{}\la\beta^!(-)$, hence the map
$J$ of the Lemma is isomorphic to the composite
\[\xymatrix@C+30pt{
\ds \beta^*\left[\prod_{\lambda\in\Lambda}A_\lambda\right]\oo\beta^!\co_W^{}\ar[r]^-{ K\oo\beta^!\co_W^{}} &
\ds \left[\prod_{\lambda\in\Lambda}\beta^* A_\lambda\right]\oo\beta^!\co_W^{}
\ar[r]^-{H} &
\ds \prod_{\lambda\in\Lambda}\left[\beta^* A_\lambda\oo\beta^!\co_W^{}\right]
}\]
Lemma~\ref{L99.-1} tells us that the map $H$ is an isomorphism, but
by the first part of the Lemma, which we already proved,
we know that $\alpha^*$ takes the composite to an
isomorphism. Hence $\alpha^*$ must take the map $K\oo\beta^!\co_W^{}$ to
an isomorphism, or to put it
differently $(-)\oo\alpha^*\beta^!\co_W^{}$ takes
$\alpha^*K$ to an isomorphism. We wish to show that
$\alpha^*K$ is an isomorphism and support theory,
more precisely \cite[Proposition~A.3(ii)]{Iyengar-Lipman-Neeman13},
tells us that
is suffices to show that every point of $U$ lies in the support
of $\alpha^*\beta^!\co_W^{}$. Since the support of $\alpha^*E$
is $\alpha^{-1}\supp(E)$ it certainly suffices
to show that $\supp(\beta^!\co_W^{})=V$.
But $\beta$ is of finite Tor-dimension, and
Lemma~\ref{L1.13}(iii) tells us that the support of $\beta^!\co_W^{}$
is equal to the support of $\beta^*\co_W^{}=\co_V^{}$, which is all of $V$.
\eprf
\lem{L99.5}
Let $U\stackrel\alpha\la V\stackrel\beta\la W$ be finite-type morphisms of
noetherian schemes
so that $\alpha$ is a closed immersion, $\beta$ is
of finite Tor-dimension, and $\beta\alpha$
proper. Suppose $E,F\in\Dqc(W)$ are any objects.
Then $\alpha^*$ and $\alpha^\times$ take the natural
map $\beta^*\HHom(E,F)\la\HHom(\beta^*E,\beta^*F)$ to an
isomorphism.
\elem
\prf
Support theory, more
concretely \cite[Proposition~A.3(ii)]{Iyengar-Lipman-Neeman13}, tells us that
$\alpha^*$ will take the map to an isomorphism if and only if
$\alpha^\times$ does. It suffices to prove the assertion for $\alpha^\times$.
Fix an object $F\in\Dqc(W)$ and let $\cl$ be the full subcategory of
all objects $E\in\Dqc(W)$ such that $\alpha^\times$ takes the map
$\p_E^{}:\beta^*\HHom(E,F)\la\HHom(\beta^*E,\beta^*F)$ to an
isomorphism. If $E$ is a perfect complex then the map $\p_E^{}$ is an
isomorphism as it stands, hence all perfect complexes belong to $\cl$.
Also $\cl$ is obviously triangulated. Because $\Dqc(W)$ is compactly
generated
it suffices to prove that $\cl$ is localizing; we need to show that the
coproduct
of any set of objects in $\cl$ lies in $\cl$.
Assume therefore that $\{E_\lambda,\,\lambda\in\Lambda\}$ is a set
of objects of $\cl$. We wish to study the map
\[\xymatrix@C+30pt{
\ds\beta^*\HHom\left[\left(\coprod_{\lambda\in\Lambda}
E_\lambda\right)\,,\,F\right]\ar[r]^-{\p_{\coprod E_\lambda}^{}} &
\ds\HHom\left[\beta^*\left(\coprod_{\lambda\in\Lambda}E_\lambda \right)\,,\,\beta^*F\right]
}\]
which, up to isomorphism, rewrites as the composite
\[\xymatrix@C+30pt{
\ds\beta^*\prod_{\lambda\in\Lambda} \HHom(E_\lambda,F)\ar[r]^-{K}&
\ds\prod_{\lambda\in\Lambda} \beta^*\HHom(E_\lambda,F)\ar[r]^-{\prod_\lambda\p_{E_\lambda}^{}}&
\ds\prod_{\lambda\in\Lambda} \HHom(\beta^*E_\lambda, \beta^*F)
}\]
The fact that $\alpha^\times K$ is an isomorphism is
by
Lemma~\ref{L99.3}. The fact that $\alpha^\times
\big[\prod_\lambda\p_{E_\lambda}^{}\big]$
is an isomorphism
is because $\alpha^\times$ respects products, and takes each
$\p_{E_\lambda}^{}$ to an isomorphism.
\eprf
\section{Some more Hochschild-style formulas}
\label{S95}
To the extent that the
results surveyed in \S\ref{S1} and \S\ref{S2} are new, they
arose out of trying to
understand the magical formulas first discovered by
Avramov and Iyengar~\cite{Avramov-Iyengar08}, and developed further
in several papers, starting with Avramov, Iyengar, Lipman and
Nayak~\cite{Avramov-Iyengar-Lipman-Nayak10}. In this section we prove
some of these formulas---what
is novel is that they are stated and proved in the unbounded derived
category.
Since the results in the section are new---at least new in this
generality---we
present complete proofs. The reader interested in the highlights can
skip ahead to Corollary~\ref{C95.13} and Example~\ref{E95.15}.
\rmd{R95.1}
Let $f:X\la Y$ be a morphism of noetherian schemes. An object $M\in\Dqc(X)$
is called \emph{$f$--perfect} if
\be
\item
$M$ belongs to $\dcoh(X)$; that is all but finitely many of the
cohomology sheaves vanish, and the non-vanishing ones are coherent.
\item
There exists an affine scheme $W$ and
a faithfully flat morphism $\rho:W\la X$ so that $(f\rho)_*\rho^*M$
is of finite Tor-amplitude.
\ee
\ermd
\lem{L95.3}
Let $Z\stackrel i\la X\stackrel f\la Y$ be composable morphisms, with
$i$ a closed immersion
and $fi$ proper. Suppose that $M$ is an $f$--perfect object
in $\Dqc(X)$ and $C\in\Dqc(X)$ is a perfect complex supported on the
image of $i$. Then $f_*(M\oo C)$ is a perfect complex in $\Dqc(Y)$.
\elem
\prf
We need to show that $f_*(M\oo C)$ has coherent cohomology and is of
finite
Tor-amplitude. The coherence of the cohomology is clear: possibly
after replacing $Z$ by an infinitesimal thickening we can assume that
the complex $C\in\dcoh(X)$, whose cohomology is supported is on $Z$, is of the form
$i_*\wt C$ for some $\wt C\in\dcoh(Z)$. But then $M\oo C=M\oo i_*\wt
C\cong i_*(i^*M\oo \wt C)$, and hence $f_*(M\oo C)\cong f_*i_*(i^*M\oo
\wt C)$. But the complex $i^*M\oo \wt C\in\Dqcmi(Z)$ has coherent
cohomology,
and as $fi:Z\la Y$ is assumed proper so does $f_*i_*(i^*M\oo
\wt C)\cong f_*(M\oo C)$.
Now for the Tor-amplitude. Choose a faithfully flat map $\rho:W\la X$
as in Reminder~\ref{R95.1}. We know that there exists an integer $n$
so that the Tor-amplitude of $(f\rho)_*\rho^*M$ lies in the interval
$[-n,\infty)$, and therefore
\[
(f\rho)_* \Big[(f\rho)^*\Dqc(Y)^{\geq 0}_{}\oo \rho^*M\Big]\eq
\Dqc(Y)^{\geq 0}_{}\oo(f\rho)_*\rho^*M
\]
is contained in
$\Dqc(Y)^{\geq -n}_{}$. The map $f\rho$ is a morphism from an affine
scheme to $Y$, therefore
$f\rho$ is quasi-affine. From the proof [not the statement]
of \cite[Corollary~2.8]{Hall-Rydh13} we
deduce that $(f\rho)^*\Dqc(Y)^{\geq 0}_{}\oo
\rho^*M=\rho^*\big[f^*\Dqc(Y)^{\geq 0}_{}\oo M\big]$
is contained in $\Dqc(W)^{\geq -n}_{}$. Since $\rho^*$ is faithfully
flat it follows that $f^*\Dqc(Y)^{\geq 0}_{}\oo M$ is contained in
$\Dqc(X)^{\geq -n}_{}$.
As $C$ is assumed to be a perfect complex on $X$ its Tor-amplitude is
contained in the interval $[-m,m]$ for some $m>0$, and hence
$f^*\Dqc(Y)^{\geq 0}_{}\oo M\oo C$ is contained in $\Dqc(X)^{\geq
-m-n}_{}$.
But then
\[
f_*\Big[f^*\Dqc(Y)^{\geq 0}_{}\oo M\oo C\big] \eq \Dqc(Y)^{\geq
0}_{}\oo f_*(M\oo C)
\]
is contained in $f_*\Dqc(X)^{\geq
-m-n}_{}\subset \Dqc(Y)^{\geq
-m-n}_{}$, and we deduce that the Tor-amplitude of $f_*(M\oo C)$ is
contained in $[-m-n,\infty)$.
\eprf
\rmd{R95.5}
Given any closed symmetric monoidal category $\cm$, that is a category $\cm$
with a symmetric tensor product and an internal Hom satisfying the usual
adjunction, there is a canonical evaluation map $\ev_{A,B}^{}:A\oo\HHom(A,B)\la
B$.
As is customary we deduce the following two
canonical maps
\[\xymatrix@C+30pt@R-20pt{
\HHom(A,B)\oo C\ar[r]^-{\alpha} & \HHom(A,B\oo C) \\
A\oo\HHom(B,C) \ar[r]^-{\beta} & \HHom\big[\HHom(A,B)\,,\,C\big]
}\]
where $\alpha=\alpha(A,B,C)$ corresponds to the map
\[\xymatrix@C+30pt@R-20pt{
A\oo \HHom(A,B)\oo C\ar[r]^-{\ev_{A,B}^{}} & B\oo C
}\]
and $\beta=\beta(A,B,C)$ corresponds to the composite
\[\xymatrix@C+30pt@R-20pt{
\HHom(A,B) \oo A\oo\HHom(B,C) \ar[r]^-{\ev_{A,B}^{}} &
B\oo\HHom(B,C)\ar[r]^-{\ev_{B,C}^{}} &C
}\]
An easy formal fact is that $\alpha(A,B,C)$ and $\beta(A,B,C)$ are isomorphisms
as long as $A$ is strongly dualizable.
\ermd
\con{C95.7}
We can of course use the basic maps of Reminder~\ref{R95.5} to
construct variants. The situation that interests us is where
$f:X\la Y$ is a morphism of schemes and $f^*\dashv f_*\dashv f^\times$
are the usual adjoint functors between $\Dqc(X)$ and $\Dqc(Y)$.
If $B,C\in\Dqc(Y)$ are any objects, then
the composite
\[\xymatrix@C+50pt@R-20pt{
f^*\HHom(B,C)\oo f^\times B
\ar[r]^-{\chi\big(\HHom(B,C),B\big)}&
f^\times\big[\HHom(B,C)\oo B\big]
\ar[r]^-{f^\times\ev_{B,C}^{}} &
f^\times C
}\]
induces by adjunction a map we
will denote $\gamma:f^*\HHom(B,C)\la\HHom(f^\times B,f^\times C)$.
If $A\in\Dqc(X)$ and $B,C\in\Dqc(Y)$ are objects, we can consider
the composites
\[\xymatrix@C+10pt@R-20pt{
\HHom(A,f^\times B)\oo f^*C\ar[r]^-{\alpha} &
\HHom(A, f^\times B\oo f^*C) \ar[r]^-{\HHom(A,\chi)} &
\HHom\big[A,f^\times(B\oo C)\big]\\
A\oo f^*\HHom(B,C)\ar[r]^-{\id\oo\gamma} &
A\oo \HHom(f^\times B,f^\times C)\ar[r]^-{\beta} &
\HHom\Big[\HHom(A,f^\times B)\,,\,f^\times C\Big]
}\]
\econ
\lem{L95.9}
Let $Z\stackrel i\la X\stackrel f\la Y$ be composable morphism of schemes
with $i$ a closed immersion and $fi$ a quasi-affine map.
Let $A\in\Dqc(X)$ be an object,
and let $K\in\Dqc(X)$ be a perfect complex supported
on the closed subset $Z$, and so that $f_*(A\oo K)$ and $f_*(A\oo K^\vee)$
are perfect in $\Dqc(Y)$, where $K^\vee=\HHom(K,\co)$
is the dual of $K$. Then, with $B,C\in\Dqc(Y)$
arbitrary, the functor $K\oo(-)$ takes
the two composites at the end of Construction~\ref{C95.7}
to isomorphisms.
\elem
\prf
Because $K$ is perfect and supported on $Z$
it is isomorphic to a bounded complex
of coherent sheaves supported on $Z$. Up to replacing $Z$
by an infinitesimal thickening we may assume there exists an
object $\wt K\in\dcoh(Z)$ with $K\cong i_*\wt K$. But then we
have isomorphisms of functors
$K\oo(-)\cong i_*\wt K\oo(-)\cong i_*\big[\wt K\oo i^*(-)\big]$.
It certainly suffices to prove that $\wt K\oo i^*(-)$ takes
the maps at the end of Construction~\ref{C95.7} to isomorphisms. The
morphism $fi$ is quasi-affine and \cite[Corollary~2.8]{Hall-Rydh13}
tells us that $(fi)_*=f_*i_*$ is conservative, hence we are
reduced to showing that
$f_*i_*\big[\wt K\oo i^*(-)\big]\cong f_*\big[K\oo(-)\big]$
takes these two morphisms of Construction~\ref{C95.7}
to isomorphisms.
But now we are in business: tensoring these two maps with the strongly
dualizable $K$ has the effect of replacing $A$ with $A\oo K^\vee$ in the
first map and with $A\oo K$ in the second. Hence we are reduced to
showing that, under the assumption that $f_*A$ is a perfect complex
in $\Dqc(Y)$, the functor $f_*$ takes both of the maps
of Construction~\ref{C95.7} to isomorphisms.
An easy exercise
with the standard isomorphisms $f_*(R\oo f^*S)\cong f_*R\oo S$
and $f_*\HHom(R,f^\times S)\cong \HHom(f_*R,S)$ allows
us to show that the functor $f_*$ takes the two maps
of Construction~\ref{C95.7}
to
\[\xymatrix@C+30pt@R-20pt{
\HHom(f_*A,B)\oo C\ar[r]^-{\alpha} & \HHom(f_*A,B\oo C) \\
f_*A\oo\HHom(B,C) \ar[r]^-{\beta} & \HHom\big[\HHom(f_*A,B)\,,\,C\big]
}\]
of Reminder~\ref{R95.5}, which are isomorphisms because $f_*A$ is perfect.
\eprf
Now we will apply our lemmas to the situation of Construction~\ref{C2.2002}.
We remind the reader:
$f:X\la Y$ is a finite-type, flat morphism of noetherian schemes,
and we formed the diagram
\[\xymatrix@C+10pt@R+10pt{
X \ar[dr]^-{\Delta} & & \\
& X\times_Y^{}X \ar[r]^-{\pi_1^{}}\ar[d]_-{\pi_2^{}} & X\ar[d]^f \\
& X \ar[r]^-{f} & Y\ar@{}[ul]|{(\diamondsuit)}
}\]
where the square is cartesian, $\pi_1^{}$ and $\pi_2^{}$ are the first
and second projections, and $\Delta:X\la X\times_Y^{}X$ is the
diagonal inclusion. We assert:
\pro{P95.11}
Let $A,B,C\in\Dqc(X)$ be objects and assume that $A$ is $f$--perfect.
Applying Construction~\ref{C95.7} to the morphism $\pi_2^{}:X\times_Y^{}X\la X$
and
the objects $\pi_1^*A\in\Dqc(X\times_Y^{}X)$ and $B,C\in\Dqc(X)$,
we obtain morphisms
\[\xymatrix@C+50pt@R-20pt{
\HHom(\pi_1^*A,\pi_2^\times B)\oo \pi_2^*C\ar[r]^-{\HHom(\pi_1^*A,\chi)\circ\alpha} & \HHom\Big[\pi_1^*A,\pi_2^\times(B\oo C)\Big] \\
\pi_1^*A\oo\pi_2^*\HHom(B,C) \ar[r]^-{\beta\circ[\id\oo\gamma]} & \HHom\big[\HHom(\pi_1^*A,\pi_2^\times B)\,,\,\pi_2^\times C\big]
}\]
We assert that, if $L\in\Dqc(X\times_Y^{}X)$
is any object supported on the diagonal, then
the functors $L\oo(-)$ and $\HHom(L,-)$
take both maps to isomorphisms, as do the functors
$\Delta^*$ and $\Delta^\times$.
\epro
\prf
Let us first observe that the statement about $\Delta^*$ and $\Delta^\times$
follows from the assertion about $L\oo(-)$ and $\HHom(L,-)$. Since
$\Delta$ is a closed immersion the functor $\Delta_*$ is conservative,
and to show that $\Delta^*$ and $\Delta^\times$ take the
two maps to isomorphisms is equivalent to showing that
the composites $\Delta_*\Delta^*$ and $\Delta_*\Delta^\times$ take
them to isomorphisms. But it is standard that
$\Delta_*\Delta^*(-)\cong\Delta_*\co_X^{}\oo(-)$
and
$\Delta_*\Delta^\times(-)\cong\HHom(\Delta_*\co_X^{},-)$.
Since $\Delta_*\co_X^{}$ is supported on the diagonal this reduces us
to the statements about $L\oo(-)$ and $\HHom(L,-)$.
Now let $\cl\subset\Dqc(X\times_Y^{}X)$ be the full subcategory of all
objects $L$ so that $L\oo(-)$ and $\HHom(L,-)$ take both maps
of the Proposition to isomorphisms. Clearly $\cl$ is a localizing
subcategory. We wish to
show that $\cl$ contains the category $\D_{\mathbf{qc},\Delta}^{}(X\times_Y^{}X)$,
that is the
full subcategory of $\Dqc(X\times_Y^{}X)$ of objects supported on the
diagonal. But the subcategory
$\D_{\mathbf{qc},\Delta}^{}(X\times_Y^{}X)$ is generated by
the objects inside it which are compact in the larger
$\Dqc(X\times_Y^{}X)$; this theorem was first proved in Thomason and
Trobaugh~\cite{ThomTro},
and for a more general, modern proof which works
for sufficienty nice algebraic stacks the reader can
see Hall and Rydh~\cite[Theorems~A, B and 4.10(2)]{Hall-Rydh13}.
This means
that any localizing subcategory, containing the compact
objects $K$ supported on the diagonal, will contain all
of $\D_{\mathbf{qc},\Delta}^{}(X\times_Y^{}X)$. It therefore suffices to
show that every compact $K$, supported on the
diagonal, belongs to $\cl$. Hence we let $K$ be a compact object
supported on the diagonal, and wish to show that $K\oo(-)$ and
$\HHom(K,-)\cong K^\vee\oo(-)$ take both maps in the
Proposition to isomorphisms.
Now the object $A\in\Dqc(X)$ is assumed $f$--perfect, and flat base-change
tells us that $\pi_1^*A$ is $\pi_2^{}$--perfect.
Consider the composable morphisms
$X\stackrel\Delta\la X\times_Y^{}X\stackrel{\pi_2^{}}\la X$; because
the composite $\id=\pi_2^{}\Delta$ is proper we may apply
Lemma~\ref{L95.3}, and because it is quasi-affine Lemma~\ref{L95.9} also
applies. More precisely: with this pair of composable morphisms
apply Lemma~\ref{L95.3} to the $\pi_2^{}$--perfect object
$\pi_1^*A\in\Dqc(X\times_Y^{}X)$ and to the perfect complexes
$K,K^\vee\in\Dqc(X\times_Y^{}X)$
supported on the image of $\Delta$, and we learn that
$\pi_{2*}^{} (\pi_1^*A\oo K)$ and $\pi_{2*}^{}(\pi_1^*A\oo K^\vee)$ are perfect in $\Dqc(X)$. But
then
Lemma~\ref{L95.9} allows us to conclude that $K\oo(-)$ and
$\HHom(K,-)\cong K^\vee\oo(-)$
take both morphisms of the Proposition to isomorphisms.
\eprf
\cor{C95.13}
Let $f:X\la Y$ be a finite-type, flat map of noetherian schemes, and
let the notation be as in Construction~\ref{C2.2002}.
For objects $A,C\in\Dqc(X)$ and $B\in\Dqc(Y)$, where $A$ is $f$--perfect, we have
isomorphisms
\begin{eqnarray*}
\HHom(A,f^!B)\oo C &\cong & \Delta^*\HHom\Big[\pi_1^*A,\pi_2^\times(f^*B\oo C)\Big] \\
\Delta^\times\Big[\pi_1^*A\oo\pi_2^*\HHom(f^*B,C)\Big] &\cong& \HHom\big[\HHom(A,f^! B)\,,\,C\big]
\big]
\end{eqnarray*}
\ecor
\prf
The classical isomorphism
$\Delta^\times\HHom(E,F)\cong\HHom(\Delta^*E,\Delta^\times F)$,
coupled with the fact that $\Delta^\times\psi(\pi_2^{}):\Delta^\times\pi_2^\times\la\Delta^\times\pi_2^!$ is an
isomorphism by Lemma~\ref{L2.2}, tell us that for any
$E\in\Dqc(X\times_Y^{}X)$ and any $G\in\Dqc(X)$ the map
$\Delta^\times\HHom\big[E,\psi(\pi_2^{})\big]:\Delta^\times\HHom(E,\pi_2^\times
G)\la \Delta^\times\HHom(E,\pi_2^!G)$
is an isomorphism. By
\cite[Proposition~A.3(ii)]{Iyengar-Lipman-Neeman13}
we deduce that
$\Delta^*\HHom\big[E,\psi(\pi_2^{})\big]:\Delta^*\HHom(E,\pi_2^\times
G)\la \Delta^*\HHom(E,\pi_2^!G)$
is also an isomorphism.
Now we turn to the proof of the Corollary. With the notation
of the Corollary, as a first step we prove
\begin{itemize}
\item
There is a natural isomorphism
$\Delta^*\Big[\HHom(\pi_1^*A,\pi_2^\times f^*B)\Big]
\cong\HHom(A,f^!B)$.
\end{itemize}
The isomorphism of $\bullet$ comes from the following string of isomorphisms
\begin{eqnarray*}
\Delta^*\Big[\HHom(\pi_1^*A,\pi_2^\times f^*B)\Big]&\cong &
\Delta^*\Big[\HHom(\pi_1^*A,\pi_2^! f^*B)\Big]\\
&\cong&\Delta^*\HHom(\pi_1^*A,\pi_1^* f^!B)\\
&\cong&\Delta^*\pi_1^* \HHom(A,f^!B) \\
&\cong& \HHom(A,f^!B)\\
\end{eqnarray*}
The first isomorphism is because the
functor
$\Delta^*\HHom(\pi_1^*A,-)$ takes the map $\psi(\pi_2^{}):\pi_2^\times
f^*B\la \pi_2^!
f^*B$
to an isomorphism. The second isomorphism is because
$\theta(\diamondsuit):\pi_1^*f^!\la\pi_2^! f^*$ is an isomorphism,
see~\ref{ST1.107.9}(ii). The third isomorphism in by Lemma~\ref{L99.5},
and the last isomorphism is because $\Delta^*\pi_1^*\cong\id^*$.
With the preliminaries out of the way, apply
Proposition~\ref{P95.11} to the objects $A,f^*B,C\in\Dqc(X)$, where
$A$ is given to be
$f$--perfect.
The Proposition tells us that the functors $\Delta^*$ and
$\Delta^\times$ take the maps below to isomorphisms
\[\xymatrix@C+50pt@R-20pt{
\HHom(\pi_1^*A,\pi_2^\times f^*B)\oo \pi_2^*C\ar[r]^-{(1)} & \HHom\Big[\pi_1^*A,\pi_2^\times(f^*B\oo C)\Big] \\
\pi_1^*A\oo\pi_2^*\HHom(f^*B,C) \ar[r]^-{(2)} & \HHom\big[\HHom(\pi_1^*A,\pi_2^\times f^*B)\,,\,\pi_2^\times C\big]
}\]
And the first isomorphism of the Corollary is by applying $\Delta^*$
to the map $(1)$ while the second isomorphism is by applying
$\Delta^\times$ to the map $(2)$. Let us take these one step at a
time, we begin be applying $\Delta^*$ to $(1)$. We obtain isomorphisms
\begin{eqnarray*}
\Delta^*\HHom\Big[\pi_1^*A,\pi_2^\times(f^*B\oo C)\Big] &\cong &
\Delta^*\Big[\HHom(\pi_1^*A,\pi_2^\times f^*B)\oo \pi_2^*C\Big] \\
&\cong&\Delta^*\Big[\HHom(\pi_1^*A,\pi_2^\times f^*B)\Big]\oo[\Delta^*\pi_2^* C]\\
&\cong& \HHom(A,f^!B)\oo C
\end{eqnarray*}
The first isomorphism is just $\Delta^*$ applied to $(1)$. The second
isomorphism is because $\Delta^*$ respects the tensor product.
The third isomorphism is the tensor product of the isomorphism in $\bullet$
with the isomorphism
$\Delta^*\pi_2^*C\cong \id^*C=C$.
A similar analysis works for $\Delta^\times$ applied to the map
$(2)$, which gives us the first isomorphism below
\begin{eqnarray*}
\Delta^\times\Big[\pi_1^*A\oo\pi_2^*\HHom(f^*B,C) \Big] &\cong &\Delta^\times\HHom\big[\HHom(\pi_1^*A,\pi_2^\times f^*B)\,,\,\pi_2^\times C\big]\\
&\cong&\HHom\big[\Delta^*\HHom(\pi_1^*A,\pi_2^\times f^*B)\,,\,\Delta^\times\pi_2^\times C\big]\\
&\cong&\HHom\big[\HHom(A,f^!B)\,,\,C\big]
\end{eqnarray*}
The second isomorphism comes from the formula
$\Delta^\times\HHom(E,F)\cong\HHom(\Delta^*E,\Delta^\times F)$.
The third isomorphism is the functor $\HHom(-,-)$ applied
to the isomorphism in $\bullet$
and the isomorphism
$\Delta^\times\pi_2^\times C\cong \id^\times C=C$.
\eprf
\exm{E95.15}
Let us work out what Corollary~\ref{C95.13} says in the affine case:
that is $f:R\la S$ will be a finite-type, flat homomorphism of noetherian
rings and, by abuse of notation, we will also write $f:\spec S\la
\spec R$ for the induced map of noetherian schemes.
We have the usual equivalences $\D(R)\cong\Dqc\big(\spec R\big)$
and $\D(S)\cong\Dqc\big(\spec S\big)$, and $f^*:\D(R)\la\D(S)$,
$f_*:\D(S)\la\D(R)$, $f^\times:\D(R)\la\D(S)$ and $f^!:\D(R)\la\D(S)$
are the affine versions of the standard functors of Grothendieck
duality. Put $\se=S\oo_R^{} S$.
In this affine case, to say that an object
$A\in\D(S)$ is $f$--perfect means that $A$ must have bounded
cohomology which is finite as $S$--modules, and
$f_*A\in\D(R)$ has finite Tor-dimension.
Let $A\in\D(S)$ be an
$f$--perfect complex, and let $B\in\D(R)$ and $C\in\D(S)$ be arbitrary.
Then the formulas of Corollary~\ref{C95.13} come down to
\begin{eqnarray*}
\Hom_S^{}(A,f^!B)\oo_S^{}C &\cong&S\oo_{\se}^{}\Hom_R^{}(A,B\oo_R^{}C)\ ,\\
\Hom_{\se}^{}\big[S,A\oo_R^{}\Hom_R^{}(B,C)\big] &\cong&
\Hom_S^{}\big[\Hom_S^{}(A,f^!B),C\big]\ .
\end{eqnarray*}
where the Homs and tensors are all derived.
The reader can find special cases of these formulas
in Avramov, Iyengar, Lipman and Nayak~\cite{Avramov-Iyengar-Lipman-Nayak10}.
The reader might note that
we have already met special cases of the
first of these formulas. If we put $A=C=S$
then
the formula specializes to
\[
f^!B \quad\cong\quad
\Hom_S^{}(S,f^!B)\oo_S^{}S\quad\cong\quad S\oo_{\se}^{}\Hom_R^{}(S,S\oo_R^{}B)
\]
of the Introduction, and if we further specialize to $B=R$ we recover
the formula $f^!R\cong S\oo_{\se}^{}\Hom_R^{}(S,S)$
of Remark~\ref{R2.9}.
\eexm
\section{A historical review}
\label{S3}
Grothendieck first mentioned that he knew how to prove a relative version of
the Serre duality theorem in his ICM talk in Edinburgh in 1958,
see~\cite{Grothendieck58C}. The first published version was
Hartshorne~\cite{Hartshorne66}; roughly speaking the construction
of $f^!$ given in \cite{Hartshorne66} is by gluing local data,
not an easy thing to do in the derived category. Three
and a half decades later
Conrad~\cite{Conrad00} expanded and filled in details missing
in~\cite{Hartshorne66}.
The presentation of the subject given here is entirely different in
spirit---it is based on early observations
by Deligne~\cite{Deligne66} and Verdier~\cite{Verdier68},
filled in and expanded greatly
in Lipman~\cite{Lipman09}. This second construction is much more global
and functorial, the usual objection to it is that it's difficult
to compute anything.
Now it is time to say what's different here from the classical literature.
Let us begin with the observation that, until the late 1980s, no one
really understood how to handle unbounded derived categories. For the
first two decades of the subject the functor
$f^*$, which involves a derived tensor product, was treated as
a functor $f^*:\Dqcmi(Y)\la\Dqcmi(X)$, while the functor $f_*$, which
involves injective resolutions, was classically viewed as a functor
$f_*:\Dqcpl(X)\la\Dqcpl(Y)$. A careful reader will note that, being defined
on different categories, these functors are not honest adjoints---there
is no counit of adjunction $f^*f_*\la\id$, and a classical version
of the treatment
of \S\ref{S1} would have had to be more delicate. Luckily for us we live
in modern times and can give the clean presentation of
the projection formula and the base-change maps of \S\ref{S1}.
The article that brought modernity
to this discipline was Spaltenstein~\cite{Spaltenstein88},
it taught us how to take injective and flat resolutions of unbounded complexes.
Spaltenstein's
article made it clear how to define the adjoint functors $f^*:\Dqc(Y)\la\Dqc(X)$ and
$f_*:\Dqc(X)\la\Dqc(Y)$.
The natural question to arise was how much of Grothendieck duality could be
developed in the unbounded derived category. The existence of a right adjoint
$f^\times:\Dqc(Y)\la\Dqc(X)$ for $f_*$
was discovered soon after, the author even
showed in~\cite{Neeman96} that it is possible to obtain this adjoint
easily and very formally using Brown representability. At the time the author
was promoting the point of view
that the right way to approach all these classical
results was to employ systematically the techniques of homotopy theory,
like Brown representability---at the time this was still a novel idea. So
Lipman challenged the author to try to use the techniques of homotopy
theory to extend Verdier's base-change theorem~\cite{Verdier68} to
the unbounded derived category. Instead of a proof
the author found a counterexample,
see~\cite[Example~6.5]{Neeman96}. There exists a cartesian square
of noetherian schemes
\[
\xymatrix@C+10pt@R+10pt{
W \ar[r]^{u}\ar[d]_{f} & X\ar[d]^{g}\\
Y \ar[r]^{v} & Z\ar@{}[ul]|{(\diamondsuit)}
}
\]
with $v$ flat (even an open immersion) and $g$ proper
(even a closed immersion), and such that the base-change map
$\Phi(\diamondsuit):u^*g^\times\la f^\times v^*$ is \emph{not} an
isomorphism. As an aside we note that the schemes in question are all
affine.
This counterexample had the unfortunate effect of
stifling the theory, for the next
twenty years it put people off trying to develop the functor $f^!$ in the
unbounded derived category. For example see Lipman's
book~\cite{Lipman09}---Lipman makes a real effort to give the results
in the greatest generality in which they were known at the time, and
for the functor $f^!$ he works
almost entirely with bounded-below complexes.
Drinfeld and Gaitsgory~\cite{Drinfeld-Gaitsgory13}
generalized a version of the theory to DG schemes,
and if the structure sheaf has negative cohomology then the category
$\Dqcpl(X)$ does not make much sense. To finesse the issue they work in
the category of Ind-coherent sheaves instead of the derived category.
In early 2013 I happened to run into Lipman at MSRI and he told me about
exciting recent work, joint with Avramov, Iyengar and Nayak, which
found a strange connection between Grothendieck duality and Hochschild
homology and cohomology. In this survey we have already met this connection
in Theorem~\ref{T2.5} and
Example~\ref{E95.15}, see also Remarks~\ref{R2.7} and \ref{R2.9}.
Theorem~\ref{T2.5} taught us about
this bizarre new map from Hochschild homology
to the dualizing complex $f^!\co_Y^{}$, and when $f$ is smooth and of relative
dimension $d$ this map happens to give an
isomorphism of $f^!\co_Y^{}$ with a shift
of the relative canonical bundle. And in \S\ref{S95} we saw that
the formulas
of \S\ref{S2} are only the tip of the iceberg, there and many
more weird and wonderful ones---we presented two of them,
together with proofs,
in Example~\ref{E95.15}.
The formulas of Example~\ref{E95.15}
are not new, special cases may be found
in~\cite{Avramov-Iyengar08,Avramov-Iyengar-Lipman-Nayak10}.
What was new in \S\ref{S95} is that we gave them as special cases of
results that hold in the unbounded derived category.
Back in 2013, when Lipman told me about the work, no one knew how
to define $f^!$ on the unbounded derived category.
Let us observe more carefully the second formula of Example~\ref{E95.15},
and for simplicity let's put $B=R$.
We remind the reader, the formula is
\[
\Hom_{\se}^{}(S,A\oo_R^{}C) \quad\cong\quad
\Hom_S^{}\big[\Hom_S^{}(A,f^!R),C\big]\ .
\]
If we fix $A$
and consider the expression on the right as a functor in $C$ then
it is clearly representable---the right hand side has the form
$\Hom_S^{}(P,-)$, where $P$ happens to be the expression $\Hom_S^{}(A,f^!R)$.
The isomorphism means that, as a functor in $C$, the expression
$\Hom_{\se}^{}(S,A\oo_R^{}C)$ is also representable, in particular it commutes
with products---which is
far from obvious. The challenge Lipman gave me was to try to use Brown
representability to prove these formulas.
There is such a proof, and Iyengar and I are working on writing it up. But
this survey is about another direction our research took: in trying to
understand better these mysterious formulas we developed the natural
transformation $\psi(f):f^\times\la f^!$---early hints of it may be found
in Lipman~\cite[Exercise~4.2.3(d)]{Lipman09}.
What was new were the naturality and functoriality properties
of $\psi$, see~\cite{Iyengar-Lipman-Neeman13} for some illustrations
of their value. Because at the time $f^!$
was defined only on the bounded-below derived category our results imposed
artificial boundedness restrictions, and it was a natural challenge to try
to remove them. Working in the category of Ind-coherent sheaves, as in
Drinfeld and Gaitsgory, is clearly wrong for this problem---the formulas
of~\cite{Avramov-Iyengar-Lipman-Nayak10} live in the derived category.
The article \cite{Neeman13} was written to address this problem, in it
Grothendieck duality is developed in the unbounded derived category,
and we gave a brief summary of some of the results in \S\ref{S1}. In
\S\ref{S2} and \S\ref{S95}
we gave illustrations of how one can approach the
unbounded versions of the formulas
of~\cite{Avramov-Iyengar08,Avramov-Iyengar-Lipman-Nayak10,Iyengar-Lipman-Neeman13} using
the techniques surveyed in this paper---we
proved the formula $f^!=\Delta^*\pi_2^\times f^*$
for unbounded complexes in Proposition~\ref{P2.202},
while Example~\ref{E95.15} showed us how to derive the
reduction formulas of Avramov and Iyengar.
These formulas occur in
\cite{Avramov-Iyengar08,Avramov-Iyengar-Lipman-Nayak10,Iyengar-Lipman-Neeman13}, but with unnatural
boundedness hypotheses.
The reader might be puzzled. We mentioned that, twenty years ago, I produced
a counterexample~\cite[Example~6.5]{Neeman96} to the unbounded
version of Verdier's base-change theorem. There exists a cartesian square
of schemes
\[
\xymatrix@C+10pt@R+10pt{
W \ar[r]^{u}\ar[d]_{f} & X\ar[d]^{g}\\
Y \ar[r]^{v} & Z\ar@{}[ul]|{(\diamondsuit)}
}
\]
with $v$ an open immersion and $g$ proper, and such that the base-change map
$\Phi(\diamondsuit):u^*g^\times\la f^\times v^*$ is not an
isomorphism. So what has changed in two decades?
What's new is Theorem~\ref{T1.7}(ii): it tells us that, as long as
we further assume that $f$ is \emph{of finite Tor-dimension,} the problem goes
away and $\Phi(\diamondsuit)$ is an isomorphism. When we compare two
compactifications of $X$ we end up with cartesian squares of the
form
\[
\xymatrix@C+10pt@R+10pt{
X \ar[r]^{u}\ar[d]_{\id} & \ov X\ar[d]^{g}\\
X \ar[r]^{v} & \ov X'\ar@{}[ul]|{(\diamondsuit)}
}
\]
and the identity $\id:X\la X$ is of finite Tor-dimension. Thus the cartesian
squares that come up in the proof that $f^!=u^*p^\times$ is independent of
the factorization of $f:X\la Y$ as $X\stackrel u\la\ov X\stackrel p\la Z$
all have base-change maps which are isomorphisms.
The place where the old counterexample rears its ugly head is when it comes
to composition. The counterexample gave a commutative square (actually,
even cartesian) and hence we have $gu=vf$. Now $u$ and $v$ are open immersions
while $f$ and $g$ are proper, hence $u^!=u^*$, $v^!=v^*$, $f^!=f^\times$ and
$g^!=g^\times$. On the other hand $u^!g^!=u^*g^\times$ is not isomorphic to
$f^!v^!=f^\times v^*$. We have already mentioned that, in
the old counterexample, the base-change map
$u^*g^\times\la f^\times v^*$ is not an isomorphism, but even more is true, the
functors are not isomorphic. The 2--functor $(-)^!$ is genuinely only
oplax---there are natural maps $\rho(f,v):(vf)^!\la f^!v^!$ and
$\rho(u,g):(gu)^!\la u^!g^!$, but clearly they cannot both be isomorphisms.
As it happens, in this
particular example $\rho(u,g)$ is an isomorphism while $\rho(f,v)$ isn't.
The situation is not hopeless: \ref{ST1.107.4} gives useful criteria for
$\rho(f,g)$ to be an isomorphism, and in \S\ref{S2}, \S\ref{S99}
and \S\ref{S95}
we illustrated
how this can be applied to obtain unbounded versions of the results
of~\cite{Avramov-Iyengar08,Avramov-Iyengar-Lipman-Nayak10,Iyengar-Lipman-Neeman13}.
The illustrations of \S\ref{S2} also
showed how, with all these new methods, the abstract
nonsense approach to the subject pioneered by Deligne, Verdier and
Lipman can produce explicit computational formulas simply and easily. The
technicalities are different: the
``residual complexes'' of Grothendieck are replaced by the more
standard tools of Hochschild homology.
In passing let me note that
Hochschild homology is a {\it K}--theoretic invariant,
and its appearance raises the question whether more sophisticated
{\it K}--theoretic invariants might shed even more light on the
subject of Grothendieck duality. This is a volume on {\it K}--theory
and its applications to algebraic geometry, and it seems the
appropriate place to raise this question.
\section{Generalizations}
\label{S4}
In March 2016 I received from the journal four referees' reports on my
article~\cite{Neeman13}. Mostly the referees' comments were simple enough
to address, but there were two difficult issues. One referee wondered what
happens if we relax the noetherian hypothesis, while another suggested that
I develop the entire theory in the generality of stacks.
This led me to think more carefully about these points. The noetherian
hypothesis seems indispensable, at least for this approach to the theory---some
of the lemmas have non-noetherian versions, but there is a point at which
the argument runs into a brick wall without the noetherian assumption.
But I'm happy to report that, under
relatively mild hypotheses, everything generalizes to noetherian
algebraic stacks.
In fact the stacky version is cleaner to state. Algebraic stacks
naturally form a 2-category, as do triangulated categories.
The clean way to think about the theory is to view $(-)^*$, $(-)^\times$
and $(-)^!$ as 2-functors from [suitably restricted] algebraic stacks
to triangulated categories, with some relations among them. These relations
can be phrased in terms of the existence of certain natural transformations
relating these functors, and the assertion that certain
pairs of composites of natural transformations agree. For example:
it turns out that $(-)^*$ has the the structure of a monoid, meaning
there is a pseudonatural transformation $(-)^*\times(-)^*\la(-)^*$,
and $(-)^\times$ and $(-)^!$ are oplax modules over it. The
map $\psi$ turns out to be an oplax natural transformation
$\psi:(-)^\times\la(-)^!$ which is a module homomorphism. Anyway: the
reader can find a thorough discussion in the introduction to
\cite{Neeman13}.
This led to an expository conundrum in writing the current survey---it was unclear what was
the right generality for the results. Avramov, Iyengar, Lipman
and Nayak work with noetherian schemes, but allow the
morphisms to be essentially of finite type (rather than the more
restrictive finite type), and sometimes of finite Tor-dimension (rather than
flat). But the methods they use don't work for noetherian stacks---at
least not yet---because one doesn't yet know that a morphism of
noetherian stacks which is essentially of finite type has a Nagata
compactification. Nayak~\cite{Nayak09} proved the existence of Nagata
compactifications for
morphisms of \emph{noetherian schemes} essentially of finite type, but so far no
one has generalized this even to algebraic spaces. In other words: the
results in this paper generalize in more than one direction, and at
present I do not know a common generalization that covers everything
that can be proved by the methods.
The compromise I made was to present the arguments in the
intersection of the known cases, that is finite-type, flat maps of noetherian
schemes, and leave to the reader the various generalizations. But I
did make an effort to give proofs that are easily adaptable, so that
the extension to (for example) algebraic stacks is straightforward.
When I gave the talk at TIFR, which amounted to a brief summary of
this survey, Geisser, Kahn, Saito
and Weibel raised the question of what portion of
the ideas might be transferrable to the six-functor formalism. Because
shortly after giving the talk I received the referees' reports, with the
questions about the non-noetherian and stacky versions of Grothendieck
duality, I haven't yet had the opportunity to think about this
other question. The six-functor
situation is another place where one defines functors like $f^!$ using
good factorizations of $f$, and it is eminently sensible to ask if there
might be an analog of the fancy, unbounded version of the base-change
theorem and of its consequences.
The question is natural enough and I would like to come back to it when
I have more time. In the interim I record it for others to study.
\def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
1706.06328
|
\section{Introduction}
Online websites are often open-ended and flexible interfaces, supporting interaction styles by users that include
exogenous actions and high noise rate. Such interfaces provide a rich framework for users and are becoming increasingly
prevalent in many web and mobile applications, but challenge conventional plan
recognition algorithms for inferring users' activities with the software.
The open-ended nature of these settings afford a rich spectrum of
interaction for clients: they can perform the same task in many different
ways (such as logging in through a website or by a mobile application),
engage in exploratory activities involving trial-and-error, they
can repeat activities indefinitely (browsing), and they can interleave between
activities.
This paper focuses on inferring customers' activities for
an Internet financial company, in which customers widely engage
in exploratory behavior, and present initial results for plan
recognition in such settings that can be the basis of future work in this direction.
Our data set consists of customer sessions of a large financial company\footnote{All interactions and examples were altered to preserve privacy}. The customer sessions are recorded as low-level activities of 'click stream' --- web page visits and user interface actions. This data is a rich source of information about customer's behavior, but cannot be used as-is to describe the intentions or plans of the customer: buy product, sell product, manage account, etc. The underlying model used for the recognition process is a set of hierarchical task networks (HTNs) called a ``plan library'', that allows hierarchical decomposition of a task, which can be used both as an explanatory accessory to an overseer and for prediction of future actions.
Our main goal is to detect frauds, validate transactions and predict future actions --- both in terms of the user's intents and in terms of the execution of these intents.
The basic stream of information can be thought of as a recording of customer's
\textit{conversation} with the system. Understanding the language of conversation would bring
benefits in various areas. For example:
\begin{itemize}
\item Predictive verification --- by following customer's actions in
real time, we can have our systems guess next actions and
alert when an agent biases from their predictive plan.
\item Proactive validation --- if we can understand the purpose of
customer's visit based on the beginning of their activity, we
can let our systems start validation of the session even
before it takes place, facilitating faster response and better
utilization of resources.
\item Computer security --- certain types of fraud are performed by
malicious software controlling customer computers. We may be able to distinguish between malicious and benign actors on the other side of the wire.
\end{itemize}
However, as a flexible interface, the system allows the user to engage in more than one task at a time (making multiple purchases), make mistakes (pressing a button twice instead of once), repeat actions (browse between different parts of the interface) and more. All of these are features that define an exploratory environment~\cite{reddyGalShieber09}.
While these traits make the interface useful and convenient to the user, it challenges conventional plan
recognition algorithms for inferring users' activities with the
software, as they need to reason about all these factors ~\cite{AG13,uzan2015plan}.
\section{Related Work}
The tasks described in the introduction are traditionally addressed by activity recognition algorithms, which are used to tokenize the stream of data into detectable actions~\cite{bao2004activity,liao2006location,Rick2017}. However, these algorithms usually cannot perform higher levels of inference to describe the agent's behavior or predict the future actions of the agent ~\cite{hammerla2016deep,Kristina2017}. Different approaches also used an HMM representation or data-driven learning to elicit tasks from low-level activities~\cite{natarajan2008logical,madani2009prediction}.
Some notable exceptions are works by ~\cite{qin2004attack,duong2005activity,talamadupula2014coordination} which try to combine low level activities and higher level domain knowledge, or ~\cite{talamadupula2014coordination} which propose a multi-agent model for robot collaboration based on plan recognition. However, the works in these lines of research tend to use a domain-theory based recognition or other probabilistic representations, which do not capture the hierarchical nature of task decomposition like Hierarchical task networks (HTNs) and plan libraries.
In order to address the above tasks, we need to use hierarchical plan
recognition, a field of research exploring algorithms that
recognize the plans of the agent based on a partial sequence of
actions, and predict future actions ~\cite{kautz87}.
Few works~\cite{Megenuzzi2017,geib2001plan} did use a plan library as the underlying domain knowledge for the task, but they did not provide predictions or used this information to formalize the output. Other works~\cite{jarvis2005identifying} do output a complete hierarchy, but the plan recognition algorithm used does not work well in exploratory environments.
In this work, we present a criteria for constructing the plan library that will then be used by the agent to perform the plan recognition task. Thus, we do not need to use an activity recognizer. We integrate instead the low-level activities directly into the plan recognition task.
\section {Definitions}
Plan recognition takes as input a plan library and a partial sequence of observations, and outputs a set of possible explanations.
\subsection{Plan Library}
The plan library can be thought of as the language of actions, as well as goals we want to be able to
recognize. The library must specify \textit{terminals}, \textit{non-terminals},
\textit{goals}, and \textit{derivation rules}.
\begin{definition} (Explanation)
\label{def:pl}
A plan library is a tuple $PL=\langle T,NT,G,R \rangle$, where $T$ is a set of are low-level, observable action,$NT$ is a set of complex level actions, composed from
either $T$s or other $NT$s, $G$ is a subset of non-terminals corresponding to the
highest level of actions, representing the goals the agent can
achieve and $R$ is a set of derivation rules which specify how each complex
action can be decomposed into other actions
of the form $ \langle nt \rangle \rightarrow \langle sequence \rangle \mid \langle order \rangle$, where:
\begin{itemize}
\item $\langle nt \rangle$ is a non-terminal, a complex action.
\item $\langle sequence \rangle$ is a sequence of actions which compose the
complex action.
\item $\langle order \rangle$ is a list of tuples defining the
the ordering of actions --- actions must appear in
the same order as in the tuple, but ordering between different
tuples is unspecified. For example, `(login, addName)` states
that the user must log-in before adding a card.
\end{itemize}
\label{def:plan-library}
\end{definition}
Consider the following very simple plan library for our domain.
\begin{itemize}
\item $T = \{ login, addName, addCredit, signup, \\
submit, home, payment, success, transfer, confirm\}$
\item $NT = \{ AddAccount, Buy \}$
\item $G = \{ AddAccount, Buy \}$
\item $R = \\
AddAccount \rightarrow login, addName, addCredit\\
\mid [(login,addName) (addName,addCredit)]\\
AddAccount \rightarrow signup, addName, submit \mid [(signup,addName) (addName,submit)]\\
Buy \rightarrow home, payment, success \mid [(home,payment) (payment,success)]\\
Buy \rightarrow home, transfer, confirm \mid [(home,transfer) (transfer,confirm)] $
\end{itemize}
\subsection{Plans and Explanations}
Based on the definitions above, we can apply an algorithm to a
sequence of actions to obtain explanations about ongoing
behaviors and anticipated future actions.
A plan is a labeled tree $p=(V,E,\mathcal{L})$, where $V$ and $E$ are the nodes and edges of the tree, respectively,
and $\mathcal{L}$ is a labeling function $\mathcal{L}: V \rightarrow NT\cup T$ mapping every node in the tree to either a basic or a complex action in the plan library. Each inner node is labeled with a complex action such that its children nodes are a decomposition of its complex action into constituent actions according to one of the refinement methods.
The set of all leaves of a plan $p$ is denoted by $leaves(p)$, and a plan is said to be {\em complete} iff
all its leaf nodes are labeled basic actions, i.e., $\forall v\in leaves(p), \mathcal{L}(v)\in T$.
An \emph{observation sequence} is an ordered set of basic actions that represents actions carried out by the observed agent.
A plan $p$ \emph{describes} an observation sequence $O$ iff every observation is mapped to a leaf in the tree. Formally, there exists an injective function $f:O\rightarrow leaves(p)\cap T$ such that $f(o)=v$.
The observed agent is assumed to plan by choosing a subset of complex actions as intended goals and then carrying out a separate plan for completing each of these goals.
An agent may pursue several goals at the same time.
Therefore, an explanation can include a set of plans, as described in the following definition:
\begin{definition} (Explanation)
\label{def:exp}
An explanation for an observation sequence
is a set of plans such that each plan describes a mutually
exclusive subset of the observation sequence and taken together the plans describe all of the observations. We then say that the explanation \emph{describes} the observation sequence.
\end{definition}
\begin{figure}[t]
\centering
\includegraphics[width=8cm]{Explanation}
\caption{An Explanation with Three Plans.}
\label{fig:exp}
\end{figure}
For the rest of this paper we will use a running example based on
the following sequence of processed observations:
$[home, login, addName, login, addCredit]$
When we submitted the sequence and the plan library to a plan
recognition algorithm, we obtained two explanations. The first one is presented in Figure~\ref{fig:exp}.
This explanation is a set of trees. In each tree the leaves
are either the actions we observed, or actions predicted
to perform in the future. A special tag $frontier$ specifies
the location in the tree from where the expansion may
continue, based on future actions. Leaves with this tags are colored
in green in the figure. Future actions that still has preceding actions
that have not been executed are colored with blue.
If a tree has no `frontier` leaf (top tree in our example),
then the goal is fully described by actions.
The output above represents the following explanation
of the input sequence: (1) The agent performed one buy transaction and two account
additions; (2) The agent performed the first action in the buy task, but
then stopped -- we expect that the next action to complete this
task is $transfer$;
(3) Regarding the account additions, one was completed
and the other was not.
For our very basic example, one can think about a brute-force
algorithm for explaining the activity. However in general, a
sophisticated algorithm is required to provide short and
meaningful explanations quickly, in a manner suitable for online
processing. The remaining of this paper talks about the very beginning of work
in collaboration with analysts and developers from the financial company.
\section{Recognition Components}
\label{sec:Rec}
We now detail the components we use to perform the plan recognition task from the raw click stream of costumer sessions.
\subsection{Preprocessing}
A session is the basic unit of interaction between a customer and the system. Efficient analysis and validation of sessions in real time is crucial for technical realization of the company's business. The raw input in our domain is a stream of costumer sessions. Each session is constructed from a list of entries, where each entry has a specific timestamp, user information and a special label describing the page on the company's site the user is visiting. The average length of a session was $80.49$ entries.
As a first step, we needed to reduce this number as many entries are irrelevant to the costumer's main tasks in the system and integrating them into terminals in the plan library would be both inefficient and less informative. We defined the following criteria for a session entry to be considered relevant to us:
\begin{definition} (Landmark in PL)
\label{def:landmark}
A raw entry $e$ in a complete session $S$ is considered a landmark in relation to a plan library $PL$ if $S \setminus \{e\}$ cannot describe a plan from $PL$.
\end{definition}
In collaboration with our partners inside the company, we elicited about 20 types of entries, defined by their page labels, which are landmarks for executing the relevant tasks. We classified each of the pages into a suitable basic action as they appear in our plan library. Counting only these pages as relevant actions we wish to observe, we received an average of $9.68$ observations per session (with stdev of $7.26$).
However, as an exploratory environment, many entries in a session are redundant even if they can be considered landmarks, as the user can make mistakes or perform a landmark action more times than necessary. To perform an elimination of these entries and to remain only with the most relevant observations, we performed another sifting. We discarded all observations that were a repetition of the earlier observation. After this elimination, we reduced the average number of relevant observations per session to $3.68$ (stdev=$2.62$).
\subsection{Recognition}
There are many possible strategies that a costumer can use to perform interactions, and variations within each due to exploratory activities and mistakes carried out by the costumer.
Even after the preprocessing we enforced on the set of actions performed by the costumer, there are still two types of exploratory behaviors which can hinder the ability of the recognizer the infer the costumer's actions correctly: (1) Exogenous actions: even after our preprocessing effort, many actions cannot be combined together when looking at the complete sequence, thus the plan recognizer cannot output any explanation the described all of the actions in the session. (2) One explanation may relate a given action to a relevant task, while another may relate this action to a failed attempt or a mistake. The space of possible explanations can become very large, even for a small number of observations.
The CRADLE algorithm~\cite{Mirsky2017} proposes solutions for both of these problems using three components:
\begin{description}
\item [Inference] it receives as input a plan library, an observation sequence, and outputs a set of explanations,
each of which is an explanation of the observation sequence in the sense of Definition~\ref{def:exp}.
\item [Filters] it filters redundant explanations according to a set of
domain-independent conditions. A filter is a function taking a
candidate explanation $e$, and returning \emph{true} or \emph{false}
depending on whether the candidate explanation does or does not pass
a certain condition. For our presented domain, we tried several values and filter types and finally set 3 filters: (1) The number of plans in the explanation is less than or equal to the average; (2) The number of frontier nodes in the explanation is less than or equal to the average; (3) The number of different plans in the explanation is less than 4. We discard all explanations which do not pass these thresholds
\item [Exogenous Actions] CRADLE can handle exogenous actions and mistakes and
can omit them from the set of explanations as needed.
\end{description}
\section{Empirical}
We used a click stream of selected sessions, labeled by the company's analysts as sessions containing the relevant tasks. In total, we tested 3 types of sessions, with 50 instances of each session type: Buy, Add account for existing user (AAExist) and Add account for new user (AANew).
For each of these sessions, we performed the process described in Section~\ref{sec:Rec}.
Figure~\ref{fig:type} presents the average number of different types of explanations outputted by CRADLE per task: Total is the total number of explanations; Full Plan is the number of explanations in which at least on plan was completed; No Open means the number of explanations in which all plans were completed and had no open frontier. Notice that even in sessions that were labeled by analysts as relevant to some task, the plan recognition process shows that only a small portion of the sessions contain a completed task. These results are similar in the PHATT runs, even though it does not discard explanations, thus might output more explanations with a completed task. We intend to validate these results with other data to evaluate if these sessions were indeed sessions of incomplete tasks.
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{chart.PNG}
\caption{Average Number of Explanations by Type.}
\label{fig:type}
\end{figure}
In order to evaluate the performance of CRADLE, we compared the runtime and number of outputted explanation in comparison to the PHATT algorithm, augmented with the exogenous actions handling of CRADLE (without this augmentation, PHATT would not be able to output any explanation at all that describes the complete sequence of observations).
Table~\ref{tab:results} summarizes these runs.
\begin{table}[b]
\centering
\label{tab:results}
\begin{tabular}{|c|r|r|r|}
\hline
& \multicolumn{1}{c|}{Buy} & \multicolumn{1}{c|}{AAExist} & \multicolumn{1}{c|}{AANew} \\ \hline
Session Entries & 147.02 & 14.13 & 80.33 \\ \hline
Observations & 5.64 & 2.06 & 3.33 \\ \hline
CRADLE Explanations & 8.74 & 1.00 & 1.33 \\ \hline
PHATT Explanations & 18.32 & 1.00 & 2.00 \\ \hline
CRADLE Time (seconds) & 0.06 & 0.01 & 0.01 \\ \hline
PHATT Time (seconds) & 0.07 & 0.01 & 0.03 \\ \hline
\end{tabular}
\caption{Runtime and Explanation Set Size for CRADLE and PHATT}
\end{table}
The first and most important thing to notice in the table, is the difference between the original length of a session entry, and the outputted set of explanations by the plan recognizers. An average decrease of $83\%$ in the number of entities representing the session. Such a decrease allows a faster analysis of the session, since the number of possible explanations is exponential with the number of observations~\cite{Mirsky2017}. For example, even with the pruning of CRADLE, the average number of explanations it outputted for the AANew sessions is $146.06$ (compared to $1.33$ with the preprocessing). Moreover, the outputted explanations have a structure imposed by the plan library that the original stream lacked.
The second point to notice it that the values of the AAExist case are similar in both algorithms. We attribute this to the fact that performing this task is always performed in the same fashion, without any options to bias in the interface of the company's website.
\section{Future Work}
This a preliminary work for using plan recognition for exploratory environments on real-world click streams. It uses a novel approach for plan recognition from bare-bone UI data, which reasons about the plan library in the lowest recognition level in order to define the relevancy of actions in our domain, and then uses it to perform plan recognition.
While we manage to process low level data using mostly tools intended for higher levels of inference, there is still much to be done from both ends of the process:
First, we expect that we can use most intelligent tools for the preprocessing stage, either from the world of activity recognition or natural language processing. As we started to show here, we wish to use the domain knowledge (e.g. the plan library) in our low-level action detection.
Second, we wish to visualize this information in a coherent manner that will allow analysts evaluate sessions in real time for fast verification and validation, or to be able to counter adversarial behavior in time.
|
1109.4477
|
\subsection{Introduction}
Ever since the original proposal of a single molecule rectifying diode by A.
Aviram and M. A. Ratner\cite{1}, there has been great interest in electron
transport (both charge and spin) at the single molecule level. Theoretical as
well as experimental work in this direction has led to the promising field of
molecular electronics\cite{6}\cite{8}. With an eye on applications it is
expected that the understanding of quantum electron transport at the molecular
scale is a key step to realizing molecular electronic devices\cite{3}.
Experiments on conduction in molecular junctions are becoming more common,
\cite{4}\cite{5} and references therein. On the experimental front, the most
common methods of contacting individual molecules are through scanning
tunneling microscope tips and mechanically controlled break junctions
\cite{6}\cite{8}. Early experiments focused on the absolute conduction and on
trends such as dependence on wire length, molecular structure, and
temperature. From a theoretical point of view, investigating electron
transport in an electrically contacted molecule is a challenging problem. In
this system, the interaction of the electronic degrees of freedom with the
vibrational ones of the molecule need to be considered. In addition, there can
be further complications arising from the electron-electron interaction on the
molecule as well as effects of the environment surrounding the system. Most of
the formal theoretical work on transport in molecular electronics has relied
on the generalized master equations approach \cite{9}\cite{10} and the non
equilibrium Green's function (NEGF) method\cite{7}.
As mentioned above, an important feature that can affect charge transport in a
single molecule is the coupling of electrons to quantized
molecular\ vibrational motion, phonons \cite{19}\cite{20}. For transport
through a single level quantum dot molecule a lot of work has already been
done taking into account vibrational degrees of freedom\cite{11
\cite{12}\cite{13}. In this paper, we consider transport through a single
molecule consisting of two coupled quantum dots in parallel configuration.
Electron transport in double quantum dots has been an area of active research,
\cite{34} and references therein. In parallel configuration electron from the
lead can tunnel through either of the two dots. We address the role of finite
coupling between the dots on the electronic transport. We also take into
account the electron-phonon interaction as well as the dissipative effects of
the environment. It has been established that it is important to take into
account electron-phonon interaction in the study of transport in single and
double dot systems\cite{35}. Here we show that the inter-dot coupling will
significantly affect transport when a single electron can occupy either dot in
the presence of electron-phonon interaction. The difference in the transport
properties with and without inter-dot coupling will be discussed in detail in
this work. For a single level molecular system with electron phonon
interaction, phonon side band peaks start disappearing with increasing
tunneling rate\cite{2} whereas we find that for a coupled dot molecule the
phonon peaks survive even if the tunneling rate from the leads is increased.
\subsection{Model}
Our system is a laterally coupled double quantum dot. It is assumed that only
a single level in each dot participates in transport. We allow finite coupling
between the single electronic levels of the two dots. An electron from the
leads can tunnel through either of the two dots.
The full Hamiltonion describing our system i
\[
H=H_{M}+H_{Leads}+H_{T}.
\]
It is the sum of the electron Hamiltonian of the coupled dot molecule $H_{M}$,
the Hamiltonian of the leads $H_{Leads}$, the tunneling Hamiltonian $H_{T}$
describing the molecule-to-lead coupling. We explain each term in the full
Hamiltonian separately
\begin{equation}
H_{M}=\sum_{i=1,\sigma}^{2}\epsilon_{i\sigma}d_{i\sigma}^{\dag}d_{i\sigma
}+\sum_{i,j=1i\neq j,\sigma}^{2}t_{i,j,\sigma}d_{i\sigma}^{\dag}d_{j\sigma}.
\end{equation}
The first term represents two discrete energy levels, one in each dot.
$d_{i\sigma}^{\dag},$ $d_{i\sigma}$ create and annihilate an electron in state
$|i\sigma>$ on the dot$.$ The second term represents inter-dot coupling where
$t_{i,j,\sigma}$ represents coupling between the electronic states of the two
dots. We have assumed $t_{i,j,\sigma}=t_{j,i,\sigma}=t$
\begin{equation}
H_{Leads}=\sum_{k}\epsilon_{\nu k\sigma}c_{\nu k\sigma}^{\dag}c_{\nu k\sigma}.
\end{equation}
This represents the leads Hamiltonian. Indices $\nu,k,\sigma$ refer to the
left/right leads, the electronic wave vector in either lead, and leads
electrons spin.
The tunneling Hamiltonian describes hopping between the leads and the
molecule. Direct hopping between the two leads is neglected
\begin{equation}
H_{T}=\sum_{i,k}(V_{i,\nu,k,\sigma}c_{\nu k\sigma}^{\dag}d_{i\sigma}+hc).
\end{equation}
The first term represents creation of electrons in the lead and annihilation
of electrons in the coupled dots, while the second term represents creation of
electrons in the coupled dots and annihilation in the lead. Here
$V_{i,\nu,k,\sigma}$ denotes lead-system coupling (hopping) amplitude and $hc$
denotes hermitian conjugation. We consider contacting the coupled dots with
two metallic leads.
Finally, phonons, electron-phonon coupling, heat bath and phonons coupling are
described by the following Hamiltonian\cite{17} (In this calculation we work
with $\hbar=1$)
\begin{align}
H_{phonon+Bath} & =\sum_{q}\omega_{q}a_{q}^{\dag}a_{q}+\sum_{\alpha=1
^{2}\sum_{q}\lambda_{\alpha\alpha}^{q}(a_{q}^{\dag}+a_{q})d_{\alpha}^{\dag
}d_{\alpha}+\nonumber\\
& \sum_{\beta}\omega_{\beta}b_{\beta}^{\dag}b_{\beta}+\sum_{q\beta}N_{q\beta
}(b_{\beta}^{\dag}+b_{\beta})(a_{q}^{\dag}+a_{q}).
\end{align}
The electron-phonon interaction is included with in the first Born
approximation, which is resonable when electron phonons coupling is weak. For
a single dot molecule this problem was studied in \cite{21}\cite{22}, whereas
we consider two dots in the molecule interacting with phonons of frequency
$\omega_{q}$. The first and the third term represents phononic and heat bath
energy. Here $\omega_{q}$, $\omega_{\beta}$ are phonon and heat bath energies.
$a_{q}^{\dag}a_{q}(b_{\beta}^{\dag}b_{\beta})$ are phonons creation and
annihilation operators (heat bath creation and annihilation operators). The
second term represents electron-phonon interaction and $\lambda_{\alpha\alpha
}^{q}$ is the coupling strength of this interaction. The last term represents
phonon and heat bath coupling and $N_{q\beta}$ is the coupling strength of
phonon heat bath interaction.
\subsection{Method}
Our approach is based on the nonequilibrium Green function technique\cite{29
\cite{30}, which is now a standard technique in mesoscopic physics as well as
molecular electronics. We follow the formulation pioneered by Meir and
Wingreen\cite{31}, Jauho and co-workers\cite{14}\cite{33}. The case of
intermediate and strong electron-phonon coupling at finite tunneling rates is
the most interesting regime but it is also the most difficult. Only the
approaches by Flensberg\cite{24}. and Galperin et al\cite{17}exist, both
starting from the exact solution for the isolated system and then switching on
tunneling as a perturbation\cite{2}.The current from lead $\nu$ is given by
the well known expression\cite{32
\begin{align}
J^{\nu}(\epsilon) & =\frac{e}{2\pi}\sum_{i,j,k}\int_{-\infty}^{\infty
}d\epsilon\lbrack\{G_{ij}^{r}(\epsilon)-G_{ij}^{a}(\epsilon)\}\Sigma_{ji\nu
k}^{<}(\epsilon)+\\
& G_{ij}^{<}(\epsilon)\{\Sigma_{ji\nu k}^{a}(\epsilon)-\Sigma_{ji\nu k
^{r}(\epsilon)\}].\nonumber
\end{align}
Here $\Sigma_{ji\nu k}^{<,r,a}$ represents lesser, retarded and advanced self
energies of leads and coupled dot molecule. $\Sigma_{ji,\nu}^{<,r,a}=\sum
_{k}\{V_{j\nu k}^{\dag}g_{\nu}^{<,r,a}(\epsilon)V_{i\nu k}\}$ where $g_{\nu
}^{<,r,a}(\epsilon)$ is the lesser,retarded and advanced Green's function of
the leads.\cite{31}.
We employ the wide-band approximation, where the self-energy of the coupled
dot molecule due to each lead is taken to be energy independent and is given
b
\[
\sum_{k}\Sigma_{ji\nu k}^{r,a}(\epsilon)=DV_{i\nu k}V_{j\nu k}^{\dag
\int_{-\infty}^{\infty}\dfrac{d\epsilon_{k}}{(\epsilon-\epsilon_{k}\pm i\eta
)}\qquad\eta\rightarrow0^{\mp}=-i2\pi D(\epsilon_{\nu})V_{i\nu k}V_{j\nu
k}^{\dag}=\mp i\dfrac{\Gamma_{ji}^{\nu}}{2}.
\]
Here $D$ is the constant energy density of the leads. Similiarly the lesser
self energy can be written a
\begin{equation}
\sum_{k}\Sigma_{ji\nu k}^{<}(\epsilon)=i\Gamma_{ji}^{\nu}f(\epsilon)
\end{equation}
where $\Gamma_{ji}^{\nu,}s$ are the tunneling rates (coupling of leads with
the molecule) and $f(\epsilon)$ is the Fermi-Dirac distribution function.
Now by employing the current symmetrization and line width proportionality
approximations\cite{14} we obtai
\begin{equation}
J^{\nu}(\epsilon)=\frac{ie}{2\pi}\sum_{i,j}\left( \frac{\Gamma_{ji}^{L
\Gamma_{ji}^{R}}{\Gamma_{ji}^{L}+\Gamma_{ji}^{R}}\right) \int_{-\infty
}^{\infty}d\epsilon\lbrack\{G_{ij}^{r}(\epsilon)-G_{ij}^{a}(\epsilon
)\}\{f^{L}(\epsilon)-f^{R}(\epsilon)\}].
\end{equation}
By using the equation of motion technique, we work out the coupled dot
molecule Green's function and fin
\begin{equation}
(g_{ii}^{r}(\epsilon))^{-1}=\left( \epsilon-\epsilon_{i}-\frac{\mid
t_{i,j}\mid^{2}}{\epsilon-\epsilon_{j}}\right) \qquad i\neq j.
\end{equation
\begin{equation}
(g_{i,j}^{r}(\epsilon))^{-1}=\frac{\left( \epsilon-\epsilon_{j}\right)
}{t_{j,i}}\left( \epsilon-\epsilon_{i}-\frac{\mid t_{i,j}\mid^{2}
{\epsilon-\epsilon_{j}}\right) \qquad i\neq j
\end{equation}
Employing Dyson's equation, we can writ
\begin{equation}
G_{ij}^{r}(\epsilon)=\left[ (g_{ij}^{r}(\epsilon))^{-1}-\Sigma_{ji\nu k
^{r}(\epsilon)\right] ^{-1}.
\end{equation}
The differential conductance in the absence of electron-phonon interaction i
\begin{equation}
\frac{dJ^{\nu}}{dV}=\frac{e^{2}}{2\pi}\sum_{i,j}\left[ \dfrac{\Gamma_{ji
^{L}\Gamma_{ji}^{R}}{\left( (g_{ij}^{r}(\epsilon))^{-1}\right) ^{2}+\left(
\dfrac{\Gamma_{ji}^{L}+\Gamma_{ji}^{R}}{2}\right) ^{2}}\right] .
\end{equation}
If the electron-phonon interaction is included then our total self energy will
becom
\[
\Sigma_{Total}=\Sigma_{Leads}+\Sigma_{el+phonon}.
\]
and in the Hartree-Fock approximatio
\begin{equation}
\Sigma_{el+ph}=\Sigma_{Hartree}+\Sigma_{Fock
\end{equation}
wher
\begin{equation}
\Sigma_{ji,Hartree}^{r}(\epsilon)=-i\sum_{q}\lambda_{q}^{2}\int_{-\infty
}^{\infty}\frac{d\epsilon}{2\pi}D_{0}^{r}(q,\epsilon=0)G_{ji}^{<}(\epsilon)
\end{equation}
an
\begin{align}
\Sigma_{ji,Fock}^{r}(\epsilon) & =-i\sum_{q}\lambda_{q}^{2}\int_{-\infty
}^{\infty}\frac{d\epsilon^{\prime}}{2\pi}\left[ D_{0}^{r}(q,\epsilon
-\epsilon^{\prime})G_{ji}^{<}(\epsilon^{\prime})+\right. \nonumber\\
& \left. D_{0}^{r}(q,\epsilon-\epsilon^{\prime})G_{ji}^{r}(\epsilon^{\prime
})+D_{0}^{<}(q,\epsilon-\epsilon^{\prime})G_{ji}^{r}(\epsilon^{\prime
})\right] .
\end{align}
Here $D_{0}^{r,<}$ represents retarded and lesser free phonon Green's
function. At this stage if we also include the effects of the heat bath then
our zeroth order phonon Green's function will b
\begin{equation}
D_{0}^{r}(\epsilon,\omega_{q})=\dfrac{1}{\epsilon-\omega_{q}+\dfrac{i\gamma
}{2}}-\dfrac{1}{\epsilon+\omega_{q}+\dfrac{i\gamma}{2}
\end{equation}
with $\gamma=2\pi\sum_{\beta}|N_{q\beta}|^{2}\delta(\epsilon-\omega_{\beta})$,
\cite{17}, represents dissipation in phonon energy due to its contact with the
heat bath
\begin{equation}
D_{0}^{<}(\epsilon)=-in(\epsilon)\dfrac{\gamma}{(\epsilon-\omega_{q
)^{2}+\left( \dfrac{\gamma}{2}\right) ^{2}}-i(1+n(\epsilon))\dfrac{\gamma
}{(\epsilon+\omega_{q})^{2}+\left( \dfrac{\gamma}{2}\right) ^{2}
\end{equation}
with $n(\epsilon)$ the Bose-Einstein distribution function. At zero
temperature the imaginary part of the electron-phonon self energy i
\begin{equation}
\text{Im}\Sigma_{ji,el+phonon}=\frac{1}{2}\sum_{q}\lambda_{q}^{2
[A_{ji}(z-\omega_{q})-A_{ji}(z+\omega_{q})]
\end{equation}
where the spectral function $A_{ji}(\epsilon)=-2\operatorname{Im}G_{ji
^{r}(\epsilon),$ whereas
\begin{equation}
\operatorname{Re}\Sigma_{ji,el+phonon}=\sum_{q}\lambda_{q}^{2}\left[
\operatorname{Re}G_{ji}^{r}(z-\omega_{q})-\operatorname{Re}G_{ji}^{r
(z+\omega_{q})+\dfrac{1}{2\omega_{q}}\right]
\end{equation}
with
\begin{equation}
z\longrightarrow\epsilon+\frac{i\gamma}{2}.
\end{equation}
In the presence of electron-phonon interaction and the dissipative effects of
the heat bath, the differential current i
\begin{equation}
\frac{dJ^{\nu}}{dV}=\frac{e^{2}}{2\pi}\sum_{i,j}\dfrac{\Gamma_{ji}^{L
\Gamma_{ji}^{R}}{\Gamma_{ji}^{L}+\Gamma_{ji}^{R}}\dfrac{\left( \Gamma
_{ji}^{L}+\Gamma_{ji}^{R}+Im\Sigma_{ji,el+phonon}\right) }{\left(
(g_{ij}^{r}(\epsilon))^{-1}+\operatorname{Re}\Sigma_{ji,el+phonon}\right)
^{2}+\left(
\genfrac{}{}{1pt}{0}{\Gamma_{ji}^{L}+\Gamma_{ji}^{R}}{2
+Im\Sigma_{ji,el+phonon}\right) ^{2}
\end{equation}
\subsection{Results And Discussions}
The system under consideration is a molecule comprising two coupled quantum
dots. This molecule is attached to two metallic leads. The primary focus of
this study is the role of inter-dot coupling in electronic transport. In
addition, the dissipative effects of the environment is taken into account by
treating the transport in this system surrounded by a heat bath of phonons.
The vibrational states of the molecule and its impact on electronic transport
is included through the electron-phonon interaction. Our results show
characteristic non-ohmic behavior in the current-voltage results presented in
Fig.(1). At this stage, we are ignoring the phononic and heat bath effects in
order to highlight the role of inter-dot coupling. When coupling of the system
to the leads, which enters through the tunneling-rate, is very small then
energy levels of the two dots are sharply peaked. Hence, no current flows
until the applied voltage bias is in resonance with the level of either of the
two dots. We see in Fig.(1) that intially no current flows on increasing the
bias voltage. But as the applied bias comes in resonance with the level of the
first dot a sharp increase in current occurs. Further increase in applied bias
does not lead to increase in current because the level of the second dot is
not in resonance. As the applied bias is further increased, it comes in
resonance with the level of the second dot leading to an abrupt increase in
current. This explains the step-like features with sharp steps and plateaus
observed in Fig.(1). Now we consider the effects of inter-dot coupling. As a
result of the coupling, the levels of the two dots are pushed apart. This
results in the plateaus becoming wider as this requires higher bias voltage
before the Fermi level in the lead is in resonance with the higher level of
the double dot molecule. This is also shown in Fig.(1) as the inter-dot
coupling is increased. Instead of the inter-dot coupling, if we increase the
tunneling rate from the lead, broadening of the electronic states in the two
dots of the molecule takes place. In this situation, current increases
linearly with applied bias and the system exhibits ohmic behavior. Now if we
increase the inter-dot coupling, step like features again begin to appear as
the applied voltage is tuned since the coupling pushes the two levels apart.
For sufficiently large tunneling rate the two states broaden to the extent
that they merge and we find that the current increases smoothly with
increasing applied bias without any step-like features, Fig.(2)
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.3287in,
width=5.3195in
{1.eps
\caption{Current as a function of applied bias. The energies of two levels are
$\epsilon_{_{1}}=0.5meV$, $\epsilon_{_{2}}=0.8meV$ . The tunneling rates from
the two states are $\Gamma_{ii}=0.04meV$, $\Gamma_{ij}=0meV,$ $where$ $i\neq
j.$
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.3287in,
width=5.3195in
{2.eps
\caption{Current as a function of applied bias. The energies of two levels are
$\epsilon_{_{1}}=0.5meV$, $\epsilon_{_{2}}=0.8meV$ . The tunneling rates from
the two states are $\Gamma_{ii}=0.4meV$, $\Gamma_{ij}=0meV,$ $where$ $i\neq
j.$
\end{center}
\end{figure}
These results are also presented in Fig(3) where it is seen that if the lead
to the system coupling is small then the energy states of the two dots are
sharp and this feature appears as two peaks in the differential conductance.
As the lead to the system coupling is increased, the electronic states get
broadened. And for sufficiently large tunneling rate (strong lead to system
coupling) both the states merge and the peaks in the differential conductance
disappear. To observe the effects of inter-dot coupling with in the molecule
even as the tunneling rate from the lead to the molecule (lead to molecule
coupling) is increased, we show in Fig.(4) that for finite inter-dot coupling,
the peaks in the differential conductance persist. The electronic states are
broadened due to the coupling of the leads and the molecule but when inter-dot
coupling is taken into account, the difference in energy between the levels
increases. This compensates the broadening of the levels and allows the two
levels to remain distinct. This results in peaks corresponding to the two
levels appearing in differential conductance inspite of broadening of the
levels, Fig.(4).
\bigski
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.3287in,
width=5.3195in
{3.eps
\caption{Differential conductance $(dJ/dV)$ as a function of applied bias and
tunneling rates. Applied bias and tunneling rates are in the units of meV. The
energies of two levels are $\epsilon_{1}=0.5meV$ ,$\epsilon_{2}=0.8meV$
,$t=0meV$.\
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.3287in,
width=5.3195in
{4.eps
\caption{Differential conductance $(dJ/dV)$ as a function of applied bias and
tunneling rates. Applied bias and tunneling rates are in the units of meV. The
energies of two levels are $\epsilon_{1}=0.5meV$ ,$\epsilon_{2}=0.8meV$
,$t=0.1meV$.\
\end{center}
\end{figure}
\bigskip
At the final stage, we consider the effects of the electron-phonon
interaction. These are shown in Figs (5) and (6). We find that in addition to
peaks in the differential conductance corresponding to the two levels of the
dots there are peaks due to phonons. These phononic peaks (side bands) occur
as the electrons can exchange energy with the phonons and contribute to
conductance. To focus on the role of inter-dot coupling on phononic peaks, we
see that as we increase the (lead to molecule coupling) tunneling rate
electronic states are broadened to the extent that phononic side bands are not
visible. On further increase in tunneling rate the two electronic states
broaden and merge into each other, Fig.(5). If we include inter-dot coupling,
not only the electronic states remain distinct but the phononic effects are
not lost either. This can be seen in Fig.(6) where peaks appear corresponding
to the two electronic states as well as the phononic side band peaks. Even
with an increase in tunneling rate from the leads to the molecule, these
features persist in the presence of inter-dot coupling.
\bigski
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.6236in,
width=5.3195in
{5.eps
\caption{Differential conductance $(dJ/dV)$ as a function of applied bias and
tunneling rates. Applied bias and tunneling rates are in the units of meV. The
energies of two levels are $\epsilon_{1}=0.5meV$ ,$\epsilon_{2}=0.8meV$
,$t=0meV$ ,$\ \omega=0.1meV$ ,$\gamma=0.013meV$.\
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.2157in,
width=3.1981in
{6.eps
\caption{Differential conductance $(dJ/dV)$ as a function of applied bias and
tunneling rates. Applied bias and tunneling rates are in the units of meV. The
energies of two levels are $\epsilon_{1}=0.5meV$ ,$\epsilon_{2}=0.8meV$
,$t=0.1meV$ ,$\ \omega=0.1meV$ ,$\gamma=0.013meV$.\
\end{center}
\end{figure}
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\bigskip
To conclude, in this work we have focused on the role of inter-dot coupling
with in the dot molecule on electron transport. We have considered a coupled
dot molecule, with inter-dot coupling, attached to two leads including
electron-phonon interaction and the coupling of the molecule with an
environment allowing dissipation of the phonons. We find that including
inter-dot coupling has profound and important role in transport. The step like
featues in current voltage characteristics of peaks in differential
conductance corresponding to the the dot energy levels is lost in the absence
of inter-dot coupling when strong coupling to the leads is considered.
Inter-dot coupling allows the two levels to remain distinct with peaks
appearing in the differential conductance even when broadening of the levels
in the dots occur for strong lead to molecule coupling. Furthermore, phononic
side bands that appear in the differential conductance also persist in the
presence of finite inter-dot coupling even for strong lead to molecule coupling.
\bigskip
\subsubsection{Acknowledgements}
M. Imran and K. Sabeeh would like to acknowledge the support of the Higher
Education Commission (HEC) of Pakistan through project No. 20-1484/R\&D/09.
$^{^{\ast}}$imran1gee@gmail.com
$^{\dagger}$kashifsabeeh@hotmail.com
\bigskip
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1109.4951
|
\section{Introduction}
An easy calculation shows that the exponential function $f(x) = e^x$ has the
somewhat `paradoxical' property that $cf$ is a translate of $f$ for
every $c>0$. It is also easy to see that every function of the form
$a+be^{kx}$ shares this property. Moreover, for every function of the form
$f(x) =
a+bx$ the graph of $cf$ is isometric to the graph of $f$. In \cite{CCR} Cain,
Clark and Rose introduced the notion of vertical rigidity, which we now
formulate for functions of several variables.
\bd
A function $f:\RR^n \to \RR$ is called \emph{vertically rigid}, if $graph(cf)$
is isometric to $graph (f)$ for all $c \in (0,\infty)$. (Clearly, $c \in \RR
\sm \{0\}$ would be the same.)
\ed
Then D.~Jankovi\'c formulated the following conjecture (see \cite{CCR}).
\bcon
\textbf(D.~Jankovi\'c)
A continuous function $f : \RR \to \RR$ is vertically rigid if and only if it
is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \in \RR$, $k \neq 0$).
\econ
This conjecture, and more, was proved in \cite{BE}.
\bt
\lab{t:Jan}
Jankovi\'c's conjecture holds. (It is actually enough to assume that $f$ is
vertically rigid for an uncountable set $C$, see Definition \ref{d:C} below.)
\et
Later C.~Richter gave generalisations of this theorem in various directions,
see \cite{Ri}.
The main goal of the present paper is to give a complete description of
the continuous vertically rigid functions of two variables.
\bt
\textbf{(Main Theorem)}
A continuous function $f : \RR^2 \to \RR$ is vertically rigid
if and only if after a suitable rotation around the $z$-axis $f(x,y)$
is of the form $a + bx + dy$, $a + s(y) e^{kx}$ or $a + b e^{kx} + dy$
($a,b,d,k \in \RR$, $k \neq 0$, $s : \RR \to \RR$ continuous).
\et
As these classes look somewhat ad hoc, we do not even have conjectures in
higher dimensions.
\bpr
Characterise the continuous vertically rigid functions of $n$ variables for $n
\ge 3$.
\epr
In fact, for the proof of the Main Theorem we need the following technical
generalisations.
\bd
\lab{d:C}
If $C$ is a subset of $(0, \infty)$ and $\iG$ is a set of isometries of
$\RR^3$ then we say that $f$ is vertically rigid \emph{for a set $C \su
(0, \infty)$ via
elements of $\iG$} if for every $c \in C$ there exists a $\varphi \in \iG$
such that $\varphi(graph(f)) = graph (cf)$.
(If we do not mention $C$ or $\iG$ then $C$ is $(0, \infty)$ and $\iG$
is the set of all isometries.)
\ed
\bd
Let us say that a set $C \su (0, \infty)$ \emph{condensates to $\infty$} if
for every $r \in \RR$ the set $C \cap (r ,\infty)$ is uncountable.
\ed
The Main Theorem will immediately follow from the following, in which we just
replace $(0,\infty)$ by a set $C$ condensating to $\infty$.
\bt
\lab{t:mainC}
\textbf{(Main Theorem, technical form)}
Let $C \su (0, \infty)$ be a set condensating to $\infty$. Then a continuous
function $f : \RR^2 \to \RR$ is vertically rigid for $C$ if and only if after
a suitable rotation around the $z$-axis $f(x,y)$ is of the form $a + bx + dy$,
$a + s(y) e^{kx}$ or $a + b e^{kx} + dy$ ($a,b,d,k \in \RR$, $k \neq 0$, $s :
\RR \to \RR$ continuous).
\et
The structure of the proof will be the following.
First we check in Section \ref{s:forms}
that functions of the above forms are rigid. (Of course, they are all
continuous.)
Then we start proving the Main
Theorem in more and more general settings. In Section
\ref{s:translations} first we show that if all the
isometries are horizontal translations then the vertically rigid function
$f(x,y)$ is of the form
$s(y) e^{kx}$ ($k \in \RR$, $k~\neq~0$, $s : \RR~\to~\RR$ continuous).
The punchline here is that we can derive a simple functional equation
from vertical rigidity (some sort of `multiplicativity', see Lemma
\ref{l:add}). Then we
conclude this section by referring to a completely algebraic proof in
\cite{BE} showing that if we allow arbitrary translations then $f(x,y)$ is of
the form $a + s(y) e^{kx}$ ($a, k \in \RR$, $k \neq 0$, $s : \RR \to \RR$
continuous).
Then we start working on the case of general isometries.
The central idea is to consider the set $S_f$ of directions of segments
connecting pairs of points on $graph(f)$ (see Definition \ref{d:S_f}). We
collect the necessary properties of this set in Section \ref{s:propS_f}. The
set $S_f$ has some sort of rigidity in that the transformation $f \mapsto cf$
distorts the shape of it,
but the resulting set has to be isometric to the
original one (see Definition \ref{d:psi} and Remark \ref{r:rigid}). Using
these we determine the possible $S_f$'s in Section \ref{s:possS_f}, then in
Section \ref{s:vertpf} we complete the proof by handling
these cases using various methods.
Finally, in Section \ref{s:open} we collect the open questions.
\section{Functions of these forms are rigid}
\lab{s:forms}
Rotation of the graph around the $z$-axis does not affect vertical rigidity,
so we can assume that $f$ is of the given form without rotations.
Functions of the form $a + bx + dy$ are clearly vertically rigid.
Let now $f(x,y) = a + s(y) e^{kx}$ ($a,k \in \RR$, $k \neq 0$, $s
: \RR \to \RR$ continuous). Then $cf(x,y) = f(x + \frac{\log
c}{k}, y) + a(c-1)$, so $f$ is actually vertically rigid via
translations in the $xz$-plane.
Before checking the third case we need a lemma.
\bl
\label{lem1}
Let $f(x,y)=g(x)+dy$, where $d>0$ and let $c>0$. If
we rotate $graph(f)$ around the $x$-axis by the angle
$\alpha_c=\arctan (cd)-\arctan(d)$ then the intersection of this
rotated graph with the $xy$-plane is the graph of a function of the form $y
= - w_{c,d} g(x) $, where $w_{c,d}>0$ and the map
$c\mapsto w_{c,d}$ is strictly monotone on $(0, \infty)$ for every fixed $d>0$.
\el
\br
By rather easy and short elementary geometric considerations one can check that
for every fixed $d>0$ the map $c \mapsto w_{c,d}$ is positive and real analytic.
It is also very easy
to see geometrically that the limit at $0$ is $\infty$, hence it is not constant,
therefore countable-to-one. This would suffice for all our purposes, but these
arguments are unfortunately very hard to write down rigorously, so we decided
to present a less instructive and longer algebraic proof.
\er
\bp
Using the matrix of the rotation we can write the rotated image
of the point of the graph $(x,y_0,g(x)+dy_0)$ as
\beq \label{e00}
\left(\begin{matrix}
1&0&0\cr
0&\cos \alpha _{c}& -\sin \alpha _{c}\cr
0&\sin \alpha _{c}& \cos \alpha _{c}
\end{matrix}\right)
\left(\begin{matrix}
x\cr
y_0\cr
g(x)+dy_0
\end{matrix}\right)
= \left(\begin{matrix}
x\cr
y_0(\cos \alpha _{c}-d\sin \alpha_{c})-g(x)\sin \alpha _{c}\cr
y_0(\sin \alpha_{c}+d\cos \alpha _{c})+g(x)\cos \alpha _{c}
\end{matrix}\right).
\eeq
Let us now determine the intersection of the rotated graph
with the $xy$-plane. This right hand side of (\ref{e00}) is in the
$xy$-plane if and only if the third coordinate vanishes, that is,
when $y_0 (\sin \alpha _{c}+d\cos \alpha_{c})+g(x)\cos \alpha
_{c}=0$. This yields
\beq
\label{e01}
y_{0}=-\frac{\cos \alpha_{c}}{\sin \alpha_{c}+d\cos \alpha _{c}}g(x).
\eeq
In order to
complete the proof of the lemma we have to calculate the
$y$-coordinate of the rotated image of the point
$(x,y_0,g(x)+dy_0)$, which is the second entry of the right hand
side of (\ref{e00}). Hence, using (\ref{e01}),
\[
y= y_{0} (\cos {\alpha _{c}}-d\sin {\alpha _{c}})-g(x)\sin \alpha
_{c}= -\frac{\cos \alpha _{c}(\cos \alpha _{c}-d\sin \alpha
_{c})}{\sin \alpha _{c}+d\cos \alpha _{c}}g(x)-
\]
\[
g(x)\sin \alpha _{c}= -\frac{\cos ^{2} \alpha _{c}-d\cos {\alpha
_{c}} \sin {\alpha _{c}}+\sin ^{2} \alpha _{c}+d\cos {\alpha _{c}}
\sin {\alpha _{c}}}{\sin \alpha _{c}+d\cos \alpha _{c}}g(x)=
\]
\[
-\frac{1}{\sin \alpha _{c}+d\cos \alpha _{c}}g(x). \]
Therefore
\[
w_{c,d}=\frac{1}{\sin \alpha _{c}+d\cos \alpha
_{c}}=\left(\sqrt{d^{2}+1}\left(\sin \alpha
_{c}\frac{1}{\sqrt{d^{2}+1}}+\cos \alpha
_{c}\frac{d}{\sqrt{d^{2}+1}}\right)\right)^{-1}.
\]
Using the identity
\beq
\label{e1:43}
\sin \alpha = \frac{\tan
\alpha}{\sqrt{\tan^2 \alpha + 1}} \ \left( \alpha \in (- \pi/2, \pi/2) \right)
\eeq
we obtain $\sin
(\arctan (d))=\frac{d}{\sqrt{d^{2}+1}}$, which easily implies $\cos (\arctan
(d))=\frac{1}{\sqrt{d^{2}+1}}$. (Note that $\arctan(d)\in (-\pi/2,\pi/2)$.) So
\[
w_{c,d}=\left(\sqrt{d^{2}+1}\big(\sin \alpha
_{c}\cos (\arctan (d))+\cos \alpha _{c}\sin (\arctan
(d))\big)\right)^{-1}.
\]
By the formula $\sin (\alpha+\beta)=\sin \alpha \cos
\beta+\cos \alpha \sin \beta$ and the definition of $\alpha _c$
this equals
\[\left(\sqrt{d^{2}+1}\sin (\alpha _{c}+ \arctan
(d))\right)^{-1}=\left(\sqrt{d^{2}+1}\sin (\arctan
(cd))\right)^{-1}.
\]
Applying \eqref{e1:43} again yields
\[
w_{c,d} = \left(\sqrt{d^{2}+1}\frac{\tan(
\arctan (cd))}{\sqrt{\tan^2 (\arctan(cd)) +
1}}\right)^{-1}=\sqrt{\frac{1}{d^{2}+1}\left(1+\frac{1}{(cd)^{2}}\right)}.
\]
From this form it is easy to see that this function is positive and
strictly monotone on $(0,\infty)$ for every fixed $d>0$.
\ep
Let now $f(x,y) = a + b e^{kx} + dy$ ($a,b,d,k \in \RR$, $k \neq 0$).
Rescaling the graph in a homothetic way does not affect vertical rigidity, so
we can consider $kf(\frac{x}{k},\frac{y}{k})$ and assume $k=1$. We may also
assume $b,d \neq 0$, otherwise our function is of one of the previous
forms. Adding a constant, reflecting the graph about the $xz$-plane (needed
only if the signs of $b$ and $d$ differ), multiplying by a nonzero constant, as
well as a translation in the $x$-direction do not affect vertical rigidity,
so by applying these in this order we can assume that $a=0$, $bd>0$, $d=1$,
and $b=1$.
Hence it suffices to check that $f(x,y) = e^x + y$ is vertically
rigid. Let us fix a $c>0$. In every vertical plane of the form
$\{x = x_0\}$ the restriction of $f$ is a straight line of slope
$1$. Rotation around the $x$-axis by angle $\alpha_c=\arctan
(c)-\frac{\pi}{4}$ takes all these lines to lines of slope $c$. By
applying Lemma \ref{lem1} with $g(x)=e^{x}$ and $d=1$, the
intersection of the rotated graph and the $xy$-plane is the graph
of the function $y = - w_{c,1} e^x$.
Now, applying a translation in the $x$-direction we can obtain a
function with still all lines of slope $c$ but now with
intersection with the $xy$-plane of the form $y = -e^x$ (note that
$w_{c,1}>0$). But then we are done, since this function clearly
agrees with $cf$. (The intersection of $graph(f)$ and the
$xy$-plane is of the form $y = -e^x$, and all lines in this graph
are of slope $1$, hence for $graph(cf)$ the intersection is still
$y = -e^x$, and all lines are of slope $c$.) This finishes the
proof of vertical rigidity.
\section{Vertical rigidity via translations}
\lab{s:translations}
\bt
\lab{t:htcont}
Let $C \su (0, \infty)$ be an uncountable set. Then a continuous function $f :
\RR^2 \to \RR$ is
vertically rigid for $C$ \emph{via horizontal translations} if and only if
after a suitable rotation around the $z$-axis $f(x,y)$ is of the form
$s(y) e^{kx}$ ($k \in \RR$, $k \neq 0$, $s : \RR \to \RR$ continuous).
\et
We already checked the easy direction in the previous section.
Before proving the other direction we need some preparation. We
will need the following result, which is Theorem 2.5 in \cite{BE}.
\bt \label{old} Let $f:\RR\rightarrow \RR$ be a continuous
vertically rigid function for an uncountable set $C\subset
(0,\infty)$ via horizontal translations. Then $f$ is of the form
$se^{kx}$ $(s\in \RR, k\in \RR\setminus \{0\}).$ \et
The following lemma will be useful throughout the paper. Sometimes
we will use it tacitly. The easy proof is left to the reader.
\bl
\lab{l:C}
Let $f : \RR^2 \to \RR$ be vertically rigid for $c_0$ via $\varphi_0$ and for
$c$ via $\varphi$. Then $c_0 f$ is vertically rigid for $\frac{c}{c_0}$ via
$\varphi\circ \varphi_0^{-1}$.
\el
From now on we will often use the notation $\vec{x}$ for two-dimensional
(and sometimes three-dimensional) vectors.
\bd
For a function $f:\RR^2\to\RR$ and a set $C \su (0, \infty)$
let $T_{f,C} \su \RR^2$ be the
additive group generated by the set $T' = \{ \vec{t} \in \RR^2 : \exists c \in C \
\forall \vec{x} \in \RR^2 \ f(\vec{x}+\vec{t})=cf(\vec{x}) \}$.
(We will usually simply write $T$ for $T_{f,C}$.)
\ed
\bl
\lab{l:add}
Let $f:\RR^2 \rightarrow \RR$ be a vertically rigid function for a set
$C\subset (0,\infty)$ via horizontal translations such that
$f(\vec{0})=1$. Then
\[
f(\vec{x}+\vec{t}) = f(\vec{x}) f(\vec{t}) \ \ \forall \vec{x} \in \RR^2 \
\forall \vec{t} \in T.
\]
Moreover, $f(\vec{t})>0$ for every $\vec{t} \in T$, and $T'$ is uncountable if
so is $C$.
\el
\bp
By assumption, for every $c \in C$ there exists $\vec{t_c} \in \RR^2$
such that $cf(\vec{x}) = f(\vec{x} + \vec{t_c})$ for every $\vec{x} \in
\RR^2$. Then $\vec{t_c} \in T'$ for every $c \in C$.
Since $T$ is the group generated by $T'$, every $\vec{t} \in T$
can be written as $\vec{t} = \sum_{i=1}^m n_i \vec{t_i}$
($\vec{t_i} \in T', n_i \in \ZZ, i=1, \dots, m$) where $f(\vec{x}
+ \vec{t_i})=c_i f(\vec{x}) \ (\vec{x} \in \RR^2, \ i=1, \dots,
m$).
From these we easily get
\beeq
\lab{2}
f(\vec{x} + \vec{t}) = c_{\vec{t}} f(\vec{x}),
\textrm{ where } c_{\vec{t}} = \prod_{i=1}^m c_i^{n_i}, \ \vec{x} \in
\RR^2, \ \vec{t} \in T.
\eeq
Note that $c_{\vec{t}}>0$
(and also that it is not necessarily a member of $C$). It suffices
to show that $c_{\vec{t}} = f(\vec{t})$ for every $\vec{t} \in T$, but this
follows if we substitute $\vec{x} = \vec{0}$ into (\ref{2}).
Since $f$ is not
identically zero, $\vec{t}_c \neq \vec{t}_{c'}$ whenever $c,c' \in C$
are distinct. Hence $\{ \vec{t}_c : c \in C \}$ is uncountable, so $T'$
is uncountable if so is $C$.
\ep
\bp (Thm. \ref{t:htcont}) If $f$ is identically zero then we are
done, so let us assume that this is not the case. The class of
continuous vertically rigid functions for some set condensating to
$\infty$ via
horizontal translations, as well as the class of functions of the
form $s(y)e^{kx}$ ($k\in \RR$, $k \neq 0$, $s : \RR \to \RR$ continuous)
are both closed
under horizontal translations and under multiplication by nonzero
constants (by Lemma \ref{l:C}).
Hence we may assume that $f(\vec{0})=1$. Then the previous lemma
yields that $f(\vec{t_1} + \vec{t_2}) = f(\vec{t_1})f(\vec{t_2})$
$(\vec{t_1}, \vec{t_2} \in T)$, and also that $f|_T>0$. Then $g(\vec{t}) =
\log f(\vec{t}) $ is defined for every $\vec{t} \in T$, and $g$ is clearly
additive on $T$.
Let us now consider $\bar{T}$, the closure of $T$, which is clearly an
uncountable closed subgroup of $\RR^2$. It is well-known that every closed
subgroup of $\RR^2$ is a nondegenerate linear image of a group of the form
$G_1 \times G_2$, where $G_1, G_2 \in \{ \{0\}, \ZZ, \RR \}$.
Hence after a suitable rotation around the origin
$\bar{T}$ is either $\RR^2$ or $\RR \times \{0\}$ or $\RR \times r\ZZ$ for
some $r>0$.
\noindent\textbf{Case 1.} $\bar{T} = \RR^2$.
In this case $T \su \RR^2$ is dense. It is well-known that a continuous
additive function on a dense subgroup is of the form $g(x,y) = \alpha x +
\beta y$, $((x,y) \in T)$ for some $\alpha, \beta \in \RR$. But then $f(x,y) =
e^{\alpha x + \beta y}$ on $T$, and by continuity this holds on the whole
plane as well. As the constant $1$ function is not vertically rigid via
horizontal translations, $\alpha = \beta = 0$ cannot hold. By applying a
rotation of angle $\frac{\pi}{2}$ if necessary we may assume that $\alpha \neq
0$. But then by choosing $k = \alpha$, $s(y) = e^{\beta y}$ we are
done.
\noindent\textbf{Case 2.} $\bar{T} = \RR \times \{0\}$.
In this case every $\vec{t}_c$ is of the form $(t_c,0)$, where $t_c \neq 0$ if
$c \neq 1$. (We may assume $1 \notin C$.)
Applying Theorem \ref{old} for every fixed $y$ we obtain that
$f(x,y) = s(y) e^{k_y x}$ ($s(y),k_y \in \RR, k_y \neq 0$). As
$s(y) = f(0,y)$, we get that $s$ is continuous. If $s(y) \neq 0$
then it is not hard to see that $k_y = \frac{\log c}{t_c}$, which
is independent of $y$, so for these $y$'s $k_y = k$ is constant.
But if $s(y) = 0$ then the value of $k_y$ is irrelevant, so it can
be chosen to be the same constant $k$. Hence without loss of
generality $k_y = k$ is constant, and we are done with this case.
\noindent\textbf{Case 3.} $\bar{T} = \RR \times r\ZZ$.
As $T'$ is uncountable, there is an $n \in \ZZ$
so that $T' \cap (\RR \times \{rn\})$
is uncountable. Fix an element $t_{c_0}$ of this set. Then Lemma \ref{l:C}
yields that $c_0 f$ is vertically rigid for an uncountable set via
translations of the form $(t,0)$. Restricting ourselves to these isometries
and $c$'s we are done using Case 2, since every uncountable set in $\RR$
generates a dense subgroup.
\ep
Now we handle the case of arbitrary translations.
\bt
\lab{t:trans}
Let $f:\RR^2 \rightarrow \RR$ be an arbitrary
function that is vertically rigid for a set $C \su (0,\infty)$ via
translations. Then there exists $a\in \RR$ such that $f-a$ is
vertically rigid for the same set $C$ via horizontal translations.
\et
\bp
The obvious modification of \cite[Thm. 2.4]{BE} works, just replace all $x$'s
and $u$'s by vectors.
\ep
This readily implies the following.
\bcor
\lab{c:trans}
Let $C \su (0, \infty)$ be an uncountable set. Then a continuous function $f :
\RR^2 \to \RR$ is
vertically rigid for $C$ \emph{via translations} if and only if
after a suitable rotation around the $z$-axis $f(x,y)$ is of the form
$a+s(y) e^{kx}$ ($a, k \in \RR$, $k \neq 0$, $s : \RR \to \RR$ continuous).
\ecor
\section{The set $S_f$}
\lab{s:propS_f}
Now we start working on the case of arbitrary isometries.
Let $\SS^2 \su \RR^3$ denote the unit sphere.
For a function $f : \RR^2 \to \RR$ let $S_f$ be the set of directions between
pairs of points on the graph of $f$, that is,
\bd
\lab{d:S_f}
\[
S_f = \left\{ \frac{p-q}{|p-q|} \in \SS^2 : p,q \in graph(f),\ p \neq q
\right\}.
\]
\ed
Recall that a \emph{great circle} is a circle line in $\RR^3$ of radius $1$
centered at the origin. We call it \emph{vertical} if it passes through the
points $(0,0,\pm 1)$.
\bl
\lab{l:S}
Let $f : \RR^2 \to \RR$ be continuous. Then
\begin{enumerate}
\item
\lab{symm}
$- S_f = S_f$ (symmetric about the origin),
\item
\lab{poles}
$(0,0,\pm 1) \notin S_f$,
\item
\lab{conn}
$S_f$ is connected,
\item
\lab{arcs}
every great circle containing $(0,0,\pm 1)$ intersects $S_f$ in two
(symmetric) nonempty arcs,
\item
\lab{comp}
$\SS^2 \sm S_f$ has exactly two connected components, one containing $(0,0,1)$
and one containing $(0,0,-1)$.
\end{enumerate}
\el
\bp
(\ref{symm}.) Obvious.
(\ref{poles}.) Obvious, since $f$ is a function.
(\ref{conn}.) $graph(f)$ is homeomorphic to $\RR^2$, hence the squared of it
minus the (2-dimensional) diagonal is a connected set. Since $S_f$ is the
continuous image of this connected set, it is itself connected.
(\ref{arcs}.) The intersection of $S_f$ with such a great circle corresponds
to restricting our attention to distinct pairs of points $(\vec{x_1}, \vec{x_2})
\in \RR^2 \times \RR^2$ so that the segment $[\vec{x_1}, \vec{x_2}]$ is
parallel to a fixed line $L \su \RR^2$. Now, given two such nondegenerate
segments it is easy to move one of them continuously to the other so that
along the way it remains nondegenerate and parallel to $L$. This shows that in
both halves of the great circle (separated by $(0,0,\pm 1)$) $S_f$ is pathwise
connected, hence it is an arc.
(\ref{comp}.) By (\ref{arcs}.) every point of $\SS^2 \sm S_f$ can be connected
with an arc of a vertical great circle either to $(0,0,1)$ or to $(0,0,-1)$ in
$\SS^2 \sm S_f$, hence there are at most two connected components.
Now we show that $(0,0,1)$ and $(0,0,-1)$ are in different ones. It suffices to
show that there exists a Jordan curve in $S_f$ so that $(0,0,1)$ and $(0,0,-1)$
are in the two distinct components of its complement.
Let $\SS^1$ denote the unit circle in $\RR^2 =
\{(x,y,z) : z=0\}$ and let $\gamma :\SS^1 \rightarrow S_{f}$ be given by
\[
\gamma(\vec{x}) =
\frac{(\vec{x},f(\vec{x}))-(-\vec{x},f(-\vec{x}))}{|(\vec{x},f(\vec{x}))-(-\vec{x},f(-\vec{x}))|}.
\]
In this paragraph the word `component' will refer to the components of $\SS^2
\sm \gamma(\SS^1)$.
One can easily check that $\gamma$ is continuous and injective, hence a Jordan curve.
Moreover, it is clearly in $S_f$, and its intersection with every vertical great
circle is a symmetric pair of points. Therefore every point of $\SS^2 \sm
\gamma(\SS^1)$ can be connected with an arc of a vertical great circle either to
$(0,0,1)$ or to $(0,0,-1)$ in $\SS^2 \sm \gamma(\SS^1)$, hence the union of the
components of $(0,0,1)$ and $(0,0,-1)$ cover $\SS^2 \sm \gamma(\SS^1)$. So
$(0,0,1)$ and $(0,0,-1)$ are in different components, otherwise
$\SS^2 \sm \gamma(\SS^1)$ would be connected, but the complement of a Jordan
curve in $\SS^2$ has two components.
\ep
The above lemma shows that $S_f$ is something like a `strip around the
sphere'. Now we make this somewhat more precise by defining the top and the
bottom `boundaries' of this strip.
\bd
\lab{d:h}
Let $h : \SS^1 \to \SS^2$ be defined as follows. Every $\vec{x} \in \SS^1$ is
in a unique half great circle connecting $(0,0,1)$ and $(0,0,-1)$. The
intersection of $S_f$ with this great circle is an arc, define $h(\vec{x})$ as
the top endpoint of this arc.
\ed
Clearly, the bottom endpoint of this arc is $-h(-\vec{x})$, so the `top
function bounding the strip $S_f$ is $h(\vec{x})$ and the bottom function is
$-h(-\vec{x})$'. The coordinate functions of $h$ are denoted by
$(h_1,h_2,h_3)$, where $h_3 : \SS^1 \to [-1, 1]$ encodes all information about
$h$.
\bl
\lab{l:h}
Let $f : \RR^2 \to \RR$ be continuous, and $h$ be defined as above. Then
\begin{enumerate}
\item
\lab{notpole}
$h(\vec{x}) \neq (0,0,-1)$ for every $\vec{x} \in \SS^1$
\item
\lab{semicont}
$h$ is lower semicontinuous (in the obvious sense, or equivalently, $h_3$ is
lower semicontinuous)
\item
\lab{convex}
$h$ is convex with respect to
great circles, that is, if $h(\vec{x})$ and $h(\vec{y})$
determine a unique nonvertical great circle (i.e.~there is a subarc of
$\SS^1$ of length~$< \pi$ connecting $\vec{x}$ and $\vec{y}$, and $h(\vec{x}),
h(\vec{y}) \neq (0,0,1)$) then on this subarc $graph(h)$ is bounded from above
by the great circle.
\end{enumerate}
\el
\bp
(\ref{notpole}.)
Obvious by Lemma \ref{l:S} (\ref{poles}.) and (\ref{arcs}.).
(\ref{semicont}.)
We have to check that if $h_3(\vec{x}) > u$ then the same holds in a
neighbourhood of $\vec{x}$. (Note that essentially $h_3$ is defined as a
supremum.) Hence $h_3(\vec{x}) > u$ if and only if there exists a segment
$[\vec{a},\vec{b}] \su \RR^2$ parallel to $\vec{x}$ over which the slope of
$f$ is bigger than $u$. But then by the continuity of $f$ the same holds for
segments close enough to $[\vec{a},\vec{b}]$, in particular to slightly
rotated copies, and we are done.
(\ref{convex}.) It is easy to see that for every $\vec{v}\in
\SS^1$ the slope of $f$ over a segment parallel to $\vec{v}$ is at most
the slope of the vector $h(\vec{v})$. Let $\vec{z} \in
\SS^1$ be an element of the shorter arc connecting $\vec{x}$ and
$\vec{y}$ in $\SS^1$, let $[\vec{a},\vec{b}] \su \RR^2$ be a
segment parallel to $\vec{z}$, and let $P:\RR^{2}\rightarrow \RR$
be the linear map whose graph passes through the origin,
$h(\vec{x})$ and $h(\vec{y})$. (Then $graph(P)$ contains the great
circle determined by $h(\vec{x})$ and $h(\vec{y})$. Moreover,
the slope of $P$ over any vector parallel to $\vec{x}$ is the slope of
$h(\vec{x})$, and similarly for $\vec{y}$.) We have to
show that the slope of $f$ between $\vec{a}$ and $\vec{b}$ is at most
that of $P$, that is, $f(\vec{b})-f(\vec{a})\leq
P(\vec{b})-P(\vec{a})$. Write $\vec{b} - \vec{a} = \al \vec{x} +
\beta \vec{y}$ for some $\al,\beta > 0$. Then by using the
definition of $P$ and our first observation for the segments
$[\vec{a},\vec{a}+\al \vec{x}]$ and $[\vec{a}+\al
\vec{x},\vec{a}+\al \vec{x}+\beta \vec{y}]$, which are parallel to
$\vec{x}$ and $\vec{y}$, respectively, we get
\[
f(\vec{b})-f(\vec{a})=f(\vec{a}+\al
\vec{x}+\beta \vec{y})-f(\vec{a})=$$ $$ \left(f(\vec{a}+\al
\vec{x}+\beta \vec{y}\right)-f(\vec{a}+\al
\vec{x}))+\left(f(\vec{a}+\al \vec{x})-f(\vec{a})\right)\leq
\]
\[
\left(P(\vec{a}+\al \vec{x}+\beta \vec{y})-P(\vec{a}+\al
\vec{x})\right)+\left(P(\vec{a}+\al
\vec{x})-P(\vec{a})\right)=P(\vec{b})-P(\vec{a}).
\]
\ep
\section{Determining the possible $S_f$'s}
\lab{s:possS_f}
\bd
\lab{d:psi}
For $c > 0$ let $\psi_c: \SS^2 \to \SS^2$ denote the map that `deforms $S_f$
according to the map $c \mapsto cf$', that is,
\[
\psi_c((x,y,z)) = \frac{(x,y,cz)}{|(x,y,cz)|} \ \ ((x,y,z) \in \SS^2).
\]
\ed
\br \lab{r:rigid} Let $\varphi_c$ be the isometry mapping
$graph(f)$ onto $graph(cf)$. Every isometry $\varphi$ is of the
form $\varphi^{trans} \circ \varphi^{ort}$, where $\varphi^{ort}$
is an orthogonal transformation and $\varphi^{trans}$ is a
translation. Moreover, if $\varphi$ is orientation-preserving then
$\varphi^{ort}$ is a rotation around a line passing through the
origin. A key observation is the following: The vertical rigidity
of $f$ for $C$ implies that $\psi_c(S_f) =\varphi_c^{ort}(S_f)$
for every $c \in C$. \er
Now we prove the main theorem of this section. For the definition of $h_3$ see
the previous section.
\bt
\lab{t:cases}
Let $C \su (0, \infty)$ be a set condensating to $\infty$, and let $f : \RR^2
\to \RR$ be a continuous function vertically rigid for $C$.
Then one of the following holds.
\begin{itemize}
\item
\textbf{Case A.} There is a vertical great circle that intersects $S_f$ in only
two points.
\item
\textbf{Case B.} $S_f = \SS^2 \sm \{ (0,0,1), (0,0,-1) \}$.
\item
\textbf{Case C.}
There exists an $\vec{x_0} \in \SS^1$ such that $h_3(\vec{x_0}) = 0$ and
$h_3(\vec{x}) = 1$ for every $\vec{x} \neq \vec{x_0}$, that is, $S_f$ is
`$\SS^2$ minus two quarters of a great circle'.
\item
\textbf{Case D.}
There exists a closed interval $I$ in $\SS^1$ with $0 < length(I) < \pi$ such
that $h_3(\vec{x}) = 0$ if $\vec{x} \in I$, and
$h_3(\vec{x}) = 1$ if $\vec{x} \notin I$, that is, $S_f$ is
`$\SS^2$ minus two spherical triangles'.
\end{itemize}
\et
\bp In this proof the word `component' will refer to the
components of $\SS^2\setminus S_{f}$. We separate two cases
according to whether $h_3 \ge 0$ everywhere or not.
First let us suppose that there exists a
$\vec{x} \in \SS^1$ such that $h_3(\vec{x}) < 0$. This implies that there is a
vertical great circle containing two arcs, one in the top component
connecting $(0,0,1)$ with $\SS^1$ and even crossing it, and an other
one (the symmetric pair in the bottom component)
running from the `South Pole to the Equator' and even
above. But then considering geometrically the action of $\psi_c$ one can
easily check that if we choose larger and larger $c$'s (tending to $\infty$)
then we obtain that $\psi_c(S_f)$ contains in the two components two
symmetrical arcs on the same great circle which are only leaving out two small
gaps of length tending to $0$. But then by Remark \ref{r:rigid} $S_f$ also
contains two such arcs in the two components on some (not necessarily
vertical) great circle, hence the distance of the components is $0$.
Let $\vec{p_n}$ and $\vec{q_n}$ be sequences in the top and bottom component,
respectively, so that $\dist(\vec{p_n}, \vec{q_n}) \to 0$. By compactness we may assume
$\vec{p_n}, \vec{q_n} \to \vec{p} \in \SS^2$. We claim that $\vec{p_n} \to \vec{p}$ implies $\vec{p} \neq
(0,0,-1)$. (And similarly $\vec{q_n} \to \vec{p}$ implies $\vec{p} \neq (0,0,1)$.) Indeed, let
$\vec{x}_n \in \SS^1$ be so that $\vec{x}_n$ and $\vec{p_n}$ lay on the same
vertical great circle, and similarly, let $\vec{x} \in \SS^1$ and $\vec{p}$ lay on
the same vertical great circle. Then $\vec{x}_n \to \vec{x}$, and using the
fact $h(\vec{x}) \neq (0,0,-1)$ and the lower semicontinuity of $h$ at
$\vec{x}$ (Lemma \ref{l:h} (\ref{notpole}.) and (\ref{semicont}.)) we are done.
Using the lower semicontinuity of $h$ at $\vec{x}$ again (and $\vec{p_n} \to \vec{p}$)
we get that
$h(\vec{x})$ cannot be above $\vec{p}$. Similarly, $-h(-\vec{x})$ cannot be below
$\vec{p}$. But $h(\vec{x})$ is always above $-h(-\vec{x})$, so the only option is
$h(\vec{x}) = -h(-\vec{x})$, hence there is a vertical
great circle whose intersection with $S_f$ is just a (symmetric) pair of
points, so Case A holds, and hence we are done with the first half of the
proof.
Now let us assume that $h_3 \ge 0$ everywhere. First we prove that
$h_3(\vec{x}) \in \{0,1\}$ for Lebesgue almost every $\vec{x} \in
\SS^1$. Indeed, fix an arbitrary $c \in C \sm \{1\}$. By rigidity the (equal)
measure of the
two components remains the same after applying $\psi_c$. Since $h_3 \ge 0$, the
intersection of the top component with the vertical great circle containing an
$\vec{x}$ shrinks if $c>1$ and grows if $c<1$, unless $h_3(\vec{x}) = 0$ or
$1$. Hence we are done, since the measure of the top component can be
calculated from the lengths of these arcs.
Now we show that $\{\vec{x} : h_3(\vec{x}) = 0 \}$ is either empty, or a pair
of points of the form $\{\vec{x_0}, -\vec{x_0}\}$, or a closed interval in
$\SS^1$ (possibly degenerate or the whole $\SS^1$). So we have to show that if
$\vec{x}, \vec{y} \in \SS^1$ are so that the shorter
arc connecting them is shorter than $\pi$, and $h_3(\vec{x}) = h_3(\vec{y}) =
0$ then $h_3(\vec{z}) = 0$ for every $\vec{z}$ in this arc. But $h_3(\vec{z})
\ge 0$ by assumption, and $h_3(\vec{z})\le 0$ by the convexity of $h$ applied
to $h(\vec{x}) = \vec{x}$ and $h(\vec{y}) = \vec{y}$.
The fact that the endpoints are also contained in $\{\vec{x} : h_3(\vec{x}) =
0 \}$ easily follows from the semicontinuity.
If $\{\vec{x} : h_3(\vec{x}) = 0 \}$ is a symmetrical pair
of points or a closed interval of length at least $\pi$ then it is easy to see
that Case A holds. Hence we may assume that it is empty, or a singleton, or a
closed interval $I$ with $0 < length(I) < \pi$.
\noindent\textbf{Case 1.} $\{\vec{x} : h_3(\vec{x}) = 0 \} = \emptyset$.
In this case, $h_3 > 0$ everywhere, and hence $h_3 = 1$ almost
everywhere. Therefore one can easily see (using the convexity) that $h_3 = 1$
everywhere but possibly at at most two points of the form $\{\vec{x_0},
-\vec{x_0}\}$. We claim that actually $h_3 = 1$ everywhere. We know already
that $S_f$ is $\SS^2$ minus two symmetric arcs on the same vertical great
circle. The arcs contain $(0,0,1)$ and $(0,0,-1)$, respectively, and they do
not reach the `Equator', since $h_3 > 0$. Let us fix an arbitrary $c \in C \sm
\{ 1 \}$. By rigidity the
(equal) length of the arcs should not change when applying $\psi_c$,
but it clearly changes, a contradiction.
Hence $S_f = \SS^2 \sm\{(0,0,1), (0,0,-1)\}$, so Case B holds.
\noindent\textbf{Case 2.} $\{\vec{x} : h_3(\vec{x}) = 0 \}$ is a singleton.
Let $\{\vec{x_0}\} = \{\vec{x} : h_3(\vec{x}) = 0 \}$. Similarly as above,
$h_3 = 1$ almost everywhere. Then convexity easily implies that $h_3(\vec{x})
= 1$ whenever $\vec{x} \notin \{\vec{x_0}, -\vec{x_0}\}$.
Again similarly, the length of the
arcs is unchanged by $\psi_c$ only if $h_3(\vec{-x_0}) = 1$, so $S_f$ is
$\SS^2$ minus two symmetric quarter arcs starting from the `Poles' on a
vertical great circle, so Case C holds.
\noindent\textbf{Case 3.} $\{\vec{x} : h_3(\vec{x}) = 0 \}$ is a closed interval
in $\SS^1$ with $0 < length(I) < \pi$.
Let $I = \{\vec{x} : h_3(\vec{x}) = 0 \}$.
As $h_3 = 0$ or $1$ almost everywhere, convexity readily implies that $h_3 =
1$ on $\SS^1 \sm I$. Hence $S_f$ is `$\SS^2$ minus two spherical
triangles', and Case D holds.
This concludes the proof.
\ep
\section{The end of the proof}
\lab{s:vertpf}
Now we complete the proof of the technical form of the
Main Theorem. We repeat the statement here.
\bt
(Main Theorem, technical form)
Let $C \su (0, \infty)$ be a set condensating to $\infty$. Then a continuous
function $f : \RR^2 \to \RR$ is vertically rigid for $C$ if and only if after
a suitable rotation around the $z$-axis $f(x,y)$ is of the form $a + bx + dy$,
$a + s(y) e^{kx}$ or $a + b e^{kx} + dy$ ($a,b,d,k \in \RR$, $k \neq 0$, $s :
\RR \to \RR$ continuous).
\et
\bp
By Theorem \ref{t:cases} it suffices to consider Cases A-D.
\noindent\textbf{Case A.} There is a vertical great circle that intersects
$S_f$ in only two points.
We may assume using a suitable rotation around the $z$-axis that the vertical
great circle is in the $yz$-plane, hence $f(x,y)$ is of the form $g(x) + dy$.
The continuity of $f$ implies that $g$ is also continuous.
\textbf{Subcase A1.} $d = 0$.
Let $c \in C$ be fixed, and let $\varphi_c$ be the corresponding isometry. The
graph of $cf$ is invariant under translations parallel to the $y$-axis. As the
same holds for $f$, by rigidity, $cf$ is also invariant under translations
parallel to the $\varphi_c$-image of the $y$-axis. If these two directions are
nonparallel, then $graph(cf)$ is a plane, and hence so is $graph(f)$, so we
are done since $f(x,y)$ is of the form $a + bx$ (note that there is no `$+
dy$' since $f$ does not depend on $y$). Therefore we may assume that all lines
parallel to the $y$-axis are taken to lines parallel to the $y$-axis, but then
all planes parallel to the $xz$-plane are taken to planes parallel to the
$xz$-plane. But this shows (by considering the intersections of the graphs
with the $xz$-plane) that $g$ is vertically
rigid for $c$, hence by Theorem \ref{t:Jan}
$g(x)$ is of the from $a + bx$ or $a + b e^{kx}$ ($a,b,k \in \RR$, $k \neq
0$), and we are done.
\textbf{Subcase A2.} $d \neq 0$.
We may assume that $d>0$, since otherwise we may consider $-f$.
For every $c \in C$ let $\varphi_c$ be the corresponding isometry.
We claim that we may assume that all these are
orientation-preserving. If $\{c \in C : \varphi_c \textrm{ is
orientation-preserving} \}$ condensates to $\infty$ then we are
done by shrinking $C$, otherwise we may assume that they are all
orientation-reversing (note that if we split $C$ into two pieces
then at least one of them still condensates to $\infty$). Let us
fix a $c_0 \in C$ and consider $c_0f$ instead of $f$. By Lemma
\ref{l:C} this function is rigid for an uncountable set with all
isometries orientation-preserving, and if it is of the desired
form then so is $f$, so we are done.
We may assume $1 \notin C$. Let us fix a $c \in C$. Similarly as
in the previous subcase, we may assume that lines parallel to
$(0,1,d)$ are taken to lines parallel to $(0,1,cd)$ as follows.
The special form of $f$ implies that $graph(f)$ is invariant under
translations in the $(0,1,d)$-direction, hence $graph(cf)$ is
invariant under translations in the $(0,1,cd)$-direction,
moreover, by rigidity, $graph(cf)$ is also invariant under
translations parallel to the $\varphi_c$-image of the lines of
direction $(0,1,d)$. If these two latter directions do not
coincide then $graph(cf)$ is a plane, and we are done.
Therefore the image of every line parallel to $(0,1,d)$ is a line
parallel to $(0,1,cd)$ under the orientation-preserving isometry
$\varphi_c$. As in Remark \ref{r:rigid}, write $\varphi_c = \varphi_c^{trans}
\circ \varphi_c^{ort}$, where $\varphi_c^{ort}$ is a rotation about
a line containing the origin and $\varphi_c^{trans}$ is a translation.
Since the translation does not affect directions, the rotation
$\varphi_c^{ort}$ takes the direction $(0,1,d)$
to the nonparallel direction $(0,1,cd)$ ($d \neq
0$), therefore the axis of the rotation has to be orthogonal to the plane
spanned by these two directions. Hence the axis has to be the
$x$-axis. Moreover, the angle of the rotation is easily seen to be
$\arctan(cd) - \arctan(d)$.
We now show that we may assume that $\varphi_c^{trans}$ is a horizontal translation. Decompose the translation as $\varphi_c^{trans} = \varphi_c^{\vec{u}} \circ \varphi_c^{\vec{v}}$,
where $\varphi_c^{\vec{v}}$ is a horizontal translation and $\varphi_c^{\vec{u}}$ is a translation in the $(0,1,cd)$-direction. Since $\varphi_c^{ort} (graph(f))$ is invariant under translations in the $(0,1,cd)$-direction, so is $\varphi_c^{\vec{v}} \circ \varphi_c^{ort} (graph(f))$, hence
\[
\varphi_c^{\vec{v}} \circ \varphi_c^{ort} (graph(f)) = \varphi_c^{\vec{u}} \circ \varphi_c^{\vec{v}} \circ \varphi_c^{ort} (graph(f)) = \varphi_c (graph(f)) = graph(cf),
\]
so we can assume $\varphi_c = \varphi_c^{\vec{v}} \circ \varphi_c^{ort}$, and we are done.
We will now complete the proof of this subcase by showing that the function $-\frac{1}{d} g$ is rigid for an uncountable set. Indeed, this suffices by Theorem \ref{t:Jan} and by the special form of $f$.
Let us denote the $xy$-plane by $\{z=0\}$ and consider the intersection of both sides of the equation $\varphi_c (graph(f)) = graph(cf)$ with $\{z=0\}$.
On the one hand, $\{z=0\} \cap \varphi_c (graph(f)) =
\{z=0\} \cap \varphi_c^{\vec{v}} \circ \varphi_c^{ort} (graph(f)) =
\varphi_c^{\vec{v}} ( \{z=0\} \cap \varphi_c^{ort} (graph(f))) =
\varphi_c^{\vec{v}} \left( graph \left( -w_{c,d} g \right)\right) =
\varphi_c^{\vec{v}} \left( graph \left( (w_{c,d}d)(-\frac{1}{d} g \right)\right)$,
where we used the fact that $\varphi_c^{\vec{v}}$ is horizontal and Lemma \ref{lem1}.
On the other hand, it is easy to see that $\{z=0\} \cap graph(cf) = graph\left( -\frac{1}{d} g \right)$. Therefore $graph\left( -\frac{1}{d} g \right) = \varphi_c^{\vec{v}} \left( graph \left( (w_{c,d}d)(-\frac{1}{d} g \right)\right)$ and hence
$-\frac{1}{d} g$
is rigid for $w_{c,d}d$ for every $c>0$. The map $c \mapsto
w_{c,d}d$ is strictly monotone for every fixed $d$, hence the range of $C$ is
uncountable. So $-\frac{1}{d} g$ is rigid for an uncountable set,
and we are done.
\noindent\textbf{Case B.} $S_f = \SS^2 \sm\{(0,0,1), (0,0,-1)\}$.
So $S_f$ is invariant under every $\psi_c$, and hence so is under every
$\varphi_c^{ort}$. Then clearly $\varphi_c^{ort}((0,0,1)) = (0,0,1)$ or
$\varphi_c^{ort}((0,0,1)) = (0,0,-1)$ for every $c \in C$. By the same
argument as above we can assume that the former holds for
every $c \in C$. Using the argument again we can assume
that all $\varphi_c$'s are orientation-preserving. But then each of these is a
rotation around the $z$-axis followed by a translation, in other words, an
orientation-preserving transformation in the $xy$-plane followed by a
translation in the $z$-direction. An orientation-preserving transformation in
the plane is either a translation or a rotation. If it is a translation for
every $c$ then we are done by Corollary \ref{c:trans}. So let us assume that
there exists a $c$ such that $\varphi_c$ is a proper rotation around $\vec{x}
\in \RR^2$ followed by a vertical translation. We claim that then $f$ is
constant, which will contradict that $S_f$ is nearly the full sphere, finishing
the proof of this case. We will actually show that
$f$ is constant on every closed disc $B(\vec{x},R)$ centered at
$\vec{x}$. Indeed, consider $\max_{B(\vec{x},R)} f - \min_{B(\vec{x},R)}
f$. This is unchanged by the rotation around $\vec{x}$ as well as by the
vertical translation, hence by $\varphi_c$. But the map $f \mapsto cf$
multiplies this amount by $c \neq 1$, so the only option is
$\max_{B(\vec{x},R)} f - \min_{B(\vec{x},R)} f = 0$, and we are done.
\noindent\textbf{Case C.} There exists an $\vec{x_0} \in \SS^1$ such that
$h_3(\vec{x_0}) = 0$ and $h_3(\vec{x}) = 1$ for every $\vec{x} \neq
\vec{x_0}$, that is, $S_f$ is `$\SS^2$ minus two quarters of a great circle'.
So $S_f$ is invariant under every $\psi_c$, and hence so is under every
$\varphi_c^{ort}$. Hence $\varphi_c^{ort}$ maps $(0,0,1)$ to one of the four
endpoints of the two arcs. Therefore we can assume by splitting $C$ into four
pieces according to the image of $(0,0,1)$ and applying Lemma \ref{l:C}
that $(0,0,1)$ is a fixed point of every $\varphi_c^{ort}$. But then the two
arcs are also fixed, and actually $\varphi_c^{ort}$ is the identity. Hence
every $\varphi_c$ is a translation, and we are done by Corollary
\ref{c:trans}.
\noindent\textbf{Case D.} There exists a closed interval $I$ in $\SS^1$ with
$0 < length(I) < \pi$ such that $h_3(\vec{x}) = 0$ if $\vec{x} \in I$ and
$h_3(\vec{x}) = 1$ if $\vec{x} \notin I$, that is, $S_f$ is `$\SS^2$ minus two
spherical triangles'.
As $S_f$ is invariant under every $\varphi_c^{ort}$, vertices of
the triangles are mapped to vertices. Hence we may assume (by splitting $C$
into six pieces) that $(0,0,1)$ is fixed. But then the triangles are also
fixed sets, and every $\varphi_c^{ort}$ is the identity, so we are done as
in the previous case.
This finishes the proof of the Main Theorem.
\ep
\section{Open questions}
\lab{s:open}
\bq
\lab{q:meas}
In the Main Theorem can we relax the assumption of continuity to Lebesgue
measurability, Baire measurability, Borel measurability, Baire class one,
separate continuity or at least one point of continuity?
\eq
\bq
Which notion of largeness of $C$ suffices for the various results of this
paper? For example, does the Main Theorem hold if we only assume that $C$
contains three elements that pairwise generate dense multiplicative subgroups
of $(0,\infty)$?
\eq
\br
It was shown in
\cite{Ri} that two such elements suffice for the analogous one-variable
result. However, two independent elements are not enough here, since if $g$
is vertically rigid for $c_1$ via a translation and $h$ is vertically rigid
for $c_2$ via a translation then
$f(x,y) = g(x) h(y)$ is vertically rigid for both.
Moreover, the main point in that proof in \cite{Ri} is to replace `splitting
$C$' by alternative arguments, and we were unable to do so here.
\er
The following question is rather vague.
\bq
\lab{q:rig}
Let us call a set $H \su \SS^2$ rigid if $\psi_c(H)$ is isometric to $H$ for
every $c > 0$. Is there a simple description of rigid sets? Or if we assume
some regularity?
\eq
And finally, the most intriguing problem.
\bq
What can we say if there are more than two variables?
\eq
|
1109.5109
|
\section{Introduction}\label{sec1}
Random matrix ensembles serve as simple models in a wide range of applications \cite{Efe97,Guhr:1997ve,Verbaarschot:2000dy,Meh04,ABF11} which can be found in number theory \cite{Mont73,BogKea}, disordered systems \cite{Efe97}, quantum chaos \cite{Guhr:1996wn}, empirical data analysis \cite{LCBP99,STRF08,Recher}, information theory \cite{Braunstein:2009gv}, and quantum chromodynamics (QCD) \cite{Shuryak:1992pi}. The complexity of most systems prevents derivations of correlation functions whereas analytic results are accessible for the corresponding random matrix model. The reason for the applicability of random matrix theory lies in the universality of spectral statistics on certain scales like the local mean level spacing \cite{ADMN97,Akemann:2003vy,MZ10} or on the global scale \cite{AJM93,Brezin:1994sq,Ambjorn:1996ga,Akemann:1996zr}. If the Lagrangian of the physical system drastically simplifies such that it is effectively described by global symmetries there might be a random matrix model fulfilling the same symmetries.
Already in the 60's and 70's \cite{Meh60,MehGau60,Dys62,Dys70,Meh71}, the $k$-point correlation functions of the Gaussian and circular ensembles for the three symmetries of orthogonal ($\beta=1$; GOE/COE), unitary ($\beta=2$; GUE/CUE) and unitary-symplectic ($\beta=4$; GSE/CSE) invariance were derived. They can be expressed as a single determinant for the unitary case and a single Pfaffian for $\beta\in\{1,4\}$ where the integrals are pulled inside of these structures. Their matrix elements only depend on two eigenvalues which is a drastic simplification of the integrand. Since then many other random matrix ensembles were studied, e.g. the Ginibre ensembles \cite{Gin65,LehSom91,Ake01,SomWie08} and the the other two rotation groups ${\rm O}(N)$ and ${\rm USp\,}(2N)$ \cite{HPZ05}. The $k$-point correlation functions as well as the averages over ratios of characteristic polynomials for many of these ensembles are determinants and Pfaffians with relatively simple entries only depending on one or two eigenvalues \cite{BreHik00,MehNor01,BorStr05}. For a long time it was thought that determinants appear for ensembles with $\beta=2$ and Pfaffians for the other two cases. In Refs.~\cite{KieGuh09a,KieGuh09b} the general conditions where derived to find these structures. Thus all these particular random matrix ensembles were unified in one procedure to derive these structures.
Very recently a random matrix model for the Wilson Dirac operator was introduced \cite{Damgaard:2010cz} in lattice QCD. It generalizes the chiral GUE which was studied in a Hermitian version \cite{Damgaard:2010cz,Akemann:2010zp,arXiv:1105.6229,arXiv:1108.3035} and a non-Hermitian one \cite{Kieburg:2011uf}. The eigenvalue correlations exhibit Pfaffians for the Hermitian \cite{arXiv:1108.3035} as well as for the non-Hermitian case \cite{KVZ11} reflecting the structure found in Ref.~\cite{KieGuh09b}. This structure has to be also valid in the continuum limit which is the chiral GUE. Hence the question arises if the Pfaffian determinants obtained for the $k$-point correlation functions and thus for the averages over ratios of characteristic polynomials are much more general than conjectured in the broad literature.
Also in other intermediate random matrix ensembles Pfaffians were found. For example a similar situation arises in the transition from GUE to GOE or GSE \cite{Pandey:1982br,KieGuh09a}. If the ensemble is purely a GUE then then the eigenvalue correlations can be cast into determinants whereas the smallest interaction with a GOE or a GSE yields a Pfaffian. It would be of theoretical, technical and numerical interest if all ensembles corresponding to $\beta=2$ exhibit this phenomenon when coupling it to another random matrix ensemble. Such a property simplifies the spectral statistics of intermediate ensembles onto the behavior of the entries of the Pfaffian which are averages of one or two characteristic polynomials only.
Recently, Forrester and Sinclair introduced Pfaffians at $\beta=2$. In Ref.~\cite{Sin} Sinclair extends the Pfaffian found for the partition function with $\beta=1,4$ to Hyperpfaffians with $\beta=L^2,L^2+1$ ($L\in\mathbb{N}$) which also comprises the $\beta=2$ case. With help of these results the authors of Ref.~\cite{ForSin} studied a ${\rm log}$-gas on a ring with two interacting species. One component of this gas is described by a $\beta=4$ ${\rm log}$ gas and the other one by a $\beta=1,2$ ${\rm log}$ gas. The Pfaffian determinants found in Refs.~\cite{Sin,ForSin} are similar to but not the same as the one derived in Sec.~\ref{sec3}.
We derive Pfaffian determinants for averages over ratios of characteristic polynomials weighted by a joint probability density function factorizing in weights of the single eigenvalues apart from a squared Vandermonde determinant. This squared Vandermonde determinant can be cast into one determinant similar to the $\beta=4$ case. Thus it fulfills the same condition as presented in Ref.~\cite{KieGuh09b} which implies a Pfaffian. This unifies all ten symmetry classes in the Cartan classification \cite{Altland:1997zz,Zir96} and exhibits a hidden universal algebraic property in all of these ensembles.
An introduction of the main idea and of the important functions for the technique used here is given in Sec.~\ref{sec0}. In Sec.~\ref{sec2}, we recall some basics known about the determinantal structure obtained for averages over ratios of characteristic polynomials with respect to chiral unitary random matrix ensembles. In contrast to this structure we derive Pfaffians for the same correlation functions in Sec.~\ref{sec3}. Thereby we discuss the Wilson-Dirac random matrix ensemble as a neat application and a good motivation of the derived Pfaffian determinant at the end of this section. The skew-orthogonal polynomials corresponding to the Pfaffian determinants are indeed closely related to the orthogonal polynomials which are found in the determinantal structures. This relation is shown in Sec.~\ref{sec4}. In Sec.~\ref{sec5}, we discuss the generalization of these results for chiral unitary ensembles to other random matrix ensembles like GUE and CUE.
\section{Preliminaries}\label{sec0}
Structures found in supersymmetry are the key ingredient for the technique used in the ensuing sections. These structures allow to derive determinants as well as Pfaffians of averaged ratios of characteristic polynomials and, thus, $k$-point correlation functions for a large class of random matrix ensembles in a direct way. The main idea is to recognize that these structures are a pure algebraic property of the random matrix ensemble and not an analytic one. By an algebraic rearrangement of the integrand one gets the determinants and Pfaffians without explicitly calculating any integrals. This idea was first proposed in Refs.~\cite{KieGuh09a,KieGuh09b}.
The requirements to obtain determinants was traced back to a factorization of the probability density of the random matrix ensemble into densities for the single eigenvalues times two Vandermonde determinants (see Ref.~\cite{KieGuh09a}), i.e. the measure for the single eigenvalues has to be
\begin{eqnarray}\label{0.1}
d\mu(z)=\prod\limits_{j=1}^Ng_1(z_j)d[z_{j}]|\Delta_N(z)|^2
\end{eqnarray}
with the Vandermonde determinant
\begin{eqnarray}\label{0.2}
\Delta_{N}(z)&=&\prod\limits_{1\leq a<b\leq N}(z_a-z_b)=(-1)^{N(N-1)/2}\det\left[z_{a}^{b-1}\right]_{1\leq a,b\leq N}.
\end{eqnarray}
The variables $z$ can be complex which correspond to ensembles related to biorthogonal polynomials \cite{Bergere:2003ht}. For Pfaffians this requirement changes to a weight for pairs of eigenvalues and a single Vandermonde determinant \cite{KieGuh09b}, i.e.
\begin{eqnarray}\label{0.3}
d\mu(z)=\prod\limits_{j=1}^Ng_2(z_{2j-1},z_{2j})d[z_{2j-1}]d[z_{2j}]\Delta_{2N}(z).
\end{eqnarray}
If one of these two conditions are fulfilled then the technique presented in Refs.~\cite{KieGuh09a,KieGuh09b} circumvents the integration theorem by Dyson and Metha \cite{Dys70,Meh71,Meh04,Gho09}. Moreover the approach of Refs.~\cite{KieGuh09a,KieGuh09b} makes an integration theorem unnecessary at the end since it is automatically fulfilled for random matrix ensembles traced back to measures of the form~\eref{0.1} or \eref{0.3}. This can be readily seen by the combination of the determinantal and Pfaffian factorization for averages over ratios of characteristic polynomials \cite{KieGuh09a,KieGuh09b}, the representation of the orthogonal and skew-orthogonal polynomials as averages of the corresponding ensemble \cite{Sze39,Eyn01,Bergere:2003ht,Meh04,Gho09,arXiv:1005.2983} and the expressions of the kernels of the determinants and Pfaffians in orthogonal and skew-orthogonal polynomials~\cite{Meh04,Gho09}. In Sections~\ref{sec2} and \ref{sec3} we derive the $k$-point correlation function without using the integration theorem by Dyson and Metha.
Although, we do not explicitly need supersymmetry, in particular a superspace, some functions are quite useful to write the algebraic expressions of the calculations in a very compact, constructive and intuitive way. These functions have their origin in the theory of supermatrices. For the interested reader, good introductions in supersymmetry are given in Ref.~\cite{Ber87} and in the appendix of Ref.~\cite{VWZ85}. Here we only recall some of these useful algebraic functions and notions.
A diagonal $(p/q)\times(p/q)$ supermatrix $x$ consists of two blocks, $x={\rm diag\,}(x_1,x_2)$. The $p\times p$ matrix $x_1$ and the $q\times q$ matrix $x_2$ are indeed diagonal, too. The supertrace ``${\rm Str\,}$'' and the superdeterminant ``${\rm Sdet\,}$'' of $s$ is then defined by
\begin{eqnarray}\label{0.4}
{\rm Str\,} x&=&\tr x_1-\tr x_2=\sum_{j=1}^p x_{j1}-\sum_{i=1}^q x_{i2},\\
{\rm Sdet\,} x&=&\frac{\det x_1}{\det x_2}=\frac{\prod_{j=1}^p x_{j1}}{\prod_{i=1}^q x_{i2}}.\nonumber
\end{eqnarray}
The crucial function of the method used here is
\begin{eqnarray}\label{0.5}
{\rm B\,}_{p/q}(x)&=&\frac{\Delta_{p}(x_1)\Delta_{q}(x_2)}{\prod\limits_{a,b}(x_{a1}-x_{b2})}\\
&=&(-1)^{q(q-1)/2+(q+1)p}\det\left[\begin{array}{c} \displaystyle\left\{\frac{1}{x_{a1}-x_{b2}}\right\}\underset{1\leq b\leq q}{\underset{1\leq a\leq p}{\ }} \\ \displaystyle\left\{x_{b2}^{a-1}\right\}\underset{1\leq b\leq q}{\underset{1\leq a\leq q-p}{\ }} \end{array}\right]\nonumber
\end{eqnarray}
for $p\leq q$. It is the square root of a Berezinian,
\begin{eqnarray}\label{0.6}
{\rm B\,}_{p/q}^2(x)&=&{\rm Ber\,}_{p/q}^{(2)}(x),
\end{eqnarray}
which is the Jacobian in superspace when diagonalizing a Hermitian $(p/q)\times(p/q)$ supermatrix. The notation on the right hand side of Eq.~\eref{0.6} refers to the one used in Refs.~\cite{KieGuh09a,KieGuh09b}.
Everything we need for the method of Refs.~\cite{KieGuh09a,KieGuh09b} are the functions ``${\rm Sdet\,}$'' and ``${\rm B\,}$'' embedded in an ordinary space like $\mathbb{R}^{p+q}$ or $\mathbb{C}^{p+q}$. Hence those readers who are not accustomed to supersymmetry may consider these functions as ordinary, rational functions.
\section{Review of chiral unitary random matrices}\label{sec2}
We consider the anti-Hermitian random matrix
\begin{eqnarray}\label{2.1}
D=\left[\begin{array}{cc} 0 & W \\ -W^\dagger & 0\end{array}\right]
\end{eqnarray}
which is distributed by the density
\begin{eqnarray}\label{2.2}
P(D)d[D]=\exp[-\alpha \tr V(WW^\dagger)]\prod\limits_{a,b}d~{\rm Re\,} W_{ab}\ d~{\rm Im\,} W_{ab}
\end{eqnarray}
with a non-zero normalization constant. In particular it serves as a model for the Dirac operator in QCD \cite{Shuryak:1992pi}. The constant $\alpha$ is proportional to $n$. The matrix $W$ is a $n\times(n+\nu)$ rectangular matrix. Each of the $n(n+\nu)$ entries of $W$ is a complex number which might be statistically coupled by the arbitrary density $P$. The parameter $\nu$ with $0\leq\nu\leq n$ is the topological charge or also known as index of the Dirac operator such that $D$ has $\nu$ generic zero eigenmodes. The potential $V$ is invariant under the group ${\rm U\,}(n)$, i.e.
\begin{eqnarray}\label{2.2b}
V(UWW^\dagger U^\dagger)=UV(WW^\dagger) U^\dagger,
\end{eqnarray}
and is chosen such that all moments of the ensemble over $\mathbb{C}^{n\times(n+\nu)}$ exist. In the simplest case $P$ is Gaussian. Nevertheless the arguments given here are also true for an arbitrary potential. We only need the property
\begin{eqnarray}\label{2.3}
P\left(\left[\begin{array}{c|cc} 0 & \Lambda & 0 \\ \hline -\Lambda & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\right)=\exp[-\alpha \tr V(\Lambda^2)]=\prod\limits_{j=1}^n\exp[-\alpha V(\lambda_j^2)]
\end{eqnarray}
for the matrix $\Lambda={\rm diag\,}(\lambda_1,\ldots,\lambda_n)$ with the singular values $0\leq\lambda_1\leq\ldots\leq\lambda_n$ of $W$, i.e. there are $U\in{\rm U\,}(n)$ and $V\in{\rm U\,}(n+\nu)$ with
\begin{eqnarray}\label{2.4}
D={\rm diag\,}(U,V)\left[\begin{array}{c|cc} 0 & \Lambda & 0 \\ \hline -\Lambda & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]{\rm diag\,}(U^\dagger,V^\dagger).
\end{eqnarray}
In this basis the measure~\eref{2.2} can be written as
\begin{eqnarray}\label{2.5}
P(D)d[D]&=&\frac{{\rm Vol\,}_n{\rm Vol\,}_{n+\nu}}{{\rm Vol\,}_1^n{\rm Vol\,}_\nu}\Delta_{n}^2(\Lambda^2)\prod\limits_{j=1}^n\exp[-\alpha V(\lambda_j^2)]\lambda_j^{2\nu+1}d\lambda_j\\
&\times&d\mu_{{\rm U\,}(n)/{\rm U\,}^n(1)}(U)d\mu_{{\rm U\,}(n+\nu)/{\rm U\,}(\nu)}(V).\nonumber
\end{eqnarray}
The abbreviation of the constant
\begin{eqnarray}\label{2.7}
{\rm Vol\,}_l=\prod\limits_{j=1}^l\frac{2\pi^j}{(j-1)!}
\end{eqnarray}
refers to the volume of the unitary group ${\rm U\,}(l)$. Thus, the prefactor in Eq.~\eref{2.5} is the volume of the coset $[{\rm U\,}(n)\times{\rm U\,}(n+\nu)]/[{\rm U\,}^n(1)\times{\rm U\,}(\nu)]$. The measure $d\mu_{\mathfrak{G}}$ is the normalized Haar measure of the coset $\mathfrak{G}$.
An important quantity to analyze the eigenvalue statistics of this ensemble is the average over ratios of characteristic polynomials with respect to $D$, i.e.
\begin{eqnarray}\label{2.8}
Z_{k_1/k_2}^{(n,\nu)}(\kappa)=\int\limits_{\mathbb{C}^{n\times(n+\nu)}}\frac{\prod\limits_{j=1}^{k_2}\det(D-\imath\kappa_{j2}\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{2n+\nu})}{\prod\limits_{j=1}^{k_1}\det(D-\imath\kappa_{j1}\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{2n+\nu})}P(D)d[D]
\end{eqnarray}
with the diagonal, non-degenerate $(k_1/k_2)\times(k_1/k_2)$ supermatrix $\kappa={\rm diag\,}(\kappa_1,\kappa_2)={\rm diag\,}(\kappa_{11},\ldots,\kappa_{k_11},$ $\kappa_{12},\ldots,\kappa_{k_22})$ and the $2n+\nu$ dimensional unit matrix $\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{2n+\nu}$. This average is also known as the partition function with $k_1$ bosonic and $k_2$ fermionic flavors in QCD \cite{Splittorff:2002eb,Fyodorov:2002wq,Akemann:2003vy}. The variables $\kappa_{j1}$ are complex numbers with a non-vanishing imaginary part such that the integral is well defined. The partition function~\eref{2.8} is simply related to the matrix Green function and, thus, to the $k$-point correlation function by derivatives with respect to $\kappa$.
The joint probability density~\eref{2.5} is of the class studied in Ref.~\cite{KieGuh09a} and can, therefore, be written as a determinant. This was derived in many articles before~\cite{Fyodorov:2002md,Splittorff:2002eb,Fyodorov:2002wq}. The crucial idea presented in Ref.~\cite{KieGuh09a} is the combination of the ratio of characteristic polynomials~\eref{2.8} with the two Vandermonde determinants~\eref{2.5} to square roots of Berezinians~\eref{0.5}, i.e.
\begin{eqnarray}\label{2.9}
\Delta_n^2(\Lambda^2)\frac{\prod\limits_{j=1}^{k_2}\det(\Lambda^2-\kappa_{j2}^2\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{n})}{\prod\limits_{j=1}^{k_1}\det(\Lambda^2-\kappa_{j1}^2\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{n})}&=&\frac{{\rm B\,}_{l_{11}/l_{21}+n}(\widetilde{\kappa}_1^2,\Lambda^2){\rm B\,}_{l_{12}/l_{22}+n}(\widetilde{\kappa}_2^2,\Lambda^2)}{{\rm B\,}_{l_{11}/l_{21}}(\widetilde{\kappa}_1^2){\rm B\,}_{l_{12}/l_{22}}(\widetilde{\kappa}_1^2)}
\end{eqnarray}
for any choice of natural numbers $l_{11}+l_{12}=k_{1}$ and $l_{21}+l_{22}=k_{2}$.
In Eq.~\eref{2.9}, we split the supermatrix $\kappa$ into the two sets $\widetilde{\kappa}_1={\rm diag\,}(\widetilde{\kappa}_{11},\widetilde{\kappa}_{21})={\rm diag\,}(\kappa_{11},\ldots,\kappa_{l_{11}1},\kappa_{12},\ldots,\kappa_{l_{21}2})$ and $\widetilde{\kappa}_2={\rm diag\,}(\widetilde{\kappa}_{12},\widetilde{\kappa}_{22})={\rm diag\,}(\kappa_{l_{11}+1,1},\ldots,\kappa_{k_{1}1},\kappa_{l_{21}+1,2},\ldots,\kappa_{k_{2}2})$. The choice how we split this set is arbitrary and, thus, we get equivalent but not trivially related results. This was already recognized by the authors of Ref.~\cite{Akemann:2002vy} for products of characteristic polynomials. Let $d_1=n+l_{21}-l_{11}$ and $d_2=n+l_{22}-l_{12}$. The interesting case is $d_1,d_2\geq0$ because we want to discuss the limit $n\to\infty$ and $k_1$, $k_2$ fixed, at the end of this section. The other cases are discussed in Ref.~\cite{KieGuh09a}.
Without loss of generality we assume $d_1\leq d_2$. We rearrange the integrand~\eref{2.8} with the help of Eq.~\eref{2.9} which yields
\begin{eqnarray}
\fl Z_{k_1/k_2}^{(n,\nu)}(\kappa)&\propto&{\rm Sdet\,}^{-\nu}\kappa\int\frac{{\rm B\,}_{l_{11}/l_{21}+n}(\widetilde{\kappa}_1^2,\Lambda^2){\rm B\,}_{l_{12}/l_{22}+n}(\widetilde{\kappa}_2^2,\Lambda^2)}{{\rm B\,}_{l_{11}/l_{21}}(\widetilde{\kappa}_1^2){\rm B\,}_{l_{12}/l_{22}}(\widetilde{\kappa}_1^2)}\prod\limits_{j=1}^n\exp[-\alpha V(\lambda_j^2)]\lambda_j^{2\nu+1}d\lambda_j\nonumber\\
\fl&\propto&{\rm Sdet\,}^{-\nu}\kappa\int\frac{\prod\limits_{j=1}^n\exp[-\alpha V(\lambda_j^2)]\lambda_j^{2\nu+1}d\lambda_j}{{\rm B\,}_{l_{11}/l_{21}}(\widetilde{\kappa}_1^2){\rm B\,}_{l_{12}/l_{22}}(\widetilde{\kappa}_1^2)}\nonumber\\
\fl&\times&\det\left[\begin{array}{cc} \displaystyle\left\{\frac{1}{\kappa^2_{a1}-\kappa^2_{b2}}\right\}\underset{1\leq b\leq l_{21}}{\underset{1\leq a\leq l_{11}}{\ }} & \displaystyle\left\{\frac{1}{\kappa^2_{a1}-\lambda^2_{b2}}\right\}\underset{1\leq b\leq n}{\underset{1\leq a\leq l_{11}}{\ }} \\ \displaystyle\left\{\kappa^{2(a-1)}_{b2}\right\}\underset{1\leq b\leq l_{21}}{\underset{1\leq a\leq d_1}{\ }} & \displaystyle\left\{\lambda^{2(a-1)}_{b2}\right\}\underset{1\leq b\leq n}{\underset{1\leq a\leq d_1}{\ }} \end{array}\right]\nonumber\\
\fl&\times&\det\left[\begin{array}{cc} \displaystyle\left\{\frac{1}{\kappa^2_{a1}-\kappa^2_{b2}}\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{ l_{11}+1\leq a\leq k_1}{\ }} & \displaystyle\left\{\frac{1}{\kappa^2_{a1}-\lambda^2_{b2}}\right\}\underset{1\leq b\leq n}{\underset{ l_{11}+1\leq a\leq k_1}{\ }} \\ \displaystyle\left\{\kappa^{2(a-1)}_{b2}\right\}\underset{l_{21}+1\leq b\leq k_{2}}{\underset{1\leq a\leq d_2}{\ }} & \displaystyle\left\{\lambda^{2(a-1)}_{b2}\right\}\underset{1\leq b\leq n}{\underset{1\leq a\leq d_2}{\ }} \end{array}\right].\label{2.10a}
\end{eqnarray}
Applying the generalized Andr{\'e}ief integration theorem \cite{And1883,KieGuh09a} we obtain
\begin{eqnarray}\label{2.10b}
\fl Z_{k_1/k_2}^{(n,\nu)}(\kappa)&\propto&\frac{{\rm Sdet\,}^{-\nu}\kappa}{{\rm B\,}_{l_{11}/l_{21}}(\widetilde{\kappa}_1^2){\rm B\,}_{l_{12}/l_{22}}(\widetilde{\kappa}_1^2)}\\
\fl&\times&\det\left[\begin{array}{ccc} 0 & \displaystyle\left\{\frac{1}{\kappa^2_{b1}-\kappa^2_{a2}}\right\}\underset{l_{11}+1\leq b\leq k_1}{\underset{ l_{21}+1\leq a\leq k_2}{\ }} & \displaystyle\left\{\kappa^{2(b-1)}_{a2}\right\}\underset{1\leq b\leq d_{2}}{\underset{l_{21}+1\leq a\leq k_2}{\ }} \\ \displaystyle\left\{\frac{1}{\kappa^2_{a1}-\kappa^2_{b2}}\right\}\underset{1\leq b\leq l_{21}}{\underset{1\leq a\leq l_{11}}{\ }} & \displaystyle\left\{F(\kappa_{a1},\kappa_{b1})\right\}\underset{l_{11}+1\leq b\leq k_{1}}{\underset{1\leq a\leq l_{11}}{\ }} & \displaystyle\left\{F_b(\kappa_{a1})\right\}\underset{1\leq b\leq d_{2}}{\underset{1\leq a\leq l_{11}}{\ }} \\ \displaystyle\left\{\kappa^{2(a-1)}_{b2}\right\}\underset{1\leq b\leq l_{21}}{\underset{1\leq a\leq d_1}{\ }} & \displaystyle\left\{F_a(\kappa_{b1})\right\}\underset{l_{11}+1\leq b\leq k_{1}}{\underset{1\leq a\leq d_1}{\ }} & \displaystyle\left\{M_{ab}\right\}\underset{1\leq b\leq d_2}{\underset{1\leq a\leq d_1}{\ }} \end{array}\right]\nonumber
\end{eqnarray}
Notice that Andr{\'e}ief's integration theorem as well as its generalization is only an algebraic rearrangement of the integrals without explicitly calculating any integral. The functions $F$ and $F_a$ are one dimensional integrals and their explicit expressions are not so important as we will see in the discussion after Eq.~\eref{2.10f}. For the interested reader we refer to Ref.~\cite{KieGuh09a} where the explicit integrals are given for general random matrix ensembles corresponding to determinants ($\beta=2$). The constant $d_1\times d_2$ matrix $M=[M_{ab}]$ is given by
\begin{eqnarray}\label{2.10c}
M_{ab}=\int_{\mathbb{R}} \lambda^{2(a+b-2)}\exp[-\alpha V(\lambda^2)]\lambda^{2\nu+1}d\lambda
\end{eqnarray}
and thus generates the moments of the measure.
In the next step we use the identity
\begin{eqnarray}\label{2.10d}
\det\left[\begin{array}{cc} A & B \\ C & D \end{array}\right]=\det D\det[ A-BD^{-1}C]
\end{eqnarray}
for arbitrary matrices $A$, $B$ and $C$ and an invertible matrix $D$. For the matrix $D$ we choose the $d_1\times d_1$ matrix
\begin{eqnarray}\label{2.10e}
D=\displaystyle[M_{ab}]\underset{1\leq a,b\leq d_1}{\ }
\end{eqnarray}
which is only a part of the full rectangular matrix $M$ appearing in Eq.~\eref{2.10b}. The determinant of $D$ is proportional to the normalization constant of the ensemble~\eref{3.2} and $M$ is therefore invertible. Employing Eq.~\eref{2.10d} we find
\begin{eqnarray}\label{2.10f}
\fl Z_{k_1/k_2}^{(n,\nu)}(\kappa)&=&\frac{1}{{\rm B\,}_{l_{11}/l_{21}}(\widetilde{\kappa}_1^2){\rm B\,}_{l_{12}/l_{22}}(\widetilde{\kappa}_2^2)}\\
\fl&\times&\det\left[\begin{array}{cc} \displaystyle\left\{G_1^{(d_1)}(\kappa_{a2},\kappa_{b2})\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{1\leq a\leq l_{21}}{\ }} & \displaystyle\left\{G_2^{(d_1)}(\kappa_{b1},\kappa_{a2})\right\}\underset{1\leq b\leq l_{11}}{\underset{1\leq a\leq l_{21}}{\ }} \\ \displaystyle\left\{G_2^{(d_1)}(\kappa_{a1},\kappa_{b2})\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{l_{11}+1\leq a\leq k_1}{\ }} & \displaystyle\left\{G_3^{(d_1)}(\kappa_{a1},\kappa_{b1})\right\}\underset{1\leq b\leq l_{11}}{\underset{l_{11}+1\leq a\leq k_1}{\ }} \\ \displaystyle\left\{H_1^{(a)}(\kappa_{b2})\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{d_1+1\leq a\leq d_2}{\ }} & \displaystyle\left\{H_2^{(a)}(\kappa_{b1})\right\}\underset{1\leq b\leq l_{11}}{\underset{d_1+1\leq a\leq d_2}{\ }} \end{array}\right]\nonumber.
\end{eqnarray}
In the last step we identify the functions $G_1^{(d_1)}$, $G_2^{(d_1)}$, $G_3^{(d_1)}$, $H_1^{(a)}$ and $H_2^{(a)}$ by considering the particular choices $(l_{11},l_{12},l_{21},l_{22})\in\{(0,0,1,1),(1,0,1,0),(1,1,0,0),(0,0,0,1),$ $(1,0,0,0)\}$. In all of these cases the determinant reduces to one of the entries. Then we obtain
\begin{eqnarray}\label{2.11}
\fl &&\frac{Z_{k_1/k_2}^{(n,\nu)}(\kappa)}{Z_{0/0}^{(n,\nu)}}=\frac{(-1)^{k_1(k_1-1)/2+(l_{21}+1)(k_1+1)+(l_{11}+1)(k_2+1)}}{{\rm B\,}_{l_{11}/l_{21}}(\widetilde{\kappa}_1^2){\rm B\,}_{l_{12}/l_{22}}(\widetilde{\kappa}_2^2)}\frac{\prod\limits_{j=0}^{d_1-1}h_j^{(\nu)}}{\prod\limits_{j=0}^{n-1}h_j^{(\nu)}}\\
\fl&\times&\det\left[\begin{array}{cc} \displaystyle\left\{-\frac{Z_{0/2}^{(d_1-1,\nu)}(\kappa_{a2},\kappa_{b2})}{h_{d_1-1}^{(\nu)}Z_{0/0}^{(d_1-1,\nu)}}\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{1\leq a\leq l_{21}}{\ }} & \displaystyle\left\{\frac{1}{Z_{0/0}^{(d_1,\nu)}}\frac{Z_{1/1}^{(d_1,\nu)}(\kappa_{b1},\kappa_{a2})}{(\kappa_{b1}^2-\kappa_{a2}^2)}\right\}\underset{1\leq b\leq l_{11}}{\underset{1\leq a\leq l_{21}}{\ }} \\ \displaystyle\left\{\frac{1}{Z_{0/0}^{(d_1,\nu)}}\frac{Z_{1/1}^{(d_1,\nu)}(\kappa_{a1},\kappa_{b2})}{(\kappa_{a1}^2-\kappa_{b2}^2)}\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{l_{11}+1\leq a\leq k_1}{\ }} & \displaystyle\left\{\frac{h_{d_1}^{(\nu)}}{Z_{0/0}^{(d_1+1,\nu)}}Z_{2/0}^{(d_1+1,\nu)}(\kappa_{a1},\kappa_{b1})\right\}\underset{1\leq b\leq l_{11}}{\underset{l_{11}+1\leq a\leq k_1}{\ }} \\ \displaystyle\left\{\frac{Z_{0/1}^{(a-1,\nu)}(\kappa_{b2})}{Z_{0/0}^{(a-1,\nu)}}\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{d_1+1\leq a\leq d_2}{\ }} & \displaystyle\left\{\frac{h_{a-1}^{(\nu)}}{Z_{0/0}^{(a,\nu)}}Z_{1/0}^{(a,\nu)}(\kappa_{b1})\right\}\underset{1\leq b\leq l_{11}}{\underset{d_1+1\leq a\leq d_2}{\ }} \end{array}\right]\nonumber
\end{eqnarray}
for the partition function~\eref{2.8} which is a particular result of the general one derived in Ref.~\cite{KieGuh09a}.
The determinant~\eref{2.11} interpolates between one-point and two-point kernels as the entries of the determinant. We emphasize again the choice of the numbers $0\leq l_{11}\leq k_1$ and $0\leq l_{21}\leq k_1$ and the splitting of $\kappa$ are arbitrary. The particular choice $l_{11}=k_1$ and $l_{21}=0$ yields the $k_1+k_2$ dimensional determinant with one-point kernels considered in Refs.~\cite{Splittorff:2002eb,Akemann:2003vy}. This choice is suitable for the microscopic limit in chiral random matrix theory. For bulk and soft edge correlations \cite{Akemann:2003vy} the representation in two point correlations are the better choice to make contact with other random matrix ensembles \cite{AJM93,Brezin:1994sq,Ambjorn:1996ga,MZ10}. This case relates to the choice $l_{11}=k_1$ and $l_{21}=k_2$ for $k_2\leq k_1$ and $l_{11}=0$ and $l_{21}=0$ for $k_2\geq k_1$.
The $k$-point correlation function at the $k$ variables $x={\rm diag\,}(x_1,\ldots,x_k)$ is given by
\begin{eqnarray}
\fl R_k^{(n,\nu)}(x)&\propto&\int_{\mathbb{R}_+^{n-k}}\Delta_n^2({\rm diag\,}(x^2,\Lambda^2))\exp[-\alpha\tr V(x^2)-\alpha\tr V(\Lambda^2)]{\det}^{2\nu+1}x\prod\limits_{j=1}^{n-k}\lambda_j^{2\nu+1}d\lambda_j\nonumber\\
\fl&\propto&\Delta_k^2(x^2){\det}\, x\exp[-\alpha\tr V(x^2)]Z_{0/2k}^{(n-k,\nu)}({\rm diag\,}(x,-x)).\label{2.12a}
\end{eqnarray}
Now we employ the formula~\eref{2.11} for $(l_{11},l_{12},l_{21},l_{22})=(0,0,k,k)$ and find the result
\begin{eqnarray}
\fl R_k^{(n,\nu)}(x)&\propto&\det\left[\sqrt{x_ax_b}\exp[-\alpha( V(x_a^2)+V(x_b^2))/2]Z_{0/2}^{(n-1,\nu)}(x_a,-x_b)\right]_{1\leq a,b\leq k}.\label{2.12b}
\end{eqnarray}
Since $Z_{0/2}^{(n-1,\nu)}(x_a,-x_b)=(-1)^\nu Z_{0/2}^{(n-1,\nu)}(x_a,x_b)$ this agrees with the general formula for $\beta=2$ ensembles \cite{Meh04}. Please notice that we derived this formula without using the integration theorem by Dyson and Mehta \cite{Dys70,Meh71,Meh04,Gho09}.
The constant $h_j^{(\nu)}$ in Eq.~\eref{2.11} is the normalization constant of the orthogonal polynomial
\begin{equation}\label{2.12}
p_j^{(\nu)}(x^2)=\frac{(-1)^{j}}{(-\imath x)^\nu}\frac{Z_{0/1}^{(j,\nu)}(x)}{Z_{0/0}^{(j,\nu)}}.
\end{equation}
These polynomials solve the orthogonality relation
\begin{equation}\label{2.13}
\int\limits_{0}^\infty p_j^{(\nu)}(x^2)p_i^{(\nu)}(x^2)x^{2\nu+1}\exp[-\alpha V(x^2)]dx=h_j^{(\nu)}\delta_{ji}.
\end{equation}
The authors of Ref.~\cite{Damgaard:1997ye} have shown that these polynomials fulfill a recursion relation with respect to the topological charge $\nu$ by
\begin{equation}\label{2.13b}
\frac{p_j^{(\nu+1)}(x)}{p_j^{(\nu+1)}(0)}=\frac{1}{x}\frac{p_j^{(\nu)}(0)p_{j+1}^{(\nu)}(x)-p_{j+1}^{(\nu)}(0)p_j^{(\nu)}(x)}{p_j^{(\nu)}(0)p_{j+1}^{(\nu)\prime}(0)-p_{j+1}^{(\nu)}(0)p_j^{(\nu)\prime}(0)}
\end{equation}
which is quite useful by taking the limit $n\to\infty$. This relation follows when setting $m=0$ in Eq.~(12) of Ref.~\cite{Damgaard:1997ye}. One can readily prove identity~\eref{2.13b} by showing the orthogonality relation~\eref{2.13} for the right hand side with respect to the $\nu+1$ measure, i.e.
\begin{eqnarray}
\fl&&\int\limits_{0}^\infty \frac{p_j^{(\nu)}(0)p_{j+1}^{(\nu)}(x^2)-p_{j+1}^{(\nu)}(0)p_j^{(\nu)}(x^2)}{x^2}\frac{p_l^{(\nu)}(0)p_{l+1}^{(\nu)}(x^2)-p_{l+1}^{(\nu)}(0)p_l^{(\nu)}(x^2)}{x^2}x^{2\nu+3}e^{-\alpha V(x^2)}dx\nonumber\\
\fl&\propto&\int\limits_{0}^\infty \sum_{a=0}^{j}\frac{p_a^{(\nu)}(0)p_a^{(\nu)}(x^2)}{h_a^{(\nu)}}(p_l^{(\nu)}(0)p_{l+1}^{(\nu)}(x^2)-p_{l+1}^{(\nu)}(0)p_l^{(\nu)}(x^2))x^{2\nu+1}e^{-\alpha V(x^2)}dx\nonumber\\
\fl&\propto&\delta_{jl}\label{2.13c},
\end{eqnarray}
where we used the Christoffel-Darboux formula. The monic normalization of $p_j^{(\nu)}(x)=x^j+\ldots$ for all $j$ and $\nu$ explains the choice of the constants.
The Cauchy transform of $p_j^{(\nu)}$ is related to the partition function with one bosonic flavor by
\begin{eqnarray}\label{2.14}
\widehat{p}_{j}^{(\nu)}(x^2)&=&\int\limits_{0}^\infty\frac{p_j^{(\nu)}(\lambda^2)}{\lambda^2-x^2}\lambda^{2\nu+1}\exp[-\alpha V(\lambda^2)]d\lambda\\
&=&(-1)^{j}(-\imath x)^\nu\frac{h_{j}^{(\nu)}}{Z_{0/0}^{(j+1,\nu)}}Z_{1/0}^{(j+1,\nu)}(x)\nonumber.
\end{eqnarray}
In the result~\eref{2.11} we recognize that the choices $(l_{11},l_{12},l_{21},l_{22})=(0,0,1,1),(1,0,1,0),$ $(1,1,0,0)$ yield the same partition functions as the choices $(l_{11},l_{12},l_{21},l_{22})=(0,0,0,2),(1,0,0,1),(2,0,0,0)$, respectively. Therefore the two-flavor partition functions in Eq.~\eref{2.11} can also be expressed in the orthogonal polynomials~\eref{2.12} and their Cauchy transforms~\eref{2.14}, i.e.
\begin{eqnarray}
\fl\frac{Z_{0/2}^{(d_1-1,\nu)}(\kappa_{a2},\kappa_{b2})}{Z_{0/0}^{(d_1-1,\nu)}}&=&-\frac{(-\kappa_{a2}\kappa_{b2})^\nu}{\kappa_{a2}^2-\kappa_{b2}^2}\det\left[\begin{array}{cc} p_{d_1-1}^{(\nu)}(\kappa_{a2}^2) & p_{d_1-1}^{(\nu)}(\kappa_{b2}^2) \\ p_{d_1}^{(\nu)}(\kappa_{a2}^2) & p_{d_1}^{(\nu)}(\kappa_{b2}^2) \end{array}\right]\label{2.15},\\
\fl\frac{Z_{2/0}^{(d_1+1,\nu)}(\kappa_{a1},\kappa_{b1})}{Z_{0/0}^{(d_1+1,\nu)}}&=&\frac{1}{h_{d_1}^{(\nu)}h_{d_1-1}^{(\nu)}}\frac{1}{(-\kappa_{a1}\kappa_{b1})^\nu(\kappa_{a1}^2-\kappa_{b1}^2)}\det\left[\begin{array}{cc} \widehat{p}_{d_1-1}^{(\nu)}(\kappa_{a1}^2) & \widehat{p}_{d_1-1}^{(\nu)}(\kappa_{b1}^2) \\ \widehat{p}_{d_1}^{(\nu)}(\kappa_{a1}^2) & \widehat{p}_{d_1}^{(\nu)}(\kappa_{b1}^2) \end{array}\right],\nonumber\\
&&\label{2.16}\\
\fl\frac{Z_{1/1}^{(d_1,\nu)}(\kappa_{a1},\kappa_{b2})}{Z_{0/0}^{(d_1,\nu)}}&=&\frac{1}{h_{d_1-1}^{(\nu)}}\left(\frac{\kappa_{b2}}{\kappa_{a1}}\right)^\nu\det\left[\begin{array}{cc} \widehat{p}_{d_1-1}^{(\nu)}(\kappa_{a1}^2) & p_{d_1-1}^{(\nu)}(\kappa_{b2}^2) \\ \widehat{p}_{d_1}^{(\nu)}(\kappa_{a1}^2) & p_{d_1}^{(\nu)}(\kappa_{b2}^2) \end{array}\right]\label{2.17}.
\end{eqnarray}
These three relations are already well known \cite{Meh04,Gho09}. They can also be derived with help of the Christoffel-Darboux formula.
The structure~\eref{2.11} is a general property of ensembles with a joint probability density including a squared Vandermonde determinant as considered in Sec.~4.2 of Ref.~\cite{KieGuh09a} whereas the relations~\eref{2.15}-\eref{2.17} have to be slightly modified for other ensembles.
In the microscopic limit the authors of Refs.~\cite{ADMN97,Akemann:2003vy} have shown that for a generic potential $V$ the orthogonal polynomials and their Cauchy transforms become
\begin{eqnarray}
p_n^{(\nu)}\left(\frac{x^2}{(cn)^2}\right)&\overset{n\gg1}{\propto}&\frac{J_{\nu}(x)}{x^{\nu}},\label{2.18}\\
\widehat{p}_n^{(\nu)}\left(\frac{x^2}{(cn)^2}\right)&\overset{n\gg1}{\propto}&x^{\nu}K_{\nu}(x),\label{2.19}
\end{eqnarray}
where $c$ is a constant depending on the potential $V$. The functions $J_{\nu}$ and $K_{\nu}$ are the Bessel function of the first kind and the modified one of the second kind, respectively. Hence in the microscopic limit the partition function~\eref{2.8} is
\begin{eqnarray}\label{2.20}
\fl Z_{k_1/k_2}^{(n,\nu)}\left(\frac{\kappa}{cn}\right)&\overset{n\gg1}{\propto}&\frac{1}{{\rm B\,}_{l_{11}/l_{21}}(\widetilde{\kappa}_1^2){\rm B\,}_{l_{12}/l_{22}}(\widetilde{\kappa}_2^2)}\\
\fl&&\times\det\left[\begin{array}{cc} \displaystyle\left\{ I^{(1)}_\nu(\kappa_{a2},\kappa_{b2})\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{1\leq a\leq l_{21}}{\ }} & \displaystyle\left\{ I^{(2)}_\nu(\kappa_{b1},\kappa_{a2})\right\}\underset{1\leq b\leq l_{11}}{\underset{1\leq a\leq l_{21}}{\ }} \\ \displaystyle\left\{ I^{(2)}_\nu(\kappa_{a1},\kappa_{b2})\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{l_{11}+1\leq a\leq k_1}{\ }} & \displaystyle\left\{ I^{(3)}_\nu(\kappa_{a1},\kappa_{b1})\right\}\underset{1\leq b\leq l_{11}}{\underset{l_{11}+1\leq a\leq k_1}{\ }} \\ \displaystyle\left\{\kappa_{b2}^{a}J_{\nu+a}(\kappa_{b2})\right\}\underset{l_{21}+1\leq b\leq k_2}{\underset{0\leq a\leq d_2-d_1-1}{\ }} & \displaystyle\left\{\kappa_{b1}^{a}K_{\nu+a}(\kappa_{b1})\right\}\underset{1\leq b\leq l_{11}}{\underset{0\leq a\leq d_2-d_1-1}{\ }} \end{array}\right]\nonumber,
\end{eqnarray}
where
\begin{eqnarray}
\fl I^{(1)}_\nu(\kappa_{a2},\kappa_{b2})&=&\left\{\begin{array}{cl} \displaystyle\frac{\kappa_{a2}J_{\nu-1}(\kappa_{a2})J_{\nu}(\kappa_{b2})-\kappa_{b2}J_{\nu}(\kappa_{a2})J_{\nu-1}(\kappa_{b2})}{\kappa_{a2}^2-\kappa_{b2}^2}, & a\neq b, \\ \displaystyle\frac{J_{\nu+1}(\kappa_{a2})J_{\nu-1}(\kappa_{a2})-J_{\nu}^2(\kappa_{a2})}{2}, & a= b, \end{array}\right.\label{2.21}\\
\fl I^{(2)}_\nu(\kappa_{a1},\kappa_{b2})&=&\frac{\kappa_{a1}K_{\nu-1}(\kappa_{a1})J_{\nu}(\kappa_{b2})-\kappa_{b2}K_{\nu}(\kappa_{a1})J_{\nu-1}(\kappa_{b2})}{\kappa_{a1}^2-\kappa_{b2}^2},\label{2.22}\\
\fl I^{(3)}_\nu(\kappa_{a1},\kappa_{b1})&=&\left\{\begin{array}{cl} \displaystyle\frac{\kappa_{a1}K_{\nu-1}(\kappa_{a1})K_{\nu}(\kappa_{b1})-\kappa_{b1}K_{\nu}(\kappa_{a1})K_{\nu-1}(\kappa_{b1})}{\kappa_{a1}^2-\kappa_{b1}^2}, & a\neq b, \\ \displaystyle\frac{K_{\nu+1}(\kappa_{a1})K_{\nu-1}(\kappa_{a1})-K_{\nu}^2(\kappa_{a1})}{2}, & a= b. \end{array}\right.\label{2.23}
\end{eqnarray}
This is the well known result found in the literature~\cite{Fyodorov:2002md,Splittorff:2002eb,Fyodorov:2002wq}.
\section{Derivation of the Pfaffian determinant}\label{sec3}
In subsection~\ref{sec3.1} we derive a Pfaffian determinant for the same class of chiral random matrix ensembles discussed in Sec.~\ref{sec2}. A neat application of this Pfaffian is presented in subsection~\ref{sec3.2}. This example is the random matrix model for the Wilson-Dirac operator in lattice QCD \cite{Damgaard:2010cz,Akemann:2010zp,arXiv:1105.6229,arXiv:1108.3035,Kieburg:2011uf}.
\subsection{Pfaffian determinants in chiral random matrix theory}\label{sec3.1}
We show that the representations in determinants~\eref{2.11} are not the only existing ones for chiral unitary ensembles. A non-trivial Pfaffian can be derived for the partition function by noticing that the square of the Vandermonde in the measure~\eref{2.5} can be rewritten as one Vandermonde determinant of the variables $\pm\lambda_j$, i.e.
\begin{equation}\label{3.1}
\Delta_{n}^2(\Lambda^2)=(-1)^{n(n-1)/2}\frac{\Delta_{2n}(\Lambda,-\Lambda)}{2^n\det\Lambda}.
\end{equation}
The determinant of $\Lambda$ will be put into the weight later on, cf. Eqs.~\eref{3.2} and \eref{3.3} below. Considering the Wilson random matrix theory \cite{Damgaard:2010cz,Akemann:2010zp,arXiv:1105.6229,arXiv:1108.3035,Kieburg:2011uf} such a splitting arises in a natural way for finite lattice spacing. Then an eigenvalue pair $\pm\imath\lambda_j$ becomes either a complex conjugated pair or two independent real eigenvalues corresponding to a pair of eigenvectors with positive and negative chirality. Hence, the Pfaffian resulting from the single Vandermonde determinant~\eref{3.1} is the one which is generalized to non-zero lattice spacing and not the determinant \cite{arXiv:1108.3035,KVZ11}.
This allows us to define an anti-symmetric two-point measure on $\mathbb{R}^2$
\begin{equation}\label{3.2}
\fl g(x_1,x_2)=\frac{|x_1x_2|^\nu}{4}\exp\left[-\alpha \frac{V(x_1^2)+V(x_2^2)}{2}\right]\delta(x_1+x_2)[\Theta(x_1)-\Theta(x_2)],
\end{equation}
where $\Theta$ is the Heaviside distribution. Then we consider the measure
\begin{equation}\label{3.3}
D[\lambda]=\frac{{\rm Vol\,}_n{\rm Vol\,}_{n+\nu}}{{\rm Vol\,}_1^n{\rm Vol\,}_\nu}\Delta_{2n}(\lambda)\prod\limits_{j=1}^ng(\lambda_{2j-1},\lambda_{2j})d\lambda_{2j}d\lambda_{2j-1}
\end{equation}
over $2n$ independent eigenvalues instead of the measure~\eref{2.5}. This measure fulfills the general condition for finding a Pfaffian, cf. Ref.~\cite{KieGuh09b} and see also Eq.~\eref{0.3}.
The partition function~\eref{2.8} can be expressed in terms of this new measure,
\begin{eqnarray}\label{3.4}
Z_{k_1/k_2}^{(n,\nu)}(\kappa)=\frac{(-1)^{n(k_1+k_2)}}{n!}{\rm Sdet\,}^{-\nu}(-\imath\kappa)\int\limits_{\mathbb{R}^{2n}}\prod\limits_{a=1}^{2n}\frac{\prod\limits_{j=1}^{k_2}(\kappa_{j2}-\lambda_a)}{\prod\limits_{j=1}^{k_1}(\kappa_{j1}-\lambda_a)}D[\lambda].
\end{eqnarray}
In the first step we extend the Vandermonde determinant~\eref{3.1} with the characteristic polynomials,
\begin{eqnarray}\label{3.5}
Z_{k_1/k_2}^{(n,\nu)}(\kappa)&=&(-1)^{n(k_1+k_2)}\frac{{\rm Vol\,}_n{\rm Vol\,}_{n+\nu}}{n!{\rm Vol\,}_1^n{\rm Vol\,}_\nu}{\rm Sdet\,}^{-\nu}(-\imath\kappa)\\
&\times&\int\limits_{\mathbb{R}^{2n}}\frac{{\rm B\,}_{k_1/k_2+2n}(\kappa,\lambda)}{{\rm B\,}_{k_1/k_2}(\kappa)}\prod\limits_{j=1}^ng(\lambda_{2j},\lambda_{2j-1})d\lambda_{2j}d\lambda_{2j-1}.\nonumber
\end{eqnarray}
This representation is apart from the $z_{2N+1}$-integral of the form as in Eq.~(3.3) in Ref.~\cite{KieGuh09b}. Notice that in this extension we do not have the same freedom as in the determinantal case~\eref{2.9} since there is only one Vandermonde determinant in the integrand~\eref{3.3}. Let $d=2n+k_2-k_1\geq0$. Then we employ the representation of the function ``${\rm B\,}$'' as a determinant, see Eq.~\eref{0.5},
\begin{eqnarray}\label{3.5a}
Z_{k_1/k_2}^{(n,\nu)}(\kappa)&\propto&\frac{{\rm Sdet\,}^{-\nu}\kappa}{{\rm B\,}_{k_1/k_2}(\kappa)}\int\limits_{\mathbb{R}^{2n}}\prod\limits_{j=1}^ng(\lambda_{2j},\lambda_{2j-1})d\lambda_{2j}d\lambda_{2j-1}\\
&\times&\det\left[\begin{array}{cc} \displaystyle\left\{\frac{1}{\kappa_{a1}-\kappa_{b2}}\right\}\underset{1\leq b\leq k_2}{\underset{ 1\leq a\leq k_1}{\ }} & \displaystyle\left\{\frac{1}{\kappa_{a1}-\lambda_{b2}}\right\}\underset{1\leq b\leq 2n}{\underset{ 1\leq a\leq k_1}{\ }} \\ \displaystyle\left\{\kappa^{a-1}_{b2}\right\}\underset{1\leq b\leq k_{2}}{\underset{1\leq a\leq d}{\ }} & \displaystyle\left\{\lambda^{a-1}_{b2}\right\}\underset{1\leq b\leq 2n}{\underset{1\leq a\leq d}{\ }} \end{array}\right]\nonumber.
\end{eqnarray}
The generalized de Bruijn integration theorem \cite{Bru55,KieGuh09a} can be applied now which yields
\begin{eqnarray}\label{3.5b}
\fl Z_{k_1/k_2}^{(n,\nu)}(\kappa)&\propto&\frac{{\rm Sdet\,}^{-\nu}\kappa}{{\rm B\,}_{k_1/k_2}(\kappa)}\\
\fl&\times&{\rm Pf\,}\left[\begin{array}{ccc} 0 & \displaystyle\left\{\frac{1}{\kappa_{b1}-\kappa_{a2}}\right\}\underset{1\leq b\leq k_1}{\underset{ 1\leq a\leq k_2}{\ }} & \displaystyle\left\{\kappa^{b-1}_{a2}\right\}\underset{1\leq b\leq d}{\underset{1\leq a\leq k_2}{\ }} \\ \displaystyle\left\{-\frac{1}{\kappa_{a1}-\kappa_{b2}}\right\}\underset{1\leq b\leq k_2}{\underset{ 1\leq a\leq k_1}{\ }} & \displaystyle\left\{\widetilde{F}(\kappa_{a1},\kappa_{b1})\right\}\underset{1\leq a,b\leq k_1}{\ } & \displaystyle\left\{\widetilde{F}_b(\kappa_{a1})\right\}\underset{1\leq b\leq d}{\underset{1\leq a\leq k_1}{\ }} \\ \displaystyle\left\{-\kappa^{a-1}_{b2}\right\}\underset{1\leq b\leq k_{2}}{\underset{1\leq a\leq d}{\ }} & \displaystyle\left\{-\widetilde{F}_a(\kappa_{b1})\right\}\underset{1\leq b\leq k_1}{\underset{1\leq a\leq d}{\ }} & \displaystyle\left\{\widetilde{M}_{ab}\right\}\underset{1\leq a,b\leq d}{\ } \end{array}\right].\nonumber
\end{eqnarray}
As Andr{\'e}ief's integration theorem the generalized de Bruijn integration theorem is only an algebraic rearrangement of the integrals without calculating any of them. The functions $\widetilde{F}$ and $\widetilde{F}_a$ are two-fold integrals and are again not much of importance, see the discussion after Eq.~\eref{3.5e}. Explicit expressions of them are given in Ref.~\cite{KieGuh09b} for general random matrix ensembles corresponding to Pfaffians comprising the measure~\eref{3.3}, too.
The $d\times d$ anti-symmetric matrix $\widetilde{M}=[\widetilde{M}_{ab}]$ consists of the moments
\begin{eqnarray}\label{3.5c}
\widetilde{M}_{ab}=\int_{\mathbb{R}^2}(\lambda_1^{a-1}\lambda_2^{b-1}-\lambda_1^{b-1}\lambda_2^{a-1})g(\lambda_1,\lambda_2)d\lambda_1d\lambda_2.
\end{eqnarray}
Analogously to Eq.~\eref{2.10d}, we employ the identity
\begin{eqnarray}\label{3.5d}
{\rm Pf\,}\left[\begin{array}{cc} A & B \\ -B^T & C \end{array}\right]={\rm Pf\,} C\, {\rm Pf\,}[A+BC^{-1}B^T]
\end{eqnarray}
with an arbitrary matrix $B$, an arbitrary antisymmetric matrix $A$ and an arbitrary even dimensional, antisymmetric matrix $C$ which has to be invertible. Let $k_1+k_2$ be even. Then $d$ is also even and the Pfaffian of the matrix $\widetilde{M}$ is proportional to the normalization constant of the ensemble~\eref{2.2}. Hence the choice $C=\widetilde{M}$ is well-defined. This yields
\begin{eqnarray}\label{3.5e}
\fl Z_{k_1/k_2}^{(n,\nu)}(\kappa)&\propto&\frac{1}{{\rm B\,}_{k_1/k_2}(\kappa)}\\
\fl&\times&{\rm Pf\,}\left[\begin{array}{cc} \displaystyle\left\{\widetilde{G}_1^{(d)}(\kappa_{a2},\kappa_{b2})\right\}\underset{1\leq a,b\leq k_2}{\ } & \displaystyle\left\{\widetilde{G}_2^{(d)}(\kappa_{b1},\kappa_{a2})\right\}\underset{1\leq b\leq k_1}{\underset{ 1\leq a\leq k_2}{\ }} \\ \displaystyle\left\{-\widetilde{G}_2^{(d)}(\kappa_{a1},\kappa_{b2})\right\}\underset{1\leq b\leq k_2}{\underset{ 1\leq a\leq k_1}{\ }} & \displaystyle\left\{\widetilde{G}_3^{(d)}(\kappa_{a1},\kappa_{b1})\right\}\underset{1\leq a,b\leq k_1}{\ } \end{array}\right]\nonumber.
\end{eqnarray}
The functions $\widetilde{G}_1^{(d)}$, $\widetilde{G}_2^{(d)}$ and $\widetilde{G}_3^{(d)}$ can be obtained by considering the cases $(k_1/k_2)=(0/2),(1/1),(2/0)$, respectively. In each of these cases the Pfaffian~\eref{3.5e} reduces to a single term.
This leads to a particular case of the general result derived in Ref.~\cite{KieGuh09b}. We find our main result of this article
\begin{eqnarray}\label{3.6}
\fl&&\frac{Z_{k_1/k_2}^{(n,\nu)}(\kappa)}{Z_{0/0}^{(n,\nu)}}=\frac{(-1)^{k_2(k_2+1)/2}}{{\rm B\,}_{k_1/k_2}(\kappa)}\frac{\prod\limits_{j=0}^{d-1}h_j^{(\nu)}}{\prod\limits_{j=0}^{n-1}h_j^{(\nu)}}\\
\fl&\times&{\rm Pf\,}\left[\begin{array}{c|c} \displaystyle\underset{}{\frac{\kappa_{b2}-\kappa_{a2}}{h_{d/2-1}^{(\nu)}Z_{0/0}^{(d/2-1,\nu)}}Z_{0/2}^{(d/2-1,\nu)}(\kappa_{a2},\kappa_{b2})} & \displaystyle\frac{1}{Z_{0/0}^{(d/2,\nu)}}\frac{Z_{1/1}^{(d/2,\nu)}(\kappa_{b1},\kappa_{a2})}{(\kappa_{a2}-\kappa_{b1})} \\ \hline \displaystyle\frac{1}{Z_{0/0}^{(d/2,\nu)}}\overset{}{\frac{Z_{1/1}^{(d/2,\nu)}(\kappa_{a1},\kappa_{b2})}{(\kappa_{a1}-\kappa_{b2})}} & \displaystyle\frac{h_{d/2}^{(\nu)}(\kappa_{a1}-\kappa_{b1})}{Z_{0/0}^{(d/2+1,\nu)}}Z_{2/0}^{(d/2+1,\nu)}(\kappa_{a1},\kappa_{b1}) \end{array}\right]\nonumber
\end{eqnarray}
for even $k_1+k_2$. The indices $a$ and $b$ run from $1$ to $k_2$ in the first columns and the first rows and from $1$ to $k_1$ in the last ones. The result for odd $k_2+k_1$ can be readily obtained by introducing an additional fermionic flavor and sending it to infinity. This shifts the parameter $d$ to $d+1$ and adds a row and a column to the matrix in the Pfaffian~\eref{3.6} with the partition functions $Z_{0/1}^{((d-1)/2,\nu)}(\kappa_{b2})$ and $Z_{1/0}^{((d+1)/2,\nu)}(\kappa_{b1})$ which are apart from a factor $\kappa^\nu$ an orthogonal polynomial and its Cauchy-transform, cf. Eqs.~\eref{2.12} and \eref{2.14}. Notice that the matrix in the Pfaffian~\eref{3.6} is indeed antisymmetric because $Z_{0/2}^{(d/2-1,\nu)}$ and $Z_{2/0}^{(d/2+1,\nu)}$ are symmetric under a permutation of the entries.
Indeed, Eq.~\eref{3.6} cannot be traced back to the identity
\begin{equation}\label{3.7}
{\rm Pf\,}\left[\begin{array}{cc} 0 & X \\ -X^T & 0 \end{array}\right]=(-1)^{p(p-1)/2}\det X
\end{equation}
with an arbitrary $p\times p$ matrix $X$. We refer to the relation~\eref{3.7} as a trivial Pfaffian extension of a determinant. The Pfaffian~\eref{3.6} seems to be the result of recursion relations of the orthogonal polynomials~\eref{2.12}. It is difficult to see how these recursions have to be performed to map the Pfaffian~\eref{3.6} to the determinant~\eref{2.11}. However the construction of this structure seems to be the same for a broad class of ensembles. This is confirmed by the fact that the result~\eref{3.6} can be extended to all factorizing ensembles with a squared Vandermonde determinant in the joint probability density~\eref{0.1}. This will be shown in Sec.~\ref{sec5}.
Again one can consider the $k$-point correlation function~\eref{2.12a} and what it looks like with the Pfaffian determinant. Using the result~\eref{3.6} we find for the $k$-point correlation function
\begin{eqnarray}
\fl R_k^{(n,\nu)}(x)&\propto&\exp[-\alpha\tr V(x)]\nonumber\\
\fl&\times&{\rm Pf\,}\left[\begin{array}{c|c} (x_a-x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) & (x_a+x_b)Z_{0/2}^{(n-1,\nu)}(x_a,-x_b) \\ \hline-(x_a+x_b)Z_{0/2}^{(n-1,\nu)}(-x_a,x_b) & -(x_a-x_b)Z_{0/2}^{(n-1,\nu)}(-x_a,-x_b) \end{array}\right]_{1\leq a,b\leq k}\nonumber\\
\fl&\propto&\exp[-\alpha\tr V(x)]\nonumber\\
\fl&\times&{\rm Pf\,}\left[\begin{array}{c|c} (x_a-x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) & (x_a+x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) \\ \hline-(x_a+x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) & -(x_a-x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) \end{array}\right]_{1\leq a,b\leq k}.\label{3.7a}
\end{eqnarray}
Again we have not employed the integration theorem by Dyson and Mehta \cite{Dys70,Meh71,Meh04,Gho09}. To see that Eq.~\eref{3.7a} indeed agrees with the determinant~\eref{2.12b} one can consider the square of the Pfaffian,
\begin{eqnarray}
\fl&&{\rm Pf\,}^2\left[\begin{array}{c|c} (x_a-x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) & (x_a+x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) \\ \hline-(x_a+x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) & -(x_a-x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) \end{array}\right]_{1\leq a,b\leq k}\nonumber\\
\fl&=&\det\left[\begin{array}{c|c} (x_a-x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) & (x_a+x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) \\ \hline-(x_a+x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) & -(x_a-x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) \end{array}\right]_{1\leq a,b\leq k}\nonumber\\
\fl&=&2^k\det\left[\begin{array}{c|c} -x_bZ_{0/2}^{(n-1,\nu)}(x_a,x_b) & x_bZ_{0/2}^{(n-1,\nu)}(x_a,x_b) \\ \hline-(x_a+x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) & -(x_a-x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) \end{array}\right]_{1\leq a,b\leq k}\nonumber\\
\fl&=&2^{2k}\det\left[\begin{array}{c|c} -x_bZ_{0/2}^{(n-1,\nu)}(x_a,x_b) & 0 \\ \hline-(x_a+x_b)Z_{0/2}^{(n-1,\nu)}(x_a,x_b) & -x_aZ_{0/2}^{(n-1,\nu)}(x_a,x_b) \end{array}\right]_{1\leq a,b\leq k}\nonumber\\
\fl&=&2^{2k}{\det}^2x\,{\det}^2\left[Z_{0/2}^{(n-1,\nu)}(x_a,x_b)\right]_{1\leq a,b\leq k}.\label{3.7b}
\end{eqnarray}
The square root of Eq.~\eref{3.7b} yields Eq.~\eref{2.12b}.
In the large $n$ limit, we employ Eqs.~(\ref{2.15}-\ref{2.19}) and (\ref{2.21}-\ref{2.23}) and obtain
\begin{eqnarray}\label{3.8}
\fl Z_{k_1/k_2}^{(n,\nu)}\left(\frac{\kappa}{cn}\right)&\overset{n\gg1}{\propto}&\frac{1}{{\rm B\,}_{k_1/k_2}(\kappa)}\\
\fl&\times&{\rm Pf\,}\left[\begin{array}{c|c} \displaystyle\underset{}{(\kappa_{a2}-\kappa_{b2})I^{(1)}_\nu(\kappa_{a2},\kappa_{b2})} & \displaystyle (\kappa_{b1}+\kappa_{a2})I^{(2)}_\nu(\kappa_{b1},\kappa_{a2}) \\ \hline \displaystyle\overset{}{-(\kappa_{a1}+\kappa_{b2})I^{(2)}_\nu(\kappa_{a1},\kappa_{b2})} & \displaystyle(\kappa_{a1}-\kappa_{b1})I^{(3)}_\nu(\kappa_{a1},\kappa_{b1}) \end{array}\right]\nonumber
\end{eqnarray}
for even $k_1+k_2$ and
\begin{eqnarray}\label{3.9}
\fl Z_{k_1/k_2}^{(n,\nu)}\left(\frac{\kappa}{cn}\right)&\overset{n\gg1}{\propto}&\frac{1}{{\rm B\,}_{k_1/k_2}(\kappa)}\\
\fl&\times&{\rm Pf\,}\left[\begin{array}{c|c|c} 0 & \displaystyle\underset{}{J_\nu(\kappa_{b2})} & \displaystyle K_\nu(\kappa_{b1})
\\ \hline -J_\nu(\kappa_{a2}) & \displaystyle\overset{}{\underset{}{(\kappa_{a2}-\kappa_{b2})I^{(1)}_\nu(\kappa_{a2},\kappa_{b2})}} & \displaystyle (\kappa_{b1}+\kappa_{a2})I^{(2)}_\nu(\kappa_{b1},\kappa_{a2}) \\ \hline -K_\nu(\kappa_{a1}) & \displaystyle\overset{}{-(\kappa_{a1}+\kappa_{b2})I^{(2)}_\nu(\kappa_{a1},\kappa_{b2})} & \displaystyle(\kappa_{a1}-\kappa_{b1})I^{(3)}_\nu(\kappa_{a1},\kappa_{b1}) \end{array}\right]\nonumber
\end{eqnarray}
for odd $k_1+k_2$. These Pfaffians carry over to the Wilson Dirac random matrix model~\cite{arXiv:1108.3035,KVZ11}. For small numbers of bosonic and fermionic flavors these results were checked by the recursion relations of the Bessel functions \cite{Ver11}.
Please notice the difference in the prefactor of Eqs.~\eref{2.20}, \eref{3.8} and \eref{3.9}. The entries of the Berezinian are the squares of the variables $\kappa$ for the determinantal structure~\eref{2.20} whereas it is only $\kappa$ for the Pfaffian. This yields a technical advantage when calculating eigenvalue correlations of the random matrix models for the Wilson Dirac operator.
\subsection{An application: Wilson-Dirac random matrix}\label{sec3.2}
The Wilson-Dirac operator is a modified Dirac operator on a lattice. In the infrared limit this operator can be modeled by the Wilson-Dirac random matrix \cite{Damgaard:2010cz,Akemann:2010zp,arXiv:1105.6229,Kieburg:2011uf} which is a $(2n+\nu)\times(2n+\nu)$ Hermitian matrix
\begin{eqnarray}\label{3.2.1}
D_{\rm W}=\left[\begin{array}{cc} a A & W \\ -W^\dagger & a B \end{array}\right]
\end{eqnarray}
distributed by the Gaussian
\begin{eqnarray}\label{3.2.2}
P(D_{\rm W})=\exp\left[-\frac{n}{2}(\tr A^2+\tr B^2)-n\tr WW^\dagger\right].
\end{eqnarray}
The variable $a$ plays the role of the lattice spacing. The chiral symmetry is explicitly broken by the Hermitian matrices $A$ and $B$, i.e.
\begin{eqnarray}\label{3.2.3}
\gamma_5\left.D_{\rm W}\right|_{m=0}\gamma_5\neq-\left.D_{\rm W}\right|_{m=0}\quad\mathrm{with}\quad\gamma_5={\rm diag\,}(\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{n},-\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{n+\nu}),
\end{eqnarray}
which have the dimensions $n\times n$ and $(n+\nu)\times(n+\nu)$, respectively.
Hence, $A$ and $B$ model the Wilson-term.
We consider the partition function with $N_f$ fermionic flavors,
\begin{eqnarray}\label{3.2.4}
Z_{N_{\rm f}}^{(n,\nu)}(m,a)=\int \prod\limits_{j=1}^{N_{\rm f}}\det(D_{\rm W}+m_j\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{2n+\nu})P(D_{\rm W})d[D_{\rm W}].
\end{eqnarray}
The external variables $m={\rm diag\,}(m_1,\ldots,m_{N_{\rm f}})$ play the role of the quark masses. Indeed one can also consider bosonic flavors. However, we restrict ourself to fermionic flavors to keep the example as simple as possible
In the microscopic limit ($n\to\infty$), $\widehat{m}=2nm$, $\widehat{a}=\sqrt{n}a/2$ and $\nu$ are kept fixed. This yields the integral
\begin{eqnarray}
\fl Z_{N_{\rm f}}^{(n,\nu)}(\widehat{m},\widehat{a})&\overset{n\gg1}{=}&\int\limits_{{\rm U\,}(N_f)}\exp\left[\frac{1}{2}\tr\widehat{m}(U+U^{-1})-\widehat{a}^2\tr\left(U^2+U^{-2}\right)\right]{\det}^\nu Ud\mu(U).\label{3.2.5}
\end{eqnarray}
For a derivation of this result we refer to Refs.~\cite{Damgaard:2010cz,Akemann:2010zp}. Exactly the integral~\eref{3.2.5} makes contact with lattice QCD \cite{ShaSin98,RS02,Sha06,necco}.
At zero lattice spacing ($\widehat{a}=0$) this partition function can be identified with the one considered in Sec.~\ref{sec2},
\begin{eqnarray}\label{3.2.6}
Z_{N_{\rm f}}^{(n,\nu)}(m,a=0)\propto Z_{0/N_{\rm f}}^{(n,\nu)}(\imath m).
\end{eqnarray}
Considering again the microscopic limit~\eref{3.2.5}, we trace the integral back to the $a=0$ result by introducing a $N_{\rm f}\times N_{\rm f}$ Hermitian random matrix $\sigma$ similar to the calculation in Ref.~\cite{arXiv:1105.6229,VerSpl11},
\begin{eqnarray}
\fl Z_{N_{\rm f}}^{(n,\nu)}(\widehat{m},\widehat{a})&\overset{n\gg1}{\propto}&\int\exp\left[-\frac{1}{4\widehat{a}^2}\tr(\sigma-\imath \widehat{m})^2-2(\widehat{a}N_{\rm f})^2\right]\nonumber\\
\fl&\times&\int\limits_{{\rm U\,}(N_f)}\exp\left[-\imath\tr\sigma(U+U^{-1})\right]{\det}^\nu Ud\mu(U)d[\sigma]\nonumber\\
\fl&\propto&\int\exp\left[-\frac{1}{4\widehat{a}^2}\tr(\sigma+\imath \widehat{m})^2-2(\widehat{a}N_{\rm f})^2\right]Z_{0/N_{\rm f}}^{(n,\nu)}(\sigma)d[\sigma].\label{3.2.7}
\end{eqnarray}
Notice that $\sigma$ is an ordinary matrix and not a supermatrix because we consider fermionic flavors, only. The constant $\exp[-2(\widehat{a}N_{\rm f})^2]$ can be shifted into the normalization constant and can, thus, be omitted in the ensuing calculations.
A diagonalization of $\sigma=VsV^\dagger$ with $V\in{\rm U\,}(N_{\rm f})$ yields a Harish-Chandra-Itzykson-Zuber-integral \cite{Har58,ItzZub80} in the Gaussian term. The partition function $Z_{0/N_{\rm f}}^{(n,\nu)}$ is invariant under ${\rm U\,}(N_{\rm f})$. We find
\begin{eqnarray}
\fl Z_{N_{\rm f}}^{(n,\nu)}(\widehat{m},\widehat{a})&\propto&\int\frac{\det\left[\exp\left[-(s_j-\imath \widehat{m}_i)^2/4\widehat{a}^2\right]\right]_{1\leq j,i\leq N_{\rm f}}}{\Delta_{N_{\rm f}}(\widehat{m})}Z_{0/N_{\rm f}}^{(n,\nu)}(s)\Delta_{N_{\rm f}}(s)d[s].\label{3.2.8}
\end{eqnarray}
Employing the result as a determinant of the microscopic limit of $\widehat{a}=0$ partition function, cf. Eq.~\eref{2.20}, we end up with a complicated expression,
\begin{eqnarray}
Z_{N_{\rm f}}^{(n,\nu)}(\widehat{m},\widehat{a})&\overset{n\gg1}{\propto}&\int\frac{\det\left[\exp\left[-(s_j-\imath \widehat{m}_i)^2/4\widehat{a}^2\right]\right]_{1\leq j,i\leq N_{\rm f}}}{\Delta_{N_{\rm f}}(\widehat{m})}\nonumber\\
&\times&\det\left[s_i^{j-1}J_{\nu-1+j}(s)\right]_{1\leq j,i\leq N_{\rm f}}\frac{\Delta_{N_{\rm f}}(s)}{\Delta_{N_{\rm f}}(s^2)}d[s].\label{3.2.9}
\end{eqnarray}
There is no obvious way to further simplify the integral~\eref{3.2.9} due to the factor $\Delta_{N_{\rm f}}(s)/\Delta_{N_{\rm f}}(s^2)$. This was not much of a problem for the authors of Refs.~\cite{arXiv:1105.6229,VerSpl11} because they only considered a small numbers of flavors. However the problem is highly non-trivial for an arbitrary number of flavors.
This problem can be solved by using the Pfaffian expressions~\eref{3.8} and~\eref{3.9} instead of the determinant. Let $N_{\rm f}$ be even to keep the expressions as simple as possible. Then we have for the microscopic limit~\eref{3.2.5}
\begin{eqnarray}
\fl Z_{N_{\rm f}}^{(n,\nu)}(\widehat{m},\widehat{a})&\overset{n\gg1}{\propto}&\int\frac{\det\left[\exp\left[-(s_j-\imath \widehat{m}_i)^2/4\widehat{a}^2\right]\right]_{1\leq j,i\leq N_{\rm f}}}{\Delta_{N_{\rm f}}(\widehat{m})}\label{3.2.10}\\
\fl&\times&{\rm Pf\,}\left[\frac{s_jJ_{\nu-1}(s_j)J_\nu(s_i)-s_iJ_{\nu}(s_j)J_{\nu-1}(s_i)}{s_j+s_i}\right]_{1\leq j,i\leq N_{\rm f}}d[s].\nonumber
\end{eqnarray}
After expanding the determinant no term hinders us to pull the integrals into the Pfaffian. We obtain the compact result
\begin{eqnarray}\label{3.2.11}
\fl Z_{N_{\rm f}}^{(n,\nu)}(\widehat{m},\widehat{a})&\propto&\frac{1}{\Delta_{N_{\rm f}}(\widehat{m})}{\rm Pf\,}\left[(\widehat{m}_j-\widehat{m}_i)Z_{2}^{(n,\nu)}(\widehat{m}_j,\widehat{m}_i,\widehat{a})\right]_{1\leq j,i\leq N_{\rm f}}\nonumber
\end{eqnarray}
with
\begin{eqnarray}\label{3.2.12}
\fl Z_{2}^{(n,\nu)}(\widehat{m}_1,\widehat{m}_2,\widehat{a})&\propto&\frac{1}{\widehat{m}_1-\widehat{m}_2}\int_{\mathbb{R}^2}\exp\left[-\frac{(s_1-\imath \widehat{m}_1)^2+(s_2-\imath \widehat{m}_2)^2}{4\widehat{a}^2}\right]\\
\fl&\times&\frac{s_1J_{\nu-1}(s_1)J_\nu(s_2)-s_2J_{\nu}(s_1)J_{\nu-1}(s_2)}{s_1+s_2}ds_1ds_2.\nonumber
\end{eqnarray}
This is a drastic simplification of the problem compared to Eq.~\eref{3.2.9}.
\section{Skew-orthogonal polynomials}\label{sec4}
What are the skew-orthogonal polynomials which correspond to the Pfaffian~\eref{3.6}? In order to solve this problem we consider the two-point measure~\eref{3.2}. The skew orthogonal polynomials $q_j$ are defined by
\begin{eqnarray}
&&\int\limits_{\mathbb{R}^2}\det\left[\begin{array}{cc} q_{2j-1}(x_1) & q_{2j-1}(x_2) \\ q_{2i-1}(x_1) & q_{2i-1}(x_2) \end{array}\right]g(x_1,x_2)dx_1dx_2\nonumber\\
&=& \int\limits_{\mathbb{R}^2}\det\left[\begin{array}{cc} q_{2j}(x_1) & q_{2j}(x_2) \\ q_{2i}(x_1) & q_{2i}(x_2) \end{array}\right]g(x_1,x_2)dx_1dx_2=0,\label{4.1}
\end{eqnarray}
and
\begin{eqnarray}
\int\limits_{\mathbb{R}^2}\det\left[\begin{array}{cc} q_{2j+1}(x_1) & q_{2j+1}(x_2) \\ q_{2i}(x_1) & q_{2i}(x_2) \end{array}\right]g(x_1,x_2)dx_1dx_2=\widehat{h}_i^{(\nu)}\delta_{ij}.\label{4.2}
\end{eqnarray}
Moreover one has to assume that $q_l$ is a polynomial of order $l$.
The integral over the measure~\eref{3.2} for two arbitrary and conveniently integrable functions $f_1$ and $f_2$ can be simplified to
\begin{eqnarray}
&&\int\limits_{\mathbb{R}^2}\det\left[\begin{array}{cc} f_1(x_1) & f_1(x_2) \\ f_2(x_1) & f_2(x_2) \end{array}\right]g(x_1,x_2)dx_1dx_2\nonumber\\
&=&\frac{1}{2}\int\limits_0^\infty \det\left[\begin{array}{cc} f_1(x) & f_1(-x) \\ f_2(x) & f_2(-x) \end{array}\right]x^{2\nu}\exp\left[-nV(x^2)\right]dx.\label{4.3}
\end{eqnarray}
Due to this identity the skew-orthogonal polynomials $q_l^{(\nu)}$ are related by the orthogonal polynomials $p_l$ in the following way
\begin{equation}
q_{2l}^{(\nu)}(x)=p_l^{(\nu)}(x^2)\label{4.4}
\end{equation}
for the even polynomials and
\begin{equation}
q_{2l+1}^{(\nu)}(x)=xp_l^{(\nu)}(x^2)+{\rm const.}\,p_l^{(\nu)}(x^2)\label{4.5}
\end{equation}
for the odd polynomials. Notice that these skew-orthogonal polynomials for $V(x)=x$ (the Laguerre ensemble) are similar to but not completely the same as the one for $\beta=1$ and $\beta=4$ shown in Ref.~\cite{Meh04,Gho09} for the Laguerre ensemble. The reason is the two point weight which is
\begin{eqnarray}
g_{\rm chGOE}(x_1,x_2)&=&(x_1x_2)^{\nu}\exp\left[-\alpha(x_1^2+x_2^2)\right]\frac{x_1-x_2}{|x_1-x_2|},\label{4.5a}\\
g_{\rm chGSE}(x_1,x_2)&=&(x_1x_2)^{2\nu+3/2}\exp\left[-\alpha(x_1^2+x_2^2)\right]\delta^\prime(x_1-x_2),\label{4.5b}
\end{eqnarray}
in comparison see Eq.~\eref{3.2} for $\beta=2$. The labels ``chGOE'' and ``chGSE'' refer to the chiral Gaussian orthogonal ensemble ($\beta=1$) and to the chiral Gaussian symplectic ensemble ($\beta=4$), respectively. The sign function $(x_1-x_2)/|x_1-x_2|$ generate the modulus of the Vandermonde determinant for $\beta=1$. The distribution $\delta^\prime$ is the first derivative of the Dirac delta function and cancels with these terms of the Vandermonde determinant which are zero at the support of the Dirac delta functions. This generates Cramers degeneracy in the quaternion case ($\beta=4$).
The solution of Eqs.~\eref{4.1} and \eref{4.2} is not unique which is reflected by the arbitrary constant in the odd polynomials~\eref{4.5}. One can readily confirm that this choice of the polynomials solves the conditions~\eref{4.1} and \eref{4.2} by recognizing the symmetry $q_{j}(-x)=(-1)^jq_j(x)$ and the orthogonality relation~\eref{2.13} for $p_j$. The normalization constant is
\begin{eqnarray}
\widehat{h}_i^{(\nu)}=h_i^{(\nu)}.\label{4.6}
\end{eqnarray}
This relation between orthogonal and skew-orthogonal polynomials seems so trivial because of the particular and simple structure of the two-point weight~\eref{3.2}.
\section{A few more ensembles with Dyson index $\beta=2$ and Pfaffians}\label{sec5}
The algebraic rearrangement for chiral unitary ensembles described in Sec.~\ref{sec3} can be extended to other random matrix ensembles which have a squared Vandermonde determinant in the joint probability density function. By the same trick as in Eq.~\eref{3.1} we write
\begin{equation}\label{5.1}
\Delta_{N}^2(z)=(-1)^{N(N-1)/2}\frac{\Delta_{2N}(\sqrt{z},-\sqrt{z})}{2^N\sqrt{\det z}},
\end{equation}
where the variables $z={\rm diag\,}(z_1,\ldots,z_N)$ might be complex. The square root is the positive one but this is without loss of generality since the right hand side of Eq.~\eref{5.1} comprises both roots. Again the determinant of $z$ will be put to the measure $d\mu$ for a single eigenvalue.
We consider an average over ratios of characteristic polynomials for random ensembles like GUE and CUE, i.e.
\begin{eqnarray}\label{5.2}
\widetilde{Z}_{k_1/k_2}^{(N)}(\kappa)=\int\limits_{\mathbb{C}^{N}}\Delta_{N}^2(z)\prod\limits_{i=1}^{N}\frac{\prod\limits_{j=1}^{k_2}(z_i-\kappa_{j2})}{\prod\limits_{j=1}^{k_1}(z_i-\kappa_{j1})}d\mu(z_i),
\end{eqnarray}
where $d\mu$ is a measure on $\mathbb{C}$ and $\kappa$ is chosen such that the integrals exist. Notice that there is no modulus of the Vandermonde determinant which is a necessary property of the following discussion. A modulus of the Vandermonde is an obstacle to map Eq.~\eref{5.2} to the general joint probability density corresponding to the Pfaffian, see Ref.~\cite{KieGuh09b}, which we have not managed yet. A modulus corresponds to the biorthogonal polynomials \cite{Bergere:2003ht} whereas the choice without the modulus corresponds to the orthogonal polynomials, only. Apart from the modulus of the Vandermonde it is exactly the correlation function discussed in Sec.~4.2 of Ref.~\cite{KieGuh09a}.
With the help of the derivation in Sec.~\ref{sec3} the integral~\eref{5.2} can be written as
\begin{eqnarray}\label{5.3}
\fl\widetilde{Z}_{k_1/k_2}^{(N)}(\kappa)&\propto&\frac{1}{{\rm B\,}_{k_1/k_2}(\sqrt{\kappa})}\\
\fl&\times&{\rm Pf\,}\left[\begin{array}{c|c} \displaystyle\underset{}{\frac{\widetilde{d}(\sqrt{\kappa_{b2}}-\sqrt{\kappa_{a2}})}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\widetilde{Z}_{0/2}^{(\widetilde{d}-1)}(\kappa_{a2},\kappa_{b2})} & \displaystyle\frac{1}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\frac{\widetilde{Z}_{1/1}^{(\widetilde{d})}(\kappa_{b1},\kappa_{a2})}{(\sqrt{\kappa_{a2}}-\sqrt{\kappa_{b1}})} \\ \hline \displaystyle\frac{1}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\overset{}{\frac{\widetilde{Z}_{1/1}^{(\widetilde{d})}(\kappa_{a1},\kappa_{b2})}{(\sqrt{\kappa_{a1}}-\sqrt{\kappa_{b2}})}} & \displaystyle\frac{\sqrt{\kappa_{a1}}-\sqrt{\kappa_{b1}}}{(\widetilde{d}+1)\widetilde{Z}_{0/0}^{(\widetilde{d})}}\widetilde{Z}_{2/0}^{(\widetilde{d}+1)}(\kappa_{a1},\kappa_{b1}) \end{array}\right]\nonumber
\end{eqnarray}
for $k_1+k_2$ even and
\begin{eqnarray}\label{5.4}
\fl&&\widetilde{Z}_{k_1/k_2}^{(N)}(\kappa)\propto\frac{1}{{\rm B\,}_{k_1/k_2}(\sqrt{\kappa})}\\
\fl&\times&{\rm Pf\,}\left[\begin{array}{c|c|c} 0 & \displaystyle\underset{}{-\frac{\widetilde{d}}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\widetilde{Z}_{0/1}^{(\widetilde{d}-1)}(\kappa_{b2})} & \displaystyle\frac{1}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\widetilde{Z}_{1/0}^{(\widetilde{d})}(\kappa_{b1}) \\ \hline \displaystyle\frac{\widetilde{d}}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\widetilde{Z}_{0/1}^{(\widetilde{d}-1)}(\kappa_{a2}) & \displaystyle\underset{}{\frac{\widetilde{d}(\sqrt{\kappa_{b2}}-\sqrt{\kappa_{a2}})}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\widetilde{Z}_{0/2}^{(\widetilde{d}-1)}(\kappa_{a2},\kappa_{b2})} & \displaystyle\frac{1}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\frac{\widetilde{Z}_{1/1}^{(\widetilde{d})}(\kappa_{b1},\kappa_{a2})}{(\sqrt{\kappa_{a2}}-\sqrt{\kappa_{b1}})} \\ \hline \displaystyle-\frac{1}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\widetilde{Z}_{1/0}^{(\widetilde{d})}(\kappa_{a1}) & \displaystyle\frac{1}{\widetilde{Z}_{0/0}^{(\widetilde{d})}}\overset{}{\frac{\widetilde{Z}_{1/1}^{(\widetilde{d})}(\kappa_{a1},\kappa_{b2})}{(\sqrt{\kappa_{a1}}-\sqrt{\kappa_{b2}})}} & \displaystyle\frac{\sqrt{\kappa_{a1}}-\sqrt{\kappa_{b1}}}{(\widetilde{d}+1)\widetilde{Z}_{0/0}^{(\widetilde{d})}}\widetilde{Z}_{2/0}^{(\widetilde{d}+1)}(\kappa_{a1},\kappa_{b1}) \end{array}\right]\nonumber
\end{eqnarray}
for $k_1+k_2$ odd. The variable $\widetilde{d}$ is
\begin{equation}\label{5.5}
\widetilde{d}=\left\{\begin{array}{cl} N+(k_2-k_1)/2, & k_2+k_1\in2\mathbb{N}, \\ N+(k_2-k_1+1)/2, & k_2+k_1+1\in2\mathbb{N}.\end{array}\right.
\end{equation}
The indices $a$ and $b$ run from $1$ to $k_1$ for $\kappa_1$ and from $1$ to $k_2$ for $\kappa_2$. Apart from the square roots of the variables $\kappa$ these structures are exactly the same as those of random matrix ensembles with Dyson index $\beta\in\{1,4\}$. Hence, it seems to be that the Pfaffian determinants~\eref{5.3} and \eref{5.4} for the average over characteristic polynomials are more general than the determinant derived in Ref.~\cite{KieGuh09a}.
Random matrix ensembles whose generating functions can be cast into the form~\eref{5.2} have this non-trivial expression as a Pfaffian. The Hermitian Gaussian unitary ensemble as well as its generalization with other potentials fulfill {\it a priori} this requirement since the joint probability density has a squared Vandermonde determinant without the modulus. More generally our derivation applies to each ensemble with a real spectrum, a squared Vandermonde determinant and a factorizing probability distribution, cf. Eq.~\eref{2.3}. Also the CUE (unitary group) can be cast into the form~\eref{5.2}. More ensembles can be found in the tables~1 and 2 of Ref.~\cite{KieGuh09a}. The Ginibre ensemble as well as its chiral counterpart are not in this class. Their joint probability density incorporates a modulus of the Vandermonde determinant and is, thus, in the class for the bi-orthogonal polynomials. Therefore it is possible that their eigenvalue correlation functions cannot be expressed in Pfaffians like Eq.~\eref{5.3} and \eref{5.4}.
The skew-orthogonal polynomials corresponding to the Pfaffians~\eref{5.3} and \eref{5.4} have the same relation to the orthogonal polynomials as chiral unitary ensembles, see Eqs.~(\ref{4.4}-\ref{4.6}). By construction this relation is so simple.
\section{Remarks and conclusions}\label{sec6}
We derived a non-trivial Pfaffian determinant for the average over ratios of characteristic polynomials of a large class of random matrix ensembles with Dyson index $\beta=2$. This structure is similar to the one for $\beta\in\{1,4\}$, cf. Ref.~\cite{KieGuh09b}. Hence, it is universal and unifies most of the symmetry classes known in the literature, particularly the Cartan classification \cite{Altland:1997zz,Zir96}. It is unclear how far beyond this classification \cite{Mag} this structure is applicable. It is only known that there are some of them which share the identity~\eref{3.6}. For example the real and quaternion Ginibre ensembles as well as there chiral counterpart fulfill an identity similar to Eq.~\eref{3.6}.
For many random matrix ensembles like the GUE it seems an academical question if one can derive a Pfaffian or not since there are no applications, yet. However, for the chiral GUE it is important to know this due to the new results obtained for the Wilson Dirac random matrix ensemble discussed in Refs.~\cite{Damgaard:2010cz,Akemann:2010zp,arXiv:1105.6229,arXiv:1108.3035,Kieburg:2011uf,KVZ11,VerSpl11}. Pfaffians were found there for finite lattice spacing. On the level of the joint probability density the authors of Ref.~\cite{arXiv:1108.3035} checked that the ensemble is the chiral GUE as well as the GUE at certain values of the lattice spacing. However for the eigenvalue correlation functions the continuum limit has not yielded the known determinant~\eref{2.20}. With this work we clarified this puzzle.
For intermediate ensembles in general our result might be helpful to understand the structure appearing by switching the interaction between the two ensembles on. It is numerically advantageous to think about spectral correlations of intermediate ensembles as kernels of Pfaffians since the integrand drastically simplifies. In combination with the supersymmetry method \cite{Guh06,Som07,LSZ08,KGG09} also the number of integrals reduces a lot.
Moreover, we derived the relation between the orthogonal polynomials and the skew-orthogonal polynomials corresponding to the determinants and the Pfaffians, respectively. This relation, see Eqs.~(\ref{4.4}-\ref{4.6}), is not only quite simple but also universal since it applies to all random matrix ensembles discussed in this work. The relation between the orthogonal and skew-orthogonal polynomials for $\beta=2$ slightly differs to those found in Ref.~\cite{Gho09} for the cases $\beta=1$ and $\beta=4$. The difference in the two-point weight is the reason for this. Based on the representations~\eref{2.11}, \eref{5.3} and \eref{5.4} shared by all random matrix ensembles with $\beta=2$ as well as checks of these representations \cite{Ver11}, we conjecture that the recursion relation of the orthogonal polynomials connects the determinant and the Pfaffian and this has to be done in a general way.
The Pfaffian found for the average over characteristic polynomials carries over to the $k$-point correlation functions. This structure is valid in the large matrix limit, too. It should not depend on which scaling limit is chosen since the Pfaffian is independent of the matrix size. Hence, the correlation functions appearing as kernels of this Pfaffian have non-trivial recursion relations mapping the determinant to the Pfaffian.
\section*{Acknowledgements}
I am grateful to Gernot Akemann, Jacobus J.M. Verbaarschot and Savvas Zafeiropoulos for fruitful discussions and helpful comments. I also thank Peter J. Forrester and Christopher D. Sinclair for pointing out their work~\cite{Sin,ForSin}. Furthermore I acknowledge financial support by the Alexander-von-Humboldt Foundation.
\section*{References}
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1109.4952
|
\section{Introduction}
The $AdS/CFT$ correspondence \cite{Maldacena:1997re}, and holographic duality in general, is a powerful, conjectured technique for the analysis of strongly coupled field theories. While originally pursued to address questions about low-energy QCD, it has expanded to include studies of a variety of strongly coupled field theories in diverse dimensions.\footnote{For older review articles see \cite{Aharony:1999ti,D'Hoker:2002aw}, while \cite{Hartnoll:2009sz,McGreevy:2009xe} are more recent with an emphasis on applications for condensed matter.}
Of much interest in recent years has been the study of defect theories and the interaction of defects. Such defects can be constructed holographically by the intersection of different stacks of $D$-branes, one of the earliest known examples being the supersymmetric $(2+1)$-dimensional intersection of the $D3/D5$ system \cite{DeWolfe:2001pq}, a defect in the ambient $(3+1)$-dimensional ${\cal N}=4$ super Yang-Mills native to the $D3$ worldvolume. A common technique for studying these systems is to consider the quenched approximation of the field theory, where one stack, say of $Dp$-branes, has parametrically more branes than the other, say of $Dq$-branes. The gravity description of this scenario can then be reliably computed at strong coupling by using a probe $Dq$-brane action in the near-horizon region of a classical $p$-brane supergravity solution \cite{Karch:2002sh}. The full dual field theory lives at the asymptotic boundary of this spacetime and the defect theory lives where the probe brane intersects the boundary.
Multiple defects may be studied by allowing several stacks of $Dq$-branes to intersect the boundary. As discussed first in \cite{Sakai:2004cn,Sakai:2005yt}, a coherent state of spatially separated defects can be achieved by a continuous probe brane configuration with a multiply connected intersection with the boundary. Since the boundary components must have opposite orientation in this scenario, it can be understood as brane/anti-brane recombination. In the scenario of \cite{Sakai:2004cn,Sakai:2005yt}, the defect degrees of freedom were $d=3+1$ chiral fermions, with those on the brane component of opposite chirality from those on the anti-brane. The coherent state where the worldvolumes join in the bulk thus describes chiral symmetry breaking. In \cite{Antonyan:2006vw,Antonyan:2006pg}, this scenario was generalized to allow for intersections of other dimension and brane species as well as for the joining process to occur dynamically.\footnote{In \cite{Sakai:2004cn,Sakai:2005yt}, topological considerations force the branes to join while in \cite{Antonyan:2006vw,Antonyan:2006pg} and later works there are multiple consistent solutions and only the minimum energy one dominates.} Further generalizations have included adding external magnetic and electric fields as well as chemical potential \cite{Parnachev:2006ev,Davis:2007ka,Bergman:2008sg,Johnson:2008vna,Johnson:2009}.
In this paper, we consider scenarios of bulk brane/anti-brane recombination in $AdS_5 \times S^5$,
\begin{equation}
ds^2\sim r^2 \left(-dt^2 + dx^2 + dy^2 + dz^2\right) + {dr^2\over r^2} + d\Omega_5^2~.
\end{equation}
As an additional ingredient to previous studies, we consider probes which are electrically charged under the background $F_5$ Ramond-Ramond field. The probe branes form two stacks, each spanning some cycle in $S^5$, the non-compact directions $(t,x,y)$ and some curve $z(r)$. The stacks have opposite orientation and are separated in the $z$ direction along the boundary.
An uncharged probe brane -- such as in the studies cited above -- experiences no force in the non-compact directions from the $F_{5}$. For such a case there are then two qualitative classes of solutions, depicted in \figref{uncharged}. The first solution is the so-called ``black hole embedding'' which reaches all the way down to the spacetime horizon. These embeddings are ``straight'' in the sense that ${dz \over dr}=0$. The second solution is a joined embedding which has two disconnected boundaries of opposite orientation although the entire worldvolume is a simply connected and oriented manifold. Only these solutions have ${dz \over dr}\ne0$. Note that since there is no Ramond-Ramond force, the brane orientation does not play a role.
On the other hand, if the probe branes are charged under the spacetime Ramond-Ramond field, the situation is somewhat different. This can occur either because the probe itself is a $D3$-brane, or the charge could be induced by worldvolume fluxes on the probe. The $D3/D5$ system where the $D5$ brane carries $q$ unit of $D3$-brane charge
was first introduced in \cite{Karch:2001}. In \cite{Myers:2008me}, black-hole embeddings of $D5$ and $D7$ probe branes with induced $D3$-brane charge were studied in $AdS_5\times S^5$. Additional $D7$ brane embeddings carrying $D3$ charge were introduced in \cite{Bergman:2010gm} and studied further in \cite{Jokela:2010nu,Davis:2011gi}. These probes are affected by the background $F_5$ and even the black hole embeddings have ${dz\over dr}\ne0$. In \figref{chargedprobe}, we see such a black-hole embedding. The brane orientation plays a major role in this situation; an oppositely oriented probe would bend in the opposite $z$-direction.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=.5\textwidth]{straight.png} &
\includegraphics[width=.5\textwidth]{skinny.png}
\end{tabular}
\caption{\label{uncharged} Straight embeddings and a joined embedding where there is no force from the background Ramond-Ramond flux. The arrows represent worldvolume orientation. There would be no change in the embedding if the arrows were reversed.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.6\textwidth]{basic_probe_brane.png}
\caption{\label{chargedprobe} A $D3$-charged probe brane in (finite-temperature) $AdS_5 \times S^5$. The probe bends in the $z$-direction as it descends from the boundary (the solid line at the top) to enter the horizon represented by the dotted line at the bottom. The arrow represents the orientation of the $D3$ charge. An oppositely oriented brane would bend in the opposite $z$-direction.}
\end{figure}
These electrically charged probe branes have a richer space of joined solutions than their uncharged cousins. Due to the force in the $z$-direction, the qualitative features of the solution depend strongly on the orientation, specifically the left-right ordering of the boundary components. The choice of orientation gives rise to the classes of solutions seen in \figref{chargedjoin}. The top left figure pictures a brane/anti-brane pair which tend toward each other despite not actually connecting, while the top right figure pictures a joined pair. These two solutions have the same boundary conditions and so it is a dynamical question which has the lower energy and is therefore stable. The figures in the bottom row also depict solutions with the same boundary conditions, but with the worldvolume orientations all opposite of the figures above. Note the surprising feature in the bottom right figure, where the joined embedding becomes wider in the bulk than at the boundary. We will call these joined solutions ``chubby'' and conversely the more typical solutions in the top right (which are widest at the boundary) we will call ``skinny.''
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=.5\textwidth]{attract.png} &
\includegraphics[width=.5\textwidth]{skinny.png} \\
\includegraphics[width=.5\textwidth]{repel.png} &
\includegraphics[width=.5\textwidth]{fat.png} \\
\end{tabular}
\caption{\label{chargedjoin} The top row pictures possible solutions of a brane/anti-brane pair in the presence of a Ramond-Ramond force. Note that the branes bend toward each other as they extend into the bulk even if they don't join. If the orientations are reversed, we have instead the bottom set of solutions. These always bend away from each other when initially leaving the boundary even if they eventually join in the bulk.}
\end{figure}
There are multiple perspectives on what these brane systems are holographically dual to. Firstly, the $(2+1)$-dimensional intersection of a probe brane with the boundary is conventionally associated with a defect in ${\cal N}=4$ super-Yang-Mills gauge theory. The field content of the defect is given by the lowest level open string modes which are localized at the $D$-brane intersection. For a $D5$-brane probe, the defect theory is supersymmetric since the intersection is $\#ND=4$; this is the spectrum studied in \cite{DeWolfe:2001pq}. For the $D7$-brane probe, the intersection is $\#ND=6$ and the spectrum is simply massless fermions \cite{Polchinski:1998rr}, in fact T-dual to the $D4/D8$ intersections of the Sakai-Sugimoto model \cite{Sakai:2004cn,Sakai:2005yt}. As a caveat, it should be mentioned that it is not clear if this picture of the spectrum still holds when internal fluxes exist on the probe, but is often nonetheless used to guide intuition.
A defect dual to a stack of $N$ branes or anti-branes is associated with a $U(N)$ global flavor symmetry inherited from the gauge field living on the brane worldvolume. Thus the recombination of an equal number of branes and anti-branes describes a breaking of symmetry $U(N)\times U(N)\to U(N)$. Since the defects are separated in space, the duals of these scenarios can be considered interacting $(2+1)$-dimensional defect bi-layer systems or as discussed in \cite{Antonyan:2006pg}, $(2+1)$-dimensional effective field theories with non-local interactions.
The dual interpretation above holds for probes with or without $D3$-charge. However, for $D3$-charged probes there are some other interesting properties of these solutions. A $D3$-charged probe brane -- even a higher dimensional brane with an induced $D3$ charge -- contributes to the overall Ramond-Ramond flux of the system. This flux is in turn related by the $AdS/CFT$ dictionary to the rank of the dual gauge group. Therefore a defect of $D3$-charge $k$ forms a domain wall in the dual gauge theory with $SU(N)$ gauge symmetry on one side and $SU(N+k)$ on the other \cite{Myers:2008me}. A cartoon representation of this situation is depicted in \figref{basiccartoon}. Once the probe enters the horizon, it is effectively parallel to the original stack of $D3$-branes sourcing the geometry, adding to the overall $D3$-brane charge as measured by a Gaussian surface outside the horizon. This is interpreted as a larger gauge symmetry existing in the region to the left. It follows that if there are multiple $D3$-charged defects, that we have a spatially non-trivial pattern of symmetry breaking in the dual theory, with a gauge group between the defects which is different from that outside. Thus the joined solutions should be considered dual to finite-width domain walls.
In this paper, we will study the thermodynamics of these domain walls, mostly from the bulk perspective. In Section 2, we introduce a class of $D3$-charged probe branes and derive a one-dimensional effective particle mechanics action that describes the entire class. The solutions of the equation of motion of this effective action are studied in Section 3 and a renormalized free energy computed in Section 4. Finally, in Section 5, we examine the phase diagram of this system in the space of external magnetic field and asymptotic separation with some comments on the phenomenon of magnetic catalysis.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.6\textwidth]{basic_cartoon_brane.png}
\end{center}
\caption{\label{basiccartoon} A cartoon representation of a probe brane (dashed line) carrying $D3$-charge $k$ bending to become parallel with the stack of $N_3$ $D3$-branes sourcing the $AdS$ geometry, represented by the solid lines at the bottom. The arrows represent brane worldvolume orientation. The dual gauge group is $SU(N_3)$ towards the right while it is enhanced to $SU(N_3+k)$ to the left. }
\end{figure}
\section{$D3$-charged probes in $AdS_5 \times S^5$}
Consider the background $IIB$ supergravity solution thermal $AdS_5 \times S^5$, the near-horizon geometry of $N_3$ $D3$-branes at finite temperature. The line-element is given by
\begin{equation}\label{ads5s5}
L^{-2} ds^2 = r^2\left(-h(r)dt^2 + dx^2+dy^2 +dz^2\right) +{dr^2 \over h(r) r^2} + d\Omega_5^2~.
\end{equation}
The $S^5$ line element is represented as a bundle over $S^2\times S^2$,
\begin{equation}
d\Omega_5^2 = d\psi^2 + \sin^2 \psi d\Omega_2^2 + \cos^2\psi d{\tilde{\Omega}_2^2}~,
\end{equation}
where $\psi\in \left(0 ,{\pi\over2}\right)$ and
\begin{equation}
d\Omega_2^2=d\theta^2 + \sin^2 \theta d\phi^2~,
\end{equation}
is the line-element for a unit $S^2$. The blackening function is
\begin{equation}
h(r) =1-{r_h^4\over r^4}~.
\end{equation}
At zero-temperature, $r_h=0$. However, any non-zero value of $r_h$ can be rescaled by a coordinate transformation. Therefore, for finite temperature, we can choose without loss of generality $r_h=1$. The scale of the geometry is related to the microscopic string theory parameters via
\begin{equation}
L^4 = 4\pi g_s N_3 (\alpha^\prime)^2~.
\end{equation}
There is also a self-dual five-form Ramond-Ramond flux
\begin{equation}
F_5 = {4 L^4\over g_s}\left(r^3 dt \wedge dx \wedge dy\wedge dz\wedge dr + \omega_5\right)~.
\end{equation}
Here $\omega_5$ is the volume form on the unit five-sphere
\begin{equation}
\omega_5 = \sin^2 \psi \cos^2 \psi d\psi \wedge \omega_2 \wedge \tilde{\omega}_2~,
\end{equation}
where $\omega_2=\sin\theta d\theta\wedge d\phi$ is the $S^2$ volume form. We encode this flux with the four-form potential
\begin{equation}\label{c4}
{g_s\over L^4}C_4 = r^4 h(r) dt\wedge dx\wedge dy\wedge dz + {1\over2} c\left(\psi\right) \omega_2 \wedge {\tilde \omega}_2~.
\end{equation}
The function $c\left(\psi\right)$ is
\begin{equation}
c(\psi) = \psi -{1\over4}\sin\left(4\psi\right) + c_0
\end{equation}
where $c_0$ is an arbitrary constant, a residual ambiguity due to the gauge symmetry of the Ramond-Ramond field. A similar constant could be added to the coefficient of $dt\wedge dx\wedge dy \wedge dz$, but we have chosen to partially fix the gauge by requiring that the first term in $C_4$ vanish at the horizon. This ensures that the term is well-defined on the Euclidean section of \eqref{ads5s5} which simplifies the treatment of the Wess-Zumino terms.
We will now consider the following set of branes
\begin{equation}\label{dbranes}
\begin{array}{rcccccccccccl}
& & t & x & y & z & r & \psi & \Omega_2 & \tilde{\Omega}_2 & \\
& D3'& - & - & - & - & \cdot & \cdot & \cdot & \cdot& \\
& D3 & - & - & - & \sim & \sim & \cdot & \cdot & \cdot& \\
& D5 & - & - & - & \sim & \sim & \cdot & - & \cdot& \\
& D7 & - & - & - & \sim & \sim & \cdot & - & -& \\
\end{array}
\end{equation}
The $D3'$ row refers to the large stack of $D3$ branes which source the $AdS$ geometry while the other rows record the configurations of the probes. A dash indicates the brane is extended in that direction, with support over the entire range of the coordinate. A dot indicates the respective brane is completely localized in that coordinate. Finally the $\sim$ symbols indicate that the brane traces a curve in those directions. For example, the $D5$-brane extends along the non-compact $(t,x,y)$ directions, wraps one of the two $S^2$ factors in the $S^5$, is localized in $\psi$ and on the other $S^2$, and finally, lies along a curve in the $(z,r)$ space.
These probes all intersect the boundary on some $2+1$ dimensional subspace at a fixed value of $z$ (although for the $D3$ probes, this will turn out to be $z=\pm\infty$). In order to induce $D3$-brane charge,\footnote{Such flux is actually required to stabilize the $D7$ probe at a non-trivial value of $\psi$ at the $AdS$ boundary \cite{Bergman:2010gm}.} the probe $D5$ and $D7$-branes will carry internal flux topologically supported on one or both $S^2$ factors, respectively. We will also allow for magnetic field in the three dimensional defect on the boundary, {\it i.e.} a non-zero $F_{xy}$ component. In \cite{Bergman:2010gm}, $D7$ branes with a more general {\it ansatz} were studied. However, our focus will be a class of solutions with different boundary conditions.
The $D5$ and $D7$ probes outlined above have $3+1$ non-compact directions and wrap some compact cycles. If one imagines integrating over these cycles, one would obtain an effective $3+1$ dimensional object which carries $D3$ charge in $AdS_5$. In other words, the higher-dimensional $D3$-charged branes act as effective $D3$-branes. These effective branes are much like excited states of a proper $D3$, they carry $D3$ charge but the effective tension is greater than the charge. This will become clearer in the next few sections. First, we will calculate an effective action for a $D3$ probe with the {\it ansatz} \eqref{dbranes}. We will then see that $D5$ and $D7$ probes will yield an effective action of the same form.
\subsection{$D3$-brane probe}
First, let us introduce a $D3$-brane probe as a model system. The action comprises the familiar DBI and Wess-Zumino terms
\begin{equation}
S_3= -T_3 \int d^{3+1} \xi e^{-\phi} \sqrt{-{\rm det}\left(g+2\pi\alpha' F\right)} - T_3 \int C_4~.
\end{equation}
The three-brane tension is
\begin{equation}
T_3 = {1\over (2\pi)^3}{1\over {\alpha'}^2}~.
\end{equation}
We choose a static gauge where $\xi^a=\left\{t,x,y,r\right\}$ are brane coordinates and the embedding is given by the function $z(r)$. The induced metric is thus
\begin{equation}
{ds_3^2 \over L^2} = r^2\left(-hdt^2+dx^2+dy^2\right) + \left(1+ r^4 h \dot{z}^2\right) {dr^2\over r^2 h}~,
\end{equation}
where a dot indicates differentiation by $r$. We also allow a magnetic field normalized as
\begin{equation}
{2\pi \alpha'\over L^2} F = B dx \wedge dy~.
\end{equation}
This information is sufficient to compute the Born-Infeld term
\begin{equation}
S_{DBI}= {\cal N}_3 \int dr \sqrt{\left(r^4+B^2\right)\left(1+ r^4 h \dot{z}^2\right)}~,
\end{equation}
where the overall constant is
\begin{equation}
{\cal N}_3={T_3 L^4 V_{2+1} \over g_s}~,
\end{equation}
with $V_{2+1}$ the infinite volume factor of the $(t,x,y)$ directions.
To compute the Wess-Zumino term we also need to specify an orientation, which we encode via an orientation parameter $\zeta=\pm1$. Evaluating,
\begin{equation}
\int C_4 = V_{2+1} \zeta \int r^4 h \dot{z} dr~.
\end{equation}
Note that while orientation is an invariant geometric feature intrinsic to the entire $D$-brane worldvolume, the parameter $\zeta$ is partly an artifact of the coordinates we use. Therefore $\zeta$ may take different values on separate branches of the same continuous brane. For example, in a brane/anti-brane recombination, the left branch has $\zeta=1$ and the right branch $\zeta=-1$, yet the worldvolume is continuous.
Putting together the terms above -- and dropping an overall constant factor -- yields an effective particle mechanics Lagrangian
\begin{equation}
L_3 = \sqrt{\left(r^4+B^2\right)\left(1+ r^4 h \dot{z}^2\right)} + \zeta r^4 h \dot{z}~.
\end{equation}
We will find similar effective Lagrangians for the $D5$ and $D7$ probes, differing only in the coefficient of the second term. Here that coefficient is of unit magnitude since physically it is the $D3$-brane charge per tension.
\subsection{$D5$ probes}
The probe action for $D5$-branes is
\begin{equation}\label{d5action}
S_5 = - T_5 \int d^{5+1} \xi e^{-\phi} \sqrt{-{\rm det}\left(g+2\pi\alpha' F\right)} - 2\pi\alpha' T_5 \int C_4 \wedge F~,
\end{equation}
where the tension is
\begin{equation}
T_5 = {1\over (2\pi)^5}{1\over {\alpha'}^3}~.
\end{equation}
We choose a static gauge with coordinates $\xi^a=\left\{t,x,y,r, \theta,\phi\right\}$ and embedding function $z(r)$. The induced metric is
\begin{equation}
{ds_5^2 \over L^2} = r^2\left(-hdt^2+dx^2+dy^2\right) + \left(1+ r^4 h \dot{z}^2\right) {dr^2\over r^2 h} + \sin^2\psi d\Omega^2_2~.
\end{equation}
The {\it ansatz} for worldvolume flux is
\begin{equation}
{2\pi \alpha'\over L^2} F = B dx \wedge dy + {f \over 2} \omega_2 ~.
\end{equation}
The magnetic field is a continuous quantity but the flux on the compact sphere is, of course, quantized
\begin{equation}
f = {2\pi \alpha' \over L^2} n~, \quad\quad\quad n\in\mathbb{Z}~.
\end{equation}
Substituting all this into the action yields
\begin{equation}
{S_5}=- {\cal N}_5 \int dr\left( \sqrt{\left(r^4+B^2\right)\left(f^2 + 4 \sin^4\psi\right) \left(1+ r^4 h \dot{z}^2\right)} + \zeta f r^4 h \dot{z}\right)~,
\end{equation}
with the normalization
\begin{equation}
{\cal N}_5={2\pi T_5 L^6 V_{2+1} \over g_s}~,
\end{equation}
and once again we have introduced an orientation parameter $\zeta=\pm1$. Our {\it ansatz} is for constant $\psi$ but we see that $\psi$ has a potential. The $\psi$ equation of motion is
\begin{equation}
{d~\over d\psi} \sqrt{f^2+4\sin^4\psi}=0~,
\end{equation}
yielding\footnote{Another solution is $\psi=0$ but it is physically trivial since the brane volume is then exactly zero.}
\begin{equation}
\psi = {\pi \over 2}~.
\end{equation}
We insert this back into the $D5$ action. Up to an overall constant we again obtain an effective particle Lagrangian for $z(r)$,
\begin{equation}
L_5 = \sqrt{\left(r^4+B^2\right)\left(1+ r^4 h \dot{z}^2\right)} + {\zeta f\over \sqrt{f^2 +4}} r^4 h \dot{z}~.
\end{equation}
The only difference from the $D3$ is in the coefficient of the second term, the effective $D3$-brane charge per unit tension. The magnitude of this ratio is less than unity here, in keeping with the picture that this $D5$ probe is a $D3$-brane in an excited state.
\subsection{$D7$ probes}
The $D7$-brane action is
\begin{equation}\label{d7action}
S_7 = - T_7 \int d^{7+1} \xi e^{-\phi} \sqrt{-{\rm det}\left(g+2\pi\alpha' F\right)} - {\left(2\pi\alpha'\right)^2\over 2} T_7 \int C_4 \wedge F\wedge F~,
\end{equation}
with tension
\begin{equation}
T_7 = {1\over (2\pi)^7}{1\over {\alpha'}^4}~.
\end{equation}
In a static gauge with coordinates $\xi^a=\left\{t,x,y,r, \theta,\phi, \tilde{\theta}, \tilde{\phi}\right\}$, we describe the embedding with the function $z(r)$. The induced metric is
\begin{equation}
{ds_7^2 \over L^2} = r^2\left(-hdt^2+dx^2+dy^2\right) + \left(1 + r^4 h \dot{z}^2\right) {dr^2\over r^2 h} + \sin^2\psi d\Omega^2_2+\cos^2\psi d\tilde{\Omega}^2_2~.
\end{equation}
For the worldvolume flux we use the {\it ansatz}
\begin{equation}
{2\pi \alpha'\over L^2} F = B dx \wedge dy + {f_1 \over 2} \omega_2 + {f_2 \over 2} \tilde{\omega}_2~.
\end{equation}
The fluxes on the $S^2$ factors are quantized
\begin{equation}
f_i = {2\pi \alpha' \over L^2} n_i~, \quad\quad\quad n_i\in\mathbb{Z}~.
\end{equation}
The DBI portion of the action is
\begin{equation}
S_{DBI} = -{\cal N}_7\int dr \sqrt{\left(r^4+B^2\right)\left(f_1^2+4\sin^4{\psi}\right)\left(f_2^2+4\cos^4{\psi}\right)\left(1 +r^4 h \dot{z}^2\right)}
\end{equation}
with
\begin{equation}
{\cal N}_7 = {4\pi^2 T_7 L^8 V_{2,1}\over g_s}~.
\end{equation}
The Wess-Zumino term is given by
\begin{equation}
S = -{\cal N}_7 \zeta f_1 f_2 \int dr r^4 h \dot{z}~,
\end{equation}
with $\zeta$ the orientation parameter. We minimize the $\psi$ potential
\begin{equation}
{d~\over d\psi} \sqrt{\left(f_1^2+4\sin^4\psi\right)\left(f_2^2+4\cos^4\psi\right)}=0~,
\end{equation}
yielding the implicit equation\footnote{While this can be solved for general $f_i$, it can be seen that fluctuations $\delta \psi$ around the solution can violate the BF bound \cite{Breitenlohner:1982bm,Breitenlohner:1982jf}. In particular, for absolutely no internal fluxes $f_i=0$, the $D7$ will be unstable \cite{Davis:2008nv}. See \cite{Bergman:2010gm} for more discussion of stabilizing this $D7$ brane embedding.}
\begin{equation}\label{psisol}
f_2^2 \sin^2 \psi -f_1^2 \cos^2 \psi +4 \cos^2 \psi \sin^2\psi \left(\cos^2\psi-\sin^2\psi\right)=0~.
\end{equation}
Substituting this back into the action yields, up to an overall constant, an effective particle Lagrangian for the $D7$-brane
\begin{equation}
L_7 = \sqrt{\left(r^4+B^2\right)\left(1+ r^4 h \dot{z}^2\right)} + {\zeta f_1 f_2\over \sqrt{\left(f_1^2 +4\sin^4\psi_0\right)\left(f_2^2+4\cos^4\psi_0\right)}} r^4 h \dot{z}~,
\end{equation}
where $\psi_0$ is a constant that solves \eqref{psisol}. This again takes the form of the effective $D3$ Lagrangian with a charge per tension smaller than unity.
\section{Solutions to effective Lagrangian}
We found that all three of the $D3$-charged probes under consideration are described by an effective particle Lagrangian of the form
\begin{equation}\label{efflag}
S_{eff}= \int dr \sqrt{r^4+B^2} \sqrt{1+ r^4 h \dot{z}^2} +\alpha \int dr r^4 h \dot{z}~.
\end{equation}
The parameter $\alpha$ is the effective $D3$-brane charge per tension and is given by
\begin{equation}\label{alphacases}
\alpha =
\begin{cases}
\zeta & D3{\rm-brane}\\
{\zeta f \over \sqrt{f^2 +4}} & D5{\rm-brane}\\
{\zeta f_1 f_2 \over \sqrt{\left(f_1^2 +4\sin^4\psi_0\right)\left(f_2^2+4\cos^4\psi_0\right)}} & D7{\rm-brane}
\end{cases}
\end{equation}
with $\psi_0$ solving \eqref{psisol} in the case of the $D7$. Note that $|\alpha|<1$ for both the $D5$ and $D7$ probes.
The equation of motion derived from \eqref{efflag} can be immediately integrated since the variable $z(r)$ is cyclic
\begin{equation}\label{pz}
P \equiv {\sqrt{r^4 +B^2\over1+r^4 h \dot{z}^2} }r^4 h \dot{z}+\alpha r^4 h = constant~.
\end{equation}
Define the intermediate function
\begin{equation}
g(r) = {P \over r^4 h} -\alpha~,
\end{equation}
then solve for $\dot{z}$ to obtain
\begin{equation}\label{zdot}
\dot{z} = {g(r) \over \sqrt{r^4 +B^2 - r^4 h g(r)^2}}~.
\end{equation}
The full profile $z(r)$ is obtained by integration. This cannot be done analytically in general, but for any choice of $B$, $P$ and $\alpha$ the integration of \eqref{zdot} is easily evaluated numerically.
These solutions are completely specified by the integration constant $P$. For any brane profile that enters the black hole horizon, substituting $r=1$ into \eqref{pz} shows that $P$ must vanish since $h\left(r=r_h=1\right)=0$,
\begin{equation}
\boxed{\Big.~P=0~{\rm for~solutions~with~support~at~} r=1~\Big.}
\end{equation}
In keeping with the literature we call these solutions {\it black hole embeddings}. Since $P=0$, we have $g(r)=-\alpha$. Thus, we see from \eqref{zdot} that for the black hole embeddings $z(r)$ is single-valued and monotonic.
The other possibility is that the profile has a minimum value of $r$. Without loss of generality, we can choose this minimum to be located at $z=0$. The signal of a minimum would be $\dot{z}$ diverging at some $r=r_0$. This yields the expression for the integration constant
\begin{equation}\label{przero}
P = r^4_0 \sqrt{h_0} \left(\sqrt{1+{B^2\over r^4_0}}{\rm sign}\left(\dot{z}_0\right) + \alpha \sqrt{h_0}\right)~,
\end{equation}
where $h_0 = h(r_0)$ and $\dot{z}_0 = \dot{z}(r\to r_0)$. The presence of an absolute minimum requires that the brane bends back up to the boundary. This other leg of the brane will have opposite orientation parameter $\zeta$ so this solution is a joined brane/anti-brane pair. We thus call the $P\ne0$ solutions {\it joined embeddings}.
The magnitude of the first term in the parentheses of \eqref{przero} is greater than unity while that of the second term is less than unity. Therefore
\begin{equation}\label{slopemin}
{\rm sign} \left(\dot{z}\left(r\to r_0\right)\right)={\rm sign}\left(P\right)~.
\end{equation}
However, $\dot{z}\to-\infty$ when approaching from the left of the minimum while $\dot{z}\to+\infty$ when approaching from the right. Furthermore, the orientation parameter $\zeta$ changes sign from one branch to the other. Therefore, $P$ changes sign as well,\footnote{The reader may find this disconcerting, since $P$ is playing the role of a conserved quantity. The resolution lies in the multi-valuedness of the function $z(r)$. $P$ need only be constant on a given single-valued branch. The minimum is precisely where the single-valued parameterization $z(r)$ breaks down and so consequently does the definition of $P$. That the magnitude of $P$ is constant follows from the continuity of the embedding. } with $P<0$ for $z<0$ and $P>0$ for $z>0$ (see \figref{minfig}). The joined configuration is clearly symmetric under parity $z\to -z$, so we can without loss of generality focus our attention to a single branch. We will therefore restrict our attention to $P\ge 0$, which includes the black hole embedding and the ``right branch'' with $\dot{z}_0>0$ of the joined solutions.
\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{minfig.png}
\caption{\label{minfig} The sign of the integration constant $P$ is the same as that of $\dot{z}$ as $r_0$ is approached and flips accordingly as the minimum at $z=0$ is crossed.}
\end{figure}
At the boundary
\begin{equation}\label{slopebound}
{\rm sign} \left(\dot{z}\left(r\to\infty\right)\right) = - {\rm sign} \left(\alpha\right)~,
\end{equation}
indicating that the direction in which the brane bends initially on its descent from infinity is given entirely by the sign of the $D3$-brane charge. Comparing \eqref{slopemin} and \eqref{slopebound} we see there are thus two qualitative classes of joined solutions, given by the relative sign of $P$ and $\alpha$. For ${\rm sign}(P)= - {\rm sign}(\alpha)$, the sign of $\dot{z}$ remains the same throughout the branch, {\it i.e.} each branch of the brane is separately monotonic. On the other hand, for ${\rm sign}(P)= {\rm sign}(\alpha)$ even a given branch is not monotonic. We call these two possibilities ``skinny'' and ``chubby,'' respectively. See \figref{skinnyfat}.
\begin{figure}[ht]
\centering
\begin{tabular}{cc}
\includegraphics[width=.5\textwidth]{skinny.png} &
\includegraphics[width=.5\textwidth]{fat.png}
\end{tabular}
\caption{\label{skinnyfat} ``Skinny'' and ``chubby'' joined embeddings.}
\end{figure}
Physically, we know that the brane and anti-brane have an attraction due to exchange of gravitons and Ramond-Ramond quanta. Further, the background $F_5$ also deflects branes and anti-branes in opposite directions. In the skinny solutions, the background $F_5$ pushes the two stacks together while in the chubby solutions the Ramond-Ramond field forces them apart.
\subsection{Asymptotics}
The asymptotic separation in $z$ of a joined brane/anti-brane pair is not independent of $r_0$. Define $L$ as
\begin{equation}\label{ldef}
L(r_0) = 2\int_{r_0}^\infty \dot{z}(r)~,
\end{equation}
where the factor of two arises since the integral is only over one branch of the brane system. For a joined solution, {\it i.e.} any solution with $r_0>1$, $L$ is the asymptotic separation in the $z$ direction of the two ends of the solution. For $r_0=1$ however, the brane and anti-brane are disconnected black hole embeddings. In this case the asymptotic separation is truly a free parameter and $L(r_0=1)$ simply records (twice) the range in $z$ that each branch of the embedding spans.
The probe branes for generic $\alpha$ have the large $r$ behavior
\begin{eqnarray}
\dot{z}\left(r\gg1\right) = -{\alpha \over \sqrt{1-\alpha^2}} {1\over r^2} + O\left({1\over r^6}\right)~,\quad\quad \left(|\alpha|<1\right)~.
\end{eqnarray}
The case $|\alpha|=1$ is non-generic. Indeed, expanding \eqref{zdot} in yields
\begin{equation}
\dot{z}\left(r\gg1\right) = -{\alpha\over \sqrt{1+B^2 +2\alpha P}}+ O\left({1\over r^4}\right)~,\quad\quad \left(|\alpha|=1\right)~.
\end{equation}
It follows that $L$ converges for $|\alpha|<1$ and diverges for $|\alpha|=1$, which means that $|\alpha|=1$ branes ({\it i.e.} $D3$-brane probes) do not intersect the $AdS$ boundary at finite $z$ while those with generic $\alpha$ do. The impossibility of the $D3$ probe to intersect the $AdS$ boundary at finite $z$ may be a symptom of the open string tachyon present at weak coupling at the $(2+1)$-dimensional intersection of $D3$-branes.\footnote{Since such a system has $\#ND=2$. See \cite{Polchinski:1998rr}.} Whatever the explanation, we will now restrict our attention to $D5$-branes and $D7$-branes so that we can study probes which intersect the boundary at a finite location.
The right-hand side of \eqref{ldef} is a complicated function of $r_0$ since $\dot{z}$ depends on it through the integration constant $P$. We do not have an analytic expression but can plot it numerically. As an example, see \figref{lvrsample}, which plots $L(r_0)$ for a $D7$-brane probe with $B=0$ and $f_1=f_2={1\over\sqrt{2}}$. Note that $L(r_0)$ is not monotonic and has a maximum. Therefore, when the brane/anti-brane pair are sufficiently separated at the boundary (with an $L\gtrsim 1.3$) there are no joined solutions, only black hole-embeddings. Further, due to the maximum there is a range of $L$ where there are two $r_0$, that is two solutions with the same boundary condition.
Another feature worth noting is the abrupt end of the curve at $r_0=1$. The $r_0=1$ solution is a black-hole embedding and $L(1)$ is (twice) the $\Delta z$ spanned by a single branch of that embedding. The curve $L(r_0)$ does not continue past this point.
\begin{figure}[ht]
\centering
\includegraphics[width=.8\textwidth]{Lvrsample.png}
\caption{\label{lvrsample} Asymptotic brane separation $L$ of a joined solution ($\alpha=-{1\over3}$ and no magnetic field) as a function of minimum radius $r_0$.}
\end{figure}
There is a class of unphysical solutions lurking within the family that we have been discussing. Some of the non-monotonic branches, {\it i.e. } those with ${\rm sign} \left(P\right) = {\rm sign} \left(\alpha\right)$, will turn out to have negative $L$. Qualitatively these solutions appear as in \figref{fish}. Note that they have the same boundary conditions as a ``skinny'' solution. These solutions are clearly unstable to brane reconnection at the intersection point and will not be considered further.
\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{fish.png}
\caption{\label{fish} An unphysical solution with negative $L$.}
\end{figure}
\section{Free energy}
Now that we have classified the solutions, we investigate the phases of a pair of brane/anti-brane probes. The dynamical problem is to find the solution in a given ensemble, with given boundary conditions, which has the lowest free energy. This solution will dominate and be thermodynamically stable. In the present case, the boundary conditions are given by the asymptotic brane positions and orientations and the values of the fluxes, including magnetic field. Without loss of generality, we can assume\footnote{Due to translation invariance in the $z$ direction.} that the center of the pair is at $z=0$, {\it i.e.} if they join, they join at $z=0$. Then the boundary conditions are given by $L$, $B$, and $\alpha$.
The free energy is conventionally given as the negative of the on-shell action. This is, up to a positive constant, simply the effective action \eqref{efflag}
\begin{equation}\label{fdef}
F (r_0)=\int_{r_0}^\infty dr \left\{ \sqrt{\left(r^4 +B^2\right)\left(1+r^4 h \dot{z}^2\right)}+\alpha r^4 h \dot{z}\right\}~,
\end{equation}
This is the free energy of a single leg of the brane/anti-brane system. In the case of $r_0=1$, \eqref{fdef} is the energy of one entire worldvolume, from horizon to boundary. For $r_0>1$, it computes the free energy of one half of the joined brane/anti-brane system. In all cases since the other branch is obtained by symmetry, the true free energy is just twice \eqref{fdef}. Substituting in the general solution \eqref{zdot} we get
\begin{eqnarray}\label{evalfree}
F(r_0) &=& \int_{r_0}^\infty dr \sqrt{r^4+B^2 \over 1 - {r^4 \over r^4+B^2} h g^2} \left(1+ {r^4\over r^4+B^2} \alpha h g\right)~,\cr
&=& \int_0^{1\over r_0} {du\over u^4} { 1+B^2 u^4+\alpha h\left({1\over u}\right) g\left({1\over u}\right)\over \sqrt{1+B^2 u^4-h\left({1\over u}\right) g\left({1\over u}\right)^2 } }~,
\end{eqnarray}
where we changed integration variables to $u=r^{-1}$ in the second line.
Note that \eqref{evalfree} is generically infinite. Indeed, placing a cut-off at the lower end of the $u$ integral yields
\begin{equation}
F (r_0) = \int_\epsilon {du\over u^4} { 1+B^2 u^4+\alpha h\left({1\over u}\right) g\left({1\over u}\right)\over \sqrt{1+B^2 u^4-h\left({1\over u}\right) g\left({1\over u}\right)^2 } } \sim {\sqrt{1-\alpha^2}\over3\epsilon^3}+{\rm finite}~.
\end{equation}
Since this divergence is independent of $r_0$, the difference in free energy between any two embeddings will be finite and numerically computable. We will thus compute a renormalized free energy
\begin{equation}
\Delta F (r_0) \equiv F(r_0) - F_0~,
\end{equation}
with $F_0$ the divergent free energy of the black hole embedding with $r_0=1$,
\begin{equation}
F_0 = \int_0^{1} {du\over u^4} \sqrt{1+B^2 u^4-\alpha^2 h\left({1\over u}\right) } ~.
\end{equation}
When $\Delta F<0$, the joined solution has less energy than the black hole embeddings and so it dominates, indicating flavor symmetry breaking in the bi-layer description.
\begin{figure}
\centering
\includegraphics[width=.7\textwidth]{FvR_samp.png}
\caption{\label{FvRsample} The renormalized free energy as a function of $r_0$ for $B=0$ and $\alpha=-{1\over3}$.}
\end{figure}
In \figref{FvRsample} we plot the renormalized free energy as a function of $r_0$ for the case $B=0$ and $\alpha=-{1\over3}$. This is the same set of solutions whose asymptotic separation versus $r_0$ is plotted in \figref{lvrsample}. The only joined solutions with negative free energy are those with $r_0 \gtrsim 1.19$ which corresponds to $L \lesssim 1.2$. For any larger $L$, the black hole embedding is less energetic or the joined embedding does not exist.
\section{Phase diagram and discussion}
In \figref{LvB}, we plot the phase diagram of the brane/anti-brane system in the $L$-$B$ plane. Each curve is at fixed $\alpha$, above the curve being the flavor symmetric phase where the stacks do not join while below the curve the symmetry is broken to the diagonal subgroup by brane recombination. We can see that for $\alpha$ negative, the stacks always join at small enough $L$. This is quite intuitive since the background $F_5$ assists the native attraction of the brane and anti-brane so there is no effect to prevent their joining. On the other hand, we see that for large enough positive $\alpha$, the stacks do not join at small $L$ unless there is also a strong enough external magnetic field. Intuitively, the force from the background $F_5$ is strong enough to overcome the brane/anti-brane attraction even at arbitrarily small separation.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{LvB.png}
\caption{\label{LvB} Phase diagrams for the defect system. Above any fixed $\alpha$ curve, the dominant solution is given by the two disconnected brane worldvolumes, {\it i.e.} the symmetric phase. The joined solutions, the broken symmetry phase, dominates below the curve. }
\end{figure}
In these types of studies, there is a general expectation of {\it magnetic catalysis}, that an external magnetic field favors the breaking of flavor symmetry. This effect has been seen both in perturbative and large-$N$ calculations in quantum field theory \cite{Gusynin:1994re}. It is also known to be a common feature in holographic scenarios of various dimension brane intersections \cite{Johnson:2008vna,Filev:2009xp,Filev:2010pm}. However, in \cite{Preis:2010cq} the Sakai-Sugimoto model was studied at finite chemical potential and magnetic field and an inverse magnetic catalysis was found in a certain region of the phase diagram, {\it i.e.} at zero temperature and fixed finite chemical potential, an increase in magnetic field can prompt a transition to a symmetric state.
We see in \figref{LvB} both catalysis and inverse catalysis, depending on the value of $\alpha$ and the region of the curve in question. One can see that all positive $\alpha$ embeddings exhibit catalysis, {\it i.e.} all of the chubby solutions. In these cases, it appears that external magnetic field always enhances the attraction of the brane/anti-brane pair. However, for $0 \gtrsim \alpha \gtrsim -.2$ the curves are similar to positive $\alpha$ so the sign of the induced $D3$ charge is not sufficient to determine the behavior with respect to magnetic field. For $\alpha\approx -.2$, we see a maximum, indicating a region of inverse catalysis for small $B$. This region expands as $\alpha$ is decreased until there is inverse catalysis for all $B$.
It is not clear from the point of view of the field theory what dictates whether the system exhibits catalysis or inverse catalysis. We will refrain from speculating on the exact mechanism here and leave this question to future work.
\begin{appendix}
\subsection*{Acknowledgments}
This work is supported in part by the Natural Sciences and Engineering Research
Council of Canada. The authors would like to thank the following individuals for fruitful discussion related to this work: Per Kraus, Thomas Levi, Hamid Omid, Gordon Semenoff, and Mark Van Raamsdonk.
\end{appendix}
|
1411.3362
|
\section{Introduction}
When considering the suprema of real-valued functions, it is often relevant
to know whether this supremum coincides with the function obtained by taking
the supremum of the real values at each point. Here we propose a natural
generalization of this notion to the pointfree setting. We first define
pointwise suprema in $\mathcal{R}L$, where the classical definition can be
naturally articulated. We then show that this concept is actually
independent of the representation of a particular $\mathbf{W}$-object as a
subobject of $\mathcal{R}L$. For it is precisely the pointwise suprema in
\mathcal{R}L$ which are preserved by every $\mathbf{W}$-morphism out of
\mathcal{R}L$. We take advantage of this unexpected information by adopting
the latter attribute as the final, purely algebraic definition of pointwise
supremum: an element $g \in G$ is the pointwise join of a subset $K
\subseteq G^+$ iff $\theta(g) = \bigvee \theta [K]$ for all $\mathbf{W}
-morphisms $\theta $ out of $G$.
The notion of pointfree pointwise suprema has several useful applications.
For example, a convex $\ell$ subgroup $K$ of a
$\mathbf{W}$-object $G$ is a $\mathbf{W}$-kernel iff it is pointwise closed,
i.e., iff $K$ contains any pointwise join of a subset of $K$ which exists in $G$.
And all existing (countable) suprema in $\mathcal{R}L$ are pointwise iff
$L$ is boolean (a $P$-frame).
This leads directly to Nakano-Stone type theorems. One of our main results is
Theorem \ref{Thm:3}: $\mathcal{R}L$ is conditionally pointwise complete
($\sigma $-complete) iff $L$ is boolean (a $P$-frame).
Unconditional pointwise completeness requires that certain unbounded subsets
of a given $\mathbf{W}$-object $G$ have pointwise suprema in $G$, but, of
course, not all subsets can have suprema in $G$. The most permissive
criterion for a subset to have a pointwise supremum in $G$ is that the
subset have a supremum in some extension of $G$. We adopt this criterion as
our definition of unconditional pointwise completeness in Section~\ref{Sec:6},
and then show that the $\mathbf{W}$-objects which enjoy this attribute are
precisely those of the form $\mathcal{R}L$ for $L$ a Boolean frame, and
those which enjoy the corresponding unconditional $\sigma$-completeness are
those of the form $\mathcal{R}L$ for $L$ a $P$-frame. Finally, we show that the
pointwise complete ($\sigma$-complete) objects form a full bireflective subcategory
of $\mathbf{W}$.
The paper is organized as follows. After a preliminary Section \ref{Sec:3},
we briefly outline Madden's pointfree representation for $\mathbf{W}$ in
Section \ref{Sec:2}. We define pointwise suprema in Section \ref{Sec:4},
first in $\mathcal{R}L$ in Subsection \ref{Subsec:5} and then in $\mathbf{W}$
in Subsection \ref{Subsec:6}. In Section \ref{Sec:8} we give several different
applications of the notion of pointwise suprema. We show that the
$\mathbf{W}$-kernels of a particular $\mathbf{W}$-object are precisely
the pointwise closed convex $\ell$-subgroups. We show that all existing
(countable) suprema in $\mathcal{R}L$ are pointwise iff $L$ is
boolean (a $P$-frame). And we show that any element of a $\mathbf{W}$-object
is the pointwise join of its truncates, we characterize the sequences that
can be realized as the truncates of an element of an extension, and we
show that every such sequence has a pointwise supremum in $\mathcal{RM}G$.
(Notation to be introduced subsequently.)
Section \ref{Sec:5} is devoted to conditional pointwise completeness; the
main result here is the pointfree Nakano-Stone Theorem for conditional
pointwise completeness, Theorem \ref{Thm:3}. This result makes heavy use of
the pointfree generalization of the classical theorem, a beautiful result of
Banaschewski and Hong (\cite{BanaschewskiHong:2003}) which appears here in
embellished form as Theorem \ref{Thm:1}.
Section \ref{Sec:6} takes up unconditional pointwise completeness. A reveiw
of the well known facts concerning essential extensions constitutes
Subsection \ref{Subsec:7}, and a review of the less well known facts
concerning cuts occupies Subsection \ref{Subsec:8}. The section culminates
in Subsection \ref{Subsec:9}, in which we summarize our findings as they
pertain to unconditional pointwise completeness in Theorem \ref{Thm:4}.
\section{Preliminaries\label{Sec:3}}
Our notation and terminology is conventional for the most part, save only
for our notation for downsets. A subset $K$ of a poset $G$ is a
\emph{downset} if $g \leq k \in K$ implies $g \in K$. We write
\begin{equation*}
\downset{K}_G \equiv \left\{ g \in G : g \leq k
\text{ for some $k \in K$}\right\}
\end{equation*}
for the downset in $G$ generated by a subset $K \subseteq G$. We drop the
subscript whenever it is unambiguous to do so. Upsets are defined and
denoted dually.
For a $\mathbf{W}$-object $G$, we denote by $\mathbb{R}^+(G)$
the set of those positive real numbers for which the corresponding
constant function is present in $G$. Thus to say that
$\bigwedge \mathbb{R}^+(G) = 0$ is to say that
$G$ contains arbitrarily small positive multiples of $1$. This is a weakening of
the condition of being divisible which plays a prominent role in our
results.
Good general references are \cite{AndersonFeil:1987} and \cite{Darnel:1994}
for $\ell$-groups, \cite{GillmanJerison:1960} for $\mathcal{C}X$, \cit
{HagerRobertson:1977} for an introduction to $\mathbf{W}$, \cit
{MaddenVermeer:1986} and \cite{BallHager:1991} for the pointfree, or Madden
representation for $\mathbf{W}$, \cite{Johnstone:1982} and \cit
{PicadoPultr:2012} for general frame theory, and the many papers of Bernhard
Banaschewski, the tireless fount of knowledge of pointfree topology.
In spite of our use of the localic terminology in the abstract and
introduction, we prefer the algebraic language of frames and frame
morphisms. Henceforth, $\mathcal{R}L$ stands for the $\mathbf{W}$-object of
frame maps $g : \mathcal{O}\mathbb{R} \to L$, where $\mathcal{O}\mathbb{R}$
is the frame of open subsets of the real numbers $\mathbb{R}$ and $L$ is a
frame, assumed completely regular unless otherwise explicitly stipulated.
\section{A brief synopsis of the Madden representation\label{Sec:2}}
We mention here some of the technical results, familiarity with which will
be assumed in the sequel. The reader may skip this section upon a first
reading, returning to it as necessary.
\subsection{Calculation in $\mathcal{R}L$}
The arithmetic operations on $\mathbb{R}$ beget corresponding operations on
\mathcal{R}L$ as follows. We write $\vec{f}$ for $(f_1,f_2,\dots,f_n) \in
\mathcal{R}L)^n$, $\vec{U}$ for $(U_1,U_2,\dots,U_n) \in (\mathcal{O}\mathbb
R})^n$. For a continuous function $w:\mathbb{R}^{n}\rightarrow\mathbb{R}$ we
write $w( \vec{U}) \subseteq U$ to mean $U_{1}\times U_{2}\times\cdots\times
U_{n}\subseteq w^{-1}\left(U\right) $.
\begin{theorem}[{\protect\cite[3.1.1]{BallWalters:2002}}]\label{Thm:2}
The canonical lifting
$w^{\prime}:\left( \mathcal{R}L\right)^{n}\rightarrow\mathcal{R}L$
of a continuous function $w:\mathbb{R}^{n}\rightarrow\mathbb{R}$
is given by the formula
\begin{equation*}
w^{\prime}( \vec{f}) \left( U\right)
= \bigvee\limits_{w( \vec{U}) \subseteq U}
\bigwedge\limits_{1\leq i\leq n} f_{i}\left( U_{i}\right),
\quad f_{i}\in \mathcal{R}L,\quad U\in\mathcal{O}\mathbb{R}.
\end{equation*}
The $U_{i}$'s in the supremum range over $\mathcal{O}\mathbb{R}$, and may be
taken to be rational intervals.
\end{theorem}
The formula also applies to constant functions; the frame map lifted from
the constant function $x\longmapsto r$ is given b
\begin{equation*}
r\left( U\right) =\left\{
\begin{array}{cc}
\top & \text{if }r\in U \\
\bot & \text{if }r\notin
\end{array}
\right. .
\end{equation*}
Theorem \ref{Thm:2} provides a ready proof of a special case of Weinberg\rq{
s Theorem (\cite{Weinberg:1963}). A term is an expression built up from
variables and constants using the operations $+$, $-$, $\vee$, and $\wedge$.
An identity is an equation with terms on either side. Weinberg\rq{}s Theorem
asserts that an equation holds in $\mathbb{R}$ iff it holds in every abelian
$\ell$-group.
\begin{corollary}
\label{Cor:2}Any identity which holds in $\mathbb{R}$ also holds in any
\mathcal{R}L$.
\end{corollary}
\begin{proof}
The terms on either side of the identity determine two functions $w_{i}
\mathbb{R}^{n}\rightarrow\mathbb{R}$, and these functions coincide because
the identity holds in $\mathbb{R}$. Therefore the liftings $w_{i}^{\prime}$
of these functions to $\mathcal{R}L$ coincide by Theorem \ref{Thm:2}.
\end{proof}
\subsection{A few useful formulas}
We record here a small number of formulas which will be especially useful in
what follows. They may be derived using from Theorem \ref{Thm:2} or even
Corollary \ref{Cor:2}. Details can be found in the literature by following
the references.
Lemma \ref{Lem:3} implies that a frame map $f:\mathcal{O}\mathbb{R
\rightarrow L$ is completely determined by its values on the right rays. For
$f$ is clearly determined by its values on the base for $\mathcal{O}\mathbb{
}$ consisting of the open intervals $\left( r,s\right) $, $r<s$, and
f\left( r,s\right) =f\left( -\infty,s\right) \wedge f\left( r,\infty\right)
, and the left ray $f(-\infty,s)$ can be expressed in terms of the right
rays using the pseudocomplementation operator in the frame:
\begin{equation*}
a^* \equiv \bigvee_{a \wedge b = \bot} b.
\end{equation*}
\begin{lemma}[{\protect\cite[3.1.1]{BallHager:1991}}]\label{Lem:3}
For any $f,g\in\mathcal{R}L$ and $r\in\mathbb{R}$,
\begin{enumerate}
\item
$f\left( -\infty,r\right)
=\bigvee\limits_{s<r}f\left( s,\infty\right) ^{\ast}$,
\item
$f\leq g$ iff $f\left( r,\infty\right) \leq g\left( r.\infty\right) $
for all $r\in\mathbb{R}$ iff $f\left( -\infty,r\right)
\geq g\left(-\infty,r\right) $ for all $r\in\mathbb{R}$.
\end{enumerate}
\end{lemma}
Lemma \ref{Lem:1} gives necessary and sufficient conditions for a function
on right rays to be extended to a frame map.
\begin{lemma}[{\protect\cite[3.1.2]{BallHager:1991}}]
\label{Lem:1} A function $f:\left\{ \left(r,\infty\right) :r\in\mathbb{R
\right\} \rightarrow L$ can be extended to an element of $\mathcal{R}L$ iff
it satisfies the following conditions for all $r,s\in\mathbb{R}$. The
extension is unique when it exists.
\begin{enumerate}
\item $f\left( s,\infty\right) \prec f\left( r,\infty\right) $ whenever $r<s
.
\item $f\left( r,\infty\right) =\bigvee_{s>r}f\left( s,\infty\right) $.
\item $\bigvee_{r}f\left( r,\infty\right) =\bigvee_{r}f\left(
r,\infty\right) ^{\ast}=\top$.
\end{enumerate}
\end{lemma}
We provide a proof of Corollary \ref{Cor:1} in order to illustrate the use
of Theorem \ref{Thm:2} in calculations. This sort of reasoning will get
heavy use in what follows. For $f \in \mathcal{R}L$, the cozero element of $f
$ is
\begin{equation*}
\coz f \equiv f(\mathbb{R}\smallsetminus \{0\}) = |f|(0,\infty).
\end{equation*}
\begin{corollary}\label{Cor:1}
For $f,g\in\mathcal{R}L$ and $c,r\in\mathbb{R}$,
\begin{enumerate}
\item $\left( f-c\right) \left( r,\infty\right) =f\left( c+r,\infty \right)
.
\item $\coz f^{+} =\left( f\vee0\right) \left( \mathbb{R}\smallsetminus\lef
\{ 0\right\} \right) =f\left( 0,\infty\right) $.
\item $\coz \left( f-c\right) ^{+} =\left( f-c\right) \left( 0,\infty\right)
=f\left( c,\infty\right) $.
\item $\left( f\wedge g\right) \left( r,\infty\right) =f\left(
r,\infty\right) \wedge g\left( r,\infty\right) $.
\item For $f,g\geq0$, $\bigvee_{\mathbb{N}}\coz \left( nf-g\right) ^{+}
\coz f$.
\end{enumerate}
\end{corollary}
\begin{proof}
To prove (1), consider $U\in\mathcal{O}\mathbb{R}$. Then by Theorem \re
{Thm:2} we have
\begin{equation*}
\left( f-c\right) \left( r,\infty\right) = \hspace{-10pt
\bigvee_{U_{1}-U_{2}\subseteq\left( r,\infty\right) } \hspace{-10pt}\left(
f\left( U_{1}\right) \wedge c\left( U_{2}\right) \right) .
\end{equation*}
But if $U_{1}-U_{2}\subseteq\left( r,\infty\right) $ then $U_{1}$ is bounded
below and $U_{2}$ is bounced above, say $U_{1}\subseteq\left( t,\infty
\right) $ and $U_{2}\subseteq\left( -\infty,t-r\right) $ for some $t\i
\mathbb{R}$. And if, in addition, and $c\left( U_{2}\right) >\bot$ then
c\in U_{2}$, hence $t>c+r$. That is to say that
\begin{equation*}
\left( f-c\right) \left( r,\infty\right) = \bigvee_{t>c+r}
f\left(t,\infty\right) =f\left( c+r,\infty\right) .
\end{equation*}
The proof of (2) is similar to the proof of (1), and (3) follows from (1)
and (2). To prove (4), again consider $U\in O\mathbb{R}$.
\begin{equation*}
\left( f\wedge g\right) \left( r,\infty\right) = \hspace{-10pt
\bigvee_{U_{1}\wedge U_{2}\subseteq\left( r,\infty\right) } \hspace{-10pt
\left( f\left( U_{1}\right) \wedge g\left( U_{2}\right) \right) .
\end{equation*}
But $U_{1}\wedge U_{2}\subseteq\left( r,\infty\right) $ iff
U_{1}\subseteq\left( r,\infty\right) $ and $U_{2}\subseteq\left( r,\infty
\right) $. Hence$\left( f\wedge g\right) \left( r,\infty\right) =f\left(
r,\infty\right) \wedge g\left( r,\infty\right) $.
To verify (5), note that
\begin{equation*}
\coz \left( nf-g\right) ^{+} =\left( nf-g\right) \left(0,\infty\right) =
\hspace{-10pt}\bigvee_{nU_{1}-U_{2}\subseteq\left( 0,\infty\right)} \hspace
-10pt}\left( f\left( U_{1}\right) \wedge g\left( U_{2}\right) \right) .
\end{equation*}
But if $nU_{1}-U_{2}\subseteq\left( 0,\infty\right) $ then
U_{1}\subseteq\left( r,\infty\right) $ and $U_{2}\subseteq\left( -\infty
,nr\right) $ for some $r\in\mathbb{R}$, so that
\begin{align*}
\bigvee_{\mathbb{N}}\coz \left( nf-g\right) ^{+} &= \bigvee_{\mathbb{N}}
\bigvee_{\mathbb{R}} \left( f\left(r,\infty\right) \wedge g\left(
-\infty,nr\right) \right) = \bigvee_{\mathbb{R}} \bigvee_{\mathbb{N}} \left(
f\left( r,\infty\right)\wedge g\left(-\infty,nr\right) \right) \\
&=\bigvee_{\mathbb{R}}\left( f\left( r,\infty\right) \wedge \bigvee_{\mathbb
N}}g\left( -\infty,nr\right) \right) = \bigvee_{\mathbb{R}}f\left(
r,\infty\right) = \coz f. \qedhere
\end{align*}
\end{proof}
\subsection{The frame of $\mathbf{W}$-kernels of $A$\label{Subsec:2}}
Most of the calculation takes place in the frame of $\mathbf{W}$-kernels of
G$. The basic facts concerning this frame are well known; we briefly review
them here to fix notation.
\begin{lemma}\label{Lem:4}
Let $K$ be a convex $\ell$-subgroup of $G$.
\begin{enumerate}
\item $G/K$ is archimedean iff
\begin{equation*}
\left( \forall~ n\in\mathbb{N~}\left( \left( nf-g\right) ^{+} \in K\right)
\Longrightarrow f\in K\right), \qquad f,g \in G^+.
\end{equation*}
\item
$K$ is a $\mathbf{W}$-kernel if, in addition,
\[
g\wedge 1\in K \implies g\in K, \qquad g \in G^+.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
(1) We have
\begin{equation*}
\left( nf-g\right) ^{+} \in K \iff K+\left( nf-g\right) ^{+}
=K\iff K+nf\vee b=K+g\iff K+nf\leq K+g.
\end{equation*}
This makes it clear that the condition displayed in (1) is equivalent to the
archimedean property of the quotient $G/K$.
(2) This is evidently a reformulation of the requirement that $K+u$ should
function as a weak unit of the quotient, i.e., that $\left( K+g\right)
\wedge \left( K+u\right) =0$ imply $K+g=0$.
\end{proof}
\begin{corollary}\label{Cor:4}
Suppose $G$ is bounded. Then a convex $\ell$-subgroup $K$
is a proper $\mathbf{W}$-kernel iff
\begin{enumerate}
\item
$
\forall~n \in \mathbb{N}~((nf - 1)^+ \in K) \implies f \in K,\ f \in G^+,$
and
\item
$1 \notin K$.
\end{enumerate}
In particular, $[g] = \{h : \forall~n~\exists~m~(n|h| - 1)^+ \leq mg\}$,
$g \in G^+$.
\end{corollary}
\begin{proof}
It is straightforward to show that in condition (1) of Lemma \ref{Lem:4},
the element $g$ may be chosen to be $1$ if $G$ is bounded. What we must
also demonstrate is that condition (2) above implies condition (2) of Lemma
\ref{Lem:4}. Given $g \in G^+$, find a positive integer $n$ such that
$g \leq n$. Then $g \wedge 1 \in K$ implies $ng \wedge n \in K$
because $K$ is a group, hence $g \wedge n \in K$ because $K$ is convex,
with the result that $g \in K$.
\end{proof}
Since $\mathbf{W}$ is closed under products,
the intersection of an
arbitrary family of $\mathbf{W}$-kernels is itself a $\mathbf{W}$-kernel. We
denote the $\mathbf{W}$-kernel generated by a subset $S\subseteq G$ by
\begin{equation*}
\left[ S\right] \equiv\bigcap\left\{ K:K\text{ is a }\mathbf{W}\text{-kernel
and }S\subseteq K\right\} .
\end{equation*}
\begin{definition}
The frame of $\mathbf{W}$-kernels of $G$ is called the \emph{Madden frame of
$G$}; we denote it by $\mathcal{M}G$.
\end{definition}
\begin{lemma}
$\mathcal{M}G$ forms a regular Lindel\"{o}f frame under the inclusion order.
Its operations are
\begin{equation*}
K_{1}\wedge K_{2} =K_{1}\cap K_{2}\text{ \ and\ \ }
\bigvee_{I}K_{i}
= \left[K_i : i \in I \right]
= \left[\bigcup_{I}K_{i}\right] .
\end{equation*}
\end{lemma}
\begin{proof}
\cite[3.2.2, 3.25]{BallHager:1991}.
\end{proof}
\subsection{The Madden representation for $\mathbf{W}$}
\label{Subsec:1}
Let $G$ be a $\mathbf{W}$-object with $L \equiv \mathcal{M}G$ its frame of
\mathbf{W}$-kernels. For each $g \in G$ and $r \in \mathbb{R}$, define
\begin{equation*}
\widehat{g}(r,\infty) \equiv \left[ \left( g - r \right)^+\right].
\end{equation*}
Thus defined, $\widehat{g}$ satisfies the requirements of Lemma \ref{Lem:1},
and thus extends to a unique frame map $\mathcal{O}\mathbb{R} \to L$, which
we also denote $\widehat{g}$. We write $\widehat{G}$ for $\{\widehat{g} : g
\in G\} $, and $\mu_G : G \to \widehat{G}$ for the mapping $g \mapsto
\widehat{g}$.
We say that a $\mathbf{W}$-morphism $\theta : H \to \mathcal{R}M$
is \emph{cozero dense} if
\begin{equation*}
a = \bigvee_{\substack{ h \in H \\ \coz \theta(h) \leq a}} \coz \theta(h),
\qquad a \in M.
\end{equation*}
Note that it is enough for this condition to hold for each $a \in \coz M$
because $M$ is assumed to be completely regular.
\begin{theorem}[\protect\cite{MaddenVermeer:1986}]\label{Thm:6}
Let $G$, $L$, $\widehat{G}$, and $\mu _{G}$ have the meaning above.
\begin{enumerate}
\item Then $\mu_G$ is a cozero dense $\mathbf{W}$-injection,
and its range restriction $G \to \widehat{G}$ is a
$\mathbf{W}$-isomorphism.
\item $L$, $\widehat{G}$, and $\mu_G$ are unique up to isomorphism with
respect to their properties in (1).
\item For any frame $M$ and $\mathbf{W}$-morphism $\theta$ there is a unique
frame map $k$ making the diagram commute.
\begin{figure}[h]
\setlength{\unitlength}{4pt}
\par
\begin{center}
\begin{picture}(36,12)(3,1)
\small
\put(0,12){\makebox(0,0){$G$}}
\put(12,12){\makebox(0,0){$\mathcal{R}L$}}
\put(12,0){\makebox(0,0){$\mathcal{R} M$}}
\put(24,12){\makebox(0,0){$L$}}
\put(24,0){\makebox(0,0){$M$}}
\put(36,12){\makebox(0,0){$\mathcal{O}\mathbb{R}$}}
\put(2,12){\vector(1,0){7.5}}
\put(2,10){\vector(1,-1){8}}
\put(12,10){\vector(0,-1){8}}
\put(24,10){\vector(0,-1){8}}
\put(33,12){\vector(-1,0){7}}
\put(34,10){\vector(-1,-1){8}}
\put(5,4.5){\makebox(0,0){$\theta$}}
\put(6,13.5){\makebox(0,0){$\mu_{G}$}}
\put(14.5,6){\makebox(0,0){$\mathcal{R} k$}}
\put(22.5,6){\makebox(0,0){$k$}}
\put(30,13.75){\makebox(0,0){$\widehat{g}$}}
\put(32.5,4.5){\makebox(0,0){$\theta(g)$}}
\end{picture}
\end{center}
\end{figure}
\item
$k$ is surjective iff $\theta$ is cozero dense, and $k$ is one-one iff, for all
$K \subseteq G^+$, $\bigvee_K \coz \theta(g) = \top $ in $M$ implies
$\bigvee_K \coz \widehat{g} = \top$ in $L$.
\end{enumerate}
\end{theorem}
\begin{proof}
A detailed proof may be found in \cite{BallHager:1991}; we comment only
on part (4). For any $\mathbf{W}$-kernel $K \subseteq G$,
\begin{align*}
k(K) &= k\left(\bigvee_{K^+}[g]\right)
= k\left(\bigvee_{K^+}\coz\widehat{g}\right)
= k\left(\bigvee_{K^+}\widehat{g}(0,\infty)\right)
= \bigvee_{K^+}k\circ \widehat{g}(0,\infty)
= \bigvee_{K^+}\theta(g)(0,\infty) \\
&= \bigvee_{K^+}\coz\theta(g).
\end{align*}
This makes the surjectivity condition clear; the injectivity condition follows from
the fact that a frame morphism between regular frames is one-one
iff it is codense, i.e., iff the only element taken to the top of the codomain
is the top element of the domain.
\end{proof}
\section{Pointwise suprema defined}\label{Sec:4}
In dealing with continuous real-valued functions on a Tychonoff space $X$,
it is often important to know whether a given function $f$ is the supremum
of a given subset $K\subseteq \mathcal{C}X$, and, if so, whether this
supremum is pointwise, i.e., whether $\bigvee_{K}k\left( x\right) =f\left(
x\right) $ for all $x\in X$. In terms of the frame $\mathcal{O}X $ of open
sets of $X$, $f$ is the pointwise supremum of $K$ iff $\bigcup_{K}k^{-1
\left( r,\infty\right) =f\left( r,\infty\right) $ for all $r\in\mathbb{R}$.
It is the latter formulation which generalizes directly to the pointfree
setting.
\subsection{Pointwise suprema in $\mathcal{R}L$\label{Subsec:5}}
\begin{definition}
Let $L$ be a frame, and let $K$ be a subset and $f$ an element of $\mathcal{
}L$. We say that $f$ is the \emph{pointwise supremum (infimum) of $K$}, and
write $f = \bigvee^\bullet K$ ($f = \bigwedge^\bullet K$), provided that
f\left(r,\infty\right) = \bigvee_{K} k \left(r,\infty\right)$ (
f\left(-\infty, r\right) = \bigvee_{K} k \left(-\infty,r\right)$ ) holds in
L$ for all $r\in\mathbb{R}$.
\end{definition}
\begin{remarks}
\label{Rem:1}A few remarks about this definition are in order.
\begin{enumerate}
\item Observe that the frame definition coincides with the spatial
definition in case $L$ is the topology of a Tychonoff space.
\item Recall that by Lemma \ref{Lem:3} a frame map is completely determined
by its values on the right or left rays alone. This makes the appearance of
only the rays in this definition less mysterious.
\item Recall that by Lemma \ref{Lem:3} an element $g\in\mathcal{R}L$ lies
above (below) each $k\in K$ iff $k\left( r,\infty\right) \leq
g\left(r,\infty\right) $ ($g\left( -\infty,r\right) \leq k\left( -\infty
,r\right) $) for all $r\in\mathbb{R}$ and all $k \in K$.
\item It follows from the preceding remarks that $f=\bigvee K$ whenever $f =
\bigvee^\bullet K$, and dually.
\item It follows from the preceding remarks that if $f=\bigvee^{\bullet }K=g$
then $f=g$.
\end{enumerate}
\end{remarks}
We list some of the nice properties of pointwise suprema and infima .
\begin{proposition}
\label{Prop:8}Let $F$ and $K$ be subsets and let $f_{0}$ and $k_{0}$ be
elements of $\mathcal{R}L$.
\begin{enumerate}
\item $f_{0} = \bigvee^\bullet F $ iff $-f_{0} = \bigwedge^\bullet \left(
-F\right) \equiv \bigwedge_{F}^\bullet \left( -f \right)$, and dually.
\item $f_{0}=\bigvee^\bullet \left\{ f_{0}\right\} = \bigwedge^\bullet
\left\{f_0\right\}$.
\item If $f_{0} = \bigvee^\bullet F$ and $k_{0} = \bigvee^\bullet K$ then
f_{0}\boxdot k_{0} = \bigvee_{F,K}^\bullet \left( f\boxdot k\right) $, where
$\boxdot$ stands for one of the $\ell$-group operations $+$, $\vee$, or
\wedge$.
\item If $f_{0} = \bigvee^\bullet F$ and $0\leq r\in\mathbb{R}$ then $rf_{0}
= \bigvee_{F}^\bullet rf$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) follows from the fact that $\left( -f\right) \left(
-\infty,r\right)=f\left( -r,\infty\right) $ for any $f\in \mathcal{R}L$ and
r\in\mathbb{R}$, as can be readily checked with the aid of Theorem \re
{Thm:2}. (2) is trivial. To prove part (3) for the $+$ operation, first
observe that for $r\in\mathbb{R}$,
\begin{gather*}
\bigvee_{F,K}\left( f+k\right) \left( r,\infty\right) =
\bigvee_{F,K}\bigvee_{ U_{1}+U_{2}\subseteq\left( r,\infty\right)}
\left(f\left( U_{1}\right) \wedge k\left( U_{2}\right) \right) .
\end{gather*}
But if $U_{1}+U_{2}\subseteq\left( r,\infty\right) $ then both $U_{i}$'s are
bounded below, say $U_{1}\subseteq\left( s,\infty\right) $ and
U_{2}\subseteq\left( r-s,\infty\right) $ for some $s\in\mathbb{R}$.
Therefore this join works out to
\begin{align*}
\bigvee_{F,K}\bigvee_s\left(f(s,\infty) \wedge k(r - s, \infty)\right) &=
\bigvee_s\bigvee_{F,K}\left(f(s,\infty) \wedge k(r - s, \infty)\right) =
\bigvee_s \left(f_0(s, \infty) \wedge k_0(r - s, \infty)\right) \\
&= \left(f_0 + k_0 \right)(r, \infty).
\end{align*}
The proofs of (3) for the join and meet operations are similar. Finally, to
verify (4) simply note that if $r>0$ then $\left( rf\right)
\left(s,\infty\right) = f\left( s/r,\infty\right) $ for $f\in \mathcal{R}L$
and $s\in\mathbb{R}$, as may be easily seen using Theorem \ref{Thm:2}.
\end{proof}
\subsection{Pointwise suprema in $\mathbf{W}$\label{Subsec:6}}
Having formulated the notion of pointwise supremum in $\mathcal{R}L$, let us
now generalize it to abstract $\mathbf{W}$-objects.
\begin{definition}[First definition of pointwise supremum in $\mathbf{W}$]
For $F\subseteq G\in\mathbf{W}$ and $f_{0}\in G$, we shall say that \emph{
f_{0}$ is the pointwise supremum (infimum) of $F$}, and write $f_{0}
=\bigvee^\bullet F$ ($f_0 = \bigwedge^\bullet F$), if the corresponding
statement holds in $\widehat{G}$, i.e., if $\widehat{f_{0}} =
\bigvee_{F}^\bullet \widehat{f}$ ($\widehat{f_{0}} = \bigwedge_{F}^\bullet
\widehat{f}$).
\end{definition}
Pointwise suprema can be characterized concretely
by use of the details of the Madden representation (see Subsection
\ref{Subsec:1}).
\begin{proposition}\label{Prop:1}
Let $F$ be a subset and $f_{0}$ an element of a $\mathbf{W}$-object $G$.
Then
\begin{align*}
f_{0} & = {\bigvee\nolimits}^\bullet F \iff \forall~r \in \mathbb{R}~\left
\left[ \left(f-r\right)^{+} :f\in F\right] = \left[ \left( f_{0}-r\right)
^{+} \right] \right), \\
f_{0} & = {\bigwedge\nolimits}^\bullet F \iff \forall~r \in \mathbb{R}~\left
\left[ \left(r-f\right)^{+} :f\in F\right] = \left[ \left( r-f_{0}\right)
^{+} \right]\right).
\end{align*}
\end{proposition}
\begin{proof}
In the Madden representation $G\rightarrow\widehat{G}$,
\[
\widehat{f}\left(r,\infty\right) =\left[ \left( f-r\right) ^{+} \right]
\quad \text{and} \quad
\widehat{f}\left( -\infty,r\right) =\left[ \left( r-f\right)^{+} \right]. \qedhere
\]
\end{proof}
$\mathbf{W}$-morphisms preserve pointwise suprema.
\begin{proposition}\label{Prop:7}
If $\theta:G\rightarrow H$ is a $\mathbf{W}$-morphism and if
f_{0} = \bigvee^\bullet F$ in $G$ then $\theta\left( f_{0}\right) = \bigvee
_{F}^\bullet \theta\left( f\right) $ in $H$.
\end{proposition}
\begin{proof}
Identify $G$ and $H$ with their Madden representations in $\mathcal{R}L$ and
$\mathcal{R}M$, where $L \equiv \mathcal{M}G$ and $M \equiv \mathcal{M} H$.
Then there is a unique frame map $\mathcal{M}\theta\equiv k:L\rightarrow M$
which realizes $\theta$ in the sense that $\theta\left(g\right) \left(
U\right) =k \circ g\left( U\right) $ for all $U\in\mathcal{O}\mathbb{R}$ and
$g\in G$. Therefore we have, for $r\in\mathbb{R}$
\begin{align*}
\bigvee_{F}\theta\left( f\right) \left( r,\infty\right) &= \bigvee_{F}k
\circ f\left( r,\infty\right) = k \left( \bigvee_{F}f\left(
r,\infty\right)\right) = k \circ f_{0}\left(r,\infty\right) = \theta\left(
f_{0}\right) \left(r,\infty\right) . \qedhere
\end{align*}
\end{proof}
It is a surprising fact that the converse of Proposition \ref{Prop:7} holds
as well. In general, the supremum of a subset $F$ of a $\mathbf{W}$-object
G $ depends on the context. If, for instance, $G$ is a subobject of $H$, it
may well happen that $f_{0}=\bigvee F$ for some $f_{0}\in G$ but
f_{0}\neq\bigvee F$ in $H$. The point of Proposition \ref{Prop:9} is that it
is precisely the pointwise suprema which are context free.
\begin{proposition}
\label{Prop:9}Let $F$ be a subset and $f_{0}$ an element in some
$\mathbf{W}$-object $G$. Then $f_{0} = \bigvee^\bullet F$ iff $\theta\left( f_{0}\right)
= \bigvee_{F}\theta\left( f\right) $ for every $\mathbf{W}$-morphism $\theta$
out of $G$, and dually.
\end{proposition}
\begin{proof}
Proposition \ref{Prop:7} is the forward implication of this equivalence. So
suppose $f_{0}$ is not the pointwise supremum of $F$, let $L$ be the Madden
frame of $G$, and identify $G$ with its Madden representation
$\widehat{G}\leq\mathcal{R}L$. We may assume that $f_{0}=0$,
since otherwise we may
replace $F$ by $F-f_{0}\equiv\left\{ f-f_{0}:f\in F\right\} $ by
Proposition~\ref{Prop:8}. We must find a $\mathbf{W}$-morphism
$\theta:G\rightarrow H$ such that $\bigvee _{F}\theta\left( f\right) \neq0$.
Let $i:L\rightarrow M$ be a frame embedding of $L$ into a boolean frame $M$.
(Such an embedding exists; see \cite[II, 2.6]{Johnstone:1982}.) Since
$\bigvee^\bullet F \neq0$ there exists some
$r\in\mathbb{R}$ such that
\begin{equation*}
a\equiv{\bigvee\nolimits}_{F}f\left( r,\infty\right) <0\left(
r,\infty\right) =\left\{
\begin{array}{ll}
\bot & \text{if }r\geq0 \\
\top & \text{if }r<
\end{array}
\right. .
\end{equation*}
Note that $0\left( r,\infty\right) $ must be $\top$, hence $r<0$.
Now $i\left( a\right) $ has complement $b$ in $M$; note that
$b>\bot$ because $i\left( a\right) <\top$ since $a<\top$ and $i$
is one-one.
Let $k:M\rightarrow{}\downarrow\!b$ designate the open quotient frame
map $c\longmapsto c\wedge b$, $c\in M$, let
$H\equiv\mathcal{R}\left({\downarrow b}\right) $, and let
$\psi\equiv\mathcal{R}(k\circ i):\mathcal{R}L\to H$.
We claim that the desired map $\theta$ is the restriction of $\psi$ to
$\widehat{G} \approx G$. For if $f\in F$ then
\begin{equation*}
\psi\left( f\right) \left( r,\infty\right) = k \circ i \circ f \left(
r,\infty\right) = i \circ f\left( r,\infty\right) \wedge b=\bot
\end{equation*}
since $i \circ f \left( r,\infty\right) \leq i\left( a\right) $ and
$i\left(a\right) \wedge b=\bot$. It follows that for $s\in\mathbb{R}$,
\begin{equation*}
\psi\left( f\right) \left( s,\infty\right)
\leq\left( r/2\right) \left(s,\infty\right)
=\begin{cases}
\bot & \text{if $s\geq r/2$} \\
\top & \text{if $s<r/2$} \end{cases},
\end{equation*}
which implies by Lemma \ref{Lem:3}(2) that $\psi\left( f\right) \leq r/2<0$
for all $f\in F$, meaning that $\bigvee_{F}\psi\left( f\right) \neq0$. This
completes the proof.
\end{proof}
Some caution is required when dealing with pointwise suprema. If $F\subseteq
G$ and $f_{0}\in G$ are such that $f_{0}=\bigvee^{\bullet }F$ in some
\mathbf{W}$-extension $H\geq G$ then $f_{0}=\bigvee F$ in $G$, of course,
but the join may not be pointwise in $G$.
\begin{example}
Let $X$ be $\omega+1$, the one-point compactification of the discrete space
of finite ordinals. Let $G$ be $\mathcal{C}X$ and let
\begin{equation*}
H\equiv\left\{ g+rh_{0}:g\in G,~r\in\mathbb{R}\right\} ,
\end{equation*}
where $h_{0}\equiv\left( n\longmapsto n\right) $ and $h_{0}\left(\omeg
\right) =\infty$. $H$ is a $\mathbf{W}$-object in $DX$. Let $F$ be the
family of functions
\begin{equation*}
f_{n}\left( k\right) \equiv
\begin{cases}
1 & \text{if $k\leq n$} \\
0 & \text{if $k>n$
\end{cases}
\qquad n<\omega.
\end{equation*}
Then it is not hard to check that $\bigvee^\bullet F=1$ in $H$ and $\bigvee
F=1$ in $G$ but the latter join is not pointwise.
\end{example}
For emphasis, we recast the definition of pointwise supremum in an arbitrary
$\mathbf{W}$-object.
\begin{definition}[Second definition of pointwise supremum in $\mathbf{W}$]
For $F\subseteq G\in\mathbf{W}$ and $f_{0}\in G$, we shall say that
\emph{$f_{0}$ is the pointwise supremum (infimum) of $F$}, and write
$f_{0}=\bigvee^\bullet F$ ($f_0 = \bigwedge^\bullet F$), if
$\bigvee\theta[F]) = \theta(f_0)$ ($\bigwedge \theta(F) = \theta(f_0)$)
for all $\mathbf{W}$-homomorphisms $\theta : G \to H$.
\end{definition}
\section{Pointwise suprema applied}\label{Sec:8}
In this section we aim to show that pointwise suprema are useful for
characterizing important attributes of a $\mathbf{W}$-object and its
Madden frame. We begin by using them to characterize
those $\mathbf{W}$-objects
in which every (countable) supremum is pointwise.
Throughout this section $G$ will represent a $\mathbf{W}$-object
with Madden frame $L$.
\subsection{When all existing (countable) suprema are pointwise}
Pointwise suprema are useful for detecting whether the Madden frame of a given
$\mathbf{W}$-object is boolean or a $P$-frame. We shall require this information
in Section \ref{Sec:6}.
\begin{theorem}\label{Thm:7}
Suppose that $\bigwedge \mathbb{R}^+(G) = 0$.
Then all existing (countable) suprema in $G$ are pointwise
iff $L$ is boolean (a $P$-frame).
\end{theorem}
\begin{proof}
We prove this theorem in the boolean case; the same proof,
mutatis mutandis, works in the $P$-frame case.
Assume $L$ is boolean, suppose $f$ is an element and $K$ is a subset of
$G$ such that $f = \bigvee K$ in $G$, and assume for the sake of argument that
$f(r, \infty) >\bigvee_K k(r, \infty) \equiv b$ for some real number $r$.
Since $\bigvee_{s> r}f(s, \infty) = f(r, \infty)$, there is some $s > r$
for which $f(s,\infty) \nleq b$. Because $L$ is boolean
$a \equiv f(s, \infty) \wedge b^* > \bot$;
define the \enquote*{characteristic function}
\begin{equation*}
\chi (t,\infty) \equiv
\begin{cases}
\top \text{ if } t < 0, \\
a \text{ if } 0 \leq t <s - r, \\
\bot \text{ if } t \geq 1
\end{cases}
\qquad t \in \mathbb{R},
\end{equation*}
and check that $\chi$ satisfies the hypotheses of Lemma \ref{Lem:1} and so
extends to a unique member of $\mathcal{R}L$, and that, moreover, $\chi > 0$.
We claim that $f - k \geq \chi$ for all $k \in K$. To verify the claim first note
that by Theorem \ref{Thm:2}
$(f - k)(t,\infty) = \bigvee_{U_i}(f(U_1) \wedge k(U_2))$,
where the join ranges over open subsets
$U_i \subseteq \mathbb{R}$ such that $U_1 - U_2 \subseteq (0,\infty)$.
This condition implies that $U_1$ is bounded below and $U_2$ is bounded above,
say $U_1 \subseteq (u,\infty)$ and $U_2 \subseteq (-\infty,u-t)$. Therefore
\begin{equation*}
(f - k)(t, \infty) = \bigvee_u (f(u,\infty) \wedge k(-\infty, u - t)),
\end{equation*}
If $t < 0$ then, since $f \geq k$ implies $f(-\infty, u-t) \leq k(-\infty, u - t)$,
we get for any choice of $u \in \mathbb{R}$ that
\begin{equation*}
(f - k)(t,\infty) \geq f(u,\infty) \wedge k(-\infty, u - t) \geq f(u,\infty)
\wedge f(-\infty, u - t) = \top.
\end{equation*}
If $0 \leq t < s - r$ then $s - t > r$, hence $k(-\infty, s - t) \geq k(r,
\infty)^*$ since
\begin{equation*}
k(-\infty, s - t) \vee k(r, \infty) = k\left((-\infty, s - t) \cup(r,
\infty) \right) = \top.
\end{equation*}
This is relevant because $b^* = \left( \bigvee_K k(r,\infty)\right)^* =
\bigwedge_K k(r,\infty)^*$ as a result of the fact that $L$ is a complete
boolean algebra, so that.
\begin{align*}
(f - k)(t,\infty) &\geq f(s,\infty) \wedge k(-\infty, s - t) \geq f(s,
\infty) \wedge k(r, \infty)^* \geq f(s, \infty) \wedge \bigwedge_L k(r,
\infty)^* \\
&= f(s, \infty) \wedge b^* = a = \chi(t,\infty).
\end{align*}
This proves the claim, which implies that $f > f - \chi \geq K$. Since
$\mu_G : G \to \mathcal{R}L$ is cozero dense, there eists $0 < g \in G$
such that $\coz g \leq a$, and, by meeting $g$ with the appropriate constant
function $r \in \mathbb{R}^+(G)$ we may assume that $g \leq \chi$. In sum,
we have $f > f-g \geq K$, a violation of the assumption that
$f = \bigvee K$ in $G$. Our only recourse is to conclude that
$f(r,\infty) = \bigvee_K k(r,\infty)$ for all $r \in \mathbb{R}$, i.e.,
$f = \bigvee^\bullet K$.
Now suppose that all
existing suprema in $G$ are pointwise;
we aim to show that an arbitrary element $a \in L \equiv \mathcal{R}L$
is complemented. For that purpose define subsets
\[
U \equiv \{g \in G : \coz g \leq a \text{ and $0\leq g \leq 1$}\},
\quad\text{and}\quad
V \equiv \{g \in G : \coz g \leq a^* \text{ and $0 \leq g \leq 1$}\}.
\]
By suitably augmenting $U$ we may assume that $0 \leq k \leq g \in U$
implies $k \in U$, and that $g \in U$ implies $ng \wedge 1 \in U$
for all $n$, and similarly for $V$.
We claim that $\bigvee (U\cup V) = 1$. If not then there exists some
$k \in G$ such that $g \leq k < 1$ for all $g \in U \cup V$.
This means that $1 - k > 0$, hence $b \equiv \coz(1 - k) > \bot$. Since
$\bigvee_{U \cup V}\coz g$ is a dense element of $L$, $b$
meets either $\bigvee_U \coz g$ or $\bigvee_V \coz g$ nontrivially,
say $0 < g \in U$ is such that $\coz g \leq b$. Since
$\coz g = g(0,\infty) = \bigvee_m g(\frac{1}{m},\infty)$,
there exists some positive integer $m$ for which
$\bot < g(\frac{1}{m},\infty) = \coz (g - \frac{1}{m})^+
= \coz (mg -1)^+$. But then
\begin{align*}
\coz(mg \wedge 1 -k)^+ &= \coz((mg - k)^+\wedge (1 - k))
=\coz(mg - k)^+ \wedge \coz(1 - k) \\
&= \coz(mg -k)^+ \wedge b
\geq \coz (mg - 1)^+ \wedge b = \coz (mg - 1)^+ > \bot.
\end{align*}
This is a contradiction, since $g \in U$ implies $mg \wedge 1 \in U$, hence
$mg \wedge 1 \leq k$. A similar argument covers the case in which
there exists some $0 < g \in V$ such that $\coz g \leq a^*$, and the two
cases together prove the claim.
Let $\bigvee_U \coz h \equiv u$ and $\bigvee_V \coz h \equiv v$.
Then because $1 = \bigvee^\bullet(U \cup V)$ we have
\[
\top = 1(0,\infty) = \bigvee_{U \cup V}g(0,\infty)
= \bigvee_{U \cup V}\coz g = \bigvee_U\coz g \cup \bigvee_V\coz g
= u \vee v.
\]
Since $u \leq a$ and $v \leq a^*$ by construction, we see that $u$ and
$v$ are complementary, and that $u = a$.
\end{proof}
\subsection{Truncate sequences}
\label{Subsec:15}
The following fact plays an important role in our analysis of unconditional
pointwise completeness in Section \ref{Sec:6}.
\begin{proposition}\label{Prop:3}
For any $f\in G$, $\bigvee _{\mathbb{N}}^\bullet \left( f\wedge n\right) = f$.
\end{proposition}
\begin{proof}
Identify $G$ with $\widehat{G} \leq \mathcal{R}L$.
For any $r\in\mathbb{R}$ we have by Corollary \ref{Cor:1}(4) that
\begin{equation*}
\left( f\wedge n\right) \left( r,\infty\right) =f\left( r,\infty\right)
\wedge n\left( r,\infty\right) =\left\{
\begin{array}{ll}
\bot & \text{if }r\geq n \\
f\left( r,\infty\right) & \text{if }r<
\end{array}
\right. .
\end{equation*}
Hence $\bigvee_{\mathbb{N}}\left( f\wedge n\right) \left( r,\infty\right) =
f\left( r,\infty\right) $ for all $r \in \mathbb{R}$.
\end{proof}
Proposition \ref{Prop:3} raises an important question: which sequences in
$G$ are sequences of truncates of a member of $\mathcal{R}L$, so
called truncate sequences?
\begin{proposition}\label{Prop:12}
Let $\{g_n\} \subseteq G^+$ be the sequence of truncates of
$h \in\mathcal{R}L^+$, i.e., $\widehat{g}_n = h \wedge n $ for all $n$. Then
\begin{enumerate}
\item $g_{n + 1} \wedge n = g_n$ in $G$, and
\item $\bigvee_n \widehat{g}_n(-\infty,n) = \top$ in $L$.
\end{enumerate}
Conversely, any sequence in $G$ having these two properties is
the sequence of truncates of some $h \in \mathcal{R}L$.
\end{proposition}
\begin{proof}
If $g_n = h \wedge n$ for all $n$ then (1) obviously holds, and
\begin{align*}
\bigvee_n g_n(-\infty,n)
&= \bigvee_n (h \wedge n)(-\infty, n)
= \bigvee_n h(-\infty,n)
= h\left( \bigvee_n (-\infty,n) \right) \\
&= h(-\infty, \infty) = \top.
\end{align*}
Suppose now that $\{ g_n \}$ is a sequence in $\mathcal{R}L^+$ satisfying
(1) and (2). Put $h(-\infty, r) \equiv g_n(-\infty,r)$ for any $n >r$.
This definition is independent of the choice of $n$ by (1). We must show
that $h$ satisfies the properties in the (up-down dual of)
Lemma \ref{Lem:1}. It is clear that $h$ satisfies the first of these properties,
namely that $h(-\infty, s) \prec h(-\infty,r)$ whenever $r < s$, because
it reduces to the same property of $g_n$ for sufficiently large $n$, and
$h$ satisfies the second property for similar reasons. Since
$\bigvee_r h(-\infty,r) = \bigvee_n h(-\infty,n) = \bigvee_n g_n(-\infty,n)$,
$h$ also satisfies half of the third property. But $h$ also satisfies the
other half because
$\bigvee_r h(-\infty,r)^* = \bigvee_r g_1(-\infty,r)^*= \top$.
\end{proof}
\begin{definition}[Truncate sequence]
We shall refer to a sequence $\{g_n\} \subseteq G$ satisfying
Proposition \ref{Prop:12} as a \emph{truncate sequence}.
\end{definition}
\begin{corollary}
\label{Cor:8}
Every truncate sequence in $G$ has a pointwise join in $\mathcal{R}L$.
\end{corollary}
In Section 10 of \cite{Hager:2013}, the second author conducted an analysis
of a construct which is closely related to truncate sequences, but stronger.
His \enquote*{expanding sequences} have the first property of truncate
sequences but satisfy $\bigcap_n (u_{n + 1} - u_n)^{\bot\bot} = 0$ instead
of the second property. The possession of a supremum for every such sequence
turns out to be equivalent to the property of being *-maximum, or *-max for
short. A $\mathbf{W}$-object is *-max if it contains a copy of every other
$\mathbf{W}$-object with the same bounded part. This interesting attribute is
not the same as requiring the truncate sequences to have joins, for it
implies, inter alia, that the classical Yosida space of $G$ be a quasi-$F$
space. As is evident from Corollary \ref{Cor:8}, no such restriction applies
to the $\mathbf{W}$-objects in which the truncate sequences have joins.
We confess ignorance of the many questions that arise naturally here,
postponing an investigation for the time being. But surely the first
question is unavoidable, as it is motivated by the characterization of the
divisible *-max $\mathbf{W}$-objects as being precisely those in which every
expanding sequence has a join (\cite[Section 10]{Hager:2013}).
See Theorem \ref{Thm:9} below for further discussion of these topics.
\begin{question}
Which $\mathbf{W}$-objects $G$ have the feature that every truncate
sequence in $G$ has a pointwise join in $G$?
\end{question}
\subsection{Pointwise closure and $\mathbf{W}$-kernels}
$\mathbf{W}$-kernels are characterized by the property of
being closed under pointwise joins. A convex $\ell$-subgroup $K \leq G$
is said to by \emph{pointwise closed} if $K_0 \subseteq K^+$ and
$\bigvee^\bullet K_0 = g$ imply $g \in K$.
\begin{proposition}
A convex $\ell$-subgroup $K$ of a $\mathbf{W}$-object $G$ is a
$\mathbf{W}$-kernel iff it is pointwise closed.
\end{proposition}
\begin{proof}
Suppose $K$ is a $\mathbf{W}$-kernel with subset $K_0$ such that
$\bigvee^\bullet K_0 = g$. According to Proposition \ref{Prop:1} we
are supposing that $[(k - r)^+ : k \in K_0]
= [(g - r)^+]$ for all $r \in \mathbb{R}$.
In particular, for $r = 0$ this says that $[K_0] = [g]$, i.e.,
any $\mathbf{W}$-kernel containing $K_0$ must also contain $g$.
But one such $\mathbf{W}$-kernel is $K$, hence $g \in K$ and $K$
is pointwise closed.
Now suppose that $K$ is a pointwise closed convex $\ell$-subgroup of $G$;
we must show that $K$ has properties (1) and (2) of Lemma \ref{Lem:4}.
To check (2), suppose that $g \wedge 1 \in K$ for some $g \in G$. Then
for each positive integer $n$ we would have $ng \wedge n \in K$
because $K$ is a group, hence $g \wedge n \in K$ because $K$ is convex,
with the result that $g \in K$ by Proposition \ref{Prop:3}.
To check property
(1) consider $a,b \in G^+$ such that $(na - b)^+ \in K$ for all $n$.
We claim that $\bigvee^\bullet_n ((na - b)^+ \wedge a) = a$.
What we will actually prove is that
\[\bigvee\nolimits^\bullet_n (((n - 1)a - b) \vee (-a)) \wedge 0) = 0,
\]
the result of subtracting $a$ from the equation claimed. Since
$0(r,\infty) = \top$ for $r < 0$ and $\bot$ otherwise, this amounts to
showing that $\bigvee_n((n - 1)a - b) \vee (-a)) (r,\infty) = \top$
for $r < 0$. According to Theorem \ref{Thm:2}, it is sufficient to
demonstrate that $\bigvee_n\bigvee_{U,V}(a(U) \wedge b(V)) = \top$,
where the inner join ranges over open
subsets $U,V \subseteq \mathbb{R}$ for which
\[
((n - 1)U - V) \vee (-U) \subseteq (r,\infty).
\]
But if $U$ and $V$ are to satisfy this containment then $U$ must be
bounded both above and below, say $U \subseteq (u,w)$, and $V$
must be bounded above, say $V \subseteq (-\infty,v)$, where
$(n-1)u > r + v$ or $-w > r$. In sum, we must show that, for $r < 0$,
\begin{align*}
\top &=
\bigvee_n\left(\bigvee_{u < w < -r}\bigvee_v(a(u,w) \wedge b(-\infty,v)) \vee
\bigvee_v\bigvee_{w>u>\frac{r + v}{n - 1}}
(a(u,w) \wedge b(-\infty,v))\right) \\
&= \bigvee_n\left(\bigvee_{u < w < -r}\left(a(u,w)
\wedge \bigvee_v b(-\infty,v)\right) \vee
\bigvee_v \left(\bigvee_{w>u>\frac{r + v}{n - 1}} a(u,w)
\wedge b(-\infty,v)\right)\right) \\
&= \bigvee_n\left(\bigvee_{u < w < -r}a(u,w) \vee
\bigvee_v \left(a(\frac{r + v}{n - 1},\infty)
\wedge b(-\infty,v)\right)\right) \\
&= a(-\infty,-r) \vee\bigvee_v\bigvee_n\left(a(\frac{r + v}{n - 1},\infty)
\wedge b(-\infty,v)\right)
\end{align*}
But for $v > -r$ we have
$\bigvee_n\left(a(\frac{r + v}{n - 1},\infty)
\wedge b(-\infty,v)\right) = a(0,\infty) \wedge b(-\infty,v)$, so that
the last join displayed reduces to
$a(-\infty,-r) \vee a(0,\infty) = a(-\infty,\infty) = \top$, thereby proving the
claim and the proposition.
\end{proof}
\section{Conditional pointwise completeness\label{Sec:5}}
The classical Nakano-Stone Theorem asserts that every bounded (countable)
subset of $\mathcal{C}X$ has a supremum in $\mathcal{C}X$ iff $X$ is
extremally disconnected (basically disconnected). In this section we prove
the corresponding result for pointwise suprema, Theorem \ref{Thm:3}. Our
analysis will be closely intertwined with the pointfree version of the
classical theorem, a result of Banaschewski and Hong \cit
{BanaschewskiHong:2003}.
\subsection{The Banaschewski-Hong Theorem}
We begin with an observation.
\begin{proposition}\label{Prop:16}
A conditionally pointwise complete ($\sigma$-complete)
\mathbf{W}$-object is conditionally complete ($\sigma$-complete).
\end{proposition}
\begin{proof}
This follows from Remark \ref{Rem:1}(4).
\end{proof}
The converse of Proposition \ref{Prop:16} does not hold, even for
$\mathbf{W}$ objects of the form $\mathcal{C} X $, $X$ a Tychonoff space.
In this case $\mathcal{C} X $ is conditionally $\sigma$-complete iff
$X$ is basically disconnected; this is the classical Nakano-Stone Theorem.
On the other hand, if $X$ is compact and basically disconnected then
$G \equiv \mathcal{C}X$ is conditionally $\sigma$-complete. But
$L \equiv \mathcal{M}G = \mathcal{O}X$ is a $P$-frame iff $X$
is a $P$ space, and a compact $P$-space is finite.
The point is that, by Theorem \ref{Thm:3}, $G$ is not conditionally
pointwise $\sigma$-complete unless $X$ is finite. See also
\cite[4N]{GillmanJerison:1960}.
Theorem \ref{Thm:1} is a modestly embellished version of the pointfree
Nakano-Stone Theorem, a beautiful result of Banaschewski and Hong \cit
{BanaschewskiHong:2003}. We prove Theorem \ref{Thm:1} in some detail, not
just because we need the result but also because the proofs provide the
basis for the corresponding result for pointwise completeness in Subsection
\ref{Subsec:4}.
Let us review the basic definitions: a frame is said to be extremally
disconnected (basically disconnected) provided that $a^{\ast}\vee
a^{\ast\ast}=\top$ for all $a\in L$ ($a\in\coz L$). And
$\mathbb{R}^{+} \left( G\right) $ stands for the set of positive real
numbers such that the corresponding constant function lies in $G$.
\begin{theorem}
\label{Thm:1}Let $G$ be a $\mathbf{W}$-object with Madden frame $L$. Then
conditions (1) and (2) together are equivalent to conditions (3) and (4).
\begin{enumerate}
\item
$\bigwedge \mathbb{R}^{+} \left( G\right) = 0$, i.e., $G$ contains
arbitrarily small positive multiples of $1$.
\item
$G^{\ast }$ is conditionally complete ($\sigma $-complete).
\item
$L$ is extremally disconnected (basically disconnected).
\item
The Madden representation carries $G^*$ onto $\mathcal{R}^*L$.
\end{enumerate}
\end{theorem}
\begin{proof}
Since it is most relevant to our purposes, we prove the version of this
theorem having to do with the conditional $\sigma $-completeness of $G^{\ast
}$ versus the basic disconnectivity of $L$. The implication from (3) and (4)
to (1) and (2) is provided by the result of Banaschewski and Hong
(\cite[Prop. 2]{BanaschewskiHong:2003}), since they prove that if
$L $ is basically disconnected then $\mathcal{R}L$, and hence
$\mathcal{R}^{\ast }L$, is conditionally $\sigma $-complete.
The opposite implication is
provided by Propositions \ref{Prop:4} and \ref{Prop:2}. The running
assumptions throughout are that $L$ is the Madden frame of $G$, and that $G$
has been identified with its Madden representation in $\mathcal{R}L$, i.e.,
$G$ is a $\mathbf{W}$ subobject of $\mathcal{R}L$.
\end{proof}
\begin{proposition}\label{Prop:4}
If $\bigwedge \mathbb{R}^+(G) = 0$ and $G^{\ast }$ is conditionally
$\sigma $-complete then $L$ is basically disconnected.
\end{proposition}
\begin{proof}
Consider a cozero element $a\in L$, say $a=f\left( 0,\infty\right) $ for
f\in C^{+} L$. By replacing $f$ by $f\wedge1$, we may assume that $f\in
\mathcal{R}^{\ast}L=G^{\ast}$. Define the sequence $\left\{ g_{n}\right\} $
in $G^{\ast}$ by setting
\begin{equation*}
g_{n}\equiv nf\wedge1,\;n\in\mathbb{N},
\end{equation*}
and let $g\in G^{\ast}$ be such that $g=\bigvee_{\mathbb{N}}g_{n}$.
We aim to show that $g$ is a component of $1$, i.e., that $\left( 1-g\right)
\wedge g=0$, by means of several claims. We first claim that $\left(
1-g\right) \wedge\left( nf-1\right) ^{+} =0$ for all $n$. Fo
\begin{equation*}
g={\bigvee\nolimits}_{\mathbb{N}}g_{n}\Longrightarrow1-g =
\bigwedge\nolimits}_{\mathbb{N}}\left(1-g_{n}\right) = {\bigwedge\nolimits}_
\mathbb{N}}\left( 1-nf\right) ^{+} ,
\end{equation*}
and, since $\left( 1-nf\right) ^{+} \wedge\left( nf-1\right) ^{+} =0$,
\begin{equation*}
\left( 1-g\right) \wedge\left( nf-1\right) ^{+} \leq\left( 1-nf\right) ^{+}
\wedge\left( nf-1\right) ^{+} =0.
\end{equation*}
We next claim that $\left( 1-g\right) \wedge f=0$. For if not, then
x\equiv\left( 1-g\right) \wedge f\wedge1>0$. Since $G$ is archimedean, there
exists $k\in\mathbb{N}$ such that $kx\nleq1$; let $k$ be the least such
integer. Then
\begin{equation*}
0<\left( kx-1\right) ^{+} \leq\left( kf-1\right) ^{+} \Longrightarrow\left(
kx-1\right) ^{+} \wedge\left( 1-g\right) =0.
\end{equation*}
Bu
\begin{equation*}
\left( k-1\right) x\leq1\Longrightarrow\left( kx-1\right) ^{+} \leq
x\leq\left( 1-g\right) ,
\end{equation*}
a contradiction. It now follows that $\left( 1-g\right) \wedge nf=0$ for all
$n$, hence $\left( 1-g\right) \wedge g_{n}=0$ for all $n$. Upon recalling a
basic fact about $\ell$-groups, namely that if $g=\bigvee_{\mathbb{N}}g_{n}$
for $\left\{ g_{n}\right\} \subseteq G^{+} $ and if $a\wedge g_{n}=0$ for
all $n$ then $a\wedge g=0$, we reach the desired conclusion: $g\wedge\left(
1-g\right) =0$.
The basic disconnectivity of $L$ follows immediately from the fact that $g$
is a component of $1$, for
\begin{equation*}
g\wedge\left( 1-g\right) =0\Longrightarrow g\vee\left( 1-g\right) =g+\left(
1-g\right) =1,
\end{equation*}
hence $\coz g\vee\coz \left( 1-g\right) =\coz 1=\top$, which is to say that
a^{\ast\ast}\vee a^{\ast }=\top$.
\end{proof}
The proof of Theorem \ref{Thm:1} is completed by Proposition \ref{Prop:2},
which requires two simple lemmas, the first of which is folklore.
We say that an $\ell$-subgroup
$H \leq G$ is \emph{order dense} in $G$ if for every $0 < g \in G$ there is some
$h \in H$ such that $0 < h \leq g$
\begin{lemma}\label{Lem:7}
Suppose $G$ is an order dense $\ell$-subgroup of $H$.
\begin{enumerate}
\item
Then suprema and infima in $G$ and $H$ agree, and
\item
$h = \bigvee \downset{h}_G$ for all $h \in H^+$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Suppose $\bigvee A = g_0$ for some $A \subseteq G^+$ and
$g_0 \in G^+$,
but that $A \leq h < g_0$ for some $h \in H$. Find $g_1 \in G$ such that
$0 < g_1 \leq g_0 - h$. Since $g_0 - g_1 < g_0$ there is some $g \in A$
for which $g \nleq g_0 - g_1$. But this flies in the face of the fact that
$g + g_1 \leq h + g_1 \leq g_0$. We conclude that $\bigvee A = g_0$ in $H$.
(2) Given $h_0 \in H^+$, let $A \equiv \downset{h_0}_{G^+}$
and suppose for
the sake of argument that $A \leq h_1 < h_0$ for some $h_1 \in H^+$.
Then find $g_0 \in G$ such
that $0 < g_0 \leq h_0 - h_1$. But for any $g \in A$ we have
$g + g_0 \leq h_1 + g_0 \leq h_0$, i.e., $A + g_0 \subseteq A$.
It follows that $ng_0 \in A$ for all $n$, which is to say that
$ng_0 \leq h_0$ for all $n$, a violation of the archimedean property
of $H$.
\end{proof}
\begin{lemma}
For $h,k \in \mathcal{R}^+L$ and $0 \leq q \in \mathbb{Q}$,
if $\coz k \leq h(q, \infty)$ and
$k \leq q$ then $k \leq h$.
\end{lemma}
\begin{proof}
By Lemma \ref{Lem:3}(2) it is sufficient to show that
for any $r \in \mathbb{R}$,
\[
h(r,\infty) \geq k(r,\infty)
= (k \wedge q)(r,\infty)
= k(r,\infty) \wedge q(r,\infty)
=\begin{cases} k(r,\infty) & \text{if $r < q$}\\
\bot &\text{if $r \geq q$}\end{cases}.
\]
But this is clear, for if $0 \leq r < q$ then
$k(r,\infty) \leq k(0,\infty)
= \coz k \leq h(q,\infty) \leq h(r,\infty)$.\qedhere
\end{proof}
We remind the reader that a cozero element $a$ of a Lindel\"{o}f frame $L$
is Lindel\"{o}f, i.e., $a = \bigvee A$ implies $a = \bigvee A_0$ for some
countable subset $A_0 \subseteq A$.
\begin{lemma}\label{Lem:5}
If $\bigwedge \mathbb{R}^+(G) = 0$ then every element of $\mathcal{R}L$
is the join of a countable subset of $\widehat{G}^*$.
\end{lemma}
\begin{proof}
Given $0 < h \in \mathcal{R}^+L$ and $q \in \mathbb{R}^+(G)$,
$h(q,\infty) = \coz(h - q)^+ $ is a cozero element of $L$ and is therefore
Lindel\"{o}f. Since $\mu_G : G \to \mathcal{R}L$ is cozero dense, this
element is the join of those of the form $\coz \widehat{g}$, $g \in G^+$.
Let $G_q$ be a countable subset of $G^+$ such that
$h(q,\infty) = \bigvee_{G_q} \coz \widehat{g}$.
By suitably restricting and augmenting
$G_q$, we may assume that $g \leq q$ for all $g \in G_q$, and that
$g \in G_q$ implies $mg \wedge q \in G_q$ for all integers $m$.
Finally, let $G_0 \equiv \bigcup_{0<q\in R}G_q$ for some countable
dense subset $R \subseteq \mathbb{R}^+(G)$.
We claim that $h = \bigvee G_0$. If not then $G_0 \leq k < h$ for some
$k \in \mathbb{R}L$, so that by Lemma \ref{Lem:3}(2) there is some
$q \in R$ such that $k(q,\infty) < h(q,\infty)$. More is true; there must
be some $s > q$ in $R$ for which $a \equiv k(-\infty,s) \wedge h(s,\infty) > \bot$,
for otherwise $h(s,\infty) \leq k(-\infty,s)^*$ for all $s > q$ would imply
\[
h(q,\infty) =
\bigvee_{q < s}h(s,\infty) \leq \bigvee_{q < s}k(-\infty,s)^*
= k(q,\infty),
\]
contrary to assumption. Now $h(s,\infty) = \bigvee_{G_s} \coz \widehat{g}$,
so we may find $g \in G_s$ such that $\coz g \wedge a > \bot$.
Since $\coz g = \bigvee_m\coz(mg - k)^+$ by Corollary \ref{Cor:1}(5),
there exists some integer $m$ for which $\coz(mg - k)^+ \wedge a > \bot$.
In sum, we have arranged that
\begin{align*}
\coz(mg\wedge s -k)^+
&= \coz ((mg - k)^+\wedge (s - k)^+)
= \coz (mg - k)^+\wedge \coz(s - k)^+ \\
&= \coz (mg - k)^+ \wedge k(-\infty,s) > \bot.
\end{align*}
But $g \in G_s$ implies $mg\wedge s \in G_s$, hence $mg \wedge s \leq k$,
contrary to the information displayed above. This completes the proof of the claim
and the lemma.
\end{proof}
\begin{proposition}\label{Prop:2}
If $\bigwedge \mathbb{R}^+(G) = 0$ and $G^*$ is conditionally
$\sigma$-complete then $\widehat{G}^* = \mathcal{R}^*L$.
\end{proposition}
\begin{proof}
Given $0 < h \in \mathcal{R}^*L$, we know from Lemma \ref{Lem:5}
that $h= \bigvee A$ for some countable subset
$A \subseteq \downset{h}_{\widehat{G}}$.
Since $h$ is bounded by some multiple of $1$, so
is $A$. By virtue of the conditional $\sigma$-completeness of $G$,
$A$ has a supremum $g_0$ in $G$. Finally, $\widehat{g}_0 = h$ by
Lemmas \ref{Lem:7} and \ref{Lem:5}.
\end{proof}
\begin{corollary}
\label{Cor:3}A conditionally $\sigma $-complete $\mathbf{W}$-object which
contains arbitrarily small positive multiples of $1$ contains all real
multiples of $1$. That is, $\bigwedge\mathbb{R}^+(G) = 0$
implies $\mathbb{R}^{+}\left( G\right) =\mathbb{R}^{+}$.
\end{corollary}
The proof of Theorem \ref{Thm:1} is complete.
It is worthwhile to restate Theorem \ref{Thm:1} in the language of regular
$\sigma$-frames. This is always possible, since the fact that $L$ is
Lindel\"{o}f means that $L$ is isomorphic to $\mathcal{H}\coz L$.
\begin{theorem}\label{Thm:8}
Let $G$ be a $\mathbf{W}$-object with Madden frame $L$. Then conditions (1)
and (2) together are equivalent to conditions (3) and (4).
\begin{enumerate}
\item
$\bigwedge \mathbb{R}^+(G) = 0$ .
\item
$G^{\ast }$ is conditionally complete ($\sigma $-complete).
\item
$L$ is extremally disconnected (basically disconnected).
\item[(3')] $L$ is isomorphic to $\mathcal{H}A$ for some regular $\sigma
-frame $A$ such that for all $C\subseteq A$ there exists a complemented
element $b\in A$ with
\begin{equation*}
\forall~d\in A~\left( \forall~c\in C~\left( d\wedge c=\bot\right)
\Longleftrightarrow d\leq b\right) .
\end{equation*}
($L$ is isomorphic to $\mathcal{H}A$ for some regular $\sigma$-frame $A$
such that for all $c\in A$ there exists a complemented element $b\in A$ with
\begin{equation*}
\forall~d\in A~\left( d\wedge c=\bot\Longleftrightarrow d\leq b\right) .)
\end{equation*}
\item[(4)] The Madden representation carries $G^{\ast}$ onto $\mathcal{R
^{\ast}L$.
\end{enumerate}
\end{theorem}
\subsection{$P$-frames and boolean frames\label{Subsec:4}}
Proposition \ref{Prop:16} holds that conditional pointwise completeness is
stronger than conditional completeness. In view of Theorem \ref{Thm:1},
then, the question naturally arises as to what condition on $\mathcal{M}G$
is equivalent to the conditional pointwise completeness of a
$\mathbf{W}$-object $G$. The answer is that $\mathcal{M}G$ must be
boolean in order for $G$ to be conditionally pointwise complete,
and $\mathcal{M}G$ must be a $P$-frame in order for $G$ to be conditionally
pointwise $\sigma $-complete.
\begin{theorem}
\label{Thm:3}Let $G$ be a $\mathbf{W}$-object with Madden frame $L$. Then
conditions (1) and (2) together are equivalent to conditions (3) and (4).
\begin{enumerate}
\item $G$ contains arbitrarily small positive multiples of $1$.
\item $G^{\ast }$ is conditionally pointwise complete (conditionally
pointwise $\sigma $-complete).
\item $L$ is boolean (a $P$-frame).
\item The Madden representation carries $G^{\ast}$ onto $\mathcal{R}^{\ast}L
.
\end{enumerate}
\end{theorem}
\begin{proof}
We first prove the countable version of this theorem. Suppose $L$ is a $P
-frame, identify $G$ with its Madden representation $\widehat{G}\leq\mathcal
R}L$, and suppose $G^{\ast}=\mathcal{R}^{\ast}L$. Then $G$ certainly
satisfies (1); in order to verify that $G^{\ast}$ is pointwise $\sigma
-complete, consider a countable subset $F\subseteq G$ with upper bound $g\in
G^{\ast}$. Define a function $f_{0}$ on the right rays by the rule
\begin{equation*}
f_{0}\left( r,\infty\right) \equiv{\bigvee\nolimits}_{F}f\left(
r,\infty\right), \quad r\in\mathbb{R}.
\end{equation*}
We claim that $f_{0}$ extends to a unique member of $\mathcal{R}L$, which
must then lie in $G^{\ast}$ by virtue of its convexity, since clearly $f\leq
f_{0}\leq g$ by Lemma \ref{Lem:3}(2). To establish this claim we need only
check the three hypotheses of Lemma \ref{Lem:1}. The first hypothesis
clearly holds, since a complemented element of any frame is rather below
itself. To verify the second, simply observe that
\begin{equation*}
\bigvee_{s>r}f_{0}\left( s,\infty\right) =\bigvee_{s>r}\bigvee_{F}f\left(
s,\infty\right) =\bigvee_{F}\bigvee_{s>r}f\left( s,\infty\right)
=\bigvee_{F}f\left( r,\infty\right) =f_{0}\left( r,\infty\right) .
\end{equation*}
To verify the third hypothesis, note that
\begin{equation*}
{\bigvee\nolimits}_{\mathbb{R}}f_{0}\left( r,\infty\right) =
\bigvee\nolimits}_{\mathbb{R}}{\bigvee\nolimits}_{F}f\left( r,\infty\right)
= {\bigvee\nolimits}_{F}{\bigvee\nolimits}_{\mathbb{R}}f\left(r,\inft
\right) = \top.
\end{equation*}
And, since $f_{0}\left( r,\infty\right) =\bigvee_{F}f\left( r,\infty \right)
\leq g\left( r,\infty\right) $ for all $r\in\mathbb{R}$,
\begin{equation*}
{\bigvee\nolimits}_{\mathbb{R}}f_{0}\left( r,\infty\right) ^{\ast}\geq
\bigvee\nolimits}_{\mathbb{R}}g\left( r,\infty\right)^{\ast} \geq
\bigvee\nolimits}_{\mathbb{R}}g\left(-\infty,r\right) =\top.
\end{equation*}
The second inequality holds because $g\left( -\infty,r\right) \wedge g\left(
r,\infty\right) =g\left( \emptyset\right) =\bot$.
Now suppose that $G$ satisfies (1) and (2), and again identify it with its
Madden representation. Then $G^{\ast}$ is $\sigma$-complete by Remark \re
{Rem:1}(5), so that Theorem \ref{Thm:1} allows us to conclude that $G^{\ast}
\mathcal{R}^{\ast}L$. To verify (3), consider a cozero element $a\in L$, say
$a=\coz f$ for some $f\in\mathcal{R}^{+} L$. By replacing $f$ by $f\wedge1$
if necessary, we may assume that $1\geq f\in G^{\ast}$. Define the sequence
\left\{ g_{n}\right\} $ in $G^{\ast}$ by setting
\begin{equation*}
g_{n}\equiv nf\wedge1,\;n\in\mathbb{N},
\end{equation*}
and let $g\in G^{\ast}$ be such that $g = \bigvee^\bullet _{\mathbb{N}}g_{n}
. Since the $g_{n}$'s here are defined exactly as in the proof of Theorem
\ref{Thm:1}, and since $g=\bigvee_{\mathbb{N}}g_{n}$, the argument given
there applies here, and shows that
\begin{equation*}
\coz g\vee\coz\left( 1-g\right) = \coz 1=\top.
\end{equation*}
But when we observe that $\coz g=g\left( 0,\infty\right) =a$ and $\coz\left(
1-g\right) =a^{\ast}$, we come to the desired conclusion: $a\vee
a^{\ast}=\top$.
It remains to prove the version of the theorem in which the pointwise joins
are of unrestricted cardinality. For the most part the argument goes along
the lines of the countable case. The only significant departure is the
implication from (1) and (2) to (3). So suppose $G$ contains arbitrarily
small positive multiples of $1$ and that $G^{\ast}$ is pointwise complete,
and consider an arbitrary element $a_{0}\in L$. Express $a_{0}$ in the form
\bigvee_{I}a_{i}$, where $\left\{ a_{i}:i\in I\right\} $ is the set of
cozero elements below $a_{0}$. For each $i\in I$ let $g_{i}$ be the
characteristic function of $a_{i}$, i.e.,
\begin{equation*}
g_{i}\left( U\right) =\left\{
\begin{array}{lll}
\bot & \text{if} & 0,1\notin U \\
a_{i} & \text{if} & 0\notin U\ni1 \\
a_{i}^{\ast} & \text{if} & 1\notin U\ni0 \\
\top & \text{if} & 0,1\in
\end{array}
\right. .
\end{equation*}
These functions lie between $0$ and $1$ and hence are in $G^{\ast}$. Let
g_{0}\equiv \bigvee^\bullet _{I}g_{i}$. By inspection one sees tha
\begin{equation*}
g_{0}\left( U\right) =\left\{
\begin{array}{lll}
\top & \text{if} & r<0 \\
a_{0} & \text{if} & 0\leq r<1 \\
\bot & \text{if} & 1\leq
\end{array}
\right. ,
\end{equation*}
the characteristic function of $a_{0}$. But since $a_{0}=g_{0}\left(
3/4,\infty\right) \prec g_{0}\left( 1/4,\infty\right) =a_{0}$, it follows
that $a_{0}$ is complemented. The shows that $L$ is boolean, and completes
the proof.
\end{proof}
A quotient of a $P$-frame need not be a $P$-frame (\cit
{BallWaltersZenk:2010}). However, a $C$-quotient of a $P$-frame is clearly a
$P$-frame, for a $C$-quotient $f:L\rightarrow M$ is coz-onto, meaning every
cozero element of $M$ is the image under $f$ of a cozero element of $L$.
Since the cozero elements of $L$ are complemented, so are their images. An
alternative argument can be made using Theorem \ref{Thm:3}.
\begin{corollary}
\label{Cor:6}A $C$-quotient of a $P$-frame is a $P$-frame.
\end{corollary}
\begin{proof}
A $C$-quotient map $f:L\rightarrow M$ induces a $\mathbf{W}$-surjection
\mathcal{R}f:\mathcal{R}L\rightarrow \mathcal{R}M$. If $L$ is a $P$-frame
then conditions (3) and (4) of Theorem \ref{Thm:3} hold, and therefore
conditions (1) and (2) are true of $\mathcal{R}L$. But the latter two
conditions are clearly inherited by any quotient of $\mathcal{R}L$, and
therefore are true of $\mathcal{R}M$. A second application of the theorem
gives the desired conclusion.
\end{proof}
\section{Unconditional pointwise completeness\label{Sec:6}}
In this section we define and analyze the ultimate, or unconditional form of
pointwise completeness. This naturally raises the question of precisely what
unconditional pointwise completeness ought to mean. Our definition comes in
Subsection \ref{Subsec:9}, but it requires a digression to review essential
extensions in Subsection \ref{Subsec:7} and cuts in Subsection \ref{Subsec:8
. The reader may wish to skip this material upon a first reading, returning
to it as necessary.
\subsection{Essential extensions and complete embeddings}
\label{Subsec:7} In this subsection we recall the basic facts concerning
essential extensions in $\mathbf{W}$. We do so not only because we will make
use of these facts in the sequel, but also for the reader\rq{}s convenience,
for these extensions appear in the literature under various names and with
various definitions. Because this material is well known (see, e.g., \cit
{BigardKeimelWolfenstein:1977}), we offer here only hints of proofs.
Recall that the \emph{booleanization} of a frame $M$ is the frame map
\begin{equation*}
b_M :M\rightarrow M^{\ast \ast }= (a\mapsto a^{\ast \ast }).
\end{equation*}
In spatial terms, this is the map which sends an open set to its
regularization, i.e., the smallest regular open subset containing it.
\begin{lemma}
\label{Lem:9}The following are equivalent for an extension $G\leq H$ in
\mathbf{W}$.
\begin{enumerate}
\item The embedding $G\rightarrow H$ is an essential monomorphism, i.e., any
morphism out of $H$ whose restriction to $G$ is one-one is also one-one on
H $.
\item Every nontrivial $\mathbf{W}$-kernel of $H$ meets $G$ nontrivially.
\item Every nontrivial polar of $H$ meets $G$ nontrivially.
\item $G$ is large in $H$, i.e., every nontrivial convex $\ell $-subgroup of
$H$ meets $G$ nontrivially.
\item If $H$ is divisible then these conditions are equivalent to $G$ being
order dense in $H$, i.e., for every $0<h\in H$ there exists some $0<g\in G$
such that $g\leq h$.
\item The frame map $f:L\equiv \mathcal{M}G\rightarrow \mathcal{M}H\equiv M$
which realizes the extension $G\leq H$ \enquote*{drops} to an isomorphism of
the booleanizations.
\begin{figure}[h]
\setlength{\unitlength}{4pt}
\par
\begin{center}
\begin{picture}(12,10)(0,2)
\small
\put(0,12){\makebox(0,0){$L$}}
\put(12,12){\makebox(0,0){$M$}}
\put(12,0){\makebox(0,0){$M^{**}$}}
\put(0,0){\makebox(0,0){$L^{**}$}}
\put(2,12){\vector(1,0){8}}
\put(0,10){\vector(0,-1){8}}
\put(12,10){\vector(0,-1){8}}
\put(2.5,0){\vector(1,0){6}}
\put(6,13.5){\makebox(0,0){$f$}}
\put(-2,6){\makebox(0,0){$b_L$}}
\put(14.5,6){\makebox(0,0){$b_M$}}
\end{picture}
\end{center}
\end{figure}
\end{enumerate}
\end{lemma}
\begin{proof}
The equivalence of (1) with (2) is evident, and the implication from (2) to
(3) is a consequence of the fact that polars are $\mathbf{W}$-kernels. To
show that (3) implies (4), one shows that $K^{\bot\bot} \cap G = 0$ for any
convex $\ell$-subgroup $K \leq H$ such that $K \cap G = 0$. And (4) clearly
implies (2).
Part (3) can be interpreted as saying that the extension is essential iff
the polars of $G$ and $H$ are in bijective correspondence by intersection.
But the polars of any $\mathbf{W}$-object are in bijective correspondence
with the elements of its booleanization via
\begin{eqnarray*}
a^{\ast \ast } &\longrightarrow &\left\{ g\in G:\coz\left\vert g\right\vert
\leq a^{\ast \ast }\right\} \\
\bigvee\nolimits_{P}\coz g &\longleftarrow &P
\end{eqnarray*}
This observation can be readily converted into a proof that (6) is
equivalent to the other conditions.
\end{proof}
From Lemma \ref{Lem:9}(6) we see that for any frame $L$ the dual $\mathcal{R
b_{L}$ of the booleanization map provides an essential extension $\mathcal{R
L\rightarrow \mathcal{R}L^{\ast \ast }$. It is this embedding which is meant
whenever we write $\mathcal{R}L\leq \mathcal{R}L^{\ast \ast }$.
We say that an object is \emph{essentially complete} if it has no proper
essential extensions. A \emph{maximal essential extension} of $G$ is an
essential extension $G \leq H$ such that $H$ is essentially complete.
\begin{proposition}
\label{Prop:17}
\begin{enumerate}
\item A $\mathbf{W}$-object is essentially complete iff it is of the form
\mathcal{R}L$ for a boolean frame $L$.
\item Every $\mathbf{W}$-object $G$ has a maximal essential extension,
namely $G \leq \mathcal{R}L \leq \mathcal{R}L^{**}$.
\item Any two maximal essential extensions of $G$ are isomorphic over $G$.
\item Let $G \leq H$ be a maximal essential extension. Then an arbitrary
extension $G \leq K$ is essential iff $K$ is isomorphic over $G$ to an $\ell
-subgroup of $H$
\end{enumerate}
\end{proposition}
\begin{proof}
(1) is a consequence of the fact that $G\leq \mathcal{R}L\leq \mathcal{R
L^{\ast \ast }$ is an essential extension which is an isomorphism iff $G
\mathcal{R}L=RL^{\ast \ast }$. (2) is Proposition 2.1 of \cit
{BanaschewskiHager:2013}. The rest is due to Conrad from his seminal article
\cite{Conrad:1970}.
\end{proof}
\cite{BanaschewskiHager:2013} provides an analysis of maximal essential
extensions in categories related to $\mathbf{W}$. See also \cit
{BanaschewskiHager:2013(2)} for a closely related analysis in the context of
completely regular frames.
Essential extensions take their importance here from the fact that any
extension may be \enquote*{reduced} to an essential extension by passage to
an appropriate quotient. This is Lemma \ref{Lem:10}, which involves an
attribute weaker than essentiality. Recall that an injective homomorphism
\tau :H\rightarrow K$ is said to be \emph{complete} if it preserves all
suprema and infima that exist in $H$. The following is folklore; see, e.g.,
\cite{Darnel:1994}.
\begin{lemma}
\label{Lem:15}An essential injection is complete.
\end{lemma}
\begin{proof}
Let $G\leq H$ be an essential extension, and let $\bigvee Z=g$ for some
subset $Z\subseteq G^{+}$ and element $g\in G^{+}$. If $G$ fails to be the
supremum of $Z$ in $H$, it is only because there is some $h\in H$ such that
Z\leq h<g$. Now the convex $\ell $-subgroup of $H$ generated by $g-h$ is
nontrival, and by Lemma \ref{Lem:9}(4) contains some $0<g^{\prime }\in G$,
say $g^{\prime }\leq n\left( g-h\right) $ for a positive integer $n$. This
rearranges to
\begin{equation*}
ng>ng-g^{\prime }\geq nh\geq nz\;\;\text{for all }z\in Z\text{.}
\end{equation*
But $\bigvee Z=g$ implies $\bigvee_{Z}nz=ng$ in $G$, and this contradicts
the displayed condition.
\end{proof}
\begin{lemma}
\label{Lem:10}For any injective homomorphism $\gamma :G\rightarrow H$ there
is a surjective homomorphism $\tau :H\rightarrow K$ such that $\tau \circ
\gamma $ is an essential injection. Moreover, $\tau $ may be chosen to be
complete.
\end{lemma}
\begin{proof}
We claim that the family $\mathcal{Q}$ of polars $Q$ such that $Q\cap \gamma
\lbrack G]=0$ contains maximal elements. For if $\mathcal{C}$ is a nonempty
chain in $\mathcal{Q}$ then, since $\gamma \lbrack G]\cap \bigcup \mathcal{C
=0$ and $\bigcup \mathcal{C}$ is convex, $\gamma \lbrack G]\subseteq
(\bigcup \mathcal{C})^{\bot }$, hence $\gamma \lbrack G]\cap (\bigcup
\mathcal{C})^{\bot \bot }=0$ and so $(\bigcup \mathcal{C})^{\bot \bot }\in
\mathcal{Q}$. If we take $\tau $ to be the quotient map $H\rightarrow H/R$
for some polar $R$ maximal in $\mathcal{Q}$ then it is clear that part (3)
of Lemma \ref{Lem:9} is satisfied by $\tau \circ \gamma $. And $\tau $ is
complete because $Q$ is order closed.
\end{proof}
\begin{lemma}
\label{Lem:14}If a subset $Z\subseteq G$ has a supremum in some extension
then it has a supremum in some essential extension.
\end{lemma}
\begin{proof}
Let $G\leq H\,$be an extension such that $\bigvee Z=h$ in $H$, and let $\tau
:H\rightarrow K$ be the complete surjection of Lemma \ref{Lem:10} such that
\tau $ is one-one on $Z$ and $K$ is an essential extension of $\tau \left[
\right] $. Then $\bigvee \tau \left[ Z\right] =\tau \left( h\right) $ in $K$
since $\tau $ is complete. Identifying $Z$ with its image under $\tau $
provides the desired extension.
\end{proof}
\subsection{Mobile downsets and cuts\label{Subsec:8}}
Many completion results are based on the technique of adjoining to $G$ a
supremum for each downset of a particular type. The downsets in play,
usually called \emph{cuts}, depend on the sort of completeness desired. The
broadest notion of a cut was introduced in Section 4 of \cite{Ball:1980}.
\begin{definition}
\label{Def:7}A downset $Z \subseteq G$ is called a \emph{cut} if it has a
supremum in some extension of $G$.
\end{definition}
Observe that by Lemma \ref{Lem:14}, a downset $Z \subseteq G^+$ is a cut iff
it has a supremum in some essential extension of $G$.
Definition \ref{Def:7} is sufficiently opaque as to appear useless, but its
utility is restored by Theorem \ref{Thm:5}, which gives a working criterion
for a subset $Z \subseteq G^+$ to be a cut. That criterion involves the
inability of the subset to remain stationary under addition by a positive
element.
\begin{definition}
A downset $Z\subseteq G$ is said to be \emph{mobile} if $Z+g\nsubseteq Z$
for all $0<g\in G$.
\end{definition}
\begin{observations}
\label{Obs:1} Let $Z$ be a downset in $G$.
\begin{enumerate}
\item $Z$ is mobile iff there is no $0 < g \in G$ for which $Z + G(g)
\subseteq Z$, where $G(g)$ designates the convex $\ell$-subgroup of $G$
generated by $g$.
\item $Z$ is mobile iff it is not the union of cosets of some nontrivial
convex $\ell$-subgroup of $G$.
\end{enumerate}
\end{observations}
\begin{proof}
If $Z + g \subseteq Z$ then $Z + ng \subseteq Z$ for all $n$, hence $Z + k
\subseteq Z$ for all $k$ such that $|k| \leq ng$ for some $n$.
\end{proof}
The next proposition hints at why mobile downsets are relevant to our
investigation, for it shows that two types of subsets which may have
pointwise joins are mobile
\begin{proposition}
\label{Lem:16}
\begin{enumerate}
\item The downset $\left\downarrow g_{0}\wedge n:n\in \mathbb{N
\right\downarrow$ generated by the truncates of an element $g_{0}\in G^{+}$ is
mobile.
\item A nonempty bounded downset is mobile.
\end{enumerate}
\end{proposition}
\begin{proof}
(2) Suppose the downset $\emptyset \neq Z\subseteq G$ is bounded above by
g_{0}$, and suppose for the sake of argument that $Z+g\subseteq Z$ for some
0<g\in G$. We may assume that $0\in Z$, for we may always replace $Z$ by
Z-z_{0}$ and $g_{0}$ by $g_{0}-z_{0}$, where $z_{0}$ is any member of $Z$.
But then $ng\in Z$ for all positive integers $n$, with the result that
ng\leq g_{0}$ for all $n$, a violation of the archimedean property of $G$.
(1) Let $Z\equiv \left\downarrow g_{0}\wedge n:n\in \mathbb{N
\right\downarrow $ be the set of lower bounds of the truncates of $g_{0}\in
G^{+}$, and suppose for the sake of argument that $Z+g\subseteq Z$ for some
0<g\in G$. Then by the archimedean property there is a positive integer $m$
such that $mg\nleq g_{0}$. It follows that $mg\nleq n\wedge g_{0}$ for any
positive integer $n$, which is to say that $mg \notin Z$, contrary to
Observation \ref{Obs:1}(1).
\end{proof}
\begin{theorem}
\label{Thm:5}The following are equivalent for a downset $Z\subseteq G$.
\begin{enumerate}
\item $Z$ is a cut in $G$.
\item $G$ has an essential extension $H$ containing an element $h$ such that
$\bigvee Z=h$ in $H$.
\item $G$ is completely embedded in an extension $H$ containing an element
h $ such that $\bigvee Z=h$ in $H$.
\item $Z$ is not a union of cosets of a nontrivial convex $\ell$-subgroup of
$G$.
\item $Z$ is mobile.
\end{enumerate}
\end{theorem}
\begin{proof}
The equivalence of (1) and (2) is Lemma \ref{Lem:14}, the implication from
(2) to (3) is Lemma \ref{Lem:15}, and the implication from (3) to (1) is
trivial. The equivalence of (3) and (5) is Proposition 4.5 of \cit
{Ball:1989}, and the equivalence of (4) and (5) is Observation \ref{Obs:1
(2) .
\end{proof}
Our main Theorem \ref{Thm:4} requires a technical lemma.
\begin{lemma}\label{Lem:2}
Let $G=\mathcal{R}L$ for a $P$-frame $L$, and let $G\leq H$ be
its maximal essential extension (Proposition \ref{Prop:17}). If all of the
truncates of an element $h \in H^+$ lie in $G$ then $h$ lies in $G$
\end{lemma}
\begin{proof}
Here $G=\mathcal{R}L\leq \mathcal{R}L^{\ast \ast }=H$, where the embedding
\mathcal{R}L\rightarrow \mathcal{R}L^{\ast \ast }$ is provided by $\mathcal{
}b_{L}=\left( g\longmapsto b_{L}\circ g\right) $. Suppose $\left\{
g_{n}\right\} \subseteq G^{+}$. The condition that $g_{n+1}\wedge n=g_{n}$
for all $n$ clearly holds in $H$ iff it holds in $G$, and the condition that
$\bigvee_{n}g_{n}\left( -\infty ,n\right) =\top $ implies that
\begin{equation*}
\top =b_{L}\left( \top \right) =b_{L}\left( \bigvee_{n}g\left( -\infty
,n\right) \right) =\bigvee_{n}b_{L}\circ g\left( -\infty ,n\right)
\end{equation*
because $b_{L}$ is a frame morphism. We must demonstrate the converse, i.e.,
that the displayed condition implies that that $\bigvee_{n}g_{n}\left(
-\infty ,n\right) =\top $.
So assume that $\bigvee_{n}b_{L}\circ g_{n}\left( -\infty ,n\right) =\top $
holds in $L^{\ast \ast }$, i.e., that $\left( \bigvee_{n}g_{n}\left( -\infty
,n\right) ^{\ast \ast }\right) ^{\ast \ast }=\top $ holds in $L$. Note that
g_{n}\left( -\infty ,n\right) $, being a cozero element of a $P$-frame, is
complemented, i.e., $g_{n}\left( -\infty ,n\right) ^{\ast \ast }=g_{n}\left(
-\infty ,n\right) $. Also note that, since the inclusion $\coz L\rightarrow L
$ is a $\sigma $-frame homomorphism, the supremum $\bigvee_{n}g_{n}\left(
-\infty ,n\right) $ in $L$ agrees with its supremum in $\coz L$. But the
latter is a cozero and is therefore complemented, so that we get $\left(
\bigvee_{n}g_{n}\left( -\infty ,n\right) \right) ^{\ast \ast
}=\bigvee_{n}g_{n}\left( -\infty ,n\right) $.
\end{proof}
The hypotheses of the preceding lemma are more generous than necessary, so
we digress briefly to tighten it up in the light of
the analysis conducted by the second author in \cite{Hager:2013}.
We refer the interested reader to that article for terminology and notation
otherwise undefined here, and omit the details of proof.
\begin{theorem}\label{Thm:9}
The following are equivalent for a $\mathbf{W}$-object $G$
with maximal essential extension $G \leq H$.
\begin{enumerate}
\item
Every element $h \in H^+$ which has all its truncates in $G$ must
lie in $G$.
\item
Every element $h \in D^+(YG)$ with all its truncates in $G$ must lie in $G$.
\item
$G$ is *-maximum, i.e., $G$ contains a copy of every $\mathbf{W}$-object
with the same bounded part as $G$.
\end{enumerate}
\end{theorem}
\subsection{Pointwise completeness \label{Subsec:9}}
\begin{definition}
A $\mathbf{W}$-object is pointwise complete ($\sigma$-complete) if every
(countably generated) cut in $G$ has a pointwise join in $G.$
\end{definition}
\begin{theorem}
\label{Thm:4}The following are equivalent for a $\mathbf{W}$-object $G$.
\begin{enumerate}
\item $G$ is pointwise complete ($\sigma $-complete).
\item Every (countably generated) mobile downset of $G$ has a pointwise join
in $G$.
\item $G$ is of the form $\mathcal{R}L$ for a boolean frame ($P$-frame) $L$.
\end{enumerate}
\end{theorem}
\begin{proof}
The equivalence of (1) with (2) follows from Theorem \ref{Thm:5}. If $G$
satisfies (2) then by Lemma \ref{Lem:16}(2) every bounded (countable) subset
of $G^{+}$ has a pointwise join in $G$, hence by Theorem \ref{Thm:3} $G$ is
a subobject of $\mathcal{R}L$, $L$ boolean (a $P$-frame), and $G$ contains
\mathcal{R}^{\ast }L$. That $G$ is actually all of $\mathcal{R}L$ follows
from Proposition \ref{Prop:3}.
Let us show that (3) implies (1) when $G$ is of the form $\mathcal{R}L$ for
some $P$-frame $L$. Let $Z$ be a countable subset of $G^{+}$ such that
\bigvee Z=h_{0}$ in some extension $G\leq H$. By Lemma \ref{Lem:14} we may
assume this extension to be essential, and, in fact, it does no harm to
assume that $H$ is the maximal essential extension of $G$. That is because
any essential extension of $G$ is isomorphic to an $\ell$-subgroup of $H$
containing $G$, and suprema in all these extensions agree by Lemma \re
{Lem:15}.
Since $G$ is conditionally pointwise $\sigma$-complete by Theorem \ref{Thm:3
, for each $n$ the subset $Z \wedge n$ has a pointwise supremum $g_n \in G$.
Of course, the same subset has supremum $h_0 \wedge n$ in $H$, and the two
suprema coincide by Proposition \ref{Prop:7}. Lemma \ref{Lem:2} then implies
that $h_0 \in G$. Since $h_0$ is the pointwise join of the $g_n$\rq{}s by
Proposition \ref{Prop:3}, and in light of the fact that each $g_n$ is the
pointwise join of $Z \wedge n$, it follows that $\bigvee^\bullet Z = h_0 \in
G$.
It remains to show that (3) implies (1) in case $G=\mathcal{R}L$ for some
boolean frame $L$. Let $Z\subseteq G^{+}$ be such that $\bigvee Z=h_{0}$ in
some extension $G\leq H$. By Lemma \ref{Lem:14} again, we may assume this
extension to be essential. Because $G$ is essentially complete by
Proposition \ref{Prop:17}, we know that $G = H$. And finally, the supremum
\bigvee Z = h_0 \in G$ is pointwise by Proposition \ref{Prop:12}.
\end{proof}
\begin{corollary}
\label{Cor:10} $G$ is pointwise $\sigma $-complete iff it is epicomplete in
\mathbf{W}$.
\end{corollary}
\begin{proof}
Both conditions are equivalent to $G$ being of the form $\mathcal{R}L$ for
L $ a $P$-frame. One equivalence is provided by Theorem 3.4 of \cit
{BallWaltersZenk:2010} and the other by Theorem \ref{Thm:4}.
\end{proof}
\begin{corollary}
The full subcategory comprised of the pointwise $\sigma $-complete objects
is reflective in $\mathbf{W}$.
\end{corollary}
\begin{corollary}
The following are equivalent for an extension $G\leq H$ in $\mathbf{W}$.
\begin{enumerate}
\item
The extension is (isomorphic to) the functorial epicompletion of $G$.
\item
The extension is (isomorphic to) $G\rightarrow \mathcal{RP}L$, where
\mathcal{P}L$ designates the $P$-frame reflection of the Madden frame $L$ of
$G$.
\end{enumerate}
\end{corollary}
\begin{proof}
A complete
description of the extension in (2) is $G \rightarrow \mathcal{RPM}G$. The
point is that the extension is the concatenation of three functors, hence it
is functorial. But there can be only one functorial epicompletion on general
grounds. See \cite{AdamekHerrlichStrecker:2004}.
\end{proof}
|
2302.10722
|
\section{Introduction}\label{sec:intro}
\input{sections/intro}
\section{Characterizing Optimal 0-1 Loss}\label{sec:mc_lb}
\input{sections/lower_bounds}
\section{Bounds on Optimal 0-1 Loss}\label{sec:bound_loss}
\input{sections/finite}
\section{Empirical Results}\label{sec:eval}
\input{sections/eval}
\section{Discussion and Related Work}\label{sec:discussion}
\input{sections/discussion}
\nocite{szegedy2013intriguing}
\nocite{huang2011adversarial}
\section{Introduction}
\subsection{Problem Formulation}
We consider a supervised classification problem where inputs are sampled from input space $\mathcal{X}$, and labels belong to $K$ classes: $y \in \mathcal{Y} = \{1...K\}$. Let $P$ be the joint probablity over $\mathcal{X} \times \mathcal{Y}$. Let $h: X \to [0, 1]^K$ denote a soft classifier, and the classification function $f: \mathcal{X} \to \mathcal{Y}$ is given by $f(x) = \argmax_i h(x)_i$. We are interested in the setting where there exists a test-time adversary that can modify any data point within a neighborhood $\tilde{x} \in N(x)$ where $N:\mathcal{X} \rightarrow 2^{\mathcal{X}}$.
The objective of the learner is:
$$\argmin_{h \in \mathcal{H}} \mathbb{E}_{(x, y) \sim P} \max_{\tilde{x} \in N(x)} \ell(h(\tilde{x}), y)$$
where $\ell$ is cross entropy loss
\subsection{Conflict Graph Construction}
Consider a conflict hypergraph $\mathcal{G} = (V, \epsilon)$
$V \subseteq X \times Y$ such that each data point $(x, y)$ with strictly positive probability in $P$ is represented as a vertex $v$
the graph is K-partite
$V_i = V \cap (X \times \{i\})$ where $i \in \{1, ..., K\}$
edge $((x, i), (x', j))$ is present if and only if $N((x, i)) \cap N((x', j))$ is nonempty (IGNORE HYPER EDGE FOR NOW)
No edge within $V_i$ for $i \in \{1, ..., K\}$
definition 1
\begin{definition}
for $h$, the correct-classification probability $q_v$ for data $v = (x, y)$ in the presence of an adversary is $q_v = inf_{\tilde{x} \in N(x)} h(\tilde{x})_y$
\end{definition}
lemma 1
\begin{lemma}
let $q \in \mathbb{R}^V$ be the vector of correct-classification probabilities, the feasible set of such probabilities is
$q \ge 0$
$Mq \le 1$ where $M = (E I) \in \mathbb{R}^{(\epsilon \cup V) \times V}$ and $E \in \mathbb{R}^{(\epsilon \cup V)}$ is the edge incidence matrix of the conflict graph
\end{lemma}
theorem1
\begin{theorem}
Let $p \in \mathbb{R}^V$ with $p_v = P(\{v\})$. Let $q^*$ be the minimizer of the following program:
\begin{align*}
\min_q & \sum_{v:p_v>0} -p_v \log q_v \\
\text{s.t.} q & \ge 0 \\
Mq & \le 1
\end{align*}
Then, there is a classifier $h^*$ that achieves the correct-classification probabilities $q^*$ and for all $h$, $\mathbb{E}_p[\ell (h^*, v)] \le \mathbb{E}_p[\ell (h, v)]$.
\end{theorem}
lemma 2
\begin{lemma}
Suppose we have $q$ and $z$ such that
\begin{align*}
q &\ge 0 \\
Mq &\le 1\\
z &\ge 0\\
\text{diag}(q)M^T &\ge p\\
1^T z &\le 1^T p
\end{align*}
Then $q$ is optimal in theorem 1.
\end{lemma}
\newpage
When $n=4$, we have
\begin{equation*}
D = \begin{pmatrix}
0 & u^2 & v^2 & w^2 \\
u^2 & 0 & x^2 & y^2 \\
v^2 & x^2 & 0 & z^2 \\
w^2 & y^2 & z^2 & 0
\end{pmatrix}
\end{equation*}
So det $D$ = $u^4 z^4 + v^4 y^4 + w^4 x^4 - 2 u^2 w^2 x^2 z^2 - 2 u^2 v^2 y^2 z^2 - 2 v^2 w^2 x^2 y^2$.
\begin{align*}
& \text{adj} D = \\
&
\begin{pmatrix}
2 x^2 y^2 z^2 & z^2 (u^2 z^2 - v^2 y^2 - w^2 x^2) & y^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) & x^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) \\
z^2 (u^2 z^2 - v^2 y^2 - w^2 x^2) & 2 v^2 w^2 z^2 & w^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) & v^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) \\
y^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) & w^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) & 2 u^2 w^2 y^2 & u^2 (u^2 z^2 - v^2 y^2 - w^2 x^2) \\
x^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) & v^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) & u^2 (u^2 z^2 - v^2 y^2 - w^2 x^2) & 2 u^2 v^2 x^2
\end{pmatrix}
\end{align*}
and
\begin{align*}
\alpha & = c D^{-1} \textbf{1}\\
& \propto \frac{(\text{adj} D) \textbf{1}}{\text{det} D}\\
& = \frac{1}{u^4 z^4 + v^4 y^4 + w^4 x^4 - 2 u^2 w^2 x^2 z^2 - 2 u^2 v^2 y^2 z^2 - 2 v^2 w^2 x^2 y^2}\\
& \times
\begin{pmatrix}
2 x^2 y^2 z^2 + z^2 (u^2 z^2 - v^2 y^2 - w^2 x^2) + y^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) + x^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) \\
z^2 (u^2 z^2 - v^2 y^2 - w^2 x^2) + 2 v^2 w^2 z^2 + w^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) + v^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) \\
y^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) + w^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) + 2 u^2 w^2 y^2 + u^2 (u^2 z^2 - v^2 y^2 - w^2 x^2) \\
x^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) + v^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) + u^2 (u^2 z^2 - v^2 y^2 - w^2 x^2) + 2 u^2 v^2 x^2
\end{pmatrix}
\end{align*}
The square of the radius of the circumsphere is
\begin{align*}
& \frac{1}{2 \textbf{1}^T D^{-1} \textbf{1}} = \frac{\text{det} D}{2 \textbf{1}^T (\text{adj} D) \textbf{1}} = \\
& \frac{u^4 z^4 + v^4 y^4 + w^4 x^4 - 2 u^2 w^2 x^2 z^2 - 2 u^2 v^2 y^2 z^2 - 2 v^2 w^2 x^2 y^2}{4T} \text{ where } \\
& T = u^2 w^2 y^2 + u^2 v^2 x^2 + v^2 w^2 z^2 + x^2 y^2 z^2 + \\
& u^2 (u^2 z^2 - v^2 y^2 - w^2 x^2) + v^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) + \\
& w^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) + x^2 (w^2 x^2 - u^2 z^2 - v^2 y^2) + \\
& y^2 (v^2 y^2 - u^2 z^2 - w^2 x^2) + z^2 (u^2 z^2 - v^2 y^2 - w^2 x^2)
\end{align*}
\end{document}
\section{Introduction}
\input{sections/intro}
\section{Formulating Lower Bounds as a Linear Program}
\input{sections/formalize}
\section{Empirical Results}
\input{sections/eval}
\section{Discussion}
\input{sections/discussion}
\begin{ack}
ANB, WD and BYZ were supported in part by NSF grants CNS-1949650, CNS-1923778, CNS-1705042, the C3.ai DTI, and the DARPA GARD program. SD was supported in part by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2039656. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. SD and PM were supported in part by the National Science Foundation under grants CNS-1553437 and CNS-1704105, the ARL’s Army Artificial Intelligence Innovation Institute (A2I2), the Office of Naval Research Young Investigator Award, the Army Research Office Young Investigator Prize, Schmidt DataX award, and Princeton E-ffiliates Award.
\end{ack}
\bibliographystyle{unsrtnat}
\section{Proofs}
\label{app:proofs}
\begin{proof}[Proof of Lemma \ref{lemma: feasible_q}]
This follows immediately from Lemma~\ref{lemma: feasible_q_side_info} with $m = K$.
\end{proof}
\label{sec:class_games}
\begin{proof}[Proof of Lemma~\ref{lemma: feasible_q_side_info}]
First, we show that for any $h \in \mathcal{H}_{soft}$, $q_N(h) \leq \mathbf{1}$
We note the first constraint hold because classification probabilities must lie in the range $[0, 1]$. We will now demonstrate that the constraint $B^{\leq m}q \leq \mathbf{1}$ must also hold. For Let $e = ((x_1, y_1), ..., (x_{\ell}, y_{\ell}))$ be a size-$\ell$ hyperedge in $\mathcal{E}^{(\le m)}$.
By construction of $\mathcal{E}$, $\exists \tilde{x} \in \bigcap_{i=1}^n N(x_i)$.
Also, there is some $S \in \binom{[K]}{m}$ with $\{y_1,\ldots,y_{\ell}\} \subseteq S$.
By definition of $q_N(h)$, for each vertex $(x_i, y_i), i \in \{1 ... n\}$, we have that $q(h)_{(x_i, y_i)} \le h(\tilde{x},S)_{y_i}$. Thus,
\[
\sum_{i=1}^{\ell} q(h)_{(x_i, y_i)} \le \sum_{i=1}^{\ell} h(\tilde{x})_{y_i} \le \sum_{i=1}^K h(\tilde{x})_i = 1.
\]
This gives $(Bq)_e \leq 1$.
For any vector $q$ in the polytope, we have a classifier $h : \mathcal{X} \times \binom{[K]}{m} \to \mathbb{R}^{[K]}$ that achieves at least those correct classification probabilities.
This mean that $h$ has the following properties. First, $h(\tilde{x},L)_y \geq 0$ and $\sum_{y \in [K]} h(\tilde{x},L)_y = 1$.
Second, for all $(x,y) \in \mathcal{V}$, all $\tilde{x} \in N(x)$, and all $L \in \binom{[K]}{m}$ such that $y \in L$, we have $h(\tilde{x},L)_y \geq q_{(x,y)}$.
To get $h$, first define the function $g : \mathcal{X} \times \binom{[K]}{m} \to \mathbb{R}^{[K]}$ so $g(\tilde{x},L)_y = 0$ for $i \not\in L$ and $g(\tilde{x},L)_y = \max(0,\sup \{q_{(x_y,y)} : x_y \in \mathcal{V}_y, \tilde{x} \in N(x_y)\})$.
Let $L' \subseteq L$ be the set of indicies where $g(\tilde{x},L)_y > 0$.
Then any list of vertices $e = (x_y : y \in L', x_y \in \mathcal{V}_y, \tilde{x} \in N(x_y))$ forms a hyperedge of size $|L'| \leq m$.
Thus
\[
\sum_{y \in [K]} g(\tilde{x},L)_y = \sum_{y \in L'} g(\tilde{x},L)_y = \sup_e \sum_{y \in L'} q_{(x_y,y)} \leq \sup_e 1 = 1.
\]
To produce $h$, allocate the remaining probability ($1 - \sum_y g(\tilde{x},L)_y$) to an arbitrary class.
\end{proof}
\bcomment{
}
\bcomment{
Define the order-$m$ array $A \in \mathbb{R}^{\mathcal{Y}^m}$ with
\[
A_{k_1,\ldots,k_{m}} = L^*(P|(y \in \{k_1,\ldots,k_{m}\}),N).
\]
\dccomment{Not sure that this is the right value for entries of $A$ with some repeated classes. This could be avoided by directly specifying that adversary uses only lists of length exactly $L$. On the other hand, if we allowed them to use shorted lists, they still shouldn't want to. If $A$ is defined the right way, it should still compute the correct loss for those bad adversarial strategies. Maybe a factor of $k_1!\ldots k_{m}!$ is needed.}
Then the optimal loss in the class-only $m$-side-information game is $\max \langle \Pi, A\rangle$ over couplings $\Pi$ of $k$ copies of $P_y$, i.e. distributions over $\mathcal{Y}^{m}$ that have $P_y$ for all $m$ marginals.
The array $A$ is symmetric, so there is a symmetric maximizer $\Pi$.
For symmetric $\Pi$, $m!\Pi_{k_1,\ldots,k_{m}}$ is the probability that $L =\{k_1,\ldots,k_{m}\}$.}
\begin{proof}[Proof of Lemma~\ref{lemma:class-only}]
The first part of this proof applies for any side-information size $m$.
The adversarial strategy for selecting $C$ is a specified by a conditional p.m.f. $p_{C|y}(C|y)$.
Thus $p_{y|C}(y|C) = p_{C|y}(C|y)p_{\mathbf{y}}(y) / \sum_{y'} p_{C|y}(C|y')p_{y}(y') $.
The optimal loss of the classifier against a particular adversarial strategy is just a mixture of the optimal losses for each class list:
$\sum_{C} p_{C|y}(C|y) \Pr[p_y(y) L^*(P|(y\in\{i,j\}),N,\mathcal{H})$.
If $p_{C|y}(C|y) = p_{C|y}(C|y')$ for all $y,y' \in C$, then $p_{y|C}(y|C) = p_{\mathbf{y}}(y)/\sum_{y' \in C}p_{\mathbf{y}}(y')$ and the adversary has not provided the classifier with extra information beyond the fact that $y \in C$.
Thus $P_{x|y}P_{y|C} = P|(y \in C)$.
Now we can spcialize to the $m=2$ case.
Any stochastic matrix $s$ with zeros on the diagonal specifies an adversarial strategy for selecting $C$ with $p_{C|y}(\{i,j\}|i) = s_{i,j}$.
Furthermore, if $s$ is also symmetric, $p_{C|y}(\{i,j\}|i) = p_{C|y}(\{i,j\}|j)$ and $p_{y|C}(i|\{i,j\}) = p_{y|C}(j|\{i,j\})$.
Then the optimal classifier for the side-information game uses the $\binom{K}{2}$ optimal classifiers for the two-class games and incurs loss $\sum_{i,j} Pr[Y=i] a_{i,j} s_{i,j}$ where $a_{i,j} = L^*(P|(y\in\{i,j\}),\mathcal{H},N)$.
Because the diagonal entries of $a$ are all zero, there is always a maximizing choice of $s$ with a zero diagonal.
Thus it is not necessary to include that constraint on $s$ when specifying the optimization.
\end{proof}
\label{app:caro-wei-proof}
\begin{proof}[Proof of Lemma~\ref{lemma:caro-wei}]
If $w=0$, then the lower bound is zero and holds trivially.
Otherwise, $\frac{1}{\mathbf{1}^T w} w$ forms a probability distribution over the vertices.
Let $X \in \mathcal{V}^{\mathbb{N}}$ be a sequence of i.i.d.\ random vertices with this distribution.
From this sequence, we define a random independent set as follows.
Include $v$ in the set if it appears in the sequence $X$ before any of its neighbors in $\mathcal{G}$.
If $v$ and $v'$ are adjacent, at most one of them can be included, so this procedure does in fact construct an independent set.
The probability that $X_i = v$ is $\frac{w_i}{\mathbf{1}^T w}$ and the probability that $X_i$ is $v$ or is adjacent to $v$ is $\frac{((A+I)w)_v}{\mathbf{1}^T w}$.
The first time that the latter event occurs, $v$ is either included in the set or ruled out.
If $w_i > 0$, the probability that $v$ is included in the set is $\frac{w_v}{((A+I)w)_v}$ and otherwise it is zero.
Thus the quantity in \eqref{eq:caro-wei} is the expected size of the random independent set and $\mathcal{G}$ must contain some independent set at least that large.
\end{proof}
\section{Hyperedge Finding}
\label{app:hyperedge_finding}
One challenge in computing lower bounds for $0-1$ loss in the multi-class setting is that we need to find hyperedges in the conflict hypergraph. In this section, we will consider an $\ell_2$ adversary: $N(x) = \{x' \in \mathcal{X} | ~||x' - x||_2 \le \epsilon \}$ and describe an algorithm for finding hyperedges within the conflict graph.
We first note that for an $n$-way hyperedge to exist between $n$ inputs $\{x_i\}_{i=1}^n$, $\{x_i\}_{i=1}^n$ must all lie on the interior of an $n-1$-dimensional hypersphere of radius $\epsilon$.
\input{sections/spheres}
\section{Experimental Setup}
\label{app:add_exp_details}
\textbf{Datasets} We compute lower bounds for MNIST \citep{lecun1998mnist}, CIFAR-10, and CIFAR-100 \citep{krizhevsky2009learning}. Since we do not know the true distribution of these datasets, we compute lower bounds based on the empirical distribution of the training set for each dataset.
\textbf{Attacker} We will consider an $\ell_2$ adversary: $N(x) = \{x' \in \mathcal{X} | ~||x' - x||_2 \le \epsilon \}$. This has been used in most prior work \cite{bhagoji2019lower,pmlr-v119-pydi20a,trillos2023multimarginal}.
\textbf{LP solver} For solving the LP in Equation \ref{eq:lp}, we primarily use Mosek LP solver \citep{mosek}. When Mosek solver did not converge, we use CVXOpt's LP solver \citep{andersen2013cvxopt}.
\paragraph{Training Details} For MNIST, we use 40 step optimization to find adversarial examples during training and use step size $\frac{\epsilon}{30}$ and train all models for 20 epochs. For CIFAR-10 and CIFAR-100, we use 10 step optimization to find adversarial examples and step size $\frac{\epsilon}{7}$ and train models for 100 epochs. For MNIST TRADES training, we use $\beta=1$ and for CIFAR-10 and CIFAR-100, we use $\beta=6$. Additionally, for CIFAR-10 and CIFAR-100, we optimize the model using SGD with learning rate and learning rate scheduling from \citet{gowal2020uncovering}. For MNIST, we use learning rate 0.01.
\paragraph{Architectures used} For CIFAR-10 and CIFAR-100, we report results from training a WRN-28-10 architecture. For MNIST, we train a small CNN architecture consisting of 2 convolutional layers, each followed by batch normalization, ReLU, and 2 by 2 max pooling. The first convolutional layer uses a 5 by 5 convolutional kernel and has 20 output channels. The second convolutional layer also uses a 5 by 5 kernel and has 50 output channels. After the set of 2 convolutional layers with batch normalization, ReLU, and pooling, the network has a fully connected layer with 500 output channels followed by a fully connected classifier (10 output channels).
\section{Additional Experimental Results}
\label{app: add_exp_results}
\subsection{Results for Gaussian data}
\label{app:gauss}
\begin{figure}[ht]
\centering
\includegraphics[width=0.25\textwidth]{plots/gauss_sample.pdf}
\caption{A sample 3-class Gaussian problem (each color pertains to a class) and a corresponding classifier for this problem shown in black. The classifier classifies a sample incorrectly when it lies over the edge of the $\epsilon$ margin (shown by the red lines) nearest the corresponding Gaussian center.}
\label{fig:gaussian_sample}
\end{figure}
\begin{figure}[]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gauss005.pdf}
\caption{$\sigma^2=0.05$}
\label{fig:sig0.05}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gauss05.pdf}
\caption{$\sigma^2=0.5$}
\label{fig:sig0.5}
\end{subfigure}
\caption{Lower bounds on error for the Gaussian 3-class problem ($\sigma^2=0.05$ and $\sigma^2 = 0.5$) computed using only constraints from edges ($L^*(2)$) and up to degree 3 hyperedges ($L^*(3)$) in comparison to the performance of the deterministic 3-way classifier depicted in Figure \ref{fig:gaussian_sample}.}
\label{fig:gauss_results}
\vspace{-20pt}
\end{figure}
We begin with a 3-way classification problem on 2D Gaussian data. To generate our Gaussian dataset, we sample 1000 points per class from 3 spherical Gaussians with means at distance 3 away from from the origin (a sample is shown in Figure \ref{fig:gaussian_sample}). We compute multiclass lower bounds via the LP in Lemma on robust accuracy at various $\ell_2$ budget $\epsilon$ and display these values in Figure \ref{fig:gauss_results} as $L^*(3)$. Additionally, we compare to a deterministic 3 way classifier. This classifier is the best performing out of the 2 strategies: 1) constantly predict a single class (thus achieving $\frac{2}{3}$ loss) or 2) is the classifier in black in Figure \ref{fig:gaussian_sample} which classifies incorrectly when a sample lies over the edge of the nearest $\epsilon$ margin of the classifier.
We observe that at smaller values of $\epsilon$, the loss achieved by the 3-way classifier matches optimal loss ($L^*(3)$); however, after $\epsilon=2.5$ for $\sigma^2 = 0.05$ and $\epsilon=2.3$ for $\sigma^2=0.5$, we find the classifier no longer achieves optimal loss. This suggests that there is a more optimal classification strategy at these larger values of $\epsilon$. In Figures \ref{fig:gauss_class_vis} and \ref{fig:gauss_class_vis_hyper}, we visualize the distribution of correct classification probabilities obtained through solving the LP with and without considering hyperedges. These figures are generated by taking a fresh sample of 1000 points from each class and coloring the point based on the correct classification probability $q_v$ assigned to its nearest neighbor that was used in the conflict hypergraph when computing the lower bound. We observe from Figure \ref{fig:gauss_class_vis}, for all classes, the data are mostly assigned classification probabilities around 0.5. In Figure \ref{fig:gauss_class_vis_hyper}, we can see that when we consider hyperedges, we some of these 0.5 assignments are reassigned values close to $\frac{2}{3}$ and $\frac{1}{3}$. Interestingly, we notice that when we do not consider hyperedges, our solver finds an asymmetric solution to the problem (strategies for class 0, 1, and 2 differ) while when considering hyperedges this solution becomes symmetric.
\begin{figure}[ht]
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gaussian_var0.5_eps2.5_classify0.pdf}
\caption{Class 0}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gaussian_var0.5_eps2.5_classify1.pdf}
\caption{Class 1}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gaussian_var0.5_eps2.5_classify2.pdf}
\caption{Class 2}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.044\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gaussian_var0.5_eps2.5_colorbar.pdf}
\end{subfigure}
\hfill
\caption{Distribution of optimal classification probabilities across samples from each class of the Gaussian obtained as a solution when computing $L^*(2)$.}
\label{fig:gauss_class_vis}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gaussian_var0.5_eps2.5_classify0_hyper.pdf}
\caption{Class 0}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gaussian_var0.5_eps2.5_classify1_hyper.pdf}
\caption{Class 1}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gaussian_var0.5_eps2.5_classify2_hyper.pdf}
\caption{Class 2}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.044\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/gaussian_var0.5_eps2.5_colorbar.pdf}
\end{subfigure}
\hfill
\caption{Distribution of optimal classification probabilities across samples from each class of the Gaussian obtained as a solution when computing $L^*(3)$.}
\label{fig:gauss_class_vis_hyper}
\end{figure}
\subsection{Additional adversarial training results}
\label{app:adv_train}
In Figure \ref{fig:full_adv_train}, we also add the loss achieved by PGD adversarial training. We find that this approach generally performs worse on MNIST compared to TRADES and is also unable to fit to CIFAR-10 data at the $\epsilon$ values tested.
\begin{figure}[]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/MNIST_AT_all.pdf}
\caption{MNIST (1, 7, 9)}
\label{fig:full_mnist_adv_train}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/CIFAR10_AT_all.pdf}
\caption{CIFAR-10 (0, 2, 8)}
\label{fig:full_cif_adv_train}
\end{subfigure}
\caption{Lower bounds on error for MNIST and CIFAR-10 3-class problems (1000 samples per class) computed using only constraints from edges ($L^*(2)$) and up to degree 3 hyperedges ($L^*(3)$) in comparison to TRADES adversarial training (TRADES-AT) and PGD adversarial training (PGD-AT) loss.}
\label{fig:full_adv_train}
\vspace{-20pt}
\end{figure}
\subsection{Truncated hypergraph lower bounds for CIFAR-100} \label{app:cifar100}
We provide results for truncated hypergraph lower bounds for the CIFAR-100 train set. We observe that similar to MNIST and CIFAR-10, including more hyperedge constraints does not influence the computed lower bound.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{plots/cifar100_loss.pdf}
\caption{Lower bounds for optimal 0-1 loss the for CIFAR-100 train set}
\label{fig:cifar100}
\end{figure}
\subsection{Impact of hyperedges}
\label{app:hyper_impact}
In Figure \ref{fig:num_hyper}, we show the count of edges, degree 3 hyperedges, and degree 4 hyperedges found in the conflict hypergraphs of the MNIST, CIFAR-10, and CIFAR-100 train sets. We note that we did not observe any increase in loss when considering degree 4 hyperedges at the $\epsilon$ with a data point for number of degree 4 hyperedges in Figure \ref{fig:num_hyper}. We find that the relative number of edges and hyperedges is not reflective of whether we expect to see an increase in loss after considering hyperedges. For example in CIFAR-10, at $\epsilon=4.0$, we there are about 100 times more hyperedges than edges, but we see no noticeable increase in the $0-1$ loss lower bound when incorporating these hyperedge constraints.
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{\textwidth}
\centering
\includegraphics[width=0.7\textwidth]{plots/legend_num.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/mnist_num_edges.pdf}
\caption{MNIST}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/cifar_num_edges.pdf}
\caption{CIFAR-10}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/cifar100_num_edges.pdf}
\caption{CIFAR-100}
\end{subfigure}
\caption{Number of edges, degree 3 hyperedges, and degree 4 hyperedges found in the conflict hypergraphs of MNIST, CIFAR-10, and CIFAR-100 train sets. The red vertical line indicates the $\epsilon$ at which we noticed an increase in the $0-1$ loss lower bound when considering degree 3 hyperedges.}
\label{fig:num_hyper}
\end{figure}
Similar to Figure \ref{fig:weights_mnist} in the main body of the paper, we plot the distribution of vertex weights $q_v$ obtained through solving the LP for $L^*(2)$ and $L^*(3)$ for CIFAR-10 in Figure \ref{fig:weights_cifar10}. Similar to trends for MNIST, we find that the gap between $L^*(2)$ and $L^*(3)$ only occurs when the frequency of 0.5 weights is higher.
\begin{figure*}[ht]
\centering
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/legend_hists.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/cifar10_3.5.pdf}
\caption{$\epsilon=3.5$}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/cifar10_4.0.pdf}
\caption{$\epsilon=4.0$}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/cifar10_4.5.pdf}
\caption{$\epsilon=4.5$}
\end{subfigure}
\caption{Distribution of optimal classification probabilities $q$ obtained by solving the LP with up to degree 2 hyperedges ($L^*(2)$) and up to degree 3 hyperedges ($L^*(3)$) on the CIFAR-10 training set.}
\label{fig:weights_cifar10}
\end{figure*}
\subsection{Computational complexity of computing lower bounds}\label{app:runtime}
Our experiments of $L^*(3)$ for higher $\epsilon$ are limited due to computation constraints. Figure~\ref{fig:timing} we see that the time taken to compute the losses grows rapidly with $\epsilon$. In future work, we are seeking algorithmic optimization to achieve more results at high $\epsilon$.
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{\textwidth}
\centering
\includegraphics[width=0.7\textwidth]{plots/legend-time.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/mnist-time.pdf}
\caption{MNIST}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.54\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/cifar10-time.pdf}
\caption{CIFAR-10}
\end{subfigure}
\caption{Time taken to compute $L^*(2)$ and $L^*(3)$ for MNIST and CIFAR-10.}
\label{fig:timing}
\end{figure}
\subsection{Pairwise optimal losses for 1v1 binary classification problems} \label{app:heat_map}
In Section \ref{sec:exp_approx}, we computed a lower bound on loss from maximum weight coupling over optimal losses for 1v1 binary classification problems. In Figure \ref{fig:heat_map}, we show the heat maps for optimal losses for each pair of 1v1 classification problems for $\epsilon=3$ on MNIST and $\epsilon=4$ on CIFAR-10. We find that for both datasets only a few pairs of classes have high losses. Specifically, for MNIST, we find that the class pairs 4-9 and 7-9 have significantly higher loss than all other pairs of classes. For CIFAR-10, we find that 2-4 has the highest loss compared to other pairs, and 2-6 and 6-4 are also quite high.
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/MNIST_pairwise.pdf}
\caption{MNIST ($\epsilon=3$)}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/CIFAR-10_pairwise.pdf}
\caption{CIFAR-10 ($\epsilon=4$)}
\end{subfigure}
\caption{Heat maps for optimal loss for each pair of 1v1 classification problems.}
\label{fig:heat_map}
\end{figure}
\subsection{Optimal loss for 3-way classification problems}
\label{sec:exact_loss}
Lemma \ref{lem:constraints} allows us to compute the optimal loss given any dataset (and its corresponding conflict hypergraph). We begin with computing the optimal loss for 3-way classification problems, due to the computational complexity of finding higher order hyperedges (see Appendix~\ref{app:hyper_impact}).
In Figure \ref{fig:img_3_class}, we plot the optimal loss $L^*(3)$ computed via the LP in Lemma \ref{lem:constraints} against the loss obtained through TRADES adversarial training \citep{zhang2019theoretically} for 3-class MNIST (classes 1, 4, 7) and 3-class CIFAR-10 (classes 0, 2, 8 which correspond to plane, bird, and ship respectively) with 1000 samples per class \footnote{We also used PGD adversarial training \cite{madry_towards_2017} but found its performance to be far worse. (See Appendix \ref{app:adv_train})}. For MNIST, we train a 3 layer CNN for 20 epochs with TRADES regularization strength $\beta=1$ and for CIFAR-10, we train a WRN-28-10 model for 100 epochs with $\beta=6$. We evaluate models using APGD-CE from AutoAttack \cite{croce2020reliable}. Additional results for a mixture of $3$ Gaussians are in Appendix~\ref{app:gauss}.
\begin{figure*}[ht]
\centering
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/legend_hists.pdf}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/mnist_qdist_2.5.pdf}
\caption{$\epsilon=2.5$}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/mnist_qdist_3.0.pdf}
\caption{$\epsilon=3.0$}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/mnist_qdist_3.5.pdf}
\caption{$\epsilon=3.5$}
\end{subfigure}
\caption{Distribution of optimal classification probabilities $q$ obtained by solving the LP with up to degree 2 hyperedges ($m = 2$) and up to degree 3 hyperedges ($m = 3$) on the MNIST training set.}
\label{fig:weights_mnist}
\vspace{-10pt}
\end{figure*}
Since the optimal loss is over the space of all soft classifiers while the trained models are members of a smaller hypothesis class, we expect some gap in performance between adversarial training and our computed bounds. From Figure \ref{fig:img_3_class}, we observe a large gap in performance of adversarial training in comparison to the computed optimal loss $L^*(3)$. In fact, we find that for CIFAR-10, adversarial training is unable to achieve loss much better than 0.6 at $\epsilon$ for which the optimal loss is near 0. This gap is much larger than observed by prior work \citep{bhagoji2019lower, bhagoji2021lower} for binary classification, suggesting that \textit{current robust training techniques struggle more to fit training data with more classes.}
\subsection{Bounds on optimal loss for 10-way classification} \label{sec:exp_approx}
As the number of classes and dataset size increases, the difficulty of solving the LP in Lemma \ref{lem:constraints} increases. In this section, we use methods discussed in Section \ref{sec:approx} in order to give a bound on the optimal loss for 10-way classification on MNIST and CIFAR-10 datasets on the full training dataset. We present results for each approximation in Figure \ref{fig:approx}. We note that for some methods, we are only able to obtain results up to a small value of $\epsilon$ due to blowup in runtime (See Appendix \ref{app:runtime} for details). We provide truncated hypergraph bounds for CIFAR-100 in Appendix \ref{app:cifar100}
\textbf{Lower bounds from truncated hypergraphs (\S \ref{sec:truncated}):} In Figure \ref{fig:approx}, we plot the loss lower bound obtained via by truncating the hypergraph to consider only edges ($L^*(2)$), up to degree 3 hyperedges ($L^*(3)$), and up to degree 4 hyperedges ($L^*(4)$). Interestingly, we find at small values of $\epsilon$, there is little difference in these bounds despite the presence of many higher degree hyperedges. For example, for CIFAR-10 at $\epsilon=3$, we observe 3M degree 3 hyperedges and 10M degree 4 hyperedges, but these edge constraints have no impact on the computed lower bound. We provide the count of hyperedges for each value of $\epsilon$ in Appendix \ref{app:hyper_impact}. In fact, from Figures \ref{fig:gauss_results} and \ref{fig:img_3_class}, we find that the difference $L^*(2)$ and $L^*(3)$ does not occur until the loss reaches above 0.4 for the 3-way classification problem.
\textit{Takeaway:} In practice, we \textit{do not lose information from computing lower bounds with only edges in the conflict hypergraph}
To understand why including more information about hyperedges does not influence the computed lower bound by much, we examine the optimal classification probability $q_v$ of each vertex obtained as a solution to the LP with ($L^*(3)$) and without considering constraints from degree 3 hyperedges ($L^*(2)$). Figure \ref{fig:weights_mnist} contains a histogram of the distributions of $q_v$ for MNIST. (A corresponding Figure for CIFAR-10 is in Appendix \ref{app:hyper_impact}.) For small $\epsilon$, there is no change in the distribution of $q_v$, which results in no change in overall loss. For example, for MNIST at $\epsilon=2.5$, we observe that the distribution of $q_v$ stays the same between $L^*(2)$ and $L^*(3)$. However at larger values of $\epsilon$, we find that for $L^*(2)$, more values are assigned $q_v$ near 0.5, and these probabilities are highly impacted after incorporating hyperedge constraints. Because the values $L^*(2)$ and $L^*(3)$ are not significantly different, very few of the 0.5 values in the $L^*(2)$ solution were in triangles of $\mathcal{G}^{\leq 2}$ that were replaced by a hyperedge in $\mathcal{G}^{\leq 3}$.
\textbf{Lower bounds from $1$v$1$ binary classification problems (\S \ref{sec:bin2multi}):}
In Section \ref{sec:bin2multi}, we introduced a way to compute lower bounds on loss based on lower bounds for binary classification problems. Since classes are equally likely for our datasets, we use maximum weight coupling over the optimal losses for all pairs of 1v1 binary classification problems. We provide heat maps of the optimal loss for each pair of classes in MNIST and CIFAR-10 in Appendix \ref{app:heat_map}. Since \citet{bhagoji2021lower} introduce an algorithm for efficiently computing lower bounds for binary classification, this technique can be more efficient than using truncated hypergraphs, allowing us to compute lower bounds at larger values of $\epsilon$ in Figure \ref{fig:approx}. However, we note that this technique scales poorly as the number of classes $K$ increases due to the need for computing lower bounds for all $\binom{K}{2}$ pairs of binary classification problems.
\textit{Takeaway:} From Figure \ref{fig:approx}, we find that \textit{while this lower bound is the most efficient to compute, the obtained bound is much looser compared to that from truncated hypergraphs.}
\textbf{Upper bounding optimal loss via Caro-Wei (\S \ref{sec:caro-wei}):} From Figure \ref{fig:approx}, we also plot the upper bound on 0-1 loss for hard classifiers (denoted by $L^*(\mathcal{H}_{\text{hard}})$) obtained via applying Lemma \ref{lemma:caro-wei} with vertex weights obtained from the solution to $L^*(2)$. When $\epsilon$ becomes large ($\epsilon \ge 3.0$ for MNIST and $\epsilon \ge 4.5$ for CIFAR-10), we find that the upper bound on loss blows up, which suggests that at larger values of $\epsilon$ higher order interactions matter more. However, these values of $\epsilon$ are much larger than studied in practice. We observe that at small values of $\epsilon$ (achieving less than 0.2 error), the lower bounds obtained through truncated hypergraphs ($L^*(2)$, $L^*(3)$, and $L^*(4)$) are close to the value of this upper bound . Also, $L^*(\mathcal{H}_{\text{hard}})$ is also an upper bound on hard classifier 0-1 loss,
\textit{Takeaways:} (i) \textit{We do not not lose much information from not including all hyperedges at small $\epsilon$ values}; (ii) At these small values of $\epsilon$, \textit{we do not expect much difference in performance between hard classifiers and soft classifiers.}
\subsection{Lower bounds on multiclass optimal loss via truncated hypergraphs}
\label{sec:truncated}
The edge set of the hypergraph $\mathcal{G}$ could be very large: there are $\prod_{i \in [K]} (1+|V_i|)$ vertex sets that are potential hyperedges.
Even when the edge set is reasonable, it is not clear that higher order hyperedges can be computed from $\mathcal{V}$ efficiently.
To work around these issues, we consider the truncated hypergraphs with bounded size hyperedges: $\mathcal{G}^{\leq m} = (\mathcal{V},\mathcal{E}^{\leq m})$ where $\mathcal{E}^{\leq m} = \{e \in \mathcal{E} : |e| \leq m\}$.
In the corresponding relaxation of $\eqref{eq:lp}$, $B$ is replaced by $B^{\leq m}$, the incidence matrix for $\mathcal{E}^{\leq m}$. Since $\mathcal{E}^{\leq m} \subseteq \mathcal{E}$, this relaxation provides a lower bound on $ L^*(P,N,\mathcal{H}_{\text{soft}})$.
\paragraph{Classification with side-information.}
This relaxation has an interpretation as the optimal loss in a variation of the classification game with side information.
In the example-dependent class list game with list length $m$, the adversary samples $(x,y) \sim P$, then selects $\tilde{x}$ and $C \subseteq \mathcal{Y}$ such that $y \in C$ and $|C| = m$.
The classifier receives both $\tilde{x}$ and $C$, so the classifier is a function $h: \mathcal{X} \times \binom{\mathcal{Y}}{m} \to \mathcal{Y}$, where $\binom{\mathcal{Y}}{m}$ is the set of $m$-element subsets of $\mathcal{Y}$.
We call $C$ the side information.
Let $L^*(m,P,\mathcal{H},N) = $ be the minimum loss in this game.
To illustrate this, consider the distribution $P'$ from Figure~\ref{fig:three_examples} with $m=2$.
The adversary can select some $\tilde{x} \in N(u') \cap N(v') \cap N(w')$, but the classifier will use the side-information to eliminate one of the three classes.
The classifier is in the same situation it would be if the distribution were $P$ and the size-three hyperedge was absent.
For classifiers using class list side-information, the correct classification probability is defined as follows: $q_{m,N}(h)_{(x,y)} = \inf_{\tilde{x} \in N(x)} \min_{C \in \binom{[K]}{m} :y \in C} h(\tilde{x},C)_y$.
When $m=K$, the minimization over $C$ is trivial and this reduces to our previous definition of $q_N(h)$.
\begin{lemma}[Feasible output probabilities in the side-information game]
\label{lemma:feasible}
The set of correct classification probability vectors for side-information of size $m$, support points $\mathcal{V}$, adversarial constraint $N$, and hypothesis class $\mathcal{H}_{soft}$ is
\begin{equation}
\mathcal{P}_{m,\mathcal{V},N,\mathcal{H}_{soft}} = \{q \in \mathbb{R}^{\mathcal{V}} : q \geq \mathbf{0},\, B^{\le m} q \leq \mathbf{1}\} \label{conflict-constaints}
\end{equation}
where $B^{\le m} \in \mathbb{R}^{\mathcal{E} \times \mathcal{V}}$ is the edge incidence matrix of the conflict graph $G_{\mathcal{V},N}^{\leq m}$.
\label{lemma: feasible_q_side_info}
\end{lemma}
The proof can be found in Appendix~\ref{sec:class_games}.
Using the feasible correct classification probabilities in Lemma \ref{lemma: feasible_q_side_info}, we can now write the LP for obtaining the optimal loss for classification with side-information:
\begin{corollary}[Optimal loss for classification with side information/ truncated hypergraph lower bound]
\begin{equation*}
1 - L^*(P,N,\mathcal{H}_{\text{soft}}) = \max_q p^T q
\;\; \text{s.t} \; q \ge 0,
\; B^{\le m}q \le 1.
\end{equation*}
\end{corollary}
\subsection{Lower bounds on multiclass optimal loss via lower bounds for binary classification}
\label{sec:bin2multi}
For large training datasets and large perturbation sizes, it may still be computationally expensive to compute lower bounds via LP even when using truncated hypergraphs due to the large number of edge constraints. Prior works \citep{bhagoji2019lower, bhagoji2021lower} proposed methods of computing lower bounds for 0-1 loss for binary classification problems and demonstrate that their algorithm is more efficient than generic LP solvers. We now ask the question: \textit{Can we use lower bounds for binary classification to efficiently compute a lower bound for multi-class classification?
Consider the setting where we obtain the optimal $0-1$ loss for all one-versus-one binary classification tasks.
Specifically, for each $C \in \binom{[K]}{2}$, the binary classification task for that class pair uses example distribution $P|Y \in C$ and the corresponding optimal loss is $L^*((P|Y \in C),N,\mathcal{H}_{soft})$.
What can we say about $L^*(P,N,\mathcal{H}_{soft})$ given these $\binom{K}{2}$ numbers?
This question turns about to be related to another variation of classification with side information that we call the \emph{class-only side-information} game.
The adversary samples $y \sim P_y$, then selects $C \in \binom{[K]}{m}$, then samples $x \sim P_{x|y}$ and selects $\tilde{x} \in N(x)$.
In the side information game from \S~\ref{sec:truncated}, which we call the "example-dependent side-information" game, the adversary's choice of $C$ can depend on both $x$ and $y$.
In the class-only variation it can only depend on $y$.
Let $L^*_{\text{co}}(m,P,\mathcal{H},N)$ be the minimum loss in the class-only class list game.
For the class only game, we will focus on the $m=2$ case.
To make the connection to the binary games, we need to add one more restriction on the adversary's choice of side information: for all $y,y'$ $\Pr[C={y,y'}|Y=y] = \Pr[C={y,y'}|Y=y']$.
This ensures that the classifier's posterior for $Y$ given $C$ is
$\Pr[Y=y|C] = \Pr[Y=y]/\Pr[Y \in C]$.
\begin{lemma}
\label{lemma:class-only}
The optimal loss in the class-only side-information game is
\[
L_{\text{co}}^*(2,P,N,\mathcal{H}) = \max_{s} \sum_{i,j} Pr[Y=i] a_{i,j} s_{i,j}
\]
where $a_{i,j} = L^*(P|(y\in\{i,j\}),\mathcal{H},N)$ and $s \in \mathbb{R}^{[K] \times [K]}$ is a symmetric doubly stochastic matrix: $s\geq 0$, $s = s^T$, $s\mathbf{1} = \mathbf{1}$.
\end{lemma}
The proof in is Appendix~\ref{sec:class_games}.
The variable $s$ represents the attacker's strategy for selecting the class side information.
When the classes are equally likely, this optimization becomes a maximum weight coupling problem: because the weights $a$ are symmetric, the constraint that $s$ be symmetric becomes irrelevant.
\subsection{Upper bounds on hard classifier loss via Caro-Wei bound on independent set probability} \label{sec:caro-wei}
In \S \ref{sec:truncated} and \ref{sec:bin2multi}, we discussed 2 ways of obtaining lower bounds for the loss of soft classifiers for the multi-class classification problem. In this section, we provide an upper bound on the loss of the optimal hard classifier (we note that this is also an upper bound for optimal loss for soft classifiers). In \S \ref{sec:hard-classifiers}, we discussed the relationship between optimal loss achievable by hard classifiers and independent set size. We upper bound the optimal loss of hard classifiers by providing a lower bound on the probability of independent set in the conflict graph.
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/MNIST_Class_1_4_7.pdf}
\caption{MNIST (1,4,7)}
\label{fig:sig0.05}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/CIFAR-10_Class_0_2_8.pdf}
\caption{CIFAR-10 (0,2,8)}
\label{fig:sig0.5}
\end{subfigure}
\caption{Optimal error for MNIST and CIFAR-10 3-class problems ($L^*(3)$). $L^*(2)$ is a lower bound computed using only constraints from edges ($L^*(2)$). \emph{AT} is the loss for a robust classifier under the strong APGD attack \cite{croce2020reliable}}
\label{fig:img_3_class}
\vspace{-10pt}
\end{figure*}
The following lemma is a generalization of the Caro-Wei theorem \cite{alon2016probabilistic} and gives a lower bound on the weight of the maximum weight independent set.
\begin{lemma}
\label{lemma:caro-wei}
Let $\mathcal{G}$ be a graph on $\mathcal{V}$ with adjacency matrix $A \in \{0,1\}^{\mathcal{V}\times\mathcal{V}}$ and let $P$ be a probability distribution on $\mathcal{V}$.
For any nonnegative vertex weights $w \in \mathbb{R}^{\mathcal{V}}$, $w \geq 0$, there is some independent set $S \subseteq \mathcal{V}$ with
\begin{equation}
P(S) \geq \sum_{v \in \mathcal{V} : w_v > 0} p_v \frac{w_v}{((A+I)w)_v}.
\label{eq:caro-wei}
\end{equation}
\end{lemma}
The proof is in Appendix~\ref{app:caro-wei-proof}.
Note that if $w$ is the indicator vector for an independent set $S'$, the bound becomes $p^T w = P(S')$.
In general, Lemma~\ref{lemma:caro-wei} can be thought of as a randomized procedure for rounding an arbitrary vector into an independent set indicator vector.
Vectors $w$ that are nearly independent set indicators yield better bounds.
Lemma~\ref{lemma:caro-wei} provides a lower bound on the size of the maximum independent set in $\mathcal{G}^{\leq 2}$ and thus an upper bound on $L^*(P,n,\mathcal{H}_{hard})$, which we call $L_{CW} = 1 - P(S)$.
\subsection{Relationship between bounds}\label{subsec:bound_relation}
When $m = K$, $C = \mathcal{Y}$ and both the example-dependent and class-only side information games are equivalent to the original game, so $L^*(P,N,\mathcal{H})=L^*(K,P,N,\mathcal{H})=L_{\text{co}}^*(K,P,N,\mathcal{H})$.
For each variation of the side-information game, the game becomes more favorable for the adversary as $m$ increases: $L^*(m,P,n,\mathcal{H}) \leq L^*(m+1,P,N,\mathcal{H})$ and $L_{\text{co}}^*(m,P,N,\mathcal{H}) \leq L_{\text{co}}^*(m+1,P,N,\mathcal{H})$.
For each $m$, it is more favorable for the adversary to see $x$ before selecting $C$, \emph{i.e.} $L_{\text{co}}^*(m,P,N,\mathcal{H}) \leq L^*(m,P,N,\mathcal{H})$.
We fix a choice of $P$ and $N$ and suppress them from our notation in the following inequalities:
\begin{align*}
0 &= L_{co}^*(1,\mathcal{H}_{soft}) = L^*(1,\mathcal{H}_{soft})\\
&\leq L_{co}^*(2,\mathcal{H}_{soft}) \leq L^*(2,\mathcal{H}_{soft})\\
&\leq L^*(3,\mathcal{H}_{soft})\\
&\leq \ldots \leq L^*(K,\mathcal{H}_{soft}) = L^*(\mathcal{H}_{soft})\\
&\leq L^*(\mathcal{H}_{hard})\\
&\leq L_{CW}
\end{align*}
\subsection{Problem Formulation}
\noindent \textbf{Notation.} We consider a supervised classification problem where inputs are sampled from input space $\mathcal{X}$, and labels belong to $K$ classes: $y \in \mathcal{Y} = [K] = \{1,...,K\}$. Let $P$ be the joint probability over $\mathcal{X} \times \mathcal{Y}$. Let $\mathcal{H}_{\text{soft}}$ denote the space of all soft classifiers; i.e. specifically, for all $h \in \mathcal{H}_{\text{soft}}$ we have that $h: \mathcal{X} \to [0, 1]^K$ and $\sum_{i=1}^K h(x)_i = 1$ for all $x \in \mathcal{X}$. Here, $h(x)_i$ represents the probability that the input $x$ belongs to the $i^{\text{th}}$ class.
We use the natural extension of 0-1 loss, or equivalently classification error probability, to soft classifiers as our loss function: $\ell(h,(x,y)) = 1-h(x)_y$. This reduces to 0-1 loss when $h(x) \in \{0,1\}^K$.
\footnote{A soft classifier can be interpreted as a randomized hard-decision classifier $f$ with $\Pr[f(x) = y] = h(\tilde{x})_y$, in which case $\ell(h,(x,y))= \Pr[f(x) \neq y]$, classification error probability of this randomized classifier.}
\noindent \textbf{Test-time adversaries.} We are interested in the setting where there exists a test-time adversary that can modify any data point $x$ to generate an adversarial example $\tilde{x}$ that lies within the neighborhood $N(x)$ of $x$.
We require that for all $x \in \mathcal{X}$, $N(x)$ always contains $x$.
The optimal loss, which depends on a distribution $P$, hypothesis class $\mathcal{H}$, and neighborhood $N$, is
\begin{multline}
\label{eq:learner_objective}
L^*(P,N,\mathcal{H})
= \inf_{h \in \mathcal{H}} \sup_{\tilde{x} \in N(x)} \ell(h,(\tilde{x},y)) \\
= 1 - \sup_{h \in \mathcal{H}} \inf_{\tilde{x} \in N(x)} h(\tilde{x})_y.
\end{multline}
\paragraph{Alternative hypothesis classes.}
In general, when $\mathcal{H}' \subseteq \mathcal{H}$, $L^*(P,N,\mathcal{H}) \leq L^*(P,N,\mathcal{H}')$.
Two particular cases of this are relevant.
First, the class of hard-decision classifiers is a subset of the class of soft classifiers ($\mathcal{H}_\text{soft}$).
Second, for any fixed architecture, the class of functions represented by some parameterization of that architecture is another subset.
Thus, optimal loss over $\mathcal{H}_\text{soft}$ provides a lower bound on loss for these settings.
\subsection{Optimal loss for distributions with finite support}
Since we would like to compute the optimal loss for distributions $P$ \textit{with finite support}, we can rewrite the expectation in Equation \ref{eq:learner_objective} as an inner product.
Let $V$ be the support of $P$, i.e. the set of points $(x, y) \in \mathcal{X} \times \mathcal{Y}$ that have positive probability in $P$.
Let $p \in [0, 1]^{V}$ be the probability mass vector for $P$: $p_v = P(\{v\})$.
For a soft classifier $h$, let $q_N(h) \in \mathbb{R}^{V}$ be the vector of correct classification probabilities for vertices $v = (x, y) \in V$, \emph{i.e.} $q_N(h)_v := \inf_{\tilde{x} \in N(x)} h(\tilde{x})_y$.
Rewriting \eqref{eq:learner_objective} with our new notation, we have $1 - L^*(P,\mathcal{H}_{\text{soft}},N) = \sup_{h \in \mathcal{H}_{\text{soft}}} p^T q_N(h)$.
This is the maximization of a linear function over all possible vectors $q_N(h)$. In fact, the convex hull of all correct classification probability vectors is a polytope and this optimization problem is a linear program, as described next.
\begin{definition}
For a soft classifier $h$, the correct-classification probability achieved on an example $v = (x,y)$ in the presence of an adversary with constraint $N$ is $q_N(h)_v=\inf_{\tilde{x} \in N(x)} h(\tilde{x})_y$.
The space of achievable correct classification probabilities is $\mathcal{P}_{\mathcal{V},N,\mathcal{H}} \subseteq [0,1]^{\mathcal{V}}$, defined as
\begin{equation*}
\mathcal{P}_{\mathcal{V},N,\mathcal{H}} = \bigcup_{h \in \mathcal{H}} \prod_{v \in \mathcal{V}} [0,q_N(h)_v]
\end{equation*}
\end{definition}
In other words we say that $q' \in [0,1]^{\mathcal{V}}$ is feasible when there exists $h \in \mathcal{H}$ such that $q' \leq q_N(h)$.
The inequality appears because we will always take nonnegative linear combinations of correct classification probabilities.
Characterizing $\mathcal{P}_{\mathcal{V},N,\mathcal{H}}$ allows the minimum adversarial loss achievable to be expressed as an optimization problem with a linear objective:\footnote{
For any loss function that is a decreasing function of $h(\tilde{x})_y$, the optimal loss can be specified as an optimization over $\mathcal{P}_{\mathcal{V},N,\mathcal{H}}$.
In fact \cite{bhagoji2021lower} focused on cross-entropy loss, which has this property.}
\begin{equation}
\label{eq:opt_over_feasible}
\sup_{h \in \mathcal{H}_{\text{soft}}} p^T q_N(h)
= \sup_{q \in \mathcal{P}_{\mathcal{V},N,\mathcal{H}}} \sum_{v \in \mathcal{V}} p_v q_v.
\end{equation}
\cite{bhagoji2021lower} characterized $\mathcal{P}_{\mathcal{V},N,\mathcal{H}_{soft}}$ in the two-class case and demonstrated that this space can be captured by linear inequalities. We now demonstrate that this also holds for the multi-class setting.
\subsection{Linear Program to obtain Optimal Loss} \label{sec:lp}
In order to characterize $\mathcal{P}_{\mathcal{V},N,\mathcal{H}_{soft}}$, we represent the structure of the classification problem with a \textit{conflict hypergraph} $\mathcal{G}_{\mathcal{V},N} = (\mathcal{V}, \mathcal{E})$, which records intersections between neighborhoods of points in $\mathcal{X}$.
The set of vertices $V$ of $\mathcal{G}_{\mathcal{V},N}$ is the support of $P$. $\mathcal{E}$ denotes the set of hyperedges of the graph. For a set $S \subseteq \mathcal{V}$, $S \in \mathcal{E}$ (i.e. $S$ is a hyperedge in $\mathcal{G}_{\mathcal{V},N}$) if all vertices in $S$ belong to different classes
and the neighborhoods of all vertices in $S$ overlap: $\bigcap_{(x,y) \in S} N(x) \ne \emptyset$.
Thus, the size of each hyperedge is at most $K$, $\mathcal{E}$ is downward-closed, and every $v \in \mathcal{V}$ is a degree 1 hyperedge.
Using the conflict hypergraph $\mathcal{G}_{\mathcal{V},N}$, we can now describe $\mathcal{P}_{\mathcal{V},N,\mathcal{H}_{soft}}$.
\begin{lemma}[Feasible output probabilities (Adapted from \citep{bhagoji2021lower})]
The set of correct classification probability vectors for support points $\mathcal{V}$, adversarial constraint $N$, and hypothesis class $\mathcal{H}_{soft}$ is
\begin{equation}
\mathcal{P}_{\mathcal{V},N,\mathcal{H}_{soft}} = \{q \in \mathbb{R}^{\mathcal{V}} : q \geq \mathbf{0},\, B q \leq \mathbf{1}\} \label{conflict-constaints}
\end{equation}
where $B \in \mathbb{R}^{\mathcal{E} \times \mathcal{V}}$ is the edge incidence matrix of the conflict graph $G_{\mathcal{V},N}$.
\label{lemma: feasible_q}
\end{lemma}
See Appendix \ref{app:proofs} for proof.
Then, using this feasible region as a constraint in our optimization problem in Equation \ref{eq:opt_over_feasible}, we arrive at the following:
\begin{corollary}[Optimal loss as an LP] \label{lem:constraints}
For any distribution $P$ with finite support,
\begin{equation} \label{eq:lp}
1 - L^*(P,N,\mathcal{H}_{\text{soft}}) = \max_q p^T q
\;\; \text{s.t} \; q \ge 0,
\; Bq \le 1.
\end{equation}
\end{corollary}
Corollary \ref{lem:constraints} allows us to compute optimal loss via LP for any dataset and adversary.
\paragraph{Duality and adversarial strategies.}
The dual linear program is
\[
\min_z \mathbf{1}^T z \quad \text{s.t.}\quad z \geq 0, \quad B^Tz \geq p.
\]
Feasible points in the dual linear program are fractional coverings of the vertices by the hyperedges.
Optimal coverings can be interpreted as adversarial strategies.
For each hyperedge $e$, there is an example $\tilde{x}$ such that $\tilde{x} \in N(x)$ for each $(x,y) \in e$.
When the adversary samples the natural example $V$, it can select the hyper edge $E$ to generate the adversarial example using any condition distribution that satifies $\Pr[E=e|V=v] \leq \frac{B_{e,v}z_e}{p_v}$.
Note that $B_{e,v}z_e$ is the amount of coverage of $v$ coming from $e$ and $\sum_e B_{e,v}z_e = (B^Tz)_v \geq p_v$.
Thus for a vertex $v$ such that $(B^Tz)_v = p_v$, there is only one choice for this distribution.
For a vertex $v$ that is over-covered, i.e. $(B^Tz)_v > p_v$, the adversary has some flexibility.
For over-covered vertices, by complementary slackness $q_v=0$ in every optimal $q$, so the optimal classifiers do not attempt to classify $v$ correctly.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\begin{scope}
\draw (-1.5,1.5) node {$\mathcal{V}$};
\fill[red] (0,0.85) circle (1pt) node[above,black] {$u$};
\fill[green] ({0.85*sqrt(3)/2},{-0.85*0.5}) circle (1pt) node[below,black] {$v$};
\fill[blue] ({-0.85*sqrt(3)/2},-{0.85*0.5}) circle (1pt) node[below,black] {$w$};
\draw (0,0.85) circle (0.8);
\draw ({0.85*sqrt(3)/2},{-0.85*0.5}) circle (0.8);
\draw ({-0.85*sqrt(3)/2},{-0.85*0.5}) circle (0.8);
\end{scope}
\begin{scope}[shift={(4,0)}]
\draw (-1.5,1.5) node {$\mathcal{V}'$};
\fill[red] (0,0.7) circle (1pt) node[above,black] {$u'$};
\fill[green] ({0.7*sqrt(3)/2},{-0.7*0.5}) circle (1pt) node[below,black] {$v'$};
\fill[blue] ({-0.7*sqrt(3)/2},-{0.7*0.5}) circle (1pt) node[below,black] {$w'$};
\draw (0,0.7) circle (0.8);
\draw ({0.7*sqrt(3)/2},{-0.7*0.5}) circle (0.8);
\draw ({-0.7*sqrt(3)/2},{-0.7*0.5}) circle (0.8);
\end{scope}
\end{tikzpicture}
\caption{Two possible conflict structures involving three examples, each from a different class. In the right case, all subsets of $\mathcal{V}' = \{u',v',w'\}$ are hyperedges in the conflict hypergraph $\mathcal{G}_{\mathcal{V}',N}$. In the left case, $\{u,v,w\}$ is not a hyperedge in $\mathcal{G}_{\mathcal{V},N}$, but all other subsets of $\mathcal{V}$ are.}
\label{fig:three_examples}
\end{figure}
\paragraph{Three-class minimal examples.}
Corollary \ref{lem:constraints} demonstrates that the optimal loss for the multi-class problem is influenced by hyperedges between vertices which reflect higher order interactions between examples. Figure~\ref{fig:three_examples} shows an important distinction between two types of three-way interactions of examples.
In case one, $L^*(P,\mathcal{H}_{soft},N) = \max(p_u,p_v,p_w,\frac{1}{2})$.
In case two, $L^*(P',\mathcal{H}_{soft},N) = \max(p'_u,p'_v,p'_w)$.
The presence or absence of the size-three hyperedge affects the optimal loss when the example probabilities are close to balanced, i.e. all at most $\frac{1}{2}$.
It is instructive to consider the optimal classifiers and adversarial strategies in the two cases.
For $\mathcal{V}$, when $\frac{1}{2} \leq p_u $, the classifier that returns $(1,0,0)$ everywhere is optimal.
The optimal cover satisfies $z_{\{u,v\}}+z_{\{u,w\}} = p_u$, $z_{\{u,v\}} \geq p_v$, $z_{\{u,w\}} \geq p_w$, $z_{\{v,w\}} = 0$.
Thus when the adversary samples $v$ or $w$, it always produces an adversarial example that could be confused for $u$.
When $\max(p_u,p_v,p_w) \leq \frac{1}{2})$, the classifier that returns $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ everywhere is optimal.
The cover $z_{\{u,v\}} = p_u+p_v - \frac{1}{2}$, $z_{\{u,w\}} = p_u+p_w - \frac{1}{2}$, and $z_{\{v,w\}} = p_v+p_w - \frac{1}{2}$ is optimal and has cost $\frac{1}{2}$.
The adversary produces examples associated with all three edges.
For $\mathcal{V}'$, things are simpler.
The cover $z_{\{u,v,w\}} = \max(p_u,p_v,p_w)$ is always optimal.
When $p_u \geq \max(p_v,p_w)$, the classifier that returns $(1,0,0)$ everywhere is optimal.
\subsection{Optimal Loss for Hard Classifiers}
\label{sec:hard-classifiers}
A hard classifier $h : \mathcal{X} \to \{0,1\}^{[K]}$ has ${0,1}$-valued correct classification probabilities.
When we apply the classifier construction procedure from the proof of Lemma~\ref{lemma: feasible_q} using an integer-valued $q$ vector, we obtain a hard classifier.
Thus the possible correct classification probabilities for hard classifiers are $\mathcal{P}_{\mathcal{V}, N, \mathcal{H}_{soft}} \cap \{0,1\}^{[K]}$.
These are exactly the indicator vectors for the independent sets in the graph on $\mathcal{V}$ formed by the hyperedges of size two in $\mathcal{G}_{\mathcal{V},N}$: the vertices included in the independent set are classified correctly and the remainder are not. Formally, we can express hard classifier loss as:
\begin{align*}
&\phantom{=}
1 - L^*(P, N, \mathcal{H}_{\text{hard}})\\
&= \max_q p^T q
\;\; \text{s.t} \; q \in \{0, 1\},
\; Bq \le 1. \\
&= \max_q p^T q
\;\; \text{s.t} \; q \in \{0, 1\},
\; B^{\leq 2}q \le 1. \\
&= \max_{S \subseteq \mathcal{V} : S \text{ independent in }\mathcal{G}^{\leq 2} } P(S).
\end{align*}
Finding the maximum weight independent set is an NP hard problem, which makes it computationally inefficient to compute optimal hard classifier loss. In Section~\ref{sec:approx}, we will discuss how to obtain an upper bound on $L^*(P, N, \mathcal{H}_{\text{hard}})$ and other truncations of $\mathcal{G}_{\mathcal{V},N}$.
\subsection{Two-class versus Multi-class hard classification}
Now we can summarize the relationships between the two-class and multi-class settings.
The facts mentioned in this section are all well-known in graph and hypergraph theory.
We interpret their consequences for adversarial classification.
When $K=2$, the conflict hypergraph is a bipartite graph.
For bipartite graphs, three different polytopes coincide:
\begin{itemize}
\itemsep0em
\item The independent set polytope, which is the convex hull of the independent set indicators.
\item The fractional independent set polytope, which has a constraint $\sum_{i \in S} q_i \leq 1$ for each clique $S$.
\item The fractional vertex packing polytope, which has a constraint $\sum_{i \in e} q_i \leq 1$ for each edge $e$.
\end{itemize}
Due to this, in the two class setting, $\mathcal{P}_{\mathcal{V}, N, \mathcal{H}_{soft}}$ is the convex hull of $\mathcal{P}_{\mathcal{V}, N, \mathcal{H}_{hard}}$, hard classifiers achieve the optimal loss, and optimal hard classifiers can be found efficiently.
For $K > 2$, all of these concepts become distinct.
An independent set in a hypergraph is a subset of vertices that induces no hyperedges.
In other words, a hyperedge of size $m$ must contain at most $m-1$ vertices from an independent set.
Because the conflict hypergraph is hereditary, only the size-two hyperedges provide binding constraints.
Thus the concept of a hypergraph independent set is not truly relevant for our application.
As seen in Lemma~\ref{lemma: feasible_q}, the fractional vertex packing polytope characterizes performance in soft classification problem.
The fractional independent set polytope of $\mathcal{G}^{\leq 2}$ has no particular significance.
Every hyperedge in $\mathcal{G}$ produces a clique in $\mathcal{G}^{\leq 2}$ but not the reverse.
Furthermore, when $K > 2$ the fractional vertex packing polytope of the conflict hypergraph, i.e. $\mathcal{P}_{\mathcal{V}, N, \mathcal{H}_{soft}}$, can have non-integral extreme points.
This means it can be strictly larger than the independent set polytope.
The first configuration in Figure~\ref{fig:three_examples} illustrates this.
Thus the soft and hard classification problems involve optimization over different spaces of correct classification probabilities.
Furthermore, maximum weight or even approximately maximum weight independent sets cannot be efficiently found in general graphs: the independent set polytope is not easy to optimize over.
In Section~\ref{sec:caro-wei}, we will use an efficiently computable lower bound on the independence number of a graph.
|
2302.10769
|
\section{Introduction}
\label{sec:introduction}
One of the most crucial and challenging steps in the development of robots intended to restore the mobility of the human body after a loss of functional movement due to neurological injuries is the IK of physiological limbs, which consists of computing joint angles configuration based on the predefined input workspace coordinates. Generally speaking, the complexity of the IK problem depends on the geometry of the manipulator and the nonlinearity of its model, which gives the corresponding relation between the task and the joint spaces. Furthermore, IK solution is essential for the real-time control. Thus, it must be precise in order to enable the robot to perform the task successfully. IK techniques can be classified into three categories, namely, analytical method, numerical method, and intelligent method.
\\
The analytical method solves IK by solving a set of closed-form equations that can give the generalized coordinate value that drives the end effector of the manipulator to the predefined target position ~\cite{b1}. Furthermore, this method takes into consideration the geometric insight into the problem and the particular structure of the robot. For an arbitrary robot kinematics, the analytic solutions for IK may not exist, or may be multiple. Whereas numerical techniques are iterative methods which converge to only one solution depending on an initial point. There are several algorithms used to solve IK problems, namely, Moore-Penrose pseudo-inverse method \cite{b2}, cyclic coordinate descent method \cite{b3}, Levenberg Marquardt damped least squares method \cite{b4}, optimization method and multi-objective optimization using genetic algorithm \cite{b6} and so forth. Moreover, the neural network approach \cite{b7} for solving the inverse kinematics is based on the idea of investigating the whole configuration workspace of the robot and selecting the optimal solution.
\\
Indeed, the main advantage of analytical method solution is its accuracy and its efficiency as it delivers results in real-time. In addition, the analytical algorithm computes the valid potential configurations. Nevertheless, the number of solutions is very large when the manipulator has more than 6 degrees of freedom. Moreover, when a solution is impossible, the algorithm would not return an approximate solution that satisfies all the constraints. However, the pseudo-inverse and damped least squares perform very poorly and are slow. Although the cyclic coordinate descent method is simple to implement and computationally fast, but it is difficult to produce smooth motions and consider non-geometric constraints as the minimum energy criteria. While optimization based methods are discrete solve the IK problem by point to point unlike neural network approach.
\\
A set of IK algorithms suitable for the lower limbs were examined in this paper in order to investigate the optimal method regarding efficiency, accurately, computational cost, energy consumption and production of realistic posture. Thus, analytical, numerical and intelligent methods have been used to solve the generalized IK problem taking into consideration position, orientation, and angular constraints.
\\
The rest of this paper is organized as follows. Section \ref{Articulated human leg model} describes the mechanical structure, elaborates the forward kinematics and investigates the workspace analysis and trajectory planning of the human lower limb respectively. While Section \ref{Inverse kinematics of the human leg} focuses on application of the different inverse kinematics methods for human leg. Finally, in sections \ref{Simulation results} and \ref{Discussion and Conclusion}, the simulation results are discussed.
\section{ARTICULATED HUMAN LEG MODEL}
\label{Articulated human leg model}
This section provides a functional description of the lower limbs based on the work of Calais Germain ~\cite{b9}. Firstly, in biomechanics, movements are usually described in a reference which has three planes: sagittal plane (median plane), frontal plane, and transverse plane and three axes named left-right axis (frontal axis), anteroposterior axis (sagittal axis), and craniocaudal axis (superior and inferior axis). In anatomy, the lower limb is decomposed into four segments, the pelvis, the thigh, the leg and the foot, linked together by three joint groups, the hip (or coxofemoral joint), the knee and the ankle. The hip joint rotates around the three perpendicular axes which have been mentioned above. Indeed this articulation has the mechanical characteristic of a spherical joint, i.e. it has three degrees of freedom (DOF). The knee joint permits movement in 1-DOF, flexion-extension in the sagittal plane around left-right axis. While, the ankle joint moves in all three planes, resulting in 3-DOF. So each side of the lower limbs has 7-DOF. However, in this study, we will only consider the movements in the sagittal plane, which means that we will only consider the movements of flexion and extension of each joint. This involves 3-DOF for each side of the lower limbs.
\\
The kinematics of the physiological lower limbs is used to find a relation between the Cartesian coordinates of the big toe E and the generalized coordinate $q=\left[\begin{array}{ccc} \theta_1 & \theta_2 & \theta_3 \end{array}\right]^T $.
Using the modified Denavit-Hartenberg (D-H) convention, the reference frame at each joint of the lower limbs is described in Figures \ref{fig:1} and \ref{fig:2}. Thus, the link parameters of the right kinematic chain is given in Table \ref{tab:1}.
\\Consequently, the relation between the coordinate system of the big toe E and the joint angular displacement is given by the following matrix.
\begin{table}
\centering
\caption{Modified D-H parameters for the right lower limb}
\label{tab:1}
\begin{tabular}{|c|c|c|c|c|}
\hline
Joint & $\alpha_{i-1}$ & $a_{i-1}$ & $d_i$ & $\theta_i$ \\
\hline
1 & 0 & 0 & b & $\theta_1$ \\
2 & 0 & $L_1$ & 0 & $-\theta_2$ \\
3 & 0 & $L_2$ & 0 & $\theta_3$\\
4 & 0 & $L_3$ & 0 & 0 \\
\hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure1.png}
\caption{Model of the right lower limb in a sagittal plane}
\label{fig:1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure2.png}
\caption{Physiological lower limbs diagram}
\label{fig:2}
\end{figure}
\begin{equation}
T_4^{0}=\left[\begin{array}{cccc}
C_{1-23} & -S_{1-23} & 0 & L_3.C_{1-23} +L_2.C_{1-2} +L_1.C_{1}\\
S_{1-23} & C_{1-23} & 0 & L_3.S_{1-23}+L_2.S_{1-2}+L_1.S_{1}\\
0 & 0 & 1 & b\\
0 & 0 & 0 & 1
\end{array}\right]
\label{eq:100}
\end{equation}
\\Where,\\
$C_{1-23} :=\cos(\theta_{1}-\theta_{2}+\theta_{3});$
\\$S_{1-23} :=\sin(\theta_{1}-\theta_{2}+\theta_{3});$
\\$C_{1-2} :=\cos(\theta_{1}-\theta_{2});$
\\$S_{1-2} :=\sin(\theta_{1}-\theta_{2});$
\\$C_{1} :=\cos(\theta_{1});$
\\$S_{1} :=\sin(\theta_{1});$\\
The Monte Carlo method \cite{b12} is used to get the feasible workspace of the lower limbs to evaluate its working ability. The principle of the method is as follows: The Monte Carlo method generates pseudo-random number in the interval $[0,1]$ which respects a uniform distribution. Thereafter, each random sampling generates a set of variable values of the generalized coordinate $q$ by using the following equation:
\begin{equation} \label{Eq:2}
q = q_{min} + \rho(q_{max} -q_{min} )
\end{equation}
Where:
\\$ q_{min}$, $ q_{max}$ : Minimum and maximum range of motion;
\\ $\rho$ : pseudo-random number in the interval $[0,1]$ which respects a uniform distribution;
\\$q $ : the generalized coordinate random value obtained by Monte Carlo method;
This method enables us to obtain a graphical representation of the workspace of the lower limb in the sagittal plane while respecting the joints constraints given in Table \ref{tab:2} as shown in Figure \ref{fig:11}.
\begin{table}
\centering
\caption{Range of motion of the lower limb joints \cite{b13}}
\begin{tabular}{|c|c|c|c|c|}
\hline
Joint & $\theta_i^{min}$ & $\theta_i^{max}$ & Comfort zone & Conditions \\
\hline
i=1 & -20° & 120° & from 15.75° to 39.55° & knee neutral 0° \\
i=2 & 0° & 118° & from 0° to 39.55° & hip neutral 0° \\
i=3 & 50° & 126° & from 77.75° to 103.3° & knee neutral 0°\\
\hline
\end{tabular}
\label{tab:2}
\end{table}
Moreover, the comfort zone is a subset of the corresponding joint range of motion. It is calculated by 35\% of the range of motion. While its center, $q^{comf}_c$, is calculated as follows ~\cite{b13}:
\begin{equation}\label{eq:3}
q^{comf}_c=\frac{q^{comf}_{max}-q^{comf}_{min}}{2}+q^{comf}_h
\end{equation}
where $q^{comf}_{max}$ and $q^{comf}_{min}$ are the maximum and the minimum values of the comfort zone, and $q^{comf}_h$ is the home position of for the generalised coordinate $q$.
\\
Moreover, a trajectory planning is proposed for which the minimum jerk criterion is applied. The jerk approach is considered as the third time derivative of the position. Limiting jerk criterion enables us to ensure the continuity of the acceleration of the joints, which leads to a reduction in the vibrations and to the avoidance of the resonance frequencies ~\cite{b14}. The minimum jerk criteria is obtained by minimizing the function (\ref{eq:4}).
\begin{equation}
J = \frac{1}{2}\int_{0}^{T}{\left(\frac{d^3x(t)}{dt^3}\right)^2}dt
\label{eq:4}
\end{equation}
To satisfy the minimum jerk criteria, the position $x$ is given by a fifth-order polynomial which is interpreted as follows:
\begin{equation}\label{eq:5}
x(t)= s_0+s_1 t+s_2 t^2+s_3 t^3+s_4 t^4+s_5 t^5
\end{equation}
Using this polynomial, it is possible to specify the position, velocity and acceleration at the beginning and the end of motion. Thus, the final formula is given in Equation (\ref{eq:6}) \cite{b27}.
\begin{equation}\label{eq:6}
\left[\begin{array}{c}
s_0 \\
s_1 \\
s_2\\
s_3\\
s_4\\
s_5\\
\end{array}\right]=
\left[\begin{array}{cccccc}
1 & t_0 & t_0^2 & t_0^3 & t_0^4 & t_0^5 \\
0 & 1 & 2.t_0 & 3.t_0^2 & 4.t_0^3 & 5.t_0^4 \\
0 & 0 & 2 & 6.t_0 & 12.t_0^2 & 20.t_0^3\\
1 & t_f & t_f^2 & t_f^3 & t_f^4 & t_f^5\\
0 & 1 & 2.t_f & 3.t_f^2 & 4.t_f^3 & 5.t_f^4 \\
0 & 0 & 2 & 6.t_f & 12.t_f^2 & 20.t_f^3\\
\end{array}\right]^{-1}
\left[\begin{array}{c}
x_0 \\
\dot{x}_0 \\
\ddot{x}_0 \\
x_f \\
\dot{x}_f \\
\ddot{x}_f \\
\end{array}\right]
\end{equation}
\section{Inverse kinematics of the human leg}
\label{Inverse kinematics of the human leg}
In ~\cite{b16}, generating a trajectory in the joint space allows not only to achieve an effective control of the lower limbs but also to avoid the problems occurring with kinematics singularities. Furthermore, it enables to deal with the problems related to the manipulator redundancy like the proposed human leg model. However, the redundancy allows the lower limbs to avoid kinematics limitations and to minimize energy consumption. Therefore, the inverse kinematics yields the corresponding trajectory in the joint space after generating it in the task space. In our case, the reference system $R_0$ is considered stationary with respect to an inertial reference frame $R_U$, in order to avoid an infinite number of solutions. Thus, this section provides different methods for solving the IK of the human leg.
\subsection{Analytical method}
After having defined the forward kinematics, it is essential to determine the IK. In other words, find the joint angles $\theta_1$, $\theta_2$ and $\theta_3$ from the predefined coordinate of the end effector $E (E_x, E_y)$ and its orientation relative to the transverse plane $\theta_0$. Thus, according to Figure \ref{fig:2}, the coordinate space of ankle joint D is given as follows:
\\
\begin{equation}
\label{eq:8}
\bigg\{\begin{array}{c}
D_x=E_x-L_3.S_0 \\
D_y=E_y-L_3.C_0
\end{array}
\end{equation}
\\
Using D-H modified method, the coordinate space of ankle joint is given by:
\begin{equation}
\label{eq:9}
\bigg\{\begin{array}{c}
D_x=L_2.C_{1-2}+L_1.C_{1} \\
D_y=L_2.S_{1-2}+L_1.S_{1}
\end{array}
\end{equation}
Equations (\ref{eq:8}) and (\ref{eq:9}), yields to:
\begin{equation}\label{eq:10}
C_2=\frac{(E_x-L_3.S_0)^2+(E_y-L_3.C_0)^2-L_1^2-L_2^2}{2.L_1.L_2}
\end{equation}
\begin{equation}\label{eq:11}
S_2=\pm\sqrt{1-{C_2}^2}
\end{equation}
Then,\\
\begin{equation}\label{eq:12}
\theta_2=\atantwo (S_2,C_2)
\end{equation}
By using this trigonometric equation, we find:
\begin{equation}
\label{eq:14}
\theta_1= \atantwo(E_x-L_3.S_0,E_y-L_3.C_0)-\atantwo(L_2.S_2,L_1+L_2.C_2)
\end{equation}
and
\begin{equation}\label{eq:15}
\theta_3= \frac{\pi}{2}-\theta_1+\theta_2-\theta_o
\end{equation}
However, if the orientation angle of the foot is not known, there are multiple possible solutions.
\subsection{Numerical methods}
\subsubsection{Cyclic coordinate descent method}
Cyclic Coordinate Descent (CCD) is an algorithm for the iterative solution used to solve inverse kinematics problem. Yotchon et al. \cite{b24} presented a combination of the differential evolution algorithm which is a meta heuristic optimization with CCD method to solve the inverse kinematics problem. This method converges to the target whatever the initial condition. CCD consists of minimizing the joint errors by varying one component of the angular vector at a time. This method can be used in real time although it takes several iterations to reach the objective point~\cite{b17}. One of the main advantages of cyclic coordinate descent is that it is easy to implement. The CCD algorithm consists of measuring the difference between the target and the end-effector vectors. Thereafter, it deduces a rotation matrix to make this difference equal to zero taking into account the joints limitation. This process is repeated for each joint, iterating from the end effector to the root joint of kinematic chains.
\subsubsection{Moore-Penrose pseudo-inverse method}
Moore-Penrose Pseudo-Inverse method (MPPI) is based on Newton Raphson method which consists on solving the following nonlinear equation:
\begin{equation}
N(q_d)\equiv f(q_d)-p_d=0
\end{equation}
with $f$ is the forward kinematics equation, $p_d$ is the target position and orientation and $q$ is the generalised coordinates. Klein et al. \cite{b25}, proved that the pseudo inverse IK method is not repeatable. That means the algorithm is not fast computationally and does not yield a minimum norm of angular joint velocities.
\\
To generalize Newton Raphson's procedure, the Taylor expansion of then forward kinematics function is used around $q_d$ as follows :
\begin{equation}
p_d= f(q_d)=f(q_k)+ \frac{\partial{f}}{\partial{q}}\bigg|_{q_k}(q_d-q_k)
\end{equation}
Thus
\begin{equation}\label{eq:50}
\Delta q= J^+(q_k)(p_d-f(q_k))
\end{equation}
Where,
\begin{equation}
J=\frac{\partial{f}}{\partial{q}}
\end{equation}
And $J^+$ is called the pseudo-inverse of the lower limb's jacobian matrix $J \in \mathbb{R}^{2\times3}$ giving by the following expression:
\begin{equation}
J^+=J^T (JJ^T)^{-1}
\end{equation}
based on (\ref{eq:50}), an iterative method can be used to calculate the joints update :
\begin{equation}
q_{k+1}=q_k+J^+(q_k)(p_d-f(q_k))
\end{equation}
Where $q_k$ and $f(q_k)$ are the generalized coordinates and the end effector position at iteration k, respectively.
\subsubsection{Levenberg Marquardt damped least squares method}
A method to handle the pseudo-inverse problem uses the Levenberg Marquardt Damped Least Squares method (LMDLS). Wampler et al. \cite{b26} proposed a method to choose the optimal damping factor in order to establish a balance between the angular joint velocity and the error.
This approach is based on finding the joints angular error vector $\Delta q$ which minimizes the tracking error and the joint velocities. This corresponds to the minimization of:
\begin{equation}\label{eq:51}
\lVert \Delta x-J\Delta q\lVert ^2-\lambda^2 \lVert\Delta q\lVert ^2
\end{equation}
Where $\lambda >0$ is the damping factor.
\\
Solving (\ref{eq:51}), the damped pseudo-inverse $J^{+\lambda}$ is as follows:
\begin{equation}
J^{+\lambda}=J^T(J^TJ+\lambda I_{(3\times3)})^{-1}
\end{equation}
\\
Thus the joints' updates are giving by the following equation:
\begin{equation}
q_{i+1}=q_i+J^{+\lambda}(q_i)(p_d-f(q_i))
\end{equation}
There are several methods that have been proposed in the literature to select the appropriate and optimum damping factor $\lambda$ ~\cite{b18}, so that the LMDLS method becomes robust against singularity and solvability problems. Among these methods, choose $\lambda$ as a constant value \cite{b19}. To take into consideration the posture of a human, ensuring the natural limits are not exceeded, $q_{min}<q<q_{max}$, the diagonal matrix $\lambda I_{(3\times3)}$ will be replaced by $D(\lambda)$, \cite{b10}.
\begin{equation}
D(\lambda)=\left[\begin{array}{cccc}
\lambda_1 & 0 & 0 \\
0 & \lambda_2 & 0\\
0 & 0 & \lambda_3 \\
\end{array}\right]
\end{equation}
Where $\lambda =\left[ \begin{array}{ccc} \lambda_1 & \lambda_2 & \lambda_3 \end{array}\right]^T$, is defined as follows:
\begin{equation}
\lambda=a{\left(\frac{2(q-q^{comf}_c)}{q_{max}-q_{min}} \right)}^b
\end{equation}
Where a and b are positive numbers, and $q^{comf}_c$ is the center of the comfort zone for joint configurations $q$. The limits of the comfort zone are calculated by $0.35\times$ the limits of the joint range of motion.
\\
Thus, $\lambda$ imposes a restriction to each joint angular value $\theta_i$, $i\in \left\{1,2,3\right\}$. If $\theta_i$ is within its range of motion mentioned in Table \ref{tab:2}, a small value of $\lambda_i$ yields accurate outcomes; and if the joint value $\theta_i$ is close to its limits, a large value of $\lambda_i$ results in a feasible solution.
\subsubsection{Optimization method}
\label{Optimization method}
Tringali et al. \cite {b22} proposed optimal inverse kinematics based on optimization method for a redundant robot manipulator with linear and nonlinear constraints by choosing the appropriate initial conditions. Lu \cite{b23} used optimization method to generate feasible and smooth joint motion while detecting collision in space.
\\
Constrained optimization is based on finding an angular vector $q=\left[\begin{array}{ccc} \theta_1 & \theta_2 & \theta_3 \end{array}\right]^T $, that is a local minimum to the objective function $f(q)$ given as follows:
\begin{equation}
\begin{array}{c}
f(q)=|\sqrt{x_d^2+y_d^2}-r_2| \\
\\r_2=\sqrt{x^2+y^2}\\
\\x=L_3C_{1-2+3}+L_2C_{1-2}+L_1C_{1}\\
\\y=L_3S_{1-2+3}+L_2S_{1-2}+L_1S_{1}\\
\end{array}
\end{equation}
Where $x_d$ and $y_d$ represent the target position coordinates.
The main goal is to identify the realistic configuration by respecting several constraints. One of them is to restrict the angular joint limits. This can be done by using the following barrier function.
\begin{equation}
B_k(q)=f(q)+\frac{1}{k}b(q)
\end{equation}
This method is called interior point algorithm. When $q$ approaches the boundary, $b(q)\rightarrow+\infty$. Where $b(q)$ is the logarithmic barrier function given by the following expression:
\begin{equation}\label{log}
b(q)=-(\log(-(q-q_{max}))+\log(-(q_{min}-q)))
\end{equation}
The other constraints can be expressed as a distance as follows:
\begin{equation}
\bigg\{\begin{array}{c}
x_d=x \\
\\y_d=y\\
\end{array}
\end{equation}
\subsubsection{ Multi-objective optimization genetic algorithm method}
A multi-objective optimization genetic algorithm (MOOGA) is a meta-heuristic algorithm used to solve optimization problems. It is inspired by the evolutionary biology process. Bjoerlykhaug \cite{b22} solved IK using optimization with genetic algorithm in real-time as the robot moves, which leads to decrease the computational time to 50\%. Solving any optimisation problem using genetic algorithm depends on three main different tasks, namely, initial random population, genetic operators and objective evaluation function.
\\
\subsection{Neural networks method}
This subsection investigates the neural networks method to find the IK solution for the human leg. Shah et al. \cite{b20} adopted deep artificial neural networks to solve the inverse kinematics of a 5-axis serial link robotic manipulator. However, the deviation between the end effector and the target resulting from actuator limitations is around 0.2 $mm$, which can be reduced by extra training. Demby’s et al. \cite{b21} evaluated the artificial neural networks performance for solving the IK problem of 4, 5, 6 and 7 DOF robots by using mean square error for the same desired outputs. The authors deduced that the error decreased as the training set increased, which took a long of time to create an effective training.
\\
The neural networks is trained by exploiting the data delivered by the forward kinematics to learn the angular joint values of the configuration space. In other words, the neural networks enables us to identify which joint configuration $q=[\theta_1,\theta_2,\theta_3]^T$ corresponds to the specified end effector position $(x_d,y_d)^T$.
Multilayer perceptron is used for solving the inverse kinematics of the lower limb. Its structure is given in Figure \ref{fig:27}. It has a two-layer feed-forward network, namely, a hidden layer with 10 interconnected sigmoid normal neurons and an output layer with 3 linear output neurons.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure27.png}
\caption{Neural network structure}
\label{fig:27}
\end{figure}
\section{Simulation results}
\label{Simulation results}
This section provides a simulation study of human IK using the methods described above. In order to generate a trajectory as an example, it is necessary to ensure that the starting and final points are in the workspace of the lower limb as described in Figure \ref{fig:11}. Based on ~\cite{b15}, the average walking speed is between $1-1.5~m/s$ for adults without mobility issues. Then, the velocity and acceleration at the beginning and at the end of motion are $1.33~m/s$ and $0~m/s^2$ respectively in this application example, as shown in Table \ref{tab:3}.
\\
\begin{table}
\centering
\caption{Initial and final values for: position, velocity and acceleration of a motion}
\label{tab:3}
\begin{tabular}{|cc|c|c|}
\hline
& & x & y \\
\hline
Position ($m$) & Initial & 0.824628 & -0.0668736 \\
& Final & 0.772227 & 0.481004 \\
\hline
Speed ($m/s$) & Initial & 1.33 & 1.33 \\
& Final & 1.33 & 1.33 \\
\hline
Acceleration ($m/s^2$) & Initial & 0 & 0 \\
& Final & 0 & 0 \\
\hline
\end{tabular}
\end{table}
Thus, assuming that the movement lasts $0.5~sec$. The results of this simulation are illustrated in Figure \ref{fig:11}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure11.PNG}
\centering
\caption{Generated trajectory of the lower limb}
\label{fig:11}
\end{figure}
Using the CCD method, the simulation results of the IK of lower limbs are illustrated in Figure \ref{fig:52}. In order to compute the position error shown in Figure \ref{fig:53}, these resulting values and the forward kinematics specified in Equation (\ref{eq:5}) are used to generate the corresponding end effector trajectory. However, according to the simulation realized in Matlab, this method makes links turn in a specific order opposite to reality which makes the movement of the lower limb appears unnatural, hence this method does not take into consideration the physiological potentiality.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure42.png}
\caption{Joints angular tracking using CCD method. For this simulation, $\theta_1^{min}=0^{\circ}$, $\theta_1^{max}=120^{\circ}$, $\theta_2^{min}=0^{\circ}$, $\theta_2^{max}=117^{\circ}$, $\theta_3^{min}=51^{\circ}$ and $\theta_3^{max}=126^{\circ}$.}
\label{fig:52}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure43.png}
\caption{Position error using CCD method}
\label{fig:53}
\end{figure}
While, Figure \ref{fig:303} shows the simulation results of IK of lower limb using MPPI algorithm. As shown in Figure \ref{fig:28}, the position error is of the order of $0.04 ~mm$. However, this algorithm does not take into consideration angular constraints.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure28.png}
\caption{Joints angular tracking using MPPI method. For this simulation, $\theta_1^{min}=42^{\circ}$, $\theta_1^{max}=63^{\circ}$, $\theta_2^{min}=79^{\circ}$, $\theta_2^{max}=102^{\circ}$, $\theta_3^{min}=61^{\circ}$ and $\theta_3^{max}=109^{\circ}$.}
\label{fig:303}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure40.png}
\caption{Position error using MPPI method}
\label{fig:28}
\end{figure}
\\
Concerning LMDLS technique, the simulation results are illustrated in Figures \ref{fig:18} and \ref{fig:41}. This method shows a high computational cost, which is due to the high complexity of the human leg structure.
\\
Figure \ref{fig:20} shows the resulting angular joint values obtained by applying the optimization algorithm. As shown in Figure \ref{fig:19}, the position error is almost zero. This confirms that the optimization technique performs well from the point of view of accuracy.
\\
Using the same objective function and constraints presented in subsection \ref{Optimization method}, the joints trajectory and the position error obtained by using the MOOGA technique are depicted in Figures \ref{fig:21} and \ref{fig:22}.
As can be seen from Figures \ref{fig:21} and \ref{fig:22}, the MOOGA technique is an accurate method, but the variation of the angular configurations are abrupt and significant.
\\
Moreover, for using neural network method, out of the 127 282 samples collected, 15\% are used for validation, 15\% to test the neural networks, and 70\% of the samples are used for the training of the neural networks. The joint angular values results and the position error are depicted in Figures \ref{fig:26} and \ref{fig:25} respectively. As illustrated in Figure \ref{fig:25}, the neural networks technique is an accurate method where the error is less than 1.6 $10^{-6}$.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure18.png}
\caption{Joints angular tracking using LMDLS method. For this simulation, $\theta_1^{min}=18^{\circ}$, $\theta_1^{max}=43^{\circ}$, $\theta_2^{min}=61^{\circ}$, $\theta_2^{max}=79^{\circ}$, $\theta_3^{min}=87^{\circ}$ and $\theta_3^{max}=118^{\circ}$.}
\label{fig:18}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure41.png}
\caption{Position error using LMDLS method}
\label{fig:41}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure20.png}
\caption{Joints angular tracking using optimisation method. For this simulation, $\theta_1^{min}=18^{\circ}$, $\theta_1^{max}=61^{\circ}$, $\theta_2^{min}=52^{\circ}$, $\theta_2^{max}=101^{\circ}$, $\theta_3^{min}=103^{\circ}$ and $\theta_3^{max}=120^{\circ}$.}
\label{fig:20}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure19.png}
\caption{Position error using optimization method}
\label{fig:19}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figure21.png}
\caption{Joints angular tracking using MOOGA method. For this simulation, $\theta_1^{min}=16^{\circ}$, $\theta_1^{max}=68^{\circ}$, $\theta_2^{min}=20^{\circ}$, $\theta_2^{max}=105^{\circ}$, $\theta_3^{min}=84^{\circ}$ and $\theta_3^{max}=120^{\circ}$.}
\label{fig:21}
\end{figure}
\begin{figure}[htp!]
\centering
\includegraphics[width=0.7\textwidth]{figure22.png}
\caption{Position error using MOOGA method}
\label{fig:22}
\end{figure}
\begin{figure}[htp!]
\centering
\includegraphics[width=0.7\textwidth]{figure26.png}
\caption{Joints angular tracking using neural network method. For this simulation, $\theta_1^{min}=15^{\circ}$, $\theta_1^{max}=59^{\circ}$, $\theta_2^{min}=48^{\circ}$, $\theta_2^{max}=95^{\circ}$, $\theta_3^{min}=89^{\circ}$ and $\theta_3^{max}=104^{\circ}$.}
\label{fig:26}
\end{figure}
\begin{figure}[htp!]
\centering
\includegraphics[width=0.7\textwidth]{figure25.png}
\caption{Position error using neural network method}
\label{fig:25}
\end{figure}
\section{Discussion}
\label{Discussion and Conclusion}
It is inferred from TABLE \ref{tab:5}, that neural networks method root mean square position error is less compared to other methods, but optimization and MOOGA methods also show accurate results where root mean square position error is less than $2.10^{-4}$. Moreover, CCD and neural networks methods are fast computationally compared to other methods.
\begin{equation}
I_{c}= mean\left(\underbrace{\xi\sum_{i=1}^{3}\bigg\vert\frac{d^3\theta_i(t)}{dt^3}\bigg\vert}_{Energy} +\underbrace{ \mu D_{CoM}+\beta\sum_{i=1}^{3}-\left(\log(-(\theta_i(t)-\theta_i^{max}))+\log(-(\theta_i^{min}-\theta_i(t)))\right)}_{Robustness}\right)
\label{eq:55}
\end{equation}
Thereafter, the comfort index $I_c$ shown in Equation (\ref{eq:55}) is used to evaluate the body posture, and it consists of two components:
\begin{itemize}
\item Energy : lower limb must satisfy during walking the minimum energy constraint which is related to the minimum jerk approach.
\item Robustness : the lower limb posture is represented by the generalized coordinates $q=\left[\begin{array}{ccc} \theta_1 & \theta_2 & \theta_3 \end{array}\right]^T $, which is constrained by upper and lower limits. Hence, the logarithmic function which is given by Equation (\ref{log}), increases remarkably when joint angles approach their respective barriers. Besides, the distance between the center of mass of the lower limb and that of whole body given by Equation (\ref{eq:06}), must be minimal in order to overcome the fatigue and the musculoskeletal discomfort. Let $M_i^{seg}$ and $CoM_i$ represent the mass and the coordinate of the center of mass of segment i, respectively, with $i\in [1,2,3]$, then:
\end{itemize}
\begin{equation}
D_{CoM}= \left\vert \frac {\sum_{i=1}^{3}M_i^{seg} CoM_i}{\sum_{i=1}^{3} M_i^{seg}} \right\vert
\label{eq:06}
\end{equation}
With $\xi$, $\mu$ and $\beta$ are homogenization of the comfort index coefficients.
The results in Tab. \ref{tab:5} show a significantly lower comfort index for neural networks method. For cyclic coordinate descent method, the comfort index goes to infinity because the joint angles are near their limits during the motion as depicted in Figure \ref{fig:52}.
\begin{figure}[htp!]
\centering
\includegraphics[width=0.7\textwidth]{figure24.png}
\caption{ Positions of the centers of mass in human body }
\label{fig:my_label}
\end{figure}
\begin{table}
\centering
\caption{Comparative table of IK methods}
\begin{tabular}{|c|c|c|c|}
\hline
IK method & Computational time (s) & RMSE & $I_c$\\
\hline
CCD method & $0.006101$ & $0.0982$ & - \\
\hline
MPPI method & $2.777852 $ & $0.0603$ & 1.1530\\
\hline
LMDLS method & $2.051712$ & $0.0299$ & 1.1551\\
\hline
Optimization method & $0.0363921$ & $9.0951.10^{-6}$ & 1.1514\\
\hline
MOOGA method & $0.5939144$ & $1.3407.10^{-4}$ & 1.1577\\
\hline
NN method & $0.0082538$ & $9.7244.10^{-7}$ & 1.0256\\
\hline
\end{tabular}
\label{tab:5}
\end{table}
\section{Conclusion}
Throughout this paper, a comparative study of human inverse kinematics techniques for lower limbs was presented. Theoretical results showed that the neural networks method is the most efficient compared to all the other methods considered in terms of root mean square position error, computational time and production of realistic posture. Indeed, human comfort during motion is affected by physiological factors including energy consumption, joint angles limits and the distance between center of mass of the lower limb and that of whole body. Also, comfort posture is impacted by environment and psychological factors that are not taken into consideration in this paper.\\
\bibliographystyle{unsrt}
|
2302.10723
|
\section*{Acknowledgments}
This work is supported by the European Union Civil Protection under grant agreement No 783299 (SWIFTERS), by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 739551 (KIOS CoE) and from the Republic of Cyprus through the Directorate General for European Programmes, Coordination and Development.
\section{Background} \label{sec:Background}
\subsection{Multi-Target Tracking}
Multi-target tracking (MTT) algorithms aim to estimate the number and states of multiple targets from noisy sensor measurements in the presence of false alarms or clutter and are commonly divided in two main categories namely data-association based and data-association free. Data-association based MTT methods require to first solve the measurements-to-tracks assignment problem and only then proceed with the multi-target state estimation. On the other hand, data-association free methods (e.g., based on random finite sets (RFSs) \cite{Mahler2003}) can bypass the data association problem and proceed directly with the multi-target state estimation. Depending on the application, this property could be highly desirable since it significantly reduces the computational complexity of the MTT algorithms.
Popular data-association based MTT algorithms include multi-scan and single-scan approaches such as the multi-hypothesis tracking algorithm (MHT) \cite{Reid1979} and the joint-probabilistic data-association filter (JPDAF) \cite{Fortmann1983} respectively, sampling-based techniques such as the Rao-Blackwellized Monte Carlo Data Association filter (RBMCDAF) \cite{Sarkka2004} and finally nonparametric Bayesian methods based on Dirichlet processes such as the work in\cite{Fox2006}. We should point out here that not all of the above methods can estimate the number of targets inside the surveillance area. For instance the JPDAF and RBMCDAF algorithms generally require a fixed and known number of targets.
Data-association free methods are generally based on random finite sets (RFS) and include the probability hypothesis density (PHD) filter \cite{Mahler2003}, the cardinalised PHD
(CPHD) filter \cite{Mahler2007} and the Multi-Bernoulli filter \cite{Vo2009}. These MTT algorithms can simultaneously estimate the number of targets and their states without solving the data-association problem.
The PHD filter \cite{Mahler2003} is the first practical approximation of the multi-target Bayes posterior which propagates in time the first-order statistical moment instead of the full multi-target distribution. The computationally more expensive CPHD filter \cite{Mahler2007} jointly propagates the first-statistical moment and the cardinality distribution. Unlike the PHD and CPHD filters the multi-Bernoulli filter \cite{Vo2009} approximates the true multi-target posterior distribution as multi-Bernoulli distribution and thus it propagates in time the parameters of a multi-Bernoulli distribution.
More recently, the RFS MTT approaches have been extended in order to handle the problem of data-association. More specifically, with the introduction of labeled RFSs \cite{Vo2013lrfs}, the generalized labeled multi-Bernoulli (GLMB) filter \cite{Vo2017lrfs} is able to simultaneously estimate the number of targets and their states from a set of noisy observations in the presence of data association uncertainty, detection uncertainty and clutter. This however comes with an increased computational cost compared to the RFS data-association free methods. A more detailed description of MTT algorithms can be found in \cite{VoBook2015}.
\subsection{Random Finite sets}
A random finite set (RFS) is a finite-set-valued random variable which exhibits the following two properties a) the number of elements in a RFS is random and b) the order of the elements in a RFS is irrelevant. The RFS $X \in \mathcal{F(X})$ is completely specified by a) its cardinality distribution $\rho(n) = p(|X|=n),~ n \in \mathbb{N}_0$ which defines a probability distribution over the number of elements in $X$ and b) by a family of joint probability distributions $p(x_1,...,x_n|n)$,~ $x_1,...,x_n \in \mathcal{X}$ that characterize the distribution of its elements over the state space $\mathcal{X}$.
Finally, the \textit{multi-target} or (\textit{multi-object}) probability density function (pdf) $f(X)$ of the RFS $X$ is given by: $f(X) = f(\{x_1,...,x_n\}) = n! \rho(n) p(x_1,...,x_n|n)$. The following RFSs are relevant in this paper:
\subsubsection{Bernoulli RFS} The Bernoulli RFS $X$ can either be empty with probability $1-r, ~ r \in (0,1)$ or be a singleton set (i.e. its set cardinality is equal to one) with (existence) probability $r$ and with its element distributed over the state space $\mathcal{X}$ according to $p(x)$. The Bernoulli multi-object pdf is given by $f(X) = 1-r, \text{if}\> X = \emptyset$, and $ f(X) = r p(x), \text{if }\> X=\{x\}$.
Thus a Bernoulli RFS can be completely characterized by the parameter set $(r, p(x))$.
\subsubsection{Poisson RFS} The poisson RFS $X$ has a cardinality distribution which is Poisson with parameter $\lambda$ i.e., $\rho(n) = \frac{e^{-\lambda}\lambda^n}{n!}, ~n =0,1,2... $ and elements which are independent and identically distributed (IID) random variables and distributed according to $p(x)$ on $\mathcal{X}$. The multi-object pdf is given by $ f(X) = e^{-\lambda}\prod_{x \in X} \lambda p(x)$.
As we have already mentioned the PHD filter \cite{Mahler2003} is an approximation of the multi-target Bayes filter in which the first-order statistical moment or the probability hypothesis density (PHD) is propagated through time instead of the full multi-target distribution. More specifically, the PHD is the conditional density function $D_k(x|Z_{1:k})$ on single targets (objects) $x \in \mathcal{X}$ which when integrated over any region $R \subseteq \mathcal{X}$ of the state space $\mathcal{X}$ gives the expected number of targets (objects) $\hat{n}_k$ contained in $R$ i.e., $\int_{R} D_k(x|Z_{1:k}) dx = \hat{n}_k(R)$.
The multi-target state $\hat{X}_k=\{\hat{x}^1_k,\cdots,\hat{x}^{\hat{n}_k(R)}_k \}$, can be estimated as the $\hat{n}_k$(R) highest local maxima of the PHD. The PHD filter operates recursively in two steps i.e., a time prediction step in which the PHD $D_{k|k-1}(x|Z_{1:k-1})$ is predicted for the next time-step and a measurement update step $D_{k}(x|Z_{1:k})$ in which the received measurement set $Z_k$ at time $k$ is incorporated into the PHD as shown below:
\begin{align} \label{eq:PHDrecur}
& D_{k|k-1}(x|Z_{1:k-1}) = b_{k}(x) ~+ \\
&\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\> \int_{\mathcal{X}} p_S(x^\prime) \pi_{k|k-1}(x|x^\prime) D_{k-1}(x^\prime|Z_{1:k-1}) d x^\prime \notag \\
& D_{k}(x|Z_{1:k}) = \!\!\Big[1 - p_D(x) \Big]D_{k|k-1}(x|Z_{1:k-1}) ~+ \\
&\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\> \Bigg[ \underset{z \in Z_k}{\sum} \frac{p_D(x) g_k(z|x)}{\kappa(z)+\tau(z)} \Bigg] D_{k|k-1}(x|Z_{1:k-1}) \notag
\end{align}
\noindent where $b_{k}(x)$ is the PHD of target births, $p_S(x)$ is the probability that a target with state $x$ will survive in the next time step, $\pi_{k|k-1}(x|x^\prime)$ is the single-target transition density, $p_D(x)$ is the target detection probability, $g_k(z|x)$ is the measurement likelihood function, $\kappa(z) = \lambda_c f_c(z)$ is the PHD of false alarms with rate $\lambda_c$ and spatial density $f_c(.)$ and finally $\tau(z) = \int p_D(x^\prime) g_k(z|x^\prime) D_{k|k-1}(x^\prime|Z_{1:k-1}) d x^\prime $.
\section{Conclusion} \label{sec:Conclusion}
In this work a novel decentralized cooperative multi-agent searching-and-tracking framework has been proposed. The proposed approach recursively computes and propagates in time the \textit{searching-and-tracking} (SAT) density which is used by the agents to devise efficient cooperative searching and tracking strategies. The proposed framework is flexible and accounts for many of the challenges present in search and rescue missions including the unknown and time varying number of targets, the noisy sensor measurements, the uncertain target dynamics and the limited sensing range of the agents. In the future we plan to investigate in more detail the communication aspects (e.g., the communication overhead vs performance) of the proposed approach. Although, the event-based strategy suggested in this work can reduce the communication overhead of the searching-and-tracking task, it sacrifices the overall performance of the team, since no global coordination is guaranteed. Future work aims to investigate how communication-efficient cooperation \cite{Leung2010} can be enabled for the problem tackled in this work and how efficient distributed transmission protocols \cite{Liu2017d} can be utilized in order to further reduce the communication burden between the agents.
\section{Evaluation}
\label{sec:Evaluation}
\subsection{Experimental Setup}
In our experimental setup we assume that the targets maneuver in an area of 100m $\times$ 100m and that the single target state at time $k$ is described by $x_k = [\text{x},\dot{\text{x}},\text{y},\dot{\text{y}}]^\top$ i.e. position and velocity components in $xy$-direction. The target dynamics are piecewise linear according to the near constant velocity model with the process noise being Gaussian. The single target transitional density is given by $\pi(x_k|x_{k-1}) = \mathcal{N}(x_k;Fx_{k-1},Q)$ where:
\begin{equation*}
F = \begin{bmatrix}
1 & T & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & T\\
0 & 0 & 0 & 1\\
\end{bmatrix}, ~
Q = \begin{bmatrix}
T/3 & T/2 & 0 & 0\\
T/2 & T & 0 & 0\\
0 & 0 & T/3 & T/2\\
0 & 0 & T/2 & T\\
\end{bmatrix}
\end{equation*}
with sampling interval $T=1$s. The target speed is initialized to $1$m/s in the $x$-direction and $1$m/s in the $y$-direction. The target survival probability from time $k-1$ to time $k$ is constant $p_{s,k}(x_{k-1})=0.99$ and does not depend on the target's state. Once an agent detects a target it receives range and bearing measurements thus the measurement model is given by $h_k(x_k,s_k) = \left[\norm{s_k -\text{H}x_k}_2,~ \text{arctan}\left(\frac{s_y-\text{y}}{s_x - \text{x}}\right) \right]$ where $\text{H}$ is a matrix which extracts the target position from its state vector. The single target likelihood function is then given by $g(z_k|x_k,s_k) = \mathcal{N}(z_k;h_k(x_k,s_k),\Sigma^\top \Sigma)$ and $\Sigma$ is defined as $\Sigma = \text{diag}(\sigma_\zeta,\sigma_\phi)$. The standard deviations $(\sigma_\zeta,\sigma_\phi)$ are range dependent and given by $\sigma_\zeta = \zeta_0 + \beta_\zeta \norm{s_k-\text{H}x_k}_2^2$ and $\sigma_\phi = \phi_0 + \beta_\phi \norm{s_k-\text{H}x_k}_2 $ respectively with $\zeta_0 = 1$m, $\beta_\zeta = 5\times10^{-5}\text{m}^{-1}$, $\phi_0 = \pi/180$rad and $\beta_\phi=10^{-5}\text{rad}/\text{m}$.
Moreover, the agent receives spurious measurements (i.e. clutter) with fixed Poisson rate $\lambda_c = 10$ uniformly distributed over the measurement space. Target births are distributed inside the agent sensing area with average rate of 3 births per time-step.
The agent's sensing model parameter $p_D^{\text{max}} = 0.99$ and the agent sensing area is $\mathcal{S}_{10}(s_k) = 10^2 ~\text{m}^2$.
The agent's dynamical model has radial displacement $\Delta_R=2$m, $N_R=2$ and $N_\theta = 8$ which gives a total of 17 control actions, including the initial position of the agent. Without loss of generality we assume in this evaluation that the function $J_k(x)$ is constant, state independent and equal to $J_k(x) = 0.999~ \forall x, k$.
The agent communication range is $C_R=50$m, the Renyi divergence parameter is set to $\alpha = 0.5$ and the tracking overlap threshold is $Q^{Th} = 0.9$m during a time-window of length 3.
Finally, we should point out that in our implementation we have used the Sequential Monte Carlo (SMC) version of the PHD filter \cite{phd_1} for which the convergence properties have been established in \cite{phd_2}. Please note that under the linear, Gaussian assumptions on the target dynamics and measurement model the more efficient Gaussian-mixture \cite{gmphd} (GM-PHD) implementation of the PHD filter can be used which does not require the computationally expensive clustering step of the SMC-PHD filter, which is used to partition the multi-target particle system into distinct target tracks. For mild non-linearities the GM-PHD filter can also be utilized by approximating the Gaussian-mixture using the extended and unscented Kalman filters. However, the severe limitation of the GM-PHD filter is that it requires a constant probability of detection (i.e., $p_D(x) = p_D$) and a constant target survival probability (i.e., $p_S(x) = p_S$). Lastly, more recently an alternative formulation of the SMC-PHD filter \cite{phd_3} has been proposed which avoids the particle system clustering step.
\subsection{Results}
\begin{figure}
\centering
\includegraphics[scale=0.4]{fig6.pdf}
\caption{The figure shows the performance of the cooperative multi-agent path-planning and searching technique proposed in this work versus a random search scheme.}
\label{fig:fig6}
\vspace{-6mm}
\end{figure}
First we compare the performance of the proposed cooperative searching technique against a baseline random search scheme. To be more specific in this random search scheme a) no path-planning is performed by the agents i.e., the search-planning technique discussed in subsection \ref{ssec:cooperative_search} is not applied and b) there is no communication between the agents and thus no joint search-plans are produced i.e., Alg. \ref{alg:oppsearch} is not applied. Instead each agent, randomly selects and applies an admissible control action $u^k \in \mathbb{U}^{k}$ towards the unvisited regions. On the other hand the proposed technique performs a coordinated search-planning with $C_R=50$m.
We have conducted 50 Monte Carlo trials where we have randomly initialized the agents (uniformly distributed) inside the surveillance area and let the system run for 100 time-steps, measuring the percentage of searched area over time. This procedure is conducted for the proposed system and for the baseline random search scheme. Figure \ref{fig:fig6} shows the results of this experiment for the case of 2 and 4 agents. The figure shows that as the number of agents increases the searching performance increases as well which is reasonable. In addition the proposed optimized cooperative search approach significantly improves with the number of agents and outperforms the baseline method. This is due to the more efficient coordinated search planning and the communication between the agents. We should note here that the searched area in this experiment never reaches the 100\%. This is due to the decay of the searched density over time. Different decay rate and/or number of agents will result in different results.
The next experiment demonstrates the impact of the communication range $(C_R)$ on the performance of the proposed cooperative multi-agent searching approach.
Two different values of the communication range i.e. $C_R=10$m and $C_R$ = 50m are investigated. More specifically, Fig. \ref{fig:fig5} shows the percentage of searched area over time during 100 time-steps for $C_R=10$m (Fig. \ref{fig:fig5}a) and $C_R=50$m (Fig. \ref{fig:fig5}b). The figure shows the average value obtained from 50 MC trials with the agents randomly spawned inside the surveillance area. As we can observe from the figure, the increased communication range between the agents significantly improves the performance of the proposed approach. This is because the agents compute search paths cooperatively and thus the search task is solved jointly.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig5.pdf}
\caption{The figure illustrates the impact of communication range on the performance of the cooperative multi-agent searching. (a) Communication range $C_R=10$m, (b) Communication range $C_R=50$m.}
\label{fig:fig5}
\vspace{-6mm}
\end{figure}
In the next experiment, we study the joint search-and-track behavior of the proposed system and we use the optimal sub-pattern assignment (OSPA) error \cite{Schuhmacher2008} to quantify its performance. The OSPA metric is defined as the distance between two sets of points. It is used to jointly characterize the dissimilarity in the number of points and the values of the points in the respective sets. Since the output of the tracking approach utilized in this work is an estimated set of points in each time-step (i.e., the targets being tracked), OSPA metric will give us the deviation of this estimated set of points from the ground-truth set of points.
More specifically, for this experiment 10 targets are randomly spawned from the center of the surveillance area with random headings, following a nearly constant-velocity motion model. The average target life-time is 60 time-steps. At time-step $k=1$ we uniformly distribute a fixed number of agents inside the surveillance area and we let the system run for 100 time-steps. Figure \ref{fig:fig7} shows the average OSPA error (OSPA order=2, $c=50$m) over 50 MC trials for 2, 3, 4 and 5 agents operating with communication ranges of $C_R=10$ in Fig. \ref{fig:fig7}a, and $C_R=50$ in Fig. \ref{fig:fig7}b. We first observe that the tracking accuracy of the proposed multi-agent system increases with the number of agents. The figure shows that the OSPA error starts high but subsequently decreases. This is because initially the agents start in search mode however, when at some point a target is detected the respective agent switches to tracking mode which results in a decrease in the OSPA error.
Finally, we observe that the increased communication range results in improved multi-agent cooperative searching which in turn results in better target discovery and thus improved tracking accuracy, as shown in Fig. \ref{fig:fig7}b.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig7.pdf}
\caption{The figure illustrates the search-and-track performance of the proposed system by means of the OSPA error for two configurations of the communication range: (a) $C_R=10$m, (b) $C_R=50$m.}
\label{fig:fig7}
\vspace{-4mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.4]{fig8.pdf}
\caption{The figure shows the percentage of the searched area for the tasks of cooperative searching versus cooperative search-and-track for the case of 2, 3, 4 and 5 agents.}
\label{fig:fig8}
\vspace{-6mm}
\end{figure}
For the next experiment, we followed a similar setup with the one described in the previous paragraph where in the first case (termed Search) a number of agents (i.e., 2, 3, 4 and 5) with $C_R=50$, are uniformly distributed inside the surveillance area. In the second case (termed SAT), in addition to the agents, we uniformly distribute 10 targets inside the surveillance area, with random headings and average life-time of 60 time-steps. We let the system run for 100 time-steps and we measure the percentage of searched area over time. Figure \ref{fig:fig8} shows the average percentage of the searched area over 60 MC trials. As we can observe, the searching performance drops during the searching-and-tracking (SAT) task compared to the Search task. When an agent detects the presence of targets, automatically switches to tracking mode which produces sub-optimal search results. One possibility to mitigate this, which is left for future work, would be with a target hand-over strategy i.e., the agents would hand over targets to their peers who have search plans that align better with respect to the target's trajectory.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig11.pdf}
\caption{The figure shows a) the average ratio of target tracking time to target lifetime as a function of the number of agents and communication range b) the performance gain from the use of tracking overlap detection and resolution.}
\label{fig:fig10}
\vspace{-6mm}
\end{figure}
Our two final experiments aim to investigate a) the expected time for which a target can be tracked successfully in a multi-agent, multi-target setting and b) the performance improvement gained with the use of the tracking overlap detection and resolution approach discussed in Sec. \ref{ssec:multi_agent_tracking}(b).
In order to investigate the expected time for which a target can be tracked successfully we have used the following procedure: We first randomly generate 20 target trajectories inside the surveillance region with average lifetime of 30 time-steps. The target birth/death times vary as the targets enter and exit the surveillance region and their birth locations are uniformly distributed inside the whole area of $100\text{m} \times 100\text{m}$. The targets move with an average speed of $1$m/s in the $x$-direction and $1$m/s in the $y$-direction. The agents are uniformly distributed inside the surveillance area. We let the system run for 100 time-steps and we measure the time for which the agents successfully track the targets. This time is then normalized with the actual lifetime of the targets (i.e., the ground-truth length of time for which a target evolves inside the surveillance area). We conduct 50 Monte Carlo trials with 2, 4, 6, 8 and 10 agents and with communication ranges of 20m and 40m. The results of this experiment is shown in Fig. \ref{fig:fig10}a. As we can observe the expected tracking time increases with the number of agents and with the communication range. The average ratio of tracking time to the total target lifetime starts from around 0.25 for 2 agents and reaches 0.83 for 10 agents operating with $C_R=40$m. A similar trend is also true for the communication range of 20m. In this setup, the increase in the communication range increases the search efficiency through cooperation which as a result improves the expected tracking time.
Finally, in order to investigate the performance gain from the proposed tracking overlap detection and resolution approach we have conducted a similar experimental setup (i.e., same parameters as before unless otherwise noted) in which we randomly spawn 15 targets and 5 agents and we fix the communication range to 20m. Figure \ref{fig:fig10}b shows a) the average ratio of tracking time to total target lifetime and b) the average ratio of searched area to total area over 50 Monte Carlo trials with and without tracking overlap detection (TOD). As we can observe the proposed tracking overlap detection and resolution approach increases both the multi-agent tracking accuracy and the searching performance of the system. This is because TOD allows the overlapping agents to disengage from tracking and switch to searching. As a result the area is searched more efficiently which in turn improves the target discovery and ultimately improves the tracking performance.
\section{Introduction} \label{sec:Introduction}
One of the biggest challenges today’s society faces is its resilience to severe disasters. First responders currently rely on a number of conventional methods to gather information that are time consuming, while the descriptive character of the collected information often lacks accuracy, eloquence and the necessary level of detail. In this work, we envision that a team of autonomous mobile agents (e.g., drones) could become an important technological tool to aid the work of the rescuers. Under this setting, the mission of one or more drone agents is to assist first responders by conducting the following important tasks: a) search the area for situational assessment, and b) detect and track victims as accurately as possible. More specifically, in a cooperative searching and tracking mission, multiple agents are tasked to cooperatively search a certain area of interest in order to discover survivors while at the same time keeping track of those survivors already detected.
This work builds upon the theory of random finite sets (RFS) and proposes a cooperative multi-agent framework for searching and tracking missions that takes into account the unknown and time varying number of survivors, the noisy sensor measurements and the limited sensing range of the agents. In addition, efficient cooperative searching-and-tracking strategies are devised which allow the agents to generate joint search-plans and detect and resolve tracking overlaps.
The rest of the paper is organized as follows. Section \ref{sec:Related_Work} reviews the existing literature on the problem of searching and tracking by single and multiple agents. Section \ref{sec:Background} provides a brief overview on multi-target tracking (MTT) and on the theory of random finite sets (RFS). Section \ref{sec:system_model} presents the modeling assumptions of the proposed framework and Section \ref{sec:proposed_approach} presents the details of the proposed approach. Finally, Section \ref{sec:Evaluation} conducts an extensive performance analysis and Section \ref{sec:Conclusion} concludes the paper.
\section{Proposed Approach} \label{sec:proposed_approach}
In this section we first describe how the proposed approach recursively propagates in time the SAT-density and then we discuss how using the SAT-density the agents cooperate to produce joint search-plans and resolve tracking overlaps.
\subsection{Searching-and-Tracking Density}
During a search and track mission, a single agent is required to be able to perform the following tasks: a) simultaneously estimate the time-varying number of targets and their states from a sequence of noisy measurements, and b) search the surveillance region for targets as efficiently as possible.
The first task can be accomplished by recursively computing and propagating in time, the PHD of the full multi-target posterior distribution of the true targets using the PHD filter \cite{Mahler2003}. In order to accomplish the second task, the agent needs to: a) keep track of the visited (i.e., searched) and unvisited regions of the surveillance area, b) estimate when and how often certain search regions need to be revisited and c) generate efficient search plans for searching the area.
In order to perform the aforementioned tasks, the agent stores a discrete representation of the environment in its memory in the form of a graph $G=\{\mathcal{V},\mathcal{E}\}$ termed as \textit{search map}, where each node $v \in \mathcal{V}$ corresponds to a region $r_v \subset \mathcal{A}$ in the surveillance area where $\bigcup_v r_v = \mathcal{A}$. The nodes $v$ of this graph correspond to virtual targets i.e., the location of each virtual target $x \in X^0$ is represented by a node $v \in \mathcal{V}$ with $|\mathcal{V}|=|X^0|$. The agent recursively computes the \textit{search value} $p^\text{search}(r_v) \in [0,1], v\in \mathcal{V}$ of each region and uses this information to decide how often to visit a particular region and how to generate search-plans for efficiently searching the surveillance area.
With this in mind, we can now discuss how the agent recursively computes the \textit{searching-and-tracking density} or SAT-density. In essence the SAT-density includes two components: a) the search density of the surveillance area and b) the target density inside the agent's sensing range. The search density is used to indicate which areas in the environment have been searched and how often certain regions need to be searched, whereas the target density is used to estimate the number and states of the targets inside the agents sensing range.
The SAT-density denoted as $\tilde{D}_{k}(\bm{x}|Z_{1:k}) ~\text{for}~ \bm{x} \in \mathcal{X} \times \{0,1\}$ is a compound density which includes the target density on true targets and the search density on virtual targets. The SAT-density is recursively computed using a prediction and update step i.e., $\tilde{D}_{k-1}(\bm{x}|Z_{1:k-1}) \rightarrow \tilde{D}_{k|k-1}(\bm{x}|Z_{1:k-1}) \rightarrow \tilde{D}_{k}(\bm{x}|Z_{1:k})$. In what follows we denote the target density on true targets as $\tilde{D}_{k}(x \in \mathcal{X}^1)$ and the search density on virtual targets as $\tilde{D}_{k}(x \in \mathcal{X}^0)$ where the dependance on the measurements $Z_{1:k}$ is dropped for notational convenience. The prediction step is given by:
\begin{align}
&\tilde{D}_{k|k-1}(x \in \mathcal{X}^1) = b_{k}(x) ~+ \label{eq:p1} \\
&\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\> \int_{\mathcal{X}^1} p_S(x^\prime) \pi_{k|k-1}(x|x^\prime) \tilde{D}_{k-1}(x^\prime) d x^\prime \notag \\
&\tilde{D}_{k|k-1}(x \in \mathcal{X}^0) = \tilde{D}_{k-1}(x\in \mathcal{X}^0) ~ \times \label{eq:p2} \\
&\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\> \Big[ \big(1 - \mathds{1}_{\mathcal{S}_a}(x ,s_{k-1})\big) J_k(x) + \mathds{1}_{\mathcal{S}_a}(x ,s_{k-1}) \Big] \notag
\end{align}
\noindent where Eqn. (\ref{eq:p1}) is due to the PHD filter i.e., the prediction of the probability hypothesis density of the true targets before the collection of target measurements. On the other hand, Eqn. (\ref{eq:p2}) computes the predicted search density on the state space of virtual targets where the term $\tilde{D}_{k-1}(x\in \mathcal{X}^0)$ is the search density of the previous time step, the indicator function $\mathds{1}_{\mathcal{S}_a}(x ,s_{k-1})$ was defined in subsection \ref{ssec:sensing_model} and finaly $J_k(x) \in [0,1]$ is a function that determines the decay value of the virtual target with state $x$. In essence, the states of all virtual targets outside the agent's sensing range are adjusted accordingly to reflect the fact that they are not being observed. This property is used to generate search plans which will guide the agent to visit areas that have not been recently visited. The updated SAT-density is given by:
\begin{align}
& \tilde{D}_{k}(x\in \mathcal{X}^1) = \Big[1 - p_D(x, s_k) \Big] \tilde{D}_{k|k-1}(x\in \mathcal{X}^1) ~+ \label{eq:u1} \\
& \>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\> \underset{z \in Z_k}{\sum} \frac{p_D(x, s_k) \cdot g_k(z|x,s_k)}{\kappa(z)+\tau(z)} \tilde{D}_{k|k-1}(x\in \mathcal{X}^1) \notag \\
&\tilde{D}_{k}(x\!\!\in\!\! \mathcal{X}^0)\!\! =\!\!\frac{\mathds{1}_{\mathcal{S}_a}(x ,s_{k})}{|\mathcal{A}|}\!\! + \!\!\Big[1\!\!-\!\!\mathds{1}_{\mathcal{S}_a}(x ,s_{k})\Big] \tilde{D}_{k|k-1}(x\!\!\in\!\! \mathcal{X}^0) \label{eq:u2}
\end{align}
\noindent In the above, Eqn. ({\ref{eq:u1}}) is the update step of the PHD filter, in which the received measurement set $Z_k$ is used to compute the posterior probability hypothesis density of the true targets whereas Eqn. ({\ref{eq:u2}}) is used to update the search density of virtual targets. In essence the search density is adjusted inside the agents sensing range to account for the agent's updated position $s_k$.
The search value $p^\text{search}_k(r_v)$ of a particular region $r_v \subset \mathcal{A},\>v\in\mathcal{V}$ can be computed by integrating the search density in $r_v$ as follows:
\begin{equation} \label{eq:search_value}
p^\text{search}_k(r_v)=\frac{\int_{r_v}\tilde{D}_k(x \in \mathcal{X}^0) dx}{|r_v||\mathcal{A}|^{-1}}
\end{equation}
where $|r_v|$ is the total area of region $r_v$. On the other hand, the number of true targets $\hat{n}_k$ (rounded to the nearest integer) inside the area $R \subseteq \mathcal{A}$ can be computed by integrating the target density in $R$ as $\hat{n}_k(R) = \int_{R} \tilde{D}_k(x \in \mathcal{X}^1) dx$ and the multi-target state $X^1_k$ can be estimated by finding the $\hat{n}_k(R)$ highest peaks of the PHD. This procedure is due to the PHD filter.
\subsection{Multi-agent Searching} \label{ssec:cooperative_search}
The search objective is to find the optimal control actions that will move the agent along areas that have not been explored for some time and could potentially reveal new targets. To address this challenge, we first discuss how searching takes into account the search map derived from the SAT-density and how low level controls employ the computed paths to steer the agent across the field.
\textbf{a) Search planning:} Given the search map $G=(\mathcal{V},\mathcal{E})$ where the set of edges in $\mathcal{E}$ connect adjacent nodes, the cost $c_{ij}$ on edge $i\mapsto j$ is defined as the Euclidean distance between the particular adjacent regions in the field. For each node on this graph, a search value $p^{\text{search}}(r_v),\>v\in\mathcal{V}$ is computed using Eqn. (\ref{eq:search_value}). This value varies between 0 and 1, where 0 indicates that the particular node (and hence region in the field) has not been searched and 1 indicates that the region has just been visited. The SAT-density recursion in Eqn. (\ref{eq:p2}) and (\ref{eq:u2}) indicates how the search value decays over time in order to steer agents to revisit particular regions in the field.
Using $p_k^\text{search}$, we then define the set of unvisited nodes $\bar{\mathcal{V}}$ as the set of nodes for which the search value goes below a certain threshold, i.e., $p_k^\text{search}(r_v)\leq\beta, v\in\bar{\mathcal{V}}$ and thus indicating that those nodes need to be revisited. Given $\bar{\mathcal{V}}$ and the current agent state $s_k$ of an agent, we would like to compute paths that visit nodes in $\bar{\mathcal{V}}$ with the least cost $c_{ij}$ in order to search the whole region for targets.
To compute optimal paths for each agent, we formulate the following integer linear program $(P1)$, with the objective of minimizing the total traversal cost.
\begin{align}
\mathrm{(P1)}&\min\>\>\sum_{(i,l)\in\mathcal{E}} c_{il} y_{il} \label{eq:objective_P1}\\
\mathrm{s.t.} & \sum_{l: (i,l) \in\mathcal{E}} y_{il} \geq \left \{ \begin{array}{ll}
0, & \forall\> i\in\mathcal{V}\\
1, & \forall\> i\in\bar{\mathcal{V}}
\end{array}
\right.\label{eq:nodes_visited}\\
& \sum_{l: (i,l) \in\mathcal{E}} y_{il} - \sum_{l: (l,i) \in\mathcal{E}} y_{li} = 0 \> i\in\mathcal{E} \label{eq:conservation_of_flow}\\
&\!\sum_{l:(s_k,l) \in\mathcal{E}} y_{s_kl} = 1 \label{eq:connected_to_s}\\
& \!\!\!\! M \!\!\!\!\sum_{(i,l) \in \mathcal{E}(\mathcal{Q})}\!\!\!\!\! y_{il} \geq \!\!\!\!\sum_{(i,l) \in \mathcal{E}'(\mathcal{Q})} \!\!\!\!\!y_{il} \> \forall\> \mathcal{Q} \subset \mathcal{E}, \mathcal{Q} \neq \emptyset, \label{eq:ASEC}\\
& y_{il}\in\mathbb{Z^+} \label{eq:domains}
\end{align}
\noindent The integer variable $y_{il}$ in $(P1)$ indicates the selected route edges of the agent's path and the number of times that agent traverses the particular edge $(i,l)$. The objective function in (\ref{eq:objective_P1}) ensures that the least-cost path is selected. Constraints (\ref{eq:nodes_visited}) ensure that all nodes in the unvisited list $\bar{\mathcal{V}}$ are visited at least once. Note that through this constraint, we allow each agent to visit a node more than once if that is necessary to minimize the total route cost. The conservation of flow constraints at each node $i$ are given in constraint equations (\ref{eq:conservation_of_flow}). In order to ensure that the computed path is connected we impose constraints \eqref{eq:connected_to_s} and \eqref{eq:ASEC}. The former constraint simply ensures that any resulting path starts from state $s_k$ and the latter constraint ensures connectness. In these constraints, for any subset $\mathcal{Q} \subset \mathcal{V}$, we define $\mathcal{E}(\mathcal{Q})$ to be the set of edges with only one end in $\mathcal{Q}$ and let $\mathcal{E}'(\mathcal{Q})$ to be the set of arcs with both ends in $\mathcal{Q}$. The right hand side of the constraint is the total flow in $\mathcal{Q}$ and the sum on the left hand side is the total flow in and out of $\mathcal{Q}$. Thus if there is some flow in $\mathcal{Q}$ then the right hand side is positive which forces the left hand side to be positive which means that the flow in $\mathcal{Q}$ is connected to nodes outside of $\mathcal{Q}$. This eliminates disconnected flow cycles but does not eliminate flow cycles that form a connected path.
When solved to optimality, $(P1)$ computes the best alternative path at the minimum cost. However due to the exponential number of constraints in Eqn. (\ref{eq:ASEC}), $(P1)$ is computationally hard to solve in practice. Hence, an alternative heuristic approach is followed in the sequel to devise a path considering the constraints as expressed in $(P1)$. Starting at $s_k$, paths are built by adding new edges with the least cost $c_{il}$, beginning from the head node of the last edge added. The process terminates when all nodes have been visited or when no more edges can be added.
\textbf{b) Search Control:} Given the computed path sequence, the objective is then to take a control action $u_k \in \mathbb{U}_k$ that will move each agent across the designated path. To achieve this, a list $\bar{\mathcal{V}}$ of nodes to-be-visited is maintained, and each node is marked as visited whenever the agent moves to a position where the particular node is within its sensing range. The search control objective can be expressed as follows: $u_{k}^{\star} = \underset{u_k}{\arg\min}~ \xi_\text{search}(u_{k},v)$ where $v\in\bar{\mathcal{V}}$ indicates the location of the next unvisited node in the list and $\xi_\text{search}$ returns the Euclidean distance between the position of the agent (when the hypothetical control action $u_k$ is applied) and the next unvisited node $v$. By iteratively visiting the path sequence, the envisioned look-ahead search control is achieved by each agent.
\textbf{c) Search Cooperation:} Whenever two or more agents are in communication range they exchange their search densities and merge their copies using a simple max operation of local and received values. A fused search map is then computed which contains the search-path histories of the cooperating agents. The agents can then compute a joint search-plan as follows: Let $\bar{S}$ be the subset of agents in communication range and assume that each agent knows the number $|\bar{S}|$ and position $s_k^j,\>j\in \bar{S}$ of cooperating agents in its vicinity. Iteratively, each agent computes $|\bar{S}|$ paths incrementally by adding one node at a time in each agent's path from the list of all unvisited nodes in $\bar{\mathcal{V}}$, until there are no more unvisited nodes. A new node is added in an agent's path only if the head node of all possible edges to traverse (starting from the edge with the least cost) is not flagged as visited and the tail node of that edge is the last node added in the particular walk. The steps followed by each agent are detailed in Algorithm \ref{alg:oppsearch}.
\begin{algorithm}
\caption{Opportunistic multi-agent search cooperation}\label{SOC}
\begin{algorithmic}
\REQUIRE $|\bar{S}|$ and $s_k^j,\>j\in \bar{S}$, $G=(\mathcal{V},\mathcal{E})$ and let $\bar{\mathcal{V}}=\mathcal{V}$
\STATE {$T_j \gets s_k^j$} \COMMENT{$T_j$ contains the path of agent $j\in \bar{S}$}
\STATE Set $\bar{\delta}=0$ \COMMENT{Indicating the tree to be uppended first}
\WHILE{($|\bar{\mathcal{V}}|\neq \emptyset$)}
\STATE $\bar{\delta}=\bar{\delta}+1, j= \bar{\delta}\>\text{mod}\> |\bar{S}|$
\STATE {$\min_{l\in\bar{\mathcal{V}}} c(i,l)$} \COMMENT{where $i$ is the last node added in $T_j$}
\STATE {$T_j\gets l$} \COMMENT{Update tree $T_j$, $\bar{\mathcal{V}} \diagdown \{l\}$}
\ENDWHILE
\end{algorithmic}
\label{alg:oppsearch}
\end{algorithm}
We should point out that Alg. \ref{alg:oppsearch}, is a greedy heuristic of the vehicle routing problem (VRP) \cite{Laporte1992}, and ensures that each computed path is minimum with respect to the cost of the edges it traverses on the graph. When executed by each agent the individual minimum cost paths will be computed but these paths are not necessarily globally optimal. Moreover, Alg. \ref{alg:oppsearch} is executed by each agent after the search density has been exchanged and merged among those agents in communication range. Alternatively, the algorithm can be executed by a single agent and then the path plans can be send out to all other agents after the computation has been completed. This however, will entail an extra communication cost. Finally, please note that the search density decays over time according to Eqn. (\ref{eq:p2}) and also captures whole surveillance area. Thus any unvisited regions in the surveillance area (or regions that have not visited recently) would have low search values pushing agents to compute path plans that include those regions and thus exploring the unvisited areas.
\textbf{d) Communication Overhead:} When the agents are in communication range they exchange information (i.e., their search densities) in order to compute joint search plans as already explained. Although, this information exchange increases the searching efficiency, it also creates a communication overhead. Assuming that the search density is implemented in this work as a discrete set of values i.e., a set of $N$ real numbers forming a grid over the surveillance area, then whenever two agents are in communication range they exchange $2N$ real numbers in total. This is the communication cost in terms of the amount of information that needs to be transmitted.
There are a number of ways which can be employed in order to reduce this communication overhead. In this work, we use the following event based strategy i.e., when two agents are in communication range they exchange their search densities and they compute a joint search plan. However, these two agents refrain to exchange their search densities with each other while executing their joint search plan. Thus the agent $i$ is allowed to exchange information only when a) is not participating in a joint search plan or b) is participating in a joint search plan and encounters an agent $j$ which is not involved in agent's $i$ joint search plan. A more detailed analysis and study of the communication aspects of this work will be investigated in the future.
\subsection{Multi-agent Tracking} \label{ssec:multi_agent_tracking}
In this subsection we discuss: a) how the agents select control actions in order to accurately track multiple targets and b) how multiple agents are cooperating to detect and resolve tracking overlaps.
\textbf{a) Tracking Control:} The objective of tracking control is to find the optimal control action $u_k \in \mathbb{U}_k$ that must be taken at time step $k$ by each agent in order to maintain tracking of the detected targets. We should point out that the control actions $u_k$ applied to the agents affect the received measurements $Z_k$ which in turn affect the multi-target state estimate $\hat{X}^1_k$ during the update step. In other words the received measurement set $Z_k$ (if any) depends on which control action $u_k$ has been applied (e.g., a target might not be detected if the wrong control action is applied). This can be seen from the agent's sensing and measurement models as discussed in Sec. \ref{sec:system_model}. Thus, ideally to optimize the control actions, would require the knowledge of the future measurement set $Z_k$.
Since, the future measurement set $Z_k$ is not available until the control action $u_k$ is applied, we generate the predicted measurement set $\bar{Z}_{k}$ and we use it in place of $Z_k$ in order to optimize the above objective function. More specifically, the predicted measurement set for each control action is generated as follows: $\bar{Z}_{k} =~ \bar{Z}_{k} ~ \cup ~\{{\arg\max}_z ~ g_k(z|x,u_k)\}$ for all $x \in \hat{X}^1_{k|k-1}$ and for all $u_k \in \mathbb{U}_k$
where $\hat{X}^1_{k|k-1}$ is the predicted multi-target state for the true targets, which can be obtained by integrating the predicted target density using Eqn. (\ref{eq:p1}) to compute the predicted number of targets and then extracting their states from the PHD. Let the tracking objective function be denoted as $\xi_\text{track}(u_k,Z_k)$. Using the predicted measurement sets $\bar{Z}_{k}$ in place of the actual measurement set, the control problem becomes: $u_{k}^{\star} = \underset{u_k}{\arg\max} ~ \xi_\text{track}(u_{k},\bar{Z}_{k})$. To optimize the tracking objective, the following steps are performed: For each admissible control action $u_k \in \mathbb{U}_k$ we generate the predicted measurement set $\bar{Z}_{k}$. For each pair $(u_k, \bar{Z}_{k})$ we perform a pseudo-update step using Eqn. (\ref{eq:u1}) to produce the (pseudo) posterior target density which we denote as $\bar{D}_k(x \in \mathcal{X}^1)$. We consider the information gain between the predicted $f_{k|k-1}(X|Z_{1:k-1})$ and the (pseudo) updated $\bar{f}_k({X|Z_{1:k-1},\bar{Z}_{k},u_k})$ multi-target distributions as a measure of decreasing the uncertainty of the estimated multi-target state. The objective is then to maximize the information gain between the two multi-target distributions. To measure the information gain, we use as $\xi_\text{track}(u_k, \bar{Z}_{k})$ the Renyi divergence which is given by:
\begin{align}\label{eq:divergence}
\frac{1}{\alpha-1} \text{log}\!\! \int [\bar{f}_k(X|\bar{Z}_{k},u_k)]^\alpha [f_{k|k-1}(X|Z_{1:k-1})]^{1-\alpha}\!\delta X
\end{align}
where $0 < \alpha < 1$ determines the emphasis given on the tails of the two distributions \cite{Ristic2011}. Finally, Eqn. (\ref{eq:divergence}) becomes for our problem:
\begin{align} \label{eq:track_control}
& \int_{\mathcal{X}^1} \tilde{D}_{k|k-1}(x) dx + \frac{\alpha}{(1-\alpha)} \int_{\mathcal{X}^1} \bar{D}_{k}(x|\bar{Z}_{k},u_k) dx ~- \notag \\
& \frac{1}{(1-\alpha)} \int_{\mathcal{X}^1} \bar{D}_{k}(x|\bar{Z}_{k},u_k)^\alpha \tilde{D}_{k|k-1}(x)^{1-\alpha} dx
\end{align}
where $\tilde{D}_{k|k-1}(x \in \mathcal{X}^1)$ is the predicted target density according to Eqn. (\ref{eq:p1}) and $\bar{D}_k(x \in \mathcal{X}^1|\bar{Z}_{k},u_k)$ is the (pseudo) updated target density according to Eqn. (\ref{eq:u1}) in which we show explicitly its dependance on the hypothetical control action ($u_k$) and predicted measurement set ($\bar{Z}_{k}$). To summarize, the agent selects the control action (before actually receiving the measurement) which makes the divergence between the predicted and updated multi-target densities as large as possible i.e., maximizing the amount of information in the posterior multi-target density with respect to the predicted density.
\textbf{b) Tracking Cooperation:} In this paragraph we discuss how our approach can handle tracking overlaps i.e., a problem in which a target (or a group of targets) is being tracked by more than one agent. The single agent tracking control strategy discussed in the previous paragraph can cause this problem, which in this work is something undesirable since valuable system resources are wasted for performing the same task i.e., tracking the exact same targets. The objective of the tracking overlap detection and resolution is to maximize the utilization of resources.
In particular, consider the scenario where 3 targets, which are being tracked by 2 different agents, approach each other over time. Eventually, the 3 targets move so close to each other which are being detected by both agents. When this happens, the local optimization of the tracking objective in Eqn. (\ref{eq:track_control}) directs each agent to track all 3 targets which causes the issue of tracking overlap. This is an unwanted behavior, which we wish to detect and resolve, in order to utilize the system resources for other tasks (e.g., searching).
In order to tackle this problem, instead of solving the joint tracking control problem which is a hard combinatorial problem that requires the enumeration of joint control actions among agents and the consideration of future multi-target states over a finite horizon, in this work we propose an alternative computationally cheaper way to tackle the tracking overlap problem. We allow any two agents to track the same targets but only for a short period of time. We consider that each agent can track multiple targets independent of other agents. When the trajectories of two or more tracking agents converge they exchange information to determine whether or not the exact same targets are being tracked. Once two agents have determined that they track exactly the same targets, one of them generates a search plan and switches to searching mode. The agent that switches to searching mode is picked at random, since each agent computes in this scenario approximately the same multi-target state as discussed next. The above procedure begins when two or more tracking agents have overlapping sensing ranges.
Two agents $i$ and $j$ with states $s^i_{k-1}$ and $s^j_{k-1}$ respectively have overlapping sensing ranges when $\mathcal{S}_a(s^i_{k-1}) \cap \mathcal{S}_a(s^j_{k-1}) \neq \emptyset$ in which case the agents exchange their multi-target state estimates. Let the predicted multi-target states (regarding the true targets) of the agents $i$ and $j$ be $\hat{X}^{1,i}_{k|k-1}$ and $\hat{X}^{1,j}_{k|k-1}$, respectively. Also, let $|\hat{X}^{1,i}_{k|k-1}| = m$ and $|\hat{X}^{1,j}_{k|k-1}| = n$ denote their cardinalities, i.e., the number of predicted targets in the set, with $n \ge m$ and $n, m \ne 0$. When $\mathcal{S}_a(s^i_{k-1}) \cap \mathcal{S}_a(s^j_{k-1}) \neq \emptyset$, the agents exchange their predicted multi-target states to compute the \textit{incremental tracking overlap score} as:
\begin{align} \label{eq:ospa}
&\Delta L^c_{k}(\hat{X}^{1,i}_{k|k-1},\hat{X}^{1,j}_{k|k-1}) = \notag \\
&\>\>\> \Bigg[~\frac{1}{n} \Bigg(\underset{\pi \in \Pi_n}{\text{min}} ~\underset{l=1}{\sum^m} ~ d_c(x^i_l,x^j_{\pi(l)})^2 + (n-m) \cdot c^2 \Bigg) ~ \Bigg]^{\frac{1}{2}}
\end{align}
where $x^i \in \hat{X}^{1,i}_{k|k-1}$, $x^j \in \hat{X}^{1,j}_{k|k-1}$ and $\Pi_n$ denotes the set of all permutations of size $m$ taken from the set $\{1,2,...,n\}$. The function $d_c(x,y) = \text{min}(c, \normvec{x-y}_2)$ where the parameter $c>0$ penalizes the cardinality mismatch between two sets. When $n<m$ Eq. (\ref{eq:ospa}) becomes $\Delta L^c_{k}(\hat{X}^{1,j}_{k|k-1},\hat{X}^{1,i}_{k|k-1})$. The above equation is called the optimal sub-pattern assignment (OSPA) \cite{Schuhmacher2008} of order 2. Then the \textit{cumulative tracking overlap score} for the time-window $[\kappa:K]$ is then defined as $Q_{\kappa:K}(s^i_{\kappa-1},s^j_{\kappa-1}) =
\sum_{k=\kappa}^{K} \mathcal{I}(\mathcal{S}_a(s^i_{k-1}),\mathcal{S}_a(s^j_{k-1})) \cdot \Delta L^c_{k}(\hat{X}^{1,i}_{k|k-1},\hat{X}^{1,j}_{k|k-1})$
\noindent where the function $\mathcal{I}(A,B)$ checks if the intersection of two regions $A$ and $B$ is non-empty and returns $1$, otherwise returns $\infty$. The cumulative tracking overlap score will generate a low score if two agents track the exact same targets over a certain period of time. In other words, when two agents have overlapping sensing ranges and they track the same number of targets with small positioning errors, the cumulative tracking overlap score is minimized. In order to determine if there is tracking overlap between two agents over a time-window the cumulative tracking overlap score is compared against a pre-determined threshold $Q^{Th}$. If $Q_{\kappa:K} \le Q^{Th}$ then the two agents track with high certainty the exact same targets. When this happen one of the two agents generates a search plan and switches to searching in the next time-step.
\section{Related Work}\label{sec:Related_Work}
Of particular interest in this work is the problem of cooperative searching and tracking for survivors during search and rescue missions. Previous works in \cite{Bourgault2003} and \cite{Liu2017} investigate the searching and tracking problem but only for the single-agent single-target case. The work in \cite{Furukawa2006} proposes a recursive Bayesian multi-agent searching and tracking solution, however the agents are required to be in communication range at all times. The work in \cite{Frew2008} proposes a task assignment algorithm that integrates area search and target tracking, however requires that the number of agents is larger than the number of targets and that a single agent can only track one target at a time. The problem of multi-agent searching and tracking is also investigated in \cite{Pitre2012} but lacks online path generation. The work in \cite{Peterson2017} proposes a cooperative search and track framework however, requires clutter free environment and perfect target detection. Finally, relevant works also include \cite{Dames2017,Dames2019,Papaioannou2019_1,Papaioannou2019_2} which implement efficient multi-agent RFS-based simultaneous coverage and tracking algorithms for tracking multiple targets. Complementary to the related work, in this paper we propose a decentralized architecture where multiple agents cooperatively search a region of interest in order to detect and track multiple targets. A preliminary work has been published in \cite{PapaioannouCDC19}. The current work is a more complete study with stronger results.
In this work, we assume that a specific obstacle-free 2D region of interest needs to be continuously searched for potential targets with the aid of a group of mobile agents. The number of targets is not known a priori and may change over time. As a consequence, target births and deaths can occur at random times and the targets can spawn from anywhere inside the surveillance area. The agents are equipped with sensors that are not perfect i.e., as a result of various sensor imperfections, the agents receive noisy target measurements and clutter (e.g., false-alarm measurements). Moreover, the agents have limited sensing range for detecting targets and limited communication range for exchanging information with other nearby agents.
We should point out that in this work we assume that the mobile agents have perfect self-localization ability and that their dynamical model and control inputs are deterministic.
The objective of each agent at an arbitrary time-step is to: a) accurately estimate the number of targets and their states from noisy measurements in the presence of clutter, and b) generate search-plans for efficiently searching the whole surveillance area. To achieve a) and b), each agent propagates in time the \textit{searching-and-tracking density} (SAT-density) which is used to a) keep track the areas in the surveillance area that need to be searched and b) estimate the number and states of all targets inside the agent's sensing range, which in this work is achieved with the PHD-filter.
Moreover, the agents opportunistically cooperate by exchanging information in order to tackle the above objectives more efficiently e.g., two or more agents cooperate to generate joint search-plans and to resolve tracking overlaps (i.e., a situation where 2 or more agents track the same targets). To summarize, the agents opportunistically exchange their search densities, estimated target states and their mode of operation i.e., \textit{searching} or \textit{tracking}. We should also note that all agents operate in \textit{searching} mode optimizing their local or joint search objective(s) (see subsection \ref{ssec:cooperative_search}) until targets are found in the surveillance area in which case the respective agents switch to \textit{tracking} mode (see subsection \ref{ssec:multi_agent_tracking}).
\section{System Model} \label{sec:system_model}
\subsection{Single Target Dynamics and Measurement Model}\label{ssec:single_target_dynamics}
Let the state of a single target is given by $\bm{x} = (x,\ell) \in \mathcal{X} \times \{0,1\}$,
where $x \in \mathcal{X}$ is the kinematic state of the target, $\mathcal{X} \subseteq R^{n_x}$ denotes the kinematic state space of the target, $n_x$ is the dimension of the state vector $x$ and $\ell \in \{0,1\}$ is the target label taken from the discrete label space $\{0,1\}$. We denote a true target with the label $\ell=1$ and a virtual target with the label $\ell=0$. True targets represent physical targets inside the surveillance region whose kinematic state $x$ needs to be estimated from a sequence of noisy measurements whereas virtual targets represent static and deterministic locations in the environment (these locations will be used as indicators to show whether specific regions in the area have been searched by the agents). Throughout this paper, the kinematic state spaces of true and virtual targets will be denoted as $\mathcal{X}^1$ and $\mathcal{X}^0$, respectively. The single target kinematic state vector $x_k, k \in \mathbb{N}$ evolves in time according to the following equation:
\begin{subnumcases}{x_k=}
\zeta(x_{k-1}) + w_k & \text{if} $~x_{k-1} \in \mathcal{X}^1$ \label{eq:single_dynamics_a} \\
x_{k-1} & \text{if} $~x_{k-1} \in \mathcal{X}^0$ \label{eq:single_dynamics_b}
\end{subnumcases}
\noindent where the function $\zeta : \mathbb{R}^{n_x} \rightarrow \mathbb{R}^{n_x}$ models the dynamical behavior of the target. Eqn. (\ref{eq:single_dynamics_a}) describes the evolution of the state vector as a first order Markov process with transitional density $\pi_{k|k-1}(x_k|x_{k-1}) = p_w(x_k - \zeta(x_{k-1}) )$. The process noise $w_{k} \in \mathbb{R}^{n_x}$ is independent and identically distributed (IID) according to the probability density function $p_w(.)$. In this paper we assume that the kinematic state vector $x_k \in \mathcal{X} \subseteq \mathbb{R}^4$ is composed of position and velocity components in Cartesian coordinates i.e., $x_k = [\text{x},\dot{\text{x}},\text{y},\dot{\text{y}}]^\top$. Since a virtual target is static, its kinematic state vector is of the form $x_k = [\text{x},0,\text{y},0]^\top$.
When an agent detects a true target i.e., $x_k \in \mathcal{X}^1$ at time $k$, it receives a measurement vector $z_k \in \mathcal{Z}$ (range and bearing observations) which is related to the target kinematic state as follows: $z_k = h(x_k,s_k) + v_k$ where the function $h(x_k,s_k)$ projects the state vector to the measurement space, $s_k$ is the state of the agent at time $k$ (described in the next sub-section) and the random process $v_{k} \in \mathbb{R}^{n_z}$ is IID, independent of $w_k$ and distributed according to $p_v(.)$. Thus, the probability density of measurement $z_k$ for a target with kinematic state $x_k$ when the agent is at state $s_k$ is given by the measurement likelihood function $g_k(z_k|x_k,s_k) = p_w(z_k - h_k(x_k,s_k))$.
On the other hand, virtual targets ($x_k\in\mathcal{X}^0$) represent fixed and known locations in the environment i.e., their states are deterministic and predetermined. Additionally, in this work we assume that the agent dynamics are deterministic and that the agents can perform perfect self-localization (as discussed in subsection \ref{ssec:AgentDynamics}). As a result the states of the virtual targets need not to be estimated. As we discuss in Sec. \ref{sec:proposed_approach} virtual targets are used to keep track whether certain regions inside the surveillance area have been searched or not.
\subsection{Agent Dynamics} \label{ssec:AgentDynamics}
Let $S = \{1,2,...,|S|\}$ be the set of all mobile agents that we have in our disposal operating in a discrete-time setting. At time $k$, the 2D surveillance region $\mathcal{A} \subseteq \mathbb{R}^2$ is monitored by $|S|$ mobile agents with states $s^1_k,s^2_k,...,s^{|S|}_k$, each taking values in $\mathcal{A}$. Each agent $j$ is subject to the following deterministic dynamics:
\begin{equation} \label{eq:controlVectors}
s^j_{k} = s^j_{k-1} + \begin{bmatrix}
l_1\Delta_R \text{cos}(l_2 \Delta_\theta)\\
l_1\Delta_R \text{sin}(l_2 \Delta_\theta)
\end{bmatrix},
\begin{array}{l}
l_2 = 0,...,N_\theta\\
l_1 = 0,...,N_R
\end{array}
\end{equation}
where $s^j_{k-1} = [s^j_x,s^j_y]^\top_{k-1}$ denotes the position (i.e., $(x, y)$ coordinates) of the $j_{\text{th}}$ agent at time $k-1$, $\Delta_R$ is the radial step size, $\Delta_\theta=2\pi/N_\theta$ and the parameters $(N_\theta,N_R)$ specify the number of possible control actions. We denote the set of all admissible control actions of agent $j$ at time $k$ as $\mathbb{U}^j_{k}=\{s^{j,1}_{k},s^{j,2}_{k},...,s^{j,|\mathbb{U}_{k}|}_{k} \}$ as computed by Eqn. (\ref{eq:controlVectors}). We should point out that the agent dynamical model is noise-free and that the agents have perfect self-localization ability (i.e., through a very accurate GPS system).
\subsection{Single Agent Sensing Model} \label{ssec:sensing_model}
The ability of an agent to sense true targets inside the surveillance area is modeled by the function $p_D(x_k,s_k)$ that measures the probability that a target with kinematic state $x_k \in \mathcal{X}^1$ at time $k$ is detected by an agent with state $s_k$. More specifically, the sensing capability of the agent is given by:
\begin{equation}\label{eq:sensing_model_true}
p_D(x_k \in \mathcal{X}^1 ,s_k) =
\begin{cases}
p_D^\text{max} & \text{if } x_k \in \mathcal{S}_a(s_k) \\
0 & \text{if } x_k \notin \mathcal{S}_a(s_k)
\end{cases}
\end{equation}
where $\mathcal{S}_a(s_k)$ denotes the agent's sensing area which in this work includes all $(x, y)$ points that satisfy the equation $\max \{|x-s_x|,|y-s_y|\}=\frac{a}{2}$; i.e., a square region with total area $a^2$ units, centered at $s_k = [s_x, s_y]^\top$ and $p_D^\text{max}$ denotes the probability that the agent's sensor detects the true targets located inside its sensing range. Although, in this work we use a square region to model the agent's sensing area, the proposed approach is not limited to square sensing areas, for instance rectangular and circular sensing areas can also be used.
On the other hand, the agent with state $s_k$ observes virtual targets $x_k \in \mathcal{X}^0$ and determines if they reside within its sensing area using the following indicator function: $\mathds{1}_{\mathcal{S}_a}(x_k \in \mathcal{X}^0 ,s_k) = 1$ if $x_k \in \mathcal{S}_a(s_k)$ and 0 otherwise.
Finally, any two agents with states $s^i_k$ and $s^j_k$ are able to communicate with each other when $\normvec{s^i_k - s^j_k}_2 \le C_R$ where $C_R$ is the communication range with $C_R \geq \sqrt{2}\frac{\alpha}{2}$ i.e., the communication range is greater than or equal to the sensing range to allow for multiple agents to exchange information to avoid tracking overlaps (see subsection \ref{ssec:multi_agent_tracking}).
\subsection{Multi-object dynamics and measurement models}
Multiple independent true targets can exist and evolve inside the surveillance region. True targets (i.e., with label $\ell=1$) can spawn from anywhere inside the surveillance region and target births and deaths occur at random times. This means that at each time $k$, there exist $n^{\ell=1}_k$ true targets with kinematic states $x^1_k, x^2_k,...,x^{n^{\ell=1}_k}_k$, each taking values in the state space $\mathcal{X}^1$, where both the number of true targets $n^{\ell=1}_k$ and their individual states $x_k^i, \forall i \in n^{\ell=1}_k$ are random and time-varying. The RFS of the multi-target state of the true targets $X^{\ell=1}_k \in \mathcal{F(X^\text{1})}$ evolves in time according to: $ X^{\ell=1}_k = [\underset{x_{k-1} \in X^{\ell=1}_{k-1}}{\cup} \Psi(x_{k-1})] \cup B_k$
where $X^{\ell=1}_{k-1}$ is the multi-target state of the true targets of previous time-step, $\Psi(x_{k-1})$ is a Bernoulli RFS which models the evolution of the set from the previous state, with parameters $(p_{S}(x_{k-1}),\pi_{k|k-1}(x_k|x_{k-1}))$. Thus a target with kinematic state $x_{k-1}$ continues to exists at time $k$ with surviving probability $p_{S}(x_{k-1})$ and moves to a new state $x_k$ with transition probability $\pi_{k|k-1}(x_k|x_{k-1})$. Otherwise, the target dies with probability $1-p_{S}(x_{k-1})$. The term $B_k$ is a Poisson RFS of spontaneous target births.
At time $k$, an agent receives a finite set of measurements from the detected true targets and from clutter denoted as $Z_k$. The multi-target measurement set is formed according to: $Z_k = [ \underset{x_{k} \in X^1_{k}}{\cup} \Theta(x_{k}) ] \cup \text{K}_k$
where $\Theta(x_{k})$ is a Bernoulli RFS with parameters $(p_{D}(x_k,s_k),g_k(z_k|x_k,s_k))$. Thus a true target with kinematic state $x_k$ at time $k$ is detected by the agent with state $s_k$ with probability $p_{D}(x_k,s_k)$. This agent then receives a measurement $z_k$ with likelihood $g_k(z_k|x_k,s_k)$. The target is not detected with probability $1-p_{D}(x_k,s_k)$ and no measurement is being received by the agent. Additionally, an agent can receive false alarms measurements i.e., the term $\text{K}_k$ is a Poisson RFS which models the set of false alarms or clutter received by an agent at time $k$ with PHD $\kappa_k(z_k) = \lambda_c f_{c}(z_k)$, where in this paper $f_c(.)$ denotes the uniform distribution over $\mathcal{Z}$ and $\lambda_c$ is the average number of clutter generated measurements per time-step.
The virtual targets i.e., $(\ell=0)$ on the other hand do not exhibit any birth and death events. Their number $n^{\ell=0}$ is fixed and their states are known. Thus the multi-target state of virtual targets is given by
$X^{\ell=0}=\{ x^1,\cdots,x^{n^{\ell=0}}\}, x \in \mathcal{X}^0$. In essence the multi-target state of virtual targets represents discrete locations inside the surveillance environment which in turn correspond to regions for which a search value is computed. The search value shows whether or not these regions have been searched by the agent. Additionally, we should point out here that the state-spaces of true and virtual targets are distinct and thus all true targets automatically received the label $l=1$ whereas virtual targets receive $l=0$.
\section{System Overview} \label{sec:system_overview}
Before discussing in detail the proposed framework we give in this section an overview of the proposed system architecture as illustrated in Fig. \ref{fig:sysarch}. The proposed system architecture can be divided into two parts namely a) independent SAT operation and b) cooperative multi-agent SAT operation.
An independent SAT operation occurs when an agent does not find any other agents inside its communication range. In this case the agent will optimize its individual SAT objective without cooperating with other agents. An agent commence in search mode will remain in searching until one or more targets have been detected inside its sensing range; in which case the agent will switch to tracking. An agent maintains a representation of the surveillance area which is used to indicate the areas already been searched. A particular location inside the surveillance area is marked as searched when it resides inside the agent's sensing range. That said, the agents can perform searching even when they are in tracking mode. In other words while in tracking mode, an agent can mark unvisited areas as visited and discover new targets to track. In this case however, tracking is optimized and not searching (i.e. the searching is a side effect of tracking a target over a particular area).
To achieve the search-and-track objective, the agent maintains and propagates in time the SAT probability hypothesis density $D_k(x|Z_{1:k})$ which encapsulates the target and search density at $x$. Please note that for the rest of the paper, $D_k(x|Z_{1:k})$ will denote the proposed SAT-PHD instead of the regular PHD. The proposed predictor-corrector recursion is shown in Fig. \ref{fig:sysarch}. The predictor step produces the predicted SAT-PHD $D_{k|k-1}(x|Z_{1:k-1})$ from which the predicted search density and target density are extracted and then the search values, the target states ($\hat{X}_{k|k-1}$) and the number of targets ($\hat{n}_{k|k-1}$) are estimated. Based on these estimates, the system generates a search plan if no targets have been detected, or a track plan otherwise. In other words, the SAT controller finds the optimal control action $u_k \in \mathbb{U}_k$ for the next time step. Then, the agent moves to the new state, where it receives the measurement set $Z_k$ which is used to compute the posterior SAT-PHD (i.e. corrector step), and from which the posterior multi-target state $\hat{X}_k$ is estimated, and the search values are updated. The previous recursion is repeated over time.
When one or more agents are in communication range they cooperate in order to solve the SAT problem more efficiently. In particular, every time that a group of agents are in communication range they exchange information (i.e. search maps) and produce a joint search plan in order to efficiently search the surveillance region (i.e. generate search trajectories that will guide the agents to search non-overlapping regions, thus maximizing the total searched area). Additionally, when one or more agents have overlapping sensing ranges they communicate in order to detect a tracking overlap i.e. a situation where two or more agents track identical targets. When this happens, the overlapping agents split and switch to search mode.
|
1105.0514
|
\section{Introduction}
In recent years, several states with charmonium-like decays were
discovered, which do not fit properly into the established charmonium
picture. These states, called $X,Y,Z$, are candidates for exotic
mesons beyond the conventional quark-antiquark model ($q\bar{q}$).
Possible interpretations are quark-gluon hybrids ($q\bar{q}g$),
four-quark states ($q\bar{q}q\bar{q}$), molecular states composed of
two usual mesons, glueballs, et cetera.
CDF reported an evidence for a narrow near-threshold structure in the
$J/\psi \phi$ mass spectrum, called $Y(4140)$, using exclusive $B^+
\to J/\psi \phi K^+$ decays.\cite{Evidence} As there was no signal
seen by Belle in a subsequent search,\cite{Belle} it is important to
investigate with a larger CDF data sample. Consisting of two vector
mesons (positive C-parity), the final state $J/\psi \phi$ is a good
channel to search for an exotic meson. The observed excess would be
the first charmonium-like structure decaying into two heavy quarkonium
states ($c\bar{c}$ and $s\bar{s}$). Because the mass in this channel
is high enough for open charm decays, an explanation as charmonium
state is very unlikely for a narrow structure. A search near the
$J/\psi\,\phi$ threshold is also motivated by the closeness of the
$Y(3930)$ to the $J/\psi\,\omega$ threshold. Some of the possible
exotic explanations for the $Y(4140)$ are discussed in
Ref.~\refcite{Theo}.
\section{Candidate Selection}
We report on an update of the search for structures in the $J/\psi
\phi$ system produced in exclusive $B^+ \to J/\psi \phi K^+$
decays.\cite{PublicNote} The employed dataset was collected by the CDF
II detector at the Tevatron and corresponds to an integrated
luminosity of $6.0\,\mathrm{fb^{-1}}$. It was accumulated using a
dedicated dimuon trigger which requires a $\mu^+ \mu^-$ pair with a
mass of $2.7 < m(\mu^+ \mu^-) < 4.0\,\mathrm{GeV}/c^2$. Due to trigger
prescales at increasing instantaneous luminosities one cannot expect a
linear increase of the sample size compared to the previous
analysis\cite{Evidence} with $2.7\,\mathrm{fb^{-1}}$.
In order to build $B^+ \to J/\psi \phi K^+$ candidates, first $J/\psi
\to \mu^+ \mu^-$ and $\phi \to K^+ K^-$ candidates are reconstructed
which are then combined with an additional charged track with kaon
mass hypothesis. Thereby, the reconstructed $J/\psi$ and $\phi$ masses
are required to lie within $50\,\mathrm{MeV/c^2}$ ($J/\psi$)
respective $7\,\mathrm{MeV/c^2}$ ($\phi$) of the corresponding world
average values. The combinatorial background can be reduced
significantly with a higher threshold of the $B^+$ decay length in the
transverse plane ($L_{xy}(B^+)$) to exploit the long $B$ meson
lifetime. In addition, a kaon identification quantity can be used to
obtain a further background reduction by separating the final state
kaons from the dominant pion background. For that purpose, the
information about the ionization energy loss $dE$/$dx$ in the drift
chamber and the information from the Time-of-Flight detector are
combined in a log-likelihood ratio $LLR_\mathrm{Kaon}$. The cuts
$L_{xy}(B^+)>500\,\mu\mathrm{m}$ and $LLR_\mathrm{Kaon}>0.2$ are
chosen by optimizing the quantity $S/\sqrt{S+B}$, where $S$ and $B$
are the numbers of $B^+$ signal and background events, respectively.
After both requirements, a background reduction factor of
approximately four orders of magnitude is accomplished.
Figure \ref{fig:B}(a) shows the resulting $J/\psi \phi K^+$ mass
spectrum. A fit with a Gaussian signal and a linear background
function yields $115 \pm 12$ signal events, corresponding to a 53\%
increase over the previous analysis. For the examination of the
$J/\psi \phi$ spectrum, only candidates within $\pm 3 \sigma$ ($\pm
17.7\,\mathrm{MeV/c^2}$) around the nominal $B^+$ mass are selected.
Furthermore, sideband events within $[-9 , -6]\sigma$ and $[+6 ,
+9]\sigma$ around the nominal $B^+$ mass are used to model the
combinatorial background in the $J/\psi \phi$ spectrum. Figure
\ref{fig:B}(b) shows the mass difference $\Delta M = m(\mu^+ \mu^- K^+
K^-) - m(\mu^+ \mu^-)$ distributions of the resulting $J/\psi \phi$
candidates. Whereas the $Y(4140)$ can be seen as narrow near-threshold
excess in the $B$ mass window, no such evidence is found from the $B$
mass sidebands. Figure \ref{fig:Valid}(a) shows the Dalitz plot of the
candidates from the $B$ mass window and figure \ref{fig:Valid}(b) the
$B$ sideband-subtracted $K^+ K^-$ mass spectrum without $\phi$ mass
window requirement. The fit function is a $P$-wave relativistic
Breit-Wigner convolved with a Gaussian to account for the detector
resolution. As there is no significant background contribution, the
$B^+ \to J/\psi K^+ K^- K^+$ final state is well described as $J/\psi
\phi K^+$. A comparison between the $J/\psi \phi$ mass difference
distributions of the dataset used for the updated analysis described
in this write-up, corresponding to an integrated luminosity of
$6.0\,\mathrm{fb}^{-1}$, and the one employed for the published
$Y(4140)$ measurement\cite{Evidence}, corresponding to
$2.7\,\mathrm{fb}^{-1}$, can be found in figure \ref{fig:DatasetComp}.
\begin{figure}
\centering
a)
\includegraphics[width=.43\textwidth]{myFig1_1.eps}
\hspace{0.05\textwidth}
b)
\includegraphics[width=.43\textwidth]{myFig3.eps}
\caption{(a) $J/\psi \phi K^+$ mass distribution with a fit to the
data represented by the solid blue line. The vertical dashed black
and red lines indicate the $B^+$ mass window and sidebands described
in the text. (b) Mass difference distributions of the resulting
$J/\psi \phi$ candidates from the $B^+$ mass window (black
histogram) and sidebands (red histogram).}
\label{fig:B}
\end{figure}
\begin{figure}
\centering
a)
\includegraphics[width=.43\textwidth]{myFig2.eps}
\hspace{0.05\textwidth}
b)
\includegraphics[width=.43\textwidth]{phi.eps}
\caption{(a) Dalitz plot of the final state $J/\psi \phi K^+$ in the
$B^+$ mass window. The boundary shows the kinematically allowed
region. (b) $K^+ K^-$ mass distribution with the fitted function
(solid blue line) as described in the text.}
\label{fig:Valid}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.43\textwidth]{myFig4.eps}
\caption{Comparison between the $J/\psi \phi$ mass difference
histograms of the dataset used for the updated analysis described in
this write-up ($6.0\,\mathrm{fb}^{-1}$) in black and the one
employed for the published $Y(4140)$ measurement
($2.7\,\mathrm{fb}^{-1}$) in dashed red.}
\label{fig:DatasetComp}
\end{figure}
\section{Fits to Data and Significance Determination}
Figure \ref{fig:Y1}(a) shows the $\Delta M$ distribution from the $B$
mass window excluding events with $\Delta M > 1.56\,\mathrm{GeV/c^2}$
in order to avoid combinatorial backgrounds from misidentified $B_s^0
\to \psi(2S)\,\phi \to (J/\psi\,\pi^+\,\pi^-)\,\phi$ decays. An
unbinned maximum likelihood fit is performed, where the enhancement is
described by the convolution of an $S$-wave relativistic Breit-Wigner
function with a Gaussian resolution of $1.7\,\mathrm{MeV/c^2}$
obtained from Monte Carlo simulations, and the background is modeled
by three-body phase space. Including systematic uncertainties, which
are estimated by varying the fit model, the fit yields $19 \pm
6(\mathrm{stat}) \pm 3(\mathrm{syst})$ signal events. It returns a
mass of $m = 4143.4^{+2.9}_{-3.0}(\mathrm{stat}) \pm
0.6(\mathrm{syst})\,\mathrm{MeV}/c^2$ after including the world
average $J/\psi$ mass and a decay width of $\Gamma =
15.3^{+10.4}_{-6.1}(\mathrm{stat}) \pm
2.5(\mathrm{syst})\,\mathrm{MeV}/c^2$, both consistent with the values
from the published measurement\cite{Evidence}. The observed width,
which is much larger than the resolution, suggests a strong decay for
the $Y(4140)$. Furthermore, the relative branching fraction to the
nonresonant $B^+ \to J/\psi\,\phi\,K^+$ decay is measured as
$\frac{\mathcal{B}(B^+ \to Y(4140)\,K^+,\,Y(4140) \to
J/\psi\,\phi)}{\mathcal{B}(B^+ \to J/\psi\,\phi\,K^+)} = 0.149 \pm
0.039(\mathrm{stat}) \pm 0.024(\mathrm{syst})$, where the relative
efficiency is determined to be $1.1$, using an $S$-wave relativistic
Breit-Wigner function with mean and width values determined from data
to represent the $Y(4140)$ structure and three-body phase space
kinematics for the nonresonant $B^+ \to J/\psi\,\phi\,K^+$ decay.
In order to estimate the probability of a creation of such a signal
due to background fluctuations, a large number of three-body phase
space $B^+$ decays are performed and the number of trials which
produce a signal with a log-likelihood ratio $-2
\ln{(\mathcal{L}_0/\mathcal{L}_{max})}$ of the null hypothesis fit and
the signal hypothesis fit larger than the value measured in data are
counted (see figure \ref{fig:Y1}(b)). Thereby, the mass can be
anywhere in the considered $\Delta M$ window and the width has to be
larger than the detector resolution and smaller than
$120\,\mathrm{MeV}/c^2$. This procedure leads to a $p$-value of $2.3
\cdot 10^{-7}$, corresponding to a significance of the enhancement of
$5.0 \sigma$.
\begin{figure}
\centering
a)
\includegraphics[width=.43\textwidth]{myFig5.eps}
\hspace{0.05\textwidth}
b)
\includegraphics[width=.43\textwidth]{myFig6.eps}
\caption{(a) $J/\psi \phi$ mass difference distribution with a fit to
the data represented by the solid red line. (b) $-2
\ln{(\mathcal{L}_0/\mathcal{L}_{max})}$ distribution for 84 million
simulation trials. The vertical red line indicates the value
obtained in data.}
\label{fig:Y1}
\end{figure}
In figure \ref{fig:Y1}(a), an additional excess above the background
appears at a mass of approximately $1.18\,\mathrm{GeV}/c^2$. With the
parameters of the $Y(4140)$ fixed to the values obtained from the fit
described above, another unbinned maximum likelihood fit assuming two
structures and the same background model as before is performed.
Thereby, the additional enhancement is described by the convolution of
an $S$-wave relativistic Breit-Wigner function with a Gaussian
resolution of $3.0\,\mathrm{MeV/c^2}$ obtained from Monte Carlo
simulations. The measured data distribution together with the fit
projection can be found in \ref{fig:Y2}(a). The fit returns a yield of
$22 \pm 8$ signal events, a mass, after including the world average
$J/\psi$ mass, of $m = 4274.4^{+8.4}_{-6.7}\,\mathrm{MeV}/c^2$ and a
decay width of $\Gamma = 32.3^{+21.9}_{-15.3}\,\mathrm{MeV}/c^2$. Just
like in the $Y(4140)$ case, the statistical significance of the
additional excess is determined by simulations, where the
log-likelihood ratio of the fit assuming only the $Y(4140)$ and the
fit assuming two signal structures is calculated (see figure
\ref{fig:Y2}(b)). This leads to a $p$-value of $1.1 \cdot 10^{-3}$,
corresponding to a significance of $3.1 \sigma$.
\begin{figure}
\centering
a)
\includegraphics[width=.43\textwidth]{myFig11.eps}
\hspace{0.05\textwidth}
b)
\includegraphics[width=.43\textwidth]{myFig14.eps}
\caption{(a) $J/\psi \phi$ mass difference distribution with a fit to
the data, where an additional signal structure is included which is
located about one pion mass higher than the $Y(4140)$. (b)
Distribution of the log-likelihood ratio of the fit assuming only
the $Y(4140)$ and the fit assuming two signal structures for the
simulation trials. The vertical red line indicates the value
obtained in data.}
\label{fig:Y2}
\end{figure}
After the first confirmation of Belle's $X(3872)$, including the
determination of its allowed quantum numbers and the most precise mass
measurement,\cite{X} CDF keeps contributing to the field of exotic
$X,Y,Z$ states with this recent observation of a narrow structure,
called $Y(4140)$, near the $J/\psi\,\phi$ threshold in exclusive $B^+
\to J/\psi\,\phi\,K^+$ decays.
|
1906.11818
|
\section{INTRODUCTION}
\label{sec:intro}
One of the most important applications of hyperspectral imaging is to the problem of detecting specific chemicals in a given scene. Unfortunately, hyperspectral images are generally much larger than traditional images even when one is collecting relatively few bands, and hyperspectral devices also generally come with a high price tag. For these reasons, any methods that allow hyperspectral images to be sampled at lower rates while at the same time retaining their discriminative ability have many applications. In particular, strategies for reduction of the number of sensors are quite valuable. Compressive sensing (CS) is exactly such a framework. In particular, CS provides methods for accurately reconstructing under-sampled signals
Applying CS to hyperspectral imaging is an active area of research, with a multitude of different approaches. These include CS sampling schemes that are performed to different extents in both the spatial and the spectral domain. Several works have gone beyond simply trying to reconstruct the image itself and instead make the process of signal unmixing, i.e. identifying the different spectral signals in an image, a component of the reconstruction process \cite{martin2012new,martin2013hyperspectral,martin2015hyca}. Other works have explored reconstruction frameworks specifically designed to utilize the structure of hyperspectral data \cite{golbabaee2012hyperspectral}. Finally, various CS strategies have been proposed for data extracted from devices specific to hyperspectral imaging \cite{rajwade2013coded}.
In this paper we explore chemical detection in CS reconstructions of hyperspectral images after low levels of sampling. We show that, surprisingly, not only can a chemical still be detected in a hyperspectral image that has been reconstructed after 10\% sampling, but at least in some examples, the chemical signature can be slightly stronger in the reconstructed image. This is a surprising result because naively, one would expect that by using 10\% sampling, one is losing 90\% of the information from the hyperspectral image. We note that signal amplification only seems to happen with low sampling levels. An interesting future direction would be to understand in which situations precisely a signal is amplified by reconstruction.
This paper is organized as follows. In Sec.~\ref{sec:compressive_senseing}, we give a brief summary of some of the ideas underlying CS. In Sec.~\ref{sec:Framework}, we set up some of the mathematical framework and notation for discussing hyperspectral imagery. In Sec.~\ref{sect:dection_algorithms}, we describe the chemical detection algorithm (ACE) used in experiments for this paper. The main results of the paper are discussed in Sec.~\ref{sec:signa_enhancement}. Finally, in Sec.~\ref{sect:explainations}, we suggest some possible explanations for the chemical signal amplification seen in reconstructed images. In Sec. \ref{sec:conclusion} we suggest some directions for future research.
\section{BACKGROUND}
In this paper we will generally use upper case letters (for example $S$, $X$, $U$, $H$, and $Y$) to denote matrices. We will use lower case letters (for example $u$, $u^*$, $x'$, $\tilde{x}$, and $y$) to denote vectors. Matrices being placed side by side always denotes standard matrix multiplication.
\subsection{Compressive sensing and the single-pixel camera framework}
\label{sec:compressive_senseing}
Compressive sensing (CS) is a collection of methods that permit highly accurate reconstruction of certain classes of signals even when they have been sampled well under Nyquist-rate~\cite{baraniuk2007compressive}. When it is applicable, CS allows one to solve the ill-posed problem of finding $\tilde{x}$ from
\begin{equation} \label{eqn-basic-CS-problem}
y = S\tilde{x} \in \mathbb{R}^k
\end{equation}
when $S$ is a $k \times n$ matrix, $\tilde{x} \in \mathbb{R}^n$, and $k < n$. A basic requirement for application of methods from CS for solution of \eqref{eqn-basic-CS-problem} is that we can make some assumption about the signal $\tilde{x}$. This allows us to choose $\tilde{x}$ (or a close approximation to $\tilde{x}$) out of the infinite number of solutions $x'$ to $Sx' = y$. One common choice of assumption about $\tilde{x}$ that is used frequently in CS is that it will be sparse in some particular basis. Although when $\tilde{x}$ is an image it will almost never be sparse in its natural basis, it is generally true that it will be compressible (that is, approximately sparse) in a wavelet basis.
Let $H, H^{-1}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be the $1$-dimensional Haar wavelet transformation and its inverse respectively. Then the CS reformulation of \eqref{eqn-basic-CS-problem} is to solve:
\begin{equation} \label{eqn-opt-problem}
u^* = \underset{u \in \mathbb{R}^{n}}{\text{argmin}}\; ||u||_{\ell_1} \quad\quad \text{such that } \quad y = SH^{-1}u,.
\end{equation}
In words, this optimization problem seeks $x$ satisfying $Sx = y$ such that $x$ is maximally sparse in the $1$-dimensional Haar wavelet basis, that is, $||u||_{\ell_1}$ is minimized for $u = Hx$.
It is important to note that even though the solution $x^* = H^{-1}u^*$ to \eqref{eqn-opt-problem} may often be a very good approximation to $\tilde{x}$, we always expect some information to be lost in the process of sampling $\tilde{x}$ with $S,$ in particular when $k \ll n$. In some cases the information lost in this process is mostly noise, leading to a reconstruction $x^*$ that is preferable to the original signal $\tilde{x}$. For example, another variant of CS reconstruction which minimizes total variation \cite{rudin1992nonlinear} is closely related to a family of denoising algorithms called total variation denoising. Such an observation may form the first step in explaining the phenomenon described in this paper.
\subsection{Framework and notation for the compressive sensing of hyperspectral data}
\label{sec:Framework}
Hyperspectral data is highly structured and it is essential to capture this structure when performing optimization routines such as \eqref{eqn-opt-problem}. In this paper we assume that a single band of our hyperspectral data has size $n$ (that is, as a 2-dimensional array it has size $n_1 \times n_2,$ where $n=n_1n_2$) and that there are $b$ bands. Given the above conventions, we will realize an $n_1 \times n_2 \times b$ hyperspectral data cube as an $n \times b$ matrix $X \in \mathbb{R}^{n \times b}$, where we have flattened each band to a column vector in $X$. Then, $U = HX$ is an $n \times b$ matrix where the columns of $U$ are the columns of $X$ transformed into the $1$-dimensional Haar wavelet basis. Our optimization problem \eqref{eqn-opt-problem} applied to a hyperspectral image is thu
\begin{equation} \label{eqn-hyper-opt-problem}
U^* = \underset{U \in \mathbb{R}^{n\times b}}{\text{argmin}}\; ||U||_{\ell_1} \quad\quad \text{such that } \quad Y = SH^{-1}U,
\end{equation}
where $S$ is again a sampling matrix in $\mathbb{R}^{k\times n}.$ The $\ell_1$-norm above is applied to $U$ in the same way as it would be to $U$ flattened to a length $nb$ vector. Note that the expression $H^{-1}U$ is equivalent to taking the inverse of the 1-dimensional Haar wavelet transform for every band in the data cube flattened to a vector. We call the $k \times b$ output $Y = SX$ a $(100\frac{k}{n})$\% sampling of $X$. In this paper we will generally sample at $10\%$, so that we are effectively throwing out $90\%$ of the information in a data cube. It is for this reason that it is surprising that chemical signals in $X$ sometimes become stronger.
There has been considerable research toward developing feasible optimization problems that reconstruct the bands of a hyperspectral image non-independently \cite{golbabaee2012hyperspectral}. Such methods make use of the correlation between bands. We chose to study \eqref{eqn-hyper-opt-problem} because in many cases hardware and sampling constraints force bands to be reconstructed independently. It would be interesting to understand if solving more hyperspectral specific optimization problems also produce reconstructions with signal enhancement.
In all the experiments described in this paper, we constructed $S$ via a modified Walsh-Hadamard matrix \cite{farnell2019sampling}. We suggest investigation into whether other sampling methods also result in signal enhancement at low sampling levels.
There are a large number of algorithms that have been developed for solving optimization problems such as \eqref{eqn-opt-problem} and \eqref{eqn-hyper-opt-problem}. We choose to utilize the split Bregman method \cite{GO09} since it is fast, lightweight, and gives reliable convergence.
\subsection{Detection algorithms}
\label{sect:dection_algorithms}
Since the point of our experiments was to demonstrate a phenomenon in which compressive sensing followed by reconstruction of a hyperspectral image amplifies chemical signals, choosing the appropriate chemical detection algorithm was a key component of our work.
In our experiments we use the \emph{adaptive coherence/cosine estimator} (ACE), a well-known technique used for chemical detection~\cite{scharf1996adaptive,kraut2001adaptive}. Let $s$ be a target spectral signature and let $x$ be a spectral signature in a specific pixel within a hyperspectral cube (in the literature, $x$ is the \emph{pixel under test} (PUT)). The ACE statistic is the square of the cosine of the angle between $s$ and $x$ relative to the background. To be precise, the ACE statistic is calculated as
$$\frac{(s^T\Gamma^{-1}x)^2}{(s^T\Gamma^{-1}s)(x^T\Gamma^{-1}x)},$$
where $\Gamma$ is the maximum likelihood estimator for the covariance matrix of background data.
In addition to using ACE for chemical detection, we compute the \emph{bulk coherence (multipulse coherence) estimator}~\cite{pakrooh2017adaptive,pakrooh2017adaptiveb,scharf2017multipulse}. The bulk coherence statistic enhances the signal in neighborhoods that have several pixels with relatively high ACE values, a property that is appropriate for chemical release settings. If $c_i$ is the ACE statistic for pixel $i$ and a neighborhood of pixels is indexed by $i=1,\ldots,M,$ then the bulk coherence statistic is $$1-\prod_{i=1}^M(1-c_i).$$ ACE values near one result in a product whose terms are close to zero, resulting in a high bulk coherence value (near one). Experimentally, the bulk coherence statistic leads to improved chemical detection. We also add one additional filter that we refer to as \emph{persistence} in some cases: we set the value associated to a pixel to zero if its bulk coherence value doesn't stay above a pre-specified threshold for at least five consecutive time steps.
To facilitate objective comparison, we further incorporate an algorithmic definition of a threshold for the ACE statistic (similarly for bulk coherence)~\cite{farnell2019TVvsL1}. We can then declare a chemical to be present or absent in a given pixel based on comparison of the ACE statistic against this threshold. The threshold definition is motivated by the idea that the threshold should be slightly larger than the ACE values observed in typical background cubes (cubes that have been collected with the intended device in which it is known that the target chemical is absent). The algorithmic determination of the threshold is responsive to the device used to sense the data, the reconstruction method, and the spectral signature of the target chemical.
\subsection{Signal enhancement in chemical detection}
\label{sec:signa_enhancement}
We demonstrate the signal enhancement phenomenon on two hyperspectral datasets. The first is the Fabry-P\'{e}rot interferometer sensor multispectral dataset\cite{cosofret2009airis} and the second is the Johns Hopkins Applied Physics Lab FTIR-based longwave infrared sensor hyperspectral dataset~\cite{broadwater2011primer}. In our analysis of each data set, we restrict to a $64\times 64$ spatial field of view in which the chemical release is observed. Figures \ref{fig_sf6_number_over}-\ref{fig-SF6_Cube90_Hist} relate to the SF6 27 Romeo release video from the Johns Hopkins dataset. This video consists of $140$ hyperspectral images. Between image $20$ and $30,$ sulfur hexafluoride (SF6) gas is released and disperses. This release is apparent in Fig.~\ref{fig_sf6_number_over}, which shows the number of pixels that the ACE algorithm indicates contain the chemical as a function of time (the number of such pixels in the raw data is indicated by the dashed line and the number of pixels in the data reconstructed from $10\%$ sampling is indicated by the solid black line). It can be seen that during the release itself, more pixels in the reconstructed data have ACE values above the threshold than do those in the raw data. The chemical returns to the scene in a dissipated form sometime around cube 70 (presumably due to a change in wind direction). In this case, the uncompressed data results in slightly stronger chemical detection, with a few exceptions for short time frames.
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{SF6_numberover.png}
\caption{Comparison of chemical detection (Johns Hopkins SF6 27 Romeo dataset): the number of pixels which the ACE algorithm shows to contain the chemical signature for SF6 as a function of video frame for both uncompressed (dashed blue line) and compressively sensed and reconstructed data (solid black line). The $x$-axis is the frame number in the video while the $y$-axis is the number of pixels above the corresponding ACE threshold (where the threshold is as described in Sec.~\ref{sect:dection_algorithms}). During the peak of the release (around frame $40$), the signal is actually stronger in the hyperspectral cube that has been reconstructed from $10\%$ sampling.}
\label{fig_sf6_number_over}
\end{figure}
The specific distributions of bulk coherence ACE values with persistence for hyperspectral image $30$ (both raw (left) and reconstructed (right)), is shown in Fig.~\ref{fig-SF6_Cube30_Hist}. As this histogram indicates, the spread between values corresponding to spatial locations containing the chemical signature and those that do not is much larger in the reconstructed image. This is useful as it makes distinguishing pixels that do and do not contain the chemical easier. For reference the threshold value for chemical presence in the raw image is $0.0077$ and the threshold value for chemical presence in the reconstructed image is $0.2656$.
A similar phenomenon is seen in Fig.~\ref{fig-SF6_Cube90_Hist} but for cube $90$ where most (but not all) the chemical has already dispersed from the scene.
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{SF6_Cube30_Hist_BCP.png}
\caption{A histogram of ACE bulk coherence values with persistence for hyperspectral image $30$ from Fig.~\ref{fig_sf6_number_over}. The $x$-axis is the ACE bulk coherence value (larger values indicate a higher likelihood of the target chemical being present in the pixel). The bin close to zero consists of pixels not containing the chemical whereas the cluster of bins to the right is the collection of pixels that contain the chemical. As can be seen, there is a much wider spread between these two classes in the reconstructed image compared to what is found in the raw image.}
\label{fig-SF6_Cube30_Hist}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{SF6_Cube90_Hist_BCP.png}
\caption{A histogram of ACE bulk coherence values with persistence for hyperspectral image $90$ from Fig.~\ref{fig_sf6_number_over}. The chemical has mostly dispersed from the scene at this point in the video. The $x$-axis is the ACE bulk coherence value (larger values indicate a higher likelihood of chemical being present in the pixel). The bin close to zero consists of pixels not containing the chemical whereas the cluster of bins to the right is the collection of pixels that contain the chemical. As can be seen, there is a much wider spread between these two classes in the reconstructed image compared to what is found in the raw image.}
\label{fig-SF6_Cube90_Hist}
\end{figure}
We next examine two hyperspectral videos in the Fabry-P\'{e}rot interferometer sensor multispectral dataset. The first contains a release of the chemical methyl salicylate (MeS). This release is shown in Fig.~\ref{fig-MESC_numberover}. As before the $x$-axis is the frame in the hyperspectral video and the solid black and dashed blue curve give the number of pixels that ACE indicates contain the chemical for reconstructed and uncompressed data, respectively. As can be seen, this is noisy data with many false positives (which appear as spikes). Despite the fact that the chemical signature in this dataset is quite weak even at the peak of the release, the signal has approximately the same strength in the reconstructed image compared to the raw image.
Figure \ref{fig-MESC_Hist} gives a histogram of ACE bulk coherence values for the MeS release at image 80 when the chemical is present in the scene. Unlike what was seen in Figs.~\ref{fig-SF6_Cube30_Hist} and~\ref{fig-SF6_Cube90_Hist}, there is no longer any clear separation between spatial locations containing the signature for MeS and spatial locations that do not. The fact that there is greater spread of values in the reconstructed image however makes classification of regions containing the chemical easier and more stable.
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{MESC_numberover.png}
\caption{Comparison of chemical detection (Fabry-P\'{e}rot MeS C dataset): the number of pixels which the ACE algorithm shows to contain the chemical signature for MeS as a function of video frame for both uncompressed (dashed blue line) and compressively sensed and reconstructed data (solid black line). The $x$-axis is the frame number in the video while the $y$-axis is the number of pixels above the corresponding ACE threshold (where the threshold is as described in Sec.~\ref{sect:dection_algorithms}). Frequently, the signal in the reconstructed image is just as strong as the signal in the raw image despite the fact that this sequence of hyperspectral images has been reconstructed from $10\%$ sampling.}
\label{fig-MESC_numberover}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{MESC_Cube80_Hist_ACE.png}
\caption{A historgram of ACE values for hyperspectral image $80$ in the MeS release (see Fig.~\ref{fig-MESC_numberover}). MeS gas is still present in the scene for this frame. The $x$-axis is the ACE value (larger values indicate a higher likelihood of chemical presence in the pixel). As can be seen, the reconstructed data (right) has a larger range of ACE bulk coherence values than the raw data. While spatial locations do not clearly separate into two classes, a wider spread in the data likely makes classification easier.}
\label{fig-MESC_Hist}
\end{figure}
Finally, the Fabry-P\'{e}rot interferometer sensor multispectral dataset also contains a hyperspectral video of release of triethyl phosphate (TEP). A plot of this chemical release as captured by ACE is shown in Fig.~\ref{fig-TEPA_numberover_BCP}. This is an example where results of detection in the raw and reconstructed images are mixed. While the signal is better detected in raw data up to the peak of the release, it appears to be the case that the reconstructed data results in better detection in later frames as the chemical dissipates.
This illustrates the point that signal amplification is not always consistent. Examples exist in which the signal is either slightly weaker or of equal strength in reconstructed images. Another important observation (which is discussed in Sec.~\ref{sect:explainations}) is that the false positives that appear in raw images are effectively eliminated in reconstructed images, pointing toward the reconstruction algorithm functioning as a denoising algorithm.
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{TEPA_numberover_BCP.png}
\caption{Comparison of chemical detection (Fabry-P\'{e}rot TEP A dataset): the number of pixels which the ACE algorithm shows to contain the chemical signature for TEP as a function of video frame for both uncompressed (dashed blue line) and compressively sensed and reconstructed data (solid black line). The $x$-axis is the frame number in the video while the $y$-axis is the number of pixels above the corresponding ACE threshold (where the threshold is as described in Sec.~\ref{sect:dection_algorithms}). After the peak of the release (around frame $160$), the signal in the reconstructed data is stronger than the signal in the raw data despite the fact that this hyperspectral data has been reconstructed from $10\%$ sampling. What is more, the uncompressed data exhibits noise throughout whereas the reconstructed data shows a strong signal when the chemical is present in the scene and does not show chemical present outside of that range.}
\label{fig-TEPA_numberover_BCP}
\end{figure}
To give the reader a sense of what these improvements look like in practice, in Fig.~\ref{fig_enhance_example} we show three pairs of frames of ACE detection for both raw data (on the left) and data reconstructed from just $10\%$ CS sampling (on the right). One can see that there are certain instances (such as (a)-(b)) where CS sampling highlights a signal that otherwise would not be noticeable. In (c)-(d), we see that the chemical signal has been amplified but there is a question about whether it is as spatially accurate. Finally, in (e)-(f), CS sampling and reconstruction simply strengthens a signal that is already present, likely showing spatial regions where the signal was too weak to have been seen before.
\begin{figure}[ht]
\centering
\includegraphics[width=10cm]{Figure_frame_evidence2.pdf}
\caption{ACE detection of plumes for a number of chemical compounds. The left column is ACE detection on raw data, the right column is ACE detection on hyperspectral images that have undergone 10\% CS sampling followed by reconstruction via \eqref{eqn-hyper-opt-problem}. (a)-(b) and (e)-(f) are detection of MeS, while (c)-(d) are detection of the chemical TEP.}
\label{fig_enhance_example}
\end{figure}
\subsection{Possible theoretical explanation and analysis}
\label{sect:explainations}
There are a number of possible explanations for the signal amplification phenomenon observed in hyperspectral data reconstruction by solving \eqref{eqn-opt-problem}. The first is that sampling $\tilde{U}$ and then using this sample to reconstruct the approximation $U$ of $\tilde{U}$ is effectively denoising $\tilde{U}$. Indeed, as was noted in Fig.~\ref{fig-TEPA_numberover_BCP}, the rate of false positives of TEP detection is strongly reduced in the reconstructed data compared to the raw data. On the other hand, the denoising effects are not supported by Fig.~\ref{fig-MESC_numberover} where high rates of false positives appear in both the raw and reconstructed data.
Another facet worth further investigation is the apparent trade-off between spatial accuracy and signal detection. As can be seen in Fig.~\ref{fig_enhance_example}, while signal strength is stronger in reconstructed cubes, the spatial accuracy may sometimes be reduced. This would be consistent with the sampling strategy used in this paper where sampling was done in the spatial domain but not the spectral domain. If, more generally, there is a tradeoff between spatial and spectral accuracy, then the phenomenon of signal amplification in reconstructed data will be more applicable to situations where detecting the presence of a chemical is valued above understanding precise spatial locations of chemicals within the field of view.
\section{Conclusion}
\label{sec:conclusion}
In this paper we described a phenomenon in which hyperspectral images sampled at very low levels and reconstructed using techniques from CS not only contain strong chemical signals, but sometimes even contain amplified signals.
This observation suggests some new directions for research, the most important of which is to explain why this is happening. While we suggest some possible explanations in Section \ref{sect:explainations}, much more work needs to be done in this direction. Some other questions that should be investigated include:
\begin{itemize}
\item Does this phenomenon occur when a different reconstruction framework (or optimization problem) is used? It would be especially interesting to know whether this occurs in some CS frameworks specifically designed for hyperspectral imaging (e.g. \cite{golbabaee2012hyperspectral}).
\item Would similar results be obtained when using a different sparsifying basis (i.e. instead of the Haar wavelet basis)?
\item In the experiments described in this paper, the same modified Walsh-Hadamard sampling basis was consistently used \cite{farnell2019sampling}. Does signal amplification still occur when different sampling strategies are used?
\end{itemize}
\acknowledgments
The authors would like to thank Louis Scharf for insightful discussions related to this work, especially with regard to content involving ACE and MPACE. This research was partially supported by
Department of Defense Army STTR Compressive Sensing Flash IR 3D Imager contract W911NF-16-C-0107
and Department of Energy STTR Compressive Spectral Video in the LWIR contract W911SR-17-C-0012.
|
1505.00753
|
\section{Introduction}
CaFe\textsubscript{2}As\textsubscript{2 } at ambient pressure and
temperature is in a paramagnetic tetragonal phase. When temperature
is lowered under 170 K it develops a collinear antiferromagnetic order
and becomes orthorhombic \cite{Ronning-CaFe2As2,Ni-CaFe2As2,Diallo-CaFe2As2-spin}.
Under pressure, this orthorhombic phase can be suppressed and replaced
by a non-magnetic collapsed-tetragonal phase in which the distance
between two FeAs layers is strongly reduced due to the formation of
covalent bonds between As atoms from two different layers. Superconductivity
can also develop from this collapsed phase \cite{Torikachvili-CaFe2As2-SC,Kreyssig-CaFe2As2-collapsed}.
Recently, it has been found that a quench of the annealing phase during
crystal synthesis can produce samples presenting similar properties
as CaFe\textsubscript{2}As\textsubscript{2} under pressure \cite{Ran-CaFe2As2-collapsed,Saparov-CaFe2As2-collapsed}.
At ambient temperature and pressure, they are in the tetragonal phase,
and when temperature is lowered there is a transition into a collapsed-tetragonal
phase, around 90 K in our samples \cite{Saparov-CaFe2As2-collapsed}.
There are also other ways to induce a collapse transition at ambient
pressure, such as isovalent substitution of As by P \cite{Coldea-CaFe2P2},
electron-doping by Rh at the Fe site \cite{Danura-Ca(FeRh)2As2} or
electron doping by rare-earth on the Ca site \cite{Saha-rare-earth-CaFe2As2}.
However, while CaFe\textsubscript{2}As\textsubscript{2} in the collapsed-tetragonal
phase can become superconductor under pressure
\footnote{Superconductivity can also occur in a non-collapsed phase, e.g. in
Ca\textsubscript{1-x}La\textsubscript{x}Fe\textsubscript{2}As\textsubscript{2}
\cite{Saha-rare-earth-CaFe2As2} or for low-doping values in CaFe\textsubscript{2}As\textsubscript{2-x}P\textsubscript{x}
\cite{Kasahara-Fermi-liquid-CaFe2As2}.
} or with rare-earth doping \cite{Torikachvili-CaFe2As2-SC,Saha-rare-earth-CaFe2As2},
it is not the case in these quenched crystals.
During the collapse, the $c$ axis of the unit cell is strongly reduced
by about 10\%, while the $a$ axis is enlarged by about 2\%. This
modification of the crystal structure is at the origin of a reorganization
of the Fermi surface and electronic structure of the compound \cite{Coldea-CaFe2P2,Danura-Ca(FeRh)2As2,Tsubota-Ca(FeRh)2As2,Dakha-CaFe2As2,Gofryk-CaFe2As2},
which has been studied within DFT \cite{Yildirim-spin-As,Tomic-CaFe2As2,Coldea-CaFe2P2,BaCo2As2-Dakha}
and very recently within combined density functional dynamical mean
field theory (``DFT+DMFT'') \cite{Mandal-CaFe2As2,Diehl-CaFe2As2}.
In particular, it was found that the electronic correlations are reduced
in the collapsed phase. Interestingly, the resistivity at the transition
changes its low-energy behavior from $\rho\propto T$ or $\rho\propto T^{1.5}$
in the tetragonal phase to $\rho\propto T^{2}$ -- as in a good Fermi
liquid -- in the collapsed-tetragonal phase \cite{Kasahara-Fermi-liquid-CaFe2As2,Danura-Ca(FeRh)2As2,Saparov-CaFe2As2-collapsed}.
In rare-earth electron-doped Ca\textsubscript{1-x}RE\textsubscript{x}Fe\textsubscript{2}As\textsubscript{2 },
this Fermi-liquid like resistivity is also observed at low temperature,
independently of the stable phase -- collapsed-tetragonal or n\textcolor{black}{on-collapsed-tetragonal
as in Ca\textsubscript{1-x}La\textsubscript{x}Fe\textsubscript{2}As\textsubscript{2 }\cite{Saha-rare-earth-CaFe2As2}.}
Recent Nuclear Magnetic Resonance data further indicate a suppression
of antiferromagnetic spin fluctuations in the collapsed-tetragonal
phase \cite{Furukawa-CaFe2As2}. Motivated by these intriguing results,
we have performed DFT+DMFT calculations and ARPES experiments on the
tetragonal and collapsed-tetragonal phase of CaFe\textsubscript{2}As\textsubscript{2}.
\section{The collapse transition as seen by ARPES}
We have performed angle-resolved photoemission measurements on samples
grown by the self-flux method that are quenched from 960\textdegree{}C
(corresponding to as-grown ``p1'' samples in \cite{Saparov-CaFe2As2-collapsed}).
Experiments were conducted at the CASSIOPEE beamline of SOLEIL synchrotron
(France) and at the Institute of Physics, Chinese Academy of Sciences
(China). Both systems are equipped with VG-Scienta R4000 electron
analyzers. All samples were cleaved \textit{in situ} at temperatures
higher than 200 K and measured in a working vacuum between $5\times10^{-10}$
and $1\times10^{-9}$ torr at SOLEIL, and better than $5\times10^{-11}$
torr at the Institute of Physics. The photon energy was varied from
20 to 80 eV in synchrotron while we used the He I$\alpha$ line of
an helium discharge lamp in the lab (21.218 eV). The angular resolution
was better than 0.5\textdegree{} and the energy resolution better
than 10 meV.
Samples were measured at 200 K, 100 K, 80 K and 30 K. The tetragonal
to collapsed-tetragonal transition is shown to occur around 90 K by
our magnetic susceptibility measurements, with a hysteresis smaller
than 5 K, in agreement with \cite{Saparov-CaFe2As2-collapsed}. Although
we have performed the temperature-dependent measurements at the Institute
of Physics, the Fermi surfaces obtained in SOLEIL are similar to those
obtained in our laboratory, indicating that the measured samples are
in the same phase.
\begin{figure}
\begin{centering}
\includegraphics[width=8.5cm]{Figure1}
\par\end{centering}
\caption{(Color online). Photon energy dependence of the ARPES spectra of CaFe\textsubscript{2}As\textsubscript{2}
in the tetragonal (T = 100 K) and collapsed-tetragonal (T = 30 K)
phases at the Fermi level around the $\Gamma$ point.\label{fig:Photon-energy-dependence}}
\end{figure}
Fig.\ \ref{fig:Photon-energy-dependence} displays the photoemission
spectra centered on the $\Gamma$ point at the Fermi level for different
photon energies, in the collapsed-tetragonal and tetragonal phases.
From the observed periodicity of the spectrum in the collapsed-tetragonal
phase we find that the $\Gamma$ point is located around 33 eV and
70 eV whereas a Z point is found around 50 eV. Using the sudden approximation
and nearly free-electron model for the final state: $k_{\perp}=\sqrt{2m\left(E_{kin}\cos^{2}\theta+V_{0}\right)}/\hbar$
and the lattice parameters of Saparov \emph{et al.} \cite{Saparov-CaFe2As2-collapsed},
we deduce an inner potential $V_{0}$ of about 15 eV, which is consistent
with other Fe-based superconductors \cite{Pierre-ARPES-review}. This
value of the inner potential also corresponds well to the data observed
by Dhaka \emph{et al.} \cite{Dakha-CaFe2As2}. In the tetragonal phase
the data are less clear, but using the same value for $V_{0}$ we
estimate the $\Gamma$ point to be around 28 eV and 60 eV and the
Z point around 43 eV. This assumption is plausible at 28 eV, even
though for higher photon energy there seems to be a slight discrepancy
with the observed spectrum. This might be due to a modification of
the inner potential since the surface will probably be different in
the tetragonal phase.
\begin{figure}
\begin{centering}
\includegraphics[width=8.5cm]{Figure2_2}
\par\end{centering}
\caption{(Color online). Fermi surface mapping of CaFe\textsubscript{2}As\textsubscript{2}
in the tetragonal (T > 90 K) and collapsed-tetragonal (T < 90 K) phases.\label{fig:CaFe2As2-Fermi-surface}}
\end{figure}
Fig.\ \ref{fig:CaFe2As2-Fermi-surface} shows the Fermi surface of
our CaFe\textsubscript{2}As\textsubscript{2} sample in the tetragonal
and collapsed-tetragonal phases recorded with a photon energy of 21.218
eV. We have lowered the temperature from 200 K to 30 K and finished
the measurements less than 30 hours after the cleave, such that the
aging of the sample was not important. Using the previously deduced
inner potential, we find that for the collapsed-tetragonal phase the
$\Gamma$ point %
\footnote{For simplicity, we name the points measured in Fig. \ref{fig:CaFe2As2-Fermi-surface}
as their projection on the $k_{z}=0$ plane.%
} has a $k_{z}$ close to 1.25 $\pi/c'$ -- with $c'=c/2$ the distance
between two FeAs layers, close to the Z point of coordinates $(0,0,\pi/c')$.
$k_{z}$ then decreases when $k_{\parallel}$ is increased, with a
value of 0.89 $\pi/c'$ at the M point and 1.01 $\pi/c'$ at the X
point. It is interesting to note that the point symmetric to $\Gamma$
with respect to the X point would have for coordinates $(2\pi/a,0,0.5\pi/c')$,
such that it would nearly correspond to the same high-symmetry point
\footnote{Indeed, the point Z with coordinates $(0,0,\pi/c')$ is equivalent
to the point with coordinates $(2\pi/a,0,0)$.%
}. For the tetragonal phase, we find $k_{z}=1.6\pi/c'$ at the $\Gamma$
point and $k_{z}=1.19\pi/c'$ at the M point.
For a more detailed analysis of the states forming the Fermi surface,
we also present three different cuts. We first show the $\Gamma$-M
direction for all temperatures (see Fig.\ \ref{fig:GM-CaFe2As2}
for the spectra and its curvature \cite{Peng-curvature}). We also
display a cut near the M point on the direction perpendicular to $\Gamma$-M
(see Fig.\ \ref{fig:GX-kperp-curvature-CaFe2As2} left panel) and
another one along the $\Gamma$-X direction (Fig.\ \ref{fig:GX-kperp-curvature-CaFe2As2}
right panel), for the collapsed-tetragonal phase at 80 K only (similar
results are obtained at 30 K). In the tetragonal phase, we can distinguish
two hole-like bands forming circular hole pockets near the $\Gamma$
point, although one may not cross the Fermi level. We also find two
electron pockets around the M point. This is similar to what is found
in many iron pnictides, and in particular in BaFe\textsubscript{2}As\textsubscript{2}.
Below the transition temperature, the Fermi surface is reorganized.
The circular hole pocket around the $\Gamma$ point shrinks drastically
-- or even disappears -- while a large square hole pocket develops.
This shape of the hole-like bands is very characteristic of the collapsed
structure and qualitatively different from what is seen in BaFe\textsubscript{2}As\textsubscript{2}
\cite{Kaminski-BaFe2As2-CaFe2As2}. This effect is due to a stronger
three-dimensional character, as can be observed from the photon-energy
dependent data of Fig.\ \ref{fig:Photon-energy-dependence}. Indeed,
the $k_{z}$ dispersion is enhanced by the strong As-As $p_{z}$ interlayer
hybridization in the collapsed phase. On the other hand, the electron
pockets near M keep a similar size.
\begin{figure}
\begin{centering}
\includegraphics[width=8.5cm]{Figure3}
\par\end{centering}
\begin{centering}
\includegraphics[width=8.5cm]{Figure4}
\par\end{centering}
\caption{(Color online). ARPES spectra (top) and curvature (bottom) of CaFe\textsubscript{2}As\textsubscript{2}
in the tetragonal (T > 90 K) and collapsed-tetragonal (T < 90 K) phases
along cut 1 of Fig.\ \ref{fig:CaFe2As2-Fermi-surface}.\label{fig:GM-CaFe2As2}}
\end{figure}
From the temperature-dependent photoemission spectra of Fig.\ \ref{fig:CaFe2As2-Fermi-surface},
it is interesting to see how the features become better defined as
temperature is lowered. Notably, there is a clear difference between
spectra above (at 100 K) and below (at 80 K) the collapse transition.
However, because the quasiparticle dispersions are also changed through
this transition, and because overall the spectrum appears to be very
sensitive to temperature, it is difficult to attribute this improvement
to the transition only.
\begin{figure}
\begin{centering}
\includegraphics[width=8.5cm]{Figure5}
\par\end{centering}
\caption{(Color online). Curvature of the ARPES spectra of CaFe\textsubscript{2}As\textsubscript{2}
in the collapsed-tetragonal phase (T < 90 K) along cut 2 (near the
M point, left) and along cut 3 (right) of Fig.\ \ref{fig:CaFe2As2-Fermi-surface}.\label{fig:GX-kperp-curvature-CaFe2As2}}
\end{figure}
\section{DFT+DMFT calculations}
We now turn to a theoretical description of the spectral properties
of CaFe\textsubscript{2}As\textsubscript{2}, using \textit{first
principles} dynamical mean field theory (DMFT) techniques. The first
step are calculations based on the by now well-established DFT+DMFT
method \cite{LDA+DMFT-licht,LDA+DMFT-anisimov-1997}. We use the DFT+DMFT
implementation of \cite{cRPA-DMFT-LaOFeAs-markus} within the Local
Density Approximation (LDA) to the exchange-correlation functional,
and Hubbard and Hund's interactions obtained from the constrained
random phase approximation (cRPA) \cite{cRPA-ferdi-2004} in the implementation
of Ref.\ \cite{TMO-vaugier}. The cRPA calculations yield $F^{0}=2.5$
eV, $F^{2}=6.0$ eV and $F^{4}=4.5$ eV, corresponding to a Hund's
rule coupling of $J=0.75$ eV.
\begin{table}
\begin{centering}
\begin{tabular}{ccccc}
\hline
& $d_{z^{2}}$ & $d_{x^{2}-y^{2}}$ & $d_{xy}$ & $d_{xz+yz}$\tabularnewline
\hline
Tetragonal & 1.43 & 1.37 & 1.64 & 1.57\tabularnewline
collapsed-tetragonal & 1.35 & 1.36 & 1.45 & 1.46\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{Mass renormalizations calculated from DFT+DMFT for the Fe-3\textit{d}
orbitals.\label{tab:Renormalizations}}
\end{table}
Fig.\ \ref{fig:LDA+DMFT-CaFe2As2} presents the superposition of
bands extracted from DFT+DMFT calculations performed at 120 K with
the ARPES spectrum along the $\Gamma$-M direction. We have taken
into account the variation of $k_{z}$ as indicated previously. Overall,
the band renormalization is correctly described by the DFT+DMFT calculations.
The theoretical quasi-particle renormalizations as extracted from
a linearization around the Fermi energy of the imaginary part of the
self-energy on the Matsubara axis are displayed in Table \ref{tab:Renormalizations}.
A caveat is however in order since the linear regime is really restricted
to the first few Matsubara frequencies only, indicating that at the
temperature of the calculation the system is at the border to an incoherent
regime. Interestingly, on larger energy scales, at least in the tetragonal
phase the self-energy can quite well be fit as a power law behavior
$\omega^{\alpha}$ with $\alpha$ around 0.75. This is reminiscent
to what was found in BaFe\textsubscript{2}As\textsubscript{2} in
\cite{udyn-werner}.
In agreement with Refs. \cite{Mandal-CaFe2As2,Diehl-CaFe2As2}, we
find the tetragonal phase to exhibit stronger electronic correlations
than the collapsed phase. Within a given phase, we observe stronger
effects on the $d_{xz+yz}$ and $d_{xy}$ orbitals than on the $d_{z^{2}}$
and $d_{x^{2}-y^{2}}$ ones.
We can also see from the calculations that there may be three hole-like
bands in total in the tetragonal phase but two are nearly degenerate
near the $\Gamma$ point. However, if we look at the precise details
of the low-energy states, we can find several discrepancies. In the
tetragonal phase at the Fermi level, one of the bands near the $\Gamma$
point is not well described. It is not clear if this is due to possible
surface effects, limitations of the calculations or other issues.
On the other hand the electron pockets are well described. In the
collapsed-tetragonal phase, the two hole-like bands near the $\Gamma$
point appear to be very close to each other from photoemission measurements,
as can be seen even more clearly on the 30 K data of Fig.\ \ref{fig:GM-CaFe2As2}
and along the $\Gamma$-X direction of Fig.\ \ref{fig:GX-kperp-curvature-CaFe2As2}.
At the M point, two bands are responsible for the electron pockets,
however the shape deviates from the experimental data due to upbending
of one of the bands. On the other hand, we consider the agreement
for the Fermi vector of the large hole pocket relatively satisfying
since this band is very sensitive to the precise value of $k_{z}$.
If we suppose that the ARPES spectrum reflects the bulk features of
the collapsed-tetragonal phase, an important test for improved calculational
schemes will be the correct prediction of the dispersion of the two
hole-like bands near $\Gamma$, and of the interesting topology found
near the M point in Fig.\ \ref{fig:GX-kperp-curvature-CaFe2As2},
which shows three bands crossing the Fermi level very close to each
other -- one of them being the large hole pocket. This last point
is very specific to this compound in the iron pnictides family and
due to the large $k_{z}$ dispersion of the collapsed phase. We will
present results beyond current DFT+DMFT techniques for CaFe$_{2}$As$_{2}$
in section 5 below.
\begin{figure}
\begin{centering}
\includegraphics[width=8.5cm]{Figure6}
\par\end{centering}
\caption{(Color online). Comparison of DFT+DMFT spectral functions with ARPES
spectra of CaFe\textsubscript{2}As\textsubscript{2} in the tetragonal
and collapsed-tetragonal phases. The parts of the spectral functions
with value higher than 4 eV\textsuperscript{-1} are superimposed
on the ARPES data of CaFe\textsubscript{2}As\textsubscript{2} in
the tetragonal (100 K) and collapsed-tetragonal (80 K) phases along
the $\Gamma$-M direction, represented using the (left) spectra or
(right) curvature.\label{fig:LDA+DMFT-CaFe2As2}}
\end{figure}
\section{Interplay of structural and electronic properties within DFT+DMFT:
interlayer versus intralayer geometries}
The origin of the reduction of correlations in the collapsed-tetragonal
phase compared to the tetragonal phase is challenging to understand
since both Fe-As and As-As bindings are modified. Indeed, in the collapsed
structure the As-As interlayer binding is much stronger, which should
increase the three-dimensional character of the band structure dispersion.
However, the transition has also another effect on the Fe-As binding
since the $c$ axis collapses so much that the As height to the Fe
plane is reduced. The result is that the Fe-As distance is shortened,
suggesting an enhancement of the hybridization between the As-4\textit{p}
and the Fe-3\textit{d} orbitals -- though the expansion of the $a$
axis limits this enhancement.
To decouple these two effects we have performed DFT+DMFT calculations
on two hypothetical ``hybrid'' compounds. In the first one, we keep
the same angle and distances between atoms within the FeAs layers
as in the tetragonal phase, while the interlayer As-As distance is
that of the collapsed-tetragonal phase. In the other one, we do the
opposite: the layer is that of the collapsed-tetragonal phase and
the interlayer distance is that of the tetragonal phase.
\begin{figure}
\begin{centering}
\includegraphics[width=8.5cm]{CaFe2As2_coherence_small}
\par\end{centering}
\caption{(Color online). Imaginary part of the self-energy in Matsubara frequencies
of the $d_{xy}$ orbital for CaFe\textsubscript{2}As\textsubscript{2}
in the tetragonal structure (T), collapsed-tetragonal structure (CT)
and in two hypothetical structures mixing the interlayer (i.e. As-As
interlayer distance) and intralayer (i.e. intralayer Fe and As angles
and distances) of the tetragonal and collapsed-tetragonal structures.
The inset shows a schematic view of the different structures (colours
correspond to the legend).\label{fig:CaFe2As2-hybrid}}
\end{figure}
The imaginary part of the self-energy of the $d_{xy}$ orbital in
Matsubara frequencies is displayed in Fig.\ \ref{fig:CaFe2As2-hybrid}.
The effect on the $d_{xz}+d_{yz}$ orbital is similar, and since those
orbitals have the highest density-of-states at the Fermi level we
expect that they control the coherence properties of the compound
\footnote{In contrast, there is no difference between the four structures for
the $d_{x^{2}-y^{2}}$ orbital and a much smaller one for the $d_{z^{2}}$
orbital.%
}. We can first see that in the collapsed phase the imaginary part
of the self-energy displays a more coherent behavior, which corresponds
to the longer lifetime of quasiparticles displayed in Fig.\ \ref{fig:LDA+DMFT-CaFe2As2}.
Furthermore, the shape of the self-energy of the hybrid compounds
depends on the structure of the FeAs layer, while it is nearly insensitive
to the interlayer As-As distance. Naturally, in reality those two
effects are linked with each other, since the deformation of the FeAs
layer is caused by the formation of As-As bonds that make the $c$
axis collapse. Still, this numerical experiment indicates that within
DFT+DMFT the improvement of coherence properties is not due to the
interlayer As-As bonding but to the increase of the Fe-As hybridization
within one single layer.
\section{Beyond DFT+DMFT: results from screened-exchange dynamical mean field
theory}
\begin{figure}
\begin{centering}
\includegraphics[width=8.5cm]{Figure8bis}
\par\end{centering}
\caption{(Color online). Comparison of SEx+DDMFT spectral function with ARPES
spectra of CaFe\textsubscript{2}As\textsubscript{2} in the collapsed-tetragonal
phase. The parts of the spectral functions with value higher than
4 eV\textsuperscript{-1} are superimposed on the ARPES data of CaFe\textsubscript{2}As\textsubscript{2}
along the $\Gamma$-M direction, represented using the (left) spectra
or (right) curvature.\label{fig:SEX+DDMFT}}
\end{figure}
Recently, some of us have proposed a new calculational scheme that
goes beyond current DFT+DMFT techniques \cite{Ambroise-BaCo2As2,Ambroise-SrVO3}:
the combination of a screened exchange Hamiltonian with ``dynamical
DMFT'', that is DMFT extended to dynamical Hubbard interactions \cite{udyn-michele,udyn-werner,SrVO3-dynU-Wang}
was shown to drastically improve upon the low-energy description of
BaCo$_{2}$As$_{2}$. In this context, the apparent success of the
standard DFT+DMFT scheme was shown to result from an error cancelation
effect: a one-body Hamiltonian where the local exchange-correlation
potential of DFT has been replaced by a non-local screened Fock term
has in fact a wider band structure than the DFT one, but including
dynamical screening effects at the level of the Hubbard interactions
leads to additional renormalizations of the electronic states, as
compared to usual DMFT. For this reason, the overall bandwidth of
LDA+DMFT calculations and ``Screened Exchange+Dynamical DMFT'' (``SEx+DDMFT'')
calculations are similar. The low-energy dispersions are however quite
strongly improved by the introduction of the non-local screened exchange
contribution.
In Fig.\ \ref{fig:SEX+DDMFT}, we present the results of this SEx+DDMFT
scheme, in the implementation of Ref.\ \cite{Ambroise-BaCo2As2},
applied to CaFe$_{2}$As$_{2}$. We use the frequency-dependent interactions
as calculated in Ref.\ \cite{Shell-folding} and a value of $\lambda=1.7$
$a_{0}^{-1}$ for the screening wavelength which corresponds to the
density-of-states at the Fermi level of the calculated result. As
anticipated, due to the antagonistic effects of the non-local screened-exchange
and of the high-frequency tail of the interactions, the overall renormalization
of SEx+DDMFT is similar to the DFT+DMFT one. Still, the details of
the quasiparticles dispersions at low energy are importantly modified.
In the collapsed-tetragonal phase of CaFe\textsubscript{2}As\textsubscript{2},
the two bands observed near the $\Gamma$ point are correctly described,
in contrast to DFT+DMFT. The third large hole pocket is at variance
with the calculations, but we stress that its precise Fermi vector
is very sensitive on the value of $k_{z}$.
\section{Conclusion}
We have performed a study of the tetragonal and collapsed-tetragonal
phases of CaFe\textsubscript{2}As\textsubscript{2} using ARPES and
electronic structure calculations. Our results support the picture
that within DMFT, the collapsed-tetragonal phase exhibits reduced
correlations and higher coherence temperature due to the higher Fe-As
hybridization, in agreement with other studies \cite{Mandal-CaFe2As2,Diehl-CaFe2As2}.
However, we note that the reduction of correlations that we observe
does not result in a dramatic change in the electronic self-energy
that could explain by itself the behavior seen by resistivity measurements.
This is confirmed by the ARPES results in the sense that the quasiparticle
lifetimes away from the $\Gamma$ point do not seem to be strongly
impacted by the transition. Around the $\Gamma$ point itself the
electronic states found by photoemission appear more coherent but
their dispersion has been largely reshaped by the collapse of the
crystal. Furthermore, the unconventional transport behavior observed
in experiments might not be the result of a change in electronic coherence
alone. The reconstruction of the Fermi surface and of the low-energy
electronic dispersion might induce geometric effects \emph{via}, e.g.
the Fermi surface nesting or dimensionality. Finally, at the temperature
where the $T$ -- or $T^{1.5}$ -- behavior of the resistivity is
observed, phonons likely also come into play.
\section*{ACKNOWLEDGMENTS}
We acknowledge useful discussions with V\'{e}ronique Brouet and the
coauthors of Ref.\ \cite{Ambroise-BaCo2As2}, in particular Thomas
Ayral, Michel Ferrero and Olivier Parcollet for support on the TRIQS
toolkit \cite{TRIQS-website}. We acknowledge SOLEIL for provision
of synchrotron radiation facilities and we would like to thank Fran\c{c}ois
Bertran, Patrick Le F\`{e}vre and Amina Taleb for assistance in using
the beamline CASSIOPEE. This work was supported by the Cai Yuanpei
program, IDRIS/GENCI Orsay under project 091393 and the European Research
Council under project 617196. We also acknowledge grants from MOST
(2010CB923000 and 2011CBA001000, 2011CBA00102, 2012CB821403) and NSFC
(10974175, 11004232, 11034011/A0402, 11234014 and 11274362) from China.
The work at Oak Ridge National Laboratory was primarily supported
by the U. S. Department of Energy, Office of Science, Basic Energy
Sciences, Materials Science and Engineering Division.
\bibliographystyle{plain}\input{CaFe2As2.bbl}
\end{document}
|
0911.5272
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\section{Introduction}
Greaves, Rodr\'{\i}guez and Ruiz-Camacho have recently published a very interesting paper in this journal, entitled ``A one-way speed of
light experiment''.\cite{Greetal2009} The authors correctly refer to the literature when stating that the problem of clock synchronization
has been used to show that the one-way speed of light is a quantity that cannot be measured. They further observe that ``any measurement
of the speed of light in one direction, from A to B, which uses only one clock to avoid the problem of synchronization, requires the
return back to A of the time information of the arrival at B''.
The experiment they propose is an attempt to avoid this situation. Nevertheless, it occurs as well.
As pointed out by
Finkelstein,\cite{Finkelstein2009} what the authors indeed measure is the two-way speed of light, if the coaxial cable of length $L$ is equivalent to a
light signal propagating in vacuum with speed $c$, as the fixed time delay is assumed to be
$\Delta t=L/c = 23.73\ \mbox{m}/3\times10^8\ \mbox{m/s}=79$ ns.
By using the value $c$ to calculate $\Delta t$,
in fact the authors have implicitly used two clocks, one at the photosensor and one at the oscilloscope,
``synchronized'' according to Einstein's procedure.
The experiment described corresponds to the example of ``synchronizing'' clocks with the speed of a $F_1$ car presented in a
previous paper.\cite{AG2008}
It is impossible to measure the speed of a $F_1$ between two points in a circuit by using its average speed along the track
to ``synchronize'' clocks. Of course the $F_1$ car
has a speed, but this speed is not the difference of times at the two positions if the clocks have been previously set using the average speed of the car.
To make the analogy precise, consider a track of length $L$ and let the $F_1$ car do several laps along the circuit, always in the same way (braking on
the same positions, accelerating on the same positions, etc.). With one clock at the start-finish line, $C_0$, we can determine the average speed of the
$F_1$ on the circuit, $\bar{v}$, by measuring the time it takes to complete one lap.
Now, if we have another clock, $C_1$, on a position of the circuit at a distance $L_1$ from the start-finish line, we can
``synchronize'' this clock with the one at the origin by setting $C_0$ to zero when the car passes at the
start-finish line and by setting $C_1$ to $L_1/\bar{v}$ when the car passes at in front of $C_1$
(note that this is exactly what it is done in the paper by Greaves \textit{et al} \cite{Greetal2009} by
assuming a constant time delay of 79 ns).
Of course we can now ``measure'' the ``speed'' of the car
on the remaining portion of the circuit, corresponding to a distance $L-L_1$. With the clocks set in the way just described, we will conclude,
without surprise, that this ``speed'' is $\bar{v}$, regardless of the value of $L_1$.
Similarily, the measurement by Greaves \textit{et al} \cite{Greetal2009} for the ``speed'' of
light gives $c$.
What has been measured is not the one-way speed of light, it is the two-way speed of light or, equivalently,
the one-way ``Einstein speed'' of light, formerly defined.\cite{GA2006b}
Of course this speed is rigorously $c$, because
the procedure has been done in such a way that it cannot be otherwise!
There is an implicit synchronization in Greaves \textit{et al},\cite{Greetal2009} where the clocks have been ``synchronized'' operationally.
There is no problem in doing so, but we cannot attribute physical meaning to a perfectly defined quantity (the Einstein speed)
other than its true meaning. If we do not know the one-way speed of light in one frame, we cannot use $c$ to cope with half of the circuit (the delay
on the coaxial cable) and then pretend we have measured the one-way speed of light in the remaining part of the circuit.
The knowledge of the one-way ``Einstein speed'' of light is just a result of a definition, based on an experimental fact: the constancy
of the \textit{two-way} speed of light. So far, this knowledge does not justify the assumption of the constancy of the one-way speed
of light in all frames, although we can work, for operational reasons, \textit{as if} the one-way speed of light was $c$.\cite{GA2006b,AG2008}
\bibliographystyle{unsrt}
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0902.0874
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\section{Introduction}
Free-energy calculations have a fundamental role in the understanding
of many natural phenomena ranging from protein folding up
to polymorphic transitions in solids.
To this end, molecular dynamics (MD)
has been extensively used together with a series of algorithms
aimed at extending its capabilities far beyond those allowed by straightforward MD.
Among some of the most popular {\it enhanced sampling} techniques are
umbrella sampling~\cite{torrie-valleau,wham1,wham2},
Jarzynski equation based methods~\cite{jarzynski},
adaptive force bias~\cite{darv-poho01jcp} and metadynamics~\cite{metad}.
\\
At the same time, a multitude of MD codes have been developed through the years
with focus on different fields of application. Some of these programs are more solid-state oriented, with
a particular attention to the variety of implemented potentials (DL\_POLY~\cite{dlpoly}).
Others are more specialized in biomolecular systems, with specific potentials developed to that scope
(CHARMM~\cite{CHARMM},GROMACS~\cite{Hess:2008p11450} and NAMD~\cite{NAMD}), or implicit solvent capabilities
(CHARMM, AMBER~\cite{amber} and GROMOS~\cite{gromos1,gromos2}).
Recently, a lot of effort has also been devoted to efficient parallelization,
allowing a linear scaling reduction of computational
time with the number of processors. In this respect, NAMD, DESMOND~\cite{desmond},
GROMACS and pmemd in AMBER are currently among the best performing programs.
\\
This wealth of codes provides the user a wide range of capabilities, but only few of them offer
interfaces for specific free-energy methods, often with a limited set of collective variables.
In particular, metadynamics has been implemented separately for some of these programs,
and so far the only one freely distributed is GROMETA~\cite{camilloni_protG} for GROMACS.
This prompted us to develop a plugin compatible with many of the aforementioned codes so as to
facilitate free-energy calculations with a unified input.
\section{Theoretical background}\label{theory}
\subsection{Free-energy methods}\label{methods}
We consider a system made of $N$ atoms characterized by microscopic coordinates
$\bm{r}\in\mathbb{R}^{3N}$ and a potential energy function $U(\bm{r})$.
We then introduce a set of $d$ collective variables (CVs), $\bm{s}(\bm{r})$, where $\bm{s}\in\mathbb{R}^{d}$.
These variables are used as order parameters, \emph{i.e.} to distinguish between
macroscopically different configurations.
The Helmholtz free energy as a function of these CVs is defined as:
\begin{equation}
F(\bm{s})=-\frac{1}{\beta}\ln\int d\bm{r}\ e^{-\beta U(\bm{r})}\delta( \bm{s}- \bm{s}(\bm{r}) ) + C,
\end{equation}
where $C$ is an immaterial constant.
The free energy $F(\bm{s})$ contains crucial informations about the thermodynamics of the system,
and allows the calculation of the ensemble average of any observable that depends on the
CVs $\bm{s}$. Moreover, when the CVs are properly chosen,
the free-energy profile can be used to model the out-of-equilibrium
behavior of the system by means of a stochastic
dynamics~\cite{zwan61pr,zwan+01book,yang+06jcp,rait+08acie}.
The simplest way to obtain the Helmholtz free energy from
an unbiased MD simulation is to evaluate the histogram of the visited configurations
in the CVs space $N(\bm{s})$, so that the estimated free energy $\tilde{F}(\bm{s})$ reads:
\begin{equation}
\label{eq:histogram}
\tilde{F}(\bm{s})=-\frac{1}{\beta}\ln N(\bm{s}).
\end{equation}
However, in presence of rare events this method
is completely impractical, since it would require an enormous
computational time.
Many methods have been proposed to tackle the rare-event problem and to calculate
free-energy profiles.
Some of them are aimed at enhancing the sampling of
the canonical ensemble, so that the free energy is still estimated
by Eq.~(\ref{eq:histogram}).
This class includes methods such as
simulated tempering~\cite{mari-pari92el},
parallel tempering~\cite{hans97cpl,sugi-okam99cpl},
Hamiltonian replica exchange~\cite{fuku+02jcp} and
solute tempering~\cite{liu+05pnas}.
Here we concentrate on a second class of methods, which are based
on collecting configurations in a biased ensemble and require one to select the
set of CVs prior to the simulation.
The prototype of all these methods is umbrella sampling~\cite{torrie-valleau}, where
the simulation is performed with a fixed additional
bias potential $V(\bm{s})$, and the unbiased free energy is recovered as
\begin{equation}
\label{eq:histogram-torrie-valleau}
\tilde{F}(\bm{s})=-\frac{1}{\beta}\ln N(\bm{s})-V(\bm{s}).
\end{equation}
While Equation~(\ref{eq:histogram-torrie-valleau}) is valid for any choice of
the bias potential $V(\bm{s})$, the efficiency of the sampling and convergence properties are strongly
dependent on $V(\bm{s})$. In particular, if an approximate free-energy estimate $\tilde{F}'(\bm{s})$
is available before the simulation, the choice
$V(\bm{s})=-\tilde{F}'(\bm{s})$ would give an approximately flat histogram, thus helping
in overcoming the free-energy barriers.
However, it is rather difficult to have a reliable free-energy estimate before
the simulation.
Various improvements have been introduced so as to refine the bias
potential on the fly (see, among others, Refs.~\cite{wang-land01prl,darv-poho01jcp,metad,mars+06jpcb}).
We focus here on metadynamics, in its standard form~\cite{metad,error} or in the recently introduced
well-tempered flavor~\cite{Barducci:2008}.
In particular, we consider the direct version of metadynamics, where the bias is acting
directly on the microscopic coordinates. For an excellent review of several variants
of metadynamics see Ref.~\cite{laio-gerv08rpp}.
In standard metadynamics the bias potential is built during the simulation
as a sum of Gaussian functions centered on the previously visited configurations
in the CVs space.
This manner of biasing the evolution by discouraging the visited configurations
was first introduced in the taboo search \cite{taboo} and, in the context of MD, by the
local elevation method~\cite{willy}. The approach is also closely related to the
Wang and Landau algorithm~\cite{wang-land01prl},
adaptive force bias~\cite{darv-poho01jcp} and
self-healing umbrella sampling~\cite{mars+06jpcb}.
In metadynamics, the bias at time $t$ is written as
an integral on the past
trajectory $\bm{r}(t)$:
\begin{equation}
V(\bm{s},t)= \int_0^t\ dt'\omega\exp\left(-
\sum_{i=1}^{d} \frac{(s_i(\bm{r})-s_i(\bm{r}(t'))^2}{2\sigma_i^2}
\right).
\end{equation}
Here $\sigma_i$ is the Gaussian width corresponding to the $i$-th CV
and represents the resolution for that CV, and $\omega$ is the rate
at which the bias grows.
As it was shown empirically~\cite{error} and analytically~\cite{bussi_noneq} for model Langevin dynamics,
in the long run the bias will converge to the negative of the free energy and then oscillate around that profile.
As a consequence, the final histogram will be approximately flat in the CVs space, allowing for an uniform exploration
in spite of the free-energy barriers.
We also observe that the bias performs a work on the system, which needs to be dissipated.
Usually during a metadynamics simulation the thermostat keeps the system in thermal
equilibrium, unless the growth rate of the bias is too large.
Metadynamics has been historically used in two different and complementary manners.
It has been used to escape free-energy minima (see \emph{e.g.} ~Ref.~\cite{ogan+05nature}).
\emph{i.e.} to find a reasonable saddle point out of a local minimum. In this case, metadynamics
should be stopped as soon as the system exits from the minimum and starts exploring
a new region of space. In other applications, it has been used to exhaustively explore the
CV space and reconstruct the free energy.
In the examples presented in this paper, we focus more on this latter application.
Its main advantage over umbrella-sampling technique is that it inherently
explores the region of low free energy first. Here the
simulation should be stopped when the
motion of the CVs becomes diffusive in this region of interest,
and the bias itself can be used as an estimate of the underlying
free energy:
\begin{equation}
\tilde{F}(\bm{s},t)=-V(\bm{s},t)
\end{equation}
(note that if $\bm{s}$ does not include all relevant order parameters the bias
may not converge in a reasonable simulation time).
The free-energy estimate at time $t$, $\tilde{F}(\bm{s},t)$,
is indeed an unbiased estimator of the exact
free energy $F(\bm{s})$~\cite{bussi_noneq}.
However, $\tilde{F}(\bm{s},t)$ fluctuates around $F(\bm{s})$ with an amplitude
which depends on both the diffusion coefficient in the CV
space and on the metadynamics parameters $\omega$ and $\sigma$, and
a more accurate calculation can be performed decreasing $\omega$.
Clearly, a smaller $\omega$ means that more time is required to reconstruct the free-energy landscape, therefore a
compromise needs to be found between speed and accuracy.
One can also exploit the fact that $\tilde{F}(\bm{s},t)$ is an unbiased estimate
of $F(\bm{s})$ at all times, and take the time
average of all the profiles as done, for instance, in Ref.~\cite{mich+04prl}.
However,
as the simulation continues,
configurations of higher and higher free energy are explored and, in order
to take the average it is necessary to force the system to remain inside
the region by a suitable restraining potential.
An alternative approach is the recently introduced well-tempered metadynamics~\cite{Barducci:2008}.
Well-tempered metadynamics
is a variant of the method that solves the problem of the fluctuations in a different way,
and is more suitable for performing free-energy calculations in several dimensions since
it allows avoiding the complication of restraining the dynamics inside a region.
In the well-tempered algorithm, the rate at which the bias is grown is decreased
during the simulation proportional to $e^{-V(\bm{s},t)/\Delta T}$,
where $\Delta T$ is a characteristic energy:
\begin{equation}
V(\bm{s},t)= \int_0^t\ dt'\omega e^{-V(\bm{s}(\bm{r}(t')),t')/\Delta T}\exp\left(-
\sum_{i=1}^{d} \frac{(s_i(\bm{r})-s_i(\bm{r}(t'))^2}{2\sigma_i^2}
\right).
\end{equation}
Over long time, it can be shown that the bias converges to a fraction of the exact free energy:
\begin{equation}
\lim_{t\rightarrow\infty} V(\bm{s},t)= -\frac{\Delta T}{T+\Delta T} F(\bm{s}).
\end{equation}
Conversely, it is possible to estimate the free energy as:
\begin{equation}
\tilde{F}(\bm{s},t)= -\frac{\Delta T+T}{\Delta T} V(\bm{s},t).
\end{equation}
This estimator does not suffer of the fluctuation problem of standard metadynamics.
Moreover at variance with standard metadynamics, the exploration for large times will not
be uniform in the CV space but instead it will satisfy the probability distribution:
\begin{equation}
P(\bm{s})\propto e^{-F(\bm{s})/(T+\Delta T)}.
\end{equation}
Thus, the CVs will be sampled at a finite but arbitrarily high temperature $T+\Delta T$.
This is a rather important feature of well-tempered metadynamics, especially
for $d>1$, since it allows to focus the exploration on the low free-energy regions,
such as the main minima and the saddle points.
The explored free-energy range can be varied by tuning $\Delta T$,
and standard metadynamics is recovered for $\Delta T\rightarrow\infty$.
In practical implementations, the bias is updated with a finite
temporal stride $\tau_G$, so that:
\begin{equation}
V(\bm{s},t)= \sum_{t'=0,\tau_G,2\tau_G,\dots}^{t'<t} W e^{-V(\bm{s}(\bm{r}(t')),t')/\Delta T} \exp\left(-
\sum_{i=1}^{d} \frac{(s_i(\bm{r})-s_i(\bm{r}(t'))^2}{2\sigma_i^2}
\right),
\end{equation}
where $W=\tau_G \omega$ is the height of a single Gaussian.
\subsection{Metadynamics implementation}
A metadynamics implementation should perform two basic tasks:
(a) keep track of the visited configurations in the CV space or, equivalently, of the shape of the bias potential and
(b) add the proper forces to the microscopic dynamics.
The first task is accomplished by maintaining a list of the Gaussians which have been added to the bias.
This list is dynamic and grows during the simulation. The list is also stored in a file
that can be used to restart a simulation, and to plot the bias with an external utility.
The second task requires evaluation of the bias forces, \emph{i.e.} the derivatives of the bias potential with
respect to the microscopic coordinates $\bm{r}$.
This derivative is calculated using the chain rule:
\begin{equation}
\label{eq:chainrule}
\frac{\partial V(\bm{s},t)}{\partial \bm{r}_i}=\sum_{j=1}^{d}\frac{\partial V(\bm{s},t)}{\partial
s_j}\frac{\partial s_j(\bm{r})}{
\partial \bm{r}_i}.
\end{equation}
The first part of the derivative is simply a sum of analytical derivatives of Gaussian functions.
This sum gets more and more expensive as the simulation proceeds and the number of Gaussians grows~\cite{babi+08jcp}.
Usually this is not a problem since, if we exclude the simplest test cases, this effort is incomparably smaller than that
of evaluating the force-field. For specific needs, an implementation based on the storage of the
bias potential on a grid could be faster.
The second part of the derivative in Eq.~(\ref{eq:chainrule}) depends on the specific choice for the CVs.
Thus, for each of the CVs that one wants to use, it is necessary to provide routines which, given
the microscopic coordinates, return the value of the CV and its gradients.
Writing and debugging these routines for a large number of CVs requires a noticeable effort.
However, it is worthwhile pointing out that this effort is the main ingredient of many other free-energy methods,
and thus our plugin has been adapted to perform other kinds of free-energy calculations,
such as umbrella sampling~\cite{wham1,wham2} or Jarzynski equation~\cite{jarzynski} based methods,
as it will be discussed in Section~\ref{usage}.
\subsection{Collective variables}
The implementation of many different
CVs is required to deal with the huge variety of problems of interest and to give a proper description of each.
Here we describe all the possibilities present in the current package.
\begin{itemize}
\item {\bf Atom position.}
The absolute position of an atom or a group of atoms.
This CV is implemented with several options that allow the user to restrict the bias to a given
direction, \emph{e.g.} $z$, or to bias the position of the particle as projected onto a
selected segment or, in analogy with the path CV, to bias the atoms distance from a segment.
This variable is not translationally invariant.
\item {\bf Distance.}
The distance between two atoms or, more generally,
the distance between the centers of mass of two groups of atoms identified as $G_1$ and $G_2$:
\begin{eqnarray}
s_{dist}&=&\left| \frac{\sum_{i\in G_1} m_i\bm{r}_i }{\sum_{i\in G_1} m_i} -
\frac{\sum_{i\in G_2} m_i \bm{r}_i}{\sum_{i\in G_2} m_i}
\right| \\
&=&\left| \bm{r}_{G_1} - \bm{r}_{G_2} \right|,
\end{eqnarray}
where $m_i$ and $\bm{r}_i$ are the mass and the position of the $i-th$ atom respectively,
and $\bm{r}_{G_1}$ and $\bm{r}_{G_2}$ are the centers of mass of the two groups.
\item {\bf Angle.}
The angle defined by three atoms or, more generally,
the angle defined by the centers of mass of three groups of atoms identified as $G_1$, $G_2$ and $G_3$:
\begin{eqnarray}
\bm{r}_c&=&\bm{r}_{G_1}-\bm{r}_{G_2}\\
\bm{r}_b&=&\bm{r}_{G_1}-\bm{r}_{G_3}\\
\bm{r}_a&=&\bm{r}_{G_2}-\bm{r}_{G_3}\\
s_{angle}&=& \cos^{-1}\left(\frac{\bm{r}_a^{2} +\bm{r}_c^{2} -\bm{r}_b^{2} }{ 2 |\bm{r}_a||\bm{r}_c|}\right).
\end{eqnarray}
\item {\bf Torsion.}
The dihedral angle defined by four atoms or, more generally,
the dihedral angle defined by the centers of mass of four groups of atoms
identified as $G_1$, $G_2$, $G_3$ and $G_4$:
\begin{eqnarray}
\bm{r}_a&=&\bm{r}_{G_4}-\bm{r}_{G_3}\\
\bm{r}_b&=&\bm{r}_{G_2}-\bm{r}_{G_3}\\
\bm{r}_c&=&\bm{r}_{G_3}-\bm{r}_{G_2}\\
\bm{r}_d&=&\bm{r}_{G_1}-\bm{r}_{G_2}\\
s_{torsion}&=& \cos^{-1}\left(\frac{(\bm{r}_a\times \bm{r}_b) \cdot ( \bm{r}_c \times\bm{r}_d) }
{ \vert \bm{r}_a\times \bm{r}_b \vert \vert \bm{r}_c \times\bm{r}_d \vert }\right).
\end{eqnarray}
\item {\bf Minimum distance.}
The distance between the two closest atoms pertaining to two different groups $G_1$ and $G_2$,
approximately obtained with the following expression:
\begin{equation}
s_{mindist}=\frac{b}{\ln\ \sum_{i\in G_1} \sum_{j\in G_2} \exp\left( \frac{b}{\vert \bm{r}_i - \bm{r}_j \vert } \right)},
\end{equation}
where $b$ is a user-supplied smoothing parameter.
\item {\bf Coordination.}
The coordination number of one atom, or more atoms, with respect to another atom or group of atoms
(\emph{e.g.} the coordination of an ion with respect to all the water molecules in the simulation box):
\begin{equation}
s_{coord}=\sum_{i\in G_1} \sum_{j\in G_2} s_{ij},
\end{equation}
where
\begin{equation}
s_{ij} = \left\{ \begin{array}{lr}
1 & \hspace{0.5cm} \textrm{if} \hspace{0.5cm} \vert \bm{r}_i - \bm{r}_j \vert < \delta \\
\frac{1-\left(\frac{\vert \bm{r}_i - \bm{r}_j \vert-\delta}{r_0}\right)^n}{ 1-\left(\frac{\vert \bm{r}_i - \bm{r}_j \vert-\delta}{r_0}\right)^m} & \hspace{0.5cm} \textrm{if} \hspace{0.5cm} \vert \bm{r}_i - \bm{r}_j \vert \ge \delta . \\
\end{array}\right.
\end{equation}
The user-supplied parameters $r_0$, $\delta$, $n$ and $m$ allow a great flexibility to fine-tune the decay of the switching function, \emph{e.g.} a more accurate counting of the atoms in the coordination shell.
In general a good guess for these parameters can be achieved by looking at the pair distribution function between the first and the second group
of atoms. A good starting point is to take $\delta$ as the position of the first peak in the pair distribution function,
$r_0$ as the full width at half maximum of the peak and $n$ and $m$ to force $s_{ij}\simeq0$ at the first minimum of the pair distribution function.
However, depending on the system properties, different choices may give better results.
\item {\bf Hydrogen bonds.} The number of intra-protein hydrogen bonds, defined as:
\begin{equation}
s_{hbonds}=\sum_{i\in G_O} \sum_{j\in G_H} f(i,j) \frac{1-\left(\frac{\vert \bm{r}_i - \bm{r}_j \vert}{r_0}\right)^n}{ 1-\left(\frac{\vert \bm{r}_i - \bm{r}_j \vert}{r_0}\right)^m},
\label{coord}
\end{equation}
where $G_O$ the group of oxygen atoms of the protein,
$G_H$ the group of hydrogen atoms of the protein.
Typically, $r_0=2.5 \ {\rm \AA}$, $n=6$ and $m=10$.
The function $f(i,j)$ selects a particular type of hydrogen bonds,
depending on the user choice: all, $\alpha$--helix pattern, $\beta$--strand
pattern (even or odd).
\item {\bf Interfacial water.}
This variable is intended to calculate the number of atoms of a certain group $G_0$ that are
in contact with atoms of both groups $G_1$ and $G_2$ at the same time (\emph{e.g.} the number of waters at the interface of two surfaces):
\begin{equation}
s_{waterbridge}= \sum_{i\in G_0} \left( \sum_{j\in G_1} \frac{1-\left(\frac{\vert \bm{r}_i - \bm{r}_j \vert}{r_0}\right)^n}{ 1-\left(\frac{\vert \bm{r}_i - \bm{r}_j \vert}{r_0}\right)^m} \right )
\left( \sum_{k\in G_2} \frac{1-\left(\frac{\vert \bm{r}_i - \bm{r}_k \vert}{r_0}\right)^n}{ 1-\left(\frac{\vert \bm{r}_i - \bm{r}_k \vert}{r_0}\right)^m} \right ).
\end{equation}
More precisely, the variable counts the number of atom triples $(i\in G_0,j\in G_1,k\in G_2)$
with $i$ in contact with both $j$ and $k$
(see also coordination parameters).
\item {\bf Radius of gyration.}
The radius of gyration of a group $G$, defined as:
\begin{equation}
s_{gyration}=\sqrt{ \frac{\sum_{i\in G} m_i \vert \bm{r}_i -\bm{r}_G \vert ^2 }{\sum_{i\in G} m_i} }.
\end{equation}
Similarly, one can be interested in the trace of the inertia tensor, which can be shown
to be equal to:
\begin{equation}
s_{\textrm{Tr}[I]}=
2
(s_{gyration})^2
{\sum_{i\in G} m_i}.
\end{equation}
\item {\bf Dipole moment.}
The electric dipole of a group of atoms:
\begin{equation}
s_{dipole}=\vert \sum_i^{n} q_i\bm{r}_i \vert,
\end{equation}
where $q_i$ is the charge of each atom $i$.
\item {\bf Dihedral correlation.}
This variable measures the degree of similarity of a list of adjacent dihedral angles.
It is defined by:
\begin{equation}
s_{dihecorr}=\sqrt{\sum_{i=2}^{n}\left( 1 + \cos^2\left( \frac{\phi_i-\phi_{i-1}}{2}\right)\right)},
\end{equation}
where $\phi_i$ is the dihedral angle defined by four atoms.
For proteins, the variable grows with the content of secondary structure.
\item {\bf Alpha-beta similarity.}
This variable measures the similarity of dihedral angles with respect to reference values:
\begin{equation}
s_{\alpha-\beta}=\frac{1}{2}{\sum_{i=1}^{n}\left( 1 + \cos\left( \phi_i-\phi_{i}^{ref}\right)\right)},
\end{equation}
where reference dihedrals $\phi_{i}^{ref}$ are given as input.
For proteins, this variable can be use to measure the amount of $\alpha$ or $\beta$
secondary structure.
\item {\bf Torsional rmsd.}
Root of mean square deviation of selected dihedral angles with respect to a reference configuration:
\begin{equation}
s_{tors rmsd}=\sqrt{ \frac{ \sum_i^{n}( \theta_i - \theta_i^{ref} )^2 }{n }},
\end{equation}
where $n$ is the number of reference dihedrals.
\item {\bf Path collective variables.}
Path collective variables are a general approach based on a previous (approximate) knowledge
of the reaction path~\cite{brand07}.
If one assumes that the transition from A to B can be described
by a set
of CVs $\mathbf{S}(\bm{r})$, which are in general
non-linear vectorial functions of the microscopic variables $\bm{r}$,
then it is possible to define two
associated variables. One aims at measuring the
progress along a parametric path in CVs space $\mathbf{S}(l)$ composed
of $P$ frames:
\begin{equation}
s\left( \bm{r}\right) = \frac{\sum_{l=1}^{P} l\ e^{-\lambda
\parallel \mathbf{S}(\bm{r})-\mathbf{S}(l)\parallel ^{2}}}{%
\sum_{l=1}^{P}e^{-\lambda \parallel \mathbf{S}(\bm{r})-\mathbf{S}%
(l)\parallel ^{2}}}, \label{discrets}
\end{equation}
and the other measures the distance from the closest point along the path:
\begin{equation}
z\left( \bm{r}\right) =-\frac{1}{\lambda }\ln \Big(\sum_{l=1}^{P}e^{-%
\lambda \parallel \mathbf{S}(\bm{r})-\mathbf{S}(l)\parallel ^{2}}\Big ),
\label{discretz}
\end{equation
where $\left\Vert \ldots \right\Vert $ is the metric that defines the
distance between two configurations.
With such definitions $s\left( \bm{r}\right) $ is a pure number that ranges from 1 to $P$ and $z\left( \bm{r}\right)$ has the dimension of the chosen distance, squared. The parameter $\lambda$ is in general chosen as
$2.3/(\Delta d)^2$ where $\Delta d$ is the average distance among adjacent frames.
The definition of the $\mathbf{S}(l)$ and of the measure $\left\Vert \ldots \right\Vert $ can be chosen by the user among the following:
\begin{itemize}
\item {\bf Root mean square displacement in Cartesian coordinates}: The path $\mathbf{S}(l)$
may be defined as a series of configuration in Cartesian space. Each configuration is made of a subset of atoms of the system and the distance is calculated as root mean square displacement (RMSD) after optimal alignment \cite{kearsley}.
\item {\bf RMSD in distances:}
the path $\mathbf{S}(l)$ must be defined as a set of pair distance among atoms. The RMSD is calculated as a difference of distances:
\begin{equation}
\left\Vert \mathbf{S}(\bm{r})-\mathbf{S}(l)) \right\Vert =
\sqrt{ \frac{\sum_{i}^{N_{dist}} ( d_{i}(\bm{r}) -d_{i}(l) )^2 }{N_{dist}}},
\end{equation}
where $d_{i}$ is the distance between the atoms of the $i$-th couple
and $N_{dist}$ is the total number of couples considered \cite{havel,torda}.
\item {\bf Contact map distance:} the path $\mathbf{S}(l)$ is defined
by a set of contact maps \cite{Bo.Bra:08}, where each contact is defined as in Eq.~(\ref{coord}).
The distance is defined as:
\begin{equation}
\Vert \mathbf{S}(\bm{r})-\mathbf{S}(l)\Vert =\sqrt{
\sum_{i}^{N_{cont}}(\mathbf{C}_{i}(\bm{r})-\mathbf{C}_{i}(l))^{2}
},
\end{equation}%
where $\mathbf{C}_{i}(\bm{r})$ is the $i$-th contact for the configuration $\bm{r}$ and $N_{cont}$ is the total number of contacts considered.
\end{itemize}
The CV $z\left( \bm{r}\right)$, when used with a single reference
frame or contact map, is equivalent to the distance between a configuration
and the reference structure, measured in the chosen metric (squared). Therefore,
this CV should be used to reproduce the standard RMSD, distance RMSD or CMAP distance.
\end{itemize}
\section{Usage examples}\label{usage}
When a given program is instructed to use PLUMED (see the manual for specific implementations),
a supplementary input file for the free-energy calculation must be provided. \\
During the calculation the main output is a file containing a record of the
values of CVs. This file is generally called {\verb COLVAR }. It may contain also additional informations depending on the chosen free-energy method.
\\
In the following we illustrate the basic use of the different methods, available for all the MD codes specified in
section \ref{instruction},
and of the algorithms currently implemented only in the GROMACS version.
Many additional examples of CVs and sampling techniques, such as multiple walkers metadynamics \cite{multiplewalkers},
are contained in the test directory distributed with the code.
\subsection{Metadynamics}
A generic input for metadynamics appears as follows:
\begin{verbatim}
# switching on metadynamics and Gaussian parameters
HILLS HEIGHT 0.1 W_STRIDE 100
# well-tempered metadynamics
WELLTEMPERED SIMTEMP 300 BIASFACTOR 10
# instruction for CVs printout
PRINT W_STRIDE 50
# a simple CV: the distance between atoms
# or group of atoms (in this case between atom 13 and atoms group <g1>)
DISTANCE LIST 13 <g1> SIGMA 0.35
g1->
17 20 22 30
g1<-
# wall potential
UWALL CV 1 LIMIT 15.0 KAPPA 100.0 EXP 4.0 EPS 1.0 OFF 0.0
# end of the input
ENDMETA
\end{verbatim}
Three kinds of keyword may exist: the \emph{directive} (needed keyword to be placed in the first position
along a line that specifies the intent of the following keywords), the \emph{parameter keyword}
(which specifies the attribute in the following field/s ) and \emph{flags} which simply turn on or off a given option.
The {\verb HILLS } is a directive and switches on metadynamics. \\
The line containing this keyword also sets up the parameters for Gaussians deposition:
{\verb HEIGHT } (parameter keyword) followed by the Gaussian height in the energy unit of the program chosen and
{\verb W_STRIDE } (parameter keyword), which specifies the time between the deposition of two consecutive Gaussians (in number of timesteps).
This input has the effect of producing an additional file, called {\verb HILLS }, which has the following layout:
\begin{verbatim}
20.000 2.78975 0.35000 0.11111 10.000
40.000 2.94914 0.35000 0.10926 10.000
60.000 2.75472 0.35000 0.10737 10.000
80.000 2.76470 0.35000 0.10542 10.000
\end{verbatim}
This is a record of the Gaussians put during the run: it displays the time step, the center of the Gaussian (one for each CV), the width (one for each CV) and the height.
In case of well-tempered metadynamics the Gaussian height is rescaled using the bias factor printed in the last column in order to
directly obtain the free energy (and not the bias), when summing all the Gaussians deposited during the run.
The {\verb WELLTEMPERED } directive switches on well-tempered metadynamics.
As explained in section \ref{methods}, CVs are sampled at a
fictitious higher temperature $T+\Delta T$ determined by the bias factor $(T+\Delta T)/T$. The user must
specify this bias factor using the keyword {\verb BIASFACTOR } and the system temperature using {\verb SIMTEMP }.
The {\verb PRINT } directive allows one to monitor, during the simulation, the evolution of the CVs between the deposition of two Gaussians.
The CVs values are printed on the {\verb COLVAR } file with a frequency, expressed in timestep units, controlled by the parameter keyword {\verb W_STRIDE }.
The file produced looks as follows:
\begin{verbatim}
0.000 2.26464 0.000 0.000
10.000 2.40452 0.000 0.000
20.000 2.78975 0.100 0.000
30.000 3.06159 0.074 0.000
40.000 2.94914 0.188 0.000
50.000 2.76442 0.185 0.000
\end{verbatim}
where the first column is the time step, the next contains the CV value (one for each CV),
the third is the bias potential and the last the potential due to a wall or a restraint.
The {\verb DISTANCE } directive selects the CV (in this case the distance between the center of mass of two groups of atoms).
{\verb LIST } (parameter keywords) specifies the two atoms or group of atoms whose distance is calculated.
The atom indices range from 1 to $N_{at}$ in the order they appear in a reference structure produced by the program.
In case of a group of atoms, the name of the group must be specified between brackets {\verb <> }.
The list of atoms belonging to the group can be placed anywhere in the input file. \\
The parameter keyword {\verb SIGMA } specifies the Gaussian width in CV units.
The {\verb UWALL } directive switches on a wall potential on the collective variable {\verb CV }. This potential
starts acting on the system when the value of the CV is greater (or lower in the case of {\verb LWALL }) then a certain
limit ({\verb LIMIT }) minus an offset ({\verb OFF }).
The functional form of this potential is the following:
\begin{equation}
V_{wall}(s)=\mathtt{KAPPA} \left (\frac{s- \mathtt{LIMIT}+ \mathtt{OFF}}{\mathtt{EPS}} \right)^{\mathtt{EXP}},
\end{equation}
where {\verb KAPPA } is an energy constant in internal unit of the code,
{\verb EPS } a rescaling factor and {\verb EXP } the exponent determining the power law.
The multiple definition of CVs is allowed.
The directive {\verb ENDMETA } specifies the end of the input. All the following text will be discarded.
The symbol {\verb # } is a comment line which is ignored.
\subsection{Umbrella Sampling}
A general input for umbrella sampling calculation is the following:
\begin{verbatim}
# a simple CV: a dihedral angle
TORSION LIST 13 15 17 1
# switching on umbrella sampling and parameters
UMBRELLA CV 1 KAPPA 200 AT -1.0
# instruction for CVs printout
PRINT W_STRIDE 100
# end of the input
ENDMETA
\end{verbatim}
The directive {\verb UMBRELLA } switches on umbrella sampling on
the collective variable specified by the parameter keyword {\verb CV } (in this case the first CV that appears in the input).
The position $s_0$ of the umbrella restraint is determined by
the keyword {\verb AT }, and the spring constant - whose energy units depend on the MD code used -
by the keyword {\verb KAPPA }.
The functional form of the potential is the following:
\begin{equation}
V_{umb}(s)=\frac{1}{2} \mathtt{KAPPA}(s- s_0)^2 \label{umbrella_potential}.
\end{equation}
\\
The directive {\verb TORSION } selects the type of CV, in
this case a dihedral angle defined by four atoms or group of atoms.
\\
The CVs value is printed on the {\verb COLVAR } file with a stride
fixed by the keyword {\verb W_STRIDE }. In case of umbrella
sampling, this file looks as follows:
\begin{verbatim}
0.000 -1.04742 0.000 0.225 RESTRAINT 1 -1.00000
20.000 -1.09302 0.000 0.865 RESTRAINT 1 -1.00000
40.000 -0.84990 0.000 2.253 RESTRAINT 1 -1.00000
60.000 -1.11383 0.000 1.296 RESTRAINT 1 -1.00000
80.000 -1.32902 0.000 10.825 RESTRAINT 1 -1.00000
\end{verbatim}
This file contains, from left to right: the time step, the CV value (one for each CV), the
potential coming from the Gaussians, the harmonic potential of umbrella sampling,
the CV on which the restraint acts and the position of the restraint.
The final calculation of the free energy as a function of this CV
can be done using the weighted histogram analysis method,
choosing one of the many possible implementations (see section \ref{umbrella_ex}).
\subsection{Thermodynamic integration and methods based on Jarzynski or Crooks relations}
PLUMED can be used to drag a system to a target value in CV space
using an harmonic potential moving at constant speed.
If the process is reversible, \emph{i.e.} for velocities tending to zero,
the work done in the dragging corresponds to the free-energy difference
between the initial and the final states.
In case of finite velocity, it is still possible to obtain an estimate of the free energy
from the work distribution using Jarzynski~\cite{jarzynski} or Crooks~\cite{Crooks98} relations.
A general input for a steered MD calculation is the following:
\begin{verbatim}
# a simple CV: a dihedral angle
ANGLE LIST 13 15 17
# switching on steered MD
STEER CV 1 TO 3.0 VEL 0.5 KAPPA 500.0
# instruction for CVs printout
PRINT W_STRIDE 100
# end of the input
ENDMETA
\end{verbatim}
The keyword {\verb STEER } activates the steering on the collective
variable specified by {\verb CV }. The target value is determined by
the parameter keyword {\verb TO }, the velocity, in unit of CV/kilostep,
by {\verb VEL } and the spring constant by {\verb KAPPA }.
The functional form of the dragging potential is the same as the one of
formula \ref{umbrella_potential}.
\\
The directive {\verb ANGLE } selects the type of CV, in
this case an angle defined by three atoms or group of atoms.
\\
The printout on {\verb COLVAR } file is analogous to umbrella
sampling with the difference that in this method the restraint position
changes during the run.
\subsection{Replica--exchange metadynamics}
When combined with GROMACS (both version 3.3 and 4), PLUMED can perform replica--exchange
simulations coupled with metadynamics in two different ways:
parallel tempering metadynamics (PTMetaD)~\cite{bussi_xc,camilloni_protG} and
bias-exchange metadynamics (BE-META)~\cite{piana}.
PTMetaD is particularly useful to increase the diffusion of the system in conformational space.
It consists in defining several replicas of the system, controlled by the same CVs but coupled with thermal baths at different temperatures $T_i$.
As in standard parallel tempering \cite{hans97cpl,sugi-okam99cpl}, pairs of replicas can exchange at a given time $t$
two conformations $\bm{r}_i$ and $\bm{r}_j$. The probability of such an exchange is given by:
\begin{align}
P_{i,j}=\min(1, \exp&[(\beta_j-\beta_i)(U(\bm{r}_j)-U(\bm{r}_i))+\beta_i(V_i(\bm{s}(\bm{r}_i),t)-V_i(\bm{s}(\bm{r}_j),t)) \nonumber \\
&+\beta_j(V_j(\bm{s}(\bm{r}_j),t)-V_j(\bm{s}(\bm{r}_i),t))]),
\end{align}
where $\beta_i=1/K_BT_i$ is the inverse temperature, $U(\bm{r})$ the internal potential, $V_i$ and $V_j$ the biasing
potentials deposited by the two replicas.
The effect of this algorithm is to sample the degrees of freedom perpendicular to the CVs more efficiently with respect
to standard metadynamics.
BE-META is designed to sample the system making use of a large number of CVs without the need of filling with Gaussians a high--dimensional space \cite{piana}.
This is done employing several replicas of the system, controlled by a few different CVs for each replica.
Usually one defines also a "neutral" replica, which evolves according to standard MD, \emph{i.e.} without metadynamics.
The temperature of the system is the same for all replicas.
The exchange probability for a pair of replicas $i$ and $j$ is:
\begin{equation}
P_{i,j}=\min(1, \exp(\beta[V_i(\bm{s}(\bm{r}_i),t)+V_j(\bm{s}(\bm{r}_j),t)-V_i(\bm{s}(\bm{r}_j),t)-V_j(\bm{s}(\bm{r}_i),t)])).
\end{equation}
To run PTMetaD simulations, one has to follow the standard
GROMACS procedure for parallel tempering (see GROMACS manual).
A binary topology file must be prepared one for each replica, while
only one PLUMED input file is required. This file looks as follows:
\begin{verbatim}
# switching on metadynamics and Gaussian parameters
HILLS HEIGHT 0.1 W_STRIDE 500
# switching on PTMetaD
PTMETAD
# instruction for CVs printout
PRINT W_STRIDE 50
# the CV: radius of gyration
RGYR LIST <CA> SIGMA 0.1
CA->
20 22 26 30 32
CA<-
# end of the input
ENDMETA
\end{verbatim}
The keyword {\verb PTMETAD } switches on parallel tempering plus metadynamics.
All replicas have the same CVs, in this case the radius of gyration defined
by the group of atoms {\verb <CA> }.
The Gaussian height set by the keyword {\verb HEIGHT } is automatically rescaled with temperature,
following $W_i=W_0\frac{T_i}{T_0}$, where $i$ is the index of a replica and $T_i$ its temperature.
The plugin will produce one {\verb COLVAR } file and one {\verb HILLS } file for each replica.
A similar procedure is used to run BE--META.
A PLUMED input and a binary topology file must be provided, one for each replica.
These files must end with the replica index (\emph{e.g.}, {\verb META_INP0 }, {\verb META_INP1 }, ...) and must contain all the CVs,
in the same order, and the keyword {\verb BIASXMD }.
The first replica ({\verb META_INP0 }) must have the {\verb NOHILLS } {\verb CV } keyword for all the CVs;
the other replicas must switch off the variables not used with a list of keywords {\verb NOHILLS } {\verb CV }.
Also in this case, PLUMED will produce one {\verb COLVAR } file and one {\verb HILLS } file for each replica.
\section{Overview of the software structure}
PLUMED performs these basic functions:
\begin{itemize}
\item Initialization and parsing of the input file;
\item Evaluation of the CVs value for a given microscopic configuration;
\item Calculation of the forces coming from the Gaussians deposited along the CVs
trajectory - in the case of metadynamics - or from a fixed/moving restraint acting on the CVs - in
case of umbrella sampling/steered MD;
\item Printout of CVs value on {\verb COLVAR } file and, in the case of metadynamics, of the Gaussians deposited
on {\verb HILLS } file.
\end{itemize}
The initialization of the plugin is done by the routine
{\verb init_metadyn } contained in {\verb metadyn.c }.
This routine is called by the main MD code, which communicates
to the plugin some critical information such as the number of atoms, masses
and charges, length of the simulation or timestep.
The parsing of the PLUMED input is performed by the routine {\verb read_restraint } in {\verb read_restraint.c }, which
reads the file and, according to the CV chosen (let's say {\verb CVname }), calls a specific parsing
routine contained in a file called {\verb restraint_CVname.c }.
\\
The second task is controlled mainly by the routine {\verb restraint } in {\verb restraint.c },
which receives by the MD code the atoms positions at every step of dynamics.
This routine calls a specific function, contained in {\verb restraint_CVname.c },
which calculates the CVs value and the derivatives with respect to the coordinates.
The same routine, depending on the free-energy method chosen, calls a
proper function to calculate the force acting on the atoms and
controls the printout of CVs on the {\verb COLVAR } file. \\
The forces calculated by the plugin are communicated back to
the main MD code and added to the internal forces before the following integration
step.
The interaction of PLUMED with the principal code is summarized
schematically in Fig. \ref{schema}.
\begin{figure}[!h]
\begin{center}
\includegraphics[height=8cm]{graf}
\end{center}
\caption{Schematic representation of the interaction of PLUMED with the main MD code.}
\label{schema}
\end{figure}
\section{Description of the individual software components}
The plugin package, distributed in a compressed tar archive, has the following directory structure:
\begin{itemize}
\item {\verb common_files }. The directory containing all the basic routines that compose PLUMED.
\item { \verb tests }. A variety of examples of different CVs and free-energy methods provided
with topology and input files for GROMACS, NAMD, AMBER (SANDER module) and DL\_POLY.
These examples, combined with a script adapted from CP2K \cite{VandeVondele:2005p10650},
work also as a regtest for the plugin.
\item {\verb patches }. A collection of patches to interface PLUMED with different codes (see section \ref{instruction} for more details).
\item {\verb utilities }. Two small utilities written in Fortran: {\verb sum_hills } and {\verb driver }.
The former is a post-processing program which reads the {\verb HILLS } file produced by the plugin
in a metadynamics simulation and returns the free energy by summing the Gaussians that have been
deposited. The latter is a tool that calculates the value of selected CVs along
a MD trajectory.
It requires a PDB file, a trajectory in DCD format and a file with the same
syntax of the PLUMED input file.
\item {\verb doc }. A complete manual with detailed installation instructions for each code.
\end{itemize}
\section{Installation instructions} \label{instruction}
The installation of PLUMED on every supported program is done through an automatic patch procedure
specific to each code. This is done on the clean code and requires its recompilation.
All the patching procedures are illustrated in detail in the manual.
Currently supported codes are NAMD 2.6, GROMACS 3.3.3 and 4.0.4, DLPOLY 2.16 and 2.19, SANDER (AMBER 9 version).
\section{Test runs description}
In the following we describe a few simple examples of the free-energy
methods implemented in PLUMED applied
to alanine dipeptide in vacuum at 300 K (see Fig. \ref{diala}).
The AMBER99SB force field \cite{Hornak:2006p11531} has been used throughout all the simulations.
These test runs have been conducted with either NAMD or AMBER (SANDER module) code.
\begin{figure}[!h]
\begin{center}
\includegraphics[height=4cm,clip]{./alanine.eps}
\end{center}
\caption{Ball and stick representation of alanine dipeptide (Ace-Ala-Nme) in vacuum. The dihedral angle $\Phi$ is defined by the
set of atoms C--N--C$_{\alpha}$--C while the angle $\Psi$ by N--C$_{\alpha}$--C--N.}
\label{diala}
\end{figure}
\subsection{Metadynamics}
Well-tempered metadynamics using the two dihedral angles $\Phi$ and $\Psi$ as CVs (see Fig. \ref{diala}) has been performed
with the SANDER code included in AMBER 9. The bias factor chosen is 10,
the initial Gaussian height 0.1 kcal/mol, the width 0.35 rad for both CVs and the deposition stride 1 ps.
The total simulation time is 5 ns.
The free energy (see Fig. \ref{metafes}) has been reconstructed from the Gaussians deposited during the run using {\verb sum_hills }, the
tool provided in the directory {\verb utilities }.
\begin{figure}[!h]
\begin{center}
\includegraphics[height=8cm,clip]{./meta-fes.eps}
\end{center}
\caption{Free-energy of the alanine dipeptide as a function of the two dihedral angles $\Phi$ and $\Psi$ obtained with
well-tempered metadynamics. Isoenergy lines are drawn every 1 kcal/mol.}
\label{metafes}
\end{figure}
\subsection{Umbrella sampling}\label{umbrella_ex}
Two-dimensional umbrella sampling on the dihedral angles $\Phi$ and $\Psi$ has been performed with NAMD
code using 676 umbrella simulations of 10 ps each and a spring constant of $100 \, kcal \, mol^{-1} \, rad^{-2}$.
Umbrellas have been chosen in an adaptive way.
The WHAM code by Alan Grossfield \cite{grossfield} has been used to produce the free-energy profile (see Fig. \ref{umbrella}).
\begin{figure}[!h]
\begin{center}
\includegraphics[height=8cm,clip]{./fes_umbrella}
\end{center}
\caption{Free-energy of the alanine dipeptide as a function of the two dihedral angles $\Phi$ and $\Psi$ obtained with
umbrella sampling. Isoenergy lines are drawn every 1 kcal/mol.}
\label{umbrella}
\end{figure}
\subsection{One dimensional umbrella sampling and thermodynamic integration}
One dimensional umbrella sampling on dihedral $\Psi$ has been performed with NAMD code using 26 windows and running
a simulation of 20 ps per umbrella (520 ps total). The spring constant used is $100 \, kcal \, mol^{-1} \, rad^{-2}$.
Thermodynamic integration has been completed dragging the dihedral $\Psi$ from $\pi$ to $-\pi$ in 504 ps.
The resulting free energies are shown in Fig. \ref{thermovsumbrella}.
\begin{figure}[!h]
\begin{center}
\includegraphics[height=8cm,clip]{./thermovsumbrella}
\end{center}
\caption{Free-energy of alanine dipeptide as a function of the dihedral angle $\Psi$ obtained from a one
dimensional umbrella sampling calculation (full line) and from thermodynamic integration (dashed line).}
\label{thermovsumbrella}
\end{figure}
\section{Availability}
The plugin can be downloaded from \url{http://merlino.mi.infn.it/plumed}.
Any questions regarding the installation and usage of PLUMED can be posted to
the users mailing list at {\verb plumed-users@googlegroups.com }.
\section{Conclusions and outlook}
In this paper we have presented PLUMED, a plugin aimed at performing
the calculation of free energy landscapes using a number of
state-of-the-art methods such as umbrella sampling, steered molecular
dynamics and metadynamics. The unique feature of PLUMED is that it
can be easily ported to four popular MD codes, namely
AMBER, DL\_POLY, GROMACS and NAMD. In the next future, we plan to
further expand this list. The possibility of using PLUMED with
different host codes will allow people to choose the proper code on
the basis of its capabilities (\emph{e.g.}, implicit solvent, parallelism,
particular force fields), and also taking into account its performance
relative to a specific application.
\section*{Acknowledgements}
This work would not have been possible without the joint effort of many people in the course of the last seven years.
Among these, we should like to thank (in alphabetical order):
Alessandro Barducci, Anna Berteotti, Rosa Bulo, Matteo Ceccarelli, Michele Ceriotti, Paolo Elvati, Antonio Fortea-Rodriguez,
Francesco Luigi Gervasio, Alessandro Laio, Matteo Masetti, Fawzi Mohamed, Ferenc Molnar, Gabriele Petraglio and Federica Trudu. \\
Francesco Marini is kindly acknowledged for his technical support,
Joost VandeVondele for permission to use his regtest script,
Jim Pfaendtner for giving precious suggestions in writing the manuscript.
\\
Alessandro Laio deserves a special acknowledgement for carefully reading the manuscript and giving
a number of useful suggestions.
|
0902.0290
|
\section{\@startsection{section}{1}{\z@}{3.5ex plus 1ex minus
.2ex}{2.3ex plus .2ex}{\large\bf}}
\defAppendix \Alph{section}{\arabic{section}}
\def\Alph{section}.\arabic{subsection}{\arabic{section}.\arabic{subsection}}
\def\arabic{subsubsection}{\arabic{subsubsection}}
\def
|
1301.7037
|
\section{Introduction}
Perturbative algebraic quantum field theory ({{p\textsc{aqft}}}) is a mathematical framework developed during the last two decades to study problems in perturbative renormalization. It proved to be very useful in constructing models in quantum field theory on curved spacetimes, because the operator algebraic approach allows one to separate the construction of the algebra of observables from the construction of a state. Research in {{p\textsc{aqft}}} is focused on two main problems: developing methods for renormalization on general globally hyperbolic backgrounds \cite{BF0,BFV,FR,FR3,H,HW,HW2,HW3,HW5}, and identifying algebraic structures appearing in perturbative renormalization on Minkowski spacetime \cite{BDF,DF,Duetsch:2000nh,DF02,DF04,Kai,FR,FR3,Rej}. An important step towards the consistent axiomatic framework for QFT on curved spacetimes was introducing the notion of \textit{general local covariance} \cite{BFV,HW}. In \cite{BFV} this notion is formulated in the language of category theory. The axiomatic framework proposed in \cite{BFV} is a generalization of the Haag-Kastler axioms \cite{HK} of local quantum field theory. In the {{p\textsc{aqft}}} approach, the main tool we use to investigate the algebraic structures appearing in renormalization is the Epstein-Glaser \cite{EG} method. It allows us to prove the existence of renormalized quantities (time-ordered products) without having to manipulate ill defined objects in the intermediate steps of quantization.
Initially the {{p\textsc{aqft}}} framework was developed for scalar fields, but recently there has been a lot of progress in constructing more complicated models. In particular, quantum electrodynamics (QED) was discussed in \cite{DF99} and Yang-Mills theory was discussed in \cite{Boas,H}. A general framework which deals with arbitrary theories with local symmetries was subsequently proposed in \cite{FR,FR3,Rej11b}. This setting makes use of the Batalin-Vilkovisky formalism, which relies on homological algebra methods. In
\cite{FR,FR3} these algebraic tools are refined by introducing functional-analytic aspects and generalizing the BV formalism to infinite dimensional spaces. In \cite{FR3} a general quantization scheme for gauge theories is proposed and some comparison with the approach of \cite{H} is made. In the present work we want to continue this line. We discuss various aspects of local gauge invariance in {{p\textsc{aqft}}}, pointing out differences and common features of existing approaches. The framework proposed in \cite{FR,FR3} is a very convenient tool for such analysis, since it is general and flexible enough. We focus our discussion on two problems:
general formulation of consistency conditions that have to be satisfied by the deformed $\star$-product in order to be compatible with structures appearing in the BV formalism, and the definition and intrinsic meaning of the free and the interacting BRST charge.
Our main result is the proof that the interacting BRST charge $R_V(Q)$ ($R_V$ denotes the derivative of the relative S-matrix) generates on-shell the quantum BV operator $\hat{s}=s-i\hbar\bigtriangleup_V$, defined in \cite{FR3}, i.e.
\begin{equation}\label{main:1}
\frac{i}{\hbar}[R_V(F),R_V(Q)]_\star=R_V(\hat{s} F)
\end{equation}
holds for local $F$, modulo the free equations of motion. Our proof generalizes results obtained in \cite{H}, since we do not restrict to $F$'s for which the renormalized BV Laplacian $\bigtriangleup_V(F)$ vanishes. Moreover, our result can be applied to a larger class of theories, including gravity \cite{BFR} and the bosonic string \cite{BRZ}.
Using the interacting star product one can formulate this result also as
\begin{equation}\label{main:2}
\frac{i}{\hbar}[V,Q]_{\star_V}=\hat{s}F\,,
\end{equation}
modulo interacting equations of motion.
The paper is organized as follows: in the first section we construct the classical theory in the framework of {{p\textsc{aqft}}}, in the second section we perform the quantization and in the last section we discuss the BRST charge. The content of the first section is essentially a brief summary of the formalism introduced in \cite{FR}, but we formulate it here in the language of graded differential geometry. We also prove some properties of the renormalized BV Laplacian $\bigtriangleup_V$. The second section starts with a detailed discussion of consistency conditions that the star product has to fulfill in order to be compatible with the free BRST operator. We show how this relates to the free quantum master equation {{\textsc{qme}}}, and we identify intrinsic reasons for the conditions to arise. Such consistency conditions are necessary, if one works with the linearized BV operator. This is an argument in favor of the approach proposed by K.~Fredenhagen and myself in \cite{FR3}, where we work with the full BV operator, instead. In \cite{FR3}, we construct interacting fields from free ones by means of the intertwining map $R_V$; we include $\theta_0$ (the part of the Lagrangian which generates the free BRST operator) in the interaction term $V$, and the free action $S_0$ doesn't contain antifields. In subsection \ref{changing} we show how the theory whose starting point action is $S_0$ relates to the theory whose starting point action is $S_0+\theta_0$. This concludes the second section of the paper. In the third section we discuss the BRST charge. We prove relations \eqref{main:1} and \eqref{main:2}, and discuss in detail differences between approaches to gauge theory quantization taken in \cite{DF99} and \cite{H}.
\section{Classical field theory}
\subsection{Kinematical structure}
We start with the kinematical structure. Let $M$ be an oriented, time-oriented globally hyperbolic spacetime. We associate to $M$ the space $\mathfrak{E}(M)$, of field configurations of the theory. $\mathfrak{E}(M)$ describes the physical content of the theory, i.e. specifies what kind of objects the theory is describing. The results of the present work can be applied to a very general class of theories, including Yang-Mills theory and gravity. We only assume that the configuration space is a space of smooth sections of some natural vector bundle $E\xrightarrow{\pi} M$ with fiber $V$ over $M$, i.e. $\mathfrak{E}(M)=\Gamma(E)\equiv\Gamma(M,V)$. Let $\mathfrak{E}_c(M)$ denote the space of compactly supported configurations and $\mathfrak{D}(M)\doteq\Ci_0(M,\RR)$. A classical measurement associates, to a configuration in $\mathfrak{E}(M)$, a real number. Therefore it is natural to identify classical observables with functionals $F:\mathfrak{E}(M)\to\RR$.
We require these functionals to be smooth in the sense of calculus on locally convex topological vector spaces. Let us briefly recall the relevant definitions. The derivative of $F$ at $\varphi\in\mathfrak{E}(M)$ in the direction of $\psi\in\mathfrak{E}(M)$ is defined by
\begin{equation}\label{de}
\left<F^{(1)}(\varphi),\psi\right> \doteq \lim_{t\rightarrow 0}\frac{1}{t}\left(F(\varphi + t\psi) - F(\varphi)\right)\,,
\end{equation}
whenever the limit exists. The function $F$ is called differentiable at $\varphi\in\mathfrak{E}(M)$ if $\left<F^{(1)}(\varphi),\psi\right>$ exists for all $\psi \in \mathfrak{E}(M)$. It is called continuously differentiable if it is differentiable at all points of $\mathfrak{E}(M)$ and
$F^{(1)}(.):\mathfrak{E}(M)\times \mathfrak{E}(M)\rightarrow \RR, (\varphi,\psi)\mapsto \left<F^{(1)}(\varphi),\psi\right>$
is a continuous map. It is called a $\mathcal{C}^1$-map if it is continuous and continuously differentiable. Higher derivatives are defined in a similar way. The continuity condition for derivatives implies that $F^{(n)}(\varphi)\in \Gamma'(M^n,V^{\otimes n})$ holds for all $\varphi\in\mathfrak{E}(M)$, $n\in\NN$, so $F^{(n)}(\varphi)$ is a distributional section with compact support.
An important property of a functional $F$ is its spacetime support. It is defined by
\begin{align}\label{support}
\supp\, F=\{ & x\in M|\forall \text{ neighbourhoods }U\text{ of }x\ \exists \varphi_1,\varphi_2\in\mathfrak{E}(M), \supp\, \varphi_2\subset U
\\ & \text{ such that }F(\varphi_1+\varphi_2)\not= F(\varphi_1)\}\ .\nonumber
\end{align}
Another crucial property of a functional is \textit{the locality}. According to the standard definition it means that the functional $F$ is of the form:
\[
F(\varphi)=\int\limits_M f(j_x(\varphi))\,d\mu(x)\,,
\]
where $f$ is a function on the jet space over $M$, $j_x(h)=(x,\varphi(x),\pa \varphi(x),\dots)$ is the jet of $\varphi$ at the point $x$ and $d\mu(x)$ denotes the invariant measure on $M$ induced by the metric. The space of compactly supported smooth local functions $F:\mathfrak{E}(M)\to\RR$ is denoted by $\mathfrak{F}_\mathrm{loc}(M)$. The algebraic completion of $\mathfrak{F}_\mathrm{loc}(M)$ with respect to the pointwise product
\begin{equation}\label{prod}
F\cdot G(h)=F(h)G(h)
\end{equation}
is a commutative algebra $\mathfrak{F}(M)$, consisting of finite sums of finite products of local functionals. We call this space \textit{the algebra of multilocal functionals}. Both $\mathfrak{F}_\mathrm{loc}$ and $\mathfrak{F}$ are covariant functors from $\Loc$ (the category of globally hyperbolic oriented and time-oriented spacetimes with causal isometric, (time)-orientation preserving embeddings as morphisms) to the category $\Vect$ of locally convex vector spaces.
\subsection{Dynamics and symmetries}
Following \cite{BDF} we introduce the dynamical principle by means of a generalized Lagrangian. Let $L$ be a natural transformation between the functor of test function spaces $\mathfrak{D}$, and the functor $\mathfrak{F}_\mathrm{loc}$. For each $M\in\mathrm{Obj}(\Loc)$ we have a morphism $L_M:\mathfrak{D}(M)\rightarrow \mathfrak{F}_\mathrm{loc}(M)$ in $\Vect$. $L$ is a generalized Lagrangian if all these morphisms, numbered by objects of $\Loc$, satisfy
\begin{equation}\label{L:supp}
\supp(L_M(f))\subseteq \supp(f)
\end{equation}
and the additivity rule
\begin{equation}\label{L:add}
L_M(f+g+h)=L_M(f+g)-L_M(g)+L_M(g+h)\,,
\end{equation}
where $f,g,h\in\mathfrak{D}(M)$ and $\supp\,f\cap\supp\,h=\emptyset$.
The action $S(L)$ is defined as an equivalence class of Lagrangians \cite{BDF}, where two Lagrangians $L_1,L_2$ are called equivalent $L_1\sim L_2$ if
\begin{equation}\label{equ}
\supp ({L_{1}}_M-{L_{2}}_M)(f)\subset\supp\, df\,,
\end{equation}
for all $f\in\mathfrak{D}(M)$. Let us fix a Lagrangian $L_{\textrm{ph}}$, defining our physical theory.
To derive the equations of motion, we follow the approach of \cite{BDF} and define the Euler-Lagrange derivative of $S_{\textrm{ph}}$ as a natural transformation ${S'_{\textrm{ph}}}:\mathfrak{E}\to\mathfrak{D}'$ given by
\begin{equation}\label{ELd}
\left<({S_{\textrm{ph}}}')_M(\varphi),\psi\right>=\left<(L_{\textrm{ph}})_M(f)^{(1)}(\varphi),\psi\right>\,,
\end{equation}
where $f\equiv 1$ on $\supp \psi$. The equation
\begin{equation}
S_{\textrm{ph}}'(\varphi)\equiv0\,.\label{eom}
\end{equation}
is called \textit{the equation of motion} ({\textsc{eom}}). The space of solutions of \eqref{eom} is a subspace of $\mathfrak{E}(M)$ denoted by $\mathfrak{E}_S(M)$. In the physics literature one calls functionals on $\mathfrak{E}_S(M)$ \textit{on-shell functionals}. Analogously, equalities that hold for functions restricted to $\mathfrak{E}_S(M)$ are called \textit{on-shell} equalities.
Local symmetries of the action $S_{\textrm{ph}}$ are described by certain vector fields on $\mathfrak{E}(M)$. We want to consider only variations in the directions of compactly supported configurations, so the corresponding space of vector fields can be identified with
\[
\mathfrak{V}(M)=\{X:\mathfrak{E}(M)\to\mathfrak{E}_c(M)| X\text{ smooth with compact support} \}\,.
\]
$X\in\mathfrak{V}(M)$ is a symmetry if
\[
(\partial_X S_{\textrm{ph}})(\varphi)\equiv 0,\ \forall\varphi\in\mathfrak{E}(M)\,,
\]
where
\[
(\partial_X S_{\textrm{ph}})(\varphi)\doteq \left<(L_{\textrm{ph}}(f))^{(1)}(\varphi),X(\varphi)\right>\,,\quad f\equiv 1\ \textrm{on}\ \supp X\,.
\]
The space of all symmetries of the given action has a structure of an infinite dimensional Lie algebra. In case of gauge theories and gravity this space can be characterized in a simple way. There exists a space of smooth sections of some vector bundle $\mathfrak{g}_c(M)=\Gamma(M,g)$ which carries a structure of a Lie algebra and there is a Lie algebra morphism $\rho:\mathfrak{g}_c(M)\rightarrow \Gamma_c (T\mathfrak{E}(M))$ such that
every symmetry $X$ can be expressed as $X(\varphi)=\rho(\xi_{\varphi})(\varphi)+I$, where $\xi_{\varphi}\in \mathfrak{g}_c(M)$ and $I$ is a trivial symmetry (a vector field that vanishes identically on $\mathfrak{E}_S(M)$). In cases which we consider, $\mathfrak{g}_c(M)$ arises as a Lie algebra of an infinite dimensional Lie group $\Gcal(M)$, called the gauge group. Since we work on a fixed background, from now on we will keep the argument ``$(M)$'' implicit, whenever this doesn't create confusion.
In order to quantize the theory we need a way to characterize the space of functionals invariant under the symmetries of $S_{\mathrm{ph}}$. In \cite{FR} it was shown how to achieve this using an appropriate extension of the Batalin-Vilkovisky (BV) formalism. In the first step one constructs the space of alternating multilinear forms (the so-called ghosts) on $\mathfrak{g}_c$ with values in $\mathfrak{F}$. In addition, we require multilocality and compact support, so we consider the space $\CE\doteq\Ci_\mathrm{ml}(\mathfrak{E},\Lambda\mathfrak{g}')$. It is a graded algebra and the corresponding grading is called the pure ghost number $\#\mathrm{pg}$. In the topology described in \cite{FR}, $\CE$ is the completion of $\mathfrak{F}\otimes\Lambda\mathfrak{g}'$ and therefore we interpret it as the space of functions on the infinite dimensional graded manifold $\overline{\mathfrak{E}}\doteq\mathfrak{E}\oplus\mathfrak{g}[1]$ (the number in square bracket denotes the shift in degree).
The Chevalley-Eilenberg differential $\gamma_{\mathrm{ce}}$ is defined in the standard way \cite{ChE} and can be identified with the exterior derivative on the space of gauge equivariant forms on the gauge group $\Gcal$. The 0-th cohomology of $\gamma_{\mathrm{ce}}$ is the space of gauge invariant functionals\footnote{In the case of gravity one needs to define this cohomology not on the level of functionals, but fields, i.e. natural transformations from $\mathfrak{D}$ to $\mathfrak{F}$. This is discussed in details in \cite{FR}.}. The Batalin-Vilkovisky algebra $\mathfrak{BV}$ is the graded symmetric tensor algebra of graded derivations of $\CE$. Again we require that elements of $\mathfrak{BV}$ considered as smooth maps on $\mathfrak{E}$ with values in a certain graded algebra are multilocal and compactly spacetime supported. The resulting space is
\[
\mathfrak{BV}\doteq\Ci_\mathrm{ml}\big(\mathfrak{E},\Lambda\mathfrak{E}_c\widehat{\otimes}\Lambda{\mathfrak{g}}'\widehat{\otimes}S^\bullet \mathfrak{g}_c\big)\,.\label{BVfix}
\]
This is again a completion of $\mathfrak{F}\otimes\Lambda\mathfrak{E}_c\otimes\Lambda{\mathfrak{g}}'\otimes S^\bullet \mathfrak{g}_c$, so
we can interpret the elements of the above space as functionals on
\[
\mathfrak{E}[0]\oplus\mathfrak{g}[1]\oplus \mathfrak{E}_c'[-1]\oplus\mathfrak{g}'_c[-2]\,,
\]
which is the odd cotangent bundle $\Pi T^*(\overline{\mathfrak{E}})$ of the extended configuration space $\overline{\mathfrak{E}}=\mathfrak{E}[0]\oplus\mathfrak{g}[1]$,
where the manifold structure on $\mathfrak{E}\oplus\mathfrak{g}[1]$ is defined by the basis of neighborhoods with the topology of $\mathfrak{E}_c\oplus\mathfrak{g}_c$. For simplicity we denote by $\varphi^\alpha$ an element of $\overline{\mathfrak{E}}$ and the index $\alpha$ runs through all the physical and ghost indices. The full field multiplet will be denoted by $\varphi$ and an evaluation functional on $\overline{\mathfrak{E}}$ will be written as $\Phi_x^\alpha$. Functions on the graded vector space $\Pi T^*(\overline{\mathfrak{E}})$ are the graded multivector fields on $\overline{\mathfrak{E}}$ and we can write example elements of $\mathfrak{BV}$ in the form
\begin{equation}
\label{Polynom}
F=\int d\mu(x_1,\dots,x_{m}) f_F(x_1, \dots ,x_{m})\Phi_{x_1}\!\dots\Phi_{x_k} \tfrac{\delta}{\delta \varphi(x_{k+1})} \dots \tfrac{\delta}{\delta \varphi(x_{m})}\,,
\end{equation}
where $d\mu(x_1,\dots,x_n)$ denotes the measure $d\mu(x_1)\dots d\mu(x_n)$, we keep the summation over the indices $\alpha$ implicit and the product denoted by juxtaposition is the graded associative product of $\mathfrak{BV}$. We can treat the functional derivatives $\tfrac{\delta}{\delta \varphi^\alpha(x)}$ as ``basis'' on the fiber $\mathfrak{E}_c'[-1]\oplus\mathfrak{g}'_c[-2]$ and we denote them by $\varphi_\alpha^{\sst\ddagger}$.
In the physics literature they are called \textit{antifields}.
In the above formula $f_F$ is a distribution with the wavefront set orthogonal to the total diagonal, and with the support that is compact and is contained in the product of partial diagonals. Later we will extend our discussion to more singular objects. In the appropriate topology (more details in \cite{FR}) elements of the form \eqref{Polynom} are dense in $\mathfrak{BV}$, so we can often restrict our discussion to such elements, without the loss of generality. Functional derivative with respect to an odd variable or an antifield can be defined on elements \eqref{Polynom} as the left derivative and extended to $\mathfrak{BV}$ by continuity. We will always assume that $\tfrac{\delta}{\delta \varphi^\alpha(x)}$, $\tfrac{\delta}{\delta \varphi_\alpha^{\sst\ddagger}(x)}$ are the left derivatives, unless stated otherwise.
The algebra $\mathfrak{BV}$ has two gradings: ghost number $\#\mathrm{gh}$ and antifield number $\#\mathrm{af}$. Functionals of physical fields have both numbers equal to 0. Functionals of ghosts have a $\#\mathrm{gh}=\#\mathrm{pg}$ and $\#\mathrm{af}=0$. All the vector fields have a non-zero antifield number and $\#\mathrm{gh}=-\#\mathrm{af}$. The space $\mathfrak{BV}$ seen as the space of graded multivector fields is equipped with a graded generalization of the Schouten bracket $\{.,.\}$, called in this context \textit{the antibracket}. The space of on-shell functionals is characterized by means of the Koszul operator. It can be written as the antibracket with the physical action $S_{\mathrm{ph}}$,
\begin{equation}\label{Koszul}
\delta_{\mathrm{ph}} F=\{F,L_{\mathrm{ph}}(f)\},\ F\in \mathfrak{BV},\,f\equiv 1\ \textrm{on }\supp\, F\,.
\end{equation}
To simplify the notation we often write $\delta_{\mathrm{ph}} F=\{F,S_{\mathrm{ph}}\}$, instead of \eqref{Koszul}. In analogy to \eqref{Koszul} one finds a generalized Lagrangian $\theta_{\mathrm{ce}}$, which implements the Chevalley-Eilenberg differential, $\gamma_{\mathrm{ce}} F=\{.,\theta_{\mathrm{ce}}\}$. The total BV differential is the sum of the Koszul-Tate differential and the Chevalley-Eilenberg differential:
\[
s_{BV}F\doteq\{F,S+\theta_{\mathrm{ce}}\}\,,
\]
which satisfies $s_{BV}^2=0$ and the 0-th cohomology of $(\mathfrak{BV},s_{BV})$ is the space of gauge invariant on-shell multilocal functionals: $\mathfrak{BV}^{\,ph}=H^0(\mathfrak{BV}, s_{BV})$.
In the next step we introduce the gauge fixing. Often one needs to extend the BV complex by adding auxiliary fields, for example antighosts $\bar{C}$ and Nakanishi-Lautrup fields $B$. One obtains a new extended configuration space and the corresponding extended space of multilocal functionals, denoted by $\mathfrak{BV}$. The full multiplet is denoted by $\varphi\in\overline{\mathfrak{E}}$ and it is a section of some graded vector bundle $\overline{E}$ over $M$. We assume that on $\overline{\mathfrak{E}}$ there exists a duality (in the sense of vector spaces, not graded vector spaces) $\left<.,.\right>_{\overline{\mathfrak{E}}}$, which allows to embed $\overline{\mathfrak{E}}$ in $\overline{\mathfrak{E}}_c'$ (in the example of the electromagnetic field this is just the Hodge duality).
To fix the gauge, we perform first an automorphism $\alpha_\Psi$ of $\mathfrak{BV}$ which leaves the antibracket invariant (see \cite{FR} for details). Performing this automorphism formally means replacing antifields $\varphi_\alpha^\ddagger(x)$ by $\varphi_\alpha^\ddagger(x)+\tfrac{\delta \Psi}{\delta\varphi^\alpha(x)}$, where $\Psi\in\mathfrak{BV}(M)$ is a functional with $\#\mathrm{gh}=-1$, called the gauge fixing fermion. The action $S+\theta_{\mathrm{ce}}$ is transformed into a new action $S_\mathrm{ext}=\alpha_\Psi(S+\theta_{\mathrm{ce}})$. One introduces also a new grading, which is sometimes called the \textit{total antifield number} $\#\mathrm{ta}$. It is equal to 1 for vector fields on $\overline{\mathfrak{E}}$, irrespective of their antifield number and is equal to 0 for functions on $\overline{\mathfrak{E}}$. Note that a functional can have a non-zero $\#\mathrm{ta}$, but have $\#\mathrm{af}=0$. This is the case in the non-minimal sector in QED, Yang-Mills or general relativity; the antifield $\bar{C}^\ddagger$ of the antighost $\bar{C}$ has $\#\mathrm{af}=0$, but, under the identification $\bar{C}^\ddagger\equiv\frac{\delta}{\delta \bar{C}(x)}$, $\bar{C}^\ddagger$ is a derivation, so $\#\mathrm{ta}(\bar{C}^\ddagger)=1$. This subtlety plays a role in the discussion of the BRST charge presented in section \ref{free:charge}. The gauge-fixing fermion has to be chosen in such a way that \textit{the gauge fixed action} $S$ (the $\#\mathrm{ta}=0$ part of $S_\mathrm{ext}$) has a well posed Cauchy problem (see \cite{FR} for details).
The transformed BV differential is given by
\[
s=\alpha_\Psi\circ s_{BV}\circ\alpha_\Psi^{-1}=\{.,S_\mathrm{ext}\}
\]
and we can expand it with respect to the total antifield number $\#\mathrm{ta}$,
\begin{equation}\label{ta:expansion}
s=\gamma+\delta\,,
\end{equation}
where the differential $\delta$ is the Koszul operator for the field equations derived from $S$ and
$\gamma$ is generated by $\theta=S_\mathrm{ext}-S$. In this context $\gamma$ is usually called \textit{the gauge-fixed BRST operator}. The uniqueness of the Cauchy problem solution for the {\textsc{eom}}'s derived from $S$ implies that $(\mathfrak{BV},\delta)$ is a resolution.
\section{Quantization}
\subsection{Free theory}\label{free:theory}
From the point of view of quantization it is convenient to split the gauge fixed action $S$ into a quadratic part and the remainder, called \textit{the interaction term}. We perform the Taylor expansion
\begin{equation}\label{Taylor1}
L(f)(\varphi_0+\varphi)=L(f)(\varphi_0)+\left<L(f)^{(1)}(\varphi_0),\varphi\right>+\frac{1}{2}
\left<L(f)^{(2)}(\varphi_0);\varphi,\varphi\right>+\dots\,,
\end{equation}
where $\left<L(f)^{(1)}(\varphi_0),\varphi\right>\doteq\sum_\alpha\left<\frac{\delta}{\delta \varphi^\alpha}(L(f))(\varphi_0),\varphi^\alpha\right>$ and
$\alpha$ runs through all the indices of the field configuration multiplet $\varphi$. The first term of the above expansion is just a constant, the second one vanishes if we choose the background configuration $\varphi_0$ to be a solution of {\textsc{eom}}'s. We denote this term by $L_{\textrm{lin}}$. The third term, denoted by $L_0$, is the quadratic part of the gauge fixed Lagrangian. Let $S_0$ denote the action corresponding to $L_0$.
We obtain an expansion:
\[
S=S(\varphi_0)+S_{\textrm{lin}}+S_0+\dots\,.
\]
The Euler-Lagrange derivative of $S_0$ induces the Euler Lagrange operator operator $P:\overline{\mathfrak{E}}\rightarrow \overline{\mathfrak{E}}_c'$. Moreover, we assume that the image of $P$ is contained in $\overline{\mathfrak{E}}$ (elements of $\overline{\mathfrak{E}}$ are identified with distributions by means of $\left<.,.\right>_{\overline{\mathfrak{E}}}$) and that
the gauge fixing is done in such a way that $P$ is normally hyperbolic (for gauge theories and gravity this was shown in \cite{FR}, the bosonic string was studied in \cite{BRZ}). This implies that $P$ has unique retarded and advanced propagators $\Delta^{A/R}$, i.e. the relations
\begin{align*}
P\circ\Delta^{A/R}&=\id_{\overline{\mathfrak{E}}_c}\,,\\
\Delta^{A/R}\circ P\big|_{\overline{\mathfrak{E}}_c}&=\id_{\overline{\mathfrak{E}}_c}
\end{align*}
hold and $\Delta^{A/R}$ fulfill the support properties
\begin{align*}
\supp(\Delta^R)&\subset\{(x,y)\in M^2| y\in J^-(x)\}\,,\\
\supp(\Delta^A)&\subset\{(x,y)\in M^2| y\in J^+(x)\}\,.
\end{align*}
The causal propagator is defined as $\Delta=\Delta^R-\Delta^A$. Let us denote $S_1=S^\mathrm{ext}-S_0$. In the first step we will quantize the free theory, i.e. the one defined by the free action $S_0$. $S_1$ is the full interaction term, with antifields included. We expand now the gauge-fixed BRST differential $\gamma$. Since $\theta(f)$ depends also on antifields, we have to take them into account as well. Actions we consider are polynomial in antifields, so the left derivative with respect to $\varphi^\ddagger$ makes sense and we can expand
\begin{multline*}
\theta(f)(\varphi_0+\varphi,\varphi^{\sst\ddagger})=\left<\frac{\delta \theta(f)}{\delta \varphi^{\sst\ddagger}}(\varphi_0,0),\varphi^{\sst\ddagger}\right>+\frac{1}{2}\left<\frac{\delta^2 \theta(f)}{\delta \varphi\delta\varphi^{\sst\ddagger}}(\varphi_0,0);\varphi,\varphi^{\sst\ddagger}\right>+\\
+\frac{1}{6}\left<\frac{\delta^3 \theta(f)}{\delta \varphi^2\delta\varphi^{\sst\ddagger}}(\varphi_0,0);\varphi,\varphi,\varphi^{\sst\ddagger}\right>+...\,.
\end{multline*}
Without any restrictions on the physical component of the background configuration $\varphi_0$, we can choose $\varphi_0$ in such a way that the first term in the above expansion vanishes and we obtain
\[
\theta=\theta_0+\theta_1+\dots\,.
\]
The first nontrivial term, denoted by $\theta_0$, generates the free BRST differential.
Its derivative $\frac{{\delta^l}^2 \theta_0(f)}{\delta\varphi^\sigma\delta\varphi_\alpha^{\sst\ddagger}}$ is an element of $\overline{\mathfrak{E}}\otimes \overline{\mathfrak{E}}'$
and it induces a differential operator $K:\overline{\mathfrak{E}}\rightarrow \overline{\mathfrak{E}}$. In local coordinates we can write the second derivative of $\theta_0(f)$ as $f(y) K^{\alpha}_{\phantom{\alpha}\sigma}(x')\delta(y-x')$ and
\begin{equation}\label{gamma0}
\gamma_0=\sum_{\sigma,\alpha}{K}^{\alpha}_{\ \sigma}\Phi_x^\sigma\frac{\delta}{\delta\varphi^\alpha(x)}\,.
\end{equation}
\begin{exa}[Free electromagnetic field]\label{ex1}
{\small As an example consider a free electromagnetic field described by the Lagrangian $L_{EM}(f)(A)=-\frac{1}{2}\int_M (F\wedge *F)f$ where $F=dA$ and $A\in\Gamma(T^*M)\equiv \Omega^1(M)$. The extended configuration space is $\overline{\mathfrak{E}}(M)=\Omega^1(M)\otimes \Ci(M)\otimes \Ci(M)[1]\otimes \Ci(M)[-1]$ and an element of this space can be written as a quadruple $\varphi=(A,B,C,\bar{C})$. Let us denote by $\left<.,.\right>$ the duality between $\overline{\mathfrak{E}}(M)$ and $\overline{\mathfrak{E}}'(M)$ and by $\left<.,.\right>_g$ the duality on the space of $p$-forms induced by the metric, i.e. $\left<u,v\right>_g\doteq\int_M u\wedge *v$, $u,v\in \Omega^p(M)$.
The extended Lagrangian takes the form
\begin{multline*}
L_M(f)(\varphi)=-\tfrac{1}{2}\left<fF,F\right>_g-\left<fdC,\tfrac{\delta}{\delta A}\right>+i\left<fB,\tfrac{\delta}{\delta \bar{C}}\right>+\\-i\left<fd\bar{C},d C\right>_g-\left<fB,\hinv d*\!A-\tfrac{1}{2}B\right>_g\,.
\end{multline*}}
Operators $P$ and $K$, written in the basis $(A,B,C,\bar{C})$, take the form
\[
P=\left(\begin{array}{cccc}
\delta d&d&0&0\\
\delta&-1&0&0\\
0&0&0&i\delta d\\
0&0&-i\delta d
\end{array}\right)\,,\quad
K=\left(\begin{array}{cccc}
0&0&d&0\\
0&0&0&0\\
0&0&0&0\\
0&i&0&0
\end{array}\right)\,.
\]
\end{exa}
The classical master equation ({\textsc{cme}}) yields $2\{\theta,S\}+\{\theta,\theta\}\sim0$, where $\sim$ is defined by \eqref{equ}. Expanding this relation in powers of field configurations, we obtain in particular (in the second order in $\varphi$ and in the 0th order in $\varphi^\ddagger$)
\begin{equation}\label{fullCME}
2\{\theta_0,S_0\}+\{\theta_1,S_{\textrm{lin}}\}\sim0\,.
\end{equation}
The first two terms of this identity correspond to classical master equation for the free action $S_0+\theta_0$. Since $\theta_0$ has to be even with respect to the ghost grading and $\#\mathrm{gh}(\varphi^\ddagger_\alpha)=-\#\mathrm{gh}(\varphi^\alpha)-1$, we obtain $\tr K=0$ and $\{\theta_0,\theta_0\}=0$. Therefore, the classical master equation of the free theory can be simply expressed as:
\begin{equation}\label{freeCME}
\{\theta_0,L_0\}\sim 0\,.
\end{equation}
and it implies that the free action $S_0$ is invariant under the free BRST operator $\gamma_0$. Relation \eqref{freeCME} is compatible with \eqref{fullCME} only if $\{\theta_1,L_{\textrm{lin}}\}\sim0$, which is true if $\varphi_0$ solves the equations of motion (i.e. $\varphi_0$ is on-shell).
We will show in subsection \ref{gauge:inv} that, in order to construct the interacting theory starting from the action $S_0+\theta_0$, one has to impose the free {{\textsc{cme}}} and this implies that $\varphi_0$ has to be a solution of the equations of motion. However, if we start from the theory with the free action $S_0$, there are no \textit{a priori} reasons to choose $\varphi_0$ to be on-shell. It could, nevertheless, happen that, for construction of states, one needs to impose restrictions on $\varphi_0$.
The general construction of the interacting theory, starting from $S_0$ as a free action, was performed in \cite{FR3}. Let us recall briefly the main ideas.
The classical linearized theory is constructed by introducing the Peierls bracket given by (to simplify the sign convention we use both the right and the left derivative):
\begin{equation*}
\Pei{F}{G} = \sum_{\alpha,\beta} \skal{\frac{\delta^r F}{\delta\varphi^\alpha}}{{\Delta}^{\alpha\beta}\frac{\delta^l G}{\delta\varphi^\beta}},
\end{equation*}
where $F, G \in\overline{\mathfrak{F}}_{\mu\mathrm{c}}(M)$ are microcausal elements of $\mathfrak{BV}$, i.e. they are smooth, compactly supported and their derivatives (with respect to both $\varphi$ and $\varphi^{\sst\ddagger}$) satisfy the WF set condition:
\begin{equation}\label{mlsc}
\WF(F^{(n)}(\varphi,\varphi^{\sst\ddagger}))\subset \Xi_n,\quad\forall n\in\NN,\ \forall\varphi\in\overline{\mathfrak{E}}(M)\,,
\end{equation}
where $\Xi_n$ is an open cone defined as
\begin{equation}\label{cone}
\Xi_n\doteq T^*M^n\setminus\{(x_1,\dots,x_n;k_1,\dots,k_n)| (k_1,\dots,k_n)\in (\overline{V}_+^n \cup \overline{V}_-^n)_{(x_1,\dots,x_n)}\}\,,
\end{equation}
The space of compactly supported vector-valued distributions on $M^n$ with the WF set contained in $\Xi_n$ will be denoted by $\Ecal'_{\Xi_n}(M^n,V)$, where $V$ is some finite dimensional vector space. The space of microcausal elements of the BV complex $\mathfrak{BV}_{{\mu\mathrm{c}}}$ is equipped with a topology $\tau_\Xi$ induced by the H\"ormander topology, as defined in \cite{FR}:
The quantized algebra of the free fields is constructed by means of the deformation quantization of the classical algebra $(\mathfrak{BV}_{{\mu\mathrm{c}}},\Pei{.}{.})$. To this end, we equip the space of formal power series $\mathfrak{BV}_{{\mu\mathrm{c}}}[[\hbar]]$ with a noncommutative star product which corresponds to the operator product of quantum observables. For this construction one needs Hadamard parametrices. A Hadamard parametrix $\omega$ is here understood as a matrix with rows and columns numbered by the indices $\alpha$ of the field configuration multiplet $\varphi$ and with entries in $\mathcal{D}'(M^2)$ which fulfill
\begin{IEEEeqnarray}{rCl}\label{parametrix}
\omega^{\alpha \beta}(x,y) - (-1)^{|\varphi^\alpha| |\varphi^\beta|} \omega^{\beta \alpha}(y,x)& =& i \Pei{\varphi^\alpha(x)}{\varphi^\beta(y)},\IEEEyessubnumber\label{classical:limit}\\
\sum_\beta P_{\alpha\beta}(x) \omega^{\beta \gamma}(x,y) & =& 0\ \textrm{mod }\Ci\textrm{ function},\IEEEyessubnumber\label{field:eq}\\
\WF(\omega^{\alpha \beta}) & \subset &C_+,\IEEEyessubnumber\label{WFset}\\
\overline{\omega^{\alpha \beta}(x,y)} & =& \omega^{\beta \alpha}(y,x).\IEEEyessubnumber\label{hermitian}
\end{IEEEeqnarray}
By $ C_+$ we denoted the following subset of the cotangent bundle $ T^*M^2$:
\[
C_+ = \{ (x_1, x_2; k_1, - k_2) \in T^*M^2 \setminus \{ 0 \} | (x_1; k_1) \sim (x_2; k_2), k_1 \in \bar V^+_{x_1} \},
\]
where $(x_1; k_1) \sim (x_2; k_2)$ if there is a lightlike geodesic from $x_1$ to $x_2$ to which $k_1$ and $k_2$ are coparallel.
If we replace the condition (\ref{field:eq}) by a stronger one
\begin{equation}\label{field:eq:s}
\sum_\beta P_{\alpha\beta}(x) \omega^{\beta \gamma}(x,y) =0\,,
\end{equation}
then the Hadamard parametrix becomes a Hadamard 2-point function. Assume that on the background manifold $M$, there exists a quasifree Hadamard state and write the corresponding 2-point function in the form $\omega=\frac{i}{2}\Delta+H$. The Feynman-like propagator is defined as $H_F=i\Delta_D+H$, where $\Delta_D\doteq \tfrac{1}{2}(\Delta_R+\Delta_A)$ is the Dirac propagator. Let $\alpha_H\doteq e^{\frac{\hbar}{2}\Gamma_H}$ be a map defined on regular functionals $\mathfrak{BV}_\mathrm{reg}(M)$ (i.e. functionals satisfying the wavefront set condition $\WF(F^{(n)}(\varphi,\varphi^{\sst\ddagger}))=\varnothing$ for all $\varphi$), where
\[
\Gamma_{H} \doteq \sum_{\alpha,\beta}\left<{H}^{\alpha\beta}, \frac{\delta^l}{\delta\varphi^\alpha}\frac{\delta^r}{\varphi^\beta}\right>\,.
\]
$\mathfrak{BV}_\mathrm{reg}$ can be completed to a larger space, the space $\mathfrak{BV}_{\mu\mathrm{c}}(M)$, of functionals that satisfy the condition
\begin{equation}\label{mlsc}
\WF(F^{(n)}(\varphi,\varphi^\ddagger))\subset \Xi_n,\quad\forall n\in\NN,\ \forall\varphi\in\overline{\mathfrak{E}}(M)\,.
\end{equation}
Functionals fulfilling this criterium are called \textit{microcausal}. On $\mathfrak{BV}_{\mu\mathrm{c}}(M)$ we can define the star product:
\begin{equation*
F\star_H G\doteq m\circ \exp({i\hbar \Gamma'_\omega})(F\otimes G),
\end{equation*}
where $\Gamma'_\omega$ is the functional differential operator
\begin{equation*
\Gamma'_\omega\doteq \sum_{\alpha, \beta} \left<{\omega}^{\alpha\beta},\frac{\delta^l}{\delta\varphi^\alpha} \otimes \frac{\delta^r}{\delta\varphi^\beta}\right>\,.
\end{equation*}
The resulting algebra is denoted by $\mathfrak{A}_H(M)$.
As there is no preferred two-point function $\omega$, and hence no preferred $H$, we have to consider all of them simultaneously. The quantum algebra (which contains in particular the Wick polynomials) is an extension of the source space $\mathfrak{BV}_\mathrm{reg}(M)$ with respect to the initial topology induced by the map $\alpha_H:\mathfrak{BV}_\mathrm{reg}\rightarrow \mathfrak{BV}_{\mu\mathrm{c}}[[\hbar]]$ (see \cite{BDF} for details). It is defined by extending $\mathfrak{BV}_\mathrm{reg}(M)$ with all elements of the form $\lim_{n\rightarrow \infty}\alpha_H^{-1}(F_n)$, where $(F_n)$ is a convergent sequence in $\mathfrak{BV}_{\mu\mathrm{c}}(M)$ with respect to $\tau_\Xi$. The resulting space, denoted by $\alpha_H^{-1}(\mathfrak{BV}_{\mu\mathrm{c}})$, is equipped with a unique continuous star product $\star$ equivalent to $\star_H$:
\[
\alpha_H^{-1}F\star \alpha_H^{-1}G\doteq \alpha_H^{-1}(F\star_H G)\,.
\]
Different choices of $H$ differ only by a smooth function, hence all the algebras $(\alpha_H^{-1}(\mathfrak{BV}_{\mu\mathrm{c}}[[\hbar]]),\star)$ are isomorphic and define an abstract algebra $\mathfrak{A}$.
We can realize $\mathfrak{A}$ more concretely as the space of families $\{ \alpha_HF \}_H$, numbered by possible choices of $H$, fulfilling the relation
\[
F_{H'} = \exp(\hbar \Gamma_{H'-H}) F_H\qquad F \in \mathfrak{A}\,,
\]
equipped with the product
\[
(F \star G)_H = F_H \star_H G_H.
\]
The support of $F \in \mathfrak{A}(M)$ is defined as $\supp(F) = \supp(\alpha_HF)$. Again, this is independent of $H$. Functional derivatives are defined by
\begin{equation}\label{derivH}
\skal{\frac{\delta F}{\delta \varphi}}{\psi} = \alpha_H^{-1}\skal{\frac{\delta \alpha_HF}{\delta \varphi}}{\psi}\,,
\end{equation}
which is well defined as $\Gamma_{H'-H}$ commutes with functional derivatives. For a fixed background $M$, the free net of local algebras is defined by assigning to each relatively compact, causally convex region $\mathcal{O}\subset(M)$, a unital $*$-algebra $\mathfrak{A}(\mathcal{O})$.
\subsection{Interacting theory}\label{int:theor}
Following \cite{FR3}, we introduce the interaction by means of renormalized time-ordered products.
Let us define operators $\mathcal{T}_{n}:\mathfrak{BV}_\mathrm{loc}^{\otimes n}\rightarrow \mathfrak{A}$ by means of
\[
\mathcal{T}_{n}(F_1,\ldots,F_n)=\alpha_{H+w}^{\sst{-1}} % power ^{-1} (F_1)\cdot_{{}^\Tcal}\ldots\cdot_{{}^\Tcal} \alpha_{H+w}^{\sst{-1}} % power ^{-1} (F_n)\,,
\]
for $F_i\in \mathfrak{BV}_\mathrm{loc}$ with disjoint supports, where $F\cdot_{{}^\Tcal} G\doteq \alpha_{i\Delta_D}(\alpha_{i\Delta_D}^{\sst{-1}} % power ^{-1}F\cdot\alpha_{i\Delta_D}^{\sst{-1}} % power ^{-1}G)$, $\Tcal_{0}=0$ and $\Tcal_1=\alpha^{-1}_{H+w}$. In the last formula, $w$ is the smooth part in the Hadamard function.
Renormalization freedom related to the choice of $w$ is discussed in \cite{HW}. Maps $\mathcal{T}_{n}$ have to be extended to functionals with coinciding supports and are required to satisfy the standard conditions given in \cite{BDF,H}. In particular, we require the graded symmetry, unitarity, scaling properties, the support property $\supp\Tcal_n(F_1,\dots,F_n)\subset\bigcup\supp F_i$ and the causal factorization property, which states that
\begin{equation}\label{CausFact}
\Tcal_{n}(F_1\otimes \dots \otimes F_n)=
\Tcal_{i}(F_1\otimes \dots \otimes F_i) \star
\Tcal_{n-i}(F_{i+1} \otimes \dots \otimes F_n) \, ,
\end{equation}
if the supports of $F_1\ldots F_i$ are later than the supports of $F_{i+1},\ldots F_n$.
Maps $\Tcal_n$ are constructed inductively, and each $\Tcal_n$ is uniquely fixed by the lower order maps $\Tcal_k$, $k<n$, up to addition of an $n$-linear map
\begin{equation}
Z_n:\mathfrak{BV}_\mathrm{loc}^n\to\alpha_{H+w}^{-1}(\mathfrak{BV}_\mathrm{loc})\cong\mathfrak{A}_\mathrm{loc}\,,
\end{equation}
which describes the possible finite renormalizations.
In \cite{FR3} it was shown that renormalized time ordered product can be extended to an associative, commutative binary product defined on the domain $\mathcal{D}_{\Tcal}\doteq\Tcal(\mathfrak{BV})$, where $\Tcal\doteq\oplus_n\Tcal_n\circ m^{-1}$.
Here $m^{-1}:\mathfrak{BV}\to S^\bullet\mathfrak{BV}^{(0)}_\mathrm{loc}$ is the inverse of the multiplication, as defined in \cite{FR3,Rej11b}. $\mathcal{D}_{\Tcal}$ contains in particular $\mathfrak{A}_\mathrm{loc}$ and is invariant under the renormalization group action. Renormalized time ordered products are defined by
\begin{equation}
A\cdot_{{}^\Tcal} B\doteq\Tcal(\Tcal^{\sst{-1}} % power ^{-1}A\cdot\Tcal^{\sst{-1}} % power ^{-1}B)\,.
\end{equation}
Using time-ordered products we can introduce the interaction.
As indicated in section \ref{free:theory}, we split the Lagrangian into $L_\mathrm{ext}=L_0+L_1$. The quantum field constructed form the interaction term $L_1$ is $\Tcal L_1$; the formal S-matrix is given by
\begin{equation}\label{Smatrix}
\mathcal{S}(L_{1}(f))\doteq e_{\sst{\Tcal}}^{i\Tcal L_{1}(f)/\hbar}=\Tcal(e^{iL_{1}(f)/\hbar})\,,
\end{equation}
which is a Laurent series in $\hbar$. Now we want to construct the interacting net of local algebras. Let $\mathcal{O}\subset M$ be an open and relatively compact subset.
The local algebra of observables associated to $\mathcal{O}$ has to be independent of an interaction switched on outside
of $\mathcal{O}$. We define
\[
\mathscr{V}_{S_1}(\mathcal{O}) \doteq \{ V\in\mathfrak{A}_{\mathrm{loc}}\ |\ \supp(V-\Tcal{L_1}(f))\cap\overline{\mathcal{O}}=\varnothing,
\text{ if } [L_1] =S_1\text{ and } f\equiv 1 \text{ on }\mathcal{O} \}\,.
\]
The relative $S$-matrix in the algebraic adiabatic limit is given by
\[
\mathcal{S}^\mathcal{O}_{S_1}(F)=(\mathcal{S}_V(F))_{V\in\mathscr{V}_{S_1}(\mathcal{O})}
\]
for $F\in\mathfrak{A}$ with $\supp\, F\subset \mathcal{O}$, where
\begin{equation}\label{Bog}
\mathcal{S}_V(F)\doteq\mathcal{S}(V)^{\star-1}\star \mathcal{S}(V+F)\,.
\end{equation}
The relative $S$-matrix defined this way is a covariantly constant section in the sense that for any $V_1,V_2\in\mathscr{V}_{S_1}(\mathcal{O})$ there exists an automorphism
$\beta$ of $\mathfrak{A}$ such that
\[
\beta(\mathcal{S}_{V_1}(F))=\mathcal{S}_{V_2}(F)\quad\ \forall F\in\mathfrak{A}_{\mathrm{loc}}\ ,\ \supp\, F\subset \mathcal{O}\,,
\]
Interacting quantum fields in $\mathcal{O}$ are generated by $\mathcal{S}_{V}^\mathcal{O}(F)$ and, for given $V\in\mathscr{V}_{S_1}(\mathcal{O})$, we can write a corresponding component as a formal power series:
\begin{equation}\label{Rv}
(R^\mathcal{O}_{S_1}(F))_V\doteq R_V(F)=\frac{d}{d\lambda}\Big|_{\lambda=0}\mathcal{S}_{V}(\lambda F)\,,
\end{equation}
Differentiation of $\mathcal{S}_{V}(\lambda F)$ yields
\begin{equation}\label{RV}
R_V(F)=\left(e_{\sst{\Tcal}}^{iV/\hbar}\right)^{\star\sst{-1}} % power ^{-1}\star\left(e_{\sst{\Tcal}}^{iV/\hbar}\cdot_{{}^\Tcal} F\right)\,,
\end{equation}
which is a formal power series in $\hbar$ and $V$. An interacting net of local algebras on a fixed spacetime $M$ is obtained by assigning to each relatively compact, causally convex region $\mathcal{O}\subset M$, an algebra $(R_V(\mathfrak{A}(\mathcal{O})),\star)$, $V\in\mathscr{V}_{S_1}(\mathcal{O})$. This definition doesn't depend on the choice of $V$, since local algebras constructed with different $V$'s belonging to $\mathscr{V}_{S_1}(\mathcal{O})$ are isomorphic.
An interacting field can be written in terms of \textit{retarded products} defined as coefficients in the following expansion:
\[
R_V(F)=\sum\limits_{n=0}^\infty\frac{i^n}{\hbar^nn!}\mathcal{R}_n(V^{\otimes n};F)\,.
\]
Time-ordered products can be normalized in such a way that retarded products satisfy a useful relation, called the GLZ identity,
\begin{equation}\label{glz}
[R_V(F),R_V(G)]_\star=i\hbar\frac{d}{d\lambda}\left(R_{V+\lambda G}(F)-R_{V+\lambda F}(G)\right)\Big|_{\lambda=0}\,.
\end{equation}
On the right hand side of \eqref{glz} we have formal derivatives of retarded products with respect to $V$, so it is convenient to use the notation
\[
R_V^{(1)}[G](F)\doteq\frac{d}{d\lambda}\left(R_{V+\lambda G}(F)\right)\Big|_{\lambda=0}\,.
\]
Retarded products satisfy an important support property, which can be conveniently written as
\begin{equation}\label{supp:prop}
\supp(\mathcal{R}_n(V_1(x_1),\dots,V_1(x_n);F(x))\subset \{(x_1,\dots,x_n;x)|x_i\in x+\bar{V}_-, \forall i=1\dots,n\}\,,
\end{equation}
where $F(x)$, $V_i(x_i)$ are local forms (non-integrated, i.e. density-valued local functionals). Using retarded products we define a new non-commutative (partial) product $\star_V$. This product was first proposed by K.~Fredenhagen in \cite{F11} as an interacting star product. It is given by
\begin{equation}\label{starV}
F\star_V G\doteq R_V^{-1}(R_V(F)\star R_V(G))\,,
\end{equation}
for $F,G\in\mathfrak{A}$ such that this expression is well defined.
Classical structures appearing in the BV formalism also have to be quantized. The renormalized time-ordered antibracket is defined by
\[
\{X,Y\}_{\sst{\Tcal}}=\Tcal\{\Tcal^{-1}X,\Tcal^{-1}Y\}\ .
\]
We can also write it in the form
\begin{equation}\label{antibracketTR}
\{X,Y\}_{\Tcal}=\sum_\alpha\int\!\left(\!\frac{\delta^r X}{\delta\varphi^\alpha}\cdot_{{}^\Tcal}\frac{\delta^l Y}{\delta\varphi_\alpha^{\sst\ddagger}}-(-1)^{|\varphi_\alpha^{\sst\ddagger}|}\frac{\delta^r X}{\delta\varphi_\alpha^{\sst\ddagger}}\cdot_{{}^\Tcal}\frac{\delta^l Y}{\delta\varphi^\alpha}\!\right)d\mu\,,
\end{equation}
where we denoted
\begin{equation}\label{antibracketTR2}
\frac{\delta^r X}{\delta\varphi^\alpha}\cdot_{{}^\Tcal}\frac{\delta^l Y}{\delta\varphi_\alpha^{\sst\ddagger}}\doteq \Tcal\Big(D^*\Big(\Tcal ^{-1}\frac{\delta X}{\delta\varphi}\otimes \Tcal ^{-1}\frac{\delta Y}{\delta\varphi^\ddagger}\Big)\Big)\,,
\end{equation}
where $D^*$ is the pullback by the diagonal map and $\big(\Tcal ^{-1}\frac{\delta X}{\delta\varphi}\big)(\varphi)$ is a compactly supported distribution (i.e. an element of $\mathfrak{E}'$) defined by
\[
\left<\big(\Tcal ^{-1}\frac{\delta X}{\delta\varphi}\big)(\varphi),f\right>\doteq\Big(\Tcal ^{-1}\Big<\frac{\delta X}{\delta\varphi},f\Big>\Big)(\varphi)=\Big<\frac{\delta}{\delta\varphi}\Tcal ^{-1}X,f\Big>(\varphi)\,,\qquad f\in\mathfrak{E}\,.
\]
In the second step we used the field independence of time ordered products. Since $X\in\Tcal(\mathfrak{BV})$, the distribution $\big(\Tcal ^{-1}\frac{\delta F}{\delta\varphi}\big)(\varphi)$ defined by the above equation is actually an element of $\mathfrak{D}$ and the pullback in \eqref{antibracketTR2} is well defined.
Similarly we define the antibracket with the $\star$-product:
\begin{equation}\label{antibracketstar}
\{X,Y\}_{\star}=\sum_\alpha\int\!\left(\!\frac{\delta^r X}{\delta\varphi^\alpha}\star\frac{\delta^l Y}{\delta\varphi_\alpha^{\sst\ddagger}}-(-1)^{|\varphi_\alpha^{\sst\ddagger}|}\frac{\delta^r X}{\delta\varphi_\alpha^{\sst\ddagger}}\star\frac{\delta^l Y}{\delta\varphi^\alpha}\!\right)d\mu\,,
\end{equation}
whenever it exists. Clearly it is well defined if one of the arguments is regular or equal to $S_0$. Moreover, the antibracket $\{.,S_0\}_\star$ with the free action defines a $\star$-derivation. Similarly, $\{.,S_0\}_{\sst{\Tcal}}$ is a $\cdot_{{}^\Tcal}$-derivation. A relation between, $\{.,S_0\}_{\sst{\Tcal}}$ and $\{.,S_0\}_\star$ is provided by the Master Ward Identity \cite{BreDue,H}:
\begin{align}\label{MWI}
\{e_{\sst{\Tcal}}^{iV/\hbar},S_0\}_\star&=\{ e_{\sst{\Tcal}}^{iV/\hbar},S_0\}_{\sst{\Tcal}}+e_{\sst{\Tcal}}^{iV/\hbar}\cdot_{{}^\Tcal}(\bigtriangleup_V+\frac{i}{2\hbar}\{V,V\}_{\sst{\Tcal}})=\\
&=\frac{i}{\hbar}e_{\sst{\Tcal}}^{iV/\hbar}\cdot_{{}^\Tcal}\big(\{V,S_0\}_{\sst{\Tcal}}+\frac{1}{2}\{V,V\}_{\sst{\Tcal}}-i\hbar\bigtriangleup(V)\big)\,,\nonumber
\end{align}
where $V\in\mathfrak{A}_\mathrm{loc}$ and $\bigtriangleup(V)$ is a local functional. One can expand it in powers of $V$ to obtain
\[
\bigtriangleup(V)=\sum_{n=0}^{\infty}\bigtriangleup^n(V^{\otimes n};V)\,,
\]
where coefficients $\bigtriangleup^n$ are linear functions $\mathfrak{A}_{\mathrm{loc}}^{\otimes n}\rightarrow \mathfrak{A}_{\mathrm{loc}}$ defined recursively by
\begin{align}
\bigtriangleup^n(V_1\otimes\ldots\otimes V_n; V)&=-\left(\tfrac{i}{\hbar}\right)^{n+1}V_1\cdot_{{}^\Tcal}\ldots\cdot_{{}^\Tcal} V_n\cdot_{{}^\Tcal}\{V,S_0\}+\nonumber\\
&-\left(\tfrac{i}{\hbar}\right)^{n}\sum_{i=1}^nV_1\ldots\cdot_{{}^\Tcal} \hat{V}_i\cdot_{{}^\Tcal}\ldots V_n\cdot_{{}^\Tcal}\int\frac{\delta V}{\delta\varphi_\alpha^\ddagger(x)}\cdot_{{}^\Tcal}\frac{\delta V_i}{\delta\varphi^\alpha(x)}d\mu(x)+\nonumber\\
&-\sum_{I\subset \{1,\dots,n\}, I\neq \varnothing}\!\!\!\!\!\!\left(\tfrac{i}{\hbar}\right)^{|I|}{\bigcirc\!\!\!\!\!{\T}}_{i\in I}V_i\cdot_{{}^\Tcal}\bigtriangleup^{|I^c|}\left(V^{\otimes(n-k)};V\right)+\nonumber\\
&+\left(\tfrac{i}{\hbar}\right)^{n+1}\int\left(V_1\cdot_{{}^\Tcal}\dots\cdot_{{}^\Tcal} V_n\cdot_{{}^\Tcal}\frac{\delta V}{\delta\varphi_\alpha^\ddagger(x)}\right)\star \frac{\delta S_0}{\delta\varphi^\alpha(x)}d\mu(x)\,,\label{Lap:coeff}
\end{align}
where ${\bigcirc\!\!\!\!\!{\T}}_{i\in I}V_i$ denotes the $\cdot_{{}^\Tcal}$-product of elements $V_i$ indexed by $i\in I$, and $I^c$ is the complement of $I$ in $\{1,\dots,n\}$. Let us now fix the interaction term $V$. Using the above relation we define the renormalized BV Laplacian on $\mathfrak{A}_\mathrm{loc}$ as:
\[
\bigtriangleup_V(X)\doteq \frac{d}{d\lambda}\Big|_{\lambda=0}\bigtriangleup({V+\lambda X})\,.
\]
Using this definition we obtain a relation:
\begin{align}\label{MWI2}
\bigtriangleup_V(X)=& \frac{d}{d\lambda}\Big|_{\lambda=0}(e_{\sst{\Tcal}}^{-i(V+\lambda X)/\hbar}\cdot_{{}^\Tcal}(\{e_{\sst{\Tcal}}^{i(V+\lambda X)/\hbar},S_0\}_\star-\{ e_{\sst{\Tcal}}^{i(V+\lambda X)/\hbar},S_0\}_{\sst{\Tcal}})+\\&-\frac{i}{2\hbar}\{V+\lambda X,V+\lambda X\}_{\sst{\Tcal}})=\nonumber\\
=&\frac{i}{\hbar}(e_{\sst{\Tcal}}^{-iV/\hbar}\cdot_{{}^\Tcal}\{e_{\sst{\Tcal}}^{iV/\hbar}\cdot_{{}^\Tcal} X,S_0\}_\star- \{X,S_0+V\}+X\cdot_{{}^\Tcal} e_{\sst{\Tcal}}^{-iV/\hbar}\cdot_{{}^\Tcal}\{e_{\sst{\Tcal}}^{iV/\hbar},S_0\}_\star)\,.\nonumber
\end{align}
Expanding the renormalized Laplacian in powers of $V$ we obtain
\begin{equation}\label{LapX:coeff}
\bigtriangleup_V(X)=\sum_{n=1}^\infty n\bigtriangleup^n\left(V^{\otimes (n-1)}\otimes X;V\right)+\sum_{n=0}^\infty\bigtriangleup^n(V^{\otimes n};X)\,.
\end{equation}
If $X$ doesn't contain antifields, only the first sum is present. Compare $\bigtriangleup_V(X)$ with the nonrenormalized graded Laplacian $\bigtriangleup_{\textrm{nren}}$ of the BV formalism, which has all the $n>0$ terms vanishing and $\bigtriangleup_{\textrm{nren}}^0$ is given by the known formula:
\begin{equation}\label{Lap:regular}
\bigtriangleup_{\textrm{nren}} X=\bigtriangleup_{\textrm{nren}}^0(X)=\sum\limits_\alpha(-1)^{|\varphi_\alpha|(1+|X|)}\int dx \frac{\delta^2 X}{\delta\varphi_\alpha^\ddagger(x)\delta\varphi^\alpha(x)}\,.
\end{equation}
Unfortunately $\bigtriangleup_{\textrm{nren}}$ is not well defined on local functionals. In the renormalized theory $\bigtriangleup_{\textrm{nren}}$ is replaced by $\bigtriangleup_V$, which is well defined on $\mathfrak{A}_\mathrm{loc}$, but contains non-vanishing higher order terms $\bigtriangleup^n$, $n>0$.
The renormalized quantum master equation QME is the condition that
\begin{equation}\label{QME}
e_{\sst{\Tcal}}^{-i\Tcal S_1/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i \Tcal S_1/\hbar},S_0\}_{\star}\right)\sim 0\,,
\end{equation}
on the level of natural transformations. Using \eqref{MWI}, condition \eqref{QME} can be expressed as:
\[
\frac{1}{2}\{S_0+S_1,S_0+S_1\}-i\hbar\bigtriangleup({S_1})\sim 0\,,
\]
where $\bigtriangleup({S_1})$ is also seen as a natural transformation. Assume that the {{\textsc{qme}}} holds for $S_1$ and let us fix
$V\in\mathscr{V}_{S_1}(\mathcal{O})$\footnote{The fulfillment of the {{\textsc{qme}}} in the algebraic adiabatic limit is guaranteed by certain cohomological conditions, see \cite{FR3} and references therein. The problem is reduced to the analysis of the Lie algebra cohomology of the gauge algebra (the Lie algebra of the local symmetries Lie group). It is well known that such cohomological conditions are fulfilled, in particular, for QED and Yang-Mills theories \cite{HennBar}, gravity \cite{BTM} and the bosonic string with the Nambu-Goto action\cite{BRZ}.}
The quantum BV operator is defined by
\begin{equation}\label{QBV}
\hat{s}(X)=e_{\sst{\Tcal}}^{-iV/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{iV/\hbar}\cdot_{{}^\Tcal} X,{S_0}(f)\}_{\star}\right)\,,
\end{equation}
where $\supp\, X\subset\mathcal{O}$ and $f\equiv 1$ on $\mathcal{O}$ and it is independent of the choice of $V\in\mathscr{V}_{S_1}(\mathcal{O})$.
If (\ref{QME}) holds, then it follows from \eqref{MWI2} that $\hat{s}$ can be expressed as
\begin{equation}\label{class:quant}
\hat{s}(X)=\{X,S_0+V\}_{\sst\Tcal}-i\hbar\bigtriangleup_V(X)=sX-i\hbar\bigtriangleup_V(X)\,,
\end{equation}
and it has the following property:
\begin{equation}\label{intertwining:s:r}
\{.,S_0\}_\star\circ R_{S_1}^\mathcal{O}=R_{S_1}^\mathcal{O}\circ\hat{s}\,.
\end{equation}
It was proven in \cite{FR3} that constructing a solution to (\ref{QME}) amounts to analyzing the cohomology $H^1(\gamma|d)$ on the space of local forms. The anomaly term $\bigtriangleup(V)$ is expressed in terms of the renormalized BV Laplacian with the use of fundamental theorem of calculus\footnote{The first version of \cite{FR3} contains a notational inconsistency which suggests that $\bigtriangleup(V)=\bigtriangleup_V(V)$. This was corrected in the erratum to that paper.}:
\[
\bigtriangleup(V)=\int\limits_0^1\bigtriangleup_{\lambda V}(V)d\lambda\,.
\]
The natural question to ask is: how to extend the operator $\bigtriangleup_V(.)$ to
multilocal functionals? The nonrenormalized counterpart satisfies:
\begin{equation}\label{Delta:Tbracket}
\bigtriangleup(X\cdot_{{}^\Tcal} Y)=\bigtriangleup(X)\cdot_{{}^\Tcal} Y+(-1)^{|X|}X\cdot_{{}^\Tcal} \bigtriangleup(Y)+\{X,Y\}_{\Tcal}\,,
\end{equation}
It would be tempting to require the same property to hold for the renormalized operator $\bigtriangleup_V(.)$, but then one would have to give up other properties. Note that since
\[
\{e_{\sst{\Tcal}}^{iV/\hbar}\cdot_{{}^\Tcal} X\cdot_{{}^\Tcal} Y,S_0\}_\star=-\hbar^2\frac{\partial^2}{\partial\lambda \partial\mu}\Big|_{\lambda=\mu=0}\{e_{\sst{\Tcal}}^{i(V+\lambda X+\mu Y)/\hbar},S_0\}_\star\,,
\]
one finds (assuming the QME and using \eqref{MWI})
\begin{multline}
\{e_{\sst{\Tcal}}^{iV/\hbar}\cdot_{{}^\Tcal} X\cdot_{{}^\Tcal} Y,S_0\}_\star=-\hbar^2\frac{\partial^2}{\partial\lambda \partial\mu}\Big|_{\lambda=\mu=0}\left(\{ e_{\sst{\Tcal}}^{i(V+\lambda X+\mu Y)/\hbar},S_0\}_{\sst{\Tcal}}+\right.\\
\left.e_{\sst{\Tcal}}^{i(V+\lambda X+\mu Y)/\hbar}\cdot_{{}^{\Tcal_H}}(\bigtriangleup({V+\lambda X+\mu Y})+\frac{i}{2\hbar}\{V+\lambda X+\mu Y,V+\lambda X+\mu Y\}_{\sst{\Tcal}})\right)=\\
e_{\sst{\Tcal}}^{iV}\cdot_{{}^\Tcal}\Big(\{X\cdot_{{}^\Tcal} Y,S_0+V\}-i\hbar\big(\bigtriangleup_V(X)\cdot_{{}^\Tcal} Y+(-1)^{|X|}X\cdot_{{}^\Tcal} \bigtriangleup_V(Y)+\{X,Y\}_{\Tcal}+\\
-i\hbar\frac{\partial^2}{\partial\lambda \partial\mu}\Big|_{\lambda=\mu=0}\bigtriangleup({V+\lambda X+\mu Y})\big)\Big)
\,.
\end{multline}
It follows that
\begin{multline*}
\hat{s}(X\cdot_{{}^\Tcal} Y)=s(X\cdot_{{}^\Tcal} Y)-i\hbar\Big(\bigtriangleup_V(X)\cdot_{{}^\Tcal} Y+(-1)^{|X|}X\cdot_{{}^\Tcal} \bigtriangleup_V(Y)+\{X,Y\}_{\Tcal}+\\-i\hbar\frac{\partial^2}{\partial\lambda \partial\mu}\Big|_{\lambda=\mu=0}\bigtriangleup({V+\lambda X+\mu Y})\Big)\,,
\end{multline*}
so in order to reproduce the relation \eqref{class:quant} also for products, it is natural to set
\begin{multline*}
\bigtriangleup_V(X\cdot_{{}^\Tcal} Y)\doteq \bigtriangleup_V(X)\cdot_{{}^\Tcal} Y+(-1)^{|X|}X\cdot_{{}^\Tcal} \bigtriangleup_V(Y)+\{X,Y\}_{\Tcal}+\\-i\hbar\frac{\partial^2}{\partial\lambda \partial\mu}\Big|_{\lambda=\mu=0}\bigtriangleup({V+\lambda X+\mu Y})\,.
\end{multline*}
Note that on regular elements $\bigtriangleup_V(.)$ is just $\bigtriangleup_{\mathrm{nren}}$ and since it doesn't depend on the interaction, the last term in the above definition vanishes for $\bigtriangleup_{\mathrm{nren}}$, so our proposal is consistent with the non-renormalized case. Moreover, the ``extra term'' is of higher order in $\hbar$, so $\bigtriangleup_V$ behaves like the graded Laplacian, modulo $\mathcal{O}(\hbar)$ corrections.
For higher powers we use an analogous definition (for simplicity of notation we assume all $X_i$ to be even
\begin{multline}\label{DeltaV:sym}
\bigtriangleup_V(X_1\cdot_{{}^\Tcal}\dots\cdot_{{}^\Tcal} X_n)\doteq\\ \sum\limits_{I\subset\{1,...,n\}\atop I\neq \varnothing, |I|\neq n}\!\!\!(-i\hbar)^{|I_c|-1}{\bigcirc\!\!\!\!\!{\T}}_{i\in I} X_{i}\cdot_{{}^\Tcal}\frac{\partial^{|I^c|}}{\partial\lambda_{j_1}\dots \partial\lambda_{j_{|I_c|}}}\Big|_{\vec{\lambda}_{I}=0}\bigtriangleup({V+\vec{\lambda}_I\cdot\vec{X}})+\\
+\sum_{i,j=1,\dots n\atop i< j}\{X_i,X_j\}_{\sst\Tcal}\cdot_{{}^\Tcal} X_{1}\cdot_{{}^\Tcal}\dots\widehat{X_i}\cdot_{{}^\Tcal}\dots\widehat{X_j}\cdot_{{}^\Tcal}\dots X_n\,,
\end{multline}
where $\vec{\lambda}_I=(\lambda_{j_1},...,\lambda_{j_{|I^c|}})$ and $\vec{\lambda}_I\cdot\vec{X}\doteq \sum_{j}\lambda_j X_j$
In \cite{FR3} we showed the existence of a map from multilocal functionals to the graded symmetric algebra over local functionals $\beta:\mathfrak{BV}\rightarrow S^\bullet\mathfrak{BV}_\mathrm{loc}$, which is the inverse of the multiplication $m:S^\bullet\mathfrak{BV}_\mathrm{loc}\rightarrow\mathfrak{BV}$. Using this map and the definition \eqref{DeltaV:sym} one can extend $\bigtriangleup_V(.)$ to $\Tcal(\mathfrak{BV})$ and, for all $X\in\Tcal(\mathfrak{BV})$,
\[
\hat{s}X=sX-i\hbar\bigtriangleup_VX
\]
holds. In the special case of the time-ordered exponential $e_{\sst \Tcal}^{iX/\hbar}$, $X\in\mathfrak{BV}_\mathrm{loc}$, we obtain
\begin{equation}\label{Lap:exp}
\bigtriangleup_V\big(e_{\sst \Tcal}^{iX/\hbar}\big)=e_{\sst \Tcal}^{iX/\hbar}\cdot_{{}^\Tcal}\Big(\tfrac{i}{\hbar}(\bigtriangleup({V+X})-\bigtriangleup(V))+\left(\tfrac{i}{\hbar}\right)^2\tfrac{1}{2}\{X,X\}_{\sst \Tcal}\Big)\,.
\end{equation}
Note that \eqref{intertwining:s:r} implies that $\hat{s}=R_V^{-1}\circ\{.,S_0\}_\star\circ R_V$, so from the nilpotency of
$\{.,S_0\}_\star$ follows that $\hat{s}$ is also nilpotent. The algebra of gauge invariant quantum fields is defined as the cohomology of the quantum BV operator $\hat{s}$. We want to stress that the quantization scheme proposed by K.~Fredenhagen and myself in \cite{FR3} doesn't require $\gamma_0$ to be a derivation with respect to $\star$. This is due to the fact that $\theta_0$ is included in the interaction term $V$ and the free action $S_0$ is just the quadratic part of the gauge fixed action $S$. Therefore, our approach is suitable for theories, like gravity, where the invariance of the star product with respect to $\gamma_0$ is not easy to establish.
\subsection{Quantization of $S_0+\theta_0$}\label{gauge:inv}
In this section we discuss the possibility to consider $S_0+\theta_0$, instead of $S_0$, as the starting point for our construction. In order to do it, we have to ensure that the $\star$-antibracket with $\theta_0$ is well defined on multilocal functionals and that $\{.,\theta_0\}_\star$ is a $\star$-derivation. We will show that this is possible only if we require additional conditions, related to the free {{\textsc{cme}}}.
We start our discussion with a slight reformulation of condition \eqref{freeCME}. Let us fix a compact region $\mathcal{K}\subset M$. From \eqref{freeCME} follows that the relation $\left<K\psi,P\psi\right>_{\overline{\mathfrak{E}}}=0$
holds for all $\psi\in\overline{\mathfrak{E}}_c$ with $\supp(\psi)\subset \mathcal{K}$. This can be also be written as
\begin{equation}\label{PKKP}
\left<P^*K\psi,\psi\right>_{\overline{\mathfrak{E}}}+\left<\psi,K^*P\psi\right>_{\overline{\mathfrak{E}}}\,,
\end{equation}
where $*$ denotes the formal adjoint with respect to $\left<.,.\right>_{\overline{\mathfrak{E}}}$. Let us define for any operator $O$ on $\overline{\mathfrak{E}}$, the following operation: $(O^\dagger)^{\alpha}_{\ \beta}=(-1)^{|\varphi^\alpha||\varphi^\beta|}(O^{\ \alpha}_{\beta})^*$ (the graded formal adjoint). Note that $P^\dagger=P$ and from \eqref{PKKP} we see that \eqref{freeCME} is satisfied, if the following, stronger, condition is fulfilled:
\begin{equation}\label{PK}
(-1)^{|\varphi^\beta|}P_{\beta\gamma}K^\gamma_{\ \sigma}+(K^\dagger)^{\ \gamma}_{\beta} P_{\gamma\sigma}=0\,.
\end{equation}
For determining the signs we used the fact that $\gamma_0$ is an odd differential, so $K^\gamma_{\ \alpha}\neq 0$ only for $|\varphi^\alpha|+|\varphi^\gamma|=1\mod2$. The same holds for $PK$, because $S_0$ is even.
Comparing with \eqref{fullCME}, we conclude that \eqref{PK} is compatible with the full {{\textsc{cme}}} only if $\varphi_0$ is on-shell. We will now study in detail the consequences of condition \eqref{PK}.
\begin{prop}
Identity \eqref{PK} implies that the linearized Koszul-Tate operator $\delta_0$ anticommutes with $\gamma_0$:
\[
\gamma_0\circ\delta_0+\delta_0\circ\gamma_0=0\,.
\]
\begin{proof}
To see this, note that for a constant derivation $X=\int X^\alpha(x)\frac{\delta}{\delta\varphi^\alpha(x)}d\mu(x)$ we have
\begin{multline*}
\gamma_0\circ\delta_0X=-\gamma_0\int X^\alpha(x)\frac{\delta_l L_0(f)}{\delta\varphi^\alpha(x)}d\mu(x)\Big|_{f\equiv 1\atop\textrm{ on }\supp X}=\\
=\int (-1)^{|\varphi^\gamma|}X^\alpha(x)({K}^{\gamma}_{\ \sigma}\Phi_z^\sigma) P_{\alpha\gamma}(x)\delta(x-z)d\mu(x,z)\,.
\end{multline*}
On the other hand
\begin{align*}
\delta_0\circ\gamma_0X&=\int X^\alpha(x) K^\gamma_{\ \alpha}(x)^*\frac{\delta L_0(f)}{\delta\varphi^\gamma(x)}d\mu(x)\Big|_{f\equiv 1\atop\textrm{ on }\supp X}=\\
&=\int X^\alpha(x) K^\gamma_{\ \alpha}(x)^*P_{\gamma\sigma}(x)\Phi^\sigma_xd\mu(x)\,.
\end{align*}
Therefore
\[
(\gamma_0\circ\delta_0+\delta_0\circ\gamma_0)X=\int X^\alpha(x)((-1)^{|\varphi^\alpha|} P_{\alpha\gamma}{K}^{\gamma}_{\ \sigma} + {K^\dagger}^{\ \gamma}_{\alpha}P_{\gamma\sigma})\Phi_x^\sigma d\mu(x)=0
\]
follows.\end{proof}\end{prop}
\begin{exa}[Free electromagnetic field]
Using results from Example \ref{ex1}, we can verify that the condition \eqref{PK} holds for the free electromagnetic field. Note that
\[
K^\dagger=\left(\begin{array}{cccc}
0&0&0&0\\
0&0&0&-i\\
-\delta&0&0&0\\
0&0&0&0
\end{array}\right)\,,
\]
and the direct computation shows that
\begin{multline*}
(-1)^{|\varphi^\beta|}P_{\beta\gamma}K^\gamma_{\ \sigma}+(K^\dagger)^{\ \gamma}_{\beta} P_{\gamma\sigma}=\\
\left(\begin{array}{cccc}
0&0&0&0\\
0&0&\delta d&0\\
0&\delta d&0&0\\
0&0&0&0
\end{array}\right)-\left(\begin{array}{cccc}
0&0&0&0\\
0&0&\delta d&0\\
0&\delta d&0&0\\
0&0&0&0
\end{array}\right)=0
\end{multline*}\,.
\end{exa}
Relation \eqref{PK} allows us to prove the so called \textit{consistency conditions}\footnote{I would like to thank Jochen Zahn for enlightening discussions about the importance of consistency conditions and for crucial remarks on the proof of proposition \ref{gauge:inv:Delta}.}, formulated first in \cite{H} in the case of Yang Mills theory and generalized in \cite{BRZ}.
\begin{prop}\label{gauge:inv:Delta}
Assume that \eqref{PK} holds and that $S_0$ induces a normally hyperbolic system of equations of motion: $P\varphi=0$. Let $\Delta^*$ be the retarded, the advanced or the causal propagator corresponding to $P$. Then $\Delta^*$ satisfies the consistency conditions:
\begin{equation}\label{const:cond}
\sum_\sigma((-1)^{|\varphi^\alpha|}K^{\alpha}_{\ \sigma}(x')\Delta^*(x',x)^{\sigma\gamma}+K^{\gamma}_{\ \sigma}(x)\Delta^*(x',x)^{\alpha\sigma})=0\,.
\end{equation}
\begin{proof} First, we prove the property \eqref{const:cond} for $\Delta^R$. We act with $\eqref{PK}$ on $\Delta^R$, which yields
\[
(-1)^{|\varphi^\alpha|}P_{\alpha\gamma}(z)\circ K^\gamma_{\ \sigma}(z)\Delta^R(z,x)^{\sigma\beta}=-K^\gamma_{\ \alpha}(z)^*\delta(z-x)\delta_\gamma^\beta\,.
\]
We multiply both sides of the above identity with $\Delta^A(z,y)^{\alpha\mu}$. The integration over $z$ results in
\[
(-1)^{|\varphi^\alpha|}\int\Delta^A(z,y)^{\alpha\mu}P_{\alpha\gamma}(z)\circ K^\gamma_{\ \sigma}(z)\Delta^R(z,x)^{\sigma\beta}d\mu(z)=-K^\beta_{\ \alpha}(x)\Delta^A(x,y)^{\alpha\mu}\,.
\]
Next, we use the integration by parts to make $P$ act on $\Delta^A$ from the left. This is possible, since (due to support properties of $\Delta^A$ and $\Delta^R$) the integrant is supported in the intersection of the past of $y$ and the future of $x$, which is a compact set. We obtain
\[
K^\mu_{\ \sigma}(y)\Delta^R(y,x)^{\sigma\beta}=-K^\beta_{\ \alpha}(x)\Delta^A(x,y)^{\alpha\mu}\,.
\]
Using the relation between the retarded and advanced propagators we rewrite the above expression as
\[
(-1)^{|\varphi^\alpha|}K^\mu_{\ \sigma}(y)\Delta^R(y,x)^{\sigma\beta}+K^\beta_{\ \alpha}(x)\Delta^R(y,x)^{\mu\alpha}\,.
\]
The same follows for $\Delta^A$ and also for the difference of the two.
\end{proof}
\end{prop}
\begin{exa}[Free electromagnetic field]
For the free electromagnetic field, the retarded and the advanced propagators take the form:
\[
\Delta^{R/A}=\left(\begin{array}{cccc}
\Delta_v^{R/A}&d\Delta_v^{R/A}&0&0\\
\delta\Delta_v^{R/A}&-d\delta\Delta_v^{R/A}&0&-i\\
-\delta&0&0&i\Delta_s^{R/A}\\
0&0&-i\Delta_s^{R/A}&0
\end{array}\right)\,,
\]
where $\Delta_v^{R/A}$ are the propagators corresponding to the Laplace operator $\delta d+d\delta$ acting on 1-forms and $\Delta_s^{R/A}$ are the propagators of $\delta d$ acting on 0-forms. Property \eqref{const:cond} is expressed as
\[
d_x\Delta_s(x,y)+\delta_y\Delta_v(x,y)=0\,.
\]
\end{exa}
We will show now that \eqref{const:cond} is a necessary condition for $\{.\theta_0\}_\star$ to be well defined on local functionals. Recall that the $\star$-antibracket with $\theta_0$ is given by
\begin{equation}\label{star:br:th0}
\{.,\theta_0(f)\}_\star=\{.,\theta_0(f)\}+\int\! d\mu(x,y,x') \Delta(y,x)^{\sigma\gamma}\frac{{\delta^l}^2 \theta_0(f)}{\delta\varphi_\sigma(y)\delta\varphi_\alpha^{\sst\ddagger}(x')}\frac{{\delta^r}^2}{\delta\varphi^\gamma(x)\delta\varphi^\alpha(x')}\,,
\end{equation}
Since $\theta_0$ is local, the second term is not well defined for local arguments, hence we require that it vanishes identically. Note that we can write this term as
\begin{multline}
\int\! d\mu(x,y,x') f(y)\Delta(y,x)^{\sigma\gamma}(-1)^{|\varphi^\alpha|+1}K^{\alpha}_{\phantom{\alpha}\sigma}(x')\delta(y-x')\frac{{\delta^r}^2}{\delta\varphi^\gamma(x)\delta\varphi^\alpha(x')}=\\=\int\! d\mu(x,x') (-1)^{|\varphi^\alpha|+1}K^{\alpha}_{\phantom{\alpha}\sigma}(x')(f(x')\Delta(x',x)^{\sigma\gamma})\frac{{\delta^r}^2}{\delta\varphi^\gamma(x)\delta\varphi^\alpha(x')}\,.
\end{multline}
We can express the above formula as a sum of two terms and in the second term we rename the indices $\alpha$ and $\gamma$. We obtain
\begin{multline*}
\int\! d\mu(x,x')\left( (-1)^{|\varphi^\alpha|+1}K^{\alpha}_{\phantom{\alpha}\sigma}(x')(f(x')\Delta(x',x)^{\sigma\gamma})\frac{{\delta^r}^2}{\delta\varphi^\gamma(x)\delta\varphi^\alpha(x')}\right.+\\
+\left.(-1)^{|\varphi^\gamma|+1}K^{\gamma}_{\phantom{\gamma}\sigma}(x')(f(x')\Delta(x',x)^{\sigma\alpha})\frac{{\delta^r}^2}{\delta\varphi^\alpha(x)\delta\varphi^\gamma(x')}\right)\,.
\end{multline*}
Next we use the graded antisymmetry of $\Delta$ and in the second term we rename the integration variables $x$ and $x'$. This results in
\begin{multline}\label{second:term}
\int\! d\mu(x,x')\left( (-1)^{|\varphi^\alpha|+1}K^{\alpha}_{\phantom{\alpha}\sigma}(x')(f(x')\Delta(x',x)^{\sigma\gamma})\right.+\\
+\left.K^{\gamma}_{\phantom{\gamma}\sigma}(x)(f(x)\Delta(x',x)^{\alpha\sigma})\right)\frac{{\delta^r}^2}{\delta\varphi^\alpha(x')\delta\varphi^\gamma(x)}\,.
\end{multline}
Now we note that in the definition of $\{X,\theta_0\}_\star$, $f$ has to be chosen to be identically 1 on the support of $X$, so \eqref{second:term} vanishes due to identity \eqref{const:cond} of lemma \ref{gauge:inv:Delta} and the following identities hold:
\[
\{.,\theta_0\}_\star=\{.,\theta_0\}=\gamma_0\,.
\]
We can say that the classical BRST symmetry survives in the free quantized theory.
Next we have to check if $\gamma_0$ is a derivation with respect to the $\star$-product. This is done in the following proposition.
\begin{prop}\label{theta:derivation}
Let $S_0$ be the quadratic term of the action with $\#\mathrm{af}=0$ and let $\gamma_0$ be the free BRST operator.
Assume that \eqref{const:cond} holds. Then, for $X,Y\in\mathfrak{BV}_{\mathrm{reg}}$:
\begin{equation}\label{theta:deriv0}
\gamma_0(X\star Y)=\gamma_0X\star Y+(-1)^{|X|}X\star\gamma_0Y\,.
\end{equation}
\begin{proof}
Recall that $X\star Y\doteq m\circ \exp({i\hbar \Gamma'_\Delta})(X\otimes Y)$. For simplicity we write the proof for $X$ and $Y$ even. The general case differs by introducing some additional signs. From the graded Leibniz rule follows that
\[
\gamma_0\circ m (X\otimes Y)=m\circ(\gamma_0\otimes 1+1\otimes\gamma_0)(X\otimes Y)\,.
\]
Clearly $\gamma_0$ is a derivation if
\[
m\circ\big((\gamma_0\otimes 1+1\otimes\gamma_0)\circ\exp({i\hbar \Gamma'_\Delta})\big)=m\circ\big(\exp({i\hbar \Gamma'_\Delta})\circ(\gamma_0\otimes 1+1\otimes\gamma_0)\big)
\]
holds. Inserting \eqref{gamma0} we obtain the condition
\begin{multline*}
\int \Delta^{\alpha\beta}(x,y){K}^{\gamma}_{\ \sigma}(z)\delta(x-z)\delta_{\alpha\sigma}\frac{\delta_l}{\delta\varphi^\gamma(z)}\otimes\frac{\delta_l}{\delta\varphi^\beta(y)}d\mu(x,y,z)+\\+\int \Delta^{\alpha\beta}(x,y){K}^{\gamma}_{\ \sigma}(z)\delta(y-z)\delta_{\beta\sigma}\frac{\delta_r}{\delta\varphi^\alpha(x)}\otimes\frac{\delta_l}{\delta\varphi^\gamma(z)}d\mu(x,y,z)=0\,.
\end{multline*}
Next we change one of the derivatives in the first term from a left to a right one and we perform the integrations over the delta distributions. The above condition becomes:
\begin{multline*}
\int {K}^{\gamma}_{\ \beta}(x)\Delta^{\beta\alpha}(x,y)(-1)^{|\varphi^\gamma|}\frac{\delta_r}{\delta\varphi^\gamma(x)}\otimes\frac{\delta_l}{\delta\varphi^\alpha(y)}d\mu(x,y)+\\+\int {K}^{\gamma}_{\ \beta}(y)\Delta^{\alpha\beta}(x,y)\frac{\delta_r}{\delta\varphi^\alpha(x)}\otimes\frac{\delta_l}{\delta\varphi^\gamma(y)}d\mu(x,y)=0\,.
\end{multline*}
Renaming the summation indices we see that the above condition is fulfilled, if $\Delta$ satisfies:
\[
(-1)^{|\varphi^\gamma|}{K}^{\gamma}_{\ \beta}(x)\Delta^{\beta\alpha}(x,y)+{K}^{\alpha}_{\ \beta}(y)\Delta^{\gamma\beta}(x,y)=0\,,
\]
which is exactly \eqref{const:cond}.
\end{proof}\end{prop}
We have seen that consistency conditions \eqref{const:cond} are necessary for the BRST construction in the free theory and that they follow automatically for the causal propagator $\Delta$. Now let $\omega=\tfrac{i}{2}\Delta+H$ be a 2-point function of some quasifree Hadamard state. We introduce a following definition
\begin{df}
A Hadamard 2-point function $\omega$ is said to be
\begin{enumerate}
\item \textbf{gauge invariant} if it satisfies the condition analogous to the one fulfilled by $\Delta$:
\begin{equation}\label{const:cond2}
\sum_\sigma((-1)^{|\varphi^\alpha|}K^{\alpha}_{\ \sigma}(x')\omega(x',x)^{\sigma\gamma}+K^{\gamma}_{\ \sigma}(x)\omega(x',x)^{\alpha\sigma})=0\,,
\end{equation}
\item \textbf{gauge invariant modulo a smooth function} if it satisfies
\begin{equation}\label{const:cond3}
\sum_\sigma((-1)^{|\varphi^\alpha|}K^{\alpha}_{\ \sigma}(x')\omega(x',x)^{\sigma\gamma}+K^{\gamma}_{\ \sigma}(x)\omega(x',x)^{\alpha\sigma})=0\ \textrm{mod }\Ci\textrm{ function}\,.
\end{equation}
\end{enumerate}
\end{df}
Let us now explain in detail why the consistency conditions are needed. Note that
\[
\{X,Y\}_{\star_H}=\alpha_H\{\alpha_H^{-1}X,\alpha_H^{-1}Y\}_{\star}\,,
\]
and since $\alpha_H^{-1}\theta_0(f)=\theta_0(f)$, we obtain
\[
\{X,\theta_0(f)\}_{\star_H}=\alpha_H\{\alpha_H^{-1}X,\theta_0(f)\}_{\star}
\]
for regular $X$. If \eqref{PK} holds, we have
\[
\{X,\theta_0(f)\}_{\star_H}=\alpha_H\{\alpha_H^{-1}X,\theta_0(f)\}=(\alpha_H\circ\gamma_0\circ\alpha_H^{-1})(X)\,.
\,,
\]
Let us denote $\gamma_0^H\doteq\alpha_H\circ \gamma_0\circ\alpha_H^{-1}$. Since
\[
\{\alpha_H^{-1}X,\theta_0\}_{\star}=\gamma_0(\alpha_H^{-1}X)=\alpha_H^{-1}\circ \gamma_0^H X\,,
\]
we can interpret $\gamma_0$ on $\mathfrak{A}$ as the normal ordered counterpart of $\gamma_0^H$ on $\mathfrak{A}_H$.
Now, let us take an arbitrary (not necessarly regular) $F\in\mathfrak{A}_H(\mathcal{O})$ and express it as a limit of the series of regular functionals $F=\lim_{n\rightarrow\infty}F_n$. Since $F_n$'s are regular,
\begin{equation}\label{gamma:H}
\gamma_0^HF_n=\{F_n,\theta_0\}+\int\frac{\delta^2F_n}{\delta\varphi^\alpha(x)\delta\varphi^\beta(y)} H^{\beta\gamma}(y,z)\frac{\delta^2\theta_0}{\delta\varphi^\gamma(z)\delta\varphi^\dag_\alpha(x)}d\mu(x,y,z)
\end{equation}
holds. The second term in the above expression is not well defined for local $F$, due to singularities of $H$. For $\gamma_0^H$ to be well defined on the full space $\mathfrak{A}_H(M)$, we have to require that $\int {H}^{\beta\gamma}(y,z)\frac{\delta^2\theta_0}{\delta\varphi^\alpha(z)\delta\varphi^\dag_\alpha(x)}d\mu(z)$ vanishes in the coinciding point limit $x\rightarrow y$, modulo a smooth function. Using the graded symmetry of the second derivative, we find that this requirement is
equivalent to the condition \eqref{const:cond3}. Let us assume that there exists at least one $H$ for which $\omega=\frac{i}{2}\Delta+H$ fulfills \eqref{const:cond3}.
Then, if we take an arbitrary parametrix $\omega'=\frac{i}{2}\Delta+H'$, such that $H-H'$ is smooth, expression \eqref{gamma:H} has a well defined limit as well and $\gamma_0^{H'}F$ is well defined for all $F\in\mathfrak{A}_{H'}(\mathcal{O})$.
We have seen that the existence of $\gamma_0^H$ on $\mathfrak{A}_H(\mathcal{O})$ requires the condition \eqref{const:cond3} to be fulfilled. One can reach exactly the same conclusion working directly with $\gamma_0$ on $\mathfrak{A}(\mathcal{O})$. By the definition of the initial topology on $\mathfrak{A}(\mathcal{O})$ we know that the limit of $\gamma_0(\alpha_H^{-1}F_n)$ exists as an element of $\mathfrak{A}(\mathcal{O})$ if there exists an $H'$ such that $(\alpha_{H'}\circ\gamma_0\circ\alpha_H^{-1})(F_n)$ converges in $\mathfrak{A}_{H'}(\mathcal{O})$. Let us write this expression in a different way:
\begin{multline*}
(\alpha_{H'}\circ\gamma_0\circ\alpha_H^{-1})(F_n)=(\alpha_{H'-H}\circ\gamma^H_0)(F_n)=\\=\{\alpha_{H'-H}F_n,\theta_0\}+\int\frac{\delta^2(\alpha_{H'-H}F_n)}{\delta\varphi^\alpha(x)\delta\varphi^\alpha(y)} {H'}^{\beta\gamma}(y,z)\frac{\delta^2\theta_0}{\delta\varphi^\alpha(z)\delta\varphi^\dag_\alpha(x)}d\mu(x,y,z)\,.
\end{multline*}
If \eqref{const:cond2} is fulfilled, then the second term in the above expression vanishes and $(\alpha_{H'}\circ\gamma_0\circ\alpha_H^{-1})(F_n)=\{\alpha_{H'-H}F_n,\theta_0\}$ converges to a microcausal functional in $\mathfrak{A}_{H'}(\mathcal{O})$, so $\lim_{n\rightarrow\infty}\gamma_0(\alpha_H^{-1}F_n)$ is a well defined element of $\mathfrak{A}(\mathcal{O})$.
Let us now discuss another possibility to define the BRST operator on $\mathfrak{A}_H(\mathcal{O})$ by using $\gamma_0$ instead of $\gamma_0^H$. In other words, we subtract ``by hand'' the singular term in \eqref{gamma:H}. For this to work, one has to prove that $\gamma_0$ is a derivation on $\mathfrak{A}_H(\mathcal{O})$. Here the consistency condition \eqref{const:cond2} enters again. We replace $\Delta$ by $\omega$ in theorem \ref{theta:derivation} and conclude that a sufficient condition for $\gamma_0$ to be a derivation with respect to $\star_H$ is \eqref{const:cond2}, which is equivalent to the requirement that $\gamma_0$ commutes with $\alpha_H$, i.e. $\alpha_H$ induces a cochain morphism. In general, this seems to be too strong, since we want to work with $\omega$ which is a parametrix but not a bisolution and one expects that \eqref{const:cond3} rather than \eqref{const:cond2} holds. Therefore it is more natural to work with $\gamma_0^H$ instead of $\gamma_0$.
To summarize, consistent BV quantization of the free theory can be performed if we can show the existence of at least one quasifree Hadamard state with a 2-point function satisfying \eqref{const:cond2}. This problem has not yet been solved in full generality\footnote{For Yang-Mills theory with a trivial principal bundle and $0$ background section one can use a deformation argument of \cite{FNW}, as it was done in \cite{H}. Problems start, however, if one allows nontrivial topology of principal bundles of the theory and considers arbitrary background connections. This issue is currently investigated by Jochen Zahn. Another example of a theory where the existence of a gauge invariant Hadamard 2-point function is not clear is perturbative quantum gravity \cite{BFR}. }. Therefore, we think that it is more convenient to use $S_0$ as the free action, since this choice doesn't require any additional conditions.
\subsection{Changing the free theory}\label{changing}
In section \ref{gauge:inv} we have shown that the {{\textsc{cme}}} of the free theory is a necessary condition that allows us to construct the free quantum theory corresponding to action $S_0+\theta_0$. Now we want to include the interaction into the discussion. It was proven in \cite{FR3} by K.~Fredenhagen and myself that the quantum master equation is a necessary condition for the gauge invariance of the interacting theory. There, we considered the perturbation around the free action $S_0$. In this section, we show that full {{\textsc{qme}}} can be equivalently formulated for $S_0+\theta_0$, provided the {{\textsc{qme}}} of the free theory holds. For the beginning, we consider only the regular functions $\mathfrak{BV}_{\mathrm{reg}}$ and the non-renormalized time ordered product. Let $V,\, \theta_0\in\mathfrak{BV}_{\mathrm{reg}}$ and denote $\tilde{V}\doteq V-\theta_0$. The {{\textsc{qme}}} is the condition that:
\begin{equation}\label{QME:nonren}
e_{\sst{\Tcal}}^{-i(\tilde{V}+\theta_0)/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i (\tilde{V}+\theta_0)/\hbar},S_0\}_{\star}\right)=0\,.
\end{equation}
Using properties of $\cdot_{{}^\Tcal}$ and $\star$ we can rewrite this condition as:
\begin{multline*}
e_{\sst{\Tcal}}^{-i(\tilde{V}+\theta_0)/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i (\tilde{V}+\theta_0)/\hbar},S_0\}_{\star}\right)=\\
e_{\sst{\Tcal}}^{-i(\tilde{V}+\theta_0)/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i\tilde{V}/\hbar}\cdot_{{}^\Tcal} e_{\sst{\Tcal}}^{i\theta_0/\hbar},S_0\}_{\sst{\Tcal}}+i\hbar\bigtriangleup_{\mathrm{nren}}(e_{\sst{\Tcal}}^{i\tilde{V}/\hbar}\cdot_{{}^\Tcal} e_{\sst{\Tcal}}^{i\theta_0/\hbar})\right)=\\
e_{\sst{\Tcal}}^{-i\theta_0/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i\theta_0/\hbar},S_0\}_{\sst{\Tcal}}+i\hbar\bigtriangleup_{\mathrm{nren}}(e_{\sst{\Tcal}}^{i\theta_0/\hbar})\right)+\\+e_{\sst{\Tcal}}^{-i\tilde{V}/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i\tilde{V}/\hbar},S_0+\theta_0\}_{\sst{\Tcal}}+i\hbar\bigtriangleup_{\mathrm{nren}}( e_{\sst{\Tcal}}^{i\tilde{V}/\hbar})\right)=\\
e_{\sst{\Tcal}}^{-i\theta_0/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i \theta_0/\hbar},S_0\}_{\star}\right)+e_{\sst{\Tcal}}^{-i\tilde{V}/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i\tilde{V}/\hbar},S_0+\theta_0\}_{\sst{\Tcal}}+i\hbar\bigtriangleup_{\mathrm{nren}}( e_{\sst{\Tcal}}^{i\tilde{V}/\hbar})\right)
\,.
\end{multline*}
If the {{\textsc{qme}}} of the free theory holds, i.e. if $e_{\sst{\Tcal}}^{-i\theta_0/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i \theta_0/\hbar},S_0\}_{\star}\right)=0$, then \eqref{QME:nonren} is equivalent to:
\begin{multline}\label{QME:modified}
e_{\sst{\Tcal}}^{-i\tilde{V}/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i\tilde{V}/\hbar},S_0+\theta_0\}_{\sst{\Tcal}}+i\hbar\bigtriangleup_{\mathrm{nren}}( e_{\sst{\Tcal}}^{i\tilde{V}/\hbar})\right)=\\= e_{\sst{\Tcal}}^{-i\tilde{V}/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i\tilde{V}/\hbar},S_0\}_{\star}+\{e_{\sst{\Tcal}}^{i\tilde{V}/\hbar},\theta_0\}_{\sst{\Tcal}}\right)=0\,.
\end{multline}
It was proven in section \ref{gauge:inv} that the free {{\textsc{cme}}} implies $\{e_{\sst{\Tcal}}^{i\tilde{V}/\hbar},\theta_0\}_{\sst{\Tcal}}=\{e_{\sst{\Tcal}}^{i\tilde{V}/\hbar},\theta_0\}_{\star}$, so finally we can write \eqref{QME:modified} as:
\[
e_{\sst{\Tcal}}^{-i\tilde{V}/\hbar}\cdot_{{}^\Tcal}\left(\{e_{\sst{\Tcal}}^{i\tilde{V}/\hbar},S_0+\theta_0\}_{\star}\right)=0\,.
\]
After this short introduction we can come back to the discussion of the renormalized time-ordered product. To distinguish it from the non-renormalized one, we denote it in this subsection by $\cdot_{{}^{\Tcal_H}}$.
First we have to check if the free {{\textsc{qme}}} can be satisfied by exploiting the renormalization freedom which we have in defining $\cdot_{{}^{\Tcal_H}}$.
Note that, since $\theta_0$ is linear in both fields and antifields and is assumed to be of degree 0, the component $\theta^\alpha(x)$ doesn't depend on field $\varphi^\alpha(x)$. It follows that $\{\theta_{0},\theta_0\}=0$ and using the anomalous {{\textsc{mwi}}} \eqref{MWI} we obtain:
\[
e_{\sst{\Tcal_H}}^{-i\theta_{0}/\hbar}\cdot_{{}^{\Tcal_H}}\{e_{\sst{\Tcal_H}}^{i\theta_{0}/\hbar},S_0\}_\star=\{\theta_{0},S_0\}_{\sst{\Tcal_H}}+i\hbar\bigtriangleup({\theta_{0}})\,,
\]
where $\bigtriangleup({\theta_0})$ is the anomaly term. Using the standard arguments of \cite{H,FR3} we can conclude that it must be constructed from elements of the relative cohomology $H^1(\gamma_0|d)$ on the space of local forms. If this cohomology is trivial, then the free theory is anomaly free, i.e. we can use the renormlization freedom to redefine the time-ordered powers of $\theta_0$ to obtain $\bigtriangleup({\theta_0})=0$, so the free {{\textsc{qme}}} holds as a consequence of free {{\textsc{cme}}}.
Now, we want to repeat the reasoning that led to equation \eqref{QME:modified} for the renormalized time-ordered product. To this end, we use the MWI \eqref{MWI} and replace $\bigtriangleup_{\textrm{nren}}$ with renormalized BV Laplacians. The anomaly term corresponding to the free action $S_0$ will be denoted by $\bigtriangleup({V})$ and the one of $S_0+\theta_0$ by $\tilde{\bigtriangleup}({\tilde{V}})$. The defining equation for $\tilde{\bigtriangleup}({\tilde{V}})$ is:
\begin{equation}\label{MWI3}
\{e_{\sst{\Tcal_H}}^{i\tilde{V}/\hbar},S_0+\theta_0\}_\star=\{ e_{\sst{\Tcal_H}}^{i\tilde{V}/\hbar},S_0+\theta_0\}_{\sst{\Tcal_H}}+e_{\sst{\Tcal_H}}^{i\tilde{V}/\hbar}\cdot_{{}^{\Tcal_H}}(\tilde{\bigtriangleup}({\tilde{V}})+\frac{i}{2\hbar}\{\tilde{V},\tilde{V}\}_{\sst{\Tcal_H}})\,.
\end{equation}
The existence and properties of the anomaly term $\tilde{\bigtriangleup}({\tilde{V}})$ were proven in the case of Yang-Mills theory in \cite{H}. Analogous arguments can be also used in a more general setting. A short calculation yields
\begin{multline*}
e_{\sst{\Tcal_H}}^{-iV/\hbar}\cdot_{{}^{\Tcal_H}}\left(\{e_{\sst{\Tcal_H}}^{i V/\hbar},S_0\}_{\star}\right)=e_{\sst{\Tcal_H}}^{-i\tilde{V}/\hbar}\cdot_{{}^{\Tcal_H}}\left(\{e_{\sst{\Tcal_H}}^{i \tilde{V}/\hbar},S_0\}_{\star}\right)+\\+e_{\sst{\Tcal_H}}^{-i\theta_0/\hbar}\cdot_{{}^{\Tcal_H}}\left(\{e_{\sst{\Tcal_H}}^{i+\theta_0/\hbar},S_0\}_{\star}\right)+\bigtriangleup({V})-\tilde{\bigtriangleup}({\tilde{V}})\,.
\end{multline*}
This shows that, if the free QME holds, and $\bigtriangleup({V+\theta_0})-\tilde{\bigtriangleup}(\tilde{V})$ can be removed with an appropriate redefinition of renormalized time-ordered products, then $e_{\sst{\Tcal_H}}^{-iV/\hbar}\cdot_{{}^{\Tcal_H}}\left(\{e_{\sst{\Tcal_H}}^{i V/\hbar},S_0\}_{\star}\right)=0$ is equivalent to $e_{\sst{\Tcal_H}}^{-i\tilde{V}/\hbar}\cdot_{{}^{\Tcal_H}}\left(\{e_{\sst{\Tcal_H}}^{i \tilde{V}/\hbar},S_0+\theta_0\}_{\star}\right)=0$.
\section{BRST charges}
\subsection{Different notions of a BRST charge}
We start this section with an overview of different approaches to quantization of gauge theories and different notions of a BRST charge in {{p\textsc{aqft}}}. We compare the approaches of \cite{DF99} and \cite{H}, using the general BV quantization framework proposed in \cite{FR3}. There is an important difference between the free BRST operator $\gamma_0$ used by Hollands in \cite{H} and the one discussed in \cite{FR}. The full BV operator
$s$ is the same, but in \cite{FR,FR3} (following \cite{Barnich:1999cy}) $s_0$ is expanded with respect to the total antifield number $\#\mathrm{ta}$ (see equation \eqref{ta:expansion}), while in
\cite{H} $s_0$ is expanded with respect to $\#\mathrm{af}$ \textit{also for the gauge fixed theory}. Explicitly, one has $s_0=\tilde{\delta}_0+\tilde{\gamma}_0$, where $\tilde{\gamma}_0$ is the term with $\#\mathrm{af}=0$ and $\tilde{\delta}_0$ has $\#\mathrm{af}<0$.
We argue that the expansion with respect to $\#\mathrm{ta}$ is physically more justified, since the $\#\mathrm{ta}=-1$ term of this expansion, denoted here by $\delta$, is the Koszul operator corresponding to the gauge fixed system of equation of motion. This allows one to view the algebra of on-shell functionals as the 0-th cohomolgy of $(\mathfrak{BV},\delta)$; this interpretation is not possible if one considers the expansion used in \cite{H}. It is, therefore, not clear how going on-shell in \cite{H} is interpreted in cohomological terms, already at the level of linearized equations of motion. Let $C^\ddagger$ be the antifield of the ghost. Following \cite{H} we obtain $\tilde{\delta_0}(C^\ddagger)=id*d\bar{C}_I-dA^\ddagger_I$, so the equations of motion corresponding to the image of $\tilde{\delta_0}$ contain a source term $dA^\ddagger_I$ which is not present in the equations of motion employed in \cite{H} for the construction of the causal propagator and the star product. This problem is not present in \cite{FR,FR3}, where we have ${\delta_0}(C^\ddagger)=id*d\bar{C}_I$ and ${\gamma_0}(C^\ddagger)=-dA^\ddagger_I$, in contrast to ${\tilde{\gamma}_0}(C^\ddagger)=0$.
It was shown in \cite{FR} that, already at the classical level, we have two (graded) Poisson brackets: the Peierls bracket $\lfloor.,.\rfloor$ and the antibracket $\{.,.\}$. The Peierls bracket is induced by the equations of motion (dynamics) and the
antibracket is of geometrical nature: it is the graded Schouten bracket. The BV operator $s$ can locally be written as the antibracket with the extended action $S_\mathrm{ext}$. In particular, the BRST operator is generated by $\theta$. Note that one uses $\{.,.\}$, rather than $\lfloor.,.\rfloor$, as the natural graded Poisson structure on $\mathfrak{BV}(M)$. This can be compared with the Hamiltonian version of the Batalin-Vilkovisky formalism (BFV formalism, \cite{Batalin:1977pb,Fradkin:1977wv,Fradkin:1975cq,Fradkin:1977hw,Fradkin:1978xi}) where everything is done with relation to one canonical structure. It was shown in \cite{FH2,BGPR} that the Lagrangian and the Hamiltonian formalism are equivalent on the formal level. We believe that a more rigorous argument can be provided within the framework of infinite dimensional symplectic geometry. \cite{BGPR} compares also the Noether charge of the Lagrangian formalism with the BRST charge of the Hamiltonian formalism. The Noether charge can be used as a generator of the BRST transformation with respect to a certain canonical structure which is defined on both fields and antifields (see formula 3.14 of \cite{BGPR}). In \cite{H} antifields are treated as external fields, so there is no dynamics associated with them and $\lfloor.,.\rfloor$ acts on them trivially. Also in \cite{FR} antifields are non-dynamical, since they are identified with geometrical objects: functional derivatives. Because $\lfloor.,.\rfloor$ is antifield-independent, the classical BRST charge $Q$ generates $\gamma$ with respect to $\lfloor.,.\rfloor$ modulo the equations of motion only on the space of functionals that don't contain antifields. Let us briefly recall the construction of $Q$. The classical BRST current is defined as
\begin{multline*}
J^\mu(x)\doteq\sum\limits_{\alpha}\Big(\gamma\varphi^\alpha\frac{\partial{L_M(x)}}{\partial(\nabla_\mu\varphi^\alpha)}+2\nabla_\nu\gamma\varphi^\alpha\frac{\partial{L_M(x)}}{\partial(\nabla_\mu\nabla_\nu\varphi^\alpha)}+\\
-\nabla_\nu\left(\gamma\varphi^\alpha\frac{\partial{L_M(x)}}{\partial(\nabla_\mu\nabla_\nu\phi^\alpha)}\right)\Big)+-K^\mu_{M}(x)\,,
\end{multline*}
where $K_{M}^\mu$ is the divergence term appearing after applying $\gamma$ to $L_M(f)$. Following \cite{H} we recall here a useful formula relating $J$ with the BV operator:
\begin{equation}\label{dJ}
dJ(x)=\sum_\alpha\{L_\mathrm{ext}(f),\varphi^\alpha(x)\}\cdot\{\varphi^{\sst\ddagger}_\alpha(x),L_\mathrm{ext}(f)\}=\sum_\alpha \theta^\alpha(x)\cdot\frac{\delta L_\mathrm{ext}(f)}{\delta\varphi^\alpha(x)}\,,
\end{equation}
where $f(x)=1$.
\subsection{The free BRST charge}\label{free:charge}
We have already shown in \ref{gauge:inv} that if $\theta_0$ is included into the free action, then additional consistency condition are needed. In particular, \eqref{const:cond2} has to hold and the background configuration $\varphi_0$ has to be a solution of the equations of motion. Here, we show that the same conditions allow us to express $\{.,\theta_0\}_\star$ as the commutator with the free BRST charge.
Let the free BRST current be denoted by $J_0$.
In a spacetime $M$ with compact Cauchy surface $\Sigma$ there exists a closed compactly supported 1-form $\alpha$ on $M$ such that $\int_M\alpha\wedge\beta=\int_\Sigma\beta$, for any closed 3-form $\beta$. In this case, we can define the free BRST charge by
\[
Q_0\doteq \int_M\alpha\wedge J_0\,.
\]
In \cite{H} the free quantum BRST operator is defined directly by giving its action on basic fields and requiring that it is a $\star$-derivation. In the formalism of \cite{FR3} this corresponds to defining the free quantum BRST as $\{.,\tilde{\theta}_0\}_\star$, where $\tilde{\theta}_0$ denotes the action that generates $\tilde{\gamma}_0$. The difference between
\cite{H} and \cite{FR3} lies again in the way in which the free quantum BRST operator acts on antifields. In \cite{H} we have $\{\varphi^\ddagger_\alpha,\tilde{\theta}_0\}=0$ for all $\varphi_\alpha^\ddagger$, whereas the formalism of \cite{FR3} applied to Yang-Mills theory yields $\{C^\ddagger,\tilde{\theta}_0\}=-dA^\ddagger$. In a proposition below we show that $\{.,\theta_0\}_\star$ is on-shell equal to $[.,Q_0]_\star$, if the argument has $\#\mathrm{ta}=0$ (i.e. it doesn't contain antifields). In general, however, $Q_0$ is not a generator for $\{.,\theta_0\}_\star$. This does not pose a problem, since Ward identities in \cite{FR3} are formulated in terms of $\{.,\theta_0\}_\star$, not $[.,Q_0]_\star$, so the results of \cite{H} can be applied. Note that $Q_0$ is not necessary for the construction of the abstract net of interacting algebras of observables. It is, however, a crucial concept in the Kugo-Ojima formalism \cite{KuOji0,KuOji}, which is a convenient method to construct states for the free theory. A deformation procedure given in \cite{DF99} allows then to construct states also on the net of local algebras of observables of the interacting theory.
From now on we work in the algebraic adiabatic limit, which means that we are interested only in constructing local algebras $\mathfrak{A}(\mathcal{O})$ and don't discuss the existence of the inductive limit. Therefore, we can apply the idea of \cite{DF99} and embed $\mathcal{O}$ into a spacetime with a compact Cauchy surface, for example into a causal completion of a spacial box. Keeping this in mind we restrict our attention to the situation where $M$ has a compact Cauchy surface.
\begin{prop}\label{Q0}
Let $F\in\mathfrak{BV}(M)$ with $\#\mathrm{ta}=0$. Assume that \eqref{PK} and \eqref{const:cond2} hold and that $\gamma_0^2=0$, then the following relation is fulfilled on-shell ($\ \stackrel{\mathrm{o.s.}}{=}\,$):
\[
\{F,\theta_0\}_{\star_H}\stackrel{\mathrm{o.s.}}{=}\frac{i}{\hbar}[F,Q_0]_{\star_H}\,.
\]
Equivalently, we can write this formula in terms of Wick-ordered expressions $\alpha_H^{-1}(F)$, $\alpha_H^{-1}(Q_0)\in \mathfrak{A}(M)$,
\[
\{\alpha_H^{-1} F,\theta_0\}_{\star}\stackrel{\mathrm{o.s.}}{=}\frac{i}{\hbar}[\alpha_H^{-1}F,\alpha_H^{-1}Q_0]_{\star}\,
\]
\end{prop}
\begin{proof}
Since we assume that $\gamma_0^2=0$, the term $\{\theta_0,\theta_0\}$ can be neglected in the definition of the conserved current $J_0$.
\[
dJ_0(x)=\sum_\alpha \theta_0^\alpha(f)(x)\cdot\frac{\delta L_0(f)}{\delta\varphi^\alpha(x)}\,,
\]
where $f(x)=1$. Since the definition of the BRST charge doesn't depend on the choice of a 1-form $\alpha$ dual to the Cauchy surface, we can choose it in a way that will facilitate the calculation. Let us take $\alpha=d\eta$, where $\alpha$ is compactly supported, its support lies in the past of the support of $F$
and $\eta=1$ on $\supp\, F$. Now we use the fact that, for the Hadamard function $\omega=$ used to define the $\star$-product, $ \omega^{\alpha \beta}(x,y) - (-1)^{|\phi^\alpha| |\phi^\beta|} \omega^{\beta \alpha}(y,x)=i\Delta^{\alpha\beta}(x,y)$ holds. Moreover, from the support properties of $\Delta_R$ and $\Delta_A$ follows that we can write the commutator with $Q_0$ as
\begin{align*}
[F,Q_0]_{\star_H}&= i\hbar\sum_{\alpha,\beta,\sigma}\int dxdz\frac{\delta F}{\delta\varphi^\beta(x)}\Delta_R^{\beta\alpha}(x,z) \frac{\delta Q_0(d\eta)}{\delta\varphi^\alpha(z)}+\\
&+ i\hbar^2\sum_{\alpha,\beta,\sigma,\atop \mu,\nu}\int dxdz\frac{\delta^2 F}{\delta\varphi^\beta(x)\varphi^\mu(x')}\Delta_R^{\mu\nu}(x',z')H^{\beta\alpha}(x,z) \frac{\delta^2 Q_0(d\eta)}{\delta\varphi^\alpha(z)\delta\varphi^\nu(z')}\,.
\end{align*}
Now let us choose $f\in\mathfrak{D}(M)$ such that $f\equiv 1$ on the support of $F$. Inserting the definition of $Q_0$ and integrating by parts we obtain:
\begin{align*}
\frac{i}{\hbar}[F,Q_0]_{\star_H}=&\sum_{\alpha,\beta,\sigma}\int\frac{\delta F}{\delta\varphi^\beta(x)}\Delta_R^{\alpha\beta}(x,z)\eta(y) \frac{\delta \theta^{\sigma}_0(f)(y)}{\delta\varphi^\alpha(z)}\frac{\delta S_0}{\delta\varphi^\sigma(y)} dxdydz\\
&\sum_{\alpha,\beta\sigma}\int\frac{\delta F}{\delta\varphi^\beta(x)}\Delta_R^{\alpha\beta}(x,z)\eta(y)\theta^{\sigma}_0(f)(y)\frac{\delta^2 S_0}{\delta\varphi^\sigma(y)\delta\varphi^\alpha(z)} dxdydz+\\
&\hbar\sum_{\alpha,\beta\sigma}\int\frac{\delta^2 F}{\delta\varphi^\sigma(y)\delta\varphi^\mu(x)}H^{\mu\nu}(x,z)\frac{\delta \theta^{\sigma}_0(f)(y)}{\delta\varphi^\nu(z)} dxdydz\,.
\end{align*}
Note that, to obtain the above result, we had to exchange the order of integration and differentiation. This is possible only if we assume
the consistency condition \eqref{const:cond3}. Without this assumption, the third term of the formula above would not be well defined for local $F$. After performing the integration over $y$ and $z$ in the second term, we arrive at:
\begin{multline*}
\frac{i}{\hbar}[F,Q_0]_{\star_H}=\sum_{\beta}\int\frac{\delta F}{\delta\varphi^\beta(x)}\theta^{\beta}_0(x) dx+\\
+\hbar\sum_{\alpha,\beta\sigma}\int\frac{\delta^2 F}{\delta\varphi^\sigma(y)\delta\varphi^\mu(x)}H^{\mu\nu}(x,z)\frac{\delta \theta^{\sigma}_0(f)(y)}{\delta\varphi^\nu(z)} dxdydz+I_0=\\
=\{F,\theta_0\}_{\star_H}+I_0\,,
\end{multline*}
where $I_0$ is an element of the ideal generated by equations of motion.
\end{proof}
The result above allows to make contact with the formalism used in \cite{DF99} and \cite{Boas}, where gauge theories are quantized in the BRST formalism, but without introducing antifields. The quantum BRST differential on free fields is defined as the commutator with $Q_0$ and, as we have just seen, this is the same as $\{.,\theta_0\}_{\star_H}$ on the space of functionals with $\#\mathrm{ta}=0$.
\subsection{The interacting BRST charge}
Up to now we have treated only the free theory, now we want to construct a charge that generates the quantum BV differential on interacting fields. We also want to drop the assumption on the total antifield number $\#\mathrm{ta}$. We work in the algebraic adiabatic limit, so we pick a bounded region $\mathcal{O}\subset M$ and choose $f\in\mathfrak{D}(M)$ such that $f\equiv 1$ on $\mathcal{O}$. We choose $\alpha=d\eta$ as in the previous section, and require that $\supp(\alpha)\subset\mathcal{O}$; let $V={S_1}_M(f)$. Interacting fields are defined by formula \eqref{RV}. For such fields we cannot make use of proposition \ref{Q0}, because $V$ in general contains antifields and $\eta$ cannot be chosen to be one on the support of $R_V(F)$. Instead, we can follow \cite{DF99,Boas,H} and use the interacting charge $R_V(Q)$.
Note that our situation is much more general than the cases studied in the literature so far. In the present setting we admit arbitrary theories with local symmetries, which satisfy the {{\textsc{qme}}}. These include in particular gravity and the free bosonic string.
Firstly, we need to prove some identities for time-ordered retarded products.
For simplicity of notation we omit the subscript ``$M$'' in ${\theta}_M$, ${S_1}_M$, etc. The conservation of the current $R_V(J(x))$ is a condition that $R_V(dJ(x))\stackrel{\mathrm{o.s.}}{=} 0$ for $x\in\mathcal{O}$. We have to prove that
\begin{equation}\label{current:cons}
e_{\sst \Tcal}^{iV/\hbar}\cdot_{{}^\Tcal} dJ(x)=e_{\sst \Tcal}^{iV/\hbar}\cdot_{{}^\Tcal} \left(\theta^\alpha(x)\cdot\frac{\delta}{\delta \varphi^\alpha(x)}(S_0+S_1)(f)\right)\stackrel{\mathrm{o.s.}}{=} 0\,.
\end{equation}
Let $h\in\mathfrak{D}(\mathcal{O})$. Using the {{\textsc{mwi}}} \eqref{Lap:coeff} we obtain an identity fulfilled by the natural transformation $dJ$:
\begin{multline*}
e_{\sst \Tcal}^{iV/\hbar}\cdot_{{}^\Tcal} dJ(h)=-i\hbar\int h(x)\left( e_{\sst \Tcal}^{iV/\hbar}\cdot_{{}^\Tcal} \theta^\alpha(x)\right)\star\frac{\delta S_0}{\delta\varphi^\alpha(x)}d\mu(x)+\\
+i\hbar\sum_{n=0}^\infty\bigtriangleup^{n}(V(f)^{\otimes n}; V(h))\,,
\end{multline*}
If the ``anomaly'' $\sum_{n=0}^\infty\bigtriangleup^{n}(V(f)^{\otimes n}; V(h))$ can be removed by a redefinition of time-ordered products, then the above identity implies \eqref{current:cons}. In \cite{H} the anomaly was removed in Yang-Mills theory. In general the interacting current is not conserved in theories which satisfy the QME with a non-zero $\bigtriangleup({V(f)})$ that cannot be removed.
It is an intuitive result, since current conservation is a classical phenomenon and to reproduce it on the quantum level one has to assume that the quantized theory ``doesn't differ too much'' from the classical one.
Recall that the map $R_V$ intertwines between the interacting and the free theory and in particular we have:
\[
R_V^{-1}\left(\frac{\delta S_0}{\delta\varphi^\alpha(x)}\right)=\frac{\delta (S_0+V)}{\delta\varphi^\alpha(x)}\,.
\]
Let us denote by $\stackrel{\mathrm{o.s._V}}{=}$ an equality that holds modulo the ideal generated, with respect to $\star_V$, by $R_V^{-1}\left(\frac{\delta S_0}{\delta\varphi^\alpha(x)}\right)$. We are now ready to prove the main result of this paper.
\begin{thm}
Assume that the {{\textsc{qme}}} holds for $V\in \mathscr{V}_{S_1}(\mathcal{O}) $ and that $\bigtriangleup(V)=0$. Let $F\in\mathfrak{A}_\mathrm{loc}(\mathcal{O})$, then
\begin{equation}\label{Q:int}
\frac{i}{\hbar}[R_V(F),R_V(Q)]_\star\stackrel{\mathrm{o.s.}}{=} R_V( s F-i\hbar\bigtriangleup_V(F))=R_V(\hat{s} F)\,.
\end{equation}
In other words, $Q$ generates, with respect to $[.,.]_{\star_V}$, the quantum BV operator,
\begin{equation}\label{Q:int2}
\frac{i}{\hbar}[F,Q]_{\star_V}\stackrel{\mathrm{o.s._V}}{=} \hat{s} F\,.
\end{equation}
\begin{proof}
Using the GLZ relation \eqref{glz} we obtain
\[
\frac{1}{i\hbar}[R_V(F),R_V(Q)]_\star=R_V^{(1)}[F](Q)-R_V^{(1)}[Q](F)\,.
\]
We can always write a local functional $F\in\mathfrak{A}_\mathrm{loc}(\mathcal{O})$ in the form $F=\int F(x) d\mu(x)$. Using \eqref{current:cons}, \eqref{MWI2} and assuming that $\bigtriangleup({V(f)})$ was already removed, we find that
\[
R_V^{(1)}[dJ_0(\eta)](F)=\frac{d}{d\lambda}(e^{iV/\hbar}_{\sst \Tcal})^{-1\star}\star\left(e^{i(V+\lambda F)/\hbar}_{\sst \Tcal}\cdot_{{}^\Tcal} dJ(\eta)\right)\Big|_{\lambda=0}\,,
\]
and we can use relation \eqref{MWI} to obtain
\begin{multline*}
e^{i(V+\lambda F)/\hbar}_{\sst \Tcal}\cdot_{{}^\Tcal} dJ(\eta)=-i\hbar\int \eta(x)\left(e^{i(V+\lambda F)/\hbar}_{\sst \Tcal}\cdot_{{}^\Tcal} \frac{\delta (\theta^\alpha+\lambda F)}{\delta\varphi_\alpha^\ddagger(x)}\right)\star\frac{\delta S_0}{\delta \varphi^\alpha(x)}d\mu(x)+\\
+i\hbar\sum_{n=0}^\infty\bigtriangleup^{n}((V(f)+\lambda F)^{\otimes n}; V(\eta)+\lambda F(\eta))+\\
-e^{i(V+\lambda F)/\hbar}_{\sst \Tcal}\cdot_{{}^\Tcal}\left(\{\lambda F(\eta),S_0+V\}_{\sst \Tcal}+\lambda^2\int \frac{\delta F(\eta)}{\delta \varphi_\alpha^\ddagger(x)}\cdot_{{}^\Tcal}\frac{\delta F}{\delta \varphi^\alpha(x)}\right)\,.
\end{multline*}
where $F(\eta)\doteq \int F(x)\eta(x) d\mu(x)$. Thus we obtain
\begin{multline}
R_V^{(1)}[dJ(\eta)](F)\stackrel{\mathrm{o.s.}}{=} - \, R_V(\{F(\eta),S_\mathrm{ext}\}_{\sst \Tcal})+\\
+i\hbar R_V\left(\sum_{n=1}^\infty n\bigtriangleup^{n}\left(V(f)^{\otimes(n-1)}\otimes F;\theta(\eta)\right)\right)+\\
+i\hbar R_V\left(\sum_{n=1}^\infty \bigtriangleup^{n}\left(V(f)^{\otimes n};F(\eta)\right)\right)\,.\label{Wid}
\end{multline}
Similarly for $R_V^{(1)}[F](dJ_0(\eta))$.
Now, we can make a particular choice for the function $\eta$ in the definition of $Q_0$. The current conservation implies that $Q_0$ is independent of this choice. We are interested in local algebras, so we can assume that $\mathcal{O}$ is embedded in a spacetime $M$ with a compact Cauchy surface $\Sigma$. We pick two other Cauchy surfaces $\Sigma_{\pm}$, such that $\Sigma_-$ is in the past of $\supp(F)$ and $\Sigma_+$ in its future. We choose a function $\eta$ such that, for any closed 3-form $\beta$, $\int_Md\eta\wedge\beta=\int_\Sigma\beta$ holds. Next, we take compactly supported functions $\eta_\pm$ such that $d\eta_\pm=d\eta+\chi_\pm$, where $\chi_\pm$ are supported in the future (past) of $\Sigma_+$ ($\Sigma_-$). Moreover we require that, on $\supp(F)$, $\eta_+=\eta$ and $\eta_-=\eta-1$ hold. An explicit construction of such functions is provided in \cite{DF99} for a flat $M$. Using the support property \eqref{supp:prop} of retarded products we find that
\begin{align*}
R_V^{(1)}[Q](F)-R_V^{(1)}[F](Q)&=-R_V^{(1)}\Big[\int_Md\eta_+\wedge J\Big](F)+R_V^{(1)}[F]\Big(\int_Md\eta_-\wedge J\Big)=\\
&=R_V^{(1)}\left[dJ(\eta_+)\right](F)-R_V^{(1)}[F]\left(dJ(\eta_-)\right)\,.
\end{align*}
Inserting \eqref{Wid} and using the properties of $\eta_\pm$ on the support of $F$ we obtain
\begin{align*}
[R_V(F),R_V(Q)]_\star&= -i\hbar\, (R_V(\{F,S_\mathrm{ext}\}_{\sst \Tcal})-i\hbar R_V\left(\bigtriangleup_V(F)\right))=\\
&=-i\hbar R_V(\hat{s}F)\,.
\end{align*}
\end{proof}
\end{thm}
Note that our result doesn't require $\bigtriangleup_V(F)$ to vanish. This means that, on-shell, $Q$ is a generator for the full \textit{quantum} BV operator. It is interesting to ask what modifications have to be made to allow for the situation in which $\bigtriangleup(V)$ is also non-zero. We consider it as a problem for future study.
\section*{Acknowledgements}
Most of the results presented in this paper were obtained during a Junior Hausdorff Trimester Program that took place during the fall 2012 at the Hausdorff Research Institute for Mathematics (HIM) in Bonn. Therefore, I would like to thank the participants of the program and the guests of HIM with whom I discussed during my stay. In particular, I gained from discussions with C. Dappiaggi, W. Dybalski, Ch. Fewster, K. Fredenhagen, S. Meinhardt, J. Schlemmer, Y. Tanimoto, M. Wrochna and J. Zahn (whom I also thank for important remarks on the first version of the manuscript).
|
1301.6580
|
\section{Introduction}
\label{intro}
\input{introA}
\section{Problem setting}
\label{problem_setting}
\input{problem_setting}
\section{Finite Element Formulation}
\label{numerical_setting}
\input{numerical_setting2}
\section{Numerical confirmation of the interface law}
\label{numerical_confirmation}
\input{numerical_confirmation_intro}
\subsection{Case I: periodic case}
\label{sec:case_I}
\input{periodic_case}
\subsection{Case II: Beavers-Joseph case}
\label{sec:case_II}
\input{bj_case}
\section{Conclusions}
\label{conclusions}
\input{conclusions}
\begin{acknowledgments}
AM-C was supported by ERC Starting Grant "Biostruct" No. 210680 and Emmy Noether Programme of German Research Council (DFG). The research of A.M. was partially supported by the Programme Inter Carnot Fraunhofer from BMBF (Grant 01SF0804) and ANR. Research visits of A.M. to the Heidelberg University were supported in part by the Romberg professorship at IWR, Heidelberg University, 2011-1013. TC was supported by the German Research Council (DFG) through project
``Modellierung, Simulation und Optimierung der Mikrostruktur mischleitender
SOFC-Kathoden'' (RA 306/17-2).
\end{acknowledgments}
\bibliographystyle{jfm}
\subsection{Finite Element Formulation of the Microscopic Problem}
\label{Numset}
To numerically solve the problems we consider the finite element method and we give exemplary in this subsection the formulation of the microscopic problem \eqref{1.3}-\eqref{1.5}. For a more thorough introduction into the theory of finite elements, we refer to standard literature such as \cite{Ciarlet:2002} or \cite{BrennS:2002}.
The natural setting of the finite element approximation of the problem is its weak formulation, shown below.
We first introduce the spaces
\begin{align}
V(\Omega^\ep) &:= \Set{\mathbf{v} \in H^1(\Omega^\ep)^2\ \big|\ \mathbf{v} =0 \text{ on } \p \Omega \setminus \Gammaper, \mathbf{v} \hbox{ is } L\text{-periodic in } x_1},\\
L_0(\Omega^\ep) &:= \Set{ p \in L^2(\Omega^\ep)\ \big | \ \int_{\Omega^\ep} p \ dx = 0},
\end{align}
where $L^2(\Omega^\ep)$ is the space of square-integrable functions in $\Omega^\ep$, i.e. for $u \in L^2(\Omega^\ep)$ holds $\displaystyle \int_{\Omega^\ep} \abs{u(x)}^2\,dx < \infty$, and $H^1(\Omega^\ep)$ is the space of square-integrable functions, with first derivatives also square-integrable.
The weak formulation of problem \eqref{1.3}-\eqref{1.5} reads as follows:
\begin{problem}[Microscopic Problem in Weak Formulation]\label{prob.micro_weak}
For given $\mathbf{f}$ find a pair $(\mathbf{v}^\ep,p^\varepsilon) \in V(\Omega^\epsilon)\times L_0(\Omega^\ep)$, such that
\begin{align}
\int_{\Omega^\ep} \big(\nabla \mathbf{v}^\ep + (\nabla \mathbf{v}^\ep)^T\big) \cdot \nabla \boldsymbol\varphi \ dx + \int_{\Omega^\ep} p^\varepsilon \nabla \cdot \boldsymbol\varphi \ dx &= \int_{\Omega^\ep} \mathbf{f} \cdot \boldsymbol\varphi \ dx &&\forall \boldsymbol\varphi \in V(\Omega^\ep),\\
\int_{\Omega^\ep} \nabla \cdot \mathbf{v}^\ep \ \psi \ dx &= 0 &&\forall \psi \in L_0(\Omega^\ep).
\end{align}
\end{problem}
We use finite elements to discretize this problem and consider a decomposition $\ensuremath{{\cal T}_{h}}$ of the domain into so called cells $T$, whose union constitutes an approximation of the problem geometry, i.e. $\ensuremath{{\cal T}_{h}} = \Set{\ensuremath{T}}$.
We consider shape regular grids.
The cells are constructed via a set of polynomial transformations $\Set{\ensuremath{\Pi}_\ensuremath{T}}_{\ensuremath{T} \in \ensuremath{{\cal T}_{h}}}$ of a unit reference cell $\hat \ensuremath{T}$, see also Remark~\ref{rem.transformation}. The diameters $h_\ensuremath{T}$ of the cells define a mesh parameter $h$ by the piecewise constant function $h_{|\ensuremath{T}} = h_\ensuremath{T}$.
On the grid we define for $s \in \mathbb N$
the finite dimensional space
\begin{align}
\mathcal{S}_h^s(\Omega^\ep):=\Set{v_h\in C^0(\overline {\Omega^\ep})\ \big| \ {v_h}_{|\ensuremath{T}} \in Q^s(\ensuremath{T}), \ensuremath{T} \in \ensuremath{{\cal T}_{h}} },
\end{align}
where $C^0(\overline \Omega^\ep)$ is the space of continuous functions on $\overline {\Omega^\ep}$. Let $P^s(\hat \ensuremath{T})$ be space of polynomials of order lower or equal to $s$, then the space
\begin{align}
Q^s(\ensuremath{T})= \Set{p: T \to \mathbb{R}\ |\ p\left(\ensuremath{\Pi}_\ensuremath{T}(\cdot)\right) \in P^s(\hat \ensuremath{T})}
\end{align}is the space of functions obtained by a transformation of bilinear ($s=1$), biquadratic ($s=2$) and in general higher order polynomials defined on the unit reference cell $\hat \ensuremath{T}$.
For convergence results with respect to $h$ we consider a family of grids obtained by either uniform or local refinement of an initial regular grid.
\begin{remark}[Boundary Approximation]\label{rem.transformation}
Since the considered domains have curved boundaries we correspondingly use cells with curved boundaries (i.e. isoparametric finite elements) to get a better approximation.
Considering the space $\mathcal{S}_h^2$ for the velocity we use biquadratic transformations of the unit cell $\hat \ensuremath{T}$.
\end{remark}
For the discretization of the Stokes system we use the Taylor-Hood element that uses the ansatz space $V_h(\Omega^\ep):=\left(\mathcal{S}^2_h(\Omega^\ep)\right)^2 \cap V(\Omega^\ep)$ for the velocity and $L_h(\Omega^\ep):=\mathcal{S}_h^1(\Omega^\ep)$ for the pressure.
This discretization is inf-sup stable (cf. \cite{BrezzF:1991}), so we do not need stabilization terms to solve the saddle point corresponding to the Stokes system, as for example in \cite*{JaegerMN:2001}.
The \textbf{finite element approximation} of the microscopic problem is obtained by replacing the (infinite dimensional) function spaces $V(\Omega^\ep)$ and $L_0(\Omega^\ep)$ by their discretized counterparts $V_h(\Omega^\ep)$ resp. $L_h(\Omega^\ep)$.
\begin{problem}[Finite Element Approximation of Microscopic Problem]\label{prob.micro_fe}
Find a pair $(\mathbf{v}^\ep_h,p^\varepsilon_h) \in V_h(\Omega^\epsilon)\times L_h(\Omega^\ep)$, such that
for all
$(\boldsymbol\varphi_h,\psi_h ) \in V_h(\Omega^\ep) \times L_{0,h}(\Omega^\ep)$
\begin{align}
\int_{\Omega^\ep} \big(\nabla \mathbf{v}^\ep_h + (\nabla \mathbf{v}^\ep_h)^T \big) \cdot \nabla \boldsymbol\varphi_h \ dx + \int_{\Omega^\ep} p^\varepsilon_h \nabla \cdot \boldsymbol\varphi_h \ dx &= \int_{\Omega^\ep} \mathbf{f} \cdot \boldsymbol\varphi_h \ dx ,\\
\int_{\Omega^\ep} \nabla \cdot \mathbf{v}^\ep_h \ \psi_h \ dx &= 0
\end{align}
and $\int_{\Omega^\ep} p^\varepsilon_h\ dx = 0$.
\end{problem}
As shown in Section~\ref{problem_setting}, see also \cite{JaegerM:1996,JaegerM:2000,JaegerM:2009}, the Navier boundary layer problem (\ref{BJ4.2})-(\ref{4.6}) has to be solved to determine the constants \ensuremath{C_1^{{bl}}}{} and \ensuremath{C_\omega^{{bl}}}{} in the interface law \eqref{4.95} and the first of \eqref{Presspm2A}.
Next subsection is thus dedicated to the numerical determination of these constants and we will show in sections \ref{sec:case_I} and \ref{sec:case_II} by direct numerical solving of the microscopic problem that the constant $\ensuremath{C_\omega^{{bl}}}$ is related to the pressure difference between the free fluid and the porous part.
As previously explained we use two different kinds of inclusion in the porous part, \textbf{circles} and \textbf{ellipses}.
The geometries of the unit cells $Y = (0.1)^2$, see figure~\ref{fig.inclusions}, for these two cases are as follows:
\begin{enumerate}
\item the solid part of the cell $Y_s$ is formed by a circle with radius $0.25$ and center $(0.5, 0.5)$.
\item $Y_s$ consists of an ellipse with center $(0.5, 0.5)$ and semi-axes $a=0.357142857$ and $b=0.192307692$, which are rotated anti-clockwise by $45^\circ$.
\end{enumerate}
\begin{figure}
\centering
\begin{subfigure}[htb]{0.3\textwidth}
\centering
\includegraphics[trim=54mm 80mm 21mm 83mm, clip, width =0.8\textwidth]{mesh_circle.pdf}
\caption{Circle}\label{subfig.circle}
\end{subfigure}
\begin{subfigure}[htb]{0.3\textwidth}
\centering
\includegraphics[trim=54mm 80mm 21mm 83mm, clip,width=0.8\textwidth]{mesh_ellipse.pdf}\caption{Ellipse}\label{subfig.ellipse}
\end{subfigure}
\caption{Mesh of the fluid part of the unit cell for the two types of inclusions: circles (\subref{subfig.circle}) and ellipses (\subref{subfig.ellipse}).}\label{fig.inclusions}
\end{figure}
The circle geometry is a case of axis symmetric geometry with respect to the axis $y$, perpendicular to the interface $\Sigma$, for which we expect from the theory that $\ensuremath{C_\omega^{{bl}}}=0$, see \cite*{JaegerMN:2001}.
All computations are done using the toolkit \texttt{DOpElib} (\cite{GollWW:2012}) based upon the C++-library \texttt{deal.II} (\cite{BangeHK:2007}).
\subsection{Finite element formulation of the Navier boundary layer problem}
\label{sec:Navier boundary problem}
The Navier boundary layer problem (\ref{BJ4.2})-(\ref{4.6}) is defined on $\ensuremath{Z^{bl}}:=Z^+ \cup \Sigma \cup Z^-$, where $\Sigma=(0,1)\times\{0\}$, $Z^+=(0,1)\times(0,+\infty)$ and $Z^-=\cup_{k=1}^\infty(Y_f - \{0,k\})$, with $Y_f$ the fluid part of the pore, see figure~\ref{subfig.unitcell}.
After \cite{JaegerM:1996} and \cite*{JaegerMN:2001}, it is known that $\boldsymbol\beta^{bl}$ converges exponentially towards $(\ensuremath{C_1^{{bl}}},0)$ and $\omega^{bl}$ towards $\ensuremath{C_\omega^{{bl}}}$ in $Z^+$ for increasing $y_2$. On the porous side it has been also shown in the same references that the pressure $\omega^{bl}$ and the velocity $\boldsymbol\beta^{bl}$ converge exponentially towards zero. In addition it has been shown therein that
\begin{subequations}\label{eq.cbl_cblw}
\begin{align}
\ensuremath{C_1^{{bl}}} &= \int_0^1 \beta^{bl}_{1}(y_1,0) \ dy_1,\\
\ensuremath{C_\omega^{{bl}}} &= \int_0^1 \omega^{bl}(y_1,0) \ dy_1= \int_0^1 \omega^{bl}(y_1,a) \ dy_1,&&\forall a\geq 0,\label{equ.cbl_cblw:cblw}
\end{align}
\end{subequations}
where $(\boldsymbol\beta^{bl}, \omega^{bl})$ is the solution of (\ref{BJ4.2})-(\ref{4.6}). Both this integrals are well defined since $\boldsymbol\beta^{bl}$ and $\omega^{bl}$ are smooth in $Z^+$ up to the interface $\Sigma$.
Since we can not deal with infinitely large domains, we consider a cut-off domain for numerical calculations defining the finite slab $\ensuremath{Z^{k}_{l}}:=\ensuremath{Z^{bl}}\cap (0,1)\times(-l,k)$, $ k, l>0$. The distance of the cut-off from the interface, determined by $k$ and $l$, has to be taken large enough taking into account the exponential decay to reduce the approximation error introduced by cutting the domain.
At the newly introduced parts of the boundary, namely $\Gamma_{ k} = (0,1)\times \{k\}$ and $\Gamma_{ l} = (0,1)\times \{-l\}$, we have to set some appropriate boundary conditions. We follow \cite*{JaegerMN:2001} and put zero Dirichlet condition for the two velocity components on $\Gamma_l$, while on $\Gamma_k$ a zero Dirichlet condition for the vertical component as well as zero normal flux of the first velocity component is imposed.
In the following we give the finite element approximation of the cut-off Navier boundary layer problem:
\begin{problem}[Cut-off Navier Boundary Layer]\label{prob.bl}
Find $\boldsymbol\beta^{bl}_h \in \tilde V_h(\ensuremath{Z^{k}_{l}})$ and $\omega^{bl}_h \in L_h(\ensuremath{Z^{k}_{l}})$, such that
\begin{align}
\int_{\ensuremath{Z^{bl}}} \left(\big(\nabla \boldsymbol\beta^{bl}_h + (\nabla \boldsymbol\beta^{bl}_h)^T\big)
\cdot \nabla \boldsymbol\varphi + \omega^{bl}_h \ \nabla \cdot \boldsymbol\varphi
\right)\ dx&= -\int_{\Sigma}\varphi_1\ dx, && \forall \boldsymbol\varphi \in\tilde V_h(\ensuremath{Z^{k}_{l}}),\\
\int_{\ensuremath{Z^{bl}}} \nabla \cdot \boldsymbol\beta^{bl}_h \ \psi\ dx&= 0, && \forall \psi \in L_h(\ensuremath{Z^{k}_{l}})
\end{align}
\end{problem}
where the space for the velocity incorporates the Dirichlet boundary conditions on $\Gamma_k$ and $\Gamma_l$ and is thus defined as follows
\begin{align}
\tilde V_h(\ensuremath{Z^{bl}}) := \{\mathbf{v}_h \in C^0(\ensuremath{Z^{bl}})~|~& {\mathbf v_h}_{|K} \in Q^2(K),K \in {\cal T}_h,\notag \\
&\mathbf v_h = (0,0) {\rm ~on} \cup_{n=1}^l (\partial Y_s - (0,n)),\notag \\
&\mathbf v_h = (0,0) \text{ on } \Gamma_l \text{ and } v_{h,1} =0 \text{ on } \Gamma_k,\notag\\
&\mathbf v_h {\rm ~is~} y_1-\mbox{periodic with period } 1 \}, \label{Space1}
\end{align}
with $\partial Y_s$ the boundary of the inclusions in the pore domain, as shown in figure~\ref{subfig.unitcell}.
With the numerical solution of Problem~\ref{prob.bl} we approximate the
constants \ensuremath{C_1^{{bl}}}{} and \ensuremath{C_\omega^{{bl}}}{} for the considered inclusions. The
approximations $\ensuremath{C_{1,h}^{{bl}}}$ and $\ensuremath{C_{\omega,h}^{{bl}}}$ are calculated by replacing in
\eqref{eq.cbl_cblw} the functions $(\boldsymbol\beta^{bl}, \omega^{bl})$ with their discretized counterparts. As usual, the index $h$ indicates the approximation due to discretization.
We observe that to enhance the numerical approximation of \ensuremath{C_\omega^{{bl}}}{} in \eqref{equ.cbl_cblw:cblw} it is beneficial to calculate the integral of the pressure along a line far enough from the interface. We calculate the integral in \eqref{equ.cbl_cblw:cblw} for $a=1$, i.e. along the line $\Set{y\in \ensuremath{Z^{bl}}\ | \ y_2=1}$.
The approximation of the cut-off problem by finite elements introduces two different sources of error: the \textbf{cut-off error} and the \textbf{discretization error}. In our computations, we set $k=l$ and compute the solution of Problem~\ref{prob.bl} for $1\leq k \leq 5$ on a family of hierarchic adaptively refined meshes.
To obtain the convergence results in Section~\ref{numerical_confirmation} with respect to $\epsilon$, it is important to control the cut-off and discretization errors and balance them to reduce the computational costs.
To balance the two errors we should cut the domain so that the order of the cut-off error equals that of the discretization error. To this aim we need to control the discretization error by a reliable estimation. Since we are interested on the calculation of \ensuremath{C_1^{{bl}}}{} and \ensuremath{C_\omega^{{bl}}}{}, we want to control directly the errors
\begin{align*}
J_1=\ensuremath{C_1^{{bl}}}{} - \ensuremath{C_{1,h}^{{bl}}}, \quad J_\omega=\ensuremath{C_\omega^{{bl}}}{} - \ensuremath{C_{\omega,h}^{{bl}}}.
\end{align*}
To this end, we employ the Dual Weighted Residual (DWR) method from \cite{BeckeR:2001} which gives an estimation of the discretization error with respect to a given functional (i.e. $J_1$ or $J_\omega$) exploiting the solution of a proper adjoint equation.
The DWR method in addition provides local error indicators to control the local mesh refinement.
The triangulation is adaptively refined until the estimated discretization error is smaller than a given tolerance.
The reliability of this estimator has been shown in different applications in the context of flow problems and other problems, see e.g. \cite{Rannacher:99}, \cite{BeckeR:2001}, \cite{BraackR:2006}, \cite{Rannacher:2010}.
Nevertheless, we have performed an additional check to assure that the order of the error is indeed the one estimated.
To check the convergence we do not have the exact solution, but we can rely on the best approximation property of Galerkin approximations on quasi-uniform meshes to perform the following verification.
On a series of uniformly refined grids we compute the approximations $C_{1,h}^{bl,unif}$ and $C_{\omega,h}^{bl,unif}$ and compare them with reference values $C_{1,h}^{bl,ref}$ and $C_{\omega,h}^{bl,ref}$ computed on a (very fine) locally refined mesh. Additionally, we evaluate the error estimator $\eta$ on the uniformly refined grids and compare it with the following approximated errors:
\[C_{1,h}^{bl,ref} - C_{1,h}^{bl,unif}, \quad C_{\omega,h}^{bl,ref} - C_{\omega,h}^{bl,unif}.\]
The results of this test show the expected reliability of the error estimator, since (cf. table~\ref{tab.results_global_mesh_refinement}) the solution on uniform meshes converges towards our reference solution and the error estimator is of the same order as the one given by the reference value. We have used grids with up to around 3.9 millions of degrees of freedom (DoF) for the verification with uniformly refined meshes.
Table~\ref{tab.results_global_mesh_refinement} shows the efficiency of the error estimator, i.e.
\begin{align*}
I_{eff}(\ensuremath{C_1^{{bl}}}) = \frac{\eta(\ensuremath{C_1^{{bl}}})}{C_{1,h}^{bl,ref} - C_{1,h}^{bl,unif}}, \quad I_{eff}(\ensuremath{C_\omega^{{bl}}}) = \frac{\eta(\ensuremath{C_\omega^{{bl}}})}{C_{\omega,h}^{bl,ref} - C_{\omega,h}^{bl,unif}}.
\end{align*}
\begin{table}
\centering
\begin{tabular}{r|ccc|ccc}
\toprule
\# DoF & $C_{1,h}^{bl,ref} - C_{1,h}^{bl,unif}$& $\eta_{\ensuremath{C_1^{{bl}}}}$ & $I_{eff}(\ensuremath{C_1^{{bl}}})$& $C_{\omega,h}^{bl,ref} - C_{\omega,h}^{bl,unif}$& $\eta_{\ensuremath{C_\omega^{{bl}}}}$ & $I_{eff}(\ensuremath{C_\omega^{{bl}}})$\\
\cmidrule(lr){1-7}
1\,096 & -4.52E-04 & -2.54E-03 & 5.61 & 2.99E-02 & -9.03E-03& -0.30 \\
4\,142 & -7.49E-05 & -2.75E-04 & 3.67 & -2.07E-04 & -1.02E-03 & 4.92 \\
16\,066 & -1.10E-05 & -1.87E-05 & 1.71 & 3.66E-06 & -9.70E-05 & -26.50\\
63\,242 & -9.60E-07 & -1.11E-06 & 1.15 & -9.72E-07 & -4.78E-06 & 4.92 \\
250\,906 & -6.83E-08 & -7.60E-08 & 1.11 & -9.67E-08 & -3.15E-07 & 3.26 \\
999\,482 & -4.54E-09 & -5.36E-09 & 1.18 & -9.03E-09 & -2.77E-08 & 3.07 \\
3\,989\,626 & -2.90E-10 & -3.82E-10 & 1.32 & -1.28E-09 & -3.67E-09 & 2.85 \\
\bottomrule
\end{tabular}
\caption{Results of the approximation of the constants \ensuremath{C_1^{{bl}}}{} and \ensuremath{C_\omega^{{bl}}}{} by uniform mesh refinement with $k=l=3$ and ellipses as inclusions. The first column gives the number of degrees of freedom (DoF).}\label{tab.results_global_mesh_refinement}
\end{table}
\begin{figure}
\centering
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[trim = 35 150 20 160,clip ,height=9cm]{grid_5crop}
\caption{Grid}
\label{fig.grid:overview}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[trim = 105 140 50 150,clip ,height=9cm]{grid_5crop_zoom}
\caption{Close-up}
\label{fig.grid:zoom}
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[trim = 130 190 420 180,clip ,height=9cm]{grid_v00000}
\caption{$\beta_{h,1}$}
\label{fig.grid:v0}
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[trim = 130 190 420 180,clip ,height=9cm]{grid_v10000}
\caption{$\beta_{h,2}$}
\label{fig.grid:v1}
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[trim = 130 190 420 180,clip ,height=9cm]{grid_pressure0001}
\caption{$\omega_h$}
\label{fig.grid:pressure}
\end{subfigure}
\caption{Example of a locally refined grid for the adaptive computation of \ensuremath{C_\omega^{{bl}}}{} with $k=l=5$ in the Navier boundary layer problem. The whole mesh is shown in (\subref{fig.grid:overview}), whereas (\subref{fig.grid:zoom}) shows a close-up around the interface. In (\subref{fig.grid:v0}), (\subref{fig.grid:v1}) and (\subref{fig.grid:pressure}), the associated solution is shown.}\label{fig.grid}
\end{figure}
\begin{table}
\centering
\begin{tabular}{r|cccc}
\toprule
k,l&\ensuremath{C_{1,h}^{{bl}}}& $\abs{\eta(\ensuremath{C_1^{{bl}}}{})}$ &\ensuremath{C_{\omega,h}^{{bl}}}&$\abs{\eta(\ensuremath{C_\omega^{{bl}}}{})}$\\
\midrule
&\multicolumn{4}{c}{\textbf{circular inclusions}}\\
1 &-0.3038181652339 & 1.9E-12 & -&-\\
2 &-0.3038219423526 & 2.0E-12 & -&-\\
3 &-0.3038219423790 & 2.0E-12 & -&-\\
4 &-0.3038219423789 & 2.0E-12 & -&-\\
5 &-0.3038219423756 & 8.9E-13 & -&-\\
\midrule
&\multicolumn{4}{c}{oval inclusions}\\
1 & -0.2694539064491 & 4.3E-12 & -0.2413211012145 & 2.1E-11\\
2 & -0.2694545953967 & 1.4E-12 & -0.2409146886571 & 7.3E-12\\
3 & -0.2694545953993 & 3.1E-12 & -0.2409148310717 & 7.2E-12\\
4 & -0.2694545953993 & 2.1E-12 & -0.2409148310975 & 8.5E-12\\
5 & -0.2694545953985 & 2.0E-12 & -0.2409148310959 & 8.6E-12 \\
\bottomrule
\end{tabular}
\caption{Results of the approximation of the constants \ensuremath{C_1^{{bl}}}{} and \ensuremath{C_\omega^{{bl}}}{} as well as the estimated discretization error $\eta$ for different domain-lengths. }\label{tab.results_c1_cw}
\end{table}
Calculations of the two constants used in Section~\ref{numerical_confirmation} have been obtained setting the following tolerances $\eta(\ensuremath{C_1^{{bl}}})$, $\eta(\ensuremath{C_\omega^{{bl}}}) < 10^{-11}$, where $\eta(\ensuremath{C_1^{{bl}}})$ and $\eta(\ensuremath{C_\omega^{{bl}}})$ are DWR error estimators respectively of $\ensuremath{C_1^{{bl}}} - C_{1,h}^{bl}$ and $\ensuremath{C_\omega^{{bl}}} - C_{\omega,h}^{bl}$. These tolerances are achieved by locally refined meshes with up to 7 millions of degrees of freedom.
In figure~\ref{fig.grid:overview} an example of a mesh generated by the error estimator for the computation of \ensuremath{C_{\omega,h}^{{bl}}}{} with $k=l=5$ is shown.
A strong refinement can be observed in the neighborhood of the line $\Set{y\in\ensuremath{Z^{bl}}\ | \ y_2=1}$, where $\omega_h$ is evaluated to compute \ensuremath{C_{\omega,h}^{{bl}}}{}, as well as in the vicinity of the first inclusion, see also figure~\ref{fig.grid:zoom} for a close-up of this region.
In this part of the domain, the associated solution has large gradients, see figures~\ref{fig.grid:v0}, \ref{fig.grid:v1} and \ref{fig.grid:pressure} for an illustration of the solution components.
\begin{figure}
\centering
\resizebox{0.5\textwidth}{!}{\input{exp_convergence.tex}}
\caption{Difference between the computed constants on domains with increasing length and $\ensuremath{C_{1,h}^{{bl}}}(Z_5^5)$ resp. $\ensuremath{C_{\omega,h}^{{bl}}}(Z_5^5)$.}\label{fig.exp_convergence}
\end{figure}
In \cite*{JaegerMN:2001} it is shown that the cut-off error decays exponentially with $k$ and $l$.
To find the optimal cut-off level $l, k$ we perform a convergence check taking as reference value the constants computed on $Z_5^5$, i.e. $\ensuremath{C_{1,h}^{{bl}}}(Z_5^5)$ resp. $\ensuremath{C_{\omega,h}^{{bl}}}(Z_5^5)$.
Figure~\ref{fig.exp_convergence} shows the error between the constants computed on $\ensuremath{Z^{k}_{l}}$ with $k=l =1,\dots, 4$ and the reference values computed on $Z_5^5$.
The exponential decay of the cut-off error with the distance from interface can be observed for both approximations. Furthermore, it can be observed that the error $|\ensuremath{C_{1,h}^{{bl}}}(Z_k^l) - \ensuremath{C_{1,h}^{{bl}}}(Z_5^5)|$ is of the order of the discretization error, i.e. $10^{-12}$, for $k,l\geq3$ for \ensuremath{C_1^{{bl}}}{} and $k,l\geq 4$ for \ensuremath{C_\omega^{{bl}}}{}.
In the following, we use approximations computed with local mesh refinement and $k,l=5$ maintaining the simplified notation \ensuremath{C_{1,h}^{{bl}}}{} and \ensuremath{C_{\omega,h}^{{bl}}}{}.
For details on the used local refinement strategy see \cite{Richter:Diss}.
The calculated constants and the respective error estimation are listed in table~\ref{tab.results_c1_cw}.
\subsection{Cell problem and determination of the permeability}
\label{subsec.cell problem}
For a numerical confirmation of
\eqref{ConcPression} we need the solution of appropriate cell problems, depending on the shape of the inclusions, to calculate the rescaled permeability $K$.
To introduce the weak form of the cell problems, we define the following function space
\begin{align}
\hat V(Y_f) &:= \Set{\mathbf{v} \in H^1(Y_f)^2\ | \ \mathbf{v} =0 \; \hbox{on } \ \p Y_s, \mathbf{v} \hbox{ is } 1\text{-periodic}}.
\end{align}
Following the derivation of Darcy's law by homogenization, the matrix $K$ is defined as
\begin{align}
K_{ij} = \int_{Y_f} w_j^i\ dy,\quad i,j=1,2,
\end{align}
where $\mathbf w$ is the velocity of the following \textbf{cell problem}.
\begin{problem}[Cell Problem]\label{prob.cell_problem}
Let $i,j=1,2$. Find a velocity field $\mathbf w^i \in \hat V(Y_f)^2$ and a pressure $\pi^i \in L_0(Y_f)$, such that,
\begin{align}
\int_{Y_f} \left(\big(\nabla \mathbf w^{i}+(\nabla \mathbf w^{i})^T \big)\cdot \nabla \boldsymbol\varphi + \pi^{i} \ \nabla \cdot \boldsymbol\varphi\right)\ dx&= \int_{Y_f}\varphi_i\ dx, && \forall \boldsymbol\varphi \in \hat V(Y_f),\\
\int_{Y_f} \nabla \cdot \mathbf w^i \ \psi\ dx&= 0, && \forall \psi \in L_0(\ensuremath{Z^{bl}})
\end{align}
\end{problem}
The cell problem is solved with Taylor-Hood elements and an adaptive algorithm based on the DWR method to compute precisely the reference values for the permeability matrix $K$, see also Subsection~\ref{sec:Navier boundary problem}. Each component $w^i$ is solved by a tailored grid refinement considering as goal functional for the a posteriori error estimation the components of $K$.
The computed reference values for the circles are
\begin{align} K^{circ}_h &= k^{circ}_h Id \approx {0.01990143534975}Id
\end{align}
with an estimated discretization error of $1.38 \ 10^{-11}$. For the case with ellipses as inclusion the following values have been calculated
\begin{align}
K^{oval}_h &= \left(
\begin{array}{cc}
K_{h,11} & K_{h,12} \\
K_{h,12} & K_{h,22} \\
\end{array}
\right) \approx \left(
\begin{array}{cc}
0.0159787174788 & 0.00303449804138 \\
0.00303449804138 & 0.0159787174788
\end{array}\right).
\end{align}
The estimated discretization errors are $2.76 \ 10^{-12}$ for $K_{h,11}$ and $1.10 \ 10^{-13}$ for $K_{h,12}$.
In the next session we use the reference values of \ensuremath{C_{1,h}^{{bl}}}{}, \ensuremath{C_{\omega,h}^{{bl}}}{} and $K_h$ to present a numerical confirmation of the interface law.
\section{}
In this study the two conditions \eqref{4.95} and the first of \eqref{Presspm2A} are numerically confirmed by a direct simulation.
To numerically verify all theoretical results we have to solve the following problems: the microscopic problem (\ref{1.3}--\ref{1.5}), the effective flow (\ref{4.91}--\ref{4.95}), the Darcy's law (\ref{PresspmA}--\ref{Presspm2A}), the boundary layer problem (\ref{BJ4.2}--\ref{4.6}) and some appropriate cell problems to compute the rescaled permeability $K$.
All these problems have to be solved for two different kinds of inclusions.
The solution of the microscopic problem has also to be computed for
different boundary conditions, particularly a periodic configuration and a
flow with an injection boundary condition, see subsections \ref{sec:case_I} and \ref{sec:case_II}.
Particular attention has to be given to the calculation of the constants \ensuremath{C_1^{{bl}}}{} and \ensuremath{C_\omega^{{bl}}}{} used in the interface condition (see Section~\ref{Interface condition}), since we are going to show converge results with $\epsilon\to0$ in Section~\ref{numerical_confirmation} and the precision of several quantities reaches quickly the discretization error.
For this reason we adopt a goal oriented adaptive scheme for the grid refinement that allows reducing the computational costs to obtain a precise evaluation of a given functional, in particular to compute the two constants.
\subsection{Microscopic equations}\label{Microgeoeq}
The geometry of the problem is given in figure~\ref{TwoDomains} and, more precisely, the periodic structure is defined as follows.
The porous part has a periodic structure and it corresponds to a repetition
of so-called cells of the characteristic size $\varepsilon$. Each cell
is made from the unit cell $Y=(0,1)^2$, rescaled by $\varepsilon$. The unit cell contains a
pore part $Y_f$ and a solid part $Y_s$ with $Y_s \subsetneq Y$ (see
figure~\ref{subfig.unitcell}). The union of all pores gives the fluid part
$\Omega_p^\varepsilon$ of porous domain $\Omega_p$. $\Gamma=(0,1)\times \{0\}$
describes an interface between the unconfined domain and the porous domain.
The flow takes place in $\Omega^\varepsilon=\Omega_p^\varepsilon \cup \Omega_f \cup
\Gamma$
\begin{figure}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\raisebox{1.2cm}{\resizebox{0.7\textwidth}{!}{\input{UnitCell}}}
\caption{Unit cell}
\label{subfig.unitcell}
\end{subfigure}
\begin{subfigure}[b]{0.6\textwidth}
\centering
\resizebox{0.8\textwidth}{!}{\input{Geometry_per}}
\caption{Flow region}
\label{TwoDomains}
\end{subfigure}
\caption{The model geometry}
\end{figure}
and it is described by the following non-dimensional steady Stokes system
in $\Omega^\varepsilon $:
\begin{gather} -
\upDelta \mathbf{v}^\varepsilon + \nabla p^{\varepsilon} = \mathbf{f} \qquad \hbox{ in } \quad \Omega^\ep
\label{1.3} \\ {\rm div} \, \mathbf{v}^\ep = 0 \qquad \hbox{ in } \quad \Omega^\ep , \qquad \int_{\Omega_f} p^\varepsilon \ dx =0,
\label{1.4} \\ \mathbf{v}^\ep =0 \; \hbox{on } \ \p \Omega^\ep \setminus \bigg( \{ x_1 = 0 \} \cup \{ x_1 = L \} \bigg)
, \qquad \{ \mathbf{v}^\ep , p^\varepsilon \} \ \hbox{ is }
L-\hbox{periodic in } \; x_1 . \label{1.5} \end{gather}
Here the non-dimensional $\mathbf{f}$ stands for the effects of external forces or an
injection at the boundary or a given pressure drop, and it corresponds to the physical forcing term
multiplied by the ratio between Reynolds' number and Froude's number squared. Specifically, if the force $\mathbf f$ is
non-constant, it corresponds to a non-constant pressure drop or to a
non-parabolic injection profile.
$\mathbf{v}^\ep $ denotes the non-dimensional velocity and $p^\varepsilon$ is the non-dimensional
pressure.
\subsection{Two-scale expansion}
The idea behind the two-scale expansion is the following: without forcing
infiltration, in the interior of the porous medium, the permeability is
$k= O(\varepsilon^2)$ and Darcy's velocity is small. Consequently, the flow is tangent to the interface $\Gamma$. The leading order approximation of system (\ref{1.3})-(\ref{1.5}) is the Stokes flow in $\Omega_f$, with no-slip condition on $\Gamma$. It is modeled by the system,
\begin{gather}
- \upDelta \mathbf{v}^0 + \nabla p^0 = \mathbf{f}
\qquad \hbox{ in } \Omega_f ,\label{4.37}\\
{\rm div} \ \mathbf{v}^0 = 0 \qquad \hbox{ in } \Omega_f , \qquad \int_{\Omega_f} p^0 \ dx =0, \label{4.38}\\
\mathbf{v}^0 = 0 \quad \hbox{ on } \p \Omega_f \setminus \bigg( \{ x_1 = 0 \} \cup \{ x_1 =L \} \bigg)
, \quad
\{ \mathbf{v}^0 , p^0 \} \quad \hbox{ is } \;
L\hbox{-periodic in } \; x_1. \label{4.40}
\end{gather}
Following the two-scale expansions from \cite{EneS:1975}, the behavior of the velocity and pressure fields in $\Omega_p$, far from the outer boundaries, is expected to be given by the following system of equations,
\begin{gather}
\mathbf{v}^\varepsilon (x) = \varepsilon^2 \sum_{j=1}^2 \mathbf{w}^j (\frac{x}{\varepsilon}) ( f_j (x) -
\frac{\partial p_D (x)}{ \partial x_j } ) + O(\varepsilon^3 )\quad x\in \Omega_p , \label{1.62} \\
p^\varepsilon (x) = p_D (x) + \varepsilon \sum_{j=1}^2 \pi^j (\frac{x}{\varepsilon}) ( f_j (x) -
\frac{\partial p_D (x)}{ \partial x_j } ) + O(\varepsilon^2) \quad x\in \Omega_p ,\label{1.63} \\
\mathbf{v}^D (x) = K
(\mathbf{f} (x) - \nabla_x p_D (x) ) \, (\hbox{\bf Darcy's law}) ,
\label{1.64} \\
\mbox{div } \mathbf{v}^D =0 \quad \mbox{ on } \; \Omega_p, \label{1.64A}\end{gather}
where $\{ \mathbf{w}^j , \pi^j \}$ are calculated using cell problems and $K$ consists of the volume averages of $\mathbf{w}^j$, $j=1,2$, see Subsection~\ref{subsec.cell problem}. Note that the dimensionless permeability is $\varepsilon^2 K$ and it is a symmetric positive definite matrix. See e.g. \cite{JaegerM:2009} or \cite{Allaire:1997} for more details.
System (\ref{1.64})-(\ref{1.64A}) describes the effective pressure $p_D$. However, we do not know the boundary condition for $p_D$ (respectively $\mathbf{v}^D$) on the interface $\Gamma$, and $\{\mathbf{v}^D, p_D\} $ are not determined.
We need \textbf{interface conditions} coupling problem (\ref{4.37})-(\ref{4.40}) with (\ref{1.64})-(\ref{1.64A}).
Natural approach to find interface conditions is by using matched asymptotic expansions (MMAE). The method is used in a number of situations arising in mechanics.
For a detailed presentation of the MMAE method we refer to the book \cite{Zeytounian:2002} and to references therein.
In the language of the MMAE, expansions in $\Omega_f$ and $\Omega_p$ are called the outer expansions. The boundary and/or interface behavior is captured by an inner expansion. In the inner expansion the independent variable is stretched out in order to describe the behavior in the neighborhood of the boundary and/or interface.
The MMAE approach matches the two expansions. In the singular perturbation
problems involving boundaries, only the function values at the boundary are
matched and the approach works well. When interfaces are involved, it is
needed to match additionally the values of the normal derivatives.
This difficulty is not easy to circumvent because imposing matching of the values of the function and its normal derivative leads to an ill posed problem for the second order equation.
This difficulty can not been easily avoided, since the simultaneous imposition of matching conditions for the values and
for the normal
derivative of the function leads to an ill posed problem for the second
order equation.
Here, Darcy's velocity in $\Omega_p$ is of order $O(\varepsilon^2)$. Therefore at the lowest order MMAE confirms the boundary condition (\ref{4.40}) on $\Gamma$, i.e. $\mathbf{v}^0 =0$.
Additional physical matching conditions would be the continuity of the
contact forces, which is not
assured by MMAE. Therefore, it is not clear if we are allowed to match
the values of the pressure on $\Gamma$.
The absence of a matching condition in the velocity gradient and in the pressure leads to a jump of the contact force.
The difficulty was solved using a boundary layer correction in \cite{JaegerM:2000}, \cite*{JaegerMN:2001} and \cite{MarciM:2012}. The applied strategy is the following:
We handle the pressure jump by adjusting the porous medium pressure and
the shear stress jump using a particular {properly
derived} boundary layer. In fact, the shear stress jump influences the pressure values as well.
At the interface $\Gamma$ we have the shear stress jump equal to $\displaystyle - {\partial
v_1^{0}}/{\partial x_2}|_{\Gamma}$. The natural stretching
variable is given by the geometry and it coincides with the fast variable $\displaystyle
y={x}/{\varepsilon}$. The correction $\{ \mathbf{w} , p_w \}$ to the zero order approximation satisfies again the Stokes system
\begin{gather} \bigl[ \mathbf{w} \bigr] (y_1 , 0)= \mathbf{w} (y_1 , 0+) - \mathbf{w} (y_1 , 0-) =0, \quad [p_w ] (y_1 , 0) =0\notag \\
\mbox{ and } \quad \bigr[ \frac{\partial w_1}{\partial y_2}
\bigl] (y_1 , 0) = \frac{\partial v_1^{0}}{\partial x_2} (x_1 ,0)
|_{\Gamma} \quad \hbox{ on } \quad
{\frac{\Gamma}{\varepsilon}} .\label{BJ5.4)}
\end{gather}
Using periodicity of the geometry and independence of $\displaystyle
\frac{\partial v_1^{0}}{\partial x_2} |_{\Gamma}$ of the fast variable $y$, we obtain
\begin{equation}\label{BJ5.6}
\mathbf{w} (y) = \frac{\partial
v_1^{0}}{\partial x_2} |_{\Gamma} \boldsymbol\beta^{bl} (y) \quad \mbox{ and }
\quad p_w (y) = \frac{\partial v_1^{0}}{\partial x_2} |_{\Gamma}
\omega^{bl} (y),
\end{equation}
where $\{ \boldsymbol\beta^{bl} , \omega^{bl} \}$ is calculated in a semi-porous column $Z_{BL} = Z^+ \cup \Sigma \cup Z^- $, with $\Sigma=(0,1)\times \{ 0\} $, $Z^+ = (0,1) \times (0, +\infty )$,
and the semi-infinite porous slab is $Z^- =
\cup_{k=1}^\infty ( Y_f -\{ 0,k \} )$. See figure~\ref{fig.boundary_layer_domain} for more details.
\begin{figure}
\center
\resizebox{0.2\textwidth}{!}{\input{GeometryBoundaryLayer}}
\caption{Domain of the Navier boundary layer problem.}
\label{fig.boundary_layer_domain}
\end{figure}
If $D_y$ denotes the symmetrized gradient, then $\{ \boldsymbol\beta^{bl} , \omega^{bl} \}$ is given by
\begin{gather}
-\upDelta _y \boldsymbol\beta^{bl} +\nabla_y \omega ^{bl} =0\qquad \hbox{ in
} Z^+ \cup Z^-, \label{BJ4.2}\\ {\rm div}_y \boldsymbol\beta^{bl} =0\qquad \hbox{ in }
Z^+ \cup Z^-, \label{4.3} \\ \bigl[ \boldsymbol\beta^{bl} \bigr]_\Sigma (\cdot , 0)= 0
\quad \mbox{ and } \quad \bigr[ \{ 2 D_y (\boldsymbol\beta^{bl} )
-\omega^{bl} I \} \mathbf{e}^2 \bigl]_\Sigma (\cdot , 0) = \mathbf{e}^1 \quad \hbox{ on }
\Sigma \label{4.5)}, \\\boldsymbol \beta^{bl} =0 \quad \hbox{ on }
\displaystyle{\bigcup_{k=1}^{\infty} ( \p Y_s} -\{ 0,k \} ), \qquad \{
\boldsymbol\beta^{bl} , \omega^{bl} \} \quad \hbox{ is } 1\hbox{-periodic in }
y_1. \label{4.6}
\end{gather}
The problem (\ref{BJ4.2})-(\ref{4.6}) was studied in \cite{JaegerM:1996} and it was proved that
\begin{itemize}
\item Gradients of $\{ \boldsymbol\beta^{bl} , \omega^{bl} \}$ stabilize exponentially fast to $0$.
\item $ \boldsymbol\beta^{bl} $ stabilizes exponentially fast to $C_1^{bl} \mathbf{e}^1$, when $y_2 \to +\infty$ and to zero when $y_2 \to -\infty$. $C_1^{bl} $ is strictly negative.
\item $\omega^{bl} $ stabilizes exponentially fast to $C_\omega^{bl} \mathbf{e}^1$, when $y_2 \to +\infty$ and to $C^{bl}_{0}$ when $y_2 \to -\infty$. Since we have liberty in adding a constant to the pressure, we choose $C^{bl}_{0}=0$.
\end{itemize}
In fact, it is the absence of stabilization of the boundary layer velocity $\boldsymbol\beta^{bl}$ to zero which yields a slip.
In addition, $\mathbf{w}$ can't be a correction
because of the stabilization of $\boldsymbol\beta^{bl }$
towards a nonzero constant velocity $ C^{bl}_1 \mathbf{e}^1 $. It creates a counterflow
at the upper boundary of $\Omega_f$, given by the following
Stokes system in $\Omega_f$:
\begin{gather}
- \upDelta \mathbf{z}^\sigma + \nabla p^\sigma = 0
\qquad \hbox{ in } \Omega_f ,\label{4.37Couette}\\
{\rm div} \ \mathbf{z}^\sigma = 0 \qquad \hbox{ in } \Omega_f ,\label{4.38Couette}\\
\mathbf{z}^\sigma = 0 \quad \hbox{ on } \{ x_2 = 1 \} \quad \mbox{and} \; \mathbf{z}^\sigma =\frac{\p v^0_1 }{ \p x_2 }
|_\Gamma \mathbf{e}^1 \quad \hbox{ on } \{ x_2 = 0 \}
,\label{4.39Couette}\\ \{ \mathbf{z}^\sigma , p^\sigma \} \qquad \hbox{ is } \;
1\hbox{-periodic in } \; x_1 .\label{4.40Couette}
\end{gather}
Now we are in the situation to propose the two-scale expansion for the velocity:
\begin{gather}
\mathbf{v}^\ep = \underbrace{\mathbf{v}^0 - \varepsilon C^{bl}_1 \mathbf{z}^\sigma
}_{\mbox{the outer expansion}} \underbrace{- \varepsilon (\boldsymbol\beta^{ bl} (\frac{x}{ \varepsilon}) - C^{bl}_1 \mathbf{e}^1 ) \frac{\p v^0_1 }{ \p x_2 }
|_\Gamma}_{\mbox{the inner expansion}} +\dots \quad \mbox{in } \; \Omega_f
\label{4.66} \\
\mathbf{v}^\ep = \underbrace{O(\varepsilon^2)}_{\mbox{the outer expansion}} \underbrace{-
\varepsilon \boldsymbol\beta^{ bl } (\frac{x}{ \varepsilon}) \frac{\p v^0_1 }{ \p x_2 }
|_\Gamma}_{\mbox{the inner expansion}} +\dots \quad \mbox{in } \; \Omega_p
\label{4.66A}
\end{gather}
For the two-scale expansions (\ref{4.66})-(\ref{4.66A}) the values at the
interface $\Gamma$ are matched exactly and the shear stresses are matched
{with an approximation of order
$O(\epsilon)$}. At the flat interface $\Gamma$, with no slip condition for $\mathbf{v}^0$ and interface continuity of the boundary layer velocity, continuity of the normal component of the normal stress (i.e. of the normal contact force) reduces to the pressure continuity.
We need the two-scale expansion for the pressure. Stabilization of the boundary layer pressure to $C^{bl}_\omega $,
when $y_2 \to +\infty$, influences strongly the pressure approximation. It reads
\begin{gather}
p^\varepsilon =\underbrace{p^0 {-}\> \varepsilon C^{bl}_1 p^\sigma}_{\mbox{the outer expansion}} \underbrace{ - \bigl( {\omega}^{ bl} (\frac{x}{ \varepsilon}) -
C^{bl}_\omega \bigr) \frac{\p v^0_1 }{ \p x_2 } |_\Gamma}_{\mbox{the inner expansion}} +\dots \quad \mbox{in } \; \Omega_f , \label{BJ47} \\
p^\varepsilon =\underbrace{p_D + \varepsilon \sum_{j=1}^2 \pi^j (\frac{x}{\varepsilon}) ( f_j (x) -
\frac{\partial p_D (x)}{ \partial x_j } ) }_{\mbox{the outer expansion}} \underbrace{ - {\omega}^{ bl} (\frac{x}{ \varepsilon}) \frac{\p v^0_1 }{ \p x_2 } |_\Gamma}_{\mbox{the inner expansion}} +\dots \quad \mbox{in } \; \Omega_p . \label{BJ47A}\notag
\end{gather}
Two-scale expansions (\ref{BJ47})-(\ref{BJ47A}) match at the interface $\Gamma$ at order $O(\varepsilon)$ if and only if
\begin{equation}\label{BJ46}
p^0 (x_1, +0) - {p}_D (x_1, -0) =- C^{bl}_\omega
\frac{\p v^0_1 }{ \p x_2 } |_\Gamma \quad \mbox{ for }
\quad x_1 \in (0,1).
\end{equation}
The conditions (\ref{BJ46}) allows to calculate Darcy's pressure $p_D$. It satisfies the equations (\ref{1.64})-(\ref{1.64A}), the condition (\ref{BJ46}), $v^D_2 =0$ on $\{ x_2 =-1 \}$ and $p_D$ is $1$-periodic in $x_1$.
{The results from \cite{MarciM:2012} (and from \cite{JaegerM:2000} in the case of Poiseuille's flow) yield in $\Omega_f \cup \Omega_p$
\begin{align*}
\mathbf{v}^\ep - \mathbf{v}^0 +\varepsilon \left( \boldsymbol\beta^{ bl}(\frac{ x}{ \varepsilon}) - C^{bl}_1 \mathbf{e}^1 H(x_2)\right) \frac{\p v^0_1
}{ \p x_2 } |_\Gamma +\varepsilon C^{bl}_1 \mathbf{z}^\sigma H(x_2 ) &=O (\varepsilon^2), \\
p^\varepsilon - p^0 H(x_2) - { p}_D H(-x_2 ) +\bigl( {\omega}^{ bl , \varepsilon} (x)
{-}\> H(x_2)C^{bl}_\omega \bigr) \frac{\p v^0_1 }{ \p x_2 } |_\Gamma +
\varepsilon C^{bl}_1 p^\sigma H(x_2)
&=O (\varepsilon ), \\
\nabla \mathbf{v}^\ep - \nabla \mathbf{v}^0 +\varepsilon \nabla \bigg( ( \boldsymbol\beta^{ bl}(\frac{ x}{ \varepsilon}) - C^{bl}_1 \mathbf{e}^1 H(x_2)) \frac{\p v^0_1
}{ \p x_2 } |_\Gamma + C^{bl}_1 \mathbf{z}^\sigma H(x_2 ) \bigg)&=O (\varepsilon),
\end{align*}
where $H(t)$ is Heaviside's function, equal to one for $t>0$ and to zero for $t<0$. Furthermore, on $\Gamma$ it holds
\begin{align}
\mathbf{v}^\ep +\varepsilon \boldsymbol\beta^{ bl}(\frac{ x}{ \varepsilon}) \frac{\p v^0_1
}{ \p x_2 } |_\Gamma =O (\varepsilon^{3/2}).\label{equ.bjs epsilon32}
\end{align}}
\subsection{Interface condition}
\label{Interface condition}
The above results allow to find the effective interface conditions. Following \cite*{JaegerM:2000, JaegerM:2009, MarciM:2012}, we compare on the interface\/ $\Gamma$ the shear stress and the tangential velocity:
\[
\frac{\p v^\varepsilon _1}{ \p x_2 } |_\Gamma = \frac{\p v^0 _1}{ \p x_2 }
|_\Gamma - \frac{\p \beta^{bl}_1 }{ \p y_2} |_{\Gamma , y=x/\varepsilon} +
O(\varepsilon) \quad \hbox{ and\/ } \quad \frac{v^\varepsilon _1}{ \varepsilon } = -
\beta^{bl}_1 ( x_1 / \varepsilon , 0) \frac{\p v^0 _1}{ \p x_2 } |_\Gamma +
O(\varepsilon) .
\]
After averaging over\/ $\Gamma$ with respect to $y_1$, we obtain the
Saffman version of the law by
Beavers and Joseph (\ref{AUX}), with
$$ \alpha = -\frac{\sqrt{k}}{\varepsilon C_1^{bl}}, \quad k=O(\varepsilon^2 ), \; C^{bl}_1 <0.$$
Therefore the effective flow in $\Omega_f$ is given by the following problem.
\begin{problem}[Effective flow]
Find a velocity field
$u^{\mathit{eff}}$ and a pressure field $p^{\mathit{eff}}$ such that
\begin{gather}
- \upDelta \mathbf{u}^{\mathit{eff}} + \nabla p^{\mathit{eff}} = \mathbf{f} \qquad \hbox{ in } \Omega_f ,\label{4.91}\\
{\rm div} \ \mathbf{u}^{\mathit{eff}} = 0 \qquad \hbox{ in } \Omega_f , \qquad \int_{\Omega_f} p^{\mathit{eff}} \ dx =0,\label{4.92}\\ \mathbf{u}^{\mathit{eff}} =
0 \qquad \hbox{ on } (0,L) \times \{1\}
; \quad \mathbf{u}^{\mathit{eff}} \; \hbox{ and } \; p^{\mathit{eff}} \quad \hbox{ are } \;
L\hbox{-periodic in} \quad x_1, \label{4.94}\\ u^{\mathit{eff}}_2 = 0 \quad
\hbox{ and } \quad u^{\mathit{eff}}_1 + \varepsilon C^{bl}_1 \frac{\p u^{\mathit{eff}}_1 }{ \p
x_2 } =0 \quad \hbox{ on } \quad \Gamma . \label{4.95}
\end{gather}
\end{problem}
After \cite{MarciM:2012}, Theorem 2, we have
\begin{align}
\int_{\Omega_f} | \mathbf{v}^\ep - \mathbf{u}^{\mathit{eff}} |^2 \ dx + \vert M^\varepsilon - M^{\mathit{eff}} \vert^2 &= O (
\varepsilon^{3}), \label{4.100A} \\
\int_{\Omega_f} \{ | p^\varepsilon - p^{\mathit{eff}} |
+ | \nabla (\mathbf{v}^\ep - \mathbf{u}^{\mathit{eff}} ) |\} \ dx &= O (\varepsilon),
\label{4.100}
\end{align}
where $\displaystyle M^{\mathit{eff}} = \int_{\Omega_f} u_1^{\mathit{eff}} \ dx $ is the mass flow.
The estimates (\ref{4.100A})-(\ref{4.100}), obtained analytically in \cite{MarciM:2012}, will be verified by a direct numerical simulation in Section~\ref{numerical_confirmation}.
Next we recall that $p_D$ is given by
\begin{gather}
\mbox{\rm div } \bigg( K (\mathbf{f} (x) - \nabla p_D )\bigg) =0\; \mbox{ in\/ } \; \Omega_p, \label{PresspmA} \\
{p}_D = p^{\mathit{eff}} + C^{bl}_\omega \frac{\p u^{\mathit{eff}}_1}{ \p x_2 } (x_1 , 0+ ) \; \mbox{ on\/ } \; \Gamma; \quad
K (\mathbf{f} (x) - \nabla { p}_D ) |_{\{ x_2 =-1 \} } \cdot \mathbf{e}^2 =0, \label{Presspm2A}
\end{gather}
and after \cite{MarciM:2012}, Theorem 3, we have
\begin{gather}
|\int_{\Omega_p} \{ \biggl( \mathbf{v}^\ep + \varepsilon \boldsymbol\beta^{bl} (\frac{x}{ \varepsilon}) \frac{\p u^{\mathit{eff}}_1}{ \p x_2 } (x_1 , 0 ) \biggr) - K (\mathbf{f} - \nabla { p}_D ) \} \varphi \ dx |= o(1), \notag \\
\mbox{ for every smooth } \; \varphi, \quad \mbox{ as } \; \varepsilon \to 0; \label{ConcDarcy}\\
\int_{\Omega_p} | p^\varepsilon - { p}_D |^2 \ dx = o(1), \quad \mbox{ as } \; \varepsilon \to 0; \label{ConcPression} \\
|\int_{\Gamma} ( p^\varepsilon - p^{\mathit{eff}} ) \varphi \ dx_1 | =O( \sqrt{\varepsilon} ) \; \mbox{ for every smooth } \; \varphi, \quad \mbox{ as } \; \varepsilon \to 0.\label{EstPressBdry}
\end{gather}
The estimate (\ref{ConcPression}) will also be verified by a direct numerical simulation in Section~\ref{numerical_confirmation}.
\begin{remark}[Extension of velocity and pressure]
Fluid velocity $\mathbf v^\varepsilon$ is extended zy zero to the solid part $\Omegap \setminus \clos{\Omega_p^\varepsilon}$ of the porous medium $\Omegap$. If $Y^\varepsilon_{f,i}$ is the $i$-th pore, then the pressure field $p^\varepsilon$ is extended to the corresponding solid part $Y^\varepsilon_{s,i}$ by setting
\begin{align}\label{equ.pressure extension}
p^\varepsilon(x) =
\begin{cases}
p^\varepsilon,& x \in \Omega^\varepsilon,\\
\frac{1}{\abs{ Y_{f,i}^\varepsilon}}\int_{ Y_{f,i}^\varepsilon} p^\varepsilon,& x\in Y_{s,i}^\varepsilon,
\end{cases}
\end{align}
where $\abs{ Y_{f,i}^\varepsilon}$ denotes the volume of $ Y_{f,i}^\varepsilon$.
The pressure extension \eqref{equ.pressure extension} is the extension of \cite{Lipton:A1989} and comes out from Tartar's construction, see \cite{Allaire:1997} for more details.
\end{remark}
|
1312.4867
|
\section{Introduction}
Micro- and nano-opto mechanical systems are the heart of refined force-sensing devices \cite{Gavartin2012,Miao2012,Poggio2013,Purdy2013}. Such systems exploit the huge susceptibility around the resonance of oscillators with excellent mechanical quality factor $Q$, combined with high sensitivity interferometric measurements. The latter are particularly efficient when the oscillator is embedded in an optical resonator with high optical quality factor, whose optical path depends on the oscillator coordinate. This kind of devices is useful both for practical applications, and in quantum optics experiments. In both cases, a frequent crucial task is detecting a weak variation of the external force (that we call signal force) on a strong background. For instance, in a quantum experiment, the signal can be due to quantum fluctuations in the radiation pressure, that are usually overwhelmed by background thermal noise (a significant exception is reported in Ref. \cite{Purdy2013}, that presents the first observation of the effect of radiation pressure shot noise on a macroscopic object).
Due to the narrow width of the resonance and, consequently, of the useful sensitive band with respect to typical input force, it is meaningful to discuss the general problem of detecting a weak signal force with flat spectral density (white spectrum) in the presence of a white background force, taking into account a given sensitivity to the oscillator displacement (i.e., a flat readout noise spectrum). This can be performed na$\ddot{\textrm{i}}$vely by measuring the area of the resonance peak emerging from the displacement noise spectrum (or, equivalently, measuring the variance of the oscillator position after band-pass filtering around the resonance). With this estimator, the rate of improvement of the statistical uncertainty for increasing measurement time $t_{meas}$ depends on the correlation time $\tau_c$ of the oscillator motion, with a relative uncertainty scaling as $\sim\sqrt{\tau_c/t_{meas}}$. It seems therefore useful to decrease $\tau_c$, i.e., enhance the damping of the oscillator. However, the fluctuation-dissipation theorem implies that such operation would increase the spectral density of thermal noise. Improved results can instead be achieved by means of a cold damping, e.g. the optical cooling \cite{Kleckner06,Gigan06,Arcizet06}, that modifies the effective susceptibility and decreases the correlation time without introducing additional noise sources. This technique does not increase the signal-to-noise ratio of input excitations, because it changes the response to both signal and background force in the same way. However, as long as the cold damped peak still emerges from the displacement spectral noise, it allows a faster accumulation of statistically independent data bringing therefore, in a given measurement time, to a smaller final uncertainty in the variance of the oscillator motion.
An important remark is that the correlation time of the signal force is by hypothesis very short, therefore the statistics can in principle be much faster than what allowed by the oscillator motion. In other words, the variance of the displacement is not a very efficient indicator, and more refined data analysis can be profitable. In the case of stationary, white input the optimal approach to the measurement is provided by the Wiener-Kolmogorov filtering theory \cite{Kolmogorov,Wiener}. This technique requires the preliminary knowledge of the exact response function to the input force, and of the signal-to-noise ratio. While the second requirement can be relaxed with a sub-optimal but robust filter using a conservative estimate of the sensitivity \cite{Astone1990}, the accurate knowledge of the susceptibility is a crucial request. Such knowledge is not trivial for micro opto-mechanical systems, where the stability of the resonance is affected by several detrimental effects, e.g., thermal phenomena and relaxations of the mechanical oscillator, and above all by the same interaction with the radiation, both due to photothermal effect and to the opto-mechanical coupling. These considerations suggest that the direct measurement of the spectral peak area could be the only applicable strategy in several kinds of opto-mechanical systems, and techniques that reduce the effective coherence time of the oscillator motion, such as cold damping or feedback, represent therefore a way to effectively improve the measurement capabilities of the system \cite{Gavartin2012}. However, it has been remarked that optimal resolution is not really improved in this way \cite{Tamayo2005,Vinante2013}, and that appropriate data filtering can completely replace these hardware techniques even in the case of non-stationary, non-Gaussian input \cite{Harris2013}. In spite of these correct remarks, the problem of the instability in the oscillator parameters and dynamics remains practically difficult to face, and the implementation of optimal analysis requires sophisticated technique of adaptive filtering. The experimental demonstration in Ref. \cite{Harris2013} keeps indeed short ($\sim$ms) measurement times. Therefore, even when willing to apply an efficient data analysis, as well as in several kinds of refined opto-mechanics experiments, stabilization and feedback techniques acting on the opto-mechanical system are crucial, and indeed this issue has been recently considered by few groups\cite{Antonio2012,Gavartin2013}.
In this work we present a micro opto-mechanical system that includes a parametric stabilization of the resonance by controlling the optical spring. We have proposed and demonstrated this technique in a recent work \cite{Pontin2013}, where the control allows to prevent instability in a parametrically modulated opto-mechanical system, thus yielding strong mechanical squeezing. Here we study the characteristics of our system as detector of stochastic force for short measurement times (for quick, high resolution monitoring) as well as for long $t_{meas}$, thus optimizing the sensitivity. We show that, thanks to the stabilization of the effective susceptibility, we can more efficiently implement Wiener filtering and investigate how this strategy improves the performance of our system.
The article is organized as follows. In Section II we describe the theoretical models for the opto-mechanical interaction, the parametric control of the oscillator, and the strategies exploitable to measure the stochastic force acting on the oscillator; in Section II we describe our experimental setup and the measurements; after the Conclusions, in the Appendix we derive the theoretical expressions for the relative uncertainty and discuss the effect of a cutoff in the measured spectra.
\section{Model}
\subsection{Opto-mechanical interaction}
In this section we recall some basic features of the opto-mechanical interaction. We consider an optical cavity where the resonance frequency depends on an effective coordinate $x$, that is kept at its rest position $x=0$ by elastic forces. The system can be sketched as a linear cavity with a rigid oscillating mirror (Fig. \ref{schema}) having position $x$, mass $M$, resonance angular frequency $\wm$, damping rate $\gm$ and susceptibility $\chi = 1/M(\wm^2-\omega^2+\mathrm{i} \omega \gm)$. The radiation pressure provides a force acting on the mirror, that depends on the detuning $\Delta= \omega_L - \omega_c$ between the input radiation at frequency $\omega_L$ and the cavity resonance at $\omega_c$. Since the latter depends on $x$, radiation pressure gives a position-dependent force that can be accounted for by defining an effective susceptibility. Its expression is given by\cite{Arcizet06,Genes2008}
\begin{equation}
\chi _{\mathrm{ eff}}(\omega )^{-1}=M\left[\omega_\mathrm{m}^{2}-\omega^{2}+\mathrm{i}\omega \gamma _\mathrm{m}+\frac{|G|^2\,\Delta\,\omega _\mathrm{m}}{\bigl(\kappa +\mathrm{i}\omega \bigr)^{2}+\Delta^{2}}\right]
\label{chieff}
\end{equation}
where $\kappa$ is cavity decay rate and $|G|^2$ is the opto-mechanical coupling, proportional to the intracavity power.
The real part of $\chi_{\mathrm{eff}}$ can be viewed as a combined effect of the mechanical stiffness and an additional spring (\emph{optical spring})\cite{Braginsky1997}. The delay in the intracavity field build-up, originating a contribution to the imaginary part of $\chi_{\mathrm{eff}}$, causes a change in the oscillator damping that allows the optical cooling of its motion \cite{Kleckner06,Gigan06,Arcizet06}.
For the case of our interest (\emph{bad cavity limit} $\kappa\gg \wm$, small detuning $\Delta\ll
\kappa$, and $\omega\approx \wm$) the expression of optical spring constant simplifies to
\begin{equation}
K_{\mathrm{opt}} \approx -\frac{M |G|^2 \wm}{\kappa^2}\,\Delta
\label{Kopt}
\end{equation}
and the optical damping rate to
\begin{equation}
\gamma_{\mathrm{opt}} \approx \frac{2 K_{\mathrm{opt}}}{M \kappa}
\label{gammaopt}
\end{equation}
allowing to write the effective susceptibility as $\,\chi_{\mathrm{eff}}^{-1}=M\left(\omega_{\mathrm{eff}}^2-\omega^2+\mathrm{i}\omega\,\gamma_{\mathrm{eff}}\right)\,$ with $\,\,\gamma_{\mathrm{eff}}=\gm+\gamma_{\mathrm{opt}}\,\,$ and
\begin{equation}
\omega_{\mathrm{eff}} = \sqrt{\wm^2-K_{\mathrm{opt}}/M}\simeq \wm + \frac{|G|^2}{2\kappa^2}\,\Delta \,.
\label{omegaeff}
\end{equation}
To our purpose, it is useful to underline that the frequency shift is approximately proportional to the detuning, and therefore a laser beam can be used to control it. Moreover, by varying the working point (detuning) we can choose the effective resonance width $\gef$ and stabilize it. On the other hand, we remark that in general the optical spring increases the uncertainty and instability of the opto-mechanical resonance frequency $\wef$ since it is influenced by the noise in the laser intensity (through $G$), in the laser frequency and in the cavity length (through $\Delta$). In addition, thermal effects due to the absorbed laser power can worsen the intrinsic stability of $\wm$.
\subsection{An oscillator with parametric control}
\begin{figure} [h]
\centering
\includegraphics[width=0.9\textwidth]{Schema.eps}
\caption{(Color online) a) Conceptual scheme of the opto-mechanical system including measurement and force terms. b) Experimental measurement of the temporal evolution of the mechanical oscillator in the phase plane, in the configuration with active parametric control. c) Experimental setup. EOM: electro-optic intensity modulator; dash-dotted lines highlight the parametric control.}
\label{schema}
\end{figure}
A conceptual scheme of the experiment is shown in Fig. \ref{schema}a. We consider an opto-mechanical oscillator excited by stochastic signal force $f_s(t)$ and thermal noise force $f_T(t)$ at temperature $T$, with respective spectral densities $S_s$ and $S_T = 2k_B T M \gm$, as well as by a coherent oscillating force of constant amplitude $F_e \cos \we t$. The oscillator position $x(t)$ is measured interferometrically by a first laser beam (\emph{signal beam}). The measurement noise $n(t)$ and the back-action force $f_{BA}(t)$ are considered uncorrelated, with white spectra $S_n$ and $S_{BA}$ bounded by
\begin{equation}
S_n\,S_{BA} \geq \hbar^2/4 \, .
\end{equation}
The evolution of the position $x(t)$ is governed by the stochastic equation
\begin{equation}
\ddot{x} + \gef \dot{x} + \wef^2 x = \frac{1}{M}\left[f_T + f_s + f_{BA} + F_e \cos \we t\right]
\label{eqx}
\end{equation}
and the result of the position measurement is $x_m(t) = x(t) + n(t)$.
The motion of the oscillator can be decomposed into two quadratures $X(t)$ and $Y(t)$ in a frame rotating at angular frequency $\we$, according to
\begin{equation}
x(t) = X(t) \cos \we t + Y(t) \sin \we t \, .
\end{equation}
Assuming $|\we-\wef|\ll \wef$, and $\gef \ll \wef$, the evolution equations for the two slowly-varying quadratures, derived from Eq. (\ref{eqx}), can be written as
\begin{subequations}
\label{eqXY}
\begin{align}
\dot{X}+\frac{\gef}{2} X - \left(\wef-\we \right)Y = \frac{1}{M \we}\left[f^{(1)}_T + f^{(1)}_s + f^{(1)}_{BA} \right] \\
\dot{Y}+\frac{\gef}{2} Y + \left(\wef-\we \right)X = \frac{1}{M \we}\left[f^{(2)}_T + f^{(2)}_s + f^{(2)}_{BA} + \frac{F_e}{2} \right]
\end{align}
\end{subequations}
where the stochastic force terms have correlation functions $\langle f_a^{(i)}(t) f_a^{(j)} (t')\rangle = \delta_{ij}\delta(t-t') S_a/2 $ (i,j=1,2 and $"a" = "T", "s", "BA"$). In the experiment, the two quadratures are measured by sending $x_m(t)$ to a lock-in amplifier whose reference signal is derived from the oscillator modulating the coherent force $F_e$. The outputs of the lock-in are $X_m = X+n^{(1)}$ and $Y_m = Y + n^{(2)}$ with $\langle n^{(i)}(t) n^{(j)} (t')\rangle = \delta_{ij}\delta(t-t') S_n/2$.
The steady state solutions of Eqs. (\ref{eqXY}) are the usual components of the oscillator response, as a function of the frequency difference between resonance and excitation $\delta \omega = \wef-\we $:
\begin{subequations}
\begin{align}
\overline{X}\left( \delta \omega \right) = \frac{F_e}{2}\frac{ \delta \omega }{\frac{\gef^2}{4}+\left( \delta \omega \right)^2} \\
\overline{Y}\left( \delta \omega \right) = \frac{F_e}{2}\frac{ \gef/2 }{\frac{\gef^2}{4}+\left( \delta \omega \right)^2} \, .
\end{align}
\end{subequations}
We remark that $\overline{X}$ is an odd function of $\delta \omega$, therefore it can be efficiently exploited to control and lock $\wef$. The $X_m$ quadrature is indeed integrated and sent to control the resonance frequency $\wef$ by modifying the optical spring constant (\emph{parametric control}). This is obtained in the experiment by acting on the detuning of a second laser beam (\emph{control beam}) according to
\begin{equation}
\omega_L(t) = \omega_L^0 - \int_{-\infty}^{t} \,\mathcal{G}(t,t')\, X_m(t') \mathrm{d}t'
\label{loop1}
\end{equation}
where $\omega_L^0$ is the initial detuning and the kernel $\mathcal{G}(t,t')$ is constant in the case of an integral feedback loop.
Given that $\omega_L$ determines the effective frequency $\wef$ via Eq. (\ref{omegaeff}), we can write
\begin{equation}
\wef(t) = \woef(t) - \int_{-\infty}^{t} \,\mathcal{\bar{G}}(t,t')\, X_m(t') \mathrm{d}t'
\label{loop}
\end{equation}
where $\woef(t)$ is the free-running opto-mechanical frequency and $\mathcal{\bar{G}} \propto \mathcal{G}$.
Eq. (\ref{gammaopt}) shows that, in the \emph{bad cavity} limit, the shift in the resonance frequency $\wef$ due to the opto-mechanical interaction is larger than the variation in the damping rate $\gef$, thus the latter can be neglected when considering small variations of $\Delta$ around the working point. We also remark that the control of the optical spring can be considered as a classical effect, and its noise neglected in a first-order treatment. In any case, such noise (for us, the radiation pressure noise of the control beam) can be included in $f_s$.
At the purpose of analyzing the effect of the control loop, we first consider slow fluctuations in the opto-mechanical resonance frequency $\wef$, that can be treated as adiabatic changes of the system, keeping the validity of Eqs. (\ref{eqXY}). In Eq. (\ref{loop}) we replace $X = \overline{X}(\delta\omega) + \delta X$ and, considering small closed-loop fluctuations, we further take $\overline{X}(\delta \omega) \propto \delta \omega$. In the absence of drift in $\woef(t)$, the steady-state solution is $\delta \omega = 0$, i.e., $\wef = \we$ (long term drifts in $\woef(t)$ can be corrected by additional integrators, as in standard servo-loop systems). In the phase plane of a reference frame rotating at $\we$, the oscillator motion
is now represented by a vector $\mathbf{R} = (X, Y)$ fluctuating around the average value $(0, Y_0)$ with $Y_0= \overline{Y}(0)=F_e/\gef$ (in Fig. \ref{schema}b we report an experimental example). The feedback loop corrects the fluctuations by counter-rotating $\mathbf{R}$ towards the $Y$ axis. If $\mathbf{R}$ remains close to $(0, Y_0)$, i.e., if $\langle X^2 + (Y-Y_0)^2 \rangle \ll Y_0^2$, we can approximate the angle $\theta$ between $\mathbf{R}$ and the $Y$ axis with $\theta \approx X/Y_0$.
In this limit, the feedback loop (that acts on $\theta$) just influence the fluctuations in the $X$ quadrature, leaving free $Y$ fluctuations. This is expressed by a linear expansion of Eqs. (\ref{eqXY}) around the steady state, with $\wef = \we + \delta \omega (t)$, $X = \overline{X}+\delta X$ and $Y = Y_0 + \delta Y$:
\begin{subequations}
\label{eqXYa}
\begin{align}
\delta \dot{X}+\frac{\gef}{2} \delta X - \delta \omega(t) \,Y_0 = \frac{1}{M \we}\left[f^{(1)}_T + f^{(1)}_s + f^{(1)}_{BA} \right] \\
\delta \dot{Y}+\frac{\gef}{2} \delta Y = \frac{1}{M \we}\left[f^{(2)}_T + f^{(2)}_s + f^{(2)}_{BA} \right] \\
\delta\omega(t) = \delta \woef(t) - \int_{-\infty}^{t} \,\mathcal{\bar{G}}(t,t')\, \left[\,\overline{X}(\delta\omega(t'))+\delta X(t')+n^{(1)}(t')\right] \,\mathrm{d}t' \, .
\end{align}
\end{subequations}
We have few important remarks on the above relations. The first one is that the equation governing the fluctuations of the $Y$ quadrature is the same that we would have without feedback, therefore $\delta Y$ behaves as in a free oscillator and, in particular, it can be used to reliably measure the external force. Second point, we have a well defined phase plane: the oscillator is not just frequency stabilized, but also phase-locked to the reference. Third issue, the response function of the $Y$ quadrature is stable, with a peak frequency defined \emph{a priori} (at $\omega=0$, corresponding to $\we$ for the evolution of $x$) and, as a consequence, stable width $\gef$ and peak signal-to noise ratio. Such parameters stability is very important for an easier application of optimal filtering.
The spectrum of the measured $Y_m$ quadrature calculated from Eq. (\ref{eqXYa}b) can be written in the form
\begin{equation}
S_{Ym} = \lor(\omega)\, S_F + S_n/2
\label{eqSY}
\end{equation}
with
\begin{equation}
\lor(\omega)=\mathcal{A} \frac{\gef}{\omega^2 + \left(\frac{\gef}{2}\right)^2} \, .
\label{eqlor}
\end{equation}
where $\mathcal{A}=\int_{-\infty}^{\infty}\lor(\omega)\,\mathrm{d}\omega/2\pi = 1/(2 \gef \,M^2 \,\we^2)$
and the total force noise spectral density is $S_F = S_s+S_T+S_{BA}$.
The treatment of this Section includes slow fluctuations of $\woef$ as well as its fast, although weak variations that can be considered as phase fluctuations. The case of strong and fast variations of $\woef$, producing trajectories in the phase plane that take $\mathbf{R}$ far from the region with $\theta < 1$, requires a numerical integration of Eqs. (\ref{eqXY}) and the approximation of a free $Y$ quadrature is no more reliable.
By excluding the coherent excitation and the frequency control, the spectrum of both quadratures, for an opto-mechanical resonance at $\woef = \we+\delta\omega$, is
\begin{equation}
S_{Xm} = S_{Ym} = \frac{1}{2}[\lor(\omega-\delta\omega)+\lor(\omega+\delta\omega)] S_F +\frac{S_n}{2}
\label{duelor}
\end{equation}
and, in case of slow fluctuations of $\delta\omega$, the spectral peaks assume the shape of a Voigt profile, maintaining a constant area.
\subsection{Force measurement strategies}
We consider two possible measurement strategies, with the aim of detecting a weak stochastic signal force $f_s$ hidden by the thermal background. In other words, we are seeking for a precise measurement of the stochastic force in order to resolve its weak variations due to changes in $S_s$. We are not dealing with measurement accuracy and reproducibility, that both depend critically on absolute calibrations.
The first strategy is simply measuring the area $\sigma^2$ of the resonance peak. The advantage of this method is that frequency stability of the opto-mechanical oscillator is not crucial: the peak area can be calculated by direct integration of the spectrum of $x$ within an appropriate frequency interval, provided that $\wef$ is well within the integration band, and the latter is extended to few $\gef$ yet maintaining a negligible contribution of the background noise $S_n$. The same measurement can be performed, with equal efficiency, on the spectrum of a quadrature. The estimated force spectral density is $E\{S_F\} = \sigma^2/\mathcal{A}$. The drawback of this method is the rather slow improvement of the statistical uncertainty, decreasing as $\propto \sqrt{\tau_c/t_{meas}}$ where the correlation time is now $\tau_c=1/\gef$. The reason is that this strategy does not exploit the full information contained in the signal, whose spectrum around resonance is dominated by the effect of the force fluctuations even well beyond the width $\gef$.
The second strategy is a close approximation of the Wiener filtering, that represents the optimal choice in case of stationary noise. The non-causal Wiener filter, applied to the spectrum $S_{Ym}$ of Eq. (\ref{eqSY}), is defined as
\begin{equation}
|W(\omega)|^2 = \frac{1}{\lor(\omega)}\left[\frac{1}{1+\Gamma\frac{\lor(0)}{\lor(\omega)}}\right]^2
\label{eqwiener}
\end{equation}
and the maximum information on $S_F$ from the experimental $S_{Ym}$ is obtained from the filtered spectrum $S_W = |W|^2 S_{Ym}$. The $1/\lor$ factor in Eq. (\ref{eqwiener}) is a whitening and calibration function, while the term between square brackets is a weight function that requires preliminary estimate of the noise-to-peak-signal ratio $\Gamma$. Its optimal value is $\Gamma_{opt} = S_n/2 \lor(0) S_F$, but an efficient, even if sub-optimum, filter can choose a $\Gamma > \Gamma_{opt}$ \cite{Astone1990}. In any case, preliminary fit of a spectrum $S_{Ym}$ allows to extract the parameters $\gef$ and $\Gamma$ for the following application of the Wiener filtering procedure. The correlation time of the filtered signal is now $\tau_c \sim \sqrt{\Gamma}/\gef$, yielding a faster improvement of the statistics with $t_{meas}$ with respect to the previous strategy. For an optimum filter (with $\Gamma = \Gamma_{opt}$), $1/\tau_c$ corresponds to the effective sensitivity bandwidth, i.e., to the frequency band where the effect of force noise falls below the measurement sensitivity (i.e., $\lor(\omega) S_F = S_n/2$). An example of the application of the whitening function and the complete Wiener filter to a real spectrum is shown in Fig. \ref{figWiener}. The force spectral density is estimated by integrating the filtered spectrum $S_W$ and dividing the result by the effective bandwidth $\sim 1/\tau_c$. In our real data some spurious peaks appear in the spectrum at few kHz from the opto-mechanical resonance, therefore the integration is truncated at $\omega_{cut}/2\pi$=3kHz, slightly below $1/\tau_c$. More details on the choice of $\omega_{cut}$ and on the consequent effective bandwidth are reported in the Appendix.
\begin{figure} [h]
\centering
\includegraphics[width=0.9\textwidth]{figWiener.eps}
\caption{(Color online) Measured spectral density in the $Y$ quadrature ($S_{Ym}$) (orange circles); whitened spectrum (violet squares); with complete Wiener filtering (green triangles).}
\label{figWiener}
\end{figure}
As we have seen, the application of the Wiener filtering requires the knowledge of the transfer function between force noise and output. For this reason, the parametric control strongly facilitates the filtering procedure, by fixing both the opto-mechanical resonance frequency at $\wef = \we$ and, as a consequence, its width $\gef$. Without control, optimal filtering would require an adaptive tuning of the parameters, that we are not trying to apply in this work.
\section{Experiment}
\subsection{Experimental apparatus}
A sketch of our experimental system is shown in Fig. \ref{schema}c. A Fabry-Perot cavity is formed between a micro-oscillator with high reflectivity dielectric coating as end mirror and a standard concave input coupler. The cavity length is 0.57~mm and its finesse is 57000 (half-linewidth $\kappa/2\pi = 2.3$~MHz). The input coupler is glued on a piezo-electric transducer for coarse tuning, and the cavity is kept in a vacuum chamber at $10^{-3}$~Pa. The low-deformation micro-mirror \cite{SerraAPL2012,SerraJMM2013} has resonance frequency $\wm/2\pi =128960$~Hz, mechanical quality factor $Q=\wm/\gm=16000$ (limited, at room temperature, by thermoelastic losses) and effective mass $M = 1.35~10^{-7}$~Kg. More details on the measurements of the opto-mechanical parameters are reported in Refs. \cite{SerraAPL2012,SerraPRA2012}.
Two laser beams derived from the same Nd:YAG source are overlapped with orthogonal polarizations and optically matched to a cavity longitudinal mode with an efficiency of $\approx96\%$. From the reflected first beam (\emph{signal beam}, with a power of $80 \mu$W) we obtain a dispersive profile of the optical resonance (PDH signal) through phase modulation at 13.3~MHz and phase-sensitive detection \cite{Drever}. Such signal is exploited for locking the laser beam to the cavity resonance. Moreover, in the approximately linear region around resonance, the PDH signal is proportional to the oscillator displacement and is used both for monitoring its motion and in the parametric control loop described below. We remark that the bandwidth of the laser locking is kept at $\sim30$~kHz (well below the mechanical frequency) and additional strong notch filters assure that the laser frequency servo loop has no effect in the frequency region of interest (around the mechanical resonance).
The second beam (\emph{control beam}), with a power of 1 mW at the cavity input, is frequency shifted with respect to the signal beam, and is used to control the optical spring. The adjustable frequency shift compensates the cavity birefringence and determines the detuning of the control beam with respect to the cavity resonance. The ratio between opto-mechanical frequency shift due to the optical spring and control beam detuning is $8 \cdot 10^{-3}$ Hz/Hz. In addition, an electro-optic intensity modulator imposes a weak sinusoidal modulation in the power of the control beam and consequently in the radiation pressure acting on the micro-mirror.
The PDH signal is calibrated by means of a modulation at $\sim20$kHz sent to the laser frequency controller. The amplitude of this modulation at the input of the controller is directly measured during the acquisitions (since it is influenced by the frequency servo loop). This measurement, as well as the measurement of the amplitude of the corresponding modulation in the PDH signal, are repeated every 1s during the data acquisition, in order to compensate for (weak) changes in the detection efficiency. The laser tuning rate (in Hz/V) had been previously calibrated with a Michelson interferometer, and the ratio between the laser frequency and the cavity length allows to convert the detuning into cavity displacement. The overall calibration has an absolute accuracy of $\sim20\%$ (we point out that such uncertainty in the calibration factor do not influence the possibility to resolve weak signal variations, that is the object of this work).
The PDH signal is also sent to a double-phase, digital lock-in amplifier and integrated with a time constant of 80$\mu$s. For the configuration with parametric control of the opto-mechanical frequency, the lock-in oscillator is sent to the intensity modulator of the control beam. A preliminary scan of its frequency allows to reconstruct the response function of the mechanical oscillator and to tune the phase of the lock-in amplifier in order to have the dispersive component at the $X$ output. The reference oscillator is then set to 127400 Hz and the $X$ output of the lock-in amplifier is integrated and sent to the drivers of the acousto-optic modulators that vary the detuning of the control beam. The opto-mechanical resonance is now phase-locked to the reference oscillator. The detuning of the control beam corresponds to about $0.09 \kappa$ and the oscillator is in rather strong optical damping condition, with a resonance width of $\gef/2\pi \simeq 200$Hz. For the configuration without parametric control, the effective opto-mechanical frequency is moved to about 127400 Hz by hand tuning the control beam, but the lock-in reference frequency is set at 127200 Hz, so that the well defined resonance peak at $\sim200$ Hz allows to measure more accurately its parameters.
\subsection{Measurements}
The signal from the $Y$ output of the lock-in amplifier is acquired by a digital scope with a resolution of 12 bit and a sampling interval of 21$\mu$s. Data are acquired by the scope in 35 consecutive time traces, each one lasting about 20 seconds (corresponding to $\sim 10^6$ data points) covering in all nearly 12 minutes, then stored in a hard disk. Several of such series are taken separated by periods of few minutes (necessary to write the data on disk), for a total observation time of several tens of minutes.
\begin{figure} [h]
\centering
\includegraphics[width=0.9\textwidth]{figSpettri.eps}
\caption{(Color online) Spectral densities of the $Y$ quadrature ($S_{Ym}$), for an oscillator without (upper panel) and with (lower panel) parametric control. With a solid line we show the respective fitting functions. In the inset, we compare spectra obtained with different values of the parametric control gain, showing that the control do not influence the dynamics of the $Y$ quadrature.}
\label{spettri}
\end{figure}
The time series are divided into 100ms long segments, a duration much larger than their correlation time. For each section the power spectrum is calculated using a FFT algorithm, and corrected for the transfer function of the lock-in amplifier. The spectra corresponding to the first 20 seconds are averaged, and the averaged spectrum is fitted to Eq. (\ref{eqSY}) (when the parametric control is active) or to Eq. (\ref{duelor}) (without control). An example of the averaged spectra and the fits are shown in Fig. \ref{spettri}. From the fitting procedure we obtain the resonance width, the signal maximum and, in the absence of the control, also the resonance frequency. The signal maximum $Max$ is just exploited to define the value of the parameter $\Gamma$ to be used for Wiener filtering. At this purpose, we consider a conservative value of the background additive noise on $Y$, at $S_{BG}=8\cdot 10^{-33} $m$^2$/Hz (one order of magnitude larger than the real $S_n$) and define $\Gamma=S_{BG}/Max$. A typical value of $\Gamma$ is $10^{-3}$.
From each of the following spectra (after the first 20s) we calculate the force spectral density $S_F$ using the different methods described in the previous Section (i.e., from the peak area and using Wiener filtering, both in the configuration with parametric feedback and with free-running oscillator). We report in Fig. \ref{media} the average $\bar{S}_F(t_{meas})$ of $S_F$ accumulated over $m$ consecutive spectra, corresponding to a measurement time $t_{meas} = m \tau$, where $\tau = 100$ms is the time interval used for calculating each spectrum. The relative standard error is given by $\sigma_{\mathrm{REL}} \simeq 2/\sqrt{t_{meas} \gef}$ for the measurement with the peak area, and $\sigma_{\mathrm{REL}} \simeq \sqrt{2\pi/t_{meas}\omega_{cut}}$ when using Wiener filtered data (these expressions refer to the configuration with parametric control where the peak is centered at null frequency, and the latter relation is valid for $\omega_{cut} \ll \gef/2\sqrt{\Gamma}$; exact calculations are reported in the Appendix). $\sigma_{\mathrm{REL}}$ is used to calculate the confidence regions $(1 \pm \sigma_{\mathrm{REL}})\bar{S}_F$, where $\bar{S}_F$ is the average at the end of the measurement period. The figure shows the expected convergence of the measured $\bar{S}_F(t_{meas})$, which is clearly faster for the filtered data.
\begin{figure} [h]
\centering
\includegraphics[width=0.9\textwidth]{figMedia.eps}
\caption{(Color online) Average over a measurement time $t_{meas}$ of the force noise spectral density $S_F$, measured on the oscillator with parametric control using the peak area (orange circles) and the Wiener filtered spectra (violet squares). The confidence bands (respectively dashed and dash-dotted lines), corresponding to one standard error, are calculated in the Appendix.}
\label{media}
\end{figure}
The calculation of the confidence region reported in Fig. \ref{media} is just valid for a stationary system. A more reliable assessment on the measurement stability on the long term and on the achievable resolution is provided by the Allan variance \cite{Allan}. In our case, its estimator is defined as
\begin{eqnarray}
\sigma_A^2(m) = \frac{1}{N-m}\sum_{k=1}^{N-2m+1} \frac{\left(\bar{x}_{k+m}-\bar{x}_k\right)^2}{2} \\
\bar{x}_k(m) =\frac{1}{m} \sum_{n = k}^{k+m-1} S_F (n)
\end{eqnarray}
where $S_F (n)$ is the value of force spectral density calculated from the $n$th spectrum and $N$ is the total number of spectra. The Allan deviation $\sigma_A(m)$ estimates the one sigma uncertainty that can be obtained with a measurement lasting $t_{meas} = m \tau$. The calculated relative Allan deviation (i.e., $\sigma_A$ divided by $\bar{S}_F$) is reported in Fig. \ref{figAllan} for the different measurement strategies. We can derive two main considerations: a) the measurement with Wiener filtering improves the statistical uncertainty much faster than the measurement from the peak area. For the former, a $1\%$ resolution is obtained after 10s and the best resolution of $0.4\%$ is achieved, thanks to the parametric stabilization, after one minute; for the latter, the necessary measurement periods are about three times longer, in agreement with the ratio between the respective $\sigma_{\mathrm{REL}}$; b) for measurement periods exceeding 1s, the parametric control is crucial for the application of Wiener filtering. The measurement resolution does not improves any more after one minute: with the parametric control it remains constant, while it becomes even worse without control. It means that the parametric control also allows a much more relaxed choice of the optimal measurement time.
\begin{figure} [h]
\centering
\includegraphics[width=0.9\textwidth]{figAllan.eps}
\caption{(Color online) Relative Allan deviation concerning the measurement of the input stochastic force $S_F$, performed with four different procedures. Solid lines, from the upper to the lower curve (as seen in the left region of the graph): measurement from the peak area, with parametric control (orange); the same, without control (red); measurement from the Wiener-filtered data, without parametric control (deep blue); the same, with control (light blue). Dashed lines display the expected behaviour in the absence of long-term effects, given by Eq. (\ref{A3}) (upper line), Eq. (\ref{A3bis}) (middle line), and Eq. (\ref{A5}) (considering an implementation of the optimal filter; lower line).}
\label{figAllan}
\end{figure}
\section{Conclusions}
We have analyzed different possible procedures for measuring the stochastic force acting on a micro opto-mechanical system. In particular, we have compared the usual strategy based on the direct measurement of the area of the resonance peak (or, equivalently, of the variance in the oscillator displacement) with a more refined data analysis that approaches the optimal Wiener filtering. For the latter case, we have introduced an abrupt bandwidth limitation that allows a near-optimal realistic measurement procedure. We have shown that, while for the former method the optical damping, decreasing the oscillator coherence time, can improve the resolution of the measurement in a given observation time, the appropriate filtering gives sensibly better results which are mostly independent on such coherence time.
The implementation of the Wiener filtering is greatly facilitated and more effective by using a parametric control of the oscillator frequency, a technique that we have recently introduced and that we have analyzed here in details. Thanks to such active stabilization, our system can reliably detect variations of the stochastic force below
1$\%$ within one minute. We remark that a correct assessment of the really achievable resolution with long integration periods cannot be simply based on the convergence of the averaged measurement. Indeed, such indicator underestimates the effect of system long-term instabilities and parameter drifts. Using the Allan variance as correct estimator, we show that parametric control plays a crucial role in the achieved performance.
The procedure for the measurement of the stochastic force that we have described in this work, including optimal filtering and parametric control, can be applied in a large variety of micro- and nano-mechanical systems, including those based on electric measurements and microwave radiation. Detecting a weak stochastic signal on a stronger background is an important task in the research field of quantum mechanics with macroscopic oscillators, in particular when exploring the properties of oscillators with low occupation number, or, e.g., in a squeezed state \cite{Pontin2013,Clerk2008,Hertzberg2009,Bowen2013} or other peculiarly quantum states. In this situation, the measurement back-action can destroy the interesting features. Particular measurement schemes can be conceived and applied \cite{chan11,Clerk2008,Hertzberg2009,Kronwald2013}, but the use of a weak measurement, where the signature of the oscillator is intrinsically weaker than the measurement noise (see, e.g., in Ref. \cite{Thompson08}), can be a useful affordable solution. The procedures investigated in this work would thus provide a valuable help.
\section{Acknowledgments}
F.M. thanks M. Prevedelli for the discussion on phase locking. This work has been supported by the European Commission (ITN-Marie Curie project cQOM), by MIUR (PRIN 2010-2011) and by INFN (HUMOR project).
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1312.5342
|
\section{NV$^{-}$ centers in high strain regime}
In this section we present the Hamiltonian for NV$^{-}$ centers under external static electric and magnetic fields. We use the Hamiltonian to derive optical transitions for ${\hat x}$ and ${\hat y}$ polarizations.
Table 1 in \cite{DohertyNJP} shows the spin-orbit and configuration basis states for a negatively charged NV. Using table 2 and 3 in \cite{DohertyNJP}, we derive the spin-orbit and spin-spin interaction Hamiltonian in the configuration basis for the excited state triplets that is represented by $\{\Phi^{c}_i\}_{i=1..6}=\{ \Phi^{c}_{E,x;1,0}, \Phi^{c}_{E,x;1,1}, \Phi^{c}_{E,x;1,-1}, \Phi^{c}_{E,y;1,0}, \Phi^{c}_{E,y;1,1}, \Phi^{c}_{E,y;1,-1}\}$.
The following Hamiltonian is the resulting spin-orbit and spin-spin Hamiltonian.
\begin{equation}\tag{S1}\label{Hsoss}
H_{so,ss}=
\begin{pmatrix}
-2D_{2A_1} & \frac{D_{2E_2}}{\sqrt{2}} & -\frac{D_{2E_2}}{\sqrt{2}} & 0 & -\frac{iD_{2E_2}}{\sqrt{2}} & -\frac{iD_{2E_2}}{\sqrt{2}} \\
\frac{D_{2E_2}}{\sqrt{2}} & D_{2A_1} & D_{2E_1} & \frac{iD_{2E_2}}{\sqrt{2}} & i\lambda_{\parallel} & -iD_{2E_1} \\
-\frac{D_{2E_2}}{\sqrt{2}} & D_{2E_1} & D_{2A_1} & \frac{iD_{2E_2}}{\sqrt{2}} & iD_{2E_1} & -i\lambda_{\parallel} \\
0 & -\frac{iD_{2E_2}}{\sqrt{2}} & -\frac{iD_{2E_2}}{\sqrt{2}} & -2D_{2A_1} & -\frac{D_{2E_2}}{\sqrt{2}} & \frac{D_{2E_2}}{\sqrt{2}} \\
\frac{iD_{2E_2}}{\sqrt{2}} & -i\lambda_{\parallel} & -iD_{2E_1} & -\frac{D_{2E_2}}{\sqrt{2}} & D_{2A_1} & -D_{2E_1}\\
\frac{iD_{2E_2}}{\sqrt{2}} & iD_{2E_1} & i\lambda_{\parallel} & \frac{D_{2E_2}}{\sqrt{2}} & -D_{2E_1} & D_{2A_1} \\
\end{pmatrix}
\end{equation}
where $D_{2A_1}=1.42/3$GHz, $D_{2E_2}=0.2/{\sqrt{2}}$GHz, $D_{2E_1}=1.55/2$GHz and $\lambda_{\parallel}=5.3$GHz are representing the spin-spin and spin-orbit interactions. Note that there are differences between some of the coefficients in this Hamiltonian and Eq. (19) in \cite{DohertyNJP}. Despite having similar eigen-energies, differences in eigenstates can be crucial for deriving the correct optical polarization selection rules.
In this basis the Hamiltonian for external electric and magnetic fields acquires the following simple form
\begin{equation}\tag{S2}\label{Helecmag}
H_{elec,mag} =
\begin{pmatrix}
-E_x^{es} & 0 & 0 & E_y^{es} & 0 & 0 \\
0 & -E_x^{es}+B_z^{es} & 0 & 0 & E_y^{es} & 0 \\
0 & 0 & -E_x^{es}-B_z^{es} & 0 & 0 & E_y^{es} \\
E_y^{es} & 0 & 0 & E_x^{es} & 0 & 0 \\
0 & E_y^{es} & 0 & 0 & E_x^{es}+B_z^{es} & 0 \\
0 & 0 & E_y^{es} & 0 & 0 & E_x^{es}-B_z^{es} \\
\end{pmatrix}
\end{equation}
where $B_z^{es}$ is the energy shift due to the axial magnetic field and $E_{x,y}^{es}$ are the excited-state energy shifts associated with external electric field components in the $x-y$ plane perpendicular to the NV axis . We diagonalize the total Hamiltonian ($H_{so,ss}+H_{elec,mag}$) in order to find the energies of the excited levels and their states as a superposition of the spin-orbit states (the basis).
\section{Optical transition (selection rules)}
Based on all of the above considerations and the results in Eq. (1) in the paper, we find the polarization selection rules from ground states to the excited states in the high electric field and low magnetic field condition, such that the $|S=1,m_s=\pm 1\rangle$ states are mixed, i.e. $E_{\perp}^{gs}\gg B_z^{gs}$. This approach allowed us to include all optical transitions due to the coupling to cavity and control fields in the present scheme.
For this purpose, we find excited eigenstates from the total Hamiltonian for the excited states of NV$^-$ centers that includes $H_{so,ss}$ and $H_{elec,mag}$, see Eqs. (\ref{Hsoss},\ref{Helecmag}). The $k^{th}$ excited eigenstate can be represented as $|k\rangle=\sum_{i=1..6} c_{i}^{k}\Phi^c_i$, where $\Phi^c_i$ denotes the configuration states shown above. In \cite{DohertyNJP}, these states are shown in terms of $|a_1 {\bar a}_1 e_x {\bar e}_x e_y {\bar e}_y\rangle$ states, where $\{a_1,e_x,e_y\}$ are single electron orbitals and the overbar denotes spin-down. Using this representation for the ground and excited eigenstates and $\langle a_1|{\hat x}\cdot{\hat r}|e_x\rangle = \langle a_1|{\hat y}\cdot{\hat r}|e_y\rangle$, we examine all possible optical transitions from ground to excited states. The following table show coupling ratios $|g_{y}(j,k)/g_{y}(-,6)|$ for transitions from ground states $|0,\pm\rangle$ to excited states $|k\rangle$ for coupling to $y$-polarized light. Similar results for coupling ratios $|g_{x}(j,k)/g_{x}(+,6)|$ for $x$-polarized light are presented in Table I in the paper.
\begin{table}[h]
\caption{The following table shows coupling ratios $|g_{y}(j,k)/g_{y}(-,9)|$, where $g_{y}(j,k)=\frac{\vec{\mu}_{jk}\cdot\hat{y}}{|\mu_{jk}|}$ and $\vec{\mu}_{jk} = \langle j|\vec{r}|k\rangle$. The electric and magnetic field splittings are, $E_{x}^{es}=120GHz, B_{z}^{es}=10kHz$ and $E_{y,z}^{es}=B_{x,y}^{es}=0$.}
\begin{tabular}{| l || c | c | c | c | c | c|}
\hline
& $|k=4\rangle$ & $|k=5\rangle$ & $|k=6\rangle$ & $|k=7\rangle$ & $|k=8\rangle$ & $|k=9\rangle$ \\ \hline
$j=1,|0\rangle$& 38.5440 & 9.3371 & $<10^{-4}$ & $<10^{-4}$ & 0.0229 & $<10^{-4}$ \\ \hline
$j=2,|+\rangle$& 9.3350 & 38.5372 & 0.0584 & $<10^{-4}$ & 0.7473 & 0.0015 \\ \hline
$j=3,|-\rangle$& 0.0137 & 0.0567 & 39.6461 & 0.0827 & 0.0011 & 1\\
\hline
\end{tabular}
\end{table}
The significant splitting of 240 GHz between the upper and lower branches of the excited suppresses off-resonant couplings to the lower branch due the relatively small detuning of ~0.8 GHz.
\section{Dynamical equations for optical polarizations and spin excitation}
The following shows examples from a larger set of dynamical equations that are derived from the Heisenberg equations. Here, we include the spin and optical inhomogeneous broadenings in the equations. A similar approach in \cite{Gorshkov} has been used to analyze storage in a 3-level configuration. The dynamics of the optical polarization for $|+\rangle\rightarrow |k=9\rangle$ transition is as follows
\begin{equation}\tag{S3}
{\dot {\hat \sigma}}_{29}=(i\Delta -\gamma/2-\gamma_{e}) {\hat \sigma}_{29}+ iG(2,9)N{\hat {\cal E}} +i G(3,9){\hat {\cal E}} {\hat \sigma}_{23}e^{i\delta_g t} + i\Omega(2,9)Ne^{-i\delta_g t} + i \Omega(3,9) {\hat \sigma}_{23},
\end{equation}
where $\delta_g=\omega_2-\omega_c$. This approach is adequate for treating the inhomogeneous broadening in the present context \cite{Gorshkov}. For treatment of echo rephasing pulses, inhomogeneous broadening must be modeled by explicitly including distribution of frequencies.
Similarly, the spin polarization dynamics is given by
\begin{equation}\tag{S4}
{\dot {\hat \sigma}}_{23}=-\gamma_{s} {\hat \sigma}_{23} - i\Sigma_{k=4..9} G(2,k){\hat {\cal E}} {\hat \sigma}_{k3} + \Omega(2,k) {\hat \sigma}_{k3}e^{-i\delta_g t} - {\hat {\cal E}}^{\dagger}G^*(3,k) {\hat \sigma}_{2k}e^{-i\delta_g t} - \Omega^*(3,k){\hat \sigma}_{2k}.
\end{equation}
For finding the single photon wavefunction of the cavity field based on the Eq. (3) in the paper, we require to find a solution to single excitation wavefunctions of spin and optical polarizations. This is performed through integrating over all dynamical equations in a this 9-level configuration and assuming that all of the NV$^-$ centers are initialized in the $|+\rangle$ ground state.
|
1312.5343
|
\section{Introduction}
Continuously improving description of the cooling of white dwarfs
(WDs) and precise measurements of their luminosity curve open a
possibility of their use as an elementary particle physics
laboratory. Thus, Isern {\it et al.} \cite{IEA} suggested to test
possible existence of axions by studying the WDs luminosity
function. Analogously, we analyze the influence of lepton number
violation on the luminosity of strongly magnetized iron WDs
(SMIWDs). The existence of Majorana type neutrino would imply the
lepton-number-violating process of electron capture by a nucleus
$X(A,Z)$:
\begin{equation}
e^- + X(A,Z) \rightarrow X(A,Z-2) + e^+\,, \label{EMEPC}
\end{equation}
which is an analogue of the netrinoless double beta-decay,
intensively studied these days \cite{VES}.
The description of the strongly magnetized white dwarfs (SMWDs: we
use this acronym for the WDs with the magnetic field in the core
higher than a critical one ${\cal B}_c$=4.414 $\times 10^{13}$ G) is
based on a strongly magnetized cold degenerate electron gas
\cite{KM}. The theory follows from the Landau quantization of a
motion of electrons in homogenous magnetic field, usually taken to
point along the z-axis \cite{LL}, with a modification to the case of
very strong ones \cite{LS}. In systems with small number of Landau
levels, which is restricted by the strength of the magnetic field
and by the Fermi energy E$_{\,\mathrm{F}}$ of the electron gas, the
mass of the SMWD can be in the range (2.3 - 2.6)\,M$_\odot$, where
M$_\odot$ is the solar mass. That is, the strong magnetic field can
enhance the energy of the electron gas to such a level that its
pressure allows the SMWD to have a mass larger than the
Chandrasekhar-Landau (ChL) limit of 1.44\,M$_\odot$ \cite{C}.
We apply the theory of SMWDs to estimate, whether the influence of
the double charge exchange reaction (\ref{EMEPC}) on cooling of the
SMIWDs can be detectable. The threshold for this reaction with the
initial nucleus being $^{56}_{26}$Fe and the final one
$^{56}_{24}$Cr is $\Delta$=6.33 MeV, thus in the SMIWDs with
E$_{\,\mathrm{F}}\,\ge\,\Delta$ this reaction can take place. We
consider
$\epsilon_{\,\mathrm{F}}$=E$_{\,\mathrm{F}}$/m$_e$\,c$^2$=20,\,46,\,90,\,150
and 200 and choose the strength of the magnetic field so that the
value of $\gamma={\cal B}/{\cal B}_c$ allows only for the ground
Landau level.
\section{Theory of strongly magnetized white dwarfs}
Theory of the SMWDs is discussed in detail in Refs.\,\cite{KM}. In
the relativistic case, one solves the Dirac equation, obtaining for
the electron energy the equation:
\begin{equation} E_\nu=m_e c^2 \left[1 + \left(\frac{p_z}{m_e c}\right)^2 +
2\nu\gamma \right]^{1/2} \, . \label{ENU} \end{equation}
Here $m_e $ is the electron mass, $c$ is a velocity of light, $p_z$
is the electron momentum along the z-axis, $\nu=l+1/2+\sigma$ labels
the Landau levels with the principal number $l$, $\sigma=\pm 1/2$.
The ground level ($\nu$=0) is obtained for $l$=0 and $\sigma$=-1/2,
and it has the degeneracy factor g$_\nu$=1. Other Landau levels
posses the degeneracy factor g$_\nu$=2. While the density of
electron states in the absence of the strong magnetic field is given
as $2/(2\pi\hbar)^3\,d^3p$, the presence of such magnetic field
modifies the number of electron states for a given level $\nu$ to $2
g_\nu e{\cal B}/[(2\pi)\hbar)^2 c]dp_z$. Then the sum over the
electron states in the presence of the strong magnetic field is
given by:
$$
\sum_E \rightarrow \sum_\nu \frac{2 e{\cal B}}{(2\pi\hbar)^2 c}\,
g_\nu \int\,dp_z = \frac{2\gamma}{(2\pi)^2\lambda^3_e}\, \sum_\nu
g_\nu \int d \frac{p_z}{m_e c} \ ,
$$
where $\lambda_e=\hbar/m_e c$ is the electron Compton wavelength.
In what follows, we consider the case of the ground Landau level,
for which the Fermi energy $\epsilon_{\,\mathrm{F}}$ and the Fermi
momentum $x_{\,\mathrm{F}}=p_{\,\mathrm{F}}/m_e c$ are obtained
directly from Eq.\,(\ref{ENU}):
$\epsilon^2_{\,\mathrm{F}}=x^2_{\,\mathrm{F}}+1$. Then one obtains
the electron number density to be
$n_{e^-}=2\gamma/[(2\pi)^2\lambda^3_e]\, x_{\,\mathrm{F}}$ and the
matter density of the system of a one sort of nuclei $\rho_m=
\mu_{e^-} m_U n_{e^-}=(n_{e^-}/Z) m_A$, where $\mu_{e^-}=A/Z$ is the
molecular weight per electron [A(Z) is the mass (atomic) number of
the nucleus], $m_U$ is the atomic mass unit and $m_A$ is the mass of
the nucleus with the mass number $A$. For the lightest nuclei,
$\mu_{e^-}=2$, but, e.g., for $^{56}_{26}$\,Fe one obtains
$\mu_{e^-}=2.15$. The pressure of the degenerate electron gas is
then
\begin{equation} P_{e^-}= \frac{\gamma m_e c^2}{(2\pi)^2\lambda^3_e}
\,\left[x_F\, \epsilon_{\,\mathrm{F}}-
ln(x_{\,\mathrm{F}}+\epsilon_F) \right]\,. \label{PNUT} \end{equation}
With the choice $\gamma$=$\epsilon^2_{\,\mathrm{F}}/2$ we stay at
the ground Landau level. Since $\epsilon^2_{\,\mathrm{F}}\gg$ 1 and
$x_{\,\mathrm{F}}\gg$ 1, the pressure of the degenerate electron gas
(\ref{PNUT}) can be written in the polytropic form:
\begin{equation}
P_{e^-}\,=\,K\, \rho_m^{\Gamma}\,, \qquad
\quad K\,= \frac{\pi^2\hbar^3}{(m_e\, m_U\,\mu_e)^2\,c\,\gamma}\,,
\qquad
\quad \Gamma\,=\,1+ \frac{1}{n} =\,2 \,,
\end{equation}
from which one obtains the value of the
polytropic index $n=1$. The input parameters of our study are
presented in Table \ref{tab:inpt}. Next we briefly describe the
calculation of the capture rate of the charge exchange reaction
(\ref{EMEPC}).
\begin{table}
\begin{tabular}{lcccc}
\hline
\tablehead{1}{c}{b}{$\epsilon_{\,\mathrm{F}} $}
& \tablehead{1}{c}{b}{${\rm n}_{e^-}/10^{33}[1/{\rm cm}^3$]}
& \tablehead{1}{c}{b} {$\rho_{e^-}/10^6 [{\rm g/cm}^3]$}
& \tablehead{1}{c}{b} {$\rho_m/10^{10} [{\rm g/cm}^3]$}
& \tablehead{1}{c}{b} {$2\, \gamma$} \\
\hline
20 & 3.52 & 3.20 & 1.26 & 400 \\
46 & 42.8 & 13.4 & 15.2 & 2116 \\
90 & 321 & 293 & 114 & 8100 \\
150 &1485 &1352 & 531 &22500 \\
200 &3519 &3206 &1250 &40000 \\
\hline
\end{tabular}
\caption{The values of the Fermi energy
$\epsilon_{\,\mathrm{F}}$=E$_{\,\mathrm{F}}$/m$_e$c$^2$, used in the
present study. Further n$_{e^-}$ is the electron number density,
$\rho_{e^-}$ is the corresponding electron density, $\rho_m$ is the
matter density, calculated for the nuclei $^{56}_{26}$\,Fe, and the
values of $\gamma$ are the smallest values allowing one to stay at
the ground Landau level.}
\label{tab:inpt}
\end{table}
\section{Reaction rate}
The $(e^-,e^+)$ conversion rate is, like the $0\nu\beta\beta$-decay
rate, proportional to the squared absolute value of the effective
mass of Majorana neutrinos $|\langle m_\nu\rangle|^2$. This quantity
is defined as $\langle m_\nu \rangle = \sum_{i=1}^{3} U^2_{ei} m_i\,$, where $U$ is
the $3\times3$ Pontecorvo-Maki-Nakagawa-Sakata unitary mixing matrix
and $m_i$ ($i=1,2,3$) is the mass of the i-th light neutrino. The
$(e^-,e^+)$ conversion on nuclei is here considered only for the
ground state to ground state transition, which is assumed to give
the dominant contribution. Both ground states of the initial
(${^{56}Fe}$) and the final (${^{56}Cr}$) nuclei have the spin and
parity $0^+$. The Coulomb interaction of electron and positron with
the nucleus is taken into account by the relativistic Fermi
functions $F(Z,E_{e^-})$ and $F(Z-2,E_{e^+})$ \cite{doi},
respectively. The leading order $(e^-,e^+)$ conversion matrix
element reads:
\begin{eqnarray}
\label{S-matrix}
\langle f \vert S^{(2)} \vert i \rangle &=& 2 \pi \delta(E_{e^+}-E_{e^-}
+ E_f - E_i) \langle f \vert T^{(2)} \vert i \rangle\,,
\label{SM}\\
\label{T-matrix}
\langle f \vert T^{(2)} \vert i \rangle &=& \mathrm{i} ~\langle
m_\nu\rangle^* ~ \frac{1}{4 \pi} G^2_\beta \sqrt{F_0(Z,E_{e^-})}
\sqrt{F_0(Z-2,E_{e^+})}~ \overline{v}(P_{e^+}) (1 + \gamma_5)
u(P_{e^-})\times \frac{ g^2_{\mathrm{A}}}{R} {M}^{(e\beta^+)}\,.
\label{TM}
\end{eqnarray}
Here $G_\beta= G_{\,\mathrm{F}}\cos\theta_c$ and $E_i$ ($E_f$) is
the energy of the initial (final) nuclear ground state. The
conventional normalization factor of the nuclear matrix element
(NME) ${M}^{(e\beta^+)}$ involves the nuclear radius $R
=1.2~A^{1/3}~{\mathrm{fm}}$. For the weak axial coupling constant
$g_A$, we adopt the value $g_A=1.269$.
From Eq.\,(\ref{TM}), one obtains the following equation for the
reaction rate in the SMWDs:
\begin{eqnarray} \Gamma^{(e\beta^+)} & =& m_e~
\frac{|\langle m_\nu\rangle|^2}{m^2_e} ~ \frac{1}{16 \pi^3}
{\left(\frac{G_{\beta} m^2_e}{~\sqrt{2}}\right)}^{{4}}
\frac{g^4_A}{(R^2 m_e^2)}~\left|{M}^{(e\beta^+)}\right|^2
\phi(\epsilon_{\,\mathrm{F}},\gamma)\,, \label{RRPNF}
\end{eqnarray}
where the function $\phi(\epsilon_{\,\mathrm{F}},\gamma)$ is defined
as:
\begin{eqnarray}
\phi(\epsilon_{\,\mathrm{F}},\gamma)&=&\frac{2\gamma}{(2\pi)^2\lambda^3_e
m^3_e}\,\int\limits_{Q+1}^{\epsilon_{\,\mathrm{F}}}\,
\left[\frac{(\epsilon_{e^-}-Q)^2-1}{\epsilon_{e^-}-1}\right]^{1/2}
\,(\epsilon_{e^-}-Q)\,\epsilon_{e^-}\,
F_0(Z,\epsilon_{e^-})~F_0(Z-2,\epsilon_{e^+}) d\epsilon_{e^-}\, ,
\label{ffi} \end{eqnarray}
with $\epsilon_{e^\pm}=E_{e^\pm}/m_e c^2$ and Q=$\Delta/m_e c^2$.\\
To calculate nuclear matrix element for the transition
(e$^-$,e$^+$) on $^{56}$Fe we use the Quasiparticle Random
Phase Approximation (QRPA)
\cite{src09} . For the A=56 system, the single-particle
model space consisted of $0-4\hbar\omega$ oscillator shells, both
for the protons and neutrons. The single particle energies are
obtained by using a Coulomb--corrected Woods--Saxon potential. We
derive the two-body G-matrix elements from the Charge Dependent Bonn
one-boson exchange potential \cite{CDB} within the Brueckner theory.
For the quantitative analysis of the (e$^-$,e$^+$) capture rate we
will consider $|{M}^{(e\beta^{+})}| \approx 3\,.$
To estimate the energy production $\bar{\varepsilon}_r$ per
one event of the reaction (\ref{EMEPC}), we calculated the
two-photon positron-electron annihilation
probability per volume normalized to unity and integrated it over the energies of
electrons $\epsilon_f=E_f/m_e$ interacting with the positron in the
final state of reaction. From this and Eq.\,(\ref{RRPNF}), one can
obtain directly the released energy per 1 second as a contribution
to the luminosity. Here we made the calculations for the case of the
SMIWD with the mass M$_{\mbox{\tiny{WD}}}$=2\,M$_\odot$ and
present them in Table \ref{tab:res} as the ratio of the calculated change
in the luminosity $\Delta\, $L to the solar luminosity L$_\odot$.
\begin{table}
\begin{tabular}{lccc}
\hline
\tablehead{1}{c}{b}{$\epsilon_{\,\mathrm{F}} $}
& \tablehead{1}{c}{b}{$\left[Log(\Delta L/L_\odot)\right]_{0.4}$ }
& \tablehead{1}{c}{b}{$\left[Log(\Delta L/L_\odot)\right]_{0.8}$}
& \tablehead{1}{c}{b}{$\bar{\varepsilon}_r\, [{\rm MeV}]$} \\
\hline
20 & -15.1 & -14.4 & 9.1 \\
46 & -12.0 & -11.4 & 25.2 \\
90 & -10.0 & -9.4 & 49.1 \\
150 & -8.6 & -8.0 & 79.7 \\
200 & -7.9 & -7.3 & 104.4 \\
\hline
\end{tabular}
\caption{The values of the Fermi energy
$\epsilon_{\,\mathrm{F}}$=E$_{\,\mathrm{F}}$/m$_e$, used in the
present study. Further $\Delta$L/L$_\odot$ is the ratio of the
change in the luminosity of the SMIWD due to the reaction
(\ref{EMEPC}) to the luminosity of the Sun, calculated for $|\langle
m_\nu\rangle|$=0.4 eV and 0.8 eV, and $\bar{\varepsilon}_r$ is the
energy, released in single reaction.}
\label{tab:res}
\end{table}
\section{Cooling of white dwarfs}
To see how the energy, produced by the process (\ref{EMEPC}),
influences the cooling of SMIWDs, one should include it into
appropriate detailed microscopic model. Unfortunately, such detailed
models of cooling for this kind of white dwarfs have not yet been
elaborated.
In order to proceed, we first calculated the luminosity of the iron
WDs within a simple cooling model, formulated by Mestel \cite{LM}.
We employed the electron pressure in the polytropic form with the
polytropic index n=3/2 and Kramers' opacity. Following the standard
calculation of the cooling rate \cite{LM,ST} one gets the relation
between the luminosity and the cooling time,
\begin{equation}
\frac{L}{L_\odot}\,=\,1.3\,\times\,10^{-4}\,\left(\frac{M}{M_\odot}\right)\,
\left(\frac{Gyr}{\tau}\right)^{7/5}\,.\label{RLT} \end{equation}
Then we calculated the luminosity of the SMIWDs in the same Mestel's
model. For the ground Landau level, the electron pressure is also of
the polytropic form, but with the polytropic index n=1; also for
this calculation we used Kramers' opacity with the result for the
luminosity,
\begin{equation}
\frac{L}{L_\odot}\,=\,3.5\,\times\,10^{-2}\,\left(\frac{M}{M_\odot}\right)\,
\gamma^{\, 4/7}\,\left(\frac{Gyr}{\tau}\right)^{9/7}\,. \label{SMILT}
\end{equation}
Numerical calculations show that both results differ significantly,
most probably because of the difference in the opacity, since the
luminosity of both types of white dwarfs are not likely to vary so
much. Thus, simple Mestel's model does not provide a reliable
estimate of the role of the considered reaction.
In order to estimate qualitatively possible effect of the reaction
(\ref{EMEPC}) on the cooling of the SMIWDs, we addressed the
asymptotic of the luminosity curves, obtained in Ref.\,\cite{PAB}
for the iron-core WDs. It is clear from Table \ref{tab:res} that the
effect of the double charge exchange reaction (\ref{EMEPC}) could
influence the cooling only at low luminosity. So extrapolating the
data, presented in Fig.\,17 \cite{PAB} for the curve, corresponding
to $M/M_\odot$=0.6 to smaller values of the luminosity, we got that
$Log(L/L_\odot)\,\approx\,$-5.0 (-7.54) is achieved after the
cooling time $\tau\,\approx\,$3.48 (3.90) Gyr\footnote{We obtained
similar results also extrapolating the data for the curve,
corresponding to $M/M_\odot$=0.8.}. One can see that the value
$Log(L/L_\odot)$=-7.54 is of the same size as are the values of
$Log(\Delta L/L_\odot)$, presented in the last line of Table
\ref{tab:res}. It means that the reaction (\ref{EMEPC}) could
effectively retard the cooling of the SMIWDs after low
enough luminosity evolves, unless are meanwhile all the iron nuclei transformed
into chrome ones. However, before making more definite conclusions, the
theory of cooling of the SMIWDs should be elaborated.
\begin{theacknowledgments}
This work was supported by the Votruba-Blokhintsev Program for
Theoretical Physics of the Committee for Cooperation of the Czech
Republic with JINR, Dubna. F. \v S. acknowledges the support by
the VEGA Grant agency
of the Slovak Republic under the contract No. 1/0876/12.
We thank L.~Althaus for providing us with
the data, presented in Fig.\,17 \cite{PAB} by the luminosity curves
for the pure iron-core DA WDs.
\end{theacknowledgments}
\bibliographystyle{aipproc}
|
2204.12923
|
\section{Introduction}
Barycentric coordinates represent a fundamental concept used in major computer graphics and geometric modeling applications such as mesh parameterization \cite{3} \cite{14}, freeform deformations \cite{15} \cite{17}, finite elements \cite{2} and shading
\cite{7} \cite{13}.
\subsection{Generalized barycentric coordinates with respect to arbitrary polytopes}
Generalized barycentric coordinates are an extension of the notion of barycentric coordinates for simplices, to general polytopes. They being too large to be discussed in detail here, we briefly review few approaches closely related to our subject. For more details on generalized barycentric coordinates see K. Hormann and N. Sukumar \cite{8}. The first generalizations were proposed by Wachspress \cite{19}, U. Pinkall and K. Polthier \cite{12} for convex polygons and Sibson \cite{16} for scattered sets of points. In 2003, Floater \cite{5} introduced mean value coordinates that are defined in convex and non-convex polygons. 3D extensions of Wachspress coordinates Ju et al. \cite{17}, Warren et al. \cite{20} and discrete harmonic coordinates Ju et al. \cite{9}, are well defined within convex polyhedra with triangular faces, while 3D mean value coordinates are well defined in arbitrary convex or non-convex polyhedra with triangular faces
Floater \cite{4}, Ju et al. \cite{18} and extended to arbitrary polyhedra with polygonal faces Langer et al \cite{10}.
\begin{definition}
Let P be a polytope in $\mathbb{R}^{d}$, with $n$ vertices $v_1,...,v_n$.
The functions $\phi_i : \mathbb{R}^{d} {\longrightarrow} \mathbb{R},\; i = 1,...,n$\; are called barycentric coordinates if they satisfy
\begin{itemize}
\item[\textbf{(a)}]\;\; \textbf{Partition of unity:}\quad $\sum_{i=1}^{n} \phi_i(x) = 1,\qquad \forall x\in P.$
\item[\textbf{(b)}]\;\;\textbf{Linear precision:}
$\sum_{i=1}^{n} \phi_i(x)\; v_i = x,\qquad \forall x\in P.$
\end{itemize}
\end{definition}
The following additional properties are often required:
\begin{itemize}
\item[\textbf{(c)}]\;\; \textbf{Non-negativity:} $\phi_i(x)\geqslant 0,\quad i=1,...,n,\qquad \forall x\in P$.
\item[\textbf{(d)}]\;\; \textbf{Lagrange property:}\quad
$\phi_i (v_j) = \delta_{ij}$\\
where $\delta_{ij}$ are the Kronecker symbols.
\item[\textbf{(e)}]\;\; \textbf{Restriction on facets of the boundary:}\\
For a facet F with vertices $v_{i1},...,v_{im}$, we have
$$\sum_{j=1}^{m} \phi_{ij}(x)\; v_{ij} = x,\qquad \forall x\in F$$
and
$$\forall j\notin {i_1,...,i_m},\quad \phi_j(x) = 0,\qquad \forall x\in F. $$
\item[\textbf{(f)}]\;\; \textbf{Smoothness:}
The coordinate functions $\phi_i$ are $\mathcal{C}^{\infty}$.
\end{itemize}
\subsection{Spherical barycentric coordinates}
Spherical barycentric coordinates represent another variant of barycentric coordinates that express a point $x$ inside an arbitrary spherical polygon $P$ as a positive linear combination of $P$'s vertices. They were studied in a spherical triangle by M\"obius \cite{11} (1846) and introduced to computer graphics by Alfeld et al \cite{1} (1996). These works are limited to triangles on the sphere or on surfaces like-sphere, where the resulting coordinates are unique because of the linear independence of the vertices. Next, Ju et al \cite{17} (2005) extended them to arbitrary convex polygons by appliying Stokes' theorem to the dual of a polyhedral cone bounded by rays whose end points are the vertices of a convex spherical polygon. These coordinates were called 'vector coordinates', and are given as ratios of areas of certain dual faces. However, they are only limited to convex polygons.
Later, Langer et al \cite{10} (2006) developed a new construction of spherical barycentric coordinates of a point x inside an arbitrary spherical polygon P by using the gnomonic projection into the tangent plane of the sphere at x. This allowed them to construct 3D Mean Value barycentric coordinates for arbitrary, closed polygonal meshes.
In all of these constructions, the linear precision property is preserved at the cost of sacrificing the partition of unity property.
However, research in this very promising field remains very limited.
In this work, we preserve the linear precision property with the resulting sacrifices of partition of unity. The relaxed property proposed by Alfred et al \cite{1}
\begin{equation}
\sum_{i=1}^{n} \phi_i(x) \geqslant 1,\qquad \forall x\in P
\label{moneq1}
\end{equation}
is rather a consequence of the linear precision property than a condition, indeed
$$x = \sum_{i=1}^{n} \phi_i(x)\; v_i$$
hence
$$1= \parallel x \parallel = \parallel \sum_{i=1}^{n} \phi_i(x)\; v_i \parallel\leqslant \sum_{i=1}^{n} \parallel \phi_i(x)\; v_i \parallel= \sum_{i=1}^{n} \phi_i(x)$$
\section{Construction}
Our goal in this section is to find barycentric coordinates with respect to spherical polygons that lie in some hemisphere.\\
A spherical polygon has the same definition as the planar one except that its edges are geodesics (arcs of great circles) connecting the vertices.
\begin{definition}
Let $P$ be a spherical polygon on the unit sphere centred at $0$, with vertices $v_1,v_2,...,v_n$,\; which are ordered anti-clockwise, viewed from outside the sphere.
We call any positive values $\psi_i,\;i=1,...,n$ spherical barycentric coordinates, if they satisfy
$$\sum_{i=1}^{n} \psi_i(x)\; v_i = x,\qquad \forall x\in P.$$
\end{definition}
Let $P$ be a spherical polygon on the unit sphere centered at $0$, with vertices $v_1,v_2,...,v_n$, cyclically indexed ($v_{i+n}=v_i$),\;and $x$ be an interior point of $P$. We consider the (non-spherical) polyhedron $Q=[ v_1, v_2,...,v_n, x, -x ]$,\;bounded by the triangular faces $[x,v_i,v_{i+1}]$\ and $[-x,v_i,v_{i+1}]$,\;$i=1,...,n$\;
(see figure~\ref{figure1}).
\begin{figure}[!h]
\includegraphics[width=14cm,height=10cm]{0.png}
\caption{The spherical polygon $P$ and the polyhedron $Q$}
\label{figure1}
\end{figure}
\newpage
Now we state the following theorem
\begin{theorem}
Spherical barycentric coordinates, for points inside the polygon P, are given by
\begin{equation}
\psi_i(x) = \frac{\phi_{i}(0)}{\phi_{n+2}(0) - \phi_{n+1}(0)}.
\label{moneq2}
\end{equation}
and on the boundary, by $\psi_i(v_j)=\delta_{ij}$ and
\begin{equation}
\left\{
\begin{array}{lll}
\psi_{j}(x)=\dfrac{\phi_{j}(0)}{\phi_{n+2}(0)} & \mbox{ } \\
\psi_{j+1}(x)=\dfrac{\phi_{j+1}(0)}{\phi_{n+2}(0)}& \mbox{} \\
\psi_k(x)=0 & \mbox{for} & k \neq j,j+1
\end{array}
\right.
\label{moneq3}
\end{equation}
where $\phi_{i},\; i=1,...,n+2$\; are any well known 3D barycentric coordinates defined on $Q$ \\
Furthermore, the $\psi_i's$ are linear on the edges of $P$.
\end{theorem}
\begin{proof}
\begin{enumerate}
\item The origine $0$ lies in the interior of $Q$ ($0\in [x,-x]$) and it can be written as a linear combination of the vertices $v_1,v_2,...,v_n,x,-x$ as follows
\begin{equation}
\sum_{i=1}^{n} \phi_{i}(0)\; v_{i} + \phi_{n+1}(0)\; x + \phi_{n+2}(0)\; (-x) = 0
\label{moneq4}
\end{equation}
Hence
\begin{equation}
\sum_{i=1}^{n} \phi_{i}(0)\; v_{i} = (\phi_{n+2}(0) - \phi_{n+1}(0))\;x.
\label{moneq5}
\end{equation}
The point $p=\sum_{i=1}^{n} \phi_{i}(0)\;v_{i}$ on the left-hand side of ~\eqref{moneq5} belongs to the polyhedral cone $P^{\prime}$ of the vertices $v_i, i=1,...,n$. This implies that\; $\phi_{n+2}(0) - \phi_{n+1}(0)\geq 0$, since the point $q=(\phi_{n+2}(0) - \phi_{n+1}(0))\;x$ on the right-hand side would otherwise be outside of $P^{\prime}$.
We claim that\;$\phi_{n+2}(0) - \phi_{n+1}(0) \neq 0$.\;On the contrary, suppose that \;$\phi_{n+2}(0) - \phi_{n+1}(0) = 0$.\;Then we would have
$$p=\sum_{i=1}^{n} \phi_{i}(0)\; v_{i}=0$$
and therefore\;$\phi_{i}(0)=0, i=1,...,n$. Indeed, suppose there is a
$k\in \left\{ 1,...,n \right\}$ such that $\phi_{k}(0)\neq0$, then we would have \;$- v_{k}=\sum_{i=1,i\neq k}^{n} \dfrac{ \phi_{i}(0)}{ \phi_{k}(0)}\; v_{i}$, but the point $q=\sum_{i=1,i\neq k}^{n} \dfrac{ \phi_{i}(0)}{ \phi_{k}(0)}\; v_{i}$ lies in the polyhedral cone $P^{\prime \prime}$ of the vertices $v_1,,...,v_{k-1},v_{k+1},...,v_n$, while $-v_k$ is outside of $P^{\prime \prime}$. A contradiction. Now we conclude from the partition of unity property that\; $\phi_{n+2}(0) = \phi_{n+1}(0)=\dfrac{1}{2}$. The restriction on facets of the boundary proprety $\textbf{(e)}$ shows that this is only possible if the edge\;$[x,-x]$\; coincides with an edge of $Q$, but this would imply that $x$ coincides with a certain vertex $v_k$ of $Q$ and therefore we would have $0=\phi_k(0)=\phi_{n+1}(0)= \dfrac{1}{2}$.\;A contradiction.\\
Since x is in the interior of P, equation~\eqref{moneq5} gives
$$x = \sum_{i=1}^{n} \frac{\phi_{i}(0)}{\phi_{n+2}(0) - \phi_{n+1}(0)}\; v_{i}$$
From\;$\phi_{i}(0)\geqslant 0$ and\; $\phi_{n+2}(0) - \phi_{n+1}(0) > 0$ we conclude that $\psi_i(x)\geqslant 0.$
We now compute these coordinates on the boundary using properties $\textbf{(d)}$ and $\textbf{(e)}$, and show that they are linear on each edge and satisfy Lagrange property.
\item \textbf{On the edges}\\
If a point $x$ inside $P$ approaches the arc $e_j$, then $0$ approaches the interior of the face (triangle) $[-x,v_j,v_{j+1}]$ of $Q$.
The restriction on facets of the boundary proprety $\textbf{(e)}$ shows that in the limit
\begin{equation}
\phi_{i}(0)\; = 0,\;\:for\; i\neq j,j+1,n+2
\label{moneq6}
\end{equation}
and therefore $\psi_i(x)=\dfrac{\phi_{i}(0)}{\phi_{n+2}(0)}$\;for\;$i=j,j+1$ $\quad(\phi_{n+1}(0)= 0\;see\ equation~\eqref{moneq6})$\\
where $\phi_{i}(0),\; i=1,...,n+2$\;are the continuous extensions to the boundary of the 3D barycentric coordinates used in equation~\eqref{moneq2}.
Therefore, equation equation~\eqref{moneq2} becomes
\begin{eqnarray}
x&=&\dfrac{\phi_{j}(0)}{\phi_{n+2}(0)}\;v_j+\dfrac{\phi_{j+1}(0)}{\phi_{n+2}(0)}\;v_{j+1}\\
&=& \psi_{j}(x)\; v_j + \psi_{j+1}(x)\; v_{j+1}
\end{eqnarray}
where $\psi_{j}(x)=\dfrac{\phi_{j}(0)}{\phi_{n+2}(0)}$ and $\psi_{j+1}(x)=\dfrac{\phi_{j+1}(0)}{\phi_{n+2}(0)}$.\\
Spherical barycentric coordinates on the edge $e_j$ are therefore given by
$$
\left\{
\begin{array}{lll}
\psi_{j}(x)=\dfrac{\phi_{j}(0)}{\phi_{n+2}(0)} & \mbox{ } \\
\psi_{j+1}(x)=\dfrac{\phi_{j+1}(0)}{\phi_{n+2}(0)}& \mbox{} \\
\psi_k(x)=0 & \mbox{for} & k \neq j,j+1
\end{array}
\right.
$$
\item \textbf{Lagrange property}\\
Equation $(2.7)$ yields
$$\displaystyle\lim_{x \rightarrow v_i}\;x= \displaystyle\lim_{x \rightarrow v_i}\;\psi_{i}(x)\; v_i + \psi_{i+1}(x)\; v_{i+1}$$
hence
$$v_i= \displaystyle\lim_{x \rightarrow v_i}\;\psi_{i}(x)\;v_i+\displaystyle\lim_{x \rightarrow v_i}\;\psi_{i+1}(x)\; v_{i+1}$$
i.e.
$$\left(1-\displaystyle\lim_{x \rightarrow v_i}\;\psi_{i}(x)\right)\;v_i=\displaystyle\lim_{x \rightarrow v_i}\;\psi_{i+1}(x)\;v_{i+1}$$
but this means that $v_i$ and $v_{i+1}$ are collinear. A contradiction.\\
So we must have\; $1-\displaystyle\lim_{x \rightarrow v_i}\;\psi_{i}(x)=0$\; and\; $\displaystyle\lim_{x \rightarrow v_i}\;\psi_{i+1}(x)=0.$\\
Finally, the fact that \; $\psi_j(x)=0$\; for \;$j\neq i,i+1$,\;(see\ equation~\eqref{moneq5})
completes the proof.
\item \textbf{Linearity on the edges}\\
We have $\forall x\in [v_j,v_{j+1}],\; x=\psi_{j}(x)\; v_j + \psi_{j+1}(x)\; v_{j+1}$. To prove the linearity on the edges, it suffices to verify that
$$\psi_i \left[\psi_{j}(x)\; v_j + \psi_{j+1}(x)\; v_{j+1}\right]=\psi_{j} (x) \psi_i (v_j) + \psi_j (x) \psi_i (v_{j+1})$$ indeed
\begin{itemize}
\item for\;$i\neq j,j+1$,\; we have\;$\psi_{i} (x)=0$,\; hence
\begin{align*}
\psi_{j} (x) \psi_i (v_j) + \psi_j (x) \psi_i (v_{j+1}) &= \psi_{j} (x) \times 0 + \psi_{j} (x) \times 0\qquad (Lagrange\; property \textbf{(d)})\\
&= 0\\
&=\psi_{i} (x)\\
&=\psi_i \left[\psi_{j}(x)\; v_j + \psi_{j+1}(x)\; v_{j+1}\right]\quad(x=\psi_{j}(x)\; v_j + \psi_{j+1}(x)\; v_{j+1})
\end{align*}
\item for $i = j$
\begin{align*}
\psi_{j} (x) \psi_i (v_j) + \psi_j (x) \psi_i (v_{j+1}) &= \psi_{j} (x) \psi_j (v_j) + \psi_j (x) \psi_j (v_{j+1})\qquad (i=j)\\ &= \psi_{j} (x) \times 1 + \psi_{j} (x) \times 0\qquad (Lagrange\; property \textbf{(d)})\\
&= \psi_j (x)\\
&= \psi_i (x)\qquad (i=j)\\
&=\psi_i \left[\psi_{j}(x)\; v_j + \psi_{j+1}(x)\; v_{j+1}\right]\quad(x=\psi_{j}(x)\; v_j + \psi_{j+1}(x)\; v_{j+1})
\end{align*}
\item same for $i = j+1.$
\end{itemize}
\end{enumerate}
\end{proof}
\begin{remark}
The new coordinates are more general than the classical ones in the sens that:
\begin{enumerate}
\item Unlike in the classical approach, they are well defined inside $P$ without need of any continuous extension to the special case where the angle $\theta_i$ between $x$ and any vertex of $P$ is half of pi (i.e $\langle x ,v_i \rangle=0$)
\item The classical approach works only in the case where $\langle x ,v_i \rangle>0$ for all i, while our approach works perfectly regardless of the signs of the $\langle x ,v_i \rangle$
\item In the limit case where all vertices $v_i$ lying on a great circle $C$, our approach computes spherical barycentric coordinates of any point $x$ on the sphere not lying on $C$
\end{enumerate}
\end{remark}
\subsection{Comparison between new and existing coordinates}
We adopt the following notation:\\
NC: The new coordinates introduced above\\
CC: Spherical coordinates introduced by Langer et al \cite{10}\\
CF: Spherical coordinates introduced by Floater \cite{6}\\
MV: Mean value coordinates\\
WC: Wachspress coordinates\\ \\
\subsubsection{Mean value coordinates}
We show that NC and CC mean value coordinates coincide in the case where $\langle x ,v_i \rangle>0$ for all i.
\begin{itemize}
\item 3D mean value coordinates of \cite{4} are given, for a point x inside the kernel of a given polyhedron, by
\begin{equation}
\psi_i(x)=\dfrac{w_i(x)}{w}\quad\quad\quad \omega_i=\dfrac{1}{\parallel v_{i}-x\parallel}\;\sum_{T\in T(v_i)}\;\;\mu_{i,T}
\label{moneq7}
\end{equation}
where $\mu_{i,T}=\dfrac{\beta_{jk}+\beta_{ij}\;\langle n_{ij},n_{jk}\rangle+\beta_{ki}\;\langle n_{ki},n_{jk}\rangle}{\langle 2e_{i},n_{jk}\rangle}$\;and $\beta_{rs}$ is the angle between the two line segments $[x , v_r]$ and $[x ,v_s]$, $n_{rs}=\dfrac{e_r \times e_s}{\parallel e_r \times e_s \parallel}$, $e_i=\dfrac{v_i-x}{\parallel v_i-x \parallel}$ and $T(v_i)$ denotes the set of faces (triangles) $T$ incident to the vertex $v_i$.\\
Now, we consider the two faces $T1=[v_i,v_{i+1},x]$ and $T2=[v_i,-x,v_{i+1}]$ and compute $\mu_{i,T1}$ and $\mu_{i,T2}$.\\
$$\mu_{i,T1}=\dfrac{\beta_{(i+1)(n+1)}+\beta_{i(i+1)}\left\langle \dfrac{v_i \times v_{i+1}}{\parallel v_i \times v_{i+1} \parallel},\dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel}\right\rangle+\beta_{(n+1)i}\left\langle \dfrac{x \times v_{i}}{\parallel x \times v_{i} \parallel},\dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel}\right\rangle}{\left\langle 2v_i,\dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel}\right\rangle}$$
and
$$\mu_{i,T2}=\dfrac{\beta_{(n+2)(i+1)}+\beta_{i(n+1)}\left\langle \dfrac{v_i \times -x}{\parallel v_i \times -x \parallel},\dfrac{-x \times v_{i+1}}{\parallel -x \times v_{i+1} \parallel}\right\rangle+\beta_{(i+1)i}\left\langle \dfrac{ v_{i+1} \times v_{i}}{\parallel v_{i+1} \times v_{i} \parallel},\dfrac{-x \times v_{i+1}}{\parallel -x \times v_{i+1} \parallel}\right\rangle}{\left\langle 2v_i,\dfrac{-x \times v_{i+1}}{\parallel -x \times v_{i+1} \parallel}\right\rangle}$$
hence
$$\mu_{i,T1}=\dfrac{\theta_{i+1}+\beta_{i(i+1)}\left\langle \dfrac{v_i \times v_{i+1}}{\parallel v_i \times v_{i+1} \parallel},\dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel}\right\rangle-\theta_i \cos \alpha_i}{\left\langle 2v_i,\dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel}\right\rangle}$$
and
$$\mu_{i,T2}=\dfrac{\pi-\theta_{i+1}-(\pi-\theta_{i}) \cos \alpha_i -\beta_{i(i+1)}\left\langle \dfrac{v_i \times v_{i+1}}{\parallel v_i \times v_{i+1} \parallel},\dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel}\right\rangle}{\left\langle 2v_i,\dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel}\right\rangle}$$
where $\theta_i=\widehat{(x,v_{i})}$ and $\alpha_i= \widehat {(x \times v_{i},x \times v_{i+1})}$. Hence
\begin{equation}
\mu_{i,T1}+\mu_{i,T2}=\dfrac{\pi (1-\cos \alpha_i)}{\dfrac{\langle 2v_i,v_{i+1} \times x \rangle}{\sin \theta_{i}}}
\label{moneq8}
\end{equation}
The term ${V=\langle v_i,v_{i+1} \times x \rangle}$ in the denominator of equation~\eqref{moneq8} is the volume of the parallelepiped determined by the vectors $v_i,v_{i+1}$ and $x$ and it is given by the formula
\begin{equation}
V=\sqrt{1+2\cos \theta_i \cos \widehat{(v_{i},v_{i+1})} \cos \theta_{i+1}- {\cos^2 \theta_i}-{\cos^2 \widehat{(v_{i},v_{i+1})}}-{\cos^2 \theta_{i+1}}}
\label{moneq9}
\end{equation}
using the identity relating the cross product to the scalar triple product
\begin{equation}
\langle a \times b , c \times d \rangle=\langle a , c \rangle \langle b , d \rangle-\langle a , d \rangle \langle b , c \rangle
\label{moneq10}
\end{equation}
we obtain
$$\langle v_{i} \times x , v_{i+1} \times x \rangle=\langle v_i , v_{i+1}\rangle-\langle v_i , x\rangle\langle v_{i+1} , x \rangle$$
i.e.
$$\parallel v_{i} \times x \parallel \parallel v_{i+1} \times x \parallel \cos \alpha_i = \cos \widehat{(v_i,v_{i+1})} - \cos \theta_i \cos \theta_{i+1}$$
therefore
$$\sin \theta_i \sin \theta_{i+1} \cos \alpha_i = \cos \widehat{(v_i,v_{i+1})} - \cos \theta_i \cos \theta_{i+1}$$
and so
\begin{equation}
\cos \widehat{(v_i,v_{i+1})}= \sin \theta_i \sin \theta_{i+1} \cos \alpha_i+ \cos \theta_i \cos \theta_{i+1}
\label{moneq11}
\end{equation}
By inserting this term into equation~\eqref{moneq9} and after a simple calculation we find
$$V=\sin \theta_i \sin \theta_{i+1} \sin \alpha_i$$
Now, we have
$$\mu_{i,T1}+\mu_{i,T2}= \dfrac{\pi\; {\sin^2 \dfrac{\alpha_i}{2}}}
{\dfrac{\sin \theta_i \sin \theta_{i+1} 2 \sin \dfrac{\alpha_i}{2} \cos \dfrac{\alpha_i}{2}}{\sin \theta_{i+1}}}$$
i.e.
$$\mu_{i,T1}+\mu_{i,T2}= \dfrac{\pi\; {\tan \dfrac{\alpha_i}{2}}} {2 \sin \theta_i}$$\\
we do the same for the faces $T3=[v_{i},x,v_{i-1}]$ and $T4=[v_i,v_{i-1},-x]$ and we find
$$\mu_{i,T3}+\mu_{i,T4}= \dfrac{\pi\; \tan \dfrac{\alpha_{i-1}}{2}} {2 \sin \theta_i}$$
Now, the weight of the origin $0$ with respect to the vertex $v_i$ is given by
$$\omega_i=\mu_{i,T1}+\mu_{i,T2}+\mu_{i,T3}+\mu_{i,T4}= \dfrac{\pi\; (\tan \dfrac{\alpha_i}{2}+\tan \dfrac{\alpha_{i-1}}{2})} {2 \sin \theta_i}$$
we need to compute the weights of the vertices $x$ and $-x$. In the same way we get
$$\mu_{n+2,T2}-\mu_{n+1,T1}=\dfrac{\pi\left\langle \dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel},\dfrac{v_{i} \times v_{i+1}}{\parallel v_{i} \times v_{i+1} \parallel}\right\rangle+\pi\left\langle \dfrac{x \times v_{i}}{\parallel x \times v_{i} \parallel},\dfrac{v_{i} \times v_{i+1}}{\parallel v_{i} \times v_{i+1} \parallel}\right\rangle}{\left\langle 2v_i,\dfrac{v_{i+1} \times x}{\parallel v_{i} \times v_{i+1} \parallel}\right\rangle}$$
i.e.
$$\mu_{n+2,T2}-\mu_{n+1,T1}=\dfrac{\pi\left\langle \dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel},v_{i} \times v_{i+1}\right\rangle+\pi\left\langle \dfrac{x \times v_{i}}{\parallel x \times v_{i} \parallel},v_{i} \times v_{i+1}\right\rangle}{\left\langle 2v_i,v_{i} \times v_{i+1}\right\rangle}.$$
By using equations ~\eqref{moneq10} and ~\eqref{moneq11}, we find
\begin{eqnarray*}
\left\langle \dfrac{v_{i+1} \times x}{\parallel v_{i+1} \times x \parallel},v_{i} \times v_{i+1}\right\rangle &=&\dfrac{\cos \theta_{i+1} \cos \widehat{(v_i,v_{i+1})}-\cos \theta_{i}}{\sin \theta_{i+1}}\\
&=&\dfrac{\cos \theta_{i+1} \sin \theta_i \sin \theta_{i+1} \cos \alpha_i-\cos \theta_{i} \cos^2 \theta_{i+1}}{\sin \theta_{i+1}}\\
&=& \dfrac{\cos \theta_{i+1} \sin \theta_i \sin \theta_{i+1} \cos \alpha_i-\cos \theta_{i} \sin^2 \theta_{i+1}}{\sin \theta_{i+1}}\\
&=&\cos \theta_{i+1} \sin \theta_i \cos \alpha_i-\cos \theta_{i} \sin \theta_{i+1}
\end{eqnarray*}
and
$$\left\langle \dfrac{x \times v_{i}}{\parallel x \times v_{i} \parallel},v_{i} \times v_{i+1}\right\rangle=\cos \theta_{i} \sin \theta_{i+1} \cos \alpha_i-\cos \theta_{i+1} \sin \theta_{i}$$
hence
$$\mu_{n+2,T2}-\mu_{n+1,T1}=\dfrac{\pi \cot \theta_{i+1}-\pi \cot \theta_{i+1}\cos \alpha_i+\pi \cot \theta_{i}-\pi \cot \theta_{i}\cos \alpha_i}{2\sin \alpha_i}$$
And then
\begin{eqnarray*}
\mu_{n+2,T2}-\mu_{n+1,T1}&=&\dfrac{\pi (\cot \theta_{i+1}+\cot \theta_{i})(1-\cos \alpha_i)}{2\sin \alpha_i} \\
&=&\dfrac{\pi}{2} (\cot \theta_{i+1}+\cot \theta_{i}) \tan \dfrac{\alpha_i}{2}
\end{eqnarray*}
now
\begin{eqnarray*}
w_{n+2}-w_{n+1}&=&\sum_{i=1}^{n} \mu_{n+2,T2}-\mu_{n+1,T1}\\
&=& \sum_{i=1}^{n} \dfrac{\pi}{2} (\cot \theta_{i+1}+\cot \theta_{i}) \tan \dfrac{\alpha_i}{2}\\
&=&\dfrac{\pi}{2} \left(\sum_{i=1}^{n} \cot \theta_{i} \tan \dfrac{\alpha_i}{2}+\sum_{i=1}^{n} \cot \theta_{i+1} \tan \dfrac{\alpha_i}{2}\right)
\end{eqnarray*}
by setting $k=i+1$ and use the fact that $v_{i+n}=v_i$, we get
\begin{eqnarray*}
w_{n+2}-w_{n+1}&=&\dfrac{\pi}{2} \left(\sum_{i=1}^{n} \cot \theta_{i} \tan \dfrac{\alpha_i}{2}+\sum_{k=2}^{n+1} \cot \theta_{k} \tan \dfrac{\alpha_{k-1}}{2}\right)\\
&=&\dfrac{\pi}{2} \sum_{i=1}^{n}\cot \theta_{i} \left(\tan \dfrac{\alpha_i}{2}+\tan \dfrac{\alpha_{i-1}}{2}\right)
\end{eqnarray*}
The new coordinates $\psi_i(x)$ are therefore given in terms of the classical coordinates $\lambda_i(x)$ by
\begin{eqnarray*}
\psi(x)=\frac{\phi_{i}(0)}{\phi_{n+2}(0) - \phi_{n+1}(0)}=\dfrac{w_i}{w_{n+2}-w_{n+1}}&=&\dfrac{\dfrac{\pi\; \left(\tan \dfrac{\alpha_i}{2}+\tan \dfrac{\alpha_{i-1}}{2}\right)} {2 \sin \theta_i}}{\dfrac{\pi}{2} \sum_{i=1}^{n} \cot \theta_{i} \left( \tan \dfrac{\alpha_i}{2}+\tan \dfrac{\alpha_{i-1}}{2}\right)}\\ \\
&=&\lambda_i(x)
\end{eqnarray*}
\item figure ~\ref{figure2} provides a visual example which confirms that both coordinates coincide.
\end{itemize}
\begin{figure}[!h]
\includegraphics[width=14cm,height=10cm]{1.png}
\caption{Contour lines of the MV with respect to the vertex $v_1$ for values of $\psi_1$ in $[0.09,0.10]$, $[0.11,0.12]$, $[0.17,0.18]$, $[0.23,0.24]$, $[0.30,0.29]$ and $[0.35,0.36]$}
\label{figure2}
\end{figure}
\subsubsection{Wachspress coordinates}
\begin{figure}[!h]
\includegraphics[width=14cm,height=10cm]{2.png}
\caption{Contour lines of the WC with respect to the vertex $v_1$ for values of $\psi_1$ in $[0.09,0.10]$, $[0.11,0.12]$, $[0.17,0.18]$, $[0.23,0.24]$, $[0.30,0.29]$ and $[0.35,0.36]$}
\label{figure3}
\end{figure}
\newpage
Figure ~\ref{figure3} (respectively ~\ref{figure4}), provides contour lines of WC for NC and CC (respectively NC and CF). The coordinates seem to differ: we intend to give the proof of the general case in the future.
\begin{figure}[!h]
\includegraphics[width=14cm,height=10cm]{3.png}
\caption{Contour lines of the WCF with respect to the vertex $v_1$ for values of $\psi_1$ in $[0.09,0.10]$, $[0.11,0.12]$, $[0.17,0.18]$, $[0.23,0.24]$, $[0.30,0.29]$ and $[0.35,0.36]$}
\label{figure4}
\end{figure}
\newpage
\section{Conclusion}
The new approach gives us a direct relationship between spherical barycentric coordinates and 3D barycentric coordinates via the origin $0$. Their resulting coordinates are more general than the classical ones.\\
\newpage
\bibliographystyle{elsarticle-num}
|
1807.00321
|
\section{Introduction}
Polynomial complementarity problems and polynomial variational inequalities have been recently investigated by many authors, see, for example
\cite{Gowda16,HYY2015a,LLP2018,SongQi2015} and the references therein. These problems are natural extensions of affine variational inequalities (see \cite{LTY2005} and the references therein) and special cases of weakly homogeneous variational inequalities introduced by Gowda and Sossa \cite{GoSo18}.
In the recent past, several authors studied different properties of polynomial complementarity problems, tensor complementarity problems and tensor variational inequalities, which are subclasses of polynomial variational inequalities. In particular, the solvability, the global uniqueness, and the boundedness of the solution sets have been studied in \cite{BHW2016,Gowda16,LHL2017,SongYu2016,WHQ18}. In \cite{Hieu18}, the author discussed the upper semicontinuity and the finite-valuedness of the solution maps of tensor complementarity problems.
Very recently, the authors \cite{LLP2018} investigated the genericity and the H\"{o}lder stability for semi-algebraic variational inequalities where the related maps are the sums of semi-algebraic maps and parametric vectors.
In this paper, we investigate several properties of the solution maps of variational inequalities with polynomial data. First, we introduce the $R_0$-property and show that it plays an important role in the investigation. Since polynomial maps are weakly homogeneous, the normalization argument (see, e.g. \cite{AusTeb2003,GoSo18,OettliYen95}) can be applied. Several facts on the local boundedness and the upper semicontinuity of the solution maps are shown. Second, we obtain a result on the existence of solutions under the copositivity. We develop the results on solution stability of copositive linear complementarity problems and affine variational inequalities in \cite{CPS1992,LTY2005} for copositive polynomial variational inequalities.
Third, when the constraints are polynomial, techniques from semi-algebraic geometry (see, e.g. \cite{BCF98}) and differential geometry (see, e.g. \cite{Loring_2010}) can be used. Under some mild
conditions of the constraint set, we prove the genericity of the $R_0$-property and the finite-valuedness of the solution maps.
The present paper is organized as follows: In the next section, we give a short introduction to variational inequalities, asymptotic cones, polynomial maps, and semi-algebraic sets. In Section \ref{sec:usc}, we investigate the $R_0$-property and the upper semicontinuity of the solution maps. In Section \ref{sec:exist}, we prove some facts on the solution existence and stability of copositive polynomial variational inequalities. In the last section, two results on genericity are shown.
\section{Preliminaries
This section gives a short introduction to variational inequalities, asymptotic cones, polynomial maps, and semi-algebraic sets.
\subsection{Variational inequalities and asymptotic cones}
The usual scalar product of two vectors $x, y\in\R^n$ is denoted by $\langle x,y\rangle$. Let $K$ be a nonempty closed convex subset of $\R^n$ and $F: \R^n\to\R^n$ be a continuous vector-valued map. The \textit{variational inequality} defined by $K$ and $F$ is the problem:
$$
{\rm Find} \ x\in K\ {\rm such\ that}\ \langle
F(x),y-x\rangle\geq 0,\ \forall y\in K.
$$
The problem and the corresponding solution set are denoted by $\VI(K,F)$ and $\Sol(K,F)$, respectively.
By the continuity of $F$ and the closedness of $K$, it is not difficult to check that $\Sol(K,F)$ is closed.
If $F$ is a polynomial map then $\VI(K,F)$ is called a \textit{semi-polynomial variational inequality}. Furthermore, if $K$ is defined by finitely many polynomial equations and inequalities, then the problem is a \textit{polynomial variational inequality}.
\begin{theorem}\label{thm:HS}{\rm(The Hartman-Stampacchia Theorem, \cite[Chapter 1, Theorem 3.1]{KindStam})}
If $K$ is compact, then the solution set $\Sol(K,F)$ is nonempty.
\end{theorem}
Let us recall that a nonempty set $C\subset \R^m$ is called a \textit{cone} if $\lambda>0$ and $x\in C$ then $\lambda x\in C$. The cone $C$ is bounded if and only if $C=\{0\}$. Note that $C$ is a cone if and only if $\R^m\setminus C$ is a cone. We denote by $\inte C$ and $C^*$ the interior and the dual cone of $C$, respectively.
When (the closed convex set) $K$ is a cone, the \textit{complementarity problem} defined by $K$ and $F$, denoted by $\CP(K,F)$, is to find a vector $x\in\R^n$
satisfying the following conditions:
\begin{equation}\label{CP} x\in K, \
F(x)\in K^*, \ \langle
F(x),x\rangle=0.
\end{equation}
In this setting, it is known that a vector $x$ solves $\CP(K,F)$ if and only if $x$ solves $\VI(K,F)$ \cite[Proposition 1.1.3]{FaPa03}. Therefore, the solution set of $\CP(K,F)$ is also denoted by $\Sol(K,F)$.
\begin{remark} It is easy to see that a vector $x$ solves $\CP(K,F)$ if and only if there is $\lambda\in\R^n$ such that
\begin{equation}\label{CP_sys1} x\in K, \ F(x)-\lambda=0, \ \langle
\lambda,x\rangle=0, \ \langle
\lambda,y\rangle\geq 0 \ \forall y\in K.
\end{equation}
\end{remark}
\begin{remark} If $K$ is a cone and $F$ is homogeneous of degree $d>0$, i.e. $F(tx)=t^{d}F(x)$ for all $t> 0$ and $x\in\R^n$, then the solution set of $\CP(K,F)$ contains $0$ and is a closed cone.
\end{remark}
The \textit{asymptotic cone} of $K$ is defined by
$$
K^{\infty}=\left\lbrace v\in\R^n:\exists t_k\to \infty, \exists x_k\in K \text{ with } \lim_{k\to\infty} \frac{x_k}{t_k}=v\right\rbrace.
$$
By the convexity of $K$, $K^{\infty}$ coincides with the recession cone of $K$ which
is defined by the set of vectors $v \in\R^n$ such that for some vector
$x\in K$ the ray $\{x+tv: t\geq 0\}$ is contained in $K$ \cite[p.158]{FaPa03}. So, one has $K=K+K^{\infty}$. Recall that the cone $K^{\infty}$ is closed and convex; $K$ is bounded if and only if $K^{\infty}=\{0\}$; if $K$ is a cone then $K^{\infty}=K$.
Let $P$ be a polynomial map, i.e. $P=(P_1,\dots,P_n)$ where $P_l$ is polynomial in $n$ variables, $l=1,\dots,n$. The maximum of the numbers $\deg P_l$ is called the \textit{degree} of the polynomial map $P$ and one denotes $\deg P=d$.
We denote $P^{\infty}=(P_1^{\infty},\dots,P_m^{\infty})$, where $P_l^{\infty}$ is the homogeneous component of degree $d$ of $P_l$, $l=1,\dots,n$. Clearly, $P^{\infty}$ is the \textit{leading term} of the polynomial map $P$, i.e.
$$
P^{\infty}(x)=\lim_{\lambda\to+\infty}\frac{P(\lambda x)}{\lambda^d}, \ \forall x\in\R^n,
$$
and the map $P^{\infty}$ is homogeneous of degree $d$.
\textit{Throughout the paper, we assume that $K$ is a nonempty, closed, and convex set, $P$ is a polynomial map of degree $d$, where $d$ is a positive integer, and let $\Sc:=\Sol(K^{\infty},P^{\infty})$}.
Let $\Po_{d}$ be the linear space of all polynomial maps $Q=(Q_1,\dots,Q_n)$ of degree at most $d$, $m$ be the dimension of $\Po_{d}$, and $X$ be the vector consisting of all monomials degree at most $d$ which is listed by the lexicographic ordering \begin{equation}\label{monomials}
X := (1,x_1,x_2,\dots,x_n, x_1^2, x_1x_2,\dots,x_1x_n,\dots,x_1^d,x_1^{d-1}x_2,\dots,x_n^d)^T.
\end{equation}
For any polynomial map $Q\in\Po_d$, there exists a unique matrix $A\in\R^{n\times m}$,
\begin{equation}\label{A}
A=\begin{bmatrix}
a_{11}& a_{12} & \cdots & a_{1m} \\
a_{21}& a_{22} & \cdots & a_{2m} \\
& & \vdots & \\
a_{n1}& a_{n2} & \cdots & a_{nm}
\end{bmatrix},
\end{equation}
such that $Q(x)=AX$. The norm $\|Q\|$ of $Q$ is defined by the Frobenius norm of the coefficients of
matrix $A$. Let $\{Q^k\}$ be a convergent sequence in $\Po_d$ with $Q^k\to Q$ and $\{x^k\}$ be a convergent sequence in $\R^n$ with $x^k\to \bar x$. Then $Q^k(x^k)$ also is convergent with $Q^k(x^k)\to Q(\bar x)$.
\begin{remark}\label{P_infty}
Let $\{Q^k\}$ be a sequence in $\Po_{d}$ with $Q^k\to P$. Assume that $Q^k(x)=A^kX$ and $P(x)=BX$. Clearly, one has $A^k \to B$. It follows that $(Q^k)^{\infty}\to P^{\infty}$.
\end{remark}
The $R_0$-property of linear complementarity problems and affine variational inequalities has been investigated in \cite{JaTi87,OettliYen95} and \cite[p.189]{FaPa03}. We introduce a generalization of this property for semi-polynomial variational inequalities.
\begin{definition} One says that the problem $\VI(K,P)$ has the $R_0$-\textit{property} or $(K,P)$ is an \textit{$R_0$-pair} if the cone $\Sc$ is trivial, i.e., $\Sol(K^{\infty},P^{\infty})=\{0\}$.
\end{definition}
\begin{remark}\label{P_Q} Let $Q\in\Po_{d-1}$. Then $(P+Q)^{\infty}=P^{\infty}$. Furthermore, if $(K,P)$ is an $R_0$-pair, then $(K,P+Q)$ also is an $R_0$-pair.
\end{remark}
\begin{remark}\label{K_compact} If $K$ is compact, then $K^{\infty}=\{0\}$, and hence $(K,P)$ is an $R_0$-pair.
\end{remark}
Let $\Ri_0(K,d)$ be the set of all polynomial maps $Q$ of degree $d$ such that $(K,Q)$ is an $R_0$-pair.
\begin{remark}The set $\Ri_0(K,d)$ is a cone. Indeed, for each $t>0$ and each polynomial map $Q$, one has
$(tQ)^{\infty} =tQ^{\infty}$.
This implies that $$\Sol(K^{\infty},(tQ)^{\infty})=\Sol(K^{\infty},tQ^{\infty}).$$
Moreover, it is easy to check that above sets coincide with $\Sol(K^{\infty},Q^{\infty})$, i.e.,
$$\Sol(K^{\infty},(tQ)^{\infty})=\Sol(K^{\infty},tQ^{\infty})=\Sol(K^{\infty},Q^{\infty}).$$
Therefore, $\Sol(K^{\infty},Q^{\infty})$ is bounded iff $\Sol(K^{\infty},(tQ)^{\infty})$ is bounded, for any $t>0$. This implies that $\Ri_0(K,d)$ is a cone in $\Po_{d}$.
\end{remark}
This paper mostly focuses on two solution maps $\Sol$ and $\Sol_P$ which are respectively defined by
\begin{equation}\label{Sol}
\Sol:\Po_{d}\rightrightarrows \R^n, \ Q\to \Sol(K,Q),
\end{equation}
and
\begin{equation}\label{Sol_P}
\Sol_{P}:\R^n\rightrightarrows\R^n, \ p\to \Sol(K,P+p).
\end{equation}
\begin{remark}\label{cl_graph}
By definition, it is easy to see that the map $\Sol$ is closed, i.e. the graph of $\Sol$, which is defined by
$$\gph(\Sol)=\big\{(Q,x)\in \Po_{d}\times \R^n: x\in \Sol(K,Q)\big\},$$
is closed in $\Po_{d}\times \R^n$. Similarly, the map $\Sol_{P}$ also is closed.
\end{remark}
\subsection{Semi-algebraic sets and the LICQ}
Recall that a set in $\R^n$ is \textit{semi-algebraic}, if it is the union of finitely many subsets of the form
\begin{equation*}\label{basicsemi}
\big\{x\in \R^n\,:\,f_1(x)=\dots=f_\ell(x)=0,\ g_{\ell+1}(x)<0,\dots,g_m(x)<0\big\},
\end{equation*}
where $\ell,m$ are natural numbers, and $f_1,\dots, f_\ell, g_{\ell+1},\dots,g_m$ are polynomials with real coefficients. For further details on semi-algebraic geometry, we refer to \cite{BCF98}.
Suppose that $S_1\subset \R^m$ and $S_2\subset \R^n$ are semi-algebraic sets. A vector-valued map $G:S_1\to S_2$ is said
to be semi-algebraic \cite[Definition 2.2.5]{BCF98}, if its graph is a semi-algebraic subset in $\R^m\times\R^n$.
Let $S\subset \R^m$ be a semi-algebraic set. Then there exists a decomposition of $S$ into a disjoint union of semi-algebraic subsets \cite[Theorem 2.3.6]{BCF98} $$S=\bigcup_{i=1}^sS_i,$$ where each $S_i$ is semi-algebraically homeomorphic to $(0,1)^{d_i}$, i.e., there is a map $h:S_i \to (0,1)^{d_i}$ such that $h$ is semi-algebraic and homeomorphic. Let $(0,1)^{0}$ be a point and $(0,1)^{d_i}\subset \R^{d_i}$ be the set of points $x=(x_1,\dots,x_{d_i})$ such that $x_j\in (0,1)$ for all $j=1,\dots,d_i$. The \textit{dimension} of $S$ is defined by
$$\dim S:=\max\{d_1,\dots,d_s\}.$$
The dimension is well-defined and does not depend on the decomposition of $S$.
If the semi-algebraic set $S$ is nonempty and $\dim S=0$, then $S$ has finitely many points. If $\dim(\R^m\setminus S)<m$, then $S$ is \textit{generic} in $\R^m$ in the sense that $S$ contains a countable intersection of dense and open sets in $\R^m$.
Let $X\subset \R^m$ and $Y\subset \R^n$ be manifolds.
The tangent spaces of $X$ at $x$ and of $Y$ at $y$ are denoted by $T_xX$ and $T_yY$, respectively. Consider the differentiable map $\Phi:X \to Y$. A point $y \in \R^n$ is called a \textit{regular value} of $\Phi$ if either the level set $\Phi^{-1}(y)$ is empty or the derivative
map
$$D\Phi(x): T_xX\to T_yY$$
is surjective at every point $x \in \Phi^{-1}(y)$. So, $y$ is a regular value of $\Phi$ if and only if $\rank D\Phi(x)=n$ for all $x\in \Phi^{-1}(y)$.
\begin{remark
Consider a differentiable semi-algebraic map $\Phi:X \to \R^n$, where $X\subset \R^n $. Assume that $y \in \R^n$ is a regular value of $\Phi$ and $\Phi^{-1}(y)$ is nonempty. Applying the regular level set theorem \cite[Theorem 9.9]{Loring_2010}, one has $\dim\Phi^{-1}(y)=0$; this implies that $\Phi^{-1}(y)$ has finitely many points.
\end{remark}
\begin{remark
Let $\Phi : \R^m\times X\to \R^n$ be a differentiable semi-algebraic map, where $X\subset \R^n $. Assume that $y \in \R^n$ is a regular value of $\Phi$. The Sard theorem with parameter \cite[Theorem 2.4]{DHP16} says that there is a generic
semi-algebraic set $\Sa\subset\R^m$ such that, for every $p\in \Sa$, $y$ is a regular value of the map
$\Phi_p:X\to \R^n$ with $x \mapsto \Phi(p,x).$
\end{remark}
Suppose that the constraint $K$ is represented by finitely many convex polynomial functions $g_i(x),i\in I,$ and finitely many affine functions $h_j(x),j\in J,$ as follows:
\begin{equation}\label{K}
K = \left\lbrace x \in \R^n : g_i(x) \leq 0, i\in I, \; h_j(x) = 0, j \in J\right\rbrace.
\end{equation}
For each index set $\alpha \subset I=\{1,\dots,s\}$, the \textit{pseudo-face} $K_{\alpha}$ of $K$ is defined by
$$
K_{\alpha}=\left\lbrace x\in \R^n:g_i(x) =0, \forall i\in\alpha, g_i(x) <0, \forall i\notin\alpha, h_j(x) = 0, \forall j\in J\right\rbrace.
$$
The number of pseudo-faces of $K$ is finite and these pseudo-faces establish a disjoint decomposition of $K$. So, we obtain
\begin{equation}\label{Sol_decom}
\Sol(K,F)=\displaystyle\bigcup_{\alpha\subset I}\left[ \Sol(K,F)\cap K_{\alpha}\right].
\end{equation}
For each $x\in K$, denote by $I(x)$ the active index set at $x$ which is defined by
$I(x)= \{i\in I : g_i(x) = 0\}.$
One says that $K$ satisfies the \textit{linearly independent constraint qualification} ($\rm LICQ$ for short), if the gradient vectors $$\{\nabla g_i(x), \nabla h_j(x), i\in I(x),j\in J\}$$ are linearly independent for all point $x\in K$. If the $\rm LICQ$ holds on $K$, then the Abadie constraint qualification (see, e.g. \cite[p.~17]{FaPa03}) also holds on $K$. Hence, the Karush-Kuhn-Tucker conditions can be applied (see, e.g. \cite[Proposition 1.3.4]{FaPa03}).
\section{Upper semicontinuity of solution maps}\label{sec:usc}
This section focuses on upper semicontinuity of the solution maps $\Sol$ and $\Sol_P$ given by \eqref{Sol} and \eqref{Sol_P}, respectively. A close relation between the upper semicontinuity and the $R_0$-property is shown.
\subsection{Local boundedness}
The following theorem describes a relation between the $R_0$-property and the boundedness of certain solution sets.
\begin{proposition}\label{bounded1} Consider the following statements:
\begin{description}
\item[\rm(a)] $(K,P)$ is an $R_0$-pair;
\item[\rm(b)] $\Sol(K,P+Q)$ is bounded, for every $Q\in\Po_{d-1}$;
\item[\rm(c)] For any bounded open set $\Oo\subset \Po_{d-1}$, the following set is bounded:
$$S_{\Oo}:=\bigcup_{Q\in \Oo} \Sol(K,P+Q).$$
\end{description}
One has $ \rm(a) \Rightarrow \rm(c) \Rightarrow \rm(b)$. Moreover, if $K$ is a cone then the three statements are equivalent.
\end{proposition}
\begin{proof} $\rm(a) \Rightarrow \rm(c)$ Assume that $(K,P)$ is an $R_0$-pair and, on the contrary, there is a bounded open set $\Oo$ such that $S_{\Oo}$ is unbounded. There exists an unbounded sequence $\{x^k\}\subset K$ and a sequence
$\{Q^k\}\subset\Oo$
such that $x^k\in\Sol(K,P+Q^k)$ for every $k$. By the unboundedness of $\{x^k\}$, without loss of generality, we can assume that $\|x^k\|^{-1}x^k\to\bar x $ with $\|\bar x\|=1.$ Because of the boundedness of $\{Q^k\}$, we can suppose that $Q^k\to \overline Q$ with $\deg \overline Q\leq d-1$.
By assumptions, one has
\begin{equation}\label{VI_k}
\left\langle (P+Q^k)(x^k),y-x^k \right\rangle\geq 0, \ \forall y\in K.
\end{equation}
Let $u\in K$ be fixed. Then, for every $v\in K^{\infty}$, we have $u+\|x^k\|v\in K$ for any $k$. From \eqref{VI_k}, we deduce that
$$\left\langle (P+Q^k)(x^k),u+\|x^k\|v-x^k \right\rangle\geq 0.$$
Dividing this inequality by $\|x^k\|^{d+1}$ and letting $k\to+\infty$, we obtain
$$\left\langle (P+\overline Q)^{\infty}(\bar x),v-\bar x \right\rangle=\left\langle P^{\infty}(\bar x),v-\bar x \right\rangle\geq 0,$$
and hence it follows that $\bar x \in \Sc=\{0\}$. As $\|\bar x\|=1$, this is a contradiction; therefore, $S_{\Oo}$ must be bounded. The assertion $\rm(c)$ is proved.
$\rm(c) \Rightarrow \rm(b)$ Assume that $\rm(c)$ holds, but there is $\overline Q\in\Po_{d-1}$ such that $\Sol(K,P+\overline Q)$ is unbounded. Then there exists a bounded and open set $\Oo$ such that $\overline Q\in \Oo$. Clearly,
$$\Sol(K,P+\overline Q)\subset S_{\Oo}.$$
This is impossible, because $S_{\Oo}$ is bounded; hence $\rm(b)$ follows.
Assuming that $K$ is a cone, we need only to prove that $\rm(b) \Rightarrow \rm(a)$. Consider $Q:=P-P^{\infty}\in\Po_{d-1}$. Then the assertion $\rm (b)$ implies that $\Sol(K,P^{\infty})$ is a bounded cone; consequently, we get $\Sol(K,P^{\infty})=\{0\}$. The proof is complete. \qed
\end{proof}
The following lemma will be used in the proof of Theorem \ref{bounded2}.
\begin{lemma}\label{open_cone} The cone $\Ri_0(K,d)$ is open in $\Po_d$.
\end{lemma}
\begin{proof} We need only to prove that $\Po_d\setminus\Ri_0(K,d)$ is closed. Let $\{Q^k\}$ be a sequence in $\Po_d\setminus\Ri_0(K,d)$ such that $Q^k\to Q$. Then $(Q^k)^{\infty}\to Q^{\infty}$ by Remark \ref{P_infty}. Moreover, for each $k$, $\Sol(K^{\infty},(Q^k)^{\infty})$ is unbounded. Hence, there exists an unbounded sequence $\{x^k\}$ such that, for each $k$, $x^k\in\Sol(K^{\infty},(Q^k)^{\infty})$. Without loss of generality we can assume that $x^k\neq 0$ for all $k$ and
$\|x^k\|^{-1}x^k\to\bar x$ with $\|\bar x\|=1.$
By \eqref{CP_sys1}, for each $y\in K^{\infty}$, we get
$$\langle
(Q^k)^{\infty}(x^k),x^k\rangle=0, \ \langle
(Q^k)^{\infty}(x^k),y\rangle\geq 0.$$
Dividing the above equation and inequality by, respectively, $\|x^{k}\|^{d_k+1}$ and $\|x^{k}\|^{d_k}$, where $d_k$ is the degree of $Q^k$, and letting $k\to+\infty$, one gets
$$\langle
Q^{\infty}(\bar x),\bar x\rangle=0, \ \langle
Q^{\infty}(\bar x),y\rangle\geq 0.$$
This leads to $\bar x \in \Sol(K^{\infty}, Q^{\infty})$.
As $\|\bar x\|=1$, we have $\bar x\neq 0$. It follows that $Q$ belongs to $\Po_d\setminus\Ri_0(K,d)$. The proof is complete. \qed
\end{proof}
Denote by $\B(0,\varepsilon)$ the open ball in $\Po_{d}$ with center at $0$ and radius $\varepsilon$. The closure of this ball is denoted by $\overline\B(0,\varepsilon)$.
The following theorem establishes a result on the local boundedness of the solution map $\Sol$ given in \eqref{Sol}.
\begin{theorem}\label{bounded2} If $(K,P)$ is an $R_0$-pair, then the map $\Sol$ is locally bounded at $P$, i.e. there exists $\varepsilon>0$ such that the set
$$O_{\varepsilon}:=\bigcup_{Q\in \B(0,\varepsilon)} \Sol(K,P+Q)$$
is bounded. Consequently, $\Sol(K,P+Q)$ is bounded for every $Q\in \B(0,\varepsilon)$.
\end{theorem}
\begin{proof} According to Lemma \ref{open_cone}, the cone $\Ri_0(K,d)$ is open in $\Po_d$. Then there is some $\varepsilon$ small enough such that
\begin{equation}\label{PB}
P+\overline\B(0,\varepsilon)\subset \Ri_0(K,d).
\end{equation}
Suppose on the contrary, $O_{\varepsilon}$ is unbounded. Then there exists an unbounded sequence $\{x^k\}$ and a sequence $\{Q^k\}\subset\B(0,\varepsilon)$ such that $x^k\in\Sol(K,P+Q^k)$, $x^k\neq 0$ for every $k$, and
$\|x^k\|^{-1}x^k\to\bar x$ with $\|\bar x\|=1$.
By the compactness of $\overline\B(0,\varepsilon)$, without loss of generality, we can assume that $Q^k\to Q$. Clearly, $P+Q^k\to P+ Q$ and
\begin{equation}\label{PQ}
P+ Q \in P+\overline\B(0,\varepsilon).
\end{equation}
By repeating the argument of the proof of Proposition \ref{bounded1}, we can show that $\bar x\in\Sol\left(K^{\infty}, (P+Q)^{\infty}\right)$. From \eqref{PB} and \eqref{PQ}, $(K, P+Q)$ is an $R_0$-pair. This gives $\bar x=0$ which contradicts $\|\bar x\|=1$. Therefore, $O_{\varepsilon}$ is bounded.
The proof is complete. \qed
\end{proof}
\subsection{Upper semicontinuity}
A set-valued map $\Psi:X\rightrightarrows Y$ between two topological spaces $X,Y$ is \textit{upper semicontinuous} at $x\in X$ iff for any open set $V\subset Y$ such that $\Psi(x)\subset V$ there is a neighborhood $U$ of $x$ such that $\Psi(x')\subset V$ for all $x'\in U$. If $\Psi$ upper semicontinuous at every $x\in X$ then one says that $\Psi$ is upper semicontinuous on $X$. Recall that if $\Psi$ is closed, i.e. its graph is a closed set in $X\times Y$, and locally bounded at $x$ then $\Psi$ is upper semicontinuous at $x$ (see, e.g., \cite[p.139]{FaPa03}).
\begin{proposition}\label{usc_1} If $(K,P)$ is an $R_0$-pair and $\Sol(K,P)\neq\emptyset$, then the map $\Sol$ is upper semicontinuous at $P$.
\end{proposition}
\begin{proof} Assume that $(K,P)$ is an $R_0$-pair and $\Sol(K,P)\neq\emptyset$. From Remark \ref{cl_graph} and Theorem \ref{bounded2}, the map $\Sol$ is closed and locally bounded at $P$. Hence, $\Sol$ is upper semicontinuous at $P$. \qed
\end{proof}
\begin{corollary
If $K$ is compact, then the solution map $\Sol$ is upper semicontinuous on $\Po_{d}$.
\end{corollary}
\begin{proof} Suppose that $K$ is compact. Let $Q\in\Po_{d}$. From Remark \ref{K_compact}, $\VI(K,Q)$ has the $R_0$-property. Besides, Theorem \ref{thm:HS} says that $\Sol(K,Q)$ is nonempty. According to Proposition \ref{usc_1}, $\Sol$ is upper semicontinuous at $Q$. \qed
\end{proof}
\begin{corollary}\label{usc_3} Assume that $(K,P)$ is an $R_0$-pair and $p\in\R^n$. If the set $\Sol(K,P+p)$ is nonempty, then the solution map $\Sol_P$ is upper semicontinuous at $p$.
\end{corollary}
\begin{proof} Let $p\in\R^n$ be given. Clearly, $(K,P+p)$ is an $R_0$-pair. According to Proposition \ref{usc_1}, $\Sol$ is upper semicontinuous at $P+p$. Then for any open set $V$ containing $\Sol(K,P+p)$, there exists an open ball $\B(0,\varepsilon)$ such that
$$V \supset\bigcup_{Q\in \B(0,\varepsilon)} \Sol(K,(P+p)+Q)\supset\bigcup_{\|q\|<\varepsilon} \Sol(K,(P+p)+q).$$
So, $U:=\{p+q\in\R^n:\|q\|<\varepsilon\}$ is an open neighbourhood of $p$ in $\R^n$ and $\Sol_{P}(U)\subset V$. It follows that $\Sol_{P}$ is upper semicontinuous at $p$. \qed
\end{proof}
The following theorem gives a sufficient condition for the $R_0$-property. The proof is a modification of one in \cite[Theorem 18.1]{LTY2005}.
\begin{theorem
Assume that $K$ is a cone. If there exists $Q\in\Po_{d-1}$ such that the following two conditions are satisfied:
\begin{description}
\item[\rm(a)] $\Sol(K,P+Q)$ is nonempty and bounded;
\item[\rm(b)] The solution map $\Sol$ is upper semicontinuous at $P+Q$;
\end{description}
then $(K,P)$ is an $R_0$-pair.
\end{theorem}
\begin{proof} Since $K$ is a cone, we have $K=K^{\infty}$. Suppose that there is $Q\in\Po_{d-1}$ such that $\rm(a)$ and $\rm(b)$ hold, but $(K,P)$ is not an $R_0$-pair. Let $0\neq z\in\Sol(K,P^{\infty})$. From \eqref{CP_sys1}, there exists $\lambda\in\R^n$ such that
\begin{equation*}\label{KKT_H0}
P^{\infty}(z)-\lambda=0,\
\langle \lambda,z \rangle=0,\ \langle \lambda,y\rangle\geq 0 \ \; \forall y\in K.
\end{equation*}
For each $t\in(0,1)$, we take $z_t:=t^{-1}z$ and $ \lambda_t:=t^{-d}\lambda$. We prove the existence of $Q_t\in\Po_d$, with $Q_t\to P+Q$ as $t\to 0$, satisfying
\begin{equation}\label{KKT_Ht}
Q_t (z_t)-\lambda_t=0,\
\langle \lambda_t,z_t \rangle=0,\ \langle\lambda_t, y \rangle \geq 0 \ \forall y\in K.
\end{equation}
Suppose that $$P=P^{\infty}+P^{d-1}+\dots+P^{1}+P^{0}$$ and $$Q= Q^{d-1}+\dots+Q^{1}+ Q^{0},$$ where $P^k, Q^k$ are homogeneous polynomial maps of degree $k$ $(k=1,\dots,d-1)$ and $P^{0}\in\R^n,Q^{0}\in\R^n$. The sum $P+Q$ can be written as
\begin{equation}\label{PQ_bar}
P+Q=P^{\infty}+[P^{d-1}+Q^{d-1}]+\cdots+[P^{1}+ Q^{1}]+[P^{0}+Q^{0}].
\end{equation}
Because $z=(z_1,\dots,z_n)$ is nonzero, there exists $l\in\{1,\dots,n\}$ such that $z_l\neq 0$; hence $z^{k}_l\neq 0$ for $k=1,\dots,d$. Take $\overline Q\in\Po_d$ with $$\overline Q(x)=\overline Q^d(x)+\cdots+\overline Q^1(x),$$
where $\overline Q^k$ is a homogeneous polynomial map of degree $k$ defined by
$$\overline Q^k(x)=\left( a_{k1}x_l^{k},\dots,a_{kn}x_l^{k}\right), \ a_{ki}=-\frac{P_i^{k-1}(z)+Q_i^{k-1}(z)}{z_l^{k}}, \ i=1,\dots,n,$$
with $P_i^{k-1}, Q_i^{k-1}$ are the $i-$th components of $P^{k-1}, Q^{k-1}$, respectively.
It is easy to check that \begin{equation}\label{FQ0}
P^{k-1}(z)+Q^{k-1}(z)+\overline Q^{k}(z)=0.
\end{equation}
Choosing
$Q_t=(P+Q)+t\overline Q$, we now prove that the system \eqref{KKT_Ht} is valid. Indeed, the last one in \eqref{KKT_Ht} is obvious. The second one in~\eqref{KKT_Ht} is obtained by
$$\langle \lambda_t,z_t \rangle=\langle t^{-d}\lambda,t^{-1}z \rangle=t^{-d-1}\langle \lambda,z \rangle=0.$$
We now prove the first equality of \eqref{KKT_Ht}. From \eqref{PQ_bar}, we get
$$ \begin{array}{cl}
Q_t(z_t)-\lambda_t&=[(P+Q)+t\overline Q](t^{-1}z) - t^{-d}\lambda \medskip \\
&= t^{-d}\left[P^{\infty} (z)-\lambda\right] +\sum_{k=1}^{d}t^{-(k-1)}\left[ P^{k-1} (z)+Q^{k-1} (z)+\overline Q^{k}(z)\right].
\end{array} $$
This and \eqref{FQ0} imply that $Q_t(z_t)-\lambda_t=0$. Hence, we get $z_t \in \Sol(K,Q_t)$. This holds for all $t\in (0,1)$.
Since $\Sol(K,P+Q)$ is bounded, there is a bounded open set $V$ containing $\Sol(K,P+Q)$. By the upper semicontinuity of $\Sol$ at $P+Q$, there is $\varepsilon>0$ such that $\Sol(K,P+Q+Q')\subset V$ for all $Q'\in\Po_n$ and $\|Q'-(P+Q)\|<\varepsilon$. Taking $t$ small enough such that $\|Q_t-(P+Q)\|<\varepsilon$, we have $\Sol(K,Q_t)\subset V$. So, $z_t\in V$ for every $t>0$ sufficiently small. This is impossible, because $V$ is bounded and $z_t$ is unbounded as $t\to 0$.
The proof is complete. \qed
\end{proof}
\section{Solution existence and stability under copositivity condition}\label{sec:exist}
In this section, we will establish some results on solution existence and stability of semi-polynomial variational inequalities whose involved maps are copositive.
\subsection{Solution existence}
Let $C$ be a nonempty and closed subset of $\R^n$. Note that
$q\in\inte C^*$ if and only if
$\left\langle v,q \right\rangle >0$ for all $v\in C$ and $v\neq 0$ (see, e.g., \cite[Lemma 6.4]{LTY2005}).
Recall that the map $F$ is copositive on $K$
if $\left\langle F(x),x \right\rangle \geq 0$ for all $x\in K$, and
monotone on $K$ if
\begin{equation}\label{mono}
\left\langle F(y)-F(x),y-x\right\rangle \geq 0,\end{equation}
for all $x,y\in K$.
If the inequality in \eqref{mono} is strict for all $y\neq x$, then $F$ is strictly monotone on $K$. If $0\in K$, $F(0)=0$, and $F$ is monotone on $K$, then $F$ is copositive on $K$.
Theorem 6.2 in \cite{GoSo18} gives a result on the solution existence under a copositivity condition along with $\inte(K^*)\neq\emptyset$. We will now improve this result by weakening the interiority condition.
\begin{theorem}\label{cop1} Assume that $0\in K$ and $P$ is copositive on $K$. If $p\in\inte(\Sc^*)$, then $\Sol(K,P+p)$ is nonempty and bounded.
\end{theorem}
\begin{proof} If $K$ is compact then the assertion is obvious. Hence, we suppose that $K$ be unbounded. Let $p\in\inte(\Sc^*)$ be given. For each $k=1,2,\dots$, we denote $$K_k=\{x\in\R^n:x\in K,\|x\|\leq k\}.$$
Clearly, the set $K_k$ is compact. Without loss of generality, we can assume that $K_k$ is nonempty. According to Theorem \ref{thm:HS}, $\VI(K_k,P+p)$ has a solution denoted by $x_k$.
We will show that the sequence $\{x^k\}$ is bounded. Suppose on the contrary that $\{x^k\}$ is unbounded with $x^k\neq 0$, for all $k$, and
$\|x^k\|^{-1}x^k\to\bar x$. Clearly, one has $\bar x \in K^{\infty}$ and $\|\bar x\|=1$.
For each $k$, it is true that
\begin{equation}\label{VI_H}
\left\langle P(x^k)+p,y-x^k \right\rangle \geq 0,
\end{equation}
for all $y\in K_k$. By fixing $y\in K_1$, dividing the inequality in \eqref{VI_H} by $\|x^k\|^{d+1}$ and letting $k\to+\infty$, we obtain
$\left\langle P^{\infty}(\bar x),\bar x \right\rangle\leq 0$. Moreover, by the copositity of $P$, one has $\left\langle P(x^k),x^k \right\rangle\geq 0$. This leads to $\left\langle P^{\infty}(\bar x),\bar x \right\rangle \geq 0$. We thus get $\left\langle P^{\infty}(\bar x),\bar x \right\rangle= 0$.
Let $v\in K^{\infty}\setminus \{0\}$ be fixed. For each $k$, we set
$y_k:=\|x^k\|\|v\|^{-1}v.$
Since $K=K+K^{\infty}$ and $0\in K$, one has $y_k\in K$ for any $k$. It is easy to see that $\|y^k\|=\|x^k\|\leq k$, hence that $y_k\in K_k$.
Now \eqref{VI_H} becomes
$$\left\langle P(x^k)+p,\|x^k\|\|v\|^{-1}v-x^k \right\rangle\geq 0.$$
Dividing this inequality by $\|x^k\|^{d+1}$ and taking $k\to+\infty$, we obtain
$$\left\langle P^{\infty}(\bar x),v \right\rangle\geq \|v\|\left\langle P^{\infty}(\bar x),\bar x \right\rangle.$$
From what has already been proved, we have
$$\bar x\in K^{\infty}, \ \left\langle P^{\infty}(\bar x),\bar x \right\rangle = 0, \ \left\langle P^{\infty}(\bar x),v \right\rangle\geq 0 \ \forall v\in K^{\infty}.$$
This means that $\bar x \in \Sc$.
Since $P$ is copositive, letting $y=0$ in \eqref{VI_H}, one has
$$-\left\langle p,x^k\right\rangle \geq \left\langle P(x^k),x^k \right\rangle\geq 0.$$
Dividing this inequality by $\|x^k\|$, as $k\to+\infty$, we get $\left\langle p,\bar x\right\rangle \leq 0$.
This contradicts the assumption $p\in\inte(\Sc^*)$. Thus, the sequence $\{x^k\}$ must be bounded.
We can assume that $x^k\to \hat x$. We now prove that $\hat x$ solves $\VI(K,P+p)$. Indeed, for any $y\in K$, from \eqref{VI_H}, taking $k\to+\infty$, one has
$$\left\langle P(\hat{x})+p,y-\hat{x} \right\rangle \geq 0.$$
Hence, the nonemptiness of $\Sol(K,P+p)$ is proved.
The boundedness of $\Sol(K,P+p)$ is proved by assuming that there exists an unbounded sequence of solutions $\{x_k\}\subset\Sol(K,P+p)$ with $x^k\neq 0$, for all $k$, and
$\|x^k\|^{-1}x^k\to\bar x$ with $\|\bar x\|=1$.
Applying the normalization argument, we can show that $\bar x \in \Sc$ and $\left\langle p,\bar x\right\rangle \leq 0$. This contradicts the assumption $p\in\inte(\Sc^*)$.
The proof is complete.
\qed
\end{proof}
To illustrate Theorem \ref{cop1}, we provide the following example.
\begin{example} Consider the polynomial variational inequalities $\VI(K,P+p)$, where $K=\R^2_+$, $p=(p_1,p_2)^T\in\R^2$, and $P$ is given by
$$P(x_1,x_2)=\begin{bmatrix}
(x_1-x_2)^2\\
(x_1-x_2)^2
\end{bmatrix}.$$
Clearly, one has $P^{\infty}=P,$ $K^{\infty}=K$, and $P$ is copositive on $K$. Since $$\Sc=\{(x_1,x_2)\in\R^2_+: x_1-x_2=0\},$$
one has
$$\inte(\Sc^*)=\{(p_1,p_2)\in\R^2: p_1+ p_2>0\}.$$
From Theorem \ref{cop1}, $\Sol(K,P+p)$ is nonempty and bounded for any $p\in \inte(\Sc^*)$. In fact, an easy computation shows that
\begin{equation*}\label{SolO}
\Sol_{P}(p_1,p_2)=\left\{\begin{array}{ccc}
L_0 & \text{ if } &p_1=0, \;p_2=0, \\
L_{-p_1}& \text{ if } & p_1=p_2, p_2<0, \\
\{(0,0)\}\cup\{(0,\sqrt{-p_2})\} & \text{ if } & p_1> p_2,p_2<0, \\
\{(0,0)\}\cup\{(\sqrt{-p_1},0)\} & \text{ if } & p_1<0,p_1< p_2,\\
\{(0,0)\} & \text{ if } & \text{ otherwise,}
\end{array}\right.
\end{equation*}
where $$L_{-p_1}=\{(x_1,x_2)\in\R^2_+: x_1-x_2=\sqrt{-p_1}\}.$$
Clearly, $ \Sol_{P}(p_1,p_2)$ is nonempty and bounded for any $p_1+ p_2>0$.
\end{example}
\subsection{Upper semicontinuity}
We now give a sufficient condition for the upper semicontinuity of $\Sol_{P}$ at $p$ under the copositivity.
\begin{proposition}\label{usc_int} Assume that $0\in K$ and $P$ is copositive on $K$. Then, $\Sol_{P}$ is upper semicontinuous on $\inte(\Sc^*)$.
\end{proposition}
\begin{proof} Let $p$ be in $\inte(\Sc^*)$. Theorem \ref{cop1} says that $\Sol(K,P+p)\neq \emptyset$. Suppose that $\Sol_P$ is not upper semicontinuous at $p$. Then there exist a nonempty open set $V$ containing $\Sol(K,P+p)$, a sequence $\{p^k\}\subset \R^n$, and a sequence $\{x^k\}\subset K$ such that $p^k\to p$ and
\begin{equation}\label{V_open1}
x^k\in\Sol(K,P+p^k)\setminus V,
\end{equation}
for each $k$. By repeating the argument of the proof of Theorem \ref{cop1}, one can prove that the sequence $\{x^k\}$ is bounded. So, without loss of generality we can assume that $x^k\to \bar x$. It is easy to check that $\bar x\in\Sol(K,P+p)$, hence that $\bar x\in V$. Besides, since $V$ is open, the relation \eqref{V_open1} implies that $\bar x\notin V$. One obtains a contradiction. Therefore, $\Sol_P$ is upper semicontinuous at $p$. \qed
\end{proof}
\begin{corollary}\label{cor:copo}
Assume that $0\in K$ and $P$ is copositive on $K$. If $(K,P)$ is an $R_0$-pair, then the two following assertions hold:
\begin{description}
\item[\rm(a)] $\Sol_{P}(p)$ is nonempty and bounded for any $p\in\R^n$.
\item[\rm(b)] $\Sol_{P}$ is upper semicontinuous on $\R^n$.
\end{description}
\end{corollary}
\begin{proof} Since $(K,P)$ is an $R_0$-pair, one has $\Sc=\{0\}$ and $\inte(\Sc^*)=\R^n$. From Theorem \ref{cop1}, the assertion $\rm(a)$ follows. From Proposition \ref{usc_int}, $\Sol_{P}$ is upper semicontinuous at $p$, for any $p\in \R^n$. \qed
\end{proof}
\subsection{Local upper-H\"{o}lder stability}
In \cite{LTY2005}, the authors gave a result for the solution stability of copositive affine variational inequalities. In this section, we extend these results for copositive polynomial variational inequalities.
Let $p\in\R^n$ be given. If there exist $L>0,c>0$ and a neighborhood $U_{p}$ of $p$ such that
$$\Sol_{P}(q)\subset \Sol_{P}(p)+L\|q-p\|^{c}\Bo(0,1) \ \ \forall q\in U_{p},$$
where $\Bo(0,1)$ is the closed unit ball in $\R^n$, then one says that $\Sol_{P}$ is \textit{locally upper-H\"{o}lder stable} at $p$.
When $K$ is semi-algebraic, the following result says that the upper semicontinuity and the local upper-H\"{o}lder stability of $\Sol_{P}$ at $p$ are equivalent.
\begin{theorem}\label{thm:LLP}{\rm(see \cite{LLP2018})} Assume that $K$ is semi-algebraic and $\Sol_{P}(p)$ is nonempty. Then the map $\Sol_{P}$ is upper semicontinuous at $p$ iff it is locally upper-H\"{o}lder stable at $p$.
\end{theorem}
\begin{proposition}\label{Holder} Assume that $K$ is semi-algebraic, $0\in K$, and $P$ is copositive on $K$. If $p\in\inte(\Sc^*)$, then $\Sol_{P}$ is locally upper-H\"{o}lder stable at $p$.
\end{proposition}
\begin{proof} Suppose that $p\in\inte(\Sc^*)$. Proposition \ref{usc_int} says that $\Sol_{P}$ is upper semicontinuous at $p$. According to Theorem \ref{thm:LLP}, $\Sol_{P}$ is locally upper-H\"{o}lder stable at $p$.
\qed
\end{proof}
\begin{theorem
Assume that $K$ is semi-algebraic, $0\in K$, and $P$ is copositive on $K$. Let $p\in\inte(\Sc^*)$ be given. Then there exist constants $\varepsilon>0, L>0$ and $c>0$ with the following property: If $Q\in\Po_d$, $Q$ is copositive on $K$, and $q\in\R^n$ satisfy
\begin{equation*}\label{eps}
\max\{\|Q-P\|,\|q- p\|\}<\varepsilon,
\end{equation*}
then $\Sol(K,Q+q)$ is nonempty and bounded; and
\begin{equation}\label{ell}\Sol(K,Q+q)\subset \Sol(K,P+p)+L(\|Q-P\|+\|q-p\|)^{c}\Bo(0,1). \end{equation}
\end{theorem}
\begin{proof} We first prove that there exists $\delta>0$ such that if $Q\in \Po_d$, where $Q$ is copositive on $K$, and $ p\in\R^n$ with \begin{equation}\label{delta}
\max\{\|Q-P\|,\|q- p\|\}<\delta,
\end{equation}
then $\Sol(K,Q+q)$ is nonempty and bounded. Suppose that the assertion is false. Then there is a sequence $\{(Q^k,q^k)\}\subset \Po_{d}\times\R^n$ such that $(Q^k,q^k) \to (P,p)$, where $Q^k$ is copositive on $K$ for each $k$, and $\Sol(K,Q^k+q^k)$ is empty or unbounded. Due to Theorem \ref{cop1}, one has
$$q^k\notin\inte(\Sol((Q^k)^{\infty},K^{\infty})^*).$$
This means that there exists $x^k\in\Sol((Q^k)^{\infty},K^{\infty})$ satisfying $x^k\neq 0$ and $\left\langle x^k,q^k \right\rangle\leq 0$. We can assume that $\|x^k\|^{-1}x^k\to\bar x\in K^{\infty}$ with $\|\bar x\|=1$.
It is not difficult to see that $\left\langle \bar x,p \right\rangle\leq 0$.
If we prove that $\bar x\in\Sc$, then this contradicts the assumption $p\in\inte(\Sc^*)$; and hence $\rm(a)$ will be proved. Thus, we only need to show that $\bar x\in \Sc$. From \eqref{CP}, one has
\begin{equation}\label{Qk}
\langle (Q^k)^{\infty}(x^k),x^k\rangle=0, \ \langle
(Q^k)^{\infty}(x^k),y\rangle\geq 0 \ \forall y\in K^{\infty}.
\end{equation}
It follows from Remark \ref{P_infty} that $(Q^k)^{\infty} \to P^{\infty}$. Let $y\in K^{\infty}$ be fixed; by dividing the equation and inequality in \eqref{Qk} by $\|x^k\|^{d+1}$ and $\|x^k\|^{d}$, respectively, and letting $k\to+\infty$, we obtain $$\langle P^{\infty}(\bar x),\bar x\rangle=0, \ \langle
P^{\infty}(\bar x),y\rangle\geq 0.$$
As this holds for every $y\in K^{\infty}$, we get $\bar x\in\Sc$.
We now prove the inclusion \eqref{ell}. According to Proposition \ref{Holder}, there exist $L_0>0,c>0$ and $\varepsilon$ such that
\begin{equation}\label{ell0}
\Sol(K,P+q)\subset \Sol(K,P+p)+L_0\|q-p\|^{c}\Bo(0,1)
\end{equation}
for all $q$ satisfying $\|q-p\|< \varepsilon$.
Suppose $Q$ (copositive on $K$) and $q$ (in $\R^n$) satisfy \eqref{delta}. As $\Sol(K,Q+q)$ is nonempty, for each $z_q\in \Sol(K,Q+q)$, by setting
\begin{equation}\label{q_hat} \widehat q:=q+\left(Q- P\right)(z_q), \end{equation}
we have
$P(z_q)+ \widehat q=Q(z_q)+q$ and
$$\left\langle P(z_q)+ \widehat q, y-z_q\right\rangle=\left\langle Q(z_q)+q,y-z_q\right\rangle \geq 0 \ \forall y\in K.$$
This gives \begin{equation}\label{z_q}
z_q\in\Sol(K,P+\widehat q).
\end{equation}
Since $\Sol(K,Q+q)$ is compact, there exists $\beta>0$ such that
\begin{equation}\label{norm_ineq}
\|(Q-P)(z)\|\leq \beta\|Q-P\|
\end{equation}
for all $z\in\Sol(K,Q+q)$.
From \eqref{q_hat}, \eqref{norm_ineq}, and \eqref{delta}, we get
$$\begin{array}{ll}
\|\widehat q-p\| &\leq \; \|\widehat q-q\|+\|q-p\| \smallskip \\
&\leq \; \|(Q-P)(z_q)\|+\|q-p\| \smallskip \\
&\leq \; \beta \|Q-P\|+\|q-p\| \smallskip \\
&\leq \; (1+\beta)\delta.
\end{array}$$
Choosing $\delta$ small enough such that $(1+\beta)\delta<\varepsilon$, we have $\|\widehat q-p\|<\varepsilon$. From \eqref{ell0}, \eqref{z_q} and \eqref{norm_ineq}, there exists $x\in\Sol(K,P+p)$ such that
$$\begin{array}{rl}
\|z_q-x\| \; & \leq \; L_0\|\widehat q-p\|^{c} \smallskip\\
&\leq \; L_0\left(\|q-p\|+\beta \|Q-P\| \right)^{c} \smallskip \\
& \leq \; L \left(\|q-p\|+\|Q-P\| \right)^{c},
\end{array} $$
where $L:=\max\left\lbrace L_0^{c}\beta,L_0^{c}\right\rbrace $.
Since the inequality holds for any $z_q$ in $\Sol(K,Q+q)$, the inclusion \eqref{ell} holds. \qed
\end{proof}
\subsection{The GUS-property}
If $\VI(K,P+p)$ has a unique solution for every $p\in\R^n$, then $\VI(K,P)$ is said to have the globally uniquely solvable property (GUS-property). The following theorem develops Theorem 4.3 in \cite{WHQ18} which concerning the GUS-property of tensor variational inequalities.
\begin{proposition}\label{GUS1}
Assume that $0 \in K$ and $P$ is strictly monotone on $K$. If $(K,P)$ is an $R_0$-pair, then $\VI(K,P)$ has the GUS-property and $\Sol_{P}$ is single-valued and continuous on $\R^n$.
\end{proposition}
\begin{proof} Suppose that $(K,P)$ is an $R_0$-pair. By the monotonicity of $P$, $Q(x):=P(x)-P(0)$ is monotone on $K$. Because $0$ belongs to $K$, the map $Q$ is copositive on $K$. From Remark \ref{P_Q},
$(K,Q)$ is an $R_0$-pair. According to the assertion $\rm(a)$ in Corollary \ref{cor:copo}, the set $$\Sol(K,Q+q)=\Sol(K,P-P(0)+q)$$
is nonempty for all $q\in\R^n$. This is equivalent to saying that $\Sol(K,P+p)\neq \emptyset$ for all $p\in\R^n$.
Since $P$ is strictly monotone, so is $P+p$. According to \cite[Theorem 2.3.3]{FaPa03}, $\VI(K,P+p)$ has at most one solution. Thus, $\VI(K,P+p)$ has a unique solution for every $p\in\R^n$.
Because $\VI(K,P)$ has the $R_0$-property, Corollary \ref{usc_3} says that $\Sol_{P}$ is upper semicontinuous on $\R^n$. Hence, the map is single-valued and continuous on $\R^n$.
The proof is complete. \qed
\end{proof}
To illustrate Proposition \ref{GUS1}, we give the following example.
\begin{example
Consider the polynomial variational inequality given by
$$K=\{x=(x_1,x_2)\in\R^2:x_1\geq 0\}, \ P(x)+p=\begin{bmatrix}
x_1^3\\
x_2^3
\end{bmatrix}+\begin{bmatrix}
p_1\\
p_2
\end{bmatrix},$$
where $(p_1,p_2)\in\R^2$. It is easy to check that $0\in K$, $P$ is strictly monotone on $K$, and $(K,P)$ is an $R_0$-pair. According to Proposition \ref{GUS1}, the problem has the GUS-property and $\Sol_{P}$ is single-valued and continuous on $\R^2$. In fact, an easy computation shows that
$$\Sol_{P}(p_1,p_2)=\left\{\begin{array}{cl}
\left\lbrace(\sqrt[3]{-p_1},\sqrt[3]{-p_2})\right\rbrace & \text{ if } p_1<0, \\
\left\lbrace (0,\sqrt[3]{-p_2} )\right\rbrace & \text{ if } p_1\geq 0.
\end{array}\right.$$
This map is single-valued and continuous on $\R^2$.
\end{example}
\section{Genericity
In this section, we first prove the genericity of the $R_0$-property of polynomial variational inequalities under some mild
conditions. Then, we show that the solution map $\Sol$ is finite-valued on a generic semi-algebraic set of the parametric space.
\subsection{Genericity of the $R_0$-property}
Let $\Hd_d$ be the vector space spanned by polynomial maps $H=(H_1,\dots,H_n)$, where all $H_l$ are homogeneous of degree $d$. The dimension of $\Hd_d$ is denoted by $\rho$. Clearly, $\R^{n\times\rho}$ and $\Hd_d$ are isomorphic. Let $X_d$ be a vector whose components are monomials of degree $d$ listed by lexicographic ordering
$$X_d=\left(x_1^d,x_1^{d-1}x_2,\dots,x_n^d\right)^T.$$
For any homogeneous polynomial map $H\in \Hd_d$, there is a unique $B\in \R^{n\times\rho}$,
$$B=\begin{bmatrix}
b_{11}& b_{12} &\cdots & b_{1\rho} \\
b_{21}& b_{22} &\cdots & b_{2\rho} \\
\vdots & \vdots & \ddots & \vdots \\
b_{n1}& b_{n2} &\cdots & b_{n\rho}
\end{bmatrix},$$
such that $H(x)=BX_d$, where $$H_l(x)=b_{l1}x_1^d+b_{l2}x_1^{d-1}x_2+\dots+b_{l\rho}x_n^d \; (l=1,\dots,n).$$
Assume that $K$ is an unbounded polyhedral convex cone, which is the intersection of finitely many half-spaces containing the origin, given by
\begin{equation}\label{K_0}
K = \left\lbrace x \in \R^n : Cx \leq 0\right\rbrace,
\end{equation}
where $C=(c_{ij})\in {\mathbb R}^{s\times n}$. The following lemma shows that the solution map of homogeneous polynomial complementarity problems,
$$\Gamma_K:\R^{n\times\rho}\rightrightarrows\R^n, \ B\mapsto \Gamma_K(B)= \Sol(K,BX_d),$$
is constant on a generic semi-algebraic set of $\R^{n\times\rho}$ provided that the $\rm LICQ$ holds on $K$.
\begin{lemma}\label{generic_1} Assume that $K$ is a polyhedral convex cone given by \eqref{K_0} and the $\rm LICQ$ holds on $K$. Then there exists a generic semi-algebraic set $\Sa\subset\R^{n\times\rho}$ such that $\Gamma_K(B)=\{0\}$ for any $B\in\Sa$.
\end{lemma}
\begin{proof} Firstly, since the $\rm LICQ$ holds on $K$, applying \cite[Proposition 1.3.4]{FaPa03} for the $\VI(K,H)$, we have $x\in\Sol(K,H)$ if and only if there exists $\lambda\in \R^{s}$ such that
\begin{equation}\label{KKT}
\left\lbrace \begin{array}{l}
H(x)+C^T\lambda=0,\\
\lambda^T(Cx)=0, \; \lambda\geq 0, \; Cx \leq 0.
\end{array}\right.
\end{equation}
Let $K_{\alpha}\neq\{0\}$ be a nonempty pseudo-face of $K$, given by
$$K_\alpha=\big\{x\in {\mathbb R}^n:C_{i}x=0\
\forall i\in\alpha,\ C_{i}x< 0\ \forall i\in I\setminus\alpha\big\},$$
where $C_{i}$ is the $i$-th row of $C$. Thus, $X_d$ is nonzero on this pseudo-face. We consider the function
$$\Phi_{\alpha}:\R^{n\times \rho}\times K_{\alpha}\times \R_+^{|\alpha|} \to \R^{n+|\alpha|},$$
which is defined by
$$\Phi_\alpha(B,x,\lambda_\alpha)=\Big( BX_d+\displaystyle\sum_{i\in \alpha}\lambda_i C_i, C_\alpha x\Big),$$
where
$C_\alpha x=(C_{i_1}x,\ldots,C_{i_{|\alpha|}x}), i_j\in\alpha.$ Clearly, $\Phi_\alpha$ is a smooth
semi-algebraic function.
The Jacobian matrix
of $\Phi_\alpha$ is given by
$$D\Phi_\alpha=\left[ \begin{array}{c|c|c}
D_{B}(BX_d) \; & \ * \ &\ C_\alpha^T \\
\hline
0_{|\alpha|\times \rho}&\ \ \ C_\alpha \ \ \ & 0_{|\alpha|\times |\alpha|} \\
\end{array}\right],$$
where $0_{u\times v}$ is the zero $u\times v$--matrix. Here the $n\times \rho$--matrix $D_{B}(BX_d)$ is given by
$$D_{B}(BX_d)=\begin{bmatrix}
X_d &0_{1\times \rho} & \cdots &0_{1\times \rho} \\
0_{1\times \rho} &X_d & \cdots &0_{1\times \rho} \\
\vdots & \vdots & \ddots & \vdots \\
0_{1\times \rho} &0_{1\times \rho} & \cdots & X_d
\end{bmatrix}.$$
Since $X_d$ is nonzero on $K_{\alpha}$, the rank of $D_{B}(BX_d)$ is $n$. By our assumptions, the rank of the matrix $D\Phi_\alpha$ is $n+|\alpha|$ for all $x\in K_\alpha$. Therefore, $0\in \R^{n+|\alpha|}$ is a regular value of $\Phi_\alpha$. The Sard theorem with parameter \cite[Theorem 2.4]{DHP16} says that there exists a generic semi-algebraic set $\Sa_{\alpha}\subset \R^{n\times\rho}$, such that if $B\in \Sa_{\alpha}$ then $0$ is a regular value of the map
$$\Phi_{\alpha,B}:K_{\alpha}\times \R^{|\alpha|} \to \R^{n+|\alpha|}, \ \Phi_{\alpha,B}(x,\lambda_\alpha) =\Phi_\alpha(B,x,\lambda_\alpha).$$
According to the regular level set theorem \cite[Theorem 9.9]{Loring_2010}, if $\Phi^{-1}_{\alpha,B}(0)$
is nonempty then it is a $0-$dimensional semi-algebraic set. It follows that $\Phi^{-1}_{\alpha,B}(0)$ is a finite set. Moreover, from \eqref{KKT}, one has
$$\Gamma_K(B)\cap K_{\alpha}=\pi(\Phi^{-1}_{\alpha,B}(0)),$$
where $\pi$ is the projection $\R^{n+|\alpha|} \to \R^n$ defined by $\pi(x,\lambda_{\alpha}) = x$. Therefore, $\Gamma_K(B)\cap K_{\alpha}$ is a finite set.
Since $0\in \Gamma_K(B)$, $\Gamma_K(B)\cap K_{\alpha}=\{0\}$ if $K_{\alpha}=\{0\}$. By the finite decomposition \eqref{Sol_decom},
$$\Gamma_K(B)=\bigcup_{\alpha\subset I}\Gamma_K(B)\cap K_{\alpha}$$
is a finite set. Taking $\Sa=\cap_{\alpha\subset I}\Sa_{\alpha},$
we see that $\Gamma_K(B)$ has finite points for any $B\in\Sa$. Recall that $\Gamma_K(B)$ is a closed cone which contains $0$. Hence, $\Gamma_K(B)=\{0\}$ for all $B$ in $\Sa$.
The proof is complete. \qed
\end{proof}
Consider the isomorphism $\varphi:\R^{n\times m} \to \Po_d$ defined by $\varphi(A)=AX$, where $X,A$ are as in \eqref{monomials} and \eqref{A}, respectively. A set
$\Sa$ is generic in $\R^{n\times m}$ iff $\varphi(\Sa)$ is generic in $\Po_d$.
\begin{remark
Suppose that the constraint $K$ is given by \eqref{K}. For each $i\in I$, $j\in J$, one denotes
$K_i=\left\lbrace x \in \R^n : g_i(x)\leq 0\right\rbrace$ and $K'_j=\left\lbrace x \in \R^n : h_j(x)= 0\right\rbrace$. Clearly, $K^{\infty}_i,K'^{\infty}_j$ are polyhedral convex cone (see \cite[p.39]{BK2002}). Since $$K=\big(\bigcap_{i\in I}K_i\big)\bigcap(\bigcap_{j\in J}K'_j\big),$$ it follows from \cite[Proposition 2.1.9]{AusTeb2003} that
$$K^{\infty}=\big(\bigcap_{i\in I}K^{\infty}_i\big) \bigcap \big(\bigcap_{j\in J}K'^{\infty}_j\big).$$
Thus, $K^{\infty}$ is a nonempty polyhedral convex cone.
\end{remark}
\begin{theorem
Suppose that $K$ is given by \eqref{K} and $K^{\infty}$ is given by \eqref{K_0}. If the $\rm LICQ$ holds on $K^{\infty}$, then the set $\Ri_0(K,d)$ is generic in $\Po_d$.
\end{theorem}
\begin{proof} Assume that $\rm LICQ$ holds on $K^{\infty}$. According to Lemma \ref{generic_1}, there exists a generic set $\A\subset \Hd_d$ such that the cone $\Sol(K^{\infty},H)$ is trivial, for all $H\in \A$.
From the direct sum $\Po_d=\Hd_d\oplus\Po_{d-1}$, where
$$\Hd_d\oplus\Po_{d-1}=\{H+Q \; | \; H\in\Hd_d, Q\in \Po_{d-1}\},$$
we can assert that $\A\oplus\Po_{d-1}$
is generic in $\Po_d$ and $(K,P)$ is an $R_0$-pair for any $P\in \A\oplus\Po_{d-1}$. It follows that $\Ri_0(K,d)$ is also generic in $\Po_d$. The proof is complete. \qed
\end{proof}
\subsection{Genericity of the finite-valuedness}
Recall that a set-valued map $\Psi$ is \textit{finite-valued} on $X$ if the cardinality of $\Psi(x)$ is finite, i.e. $|\Psi(x)|<+\infty$ for every $x\in X$.
The finite-valuedness of $\Sol_{P}$ on a generic set of $\R^n$ was announced and proved in \cite[Theorem 3.2]{LLP2018}. The following theorem mention the genericity of the finite-valuedness of the map $\Sol$.
\begin{theorem}\label{generic_3}
Assume that $K$ is a semi-algebraic set given by \eqref{K} and the $\rm LICQ$ holds on $K$. Then, the map $\Sol$ is finite-valued on a generic set of $\Po_d$.
\end{theorem}
\begin{proof} For $Q\in\Po_{d}$, one has $Q(x)=AX$, where $X$ and $A$ are given by \eqref{monomials} and \eqref{A}, respectively. For each nonempty pseudo-face $K_{\alpha}$ of $K$, we consider the function
$$\Phi_\alpha:\R^{n\times m}\times K_{\alpha}\times \R^{|\alpha|+|J|} \to \R^{n+|\alpha|+|J|},$$
which is defined by
$$\Phi_\alpha(A,x,\lambda_\alpha,\mu)=\Big(AX+\displaystyle\sum_{i\in \alpha}\lambda_i\nabla g_i(x)+\displaystyle\sum_{j\in J}\mu_i\nabla h_j(x), g_\alpha(x),h(x)\Big),$$
where
$g_\alpha(x)=(g_{i_1}(x),\dots,g_{i_{|\alpha|}}(x)), i_j\in\alpha$.
An easy computation shows that the Jacobian matrix $D_{A}(AX)$ is the following $n\times m-$matrix
\begin{equation}\label{Da_P}
D_{A}(AX)=\left[ \begin{array}{cccc}
X&0_{1\times m}&\cdots & 0_{1\times m} \\
0_{1\times m}&X&\cdots & 0_{1\times m}\\
\vdots&\vdots&\ddots& \vdots \\
0_{1\times m}&0_{1\times m}&\cdots & X\\
\end{array}\right].
\end{equation}
From \eqref{Da_P}, the rank of $D_{A}(AX)$ is $n$.
Since $K$ has the $\rm LICQ$ property, the rank of the Jacobian matrix $D\Phi_{\alpha}$ is $n+|\alpha|+|J|$ for all $x\in K_\alpha$.
Repeating the argument of the proof of Lemma \ref{generic_1} (by using the Sard theorem with parameter and the regular level set theorem), one can assert that there exists a semi-algebraic set $\Sa$, which is generic in $\R^{n\times m}$, such that $\Sol(K,AX)$ is a finite set for every $A\in\Sa$. Hence, we can conclude that there is a generic set in $\Po_d$ such that $\Sol$ is finite-valued on it. \qed
\end{proof}
\begin{remark} Suppose that the assumptions in Theorem \ref{generic_3} are satisfied. We can show that if the map $\Sol$ is lower semicontinous at $P$, then $\Sol(K,P)$ has finitely many points. Hence, if $\dim\Sol(K,P)\geq 1$, then $\Sol$ is not lower semicontinuous at $P$.
\end{remark}
\begin{acknowledgements}
The author would like to thank Prof. Nguyen Dong Yen, Dr. Pradeep Kumar Sharma, Dr. Vu Thi Huong, and the first anonymous referee for their corrections. The author is indebted to the third anonymous referee for numerous valuable comments and suggestions.
\end{acknowledgements}
|
1807.00365
|
\section{Notations}\label{sec:notations}
$A$ = scalar parameter of interest\\
$B$ = other parameters defining "theoretical" distribution corresponding current design\\
$a$ = assumed "true" value of A \\
$b$ = vector of the assumed values of B\\
$\alpha$ = required significance level\\
$CI= CI(\alpha) = (1-\alpha)$ confidence interval for $ a $\\
$h$ = a procedure for defining $d(CI)$\\
$C$ = full set of statistics for $h$\\
$ \Psi(C) $ - distribution of C ||
$d=d(CI)$ = width of $CI$ as defined by $h$ \\
$G(d)$ = distribution of $d(CI)$\\
$d_0$ = required width of $CI$\\
$\psi $ = probability of $d(CI)<d_0$ (analogue of power)\\
$\psi_0 $ = required probability of $d(CI)<d_0$ \\
$n_0$ =minimum sample size providing $ \psi\ge\psi_0 $ \\
\section{Introduction}\label{sec:intro}
Each study is (or should be) carefully designed. The design should specify the goals of the study, measured parameters, method of measurement, sampling scheme, analysis, and at last, sample size $N$. For estimation studies, the goal is to obtain an estimation of the parameter of interest with a given precision. The precision is usually estimated by width of an empirical Confidence Interval ($CI$) that uses the estimation(s) of the parameter(s) from the study. The design should specify all details of the confidence interval: type (symmetric, the shortest, mid-point, etc.), one-sided or two-sided, and a confidence level $1- \alpha$ (usually $\alpha$ is $0.1$, $0.05$, or $0.01$). The width of an empirical $CI$ depends on the assumed distribution of the parameter of interest, and on the sample estimates of the parameters. These estimates are never known at the stage of design. However, a researcher assume some "true" values of the parameters using information from similar studies. This gives full definition of distribution function.
Many software tools, like PASS \cite{PASS}, MINITAB \cite{Minitab}, STATA \cite{STATA}, WinPepi \cite{WinPepi}, SURVEYSYSTEM \cite{SurveySystem}, QUATRICS \cite{QUATRICS}, R packages binomSamSize \cite{binomSamSize}, samplingbook \cite{samplingbook} and others use the assumed values instead of future estimates of the parameter(s) by substituting them in the corresponding formula for width of empiric $CI$as it is recommended in many popular text-books Yau \cite{Yau}, Machin \cite{Machin}, Lemeshow \cite{Lemeshow}.
These procedures use only the confidence level, the desired width of corresponding confidence interval and expected values of the parameter(s) and their SDs (some of the procedures also permit finite population correction.) The resulting sample size looks like it guarantees that the $CI$ that will be obtained in the study will be definitely as small as required. Clearly, this is logically wrong because the empirical $CI$ is based not on the assumed values of the parameters but on their sample estimates. Thus the width of the empirical $CI$ is random. It differs from the assumed one and may be narrower or wider than the assumed value. Therefore the required precision may be not achieved.The correct approach is known for at least for 30 years (Greenland\cite{Greenland},Bristol\cite{Bristol},Beal\cite{Beal},Moore\cite{Moore},Grieve\cite{Grieve} ) but evidently much earlier. In 2003 it was revived by Jirotek et al.\cite{Jirotek} and especially by Kelley and Maxwell \cite{Kelley2003a} under the name AIPE (Accuracy In Parameter Estimation). Nevertheless, the program realization in commercial software are rare. The happy exception is SAS Proc POWER\cite{SAS}. It defines the goal as "computing the probability of achieving the desired precision of a confidence interval, or the sample size required to ensure this probability" \cite[Proc POWER]{SAS}. In explaining the analysis of Confidence Interval in Proc POWER, SAS states: "An analysis of confidence interval precision is analogous to a traditional power analysis, with $CI$ Half-Width taking the place of effect size and Prob(Width) taking the place of power". Unfortunately, in the current version 9.4, Proc POWER calculates the necessary sample size only for means, but not even for proportion and other parameters \cite[STAT 14.1]{SAS}.In R the package MBESS realized AIPE\cite. these ideas were revivedAlso R package
In this text we describe the general approach to sample size evaluation for obtaining the width of the empirical $CI$ with a predefined probability, describe its implementation in R, and give examples for Normal, Poisson and Binomial distributions.
\section{Approach}
We are looking for an estimate of a true value $a$ of a scalar parameter $A$ of a distribution $F(A,B)$, where $B$ is a set of additional parameters. The precision of the estimate is measured by the width $d$ of the confidence interval $CI=CI(\alpha)$ with confidence level $(1-\alpha) $. The $CI(\alpha)$ should be estimated using a sample $S$ of $N(S)$ observations from distribution $F(A,B)$. Let $h$ be a procedure for calculating the width $ d $ of $CI(\alpha)$ using the sample $S$. Usually there are several procedures to select from. For example, we can choose between several formula for $CI$ for a parameter $p$ of Binomial distribution $Bin(N,p)$: Clopper--Pearson, Normal, Wilson etc. To calculate the width of $CI$, procedure $h$ uses some statistics, like sample mean, sample standard deviation etc. Let $C\in \R^k$ be a set of these statistics. Given $h(C)$ and fixing $N=N(S)$, we obtain the width $d=d(N,C)$ as a statistic with some distribution $G(d)$ .
For a given precision $d_0$ and probability $\psi_0$, we aim to find a sample size $n_0$ such that $d(N,C)\le d_0$ for any $N>n_0$ with probability $\psi_0$ or higher. Parameter $\psi_0$ plays a role similar to power in hypotheses testing.
Let $N$ be given.
\begin{Def}
For a subset $R\subset\R^k$ define
\begin{equation}\label{eq:def psi(R)}
\psi_N(R)=Prob\left[C(S) \in R|\#S=N\right].
\end{equation}
Define $d_{min}(N)$ as
\begin{equation}\label{def:dmin}
d_{min}(N)=\min\left\{d| \psi_N\left(\{d(N,C)\le d\}\right)\ge \psi_0 \right\}.
\end{equation}
\end{Def}
Evidently, it is enough to find $ n_0 $ such that $ d_{min}(N)<d_0 $ for all $ N>n_0 $.
\begin{Prop}
Let $ n_0 $ such that $ d_{min}(N)<d_0 $ for all $ N>n_0 $. Then $d(N,C)\le d_0$ for any $N>n_0$ with probability $\psi_0$ or higher.
\end{Prop}
\begin{proof}
Indeed, let $N>n_0$. Then $d_{min}(N)<d_0$, and, therefore the probability that $d(N,C)\le d_0$ is at least the probability that $d(N,C)\le d_{min}(N)$. By definition of $d_{min}(N)$, the latter is at least $\psi_0$.
\end{proof}
\subsubsection{The set $R_0(N)$}\label{sssec:def of R0}
The main difficulty consists of finding $d_{min}(N)$ in \eqref{def:dmin}.
We solve it by considering explicitly the corresponding set $R_0(N)$.
\begin{Def}\label{def:def R0}
We define $R_0(N)$ as
\begin{equation}\label{def:def2 R_0}
R_0(N)=\{d(N,C)\le d_{min}(N)\}.
\end{equation}
\end{Def}
In other words, first, $R_0(N)$ must be "big" in a sense that the probability to obtain a result in $R_0(N)$ should be high (above $ \psi_0 $), and, simultaneously, "greedy" in a sense that
for any point outside of $R_0(N)$, the width of an empiric $CI(C,N)$ is larger than the maximum of the width for any point in $R_0(N)$.
\begin{Rem}
In Definition~\ref{def:def R0} the statistic $C$ can be multidimensional, $C\in \R^k$, and, correspondingly, the set $R_0(N)$ will be a subset of $\R^k$.
If the statistic $C$ has the form $C=(c,B)$, where $c$ is a scalar and $B$ is the set of additional statistics, then one can consider a majorant \begin{equation}\label{eq:d*min}
d^*_{min}(N)=\min\left\{d| \psi_N\left(\{C|\max_B \left(d(N,(c,B))\right)\le d\}\right)\ge \psi_0 \right\}.
\end{equation}
In other words, consider the function $d^*(N,c)=\max_B {d(N,(c,B))}$. Then, considering $d^*(N,c)$ as a function on $\R^k$, we get
$$
\{C|d(N,(c,B))\le d\}\subset\{C|d^*(N,c)\le d\},
$$
so $d_{min}(N)\le d^*_{min}(N)$.
Alternatively, one can define $d^*_{min}(N)$ by using one-dimensional settings, with $d^*(N,c)$ (considered as univariate function) and the marginal distribution $\tilde{\psi}_N(c)$ of $\psi_N(c,B)$.
\end{Rem}
In most practically important cases, the statistic $C$ is a scalar and the function $d(N,C)$ has a simple structure, e.g. monotone, bell-shaped etc., and the set $R_0(N)$ is an interval (for monotone $d(N,C)$) or two intervals (for bell-shaped case).
The described approach eliminates contradictions in the logic of calculating the sample size between hypotheses testing and estimation studies. The probability $\psi_0$ plays the role analogous to the power in hypothesis testing studies.
Here, we address two immediate questions. First, we show how to implement it. Second, we compare between the sample sizes calculated by the proposed approach and the ones calculated by the current approaches.
\section{Implementation.}
The implementation is simple. The common structure of a procedure for any distribution and any sampling is straightforward.
1. Find initial estimate of $N$ . For many cases the good choice for starting value of $N$ is the sample size corresponding to the required width and the hypothesized value(s) of the parameter(s).
2. Find subset $R_0(N)$ for the estimated sample size N.
3. Find $d(N)=\max_{C\in R_0(N)}(d(C,N))$
4. If $ d(N) > d_0 $ increase $N$ and repeat steps 2 and 3.
5. Put $n_0$ equal to the last value of $ N $
\section{Examples.}
\subsection{Normal distribution.}
Standard expression of CI for expectation $\mu $ of the Normal distribution $\textit{N}(\mu,\sigma)$ using I.i.d. sample of size $N$ leads to the following expression for width of the empiric confidence interval
\begin{equation}label{eq:def widthnorm}
d=2s \frac{t(1-\alpha/2,(N-1))}{\sqrt{N}}
\end{equation}
where $ t(1-\alpha/2, (N-1)) $ is the $ 1-\alpha/2 $ quantile of Student $ t $ distribution with $ (N-1)$ degrees of freedom, $ s $ is an estimated standard deviation, sample variance $ s^2 $ is defined as $ Q/N $, where $ Q=\Sigma(x_i- m)^2 $ , and $ m $ is the sample mean. The sum of squares $ Q $ is distributed as $ \sigma^2\chi^2(N-1) $ where $ \sigma^2 $ is the assumed variance and $ \chi^2(N-1) $ is a chi-sqaure distribution with $ (N-1) $ degrees of freedom. The variance of the sample mean $ m $ is $ s^2/N $.
Thus $ d $ is a monotone increasing function of $ s $.
To be sure with probability $ \psi $ that the empirical interval will be shorter than $ d_0 $ for a fixed sample size $ N $ we should use $ s_0 = \sigma\sqrt{\chi^2(\psi,N) } $ , where $ \chi^2(\psi,N) $ is the $ \psi $ quantile of the $\chi^2(N-1)) $ distribution.
The given width $ d_0 $ will be guaranteed with the probability $ \psi $ if
\begin{equation}label{eq:def dnorm1}
Prob\Big( 2s \frac{t(1-\alpha/2,N-1)}{\sqrt{N}} < d_0 \Big) \geq \psi,
\end{equation}
i.e.
\begin{equation}label{eq:def dnorm2}
Prob\Big(s < \frac{d_0\sqrt{N}}{2t(1-\alpha/2, N-1)} \Big) \geq \psi
\end{equation}
and
\begin{equation}label{eq:def dnorm3}
Prob\Big(s^2 < \frac{Nd_0^2}{4t^2(1-\alpha/2, N-1)} \Big) \geq \psi
\end{equation}
but $s^2 $ is distributed as $ \sigma^2\chi^2(N-1)/N $
For any fixed $ \psi $, $ \chi^2(\psi,N)/N^2 $ is a monotone decreasing function of $ N $, and $ qt(1-\alpha/2, (N-1)) $ is a monotone decreasing function of N for any $\alpha$.
Therefore the necessary sample size $ n_0 $ is the smallest solution of the inequality
\begin{equation}label{eq:def dnorm4}
\frac{t^2(1-\alpha/2, N-1)\chi^2(\psi,N-1)}{N^2} < \frac{d_0^2}{4\sigma^2}
\end{equation}
In Table 1 we present sample size $ (n_0) $ with corresponding coverage probability (Cov) and percent (Pow) of $CI$ with width less than the required (Width0) for postulated value of variance (named "Expected") and the variance found by the described algorithm (named "Exact"). The should be compared with the required coverage (named (1-alpha)) and the required proportion (named "Power")
\begin{landscape}
\begin{center}
Table 1.Normal distribution\vskip1cm
\begin{tabular}{ |c|c|c|c|c|c|c|c|c|c }
\hline
\phantom{\Bigg(}Width0\phantom{\Bigg)} &1-alpha&Power&$n_0$Exp&CovExp&PowExp&$n_0$Exa&CovExa&PowExa\\
\hline
0.50000 &0.95 &0.8 &62 &0.9478 &0.4552 &73 &0.9489 &0.8135 \\
0.50000 &0.95 &0.9 &62 &0.9491 &0.4422 &78 &0.9456 &0.9177 \\
0.50000 &0.90 &0.8 &44 &0.9037 &0.4807 &53 &0.9001 &0.8299 \\
0.50000 &0.90 &0.9 &44 &0.8921 &0.4791 &57 &0.9016 &0.9170 \\
0.25000 &0.95 &0.8 &246 &0.9518 &0.4745 &267 &0.9453 &0.8144 \\
0.25000 &0.95 &0.9 &246 &0.9526 &0.4682 &276 &0.9507 &0.9026 \\
0.25000 &0.90 &0.8 &174 &0.8961 &0.4883 &190 &0.9036 &0.7972 \\
0.25000 &0.90 &0.9 &174 &0.9007 &0.4871 &198 &0.9062 &0.8998 \\
0.12500 &0.95 &0.8 &984 &0.9504 &0.4839 &1023 &0.9527 &0.8077 \\
0.12500 &0.95 &0.9 &984 &0.9494 &0.4983 &1042 &0.9491 &0.9046 \\
0.12500 &0.90 &0.8 &693 &0.8985 &0.4848 &725 &0.9041 &0.8022 \\
0.12500 &0.90 &0.9 &693 &0.9007 &0.4886 &742 &0.8965 &0.9075 \\
0.06250 &0.95 &0.8 &3934 &0.9501 &0.4899 &4010 &0.9479 &0.7986 \\
0.06250 &0.95 &0.9 &3934 &0.9484 &0.4942 &4049 &0.9488 &0.9025 \\
0.06250 &0.90 &0.8 &2771 &0.9055 &0.4979 &2835 &0.8974 &0.8035 \\
0.06250 &0.90 &0.9 &2771 &0.9004 &0.4949 &2867 &0.9004 &0.9043 \\
\hline
\end{tabular}
\end{center}
\end{landscape}
\subsubsection{Comments}
It can be seen that the “expected” sample size substantially underestimates the necessary sample size. Relative difference varies from 0.5 for power=0.9 and large width to 0.015 for power=0.8 and small width. This is due to the decrease of the ratio of standard deviation of chi-square distribution to its expectation when d.f. grows. Thus the sample size calculated for the expected value of the variance for D=1 and 1-beta=0.9 should be increased by 50\% from 16 to 24 to provide the declared probability of 0.9 to obtain the required width of 95\% CI.
\subsection{Poisson distribution.}
Poisson distribution is a one parametric distribution on the set of all non-negative integers with probability function
\begin{equation}label{eq:def Poiss1}
P(n|\lambda)=\frac{\lambda^n}{n!}e^{-\lambda}
\end{equation}
Poisson random variable mostly appears as a result of observing number of events in Poisson process. The intensity $ e $ of a process is assumed to be known and the question of “sample size” takes a form of “how long we have to observe the process to get a good interval estimation of the intensity?”. For example N is the total number of person-years of follow up and $ e $ is the incidence of a disease. The expected number of outcomes will be $ \lambda = e*N $. Under some well-known assumptions, the observed number of events will have Poisson distribution with parameter $ \lambda $. Our goal is to find a sample size $ n_0 $, such that $ 1-\alpha $ confidence interval for the parameter $ e $ will have width $ d $ less or equal to $ d_0 $ with probability $ \psi_0 $. In a standard approach to calculating the necessary sample size, we assume some fixed value $ e_0 $ of the parameter of interest. After obtaining a sample with size $ N $ we consider the observed number of outcomes x as an estimate of expected number of outcomes $ \lambda $ and calculate the sampling estimation of $e $ as $ x/N $. Correspondingly the width $ d(e)=d(x)/N $. Thus we are looking for a solution of $ n_0=min(N) $ such that
\begin{equation}label{eq:def Poiss2}
P\Big(\frac{d(x,N)}{N} < d_0(e)\Big)\geq\psi
\end{equation}
for any $ N>n_0 $ .
There are many expressions for $CI$ of the parameter x of Poisson distribution (Patil and Kulkarni \cite{Patil} considers 19).
One of expressions, recommended by \cite{Patil} for $CI$ of the parameter $ \lambda $ of Poisson distribution is
Garwood (1994)\cite{Garwood},
\begin{equation}label{eq:def Poiss3}
(\chi^{2}(2x,\alpha_1),\chi^{2}(2x+2,\alpha_2))
\end{equation}
Usually, $ \alpha_2=1-\alpha_1, \alpha_1=\alpha/2. $
Then
\begin{equation}label{eq:def Poiss4}
d(x,\alpha)=(\chi^{2}(2x+2,1-\alpha/2) - \chi^{2}(2x,\alpha/2))
\end{equation}
and correspondingly
\begin{equation}label{eq:def Poiss5}
d(e,\alpha)=(\chi^{2}(2x+2,1-\alpha/2) - \chi^{2}(2x,\alpha/2))/N
\end{equation}
For any fixed $ x $, $ d(e,N) $ is a monotone decreasing function of $ N $ and for any fixed $ N $, $ d(e,N) $ is a monotone increasing function of $ x $. Therefore we are looking for a set $ R_0(\lambda,\psi) $ such that $ Prob(\lambda)(R_0)\geqq \psi $ with the lowest $ max{(x \in R_0)}$.Thus the set $ R_0(\lambda,\psi) $ is an interval $ {0,1,2,..Q(\psi,\lambda) } $
From the definition of Poisson(m) we can see that for any $ \lambda >=1 $, the probability to observe number of events in Poisson(m) not greater than $\lambda $ is below 0.8 (for $\lambda =1 P(x<=1)= .73575888 $ and this probability monotone decreasing with $\lambda $).
Thus the $ Q(\psi,\lambda )>\lambda $ for $ \psi > 0.75 $ and $ m\geqq1 $ , $ d(Q(\psi,\lambda),N)>d(\lambda,N) $ leading to "expected" sample size will be smaller than the "empiric" one.
\subsubsection{Results}
Table 2 has the same structure as Table 1. It presents sample size ($n_0$) with corresponding coverage probability (Cov) and percent (Pow) of $CI$ with width less than the required (Width0) for postulated value of variance (named "Expected") and the variance found by the described algorithm (named "Exact"). The should be compared with the required coverage (0.95) and the required proportion (named "Power")
\begin{landscape}
\begin{center}
Table 2. Poisson distribution.\vskip1cm
\begin{tabular}{ |c|c|c|c|c|c|c|c|c| }
\hline
\phantom{\Bigg(}Rate\phantom{\Bigg)} & Width0 & Power & $n_0$Exp & CovExp & PowExp & $n_0$Exa & CovExa & PowExa \\
\hline
0.01 & 0.002 & 0.8 & 39439 & 0.9532 & 0.5055 & 41064 & 0.9589 & 0.8070 \\
0.01 & 0.002 & 0.9 & 39439 & 0.9534 & 0.5092 & 41861 & 0.9515 & 0.9045 \\
0.01 & 0.001 & 0.8 & 155683 & 0.9479 & 0.4929 & 158936 & 0.9491 & 0.8072 \\
0.01 & 0.001 & 0.9 & 155683 & 0.9529 & 0.4917 & 160630 & 0.9537 & 0.9050 \\
0.02 & 0.004 & 0.8 & 19719 & 0.9538 & 0.5064 & 20532 & 0.9561 & 0.8162 \\
0.02 & 0.004 & 0.9 & 19719 & 0.9548 & 0.4972 & 20931 & 0.9512 & 0.9042 \\
0.02 & 0.002 & 0.8 & 77841 & 0.9562 & 0.5008 & 79468 & 0.9500 & 0.8069 \\
0.02 & 0.002 & 0.9 & 77841 & 0.9499 & 0.5040 & 80315 & 0.9540 & 0.9047 \\
0.04 & 0.008 & 0.8 & 9859 & 0.9521 & 0.5120 & 10266 & 0.9513 & 0.8112 \\
0.04 & 0.008 & 0.9 & 9859 & 0.9535 & 0.5066 & 10466 & 0.9530 & 0.9014 \\
0.04 & 0.004 & 0.8 & 38920 & 0.9498 & 0.5011 & 39734 & 0.9490 & 0.7996 \\
0.04 & 0.004 & 0.9 & 38920 & 0.9502 & 0.4944 & 40158 & 0.9509 & 0.8971 \\
0.08 & 0.016 & 0.8 & 4929 & 0.9534 & 0.5072 & 5133 & 0.9559 & 0.8047 \\
0.08 & 0.016 & 0.9 & 4929 & 0.9532 & 0.5034 & 5233 & 0.9490 & 0.9037 \\
0.08 & 0.008 & 0.8 & 19460 & 0.9532 & 0.5021 & 19867 & 0.9510 & 0.8057 \\
0.08 & 0.008 & 0.9 & 19460 & 0.9510 & 0.4965 & 20079 & 0.9492 & 0.9075 \\
\hline
\end{tabular}
\end{center}
\end{landscape}
\subsubsection{Comments}
It can be seen that the Garwood formula provides perfect coverage. However, the "expected" sample size is lower that the "exact" by 2\% - 5\% and provides the "expected power" around 50\% while the "exact" sample size guarantees the requested values of power.
If one prefers to avoid using statistical software, it is possible to use simple algebraic approximations for $ Q(\psi,\lambda) $ and $ CI $.
The value $ Q(\psi,\lambda) $ may be approximately estimated by using direct and inverse transformation of Poisson variable to Normal using Anscombe \cite{Anscombe} approximation
\begin{equation}
z=2\sqrt{(x+0.375)}~{}N(\sqrt(x0),1)
\end{equation}
and recommended inverse transformation
\begin{equation}
x=\frac{z^2}{4}-0.125
\end{equation}
From these we obtain
\begin{equation}
Q(\psi,\lambda)\approx(2\sqrt{(Ne_0 +0.375)}+Z(\psi )^2-0.125
\end{equation}
\subsection{Binomial distribution.}
Here we are interested in finding the sample size $ n_0 $ that provides the desired width $ d_0 $ for assumed proportion $ p_0 $ of the number $ x $ of outcomes 1 in a sample of size $ N $.
There are many formula for $CI$ of the parameter $ p $ of the Binomial distribution. One of the most usable\cite{Newcombe} is the Wilson’s interval\cite{Wilson}
\begin{equation}label{eq: Wilson1}
d(CI)=\frac{ 2N\hat{p}+Z_{\alpha}^2 \pm Z_{\alpha}^2\sqrt{4N\hat{p}(1-\hat{p}) +Z_{\alpha}^2 }}{2N+ 2Z_{\alpha}^2}
\end{equation}
where $ \hat{p} $ is the observed proportion $ o=x/N $ and x is the number of outcomes 1. Its width is
\begin{equation}label{eq: Wilson2}
D(\hat{p},N)=\frac{ 2Z_{\alpha}^2\sqrt{4N\hat{p}(1-\hat{p}) +Z_{\alpha}^2 }}{2N+ 2Z_{\alpha}^2}
\end{equation}
The maximum width corresponds to maximum $ \hat{p}*(1-\hat{p}) $. Thus for any $ N $, if $ abs(\hat{p}_1-0.5)>abs(\hat{p}_2-0.5) $ then $ d(\hat{p}_1,N)< d(\hat{p}_2,N) $. Our goal is to guarantee that $ Prob(D<d_0)\geqq \psi $ with the smallest $ N $. Following the “worst – best” scenario, we are going to find a set $ R_0 $, such that
\begin{equation}label{eq: Wilson3}
Prob(R_0|p,N)\geqq \psi
\end{equation}
and
\begin{equation}label{eq: Wilson4}
R_0=\argmin_R \max_{\hat{p} \in R}(abs(\hat{p}-0.5))).
\end{equation}
Evidently, this set has a form ${0,q}\bigcup{(1-q),1} $ where $q$ is defined from equation \ref{Wilson4}. However, for $p$ not too close to $0.5$ and reasonable sample size $N$ one of intervals has negligible probability under $Bin(p,N)$. For examples, the upper interval $\left((1-q),1\right)$ will have the probability below $0.001$ for $p_0 =0.2$ if $N>20$ , for $p_0=0.35$ if $N>40$, for $p_0=0.4$ if $N>70$ and for $p_0= 0.45$ if $N>260$. Our simulations demonstrated that for $p<0.45$ and $R_0$ of the form $\{0,q\}$ the probability $P(N,q)$ will be greater than the probability $P({(1-q),1})$ for $N$ equal to necessary sample size. Therefore we will use $R_0$ of the form of one interval.
\subsubsection{Results}
Table 3 has the same structure as table 1. It presents the sample size ($n_0$) with corresponding coverage probability (Cov) and percent (Pow) of $CI$ with width less than the required (Width0) for postulated value of the parameter $p_0$ (named "Expected") and the variance found by the described algorithm (named "Exact"). The should be compared with the required coverage (0.95) and the required proportion (named "Power")
\begin{landscape}
\begin{center}
Table 3. Binomial distribution.\vskip1cm
\begin{tabular}{ |c|c|c|c|c|c|c|c|c| }
\hline
\phantom{\Bigg(}$p_0$\phantom{\Bigg)}&Width0&Power&$n_0$Exp&PowExp&CovExp&S$n_0$Exa&PowExa&CovExa\\
\hline
0.5000 & 0.10& 0.8 & 381 & 1.0000 & 0.944 & 381 & 1.0000 & 0.947\\
0.5000 & 0.10& 0.9 & 381 & 1.0000 & 0.943 & 381 & 1.0000 & 0.948\\
0.5000 & 0.05& 0.8 & 1533 & 1.0000 & 0.951 & 1533 & 1.0000 & 0.950\\
0.5000 & 0.05& 0.9 & 1533 & 1.0000 & 0.947 & 1533 & 1.0000 & 0.949\\
0.2500 & 0.10& 0.8 & 286 & 0.5078 & 0.945 & 302 & 0.8253 & 0.955\\
0.2500 & 0.10& 0.9 & 286 & 0.5031 & 0.944 & 309 & 0.9061 & 0.957\\
0.2500 & 0.05& 0.8 & 1150 & 0.5005 & 0.951 & 1182 & 0.8134 & 0.951\\
0.2500 & 0.05& 0.9 & 1150 & 0.4961 & 0.949 & 1199 & 0.8989 & 0.952\\
0.1250 & 0.10& 0.8 & 170 & 0.5351 & 0.953 & 192 & 0.8397 & 0.938\\
0.1250 & 0.10& 0.9 & 170 & 0.5378 & 0.950 & 201 & 0.9097 & 0.957\\
0.1250 & 0.05& 0.8 & 674 & 0.5091 & 0.952 & 722 & 0.8227 & 0.952\\
0.1250 & 0.05& 0.9 & 674 & 0.5203 & 0.953 & 745 & 0.9145 & 0.949\\
0.0625 & 0.10& 0.8 & 98 & 0.5850 & 0.945 & 121 & 0.8624 & 0.944\\
0.0625 & 0.10& 0.9 & 98 & 0.5839 & 0.948 & 132 & 0.9326 & 0.953\\
0.0625 & 0.05& 0.8 & 369 & 0.5564 & 0.961 & 424 & 0.8479 & 0.958\\
0.0625 & 0.05& 0.9 & 369 & 0.5484 & 0.960 & 449 & 0.9293 & 0.952\\
\hline
\end{tabular}
\end{center}
\end{landscape}
\subsubsection{Comments}
The first fact that we can see from table 3 is that for expected probability $p_0=0.5$ the results strongly differ from all others. The thing it that in our best-worst scenario the set $R_0$ always contains the value $0.5$.Therefore the sample size does not depend on power $0.8$ or $0.9$ and coincides with expected power for $p_0=0.5$. The power, i.e the proportion of $CI$ with width less than the required, is $1.0$ because the width of $CI$ for $p=0.5$ is bigger than for any other $p$. The coverage is close to nominal value of $0.95$ supporting good properties of Wilson $CI$. However, the situation is different for other values of $p_0$. The "exact" sample size is bigger than the "expected" one. The relative difference may be more than 20\% (for $p_0=0.0625$, $d_0=0.05$ and power$=0.9$). The "exact' sample size provides power not less than the required, while the power for "expected" sample size is around $0.5$.
|
2009.13743
|
\section{INTRODUCTION}\label{sec:introduction}}
\IEEEPARstart{A}{rtificial} intelligence, human attempts to make computers think as we do, has experienced noticeable growth in the past few years. However, these models have been around since the 1940s \cite{Walczak2019}.\\
\indent One important feature of the human brain is its ability to detect faces at a glance by quickly processing images perceived by the eyes. Based on the behavior of the brain’s neurons, artificial neural networks were born and, as early as the 1960s were being used for facial detection when Bledsoe, Wolf, and Bisson started using computers to identify the characteristic features of the human face \cite{Andreopoulos2013}. Following Bledsoe, Wolf, and Bisson’s work, in the 1970s, Goldstein, Harmon, and Lesk worked on an even more extensive and specific list of 22 features to be used as markers to detect and then identify human faces from a pool of photographs \cite{Goldstein1971}. A few decades later, Turk and Pentland developed a “near-real-time computer system” able to notice, locate and track a subject’s head movement by detecting characteristic features of human's face \cite{Turk1991}.
\begin{figure}[!htb]
\centering
\includegraphics[width=3.5in,height=2.5in]{detection_examples}
\captionsetup{justification=centering,margin=0.3in}
\caption{Different face detection samples.}
\label{fig_1}
\end{figure}
\indent These early advances on facial detection were hindered by the limitations of the technology of that time but served as the basis for future research in the area. Hence, as the internet grew through the early 2000s and made available even bigger sets of images that could be used for analysis and training, facial detection models flourished rapidly, making it possible for machines to accurately and quickly locate faces in both photos and videos. Now, a couple of decades later, we have cellphones with integrated cameras, social media that encourage users to take and share numerous photos and videos, and incredibly powerful computer devices have become easily accessible for a great number of people \cite{Anil2016}.\\
\indent With the growth of resources available to make experiments in facial detection, so has grown the number of fields on which its application means an increase in productivity and/or ease of use for different practices in daily living \cite{Kumar2018}. These developments in technology mean that precision and accuracy are no longer the only relevant characteristics when choosing an image detection model to implement. A model’s capacity to detect faces quickly is necessary to accommodate the day-to-day usage of different practices. A detection algorithm needs then not only to be accurate but to be able to respond fast in order to successfully play its role in systems such as face-identification log-in features, social networks image tagging, photography and entertainment apps filters, augmented reality devices, surveillance and security, market research, among others.\\
\indent Top-performing real-time face detection models have achieved performance rates of over 99\% in 2020, compared to the 96\% for leading algorithms in 2014, which have allowed for facial detection models to be applied successfully even in emergencies \cite{Tikoo2017}. Let’s take, for example, our current situation with the COVID-19 pandemic. Face detection has reached such impressive performance rates that, even when people wear masks, facial detection can efficiently locate individual’s faces, making it possible to track social interactions of potentially contagious individuals, allowing for an almost immediate response, hence minimizing the impact of the virus \cite{findfacepro2020}.\\
\indent The general trend in computer vision is to make deeper artificial networks, with numerous blocks and layers to achieve higher accuracy \cite{Szegedy2015}. However, those higher accuracy rates are paired to slower responses due to the heavier computational cost. These accurate but slow face detection models might fall short when facing real-world applications, which require real-time performance in equipment with limited computational capacity. \\
\indent In section two of this article some of the current state-of-the art image detection models are presented, with focus in performance comparisions, specifically those related to the model’s accuracy, recall, and speed.\\
\indent In section three, SwiftFace is presented. A faster approach to face detection based on previous image detection models thought to maintain accuracy while improving the model’s speed.\\
\indent In section four, SwiftFace’s performance is brought into comparison with two image detection models, widely used due to their high accuracy rates and out-standing speed even when used in low-end devices.\\
\indent In section five, possible real applications of SwiftFace are presented, with a focus on SwiftFace’s strong point: top-performing speed without loss of accuracy.\\
\indent In section six, the concluding remarks from this article are presented and related future research topics are proposed.
\section{RELATED WORK}
With the growth of available computing power and images for training and testing data, the number of works in developing object detection algorithms has risen.
Face detection, as a special case of object detection, is usually improved based on algorithms originally meant to be used in multi-class object detection. These algorithms are usually based on deep learning and artificial neural networks and, as such, can handle and process large amounts of data in a relatively short time \cite{Al-allaf2014}.\\
\indent Convolutional Neural Networks (CNNs) is one of the main architectures used in computer vision. As opposed to Multi-Layer Perceptrons, CNN has filters that in turn generate convoluted layers from which information is extracted, instead of just fully-connected layers. The use of a convolution layer allows for the model to extract relational information and identify patterns drawn from an input. Also, since the layers are not fully-connected, CNN has fewer values that need to be learned and updated than a regular MLP neural network, as filters perform much better in image recognition problems.\\
\indent Based on Convolutional Neural Networks, some of the most used image recognition models are presented in the following subsections.
\begin{figure}[!htb]
\centering
\includegraphics[width=3.5in]{competitor_image}
\captionsetup{justification=centering,margin=0.3in}
\caption{Graphic comparison between the main performance indicators of current state-of-the-art image detection models.}
\label{fig_2}
\end{figure}
\subsection{YOLO}
You Only Look Once (YOLO) \cite{Redmon2016} is an algorithm based on CNN designed to operate with fast response speeds as an object detector in production systems. It was developed as a one-step process involving detection and classification. YOLO uses a single-step neural net-work to predict class probabilities from the input images directly in one evaluation \cite{Bochovsky2020}. The CNN uses several filters to divide the input image into weighted grid cells and then predicts the and classification values for the image by studying batches of cells and their relationship with each other. At 78.6 FPS, YOLO’s speed is several times bigger than the most common two-stage detectors, and the model’s accuracy at mAP of 91 looks relatively high when compared with direct counterparts \cite{Chen2020}.\\
\indent Since the first YOLO model came out, several variations have been developed; the Tiny YOLO version being one of the fastest among them. The Tiny-YOLO architecture is around 400\% faster than the original version, being able to achieve 244 FPS on a computer with a single GPU \cite{Bonn2020}.
\subsection{RCNN}
Regional-based Convolutional Neural Networks (R-CNN) is also a CNN based algorithm used for image detection \cite{Zang}. R-CNN uses selective search to extract certain regions from the target image. Therefore, the problem is reduced from the full number of cells in which the image was originally divided, to just regions that group some adjacent cells. From each region proposal, a feature vector is extracted and fed into a regular CNN and then evaluated \cite{Jiang2017}. R-CNN results in high accuracy, but even reducing the image analysis according to the regions, this model doesn’t achieve acceptable real-time performance, which makes it unviable for real-time applications.\\
\indent To correct these drawbacks of R-CNN, several variations have been developed in order to obtain similar object detection algorithms but with higher speeds. Fast R-CNN and Faster R-CNN are two of the most notables. As the name suggests, Faster R-CNN has the best performance in terms of speed, with 70.4 FPS, but with a lower accuracy at mAP 17. \\
\subsection{SSD}
Single Shot Detection (SSD) is a feed-forward CNN based architecture that produces a fixed-size group of bounding boxes and classification values for the presence of object class instances within those boxes \cite{Liu2016}. After the boxing analysis, a non-maximum suppression step is applied in order to generate the output values. The network layers are based on a standard CNN architecture used for high-quality image classification, where convolution filters for each cell are used to make the predictions \cite{Li2018}.\\
\indent SSD has remarkable performance indicators at both speed (78.5 FPS) and accuracy (mAP 59).\\
\subsection{R-FCN}
Region-based Fully Convolutional Networks (R-FCN) was devised as an accurate and efficient object detection model \cite{Dai2016}. R-FCN addresses the computational cost of region-based networks and proposes a solution by using position-sensitive score maps, or regional feature maps where each detects and scores their corresponding region of the image \cite{Tang2020}. With this, combining the scores results in the image accurately being located.\\
\indent By reducing the computations needed compared to other regional-based convolutional neural networks, R-FCN tends to perform faster than its counterparts at 77.6 FPS, but has an even lower accuracy given by a mAP of 6.\\
\begin{table}[!htb]
\renewcommand{\arraystretch}{2.5}
\captionsetup{justification=centering,margin=0.3in}
\caption{Main Performance Indicators of Current State-of-the-Art Image Detection Models}
\label{table_1}
\centering
\begin{tabular}{p{3.5cm} C{2cm} C{2cm}}
\hline
Model & Accuracy & FPS\\
\hline
Faster R-CNN & 70.4 & 17\\
R-FCN & 77.6 & 6\\
SSD & 78.5 & 59\\
YOLOv3 & 78.6 & 91\\
\hline
\end{tabular}
\end{table}
\section{SWIFTFACE, A FASTER APPROACH}
SwiftFace is an architecture based on the tinyYOLO model, specially devised to focus on face detection, maintaining its accuracy while improving its speed. SwiftFace was trained using the WIDERFACE \cite{yang2016} dataset, making improvements to the original Tiny-YOLO architecture and tuning-up the model to improve greatly the detection speed and still achieve similar accuracy rates when detecting faces.
\begin{figure}[!htb]
\centering
\includegraphics[width=3.5in,height=5.5in]{swiftface_architecture}
\captionsetup{justification=centering,margin=0.3in}
\caption{A visual description of SwiftFace’s architecture.}
\label{fig_3}
\end{figure}
\begin{table}[!htb]
\renewcommand{\arraystretch}{2.5}
\captionsetup{justification=centering}
\caption{SwiftFace layers, including the number of filters, the filter’s size and stride and the layer’s input and output dimensions.}
\label{table_1}
\centering
\begin{tabular}{ccccc}
\hline
Layer & Filters & Size/Stride & Input & Output\\
\hline
0 conv & 16 & 3x3 / 1 &512x512x3 & 512x512x16\\
1 max & & 2x2 / 2 &512x512x16 &256x256x16\\
2 conv & 32 & 3x3 / 1 &256x256x16 & 256x256x32\\
3 max & & 2x2 / 2 &256x256x32 & 128x128x32\\
4 conv & 64 & 3x3 / 1 & 128x128x32 & 128x128x64\\
5 max & & 2x2 / 2 & 128x128x64 & 64x64x128\\
6 conv & 128 & 3x3 / 1 &64x64x64 & 64x64x64\\
7 max & & 2x2 / 2 &64x64x128 & 32x32x128\\
8 conv & 256 & 3x3 / 1 &32x32x128 & 32x32x256\\
9 max & & 2x2 / 2 &32x32x256 & 16x16x256\\
10 conv & 128 & 3x3 / 1 &16x16x256 & 16x16x512\\
11 conv & 18 & 1x1 / 1 &16x16x512 & 16x16x18\\
12 yolo & \multicolumn{4}{p{6cm}}{mask = 3,4,5; anchors = 10,14,23,27,37,58,81,82,135,169,344,319 classes = 1; num = 6; jitter = 0.3; ignorethresh = 0.7; truththresh = 1; random = 1} \\
13 route & 9 & & & 16x16x256\\
14 conv & 128 & 1x1 / 1 &16x16x256 & 16x16x128\\
15 upsample & & 2x / 1 &16x16x128 & 32x32x128\\
16 route & 158 & & & 32x32x384\\
17 conv & 256 & 3x3 / 1 &32x32x384 & 32x32x256\\
18 conv & 18 & 1x1 / 1 &32x32x256 & 32x32x18\\
\hline
\end{tabular}
\end{table}
\indent Swiftace is made by 18 layers, as opposed to Tiny-YOLO’s 23. The first ten layers are alternating pairs of convolution layers and pools, then there’s an additional convolution layer right before one that runs the same as the original Tiny-YOLO algorithm. Then a routing layer concatenates with the output of a previously defined layer. The upsample layer doubles the dimension of the input right before another route layer. Finally, two additional convolution layers extract information from their respective inputs and sending the data towards the final YOLO layer which then delivers the corresponding output.\\
\indent Being CNN based, SwiftFace's core feature is its convolutional layers. Their role is to extract features from the input image while preserving the information provided by the dimensional relationship between the cells of that image.\\
\indent In order to obtain the best results from the convolution neural network, specific parameters were set based on modifications on the YOLO original structure:\\
\indent The total amount of classes was set to one since the goal of SwifFace is face detection only. The maximum batches size parameter was established following the recommendation that it should be larger than the number of training images (~13.000), hence, it was set at 15.000. Steps then were set in the range of 80\%~90\% of the maximum batches, which meant between 12.000 and 13.500. The input image size was 512x512. The last layer function was set to be linear activation since only one class is expected as an output.\\
\indent SwiftFace was then trained and re-trained several times using the WIDERFACE dataset in order to come up with the best performant model for the face detection task. Paired with that, since SwiftFace focuses only on one class, as opposed to YOLO’s 80 classes, it performs faster than its counterpart.
\section{BENCHMARKS}
We tested our SwiftFace model against two of its direct competitors: YOLOv4 and Tiny-YOLOv4.\\
\begin{figure}[!htb]
\centering
\includegraphics[width=3.5in,height=2.5in]{speed_comparison}
\captionsetup{justification=centering,margin=0.3in,belowskip=15pt}
\caption{Speed comparison against YOLOv4 and TinyYOLOv4.\\}
\label{fig_4}
\end{figure}
\indent In terms of speed, YOLOv4 took 1329 seconds to process the entire WIDERFACE dataset, for a FPS 0f 12.1; while SwiftFace managed to do it in just 470 seconds, achieving a FPS of 39.5, making our model 69\% faster than the widely used YOLOv4. When testing Tiny-YOLOv4, a model designed specifically to yield high-performance rates in terms of speed, the time spent to process the entire WIDERFACE dataset was 533 seconds, yielding a FPS of 30.1, which makes it 24\% slower than SwiftFace. Figure 4 shows a graphic comparison between the processing speed of all three tested models.\\
\begin{table}[!htb]
\renewcommand{\arraystretch}{2.5}
\captionsetup{justification=centering,margin=0.3in}
\caption{Time to Process the Entire WIDERFACE Dataset}
\label{table_1}
\centering
\begin{tabular}{p{3.5cm} C{2cm} C{2cm}}
\hline
Model & Time (s) & Number of images\\
\hline
YOLOv4 & 1329 & 16067\\
Tiny YOLOv4 & 533 & 16067\\
SwiftFace & 407 & 16067\\
\hline
\end{tabular}
\end{table}
\indent In order to compare the models in terms of accuracy, we calculated their mean average precision (mAP). For its mAP (.5:.95), Tiny-YOLOv4 had a mAP of 54\%, versus SwiftFace’s 51\%. Despite the difference in speed performance, accuracy-wise both models perform quite similarly, which puts SwiftFace in the front line of models that could be used in different real-time applications without falling behind top-tiers object detection algorithms.
\begin{figure}[!htb]
\centering
\includegraphics[width=3.5in]{accuracy_comparison}
\captionsetup{justification=centering,margin=0.3in}
\caption{Speed comparison against YOLOv4 and TinyYOLOv4.}
\label{fig_5}
\end{figure}
\section{REAL-LIFE APPLICATIONS}
The range of applications for software such as SwiftFace is wide; and it’s getting wider every year thanks to technological advances. Being a fast-performing face detection algorithm that competes in terms of accuracy with top-of-the-line image detection models, SwiftFace provides state-of-the-art performance for mobile devices, low-end devices, and edge computing. It can be used in real-time applications and efficiently reduce the amount of human input in different fields. Applications involving low-end devices, which are more affordable and easily available, especially benefit from algorithms like SwiftFace, since it reduces the hardware and deployment costs of face detection applications in production. Among the most promising uses, we find:
\subsection{Human-Machine interaction}
Human-computer interaction systems are evolving to be more independent and to need less and less human input. One way of achieving this is making human-machine interactions control schemes dependent on intuitive human features, such as the face. Cameras that automatically take a picture when detecting a smiling face are an example of this.
\subsection{Mobile apps and entertainment}
Talking a bit deeper about cameras, some cellphone cameras use face detection for autofocus. This type of face detection is also useful for selecting regions of interest in photo slideshows, automatic face-tagging in photo-graphs is social media, face-priority algorithms when displaying images previews in profile pictures and in-ternet posts, etcetera.
\subsection{Work from Home and online education}
With the current worldwide-pandemic situation, online solution to daily activities like work and education have been rapidly growing. Fast-performing and accurate face detection models work as a as well as the main step in relevant processes such as quick and easy attendance-check, face-focusing video and automatic background changing.
\subsection{Market research}
Face detection will have an impact on marketing thanks to its application in market research. Instead of having a human performing the tedious task of recognizing and counting how many possible buyers’ glance at a product in a display, a face detection algorithm can do the same at a lower cost and possibly even with higher accuracy, by just integrating a camera to detect face that walks by. With further processing, additional information can be gathered from the public, such as age and gender, to build even more specific buyer personas.
\subsection{Access control}
A quick and efficient face detection phase in biometric access control, whether it is face identification to unlock mobile devices or a company’s biometric access to its office building, is essential for the proper performance of the access system.
\section{CONCLUSION AND FUTURE RESEARCH}
In a world where face detection and face recognition are going to be a central part of the technological develop-ment, affecting in fields as diverse as business, entertainment, and security, SwiftFace is a lighter face detec-tion model that performs 30\% faster than state-of-the-art object detection models. This will help future research-ers, engineers, and entrepreneurs to build world-class applications to advance even more the growing world-wide technological development.\\
\indent Face recognition in low-end or edge devices is still an unsolved problem with the current models. SwiftFace could provide an affordable, easy way of bringing face detection to these devices. In addition, SwiftFace offers numerous possible contributions to the field of face recognition. With similar accuracy performance to top-of-the-line models, but with better response times, SwiftFace has great potential for real-time face detection phases in face recognition models. \\
\indent In general, faster object detection algorithms tend to have lower accuracy rates than their slower counterparts. Nevertheless, with the impressive rise in the accuracy of computer vision models in the past years, this gap is sure to close with further improvements. As such, even SwiftFace could improve its rates compared to top-accuracy models like faster RCNN to provide better detections.\\
Future research should be carried in order to translate the results from SwiftFace to a face recognition model, to improve and compare inference time against state-of-the-art models such as RetinaNet, OpenFace, and FaceNet.
|
2110.10363
|
\section{Introduction}
Optimal transport theory concerns the minimum cost, called the \textit{transportation distance}, of moving mass from one configuration to another. In this paper, the notion of transportation distance that we are concerned with is the $L^1$ transportation distance, which we refer to as the \textit{Wasserstein distance}. The Wasserstein distance has applications in fields such as image processing, where a goal is to efficiently transform one image into another (e.g., \cite{rubner2000earth}), and machine learning, where a goal is to minimize some transport-related cost (e.g., \cite{frogner2015learning}).
The application of Wasserstein distance that motivates this paper is the definition of \textit{$\alpha$-Ricci curvature} $\kappa_\alpha$ on graphs introduced by Lin, Lu, and Yau in \cite{lin2011ricci}:
$$\kappa_\alpha = 1 - \frac{W(m_x^\alpha,m_y^\alpha)}{\textrm{d}(x,y)}.$$
Here $\textrm{d}(x,y)$ is the graph distance between vertices $x$ and $y$, while $m_v^\alpha$ is the 1-step transition probability measure of a random walk starting at vertex $v$ with laziness $\alpha$, and $W(m_x^\alpha,m_y^\alpha)$ is the Wasserstein distance between $m_x^\alpha$ and $m_y^\alpha$.
The $\alpha$-Ricci curvature is a generalization of classical Ricci curvature, an object from Riemannian geometry that captures how volumes change as they flow along geodesics (\cite{ollivier2011visual}).
In \cite{ollivier2009ricci}, Ollivier created the Ollivier-Ricci curvature to generalize the idea of Ricci curvature to discrete spaces, such as graphs. The Ollivier-Ricci curvature between $X$ and $Y$ is defined via the Wasserstein distance between the $1$-step transition probability measures of random walks starting at $X$ and $Y$. It captures roughly whether the neighborhoods of $X$ and $Y$ are closer together than $X$ and $Y$ themselves. The Ollivier-Ricci curvature is well-studied in geometry and graph theory (\cite{jiradilok2021transportation}, \cite{CushingKamtue+2019+22+44}, \cite{bourne2018ollivier}, \cite{cushing2020rigidity}, \cite{van2021ollivier}), and is also used to study economic risk, cancer networks, and drug design, among other applications (\cite{sandhu2015graph}, \cite{sandhu2016ricci}, \cite{sia2019ollivier}, \cite{wang2016interference}, \cite{wee2021ollivier}, \cite{jiradilok2021transportation}). Lin, Lu, and Yau further generalized the Ollivier-Ricci curvature to $\alpha$-Ricci curvature (\cite{lin2011ricci}), allowing for the laziness $\alpha$ of the random walks considered to be greater than zero.
In \cite{ollivier2009ricci}, Ollivier suggested exploring Ollivier-Ricci curvature on graphs at ``larger and larger scales." Thus, in this paper, we study the Wasserstein distance between $k$-step probability measures of random walks with potentially nonzero laziness as $k$ gets larger and larger. Since $1$-step probability distributions of random walks were used to study the initial ``small-scale" $\alpha$-Ricci curvature, these $k$-step probability distributions are a natural way to understand curvature at ``larger and larger scales." Jiradilok and Kamtue (\cite{jiradilok2021transportation}) study these $k$-step distributions for larger and larger $k$ on infinite regular trees; in this paper, we study them instead on finite graphs.
Given a finite, connected, simple graph, we consider a random walk with starting vertex $w$ and laziness $\alpha$. The random walk is defined to be a Markov chain where at each step, we either stay at the current vertex with probability $\alpha$ or pick a neighboring vertex at random and move there. We then consider the probability distribution encoding the likelihood of being at each possible vertex after $k$ steps of this random walk, which is called a $k$-step probability distribution, or $k$-step probability measure.
Given two such random walks on one graph, starting at vertices $u,v$ and with respective lazinesses $\alpha,\beta$, we define the Wasserstein distance between their two $k$-step probability measures to be the minimum cost of moving between the two distributions. Here, moving 1 unit of mass across 1 edge costs 1 unit.
We can ask many questions about the Wasserstein distance at ``larger and larger scales." For instance, does the Wasserstein distance between the two $k$-step probability distributions always converge as $k \to \infty$? Also, what does it converge to in different cases? Even more interestingly, what can we say about the rate of convergence? In particular, when does the distance eventually remain constant, and how long could it take to reach constancy?
In this paper, we show in all cases that either the Wasserstein distance converges or the Wasserstein distance at every other step converges. We also classify what the distance converges to in all cases, addressing the first and second questions.
We then seek to understand the rate of convergence of the Wasserstein distance. We reach two main results. First, addressing the third question, we show that unless the Wasserstein distance at every other step is eventually constant, its rate of convergence is exponential (Theorem \ref{thm: Guvab Convergence Theorem}). We also address the fourth question by providing a partial characterization of exactly when the Wasserstein distance is eventually constant (Theorem \ref{thm: Characterization of Constancy}).
In Section 2, we provide formal definitions of key concepts used throughout the paper. In particular, we recall the definition of the Wasserstein distance and introduce the notion of a Guvab. A \textit{Guvab} refers to a pair of random walks on a finite connected simple graph, and these Guvabs are the primary object we study in this paper. In Section 3, we classify for all possible Guvabs the limiting behavior of the Wasserstein distance, when the distance converges, and what the distance converges to. This characterization provides a natural way to classify the Guvabs into four categories based on their limiting behavior: $W=1$; $W=0$; $W=\frac{1}{2}$; and $\beta = 1$. In each of Sections 4, 5, 6, and 7, we consider one of these four categories of Guvabs and determine when the Wasserstein distance is eventually constant as well as examine the rate of convergence if the Wasserstein distance is not constant. Along the way, we encounter various interesting results about the different cases. Finally, in Section 8, we present main results about constancy and rate of convergence in general, obtained by considering each of these four cases individually.
\section{Preliminaries}
We begin with several formal definitions that we use in the remainder of the paper. We start by recalling graph theory terminology and the definition of Wasserstein distance on graphs. Then, we review random walks on graphs and define Guvabs. Finally, we briefly discuss terminology used to describe convergence.
In this paper, all graphs we consider are finite, connected, simple graphs. For a graph $G$, let $V(G)$ be the vertex set of $G$ and $E(G)$ be the edge set of $G$, i.e., the set of unordered pairs $\{v_1,v_2\}$ where $v_1,v_2$ are adjacent vertices in $G$. Further, for any $v\in V(G)$, let $N(v)$ be the neighbor set of $v$. Finally, denote by $\textrm{d}(w_1,w_2)$ the graph distance between vertices $w_1$ and $w_2$.
\begin{definition}
Define a \textbf{distribution} on the graph $G$ to be a function $\mu:V(G)\to \mathbb{R}$. We say $\mu$ is a \textbf{nonnegative distribution} if, for all $v\in V(G)$, we have $\mu(v)\geq 0$. A nonnegative distribution $\mu$ is a \textbf{probability distribution} if $\sum_{w \in V(G)} \mu(w) = 1$.
\end{definition}
For convenience, we will denote by $\Tilde{\textbf{0}}$ the distribution with value $0$ at all vertices, (i.e., for all $v\in V(G)$, we have $\Tilde{\textbf{0}}(v) = 0$). In addition, we will refer to a distribution $\mu$ for which $\sum_{w \in V(G)} \mu(w) = 0$ as a \textbf{zero-sum distribution}.
Given a graph $G$, let $\{\mu_i\}_{i=0}^\infty$ be an infinite sequence of distributions. Suppose that $f: \mathbb{Z}_{\geq0} \to \mathbb{Z}_{\geq0}$ is a strictly increasing function such that for all vertices $w\in G$, $\displaystyle\lim_{k\to\infty}\mu_{f(k)}(w)$ exists. Then denote by $\displaystyle\lim_{k\to\infty}\mu_{f(k)}$ the pointwise limit. Namely, for all $w \in V(G)$, let $\displaystyle \left( \lim_{k\to\infty}\mu_{f(k)} \right)(w)$ be $\displaystyle\lim_{k\to\infty}(\mu_{f(k)}(w))$.
For a given graph $G$, let $D = D(G)$
be the set of all ordered pairs $(\mu,\nu)$ of distributions on $V(G)$ that satisfy $\sum_{w \in V(G)} \mu(w) = \sum_{w \in V(G)} \nu(w)$. Further, let $D_{\geq0}$ be the set of all ordered pairs $(\mu,\nu) \in D$ with $\mu,\nu$ nonnegative distributions.
We now introduce some terminology from optimal transport theory. We follow definitions equivalent to those in the book of Peyre and Cuturi \cite{COTFNT}.
In Definitions~\ref{def: terry the transportation plan fairy},~\ref{def: carrie the cost function fairy}, and~\ref{def: warry the wasserstein distance fairy}, we let $G$ be a graph with two nonnegative distributions $\mu, \nu$ on $V(G)$ such that $(\mu,\nu)\in D_{\geq0}$.
\begin{definition}[c.f. \cite{COTFNT}]\label{def: terry the transportation plan fairy}
Define a \textbf{transportation plan} from $\mu$ to $\nu$ for $(\mu,\nu)\in D_{\geq0}$ to be a function $T_{\mu,\nu}: V(G)\times V(G) \to \mathbb{R}$ such that
\begin{itemize}
\item for any vertices $w_1, w_2 \in V(G)$, we have that $T_{\mu,\nu}(w_1,w_2) \geq 0$,
\item for all vertices $w\in V(G)$, we have that $\sum_{i \in V(G)} T_{\mu,\nu}(w,i) = \mu(w)$,
\item for all vertices $w\in V(G)$, we have that $\sum_{i \in V(G)} T_{\mu,\nu}(i,w) = \nu(w)$.
\end{itemize}
Denote by $\mathcal{T}_{\mu,\nu}$ the set of all transportation plans from $\mu$ to $\nu$.
\end{definition}
Following \cite{kantorovich2006translocation}, we can intuitively visualize a transportation plan $T_{\mu,\nu}$ as a way to move mass distributed over the vertices of $G$ according to $\mu$ along the edges of $G$ to an arrangement according to $\nu$. We now consider the \textit{cost} of a given transportation plan $T_{\mu,\nu}$: if moving 1 unit of mass across 1 edge has a cost of 1, how much does it cost to move the mass distribution of $\mu$ to that of $\nu$ according to $T_{\mu,\nu}$?
\begin{definition}[c.f. \cite{COTFNT}]\label{def: carrie the cost function fairy}
Define the \textbf{cost function} $C: \mathcal{T}_{\mu,\nu}\to\mathbb{R}$ to take any transportation plan $T$ to its cost $$C(T) = \sum_{(w_1,w_2) \in V(G)\times V(G)} \textrm{d}(w_1,w_2) \cdot T(w_1,w_2).$$
\end{definition}
\begin{definition} [c.f. \cite{COTFNT}] \label{def: warry the wasserstein distance fairy}
Define the \textbf{Wasserstein distance} $W_{\geq 0}: D_{\geq 0} \to \mathbb{R}_{\geq 0}$ by $W_{\geq 0}(\mu,\nu) := \displaystyle \min_{T \in \mathcal{T}_{u,v}} C(T)$
\end{definition}
We can thus interpret the Wasserstein distance as the minimum cost of transporting mass from its arrangement in distribution $\mu$ to an arrangement in distribution $\nu$.
\begin{remark} \label{rem: addy the addition fairy}
Note that for any distribution $\psi$ on $V(G)$, if $\mu$, $\nu$, $\mu+\psi$, and $\nu+\psi$ are all nonnegative, then $W_{\geq0}(\mu,\nu) = W_{\geq0}(\mu+\psi,\nu+\psi)$ (for a proof, see for example \cite{jiradilok2021transportation}, which notes that the Wasserstein distance between $\mu$ and $\nu$ can be defined in terms of $\mu-\nu$).
\end{remark}
Let $G$ be a graph with two distributions $\mu, \nu$ on $V(G)$ such that $$\sum_{w \in V(G)} \mu(w) = \sum_{w \in V(G)} \nu(w).$$ Let $\psi$ be a distribution such that $\mu+\psi$ and $\nu+\psi$ are both nonnegative. We extend the domain of the Wasserstein distance to include distributions $\mu,\nu$ with negative entries by defining $W(\mu,\nu): D\to \mathbb{R}_{\geq0}$ to be $W_{\geq0}(\mu+\psi,\nu+\psi)$. By Remark~\ref{rem: addy the addition fairy}, $W(\mu,\nu)$ is well-defined.
Even if $\mu$ and $\nu$ have negative entries, we can interpret $W(\mu,\nu)$ as the cost of some optimal ``transportation plan" that moves mass from distribution $\mu$ to distribution $\nu$.
Thus, in the rest of the paper, ``transportation plans" between distributions $\mu$ and $\nu$ allow for negative entries in $\mu$ and $\nu$. In this case, a transportation plan rigorously refers to a transportation plan from $\mu + \psi$ to $\nu+\psi$ for some $\psi$ large enough that $\mu+\psi$ and $\nu+\psi$ are both nonnegative. In particular, the movement of mass between $\mu$ and $\nu$ from a vertex $w_1$ to a different vertex $w_2$ actually refers to that same movement of mass from $w_1$ to $w_2$ between the distributions $\mu+\psi$ and $\nu+\psi$.
We now discuss a different way of calculating the Wasserstein distance.
\begin{definition}[c.f. \cite{COTFNT}]
Given a graph $G$, a \textbf{1-Lipschitz function} $\ell: V(G) \to \mathbb{R}$ is a function on the vertices of G where for any $w_1, w_2 \in V(G)$, we have that $|\ell(w_1) - \ell(w_2)| \leq \textrm{d}(w_1,w_2)$. Let $L(G)$ be the set of all 1-Lipschitz functions on $G$.
\end{definition}
\begin{theorem}[Kantorovich Duality, c.f. \cite{COTFNT}]
Let $G$ be a graph with two distributions $\mu, \nu$ on $V(G)$ such that $\sum_{w \in V(G)} \mu(w) = \sum_{w \in V(G)} \nu(w)$. Then $$W(\mu,\nu) = \max_{\ell \in L} \sum_{w \in G} \ell(w)(\mu(w) - \nu(w)).$$
\end{theorem}
We now seek a way to refer to a pair of random walks on a graph, as these pairs of random walks are the objects we study. The information needed to define such a pair consists of the graph $G$, the starting vertices $u$ and $v$ of the two random walks, and the respective lazinesses $\alpha$ and $\beta$ of the random walks. We thus define a \textit{Guvab} comprised of this information.
\begin{definition}
We define a \textbf{Guvab} to be a tuple $(G,u,v,\alpha,\beta)$ where $G$ is a finite, connected, simple graph, $u,v \in V(G)$, and $\alpha, \beta \in [0,1]$ with $\alpha \leq \beta$.
\end{definition}
\begin{definition}
Consider a graph $G$. For any starting vertex $u\in V(G)$ and laziness $\alpha\in[0,1]$, consider the random walk $R = \{R_k\}_{k=0}^{\infty}$ such that $R_0=u$, and, for $i\geq 1$, we have $R_i=R_{i-1}$ with probability $\alpha$, and $R_i=t$ with probability $\frac{1-\alpha}{\deg(R_{i-1})}$ for any $t\in N(R_{i-1})$. We say the probability distribution $\mu_k$ for $R_k$ is a \textbf{k-step probability measure}.
\end{definition}
Consider some Guvab $\mathcal{G} = (G,u,v,\alpha,\beta)$. We let $X(\mathcal{G}) = \{X_k\}_{k=0}^{\infty}$ be the Markov chain corresponding to a random walk with laziness $\alpha$ starting from vertex $u$ and we let $Y(\mathcal{G}) = \{Y_k\}_{k=0}^{\infty}$ be the Markov chain corresponding to a random walk with laziness $\beta$ starting from vertex $v$. When it is clear which Guvab $\mathcal{G}$ we are referring to, we write $X,Y$ instead of $X(\mathcal{G}), Y(\mathcal{G})$, respectively.
Consider some Guvab $\mathcal{G} = (G,u,v,\alpha,\beta)$. For all $k \geq 0$ we let $\mu_k(\mathcal{G}),\nu_k(\mathcal{G})$ be the k-step probability measures of $X(\mathcal{G}), Y(\mathcal{G})$ respectively. We let $\xi_k(\mathcal{G}) = \mu_k(\mathcal{G}) - \nu_k(\mathcal{G})$ and $W_k(\mathcal{G}) = W(\mu_k(\mathcal{G}), \nu_k(\mathcal{G}))$. When it is clear which Guvab $\mathcal{G}$ we are referring to, we write $\mu_k,\nu_k,\xi_k,W_k$ instead of $\mu_k(\mathcal{G}),\nu_k(\mathcal{G}),\xi_k(\mathcal{G}),W_k(\mathcal{G})$, respectively.
Given a Guvab $\mathcal{G}$, we define $P_{\alpha}$ and $P_{\beta}$ to be the transition probability matrices of $X$ and $Y$, respectively. In particular, for all $k$, we have that $\mu_k = \mu_0 P_{\alpha}^k$ and $\nu_k = \nu_0 P_{\beta}^k$, where the distributions are row vectors. We also define $P$ to be the transition probability matrix of a random walk with zero laziness on $G$ (note that $P$ does not depend on the starting vertex of the random walk). We note that $P_\alpha$ and $P_\beta$ only depend on $\alpha$ and $\beta$, not $u$ and $v$. In particular, $P_\alpha = \alpha I + (1-\alpha)P$ and $P_\beta = \beta I +(1-\beta)P$.
\begin{lemma} \label{lem: eileen the eigval sum fairy}
Let $\{\lambda_1,\ldots, \lambda_n\}$ be the union of the set of eigenvalues of $P_{\alpha}$ and the set of eigenvalues of $P_{\beta}$. For all vertices $w$, there exist some constants $c^w_i$ such that for all $k \geq 1$, we have $\xi_k(w) = \sum_{i = 1}^n c^w_i \lambda_i^k$.
\end{lemma}
\begin{proof}
This follows from the fact that $P_{\alpha}$ and $P_{\beta}$ are diagonalizable (since random walks are reversible (\cite{levin2017markov}) and thus have diagonalizable matrices (\cite{levin2017markov}, Chapter 12)). Say $P_{\alpha}$ has eigenvalues $\lambda_1, \ldots, \lambda_m$ and $P_{\beta}$ has eigenvalues $\lambda_{m+1},\ldots, \lambda_{m'}$. Since $P_{\alpha}$ is diagonalizable, we can write it as $ADA^{-1}$ for invertible matrix $A$ and diagonal matrix $D$ with diagonal entries $\lambda_1, \ldots, \lambda_m$.
Then $\mu_k = \mu_0 P_{\alpha}^k = \mu_0 A D^k A^{-1}$, so for all $w$ there exist constants $x^w_1, \ldots, x^w_m$ such that for all $k \geq 1$ we have $\mu_k(w) = \sum_{i = 1}^m x^w_i \lambda_i^k$. By similar reasoning, for all $w$ there exist constants $y^w_{m+1}, \ldots, y^w_{m'}$ such that for all $k \geq 1$ we have $\nu_k(w) = \sum_{i = m+1}^{m'} y^w_i \lambda_i^k$. Therefore, for all $w$, there exist some constants $c^w_i$ such that for all $k \geq 1$, we have that $\xi_k(w) = \sum_{i = 1}^{m'} c^w_i \lambda_i^k$. If for any $i$ and $j$ we have $\lambda_i = \lambda_j$, we can collect these like terms and thus create a list of distinct eigenvalues $\lambda_1,\ldots, \lambda_n$ and constants $c^w_1,\ldots c^w_n$ such that for all $k \geq 1$, we have $\xi_k(w) = \sum_{i = 1}^n c^w_i \lambda_i^k$. In particular, $\lambda_1,\ldots, \lambda_n$ will be exactly the elements of the union of the set of eigenvalues of $P_\alpha$ and the set of eigenvalues of $P_\beta$.
\end{proof}
In the next section, we discuss when and how the Wasserstein distance converges, which is related to the convergence of probability distributions of random walks. Since random walks can be viewed as Markov chains, we reference some classical Markov chain theory, using the same definitions as in \cite{levin2017markov}. We also use the following well-known Markov chain theorem.
\begin{theorem} [c.f. \cite{levin2017markov}] \label{thm: aperiodic irreducible}
Suppose that a Markov chain $X$ is aperiodic and irreducible with probability distributions $(\mu_0,\mu_1,\ldots)$ and stationary distribution $\pi$. Then $\lim_{k\to\infty}\mu_k = \pi$.
\end{theorem}
Finally, in our discussion of convergence, we encounter cases where the Wasserstein distance is eventually constant. To quantify this precisely, we provide the following definition.
\begin{definition}
We call an infinite sequence $\{S_i\}_{i=0}^{\infty}$ for $S_i \in \mathbb{R}$ \textbf{eventually constant} if there exists $N \geq 0$ such that for all $k \geq N$, we have that $S_k = S_N$.
\end{definition}
\section{Classifying End Behavior of $W_k$}
In this section, we seek to enumerate the possible end behaviors of the Wasserstein distance for a Guvab. In particular, we prove results about when the Wasserstein distance converges and what it converges to for different Guvabs. The classification of Guvabs by end behavior paves the way for our later discussion of the rate of convergence of the Wasserstein distance.
We begin with a technical lemma showing that the limit of the Wasserstein distance is the Wasserstein distance of the limit, as we expect.
\begin{lemma}\label{lem: converges to stationary distance}
Let $f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ be a strictly increasing function. If $\displaystyle\lim_{k\to\infty}\mu_{f(k)}=\mu$ and $\displaystyle\lim_{k\to\infty}\nu_{f(k)}=\nu$ (and, in particular, both limits exist), then $$\lim_{k\to\infty}W(\mu_{f(k)},\nu_{f(k)})=W(\mu,\nu).$$
\end{lemma}
\begin{proof}
Note that, by the triangle inequality, $$W(\mu_{f(k)},\nu_{f(k)})\leq W(\mu_{f(k)},\mu)+W(\mu,\nu)+W(\nu,\nu_{f(k)})$$ and $$W(\mu,\mu_{f(k)})+W(\mu_{f(k)},\nu_{f(k)})+W(\nu_{f(k)},\nu)\geq W(\mu,\nu).$$ This implies that
\begin{align*}
W(\mu,\nu)-W(\mu,\mu_{f(k)})-W(\nu_{f(k)},\nu)
&\leq W(\mu_{f(k)},\nu_{f(k)})\\
&\leq W(\mu_{f(k)},\mu)+W(\mu,\nu)+W(\nu,\nu_{f(k)}).
\end{align*}
However, $$\displaystyle\lim_{k\to\infty}W(\mu,\mu_{f(k)})=\lim_{k\to\infty}W(\mu-\mu_{f(k)},0)=0$$ (and similarly for $W(\nu,\nu_{f(k)})$). The above inequality implies that $$\lim_{k\to\infty}W(\mu_{f(k)},\nu_{f(k)})=W(\mu,\nu),$$ as desired.
\end{proof}
Due to classical Markov chain theory, we expect that in most cases, the probability distributions of both random walks converge to the same stationary distribution, and thus $\lim_{k\to\infty} W_k = 0$. The following definition and lemma quantify the stationary distribution that most random walks converge to. The subsequent theorem specifies what the ``most cases'' in which the distance goes to zero are.
\begin{definition}
For any graph $G$, we define the distribution $\pi$ to be such that for any $i \in G$, we have $\pi_i=\displaystyle \frac{\deg(i)}{\sum_{j\in G}\deg(j)}.$
\end{definition}
\begin{lemma} \label{lem: pippa the pi fairy}
When $0 < \alpha < 1$, the k-step probability measure $\mu_k$ converges to the stationary distribution $\pi$.
\end{lemma}
\begin{proof}
Recall that $X$ is the Markov chain of the random walk. We have that $X$ is aperiodic (we can return from a vertex to itself in one step) and irreducible ($G$ is connected). We have that for any vertex $w\in G$, $$\pi_w = \displaystyle \sum_{i \sim w}\pi_i\frac{1}{\deg(i)} = \displaystyle \alpha\pi_w + \sum_{i \sim w}\pi_i\frac{1-\alpha}{\deg(i)}.$$ Thus, $\pi$ is a stationary distribution of $X$. Hence, by Theorem \ref{thm: aperiodic irreducible}, $\pi$ is a limiting distribution for $X$ and thus $\lim_{k\to\infty}\mu_k = \pi$.
\end{proof}
\begin{theorem}\label{thm: 0-convergence}
The value $W(\mu_k,\nu_k)$ converges to $0$ as $k \to \infty$ if and only if one of the following conditions is true:
\begin{itemize}
\item $0 < \alpha\leq \beta < 1$,
\item $\alpha = \beta = 1$ and $u = v$,
\item $G$ is not bipartite and $0=\alpha\leq \beta < 1$,
\item $\alpha = \beta = 0$ and there exists a path from $u$ to $v$ with an even number of steps.
\end{itemize}
\end{theorem}
\begin{proof}
Note that for $0 < \alpha\leq \beta < 1$ , we have by Lemma \ref{lem: pippa the pi fairy} that $$\lim_{k\to\infty}\mu_k = \lim_{k\to\infty}\nu_k = \pi.$$ Thus, $\lim_{k\to\infty} W(\mu_k,\nu_k)=0$ in this case.
We now consider the cases where $\alpha=0$ or $\beta = 1$. If $\beta=1$, then $Y$ stays at $v$ forever. Thus, in order to have $\lim_{k\to\infty} W(\mu_k,\nu_k)=0$, we need $\alpha=1$ and $u=v$. This is sufficient to imply $\lim_{k\to\infty} W(\mu_k,\nu_k)=0$.
It remains to look at the case where $\alpha=0$ and $\beta< 1$, which we break into subcases based on whether $G$ is bipartite.
We first tackle the subcase where $G$ is not bipartite, i.e., $G$ contains an odd cycle. Since $\alpha,\beta< 1$, both $X,Y$ are aperiodic (there is a path from any vertex to itself in both an odd number of steps and an even number of steps via the odd cycle) and irreducible ($G$ is connected). Thus, $\lim_{k\to\infty}\mu_k=\lim_{k\to\infty}\nu_k=\pi$ and $\lim_{k\to\infty} W(\mu_k,\nu_k)=0$ as before.
Finally, we address the subcase where $G$ is bipartite with sides $S_1,S_2$. Here, $X$ is periodic (with period 2), so $Y$ must be periodic as well to have $\lim_{k\to\infty} W(\mu_k,\nu_k)=0$. Thus, $\beta=0$. If $u,v$ are on different sides of $G$, then $X$ and $Y$ will never be on the same side, so we cannot have $\lim_{k\to\infty} W(\mu_k,\nu_k)=0$. Otherwise, without loss of generality let $u,v \in S_1$. Consider the Markov chains $X' =\{X_{2k}\}_{k=0}^{\infty}$ and $Y' =\{Y_{2k}\}_{k=0}^{\infty}$ with vertex set $S_1$. Since $X'$ and $Y'$ are aperiodic (we can get from a vertex to itself in one step of $X'$ or $Y'$ by moving back and forth along the same edge) and irreducible ($G$ is connected), and they both have the same transition matrix, the Markov chains converge to the same stationary distribution. Similar reasoning applies for $\{X_{2k+1}\}$ and $\{Y_{2k+1}\}$. This finishes the proof for this case, hence completing the proof of Theorem~\ref{thm: 0-convergence}.
\end{proof}
In the next part of this section, we specify what the stationary distributions look like for any possible Guvab, particularly considering Guvabs with more than one stationary distribution $\pi$. We show that all Guvabs either have one set of end behaviors they converge to or switch back and forth between two sets of end behaviors.
Suppose $\alpha = 0$. Let $G$ be bipartite with sides $S_1,S_2$, and without loss of generality let $u\in S_1$. Let $X_1 = \{X_{2k}\}_0^{\infty}$ and $X_2 = \{X_{2k+1}\}_0^{\infty}$.
Let $X_i'$ denote $X_i$ restricted to $S_i$ for $i\in\{1,2\}.$ For $i\in\{1,2\}$, let $\tau_i'$ be a distribution on $S_i$ such that $$(\tau_i')_w=\frac{2\deg(w)}{\sum_{j \in G}\deg(j)}$$ for $w \in S_i$. Further, for $i \in \{1,2\}$, let $\tau_i$ be a distribution on $G$ that is $\tau_i'$ on $S_i$ and has value 0 elsewhere.
\begin{lemma} \label{lem: bipartite limiting distribution}
For $i\in\{1,2\}$, the distribution $\tau_i'$ is the limiting distribution of $X_i'$.
\end{lemma}
\begin{proof}
First, we claim that $\tau_1P^2=\tau_1$ and $\tau_2P^2=\tau_2$, where $P$ is the transition matrix of $X$. Note that for $w \in S_2$, we have
$$\displaystyle(\tau_1P)_w= \sum_{i\sim w} \frac{1}{\deg(i)}(\tau_1)_i = \sum_{i\sim w} \left(\frac{1}{\deg(i)}\right)\left(\frac{\deg(i)}{\sum_{j\in G}\deg(j)}\right)=\frac{\deg(w)}{\sum_{j\in G}\deg(j)}=(\tau_2)_w.$$
This is because for all $i \sim w$, we have $i \in S_1$, which implies $(\tau_1)_i = \frac{\deg(i)}{\sum_{j\in G}\deg(j)}$. For $w\in S_1$, we have $\displaystyle(\tau_1P)_w=\sum_{i\sim w} \frac{1}{\deg(i)}(\tau_1)_i=0=(\tau_2)_w$. This is because for all $i \sim w$, we have $i \in S_2$, which implies $(\tau_1)_i = 0$. Hence, $\tau_1P=\tau_2$ and, by similar reasoning, $\tau_2P=\tau_1$. Thus, $\tau_1P^2=\tau_1$ and $\tau_2P^2=\tau_2$ as desired.
Also, we note that $\displaystyle \sum_{w\in S_i} (\tau_i')_w = \sum_{w\in S_i}\frac{2\deg(w)}{\sum_{j \in G}\deg(j)} = \frac{2|E(G)|}{\sum_{j \in G}\deg(j)} = 1.$
We now see that $\tau_i'$ is a stationary distribution of $X_i'$ for $i\in\{1,2\}$. Since $X_1'$ and $X_2'$ are irreducible and aperiodic (as shown in the proof of Theorem~\ref{thm: 0-convergence}), we have that $\tau_i'$ is a limiting distribution of $X_i'$ for $i\in\{1,2\}$.
\end{proof}
\begin{corollary}\label{cor: bipartite convergence}
If $u\in S_1$, then as $k \to \infty$, we have $\mu_{2k}$ converges to $\tau_1$ and $\mu_{2k+1}$ converges to $\tau_2$. Analogously, if $u\in S_2$ then as $k \to \infty$, we have $\mu_{2k}$ converges to $\tau_2$ and $\mu_{2k+1}$ converges to $\tau_1$.
\end{corollary}
\begin{proof}
Suppose $u\in S_1$; the proof will proceed analogously if $u\in S_2$. Then, $\mu_{2k}$ will always be 0 on $S_2$ and, by Lemma \ref{lem: bipartite limiting distribution}, it will converge to $\tau_1'$ on $S_1$ because $\mu_{2k}$ is the probability distribution of $X_1'$ on $S_1$. Similarly, $\mu_{2k+1}$ will always be 0 on $S_1$ and it will converge to $\tau_2'$ on $S_2$. Thus, $\mu_{2k}$ converges to $\tau_1$ and $\mu_{2k+1}$ converges to $\tau_2$.
\end{proof}
\begin{corollary}\label{corolawrence: xi0,xi1 are well defined}
For any Guvab, $\displaystyle \lim_{k \to \infty} \xi_{2k}$ and $\displaystyle \lim_{k \to \infty} \xi_{2k+1}$ are well-defined.
\end{corollary}
\begin{proof}
We show that for any $\mu$, we have that $\lim_{k\to\infty} \mu_{2k}$ and $\lim_{k\to\infty} \mu_{2k+1}$ are well-defined; this implies the statement of the corollary. When $G$ is bipartite and $\alpha = 0$, we know that $\lim_{k\to\infty} \mu_{2k} = \tau_1$ and $\lim_{k\to\infty} \mu_{2k+1} = \tau_2$ (assuming, without loss of generality, that $u\in S_1$). When $\alpha = 0$ and $G$ is not bipartite or when $0 < \alpha < 1$, we have
$$\lim_{k\to\infty} \mu_{2k} = \lim_{k\to\infty} \mu_{2k+1} = \pi.$$ Finally, when $\alpha = 1$, we know $\lim_{k\to\infty} \mu_{2k} = \lim_{k\to\infty} \mu_{2k+1} = \mathbbm{1}_u$. This covers all possible cases for $\alpha$ and $G$, so we are done.
\end{proof}
For any Guvab, we refer to $\displaystyle \lim_{k \to \infty} \xi_{2k}$ as $\xi^0$ and $\displaystyle \lim_{k \to \infty} \xi_{2k+1}$ as $\xi^1$.
The following corollary is quite important for the rest of this section and the remainder of this paper. Its relevance to this section is that $\lim_{k\to\infty} W_k$ will be well-defined unless $\lim_{k\to\infty} W_{2k} \neq \lim_{k\to\infty} W_{2k+1}$. The corollary is pertinent to the rest of the paper because it indicates that the rates of convergence of $\{W_{2k}\}$ and $\{W_{2k+1}\}$ are always well-defined. Thus, for any possible Guvab, we can study and state results about the rates of convergence of $\{W_{2k}\}$ and $\{W_{2k+1}\}$.
\begin{corollary}
We have that $\lim_{k\to\infty} W_{2k}$ and $\lim_{k\to\infty} W_{2k+1}$ are always well-defined.
\end{corollary}
\begin{proof}
We know that $\lim_{k\to\infty} W_{2k} = W(\xi^0,\Tilde{\textbf{0}})$ and $\lim_{k\to\infty} W_{2k+1} = W(\xi^1,\Tilde{\textbf{0}})$.
\end{proof}
We soon discuss many cases where $\lim_{k\to\infty} W_k$ exists, so we designate a way to refer to this limit. For any Guvab $\mathcal{G}$ where $\lim_{k\to\infty} W_k$ exists, we denote by $W$ the limit $\lim_{k\to\infty} W_k$.
We can now state and prove our main theorems about whether the Wasserstein distance converges and the values it converges to. For any possible Guvab, Theorem \ref{thm: convergence condition on mildly vexing scenarios} allows us to determine whether the Wasserstein distance converges. Furthermore, Theorem \ref{thm: convergence values} allows us to, in most cases, quickly and easily determine what value the Wasserstein distance will converge to. Finally, these theorems provide a framework for us to classify the Guvabs into four categories so we can use casework to understand the rate of convergence.
\begin{theorem}\label{thm: convergence values}
Unless $G$ is bipartite, $\alpha=0$, and $\beta=1$, we have that $\displaystyle W=\lim_{k\to\infty}W(\mu_k,\nu_k)$ is always well defined, and furthermore
\begin{itemize}
\item $W=0$ under the conditions specified in Theorem~\ref{thm: 0-convergence},
\item $W=1$ if $\alpha=\beta=0$ and $W\neq 0$,
\item $W=\frac{1}{2}$ if $0=\alpha< \beta < 1$ and $G$ is bipartite.
\end{itemize}
\end{theorem}
\begin{proof}
The first condition is clear by Theorem~\ref{thm: 0-convergence}. Next, we look at the case where $\alpha=\beta=0$ and $W\neq 0$. By Theorem~\ref{thm: 0-convergence}, this corresponds to the case where $G$ is bipartite and $u,v$ are on opposite sides of $G$. Without loss of generality, let $u\in S_1$ and $v\in S_2$. Then, as $k \to \infty$, we have $\mu_{2k}$ converges to $\tau_1$ and $\nu_{2k}$ converges to $\tau_2$ by Corollary~\ref{cor: bipartite convergence}. Analogously, $\mu_{2k+1}$ converges to $\tau_2$ and $\nu_{2k+1}$ converges to $\tau_1$. Thus, $\displaystyle \lim_{k \to \infty} W(\mu_k, \nu_k) = W(\tau_1, \tau_2)$. We have that $W(\tau_1, \tau_2) \geq 1$ because to get from $\tau_1$ to $\tau_2$, we must move all the mass from $S_1$ across at least one edge to $S_2$. Also, $W(\tau_1, \tau_2) \leq 1$ because we can achieve a distance of 1 by, for any given edge $ab$ with $a \in S_1$ and $b \in S_2$, moving a mass of $\displaystyle \frac{2}{\sum_{j \in G}\deg(j)}$ from $a$ to $b$.
We now consider the case when $0 = \alpha < \beta < 1$ and $G$ is bipartite. Without loss of generality, let $u\in S_1$. Since $\alpha=0$, we have that $\lim_{k\to\infty}\mu_{2k} = \tau_1$ and $\lim_{k\to\infty}\mu_{2k+1} = \tau_2$. Since $\beta > 0$, we have that $\lim_{k\to\infty}\nu_k = \pi$. Thus, we have that $\displaystyle \lim_{k \to \infty} W(\mu_{2k}, \nu_{2k}) = W(\tau_1,\pi)$ and $\displaystyle \lim_{k \to \infty} W(\mu_{2k+1}, \nu_{2k+1}) = W(\tau_2,\pi)$. If we show that $W(\tau_1,\pi) = W(\tau_2,\pi) = \frac{1}{2}$, we will have shown the third condition. We know that $\pi$ will have half its mass on $S_1$ and half its mass on $S_2$ because $$\sum_{v \in S_1} \pi_v = \sum_{v \in S_1} \frac{\deg(v)}{\sum_{j \in G}\deg(j)} = \frac{|E(G)|}{\sum_{j \in G}\deg(j)} = \frac{1}{2}.$$ Thus, half the mass must move from $S_2$ to $S_1$, so $W(\pi, \tau_1) \geq \frac{1}{2}$. We can also achieve a distance of exactly $\frac{1}{2}$ from $\pi$ to $\tau_1$ by, for any given edge $ab$ with $a \in S_1$ and $b \in S_2$, moving $\displaystyle \frac{1}{\sum_{j \in G}\deg(j)}$ mass from $b$ to $a$. Thus, $W(\pi,\tau_1) = \frac{1}{2}$ and by an analogous argument, $W(\pi,\tau_2) = \frac{1}{2}$.
We have now considered all cases where $\alpha,\beta < 1$ and where $\alpha = \beta = 1$. The only case left is where $0 < \alpha < 1$ and $\beta = 1$. Here, $\lim_{k \to \infty} \mu_k = \pi$ and $\nu_k = \mathbbm{1}_v$ where $\mathbbm{1}_v$ is the distribution with 1 at $v$ and 0 elsewhere. Thus, $\lim_{k \to \infty} W(\mu_{k}, \nu_{k}) = W(\pi, \mathbbm{1}_v)$, which is a constant.
\end{proof}
\begin{theorem} \label{thm: convergence condition on mildly vexing scenarios}
The distance $W(\mu_k,\nu_k)$ does not converge as $k \to \infty$ if and only if $G$ is bipartite, $\alpha = 0$ and $\beta = 1$, and $$\sum_{w \in V(G)} (-1)^{\emph{\textrm{d}}(v,w)}\emph{\textrm{d}}(v,w)\deg(w) \neq 0.$$
\end{theorem}
\begin{proof}
By Theorem~\ref{thm: convergence values}, we know that the only case where it is possible for $W(\mu_k,\nu_k)$ not to converge is when $G$ is bipartite, $\alpha = 0$, and $\beta = 1$. In this case, $\nu_k = \mathbbm{1}_v$. Additionally, assuming without loss of generality that $u \in S_1$, we have that
$$\lim_{k\to\infty}\mu_{2k} = \tau_1 \text{ and } \lim_{k\to\infty}\mu_{2k+1} = \tau_2.$$ Thus, $W(\mu_k,\nu_k)$ converges as $k \to \infty$ if and only if $W(\mathbbm{1}_v,\tau_1) = W(\mathbbm{1}_v,\tau_2)$.
To calculate $W(\mathbbm{1}_v,\tau_1)$, we note that we must move all the mass of $\tau_1$ to vertex $v$. To move all the mass at some vertex $w$ to $v$, we necessarily move a mass of $(\tau_1)_w$ over a distance of $\textrm{d}(w,v)$. Thus the total transportation cost, and thus the total Wasserstein distance $W(\mathbbm{1}_v,\tau_1)$, is given by
\begin{align*}
\sum_{w \in G} \textrm{d}(v,w)(\tau_1)_w &= \sum_{w \in S_1} \textrm{d}(v,w)\frac{2\deg(w)}{\sum_{j \in G}\deg(j)} + \sum_{w \in S_2} \textrm{d}(v,w)\cdot 0\\ &= \frac{2}{\sum_{j \in G}\deg(j)}\sum_{w \in S_1} \textrm{d}(v,w)\deg(w).
\end{align*}
By the same reasoning, we have that $$ W(\mathbbm{1}_v,\tau_2) = \frac{2}{\sum_{j \in G}\deg(j)}\sum_{w \in S_2} \textrm{d}(v,w)\deg(w).$$ Given that $G$ is bipartite, $\alpha = 0$, and $\beta = 1$, we know that the Wasserstein distance converges if and only if $W(\mathbbm{1}_v,\tau_1) - W(\mathbbm{1}_v,\tau_2) = 0$, which is true if and only if $$\sum_{w \in V(G)} (-1)^{\textrm{d}(v,w)}\textrm{d}(v,w)\deg(w) = 0$$
since the parity of $\textrm{d}(v,w)$ depends only on the side of $G$ that $w$ is on. Thus, the theorem statement follows.
\end{proof}
We now present a table summarizing much of the information about convergence discussed in this section.
\begin{center}
\begin{table}[H]
\small\addtolength{\tabcolsep}{-5pt}
\begin{tabular}{c|c|c|c|c|c}
Conditions on $\mathcal{G}$ & $W = 0$ & $W = \frac{1}{2}$ & $W = 1$ & $W = C\neq 0,\frac{1}{2},1$ & $W_k$ does not converge \\ \hline
$G$ bipartite, $\beta=1$ & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ \\
$G$ bipartite, $\beta<1$ & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\times$ & $\times$ \\
$G$ non-bipartite, $\beta=1$ & $\checkmark$ & $\times$ & $\checkmark$ & $\checkmark$ & $\times$ \\
$G$ non-bipartite, $\beta<1$ & $\checkmark$ & $\times$ & $\times$ & $\times$ & $\times$ \\
\end{tabular}
\caption{Is it possible for the Wasserstein distance to converge to particular limits in different cases of conditions on $\mathcal{G}$?}
\label{tab: The Guvable}
\end{table}
\end{center}
\vspace{-0.4in}
\begin{remark}
We know that the case of $G$ bipartite, $\beta = 1$, and $W = \frac{1}{2}$ is possible by considering a star with $v$ at the center and $0 < \alpha < 1$. We know that the case of $G$ non-bipartite, $\beta = 1$, and $W = \frac{1}{2}$ is impossible because in order for it to be possible, $\lim_{k\to\infty}\mu_k$ would need half of its mass to be at $v$. Since mass of $\lim_{k\to\infty}\mu_k$ is proportional to degree, every edge would have to be incident to $v$, making the graph bipartite.
\end{remark}
The following corollary provides a categorization of the Guvabs into four types. In the next four sections of this paper, we examine each of these categories in turn.
\begin{corollary} \label{cor: four guvab nations}
Each Guvab satisfies exactly one of the following four conditions:
\begin{itemize}
\item $W=1$ and $\beta < 1$,
\item $W=\frac{1}{2}$ and $\beta < 1$,
\item $W = 0$ and $\beta < 1$,
\item $\beta = 1$.
\end{itemize}
\end{corollary}
\begin{proof}
If $\beta < 1$, we have $W=0$ under the conditions in Theorem \ref{thm: 0-convergence} and $W = 1$ or $W =\frac{1}{2}$ otherwise, since the conditions in Theorem \ref{thm: convergence values} cover all possible cases where $\beta < 1$ and $W \neq 0$.
\end{proof}
If we understand the convergence of the Wasserstein distance in all four of these cases, then we understand the convergence for all Guvabs. The subsequent four sections each discuss the convergence of the Wasserstein distance in one of these cases. Our two main convergence theorems, presented in Section 8, put together the general results obtained by examining these four cases individually.
\section{Convergence when $W = 1$}
In this section we consider Guvabs with $W = 1$ and $\beta < 1$. Recall that these are exactly the Guvabs for which $G$ is bipartite, $u$ and $v$ are on different sides of the bipartite graph, and $\alpha = \beta = 0$. We show that all such Guvabs have a Wasserstein distance that is eventually constant. We also begin to understand how long it takes for the Wasserstein distance to reach constancy.
We first recall that the Wasserstein distance between two distributions $\mu$ and $\nu$ with potentially negative entries is the cost of an optimal transportation plan for moving mass\footnote{As discussed in section 2, the mass of a distribution $\mu$ at a vertex $w$ is $\mu(w)$, the value of the distribution at that vertex.} from $\mu$ to $\nu$. Thus, to prove the eventual constancy of the Wasserstein distance, we construct an algorithm that produces a transportation plan between any two distributions. Then, we show that when certain inequalities are satisfied, this transportation plan has a cost of exactly 1 and is optimal. Finally, we prove that when $\xi_k$ is eventually sufficiently close to either of the stationary distributions $\xi^0$ or $\xi^1$, these inequalities are satisfied.
We start by constructing the algorithm.
Pick a spanning tree $T$ of $G$ and let $L$ be the set of leaves of $T$. Define a function $r: V(G)\to \mathbb{Z}$
such that $r(w) = \min_{\ell \in L}\textrm{d}(w,\ell)$.
For any finite set $S$, let $\textrm{Perm}(S)$ denote the set of all permutations of $S$. We say that an \textbf{$r$-monotone ordering} $\mathcal{O} = (w_1, \ldots, w_n)\in \textrm{Perm}(V(G))$ is a permutation of $V(G)$ such that $r(w_1), \ldots, r(w_n)$ is a non-decreasing sequence.
\begin{definition} \label{Allie the Albatross Algorithm Fairy}
Given a graph $G$, a spanning tree $T$ of $G$, an $r$-monotone ordering $\mathcal{O}$ and zero-sum distribution $\xi$, we define the \textbf{tree-based transport algorithm}, which transports mass from $\xi$ to $\Tilde{\textbf{0}}$, to be an $(n-1)$-step algorithm in which at the $i$th step,
\begin{itemize}
\item if the current mass at $w_i$ is nonnegative, we distribute it evenly among all $v\sim w_i$ with indices greater than $i$,
\item if the current mass at $w_i$ is negative, we take an equal amount of mass to vertex $w_i$ from all $v\in N(w_i)$ with indices greater than $i$, so that the mass at $w_i$ is now $0$.
\end{itemize}
\end{definition}
In Lemma \ref{aggie the agtoe fairy}, we see that this algorithm produces a valid transportation plan from $\xi$ to $\Tilde{\textbf{0}}$. We refer to this \textbf{tree-based transportation plan} as $A(G,T,\mathcal{O},\xi)$. Given $G, T$ and $\mathcal{O}$, we let $A_i(\xi)$ denote the distribution of mass on the vertices of $G$ after $i$ steps of the algorithm.
\begin{lemma}\label{aggie the agtoe fairy}
The tree-based transport algorithm on $G, T, \mathcal{O}, \xi$ always produces a valid transportation plan from $\xi$ to $\Tilde{\textbf{0}}$.
\end{lemma}
\begin{proof}
After the $i$th step of the tree-based transport algorithm, the mass at each of the vertices $w_1, \ldots, w_i$ is $0$, since the mass at $w_j$ becomes zero at the $j$th step, and thereafter no mass is moved to or from $w_j$. Thus, after the $(n-2)$th step, the only vertices of $G$ with nonzero mass will be $w_{n-1}$ and $w_n$. Since the total mass sums to $0$ and $w_{n-1}$ is adjacent to $w_n$, the $(n-1)$th step of the algorithm simply moves the positive mass to the negative mass so that all vertices have mass $0$.
\end{proof}
We now prove a useful property of this algorithm.
\begin{lemma} \label{lem: linear}
Given a graph $G$, tree $T$ and $r$-monotone ordering $\mathcal{O}$, for all $i$, we have that $A_i$ is a linear function on the space of zero-sum distributions.
\end{lemma}
\begin{proof}
It suffices to show that for any two zero-sum distributions $\xi$ and $\xi'$, we have $A_i(\xi + \xi') = A_i(\xi) + A_i(\xi')$.
We prove this by induction on $i$.
Base case: When $i=0$, we have that $A_0(\xi + \xi') = \xi + \xi' = A_0(\xi) + A_0(\xi')$.
Inductive step: For the inductive hypothesis, we assume that $A_i(\xi + \xi') = A_i(\xi) + A_i(\xi')$. We want to show that $A_{i+1}(\xi + \xi') = A_{i+1}(\xi) + A_{i+1}(\xi')$. For any distribution $\xi$, if $n$ denotes the number of neighbors of $w_{i+1}$ with indices greater than $i+1$, then all of the following are true:
\begin{itemize}
\item $A_{i+1}(\xi)(w_{i+1}) = 0$,
\item for $w_j\in N(w_{i+1})$ with $j > i+1$, we have $A_{i+1}(\xi)(w_j) = A_i(\xi)(w_j) + \frac{1}{n} A_i(\xi)(w_{i+1})$,
\item for all other vertices $w$, we have $A_{i+1}(\xi)(w) = A_i(\xi)(w)$.
\end{itemize}
Thus, $A_{i+1}(\xi + \xi')(w_{i+1}) = 0 = 0 +0 = A_{i+1}(\xi)(w_{i+1}) + A_{i+1}(\xi')(w_{i+1})$. For $w_j\in N(w_{i+1})$ with $j > i+1$, we have that
\begin{align*}
A_{i+1}(\xi + \xi')(w_j) &= A_i(\xi + \xi')(w_j) + \frac{1}{n} A_i(\xi + \xi')(w_{i+1})\\
&= A_i(\xi)(w_j) + \frac{1}{n} A_i(\xi)(w_{i+1}) + A_i(\xi')(w_j) + \frac{1}{n} A_i(\xi')(w_{i+1})\\
&= A_{i+1}(\xi)(w_j) + A_{i+1}(\xi')(w_j).
\end{align*}
Finally, for all other vertices $w$, we have that
$$A_{i+1}(\xi + \xi')(w) = A_i(\xi + \xi')(w) = A_i(\xi)(w) + A_i(\xi')(w) = A_{i+1}(\xi)(w) + A_{i+1}(\xi')(w).$$ We have shown $A_{i+1}(\xi + \xi') = A_{i+1}(\xi) + A_{i+1}(\xi')$ for all the vertices, so we have proven the inductive step and thus the lemma.
\end{proof}
In Definition \ref{def: inequalities} and the subsequent results, we define the inequalities used in conjunction with the tree-based transport algorithm and show that when these inequalities are satisfied, the Wasserstein distance between $\xi$ and $\Tilde{\textbf{0}}$ will be 1.
\begin{definition} \label{def: inequalities}
For any graph $G$, zero-sum distribution $\xi$, spanning tree $T$, and $r$-monotone ordering $\mathcal{O} = (w_1,\ldots w_n)$ on $V(G)$, define the \textbf{tree-based transport inequalities} $\mathcal{I}(G,T,\mathcal{O},\xi)$ to be the union of the following two sets of inequalities:
\begin{itemize}
\item $I_1$: the set of inequalities of the form $\xi(w_j)A_i(\xi)(w_j) > 0$ for all $0 \leq i \leq |V(G)|-2$ and $i < j \leq V(G)$,
\item $I_2$: the set of inequalities of the form $\xi(t)\xi(w) < 0$ for all $t \sim w$
\end{itemize}
\end{definition}
\begin{lemma} \label{lem: dist is sum of pos mass}
If the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi)$ are satisfied, then the cost of the transportation plan is at most the sum of positive mass in $\xi$, i.e., $\displaystyle C(A(G,T,\mathcal{O},\xi))\leq\frac{1}{2} \sum_{w \in G} |\xi(w)|.$
\end{lemma}
\begin{proof}
We note that the inequalities in $I_1$ mean that for any vertex $w_j$, the sign of $A_i(\xi)(w_j)$ stays the same until the mass becomes 0 at the $j$th step, at which point it remains 0 for the rest of the algorithm.
Since only positive mass moves, it suffices to show that all the positive mass of $\xi$ moves a distance of at most 1. At each step of the tree-based transport algorithm, any mass that moves must move a distance of exactly 1. Thus, it suffices to show that all mass moves at most one time in $A(G,T,\mathcal{O},\xi)$.
To show this, we demonstrate that the mass that moves at each step of the algorithm has not moved before, since this means all mass moves at most once overall. We begin by demonstrating that every time positive mass moves, it moves from a vertex $w$ for which $\xi(w) > 0$.
The only way for mass to move is via the $i$th step of the tree-based transport algorithm, which starts from the distribution $A_{i-1}(\xi)$. Suppose the vertices are $w_1,\ldots w_n$. If $A_{i-1}(\xi)(w_i)$ is zero, then no mass moves on the $i$th step. If $A_{i-1}(\xi)(w_i)$ is positive, then at the $i$th step, mass moves away from $w_i$. In addition, by the inequalities in $I_1$, if $A_{i-1}(\xi)(w_i)$ is positive then $\xi(w_i)$ is positive and if $A_{i-1}(\xi)(w_i)$ is negative then $\xi(w_i)$ is negative. Thus, by the inequalities in $I_2$, we have $\xi(t) > 0$ for all $t \sim w_i$. If $A_{i-1}(\xi)(w_i)$ is negative, mass moves from these $t \sim w_i$ to $w_i$, so all mass movements are from a vertex $t$ for which $\xi(t) > 0$. Thus, in all three of these cases, every time positive mass moves, it moves from a vertex $w$ for which $\xi(w) > 0$.
Thus, consider the $i$th vertex, call this $w$, and suppose that $\xi(w) > 0$. Then, by the inequalities in $I_2$, we know $\xi$ has negative mass at the neighbors of $w$, so throughout all steps of the algorithm, the mass at the neighbors of $w$ was nonpositive. This means that anytime executing a step for one of the neighbors of $w$ changed the mass at $w$, mass moved from $w$ to its neighbors. Because no mass moved from another vertex to $w$, any remaining positive mass at $w$ has not yet moved. We also know that the remaining mass at $w$ is always nonnegative by the inequalities in $I_1$. Thus, whenever we execute a step of the algorithm for one of the neighbors of $w$, the nonnegative mass that moves from $w$ has not yet moved.
Furthermore, during the $i$th step of the algorithm, all the remaining nonnegative mass at $w$ moves away from it, and this mass has not yet moved. Mass movements due to steps of the algorithm for neighbors of $w$ and due to the $i$th step, which is for $w$, make up all possible movements of the mass initially at $w$. This argument holds for all vertices $w$ for which $\xi(w) > 0$, so all possible movements of positive mass move mass that has not been moved before. Thus, we are done.
\end{proof}
\begin{corollary} \label{cor: when ineqs, dist is sum of pos mass}
For any graph $G$ and zero-sum distribution $\xi$, if for some $T$ and $\mathcal{O}$ the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi)$ are satisfied, then
$$\displaystyle W(\xi,0) = \frac{1}{2} \sum_{w \in G} |\xi(w)|.$$
\end{corollary}
\begin{proof}
By Lemma \ref{lem: dist is sum of pos mass}, we have that $\frac{1}{2} \sum_{w \in G} |\xi(w)|$, the sum of positive mass, is the upper bound. For the lower bound, we note that all positive mass must move because we only move positive mass. Thus, all positive mass must move at least a distance of 1, so $W(\xi,0)$ will be at least the sum of positive mass.
\end{proof}
\begin{corollary} \label{cor: when ineqs, dist is 1}
For a Guvab $\mathcal{G}$ where $W = 1$ and $\beta < 1$, suppose that there exists some spanning tree $T$ of $G$ and $r$-monotone ordering $\mathcal{O}$ such that $\xi_k$ satisfies the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi_k)$. Then $W_k(\mathcal{G}) = 1$.
\end{corollary}
\begin{proof}
Recall that when $W = 1$ and $\beta < 1$, we must have that $G$ is bipartite, $u$ and $v$ are on different sides of the bipartite graph, and $\alpha = \beta = 0$. Thus, for all $k$ we have that $\mu_k$ and $\nu_k$ are nonzero on disjoint sets of vertices, since at all times $\mu_k$ is nonzero only on one side and $\nu_k$ is nonzero only on the other side. Thus $\sum_{w \in G} |\xi(w)| = 2$, so by Corollary \ref{cor: when ineqs, dist is sum of pos mass} we have that $$W_k(\mathcal{G}) = W(\xi_k,0) = \frac{1}{2} \sum_{w \in G} |\xi(w)| = 1.$$
\end{proof}
Now all that remains to be shown is that once $\xi_k$ is sufficiently close to either of the stationary distributions $\xi^0$ or $\xi^1$, the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi_k)$ will be satisfied. To prove this, we will first show that $\xi^0$ and $\xi^1$ lie on the interior of the region of distributions that satisfy the inequalities. The next lemma helps show that $\xi^0$ and $\xi^1$ satisfy the inequalities.
\begin{lemma} \label{lem: algorithm nice on stat dib}
Suppose we have a bipartite graph $G$ with sides $S_0$ and $S_1$ and a distribution $\xi$ such that for $w \in S_0$ we have $\xi(w) = \frac{\deg(w)}{|E(G)|}$ and for $w \in S_1$ we have $\xi(w) = -\frac{\deg(w)}{|E(G)|}$. Then pick an arbitrary spanning tree T and $r$-monotone ordering $\mathcal{O}$ on $V(G)$. Consider the tree-based transport plan $A(G,T,\mathcal{O},\xi)$. After each step, for each $i$ for $i \leq n-2$, we have that $A_i(\xi)(w_j) \geq \frac{1}{|E(G)|}$ for $w_j\in S_0$ with $j > i$ and that $A_i(\xi)(w_j) \leq -\frac{1}{|E(G)|}$ for $w_j\in S_1$ with $j > i$.
\end{lemma}
\begin{proof}
We know by Lemma \ref{lem: linear} that for all $w \in G$ and for all $0 \leq i \leq n-2$, we have $A_i(|E(G)|\xi)(w) = |E(G)|((A_i(\xi)(w))$. Thus, it suffices to show that after all steps $i$ for $i \leq n-2$, we have that $A_i(|E(G)|\xi)(w_j) \geq 1$ for $w_j\in S_0$ with $j > i$ and that $A_i(|E(G)|\xi)(w_j) \leq -1$ for $w_j\in S_1$ with $j > i$.
To prove this, for all $0 \leq i \leq n-2$, we define the graph $G_i$ to consist of the vertex set $V(G_i) = \{w_{i+1},\ldots w_n\}$ and all the edges of $E(G)$ that have both endpoints in $V(G_i)$. It suffices to show by induction on $i$ that for $i \leq n-2$, we have $A_i(|E(G)|\xi)(w_j) = \deg_{G_i}w_j$ for $w_j\in S_0$ with $j > i$ and we have $A_i(|E(G)|\xi)(w_j) = -\deg_{G_i}w_j$ for $w_j\in S_1$ with $j > i$.
Base case: When $i=0$, we note that $G_0 = G$. When $i = 0$, by the definition of $\xi$, we have that $A_i(|E(G)|\xi)(w_j) = \deg_{G}w_j$ for $w_j\in S_0$ with $j > i$ and that $A_i(|E(G)|\xi)(w_j) = -\deg_{G}w_j$ for $w_j\in S_1$ with $j > i$.
Inductive step: The inductive hypothesis is that $A_i(|E(G)|\xi)(w_j) = \deg_{G_i}w_j$ for $w_j\in S_0$ with $j > i$ and that $A_i(|E(G)|\xi)(w_j) = -\deg_{G_i}w_j$ for $w_j\in S_1$ with $j > i$. Given that this is true for $i-1$, we want to show that it is true for $i$.
We suppose that $w_i\in S_0$; the case where $w_i\in S_1$ will proceed analogously. After $i-1$ steps, $w_i$ has a mass of $\deg_{G_{i-1}}w_i$. During the $i$th step, this mass is distributed evenly among $w_j\sim w_i$ with $j > i$; we note that there are exactly $\deg_{G_{i-1}}w_i$ of these neighbors. Thus, each $w_j$ will receive $+1$ mass. By the inductive hypothesis we have that before step $i$, each of these neighbors $w_j$ had $-\deg_{G_{i-1}}w_j$ mass, since each of the neighbors of $w_i$ is in $S_1$, the opposite side of the bipartite graph. Then, after step $i$, each $w_j$ has mass $-(\deg_{G_{i-1}}w_j - 1)$, and the remaining vertices with indices greater than $i$ have the same mass as before. We note that for all $\ell > i$, if $w_{\ell} \sim w_i$ then $\deg_{G_i}w_\ell = \deg_{G_{i-1}}w_\ell - 1$ because the edge $\{(w_\ell, w_i\}$ is being removed, and otherwise $\deg_{G_i}w_\ell = \deg_{G_{i-1}}w_\ell$. We have just shown that this is exactly the mass at all vertices with indices greater than $i$ after the $i$th step of the algorithm. Thus, at each vertex $w_\ell$ with $\ell > i$, we have that for $w_\ell\in S_0$, the mass at $w_\ell$ after $i$ steps is $\deg_{G_i}w_\ell$ and for $w_\ell\in S_1$, the mass at $w_\ell$ after $i$ steps is $-\deg_{G_i}w_\ell$. We proceed analogously in the case where $w_i\in S_1$. This proves the inductive hypothesis, and therefore proves the lemma.
\end{proof}
We are now ready to show that $\xi^0$ and $\xi^1$ lie on the interior of the region of distributions that satisfy the inequalities.
\begin{corollary}\label{cor: xi interior}
For any Guvab $\mathcal{G}$ where $W = 1$ and $\beta < 1$, we have that $\xi^0$ and $\xi^1$ lie strictly on the interior of the region $R \subset \mathbb{R}^{|V(G)|}$ of distributions $\xi$ that satisfy the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi)$.
\end{corollary}
\begin{proof}
We prove this for $\xi^0$; by symmetry it will hold for $\xi^1$ as well since $\xi^1 = -\xi^0$. If the sides of $G$ are $S_0$ and $S_1$, with $u\in S_0$ and $v\in S_1$, then for $w\in S_0$, we have that $$\xi^0(w) = \lim_{k\to\infty} \mu_{2k}(w) - \lim_{k\to\infty} \nu_{2k}(w) = \frac{\deg(w)}{|E(G)|} - 0 = \frac{\deg(w)}{|E(G)|}$$ and for $w\in S_1$ we have that $$\xi^0(w) = \lim_{k\to\infty} \mu_{2k+1}(w) - \lim_{k\to\infty} \nu_{2k+1}(w) = 0 - \frac{\deg(w)}{|E(G)|} = - \frac{\deg(w)}{|E(G)|}.$$ Then for all $t, w \in G$ such that $t \sim w$, we have that $\xi(t)\xi(w) < 0$. We also have that by Lemma~\ref{lem: algorithm nice on stat dib}, $\xi^0(w_j)A_i(\xi^0)(w_j) \geq \frac{1}{|E(G)|^2} > 0$ for all $0 \leq i \leq |V(G)|- 2$ and $i < j \leq V(G)$. Thus, $\xi^0$ and $\xi^1$ lie strictly on the interior of the region $R \subset \mathbb{R}^{|V(G)|}$ of distributions $\xi$ that satisfy the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi)$.
\end{proof}
Using these results, we are now ready to prove the main claim that the Wasserstein distance is eventually constant when $W = 1$ and $\beta < 1$.
We first define a variable that corresponds to how long $\{W_k\}$ takes to reach constancy. Note that this variable can be infinity if $\{W_k\}$ is not eventually constant.
\begin{definition} \label{rho}
For any Guvab $\mathcal{G}$ where $W_k \to 1$, define $\rho(\mathcal{G})$ to be $$\inf \{N \in \mathbb{Z} : \{W(\mu_k,\nu_k)\}_{k \geq N} = (1,1,1,\ldots) \}.$$
\end{definition}
\begin{theorem}\label{thm:winnie the when it's constant for W = 1 fairy}
For any Guvab $\mathcal{G}$ with $W = 1$ and $\beta < 1$, we have $\rho(\mathcal{G}) < \infty$.
\end{theorem}
\begin{proof}
Pick an arbitrary spanning tree $T$ of $G$ and $r$-monotone ordering $\mathcal{O}$. By Corollary \ref{cor: xi interior}, $\xi^0$ and $\xi^1$ are on the interior of the region $R \subset \mathbb{R}^{|V(G)|}$ of distributions $\xi$ that satisfy the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi)$. We note that all the inequalities in $\mathcal{I}(G,T,\mathcal{O},\xi)$ can be written in the form $f(\xi) > 0$, where $f: \mathbb{R}^{|V(G)|} \to \mathbb{R}$ is a continuous function. Thus, by the definition of a continuous function, there exists some $\varepsilon > 0$ such that for all $\xi \in \mathbb{R}^{|V(G)|}$ that satisfy $|\xi(w) - \xi^0(w)| < \varepsilon$ for all $w \in G$ or satisfy $|\xi(w) - \xi^1(w)| < \varepsilon$ for all $w \in G$, we have that $\xi \in R$. We also know, by the formal definition of a limit, that there exists some $N$ such that for all $k \geq N$ and all $w \in G$, we have $|\xi_{2k}(w) - \xi^0(w)| < \varepsilon$ and $|\xi_{2k+1}(w) - \xi^1(w)| < \varepsilon$. Thus, for all $k \geq 2N$, we have $\xi_k \in R$. By Corollary~\ref{cor: when ineqs, dist is 1}, for all $k \geq 2N$, we have that $W_k = 1$. Hence $\rho(\mathcal{G}) \leq 2N < \infty$.
\end{proof}
We next hope to characterize how long it takes the Wasserstein distance of these Guvabs with $W=1$ and $\beta < 1$ to become constant. In particular, we prove upper and lower bounds for $\rho(\mathcal{G})$. We start with the upper bound. To prove this upper bound, we first prove a lemma quantifying exactly how close to $\xi^0$ or $\xi^1$ a distribution must be in order for the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi)$ to be satisfied.
\begin{lemma} \label{lem: explicit epsilon bounds}
Consider a Guvab with $W = 1$ and $\beta < 1$. Pick an arbitrary spanning tree $T$ and $r$-monotone ordering $\mathcal{O}$. Let $\varepsilon(G) = \frac{1}{|V||E|}$. If for a distribution $\xi$ it is true that for all vertices $w$ we have that $|\xi(w) - \xi^0(w)| < \varepsilon(G)$ or it is true that for all vertices $w$ we have that $|\xi(w) - \xi^1(w)| < \varepsilon(G)$, then $\xi$ satisfies the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi)$.
\end{lemma}
\begin{proof}
We prove this for $\xi^0$, and an analogous argument will hold for $\xi^1$.
We note that by Lemma \ref{lem: algorithm nice on stat dib} we have that if we start with $\xi^0$, then at any point in the tree-based transport algorithm through step $|V| - 2$, the absolute value of the mass at any vertex is at least $\frac{1}{|E|}$. Thus, if at any point $i$ in the algorithm through step $|V| - 2$ the mass at a vertex differs by at most $\frac{1}{|E|}$ from $A_i(\xi^0)$, then the tree-based transport inequalities $\mathcal{I}$ are satisfied because mass is never the wrong sign.
It thus suffices to show that for all $0 \leq i \leq |V| - 2$ and for all $w \in G$, we have $|A_i(\xi)(w) - A_i(\xi^0)(w)| \leq \frac{1}{|E|}$. To prove this, we note that $\xi(w) - \xi^0(w)$ is a zero-sum distribution, and by Lemma \ref{lem: linear} for all $i$ and for all $w$, we have $A_i(\xi)(w) = A_i(\xi^0)(w) + A_i(\xi - \xi^0)(w)$.
We consider the quantity $\sum_{w \in G} |A_i(\xi)(w) - A_i(\xi^0)(w)| = \sum_{w\in G} |A_i(\xi -\xi^0)(w)|$. This will be nonincreasing as $i$ gets larger, since at step $i$ of the algorithm the absolute value of the mass at $w_i$ decreases by exactly $|A_{i-1}(\xi-\xi^0)(w_i)|$ while the sum of absolute values at $w_i$'s neighbors cannot increase by more than $|A_{i-1}(\xi-\xi^0)(w_i)|$. The maximum value of this sum is $|V|\varepsilon(G)$ (since this is an upper bound for the value at the beginning). We know that $\max_{w \in G} |A_i(\xi)(w) - A_i(\xi^0)(w)| \leq \sum_{w \in G} |A_i(\xi)(w) - A_i(\xi^0)(w)|$ so $\max_{w \in G} |A_i(\xi)(w) - A_i(\xi^0)(w)| \leq |V|\varepsilon(G) = \frac{1}{|E|}$, which is exactly what we wanted to show, so we are done.
\end{proof}
With this lemma established, we can now prove our upper bound for $\rho(\mathcal{G})$.
\begin{lemma}
Let $\lambda_{\max}$ be $\displaystyle \max_{|\lambda| \in L: |\lambda| < 1} |\lambda|$ where L is the set of all eigenvalues of $X$ and $Y$. Then for a Guvab $\mathcal{G}$ where $W = 1$ and $\beta < 1$, we have $\displaystyle \rho(\mathcal{G}) \leq \frac{10\ln|V|}{1-\lambda_{\max}^2}$.
\end{lemma}
\begin{proof}
We use \cite[Prop. 3]{diaconis1991geometric}.
The Markov chains $X_{2k}$ and $Y_{2k}$ are both converging to their even stationary distributions $\lim_{k\to\infty} \mu_{2k}$ and $\lim_{k\to\infty} \nu_{2k}$. For convenience, denote $\lim_{k\to\infty} \mu_{2k}$ by $\gamma_u$ and $\lim_{k\to\infty} \nu_{2k}$ by $\gamma_v$. Once at all vertices, $\mu_{2k}$ and $\nu_{2k}$ are both less than or equal to $\frac{1}{2|V||E|}$ away from their respective stationary distributions, $\xi_{2k}$ will satisfy the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi_{2k})$ by Lemma \ref{lem: explicit epsilon bounds}. Since $X_{2k}$ and $Y_{2k}$ are both Markov chains with limiting distributions, we use notation analogous to that of \cite{Sinclair92improvedbounds} and say that $\Delta_{\textrm{even}\,u}(k) = \displaystyle \frac{1}{2}\sum_{w\in G}|\mu_{2k}(w) - \gamma_u(w)|$. Similarly, $\Delta_{\textrm{even}\,v}(k) = \displaystyle \frac{1}{2}\sum_{w\in G}|\nu_{2k}(w) - \gamma_v(w)|$. Then for $\varepsilon > 0$ and $x \in \{u,v\}$ we let $\tau_{\textrm{even} \, x}(\varepsilon)$ be the minimum nonnegative integer $k$ such that $\Delta_{\textrm{even}\,x}(k') \leq \varepsilon$ for all $k' \geq k$. Thus, by \cite[Prop. 3]{diaconis1991geometric}, the time $\rho_{\textrm{even}}$ it takes for $W_{2k}$ to eventually have distance $1$ satisfies $$\rho_{\textrm{even}}\leq\max_{x \in \{u,v\}} 2\tau_{\textrm{even} \,x}(\frac{1}{2|V||E|}) \leq \max_{x \in \{u,v\}} \frac{2}{1-\lambda_{\max}^2}(\ln{\frac{1}{\gamma_x(x)}} + \ln{\cfrac{1}{\frac{1}{2|V||E|}}}).$$
Then we just need to bound the right-hand side above. This gives
\begin{align*}
\displaystyle \rho_{\textrm{even}}&\leq\frac{2}{1-\lambda_{\max}^2}(\ln{|E|} + \ln{2|V||E|})
=\frac{2}{1-\lambda_{\max}^2}(\ln{2|V||E|^2})\\
&\leq \frac{2}{1-\lambda_{\max}^2}(\ln{|V|^3(|V|-1)^2})\\
&< \frac{2}{1-\lambda_{\max}^2}\cdot 5 \ln(|V|)
= \frac{10\ln|V|}{1-\lambda_{\max}^2}.
\end{align*}
By similar reasoning, the same bound works for $\rho_{\textrm{odd}}$,
the time it takes for $W_{2k+1}$ to eventually have distance $1$. Thus, $\displaystyle\frac{10\ln|V|}{1-\lambda_{\max}^2}$ is an upper bound for $\rho(\mathcal{G})$.
\end{proof}
We now establish a lower bound for $\rho(\mathcal{G})$.
\begin{lemma}
For a Guvab $\mathcal{G}$ where $W = 1$ and $\beta < 1$, we have $\displaystyle \rho(\mathcal{G}) \geq \frac{\emph{\textrm{d}(u,v)}}{2} - 1$.
\end{lemma}
\begin{proof}
We note that $\mu_k(t) = 0$ for $t \in V$ if $\textrm{d}(t,u) > k$. Similarly, $\nu_k(w) = 0$ for $w \in V$ if $\textrm{d}(w,v) > k$. Suppose $k < \frac{\textrm{d}(u,v)}{2} - 1$ and consider any pair of vertices $t,w$ such that $\mu_k(t) > 0$ and $\nu_k(w) > 0$. Then $\textrm{d}(t,u) \leq \frac{\textrm{d}(u,v)}{2} - 1$ and $\textrm{d}(w,v) \leq \frac{\textrm{d}(u,v)}{2} - 1$, so $\textrm{d}(t,w) \geq \textrm{d}(u,v) - (\textrm{d}(t,u) +\textrm{d}(w,v)) = 2$. Therefore all mass will have to move a distance of at least 2 to get from $\mu_k$ to $\nu_k$, so $W_k \geq 2 > 1$.
\end{proof}
\section{Convergence when $W = \frac{1}{2}$}
In this section, we consider Guvabs where $W = \frac{1}{2}$ and $\beta < 1$. Recall that these are exactly the Guvabs for which $G$ is bipartite and $0 = \alpha < \beta < 1$. As in the previous section, and for similar reasons, the Wasserstein distance will eventually be the sum of positive mass. In this case, however, the Wasserstein distance is not eventually constant but rather an exponential that we can express explicitly. To prove this, we proceed by a similar strategy as in the $W = 1$ case. In particular, we show that the tree-based transport inequalities will eventually be satisfied, and compute the Wasserstein distance when these inequalities are satisfied.
In the next three results, we show that the tree-based transport inequalities will eventually be satisfied, and provide an initial expression for what the Wasserstein distance will be when the tree-based transport inequalities are satisfied. Later, we will calculate exactly what this expression for the Wasserstein distance evaluates to.
We begin by showing in the next two results that, analogously to before, $\xi^0$ and $\xi^1$ lie on the interior of the region of distributions that satisfy the inequalities.
\begin{lemma} \label{lem: for 1/2 algorithm nice on stat dib}
Suppose we have a bipartite graph $G$ with sides $S_0$ and $S_1$ and a distribution $\xi$ such that for $w\in S_0$ $\xi(w) = \frac{\deg(w)}{2|E(G)|}$ and for $w\in S_1$ $\xi(w) = -\frac{\deg(w)}{2|E(G)|}$. Then pick an arbitrary spanning tree T and $r$-monotone ordering $\mathcal{O}$ on $V(G)$. Consider the tree-based transport plan $A(G,T,\mathcal{O},\xi)$. We have that after each step $i$ for $i \leq n-2$, for $w_j\in S_0$ with $j > i$, we have that $A_i(\xi)(w_j) \geq \frac{1}{2|E(G)|}$ and for $w_j\in S_1$ with $j > i$ we have that $A_i(\xi)(w_j) \leq -\frac{1}{2|E(G)|}$.
\end{lemma}
\begin{proof}
We note that this is nearly the same as Lemma~\ref{lem: algorithm nice on stat dib}, but differs by a constant factor of $\frac{1}{2}$. Given the distribution $\xi$, we know by Lemma~\ref{lem: algorithm nice on stat dib} that after all steps $i$ for $i \leq n-2$, for $w_j\in S_0$ with $j>i$, we have that $A_i(2\xi)(w_j) \geq \frac{1}{|E(G)|}$ and for $w_j\in S_1$ with $j>i$ we have that $A_i(2\xi)(w_j) \leq -\frac{1}{|E(G)|}$. We also know that $A_i(2\xi) = 2A_i(\xi)$ by Lemma \ref{lem: linear} so this means for $w_j\in S_0$ with $j>i$, we have that $2A_i(\xi)(w_j) \geq \frac{1}{|E(G)|}$ and for $w\in S_1$ with $j>i$, we have that $2A_i(\xi)(w_j) \leq -\frac{1}{|E(G)|}$. Dividing both sides by 2, we get that after all steps $i$ for $i \leq n-2$, for $w_j\in S_0$ with $j>i$, we have that $A_i(\xi)(w_j) \geq \frac{1}{2|E(G)|}$ and for $w_j\in S_1$ with $j > i$, we have that $A_i(\xi)(w_j) \leq -\frac{1}{2|E(G)|}$.
\end{proof}
\begin{corollary} \label{cor: stat dib on interior for w=1/2}
For any Guvab $\mathcal{G}$ where $W = \frac{1}{2}$ and $\beta < 1$, we have that $\xi^0$ and $\xi^1$ lie strictly on the interior of the region $R \subset \mathbb{R}^{|V(G)|}$ of distributions $\xi$ that satisfy the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi)$.
\end{corollary}
\begin{proof}
We note that when $W = \frac{1}{2}$ and $\beta < 1$, we have that $\beta > 0$ so for all $w \in G$, we have that $\lim_{k\to\infty} \nu_k(w) = \frac{\deg(w)}{2|E(G)|}$. We also know that if $G$ has sides $S_0$ and $S_1$ with $u\in S_0$, for all $w \in S_0$ we have that $\displaystyle \lim_{k\to\infty} \mu_{2k}(w) = \frac{\deg(w)}{|E(G)|}$ and for all $w \in S_1$ we have that $\displaystyle \lim_{k\to\infty} \mu_{2k}(w) = 0$. Similarly, for all $w \in S_1$ we have that $\displaystyle \lim_{k\to\infty} \mu_{2k+1}(w) = \frac{\deg(w)}{|E(G)|}$ and for all $w \in S_0$ we have that $\displaystyle \lim_{k\to\infty} \mu_{2k+1}(w) = 0$. Thus for all $w \in S_0$ we have that $\xi^0(w) = \frac{\deg(w)}{2|E(G)|}$ and for all $w \in S_1$ we have that $\xi^0(w) = \frac{-\deg(w)}{2|E(G)|}$. Also $\xi^1 = -\xi^0$.
We prove the claim for $\xi^0$ - by symmetry it will hold for $\xi^1$ as well since $\xi^1 = -\xi^0$. If the sides of $G$ are $S_0$ and $S_1$, then for $w\in S_0$ $\xi^0(w) = \frac{\deg(w)}{2|E(G)|}$ and for $w\in S_1$ $\xi^0(w) = -\frac{\deg(w)}{2|E(G)|}$. Then we have that for all $t, w \in G$ such that $t \sim w$, the product $\xi(t)\xi(w) < 0$. We also have that by Lemma~\ref{lem: for 1/2 algorithm nice on stat dib}, $\xi^0(w)A_i(\xi^0)(w) \geq \frac{1}{4|E(G)|^2} > 0$ holds for all $w \in G$ and for all $0 \leq i \leq |V(G)|- 2$.
\end{proof}
We now know that $\xi^0$ and $\xi^1$ are on the interior of the region satisfying the inequalities. We can hence proceed similarly to section 4 to show that $\xi_k$ will eventually satisfy the inequalities and thus $W_k$ will be the sum of positive mass.
\begin{corollary} \label{cor: for 1/2, dist is eventually sum of pos mass}
For any Guvab where $W = \frac{1}{2}$ and $\beta < 1$, there exists $N$ such that for all $k \geq N$,
$$W_k = \frac{1}{2}\sum_{w \in G} |\xi_k(w)|.$$
\end{corollary}
\begin{proof}
We know by Corollary \ref{cor: stat dib on interior for w=1/2} that $\xi^0$ and $\xi^1$ lie on the interior of $R$. Therefore, as in the proof of Theorem \ref{thm:winnie the when it's constant for W = 1 fairy}, by the formal definition of a limit there exists some $N$ such that for all $k \geq N$, we have that $\xi_k \in R$ and thus $\mathcal{I}(G,T,\mathcal{O},\xi_k)$ are satisfied. We note that Corollary \ref{cor: when ineqs, dist is sum of pos mass} holds for any Guvab, including the ones we are currently inspecting, so if the tree-based transport inequalities $\mathcal{I}(G,T,\mathcal{O},\xi_k)$ are satisfied, $W_k = \frac{1}{2}\sum_{w \in G} |\xi_k(w)|$. Thus for all $k \geq N$, we have that $W_k = \frac{1}{2}\sum_{w \in G} |\xi_k(w)|.$
\end{proof}
We now know that eventually, the Wasserstein distance will be the sum of positive mass, so it remains to calculate the sum of positive mass. To do this, we will first need to define an auxiliary Markov chain and prove some properties of this Markov chain.
\begin{definition}
Let $s(\alpha)$ be a two-state Markov chain with states $s_0$ and $s_1$, where we start at $s_0$, and at all times we have an $\alpha$ chance of staying at our current state and a $1-\alpha$ chance of switching to the other state. Then define $(\sigma_\alpha)_k$ to be the probability distribution after $k$ steps of this Markov chain.
\end{definition}
\begin{lemma} \label{lem: mark the markov chain fairy}
For the Markov chain defined above, $(\sigma_\alpha)_k(s_0) = 0.5 + 0.5(2\alpha-1)^k$ and $(\sigma_\alpha)_k(s_1) = 0.5-0.5(2\alpha-1)^k$.
\end{lemma}
\begin{proof}
We will proceed by induction on $k$, using the transition probabilities to go from $(\sigma_\alpha)_k$ to $(\sigma_\alpha)_{k+1}$.
Base case: When $k=0$, we know that, since the Markov chain starts at $s_0$, we have $(\sigma_\alpha)_0(s_0) = 1 = 0.5+0.5(2\alpha-1)^0$ and $(\sigma_\alpha)_0(s_1) = 0 = 0.5-0.5(2\alpha-1)^0$.
Inductive step: Suppose $(\sigma_\alpha)_k(s_0) = 0.5+0.5(2\alpha-1)^k$ and $(\sigma_\alpha)_k(s_1) = 0.5-0.5(2\alpha-1)^k$. We know that
\begin{align*}
(\sigma_\alpha)_{k+1}(s_0) &= \alpha(\sigma_\alpha)_k(s_0) + (1-\alpha)(\sigma_\alpha)_k(s_1)\\ &= \alpha(0.5+0.5(2\alpha-1)^k) + (1-\alpha)(0.5-0.5(2\alpha-1)^k)\\ &= 0.5 + (2\alpha-1)\cdot 0.5(2\alpha-1)^k\\ &= 0.5 + 0.5(2\alpha-1)^{k+1}.
\end{align*}
Similarly,
\begin{align*}
(\sigma_\alpha)_{k+1}(s_1) &= (1-\alpha)(\sigma_\alpha)_k(s_0) + \alpha(\sigma_\alpha)_k(s_1)\\ &= (1-\alpha)(0.5+0.5(2\alpha-1)^k) + \alpha(0.5-0.5(2\alpha-1)^k)\\ &= 0.5+(2\alpha-1)\cdot (-0.5)(2\alpha-1)^k\\ &= 0.5 - 0.5(2\alpha-1)^{k+1}.
\end{align*}
\end{proof}
We now have all the tools we need to explicitly calculate the sum of positive mass. The next lemma tells us what the sum of positive mass will be.
\begin{lemma} \label{lem: sum of pos mass for 1/2}
When $\beta < 1$ and $W = \frac{1}{2}$, there exists some $N$ such that for all $k \geq N$, we either have that $$\frac{1}{2}\sum_{w \in G} |\xi_k(w)| = 0.5 + 0.5(1-2\beta)^k$$ or that $$\frac{1}{2}\sum_{w \in G} |\xi_k(w)| = 0.5 - 0.5(1-2\beta)^k.$$
\end{lemma}
\begin{proof}
We know $G$ is bipartite; say it has sides $S_0$ and $S_1$. We know $0 = \alpha < \beta < 1$. Assume without loss of generality that $v \in S_0$. If $u$ is on side $S_0$, then eventually for $w\in S_0$, we have that $\xi_{2k}(w)$ gets arbitrarily close to $\displaystyle\frac{\deg(w)}{2|E(G)|}$ and for $w\in S_1$, we have that $\xi_{2k}(w)$ gets arbitrarily close to $\displaystyle-\frac{\deg(w)}{2|E(G)|}$. In particular, for some $N$, for all $k \geq N$ we have $\xi_{2k}(w) > 0$ if and only if $w\in S_0$. Then for some $N$, for all $k \geq N$, when $u\in S_0$, the total positive mass of $\xi_{2k}$ is $\displaystyle\sum_{w \in S_0} \xi_{2k}(w)$. Similarly, for some $N$, for all $k \geq N$ we have $\xi_{2k+1}(w) > 0$ if and only if $w\in S_1$. Thus, for some $N$, for all $k \geq N$, when $u\in S_0$, the total positive mass of $\xi_{2k+1}$ is $\displaystyle\sum_{w \in S_1} \xi_{2k+1}(w)$.
By an analogous argument, when $u\in S_1$, for some $N$, for all $k \geq N$, the total positive mass of $\xi_{2k}$ is $\displaystyle\sum_{w \in S_0} \xi_{2k}(w)$. Similarly, for some $N$, for all $k \geq N$, the total positive mass of $\xi_{2k+1}$ is $\displaystyle\sum_{w \in S_1} \xi_{2k+1}(w)$.
Thus, to calculate what the sum of positive mass eventually equals, we simply consider how much mass of $\mu$ and $\nu$ is on each side of the bipartite graph so that we know how much mass of $\xi$ is on each side. We note that for any random walk with laziness $\beta$, at all steps the mass on a given side has a probability $\beta$ of staying on that side and a probability $1-\beta$ of moving to the other side, since any mass that moves along an edge moves to the other side. Thus the mass of $\nu_k$ on $S_0$ and $S_1$ behaves identically to the mass of $s(\beta)$ on $s_0$ and $s_1$. In other words, the amount of mass of $\nu_k$ on $S_0$ is $(\sigma_\beta)_k(s_0) = 0.5 + 0.5(2\beta-1)^k$ and the amount of mass of $\nu_k$ on $S_1$ is $(\sigma_\beta)_k(s_1) = 0.5 - 0.5(2\beta-1)^k$ by Lemma \ref{lem: mark the markov chain fairy}. Similarly, if $u\in S_0$, then the amount of mass of $\mu_k$ on $S_0$ is $(\sigma_0)_k(s_0) = 0.5 + 0.5(-1)^k$ and the amount of mass of $\mu_k$ on $S_1$ is $(\sigma_0)_k(s_1) = 0.5 - 0.5(-1)^k$. By symmetry, if $u\in S_1$, then the amount of mass of $\mu_k$ on $S_0$ is $(\sigma_0)_k(s_1) = 0.5 - 0.5(-1)^k$ and the amount of mass of $\mu_k$ on $S_1$ is $(\sigma_0)_k(s_0) = 0.5 + 0.5(-1)^k$.
This means that if $u\in S_0$,
\begin{align*}
\sum_{w \in S_0} \xi_{k}(w) &= \sum_{w \in S_0} \mu_{k}(w) - \sum_{w \in S_0} \nu_{k}(w)\\ &= (\sigma_0)_k(s_0) - (\sigma_\beta)_k(s_0)\\ &= 0.5 + 0.5(-1)^k - (0.5 + 0.5(2\beta-1)^k)
\end{align*}
and
\begin{align*}
\sum_{w \in S_1} \xi_{k}(w) &= \sum_{w \in S_1} \mu_{k}(w) - \sum_{w \in S_1} \nu_{k}(w)\\ &= (\sigma_0)_k(s_1) - (\sigma_\beta)_k(s_1)\\ &= 0.5 - 0.5(-1)^k - (0.5 - 0.5(2\beta-1)^k).
\end{align*}
Then the total positive mass of $\xi_{2k}$ is $$\sum_{w \in S_0} \xi_{2k}(w) = 0.5 + 0.5(-1)^{2k} - (0.5 + 0.5(2\beta-1)^{2k}) = 0.5 - 0.5(1-2\beta)^{2k}$$ and the total positive mass of $\xi_{2k+1}$ is $$\sum_{w \in S_1} \xi_{2k+1}(w) = 0.5 - 0.5(-1)^{2k+1} - (0.5 - 0.5(2\beta-1)^{2k+1}) = 0.5 - 0.5(1-2\beta)^{2k+1}.$$ Thus, for some $N$, the sum of the positive mass of $\xi_k$ is $0.5 - 0.5(1-2\beta)^k$ for all $k \geq N$.
If $u\in S_1$, then we have that $\displaystyle\sum_{w \in S_0} \xi_{k}(w)= (\sigma_0)_k(s_1) - (\sigma_\beta)_k(s_0)$ and we have that $\displaystyle\sum_{w \in S_1} \xi_{k}(w) = (\sigma_0)_k(s_0) - (\sigma_\beta)_k(s_1)$. By calculating this out analogously to above, we see that if $u\in S_1$ there exists some $N$ such that the sum of positive mass of $\xi_k$ is $0.5 + 0.5(1-2\beta)^k$ for all $k \geq N$.
\end{proof}
We now know that the Wasserstein distance will be the sum of positive mass, and we know exactly what the sum of positive mass will eventually be. Thus, we know exactly what the Wasserstein distance will eventually be. The next theorem therefore states explicitly the rate of convergence of the Wasserstein distance when $\beta < 1$ and $W = \frac{1}{2}$.
\begin{theorem} \label{thm: washington the w=1/2 convergence theorem}
For any Guvab where $W = \frac{1}{2}$ and $\beta < 1$, for some $N$ it will be true that for all $k \geq N$, we have that $|W_k - \frac{1}{2}| = 0.5|1-2\beta|^k$.
\end{theorem}
\begin{proof}
Corollary \ref{cor: for 1/2, dist is eventually sum of pos mass} tells us that for some $N_1$, we will have $W_k = \frac{1}{2}\sum_{w\in G} |\xi_k(w)|$ for all $k \geq N_1$. Lemma \ref{lem: sum of pos mass for 1/2} tells us that for some $N_2$, we will have $\frac{1}{2}\sum_{w\in G} |\xi_k(w)| = 0.5 + 0.5(1-2\beta)^k$ for all $k \geq N_2$ or we will have $\frac{1}{2}\sum_{w\in G} |\xi_k(w)| = 0.5 - 0.5(1-2\beta)^k$ for all $k \geq N_2$. This means that for all $k \geq N_2$, we have $|(\frac{1}{2}\sum_{w\in G} |\xi_k(w)|) - \frac{1}{2}| = 0.5|1-2\beta|^k$. Thus, for all $k \geq \max(N_1,N_2)$, we have $$\left|W_k - \frac{1}{2}\right| = \left|\left(\frac{1}{2}\sum_{w\in G} |\xi_k(w)|\right) - \frac{1}{2}\right| = 0.5|1-2\beta|^k.$$
\end{proof}
Finally, we want to characterize when the Wasserstein distance is eventually constant when $W = \frac{1}{2}$ and $\beta < 1$. This will fit into our larger characterization of eventual constancy for all Guvabs with $\beta < 1$.
\begin{corollary}\label{cor: wendy the when it's constant for W = 1/2 fairy}
When $W = \frac{1}{2}$ and $\beta < 1$, we have that $\rho < \infty$ if and only if $\beta = \frac{1}{2}$.
\end{corollary}
\begin{proof}
This follows directly from Theorem \ref{thm: washington the w=1/2 convergence theorem}.
\end{proof}
\section{Convergence when $W = 0$}
In this section we consider the case of Guvabs where $W = 0$ and $\beta < 1$. Recall that these are exactly the Guvabs enumerated in Theorem \ref{thm: 0-convergence} for which $\beta < 1$. We start by showing that the rate of convergence of $\{W_{2k}\}$ is exponential when it is not eventually constant. By an analogous argument, the rate of convergence of $\{W_{2k+1}\}$ is exponential when it is not eventually constant. We will then investigate exactly when $\{W_k\}$ is eventually constant.
Theorem \ref{thm: Simba the sim-an-exponential when W=0 fairy} states that unless it is eventually constant, the rate of convergence of $\{W_{2k}\}$ is exponential, and in particular $W_{2k} \sim c\cdot \lambda_{\textrm{even}}^{2k}$. We go about proving this by showing in the next two lemmas that $W_{2k}$ must be one of finitely many expressions, all of which are approximately some exponential.
The next lemma shows that $W_{2k}$ must be one of finitely many expressions.
\begin{lemma}\label{lem: finite function set}
For any Guvab $\mathcal{G}$, there exists a finite set $F = \{f_i,f_2,\ldots, f_m\}$ of 1-Lipschitz functions $f_i: V(G) \to \mathbb{R}$ such that for all $k$ there exists $f \in F$ such that $\displaystyle W_k = \sum_{w\in G} f(w)\xi_k(w).$
\end{lemma}
\begin{proof}
We consider the set $L$ of possible 1-Lipschitz functions $\ell$ on the graph $G$ such that $\sum_{w \in V(G)} \ell(w) = 0$ (any other 1-Lipschitz function can be transformed into such a 1-Lipschitz function by adding some value to all entries). The criteria for a function $\ell$ to be a 1-Lipschitz function are that for each pair of vertices $w_1$ and $w_2$ we have that $\ell(w_1) - \ell(w_2) \leq d(w_1,w_2)$ and $\ell(w_2) - \ell(w_1) \leq d(w_1,w_2)$. We also have that $\sum_{w\in G} \ell(w) = 0$. These each form hyperplanes in $\mathbb R^{|V(G)|}$. Additionally, from these criteria we know that none of the entries of $\ell$ can be more than $|V(G)|$ because then since the max distance between any two vertices is $|V(G)|$ there would be no negative entries. Thus, the set of 1-Lipschitz functions forms a closed set bounded by a polytope in $\mathbb R^{|V(G)|}$.
For any cost function $C$ on the graph $G$, we have that $\sum_{w \in V(G)} C(w)\ell(w)$ is a linear function on $L$. Thus $\displaystyle \textrm{argmax}_{\ell \in L} \sum_{w \in V(G)} C(w)\ell(w)$ is one of the corners of the polytope. There are finitely many of these corners, corresponding to finitely many 1-Lipschitz functions $\{f_1,f_2,\ldots, f_m\}$. We also know that $W_k = \max_{\ell \in L} \sum_{w \in V(G)} \xi_k(w)\ell(w)$, so it is thus maximizing the cost function $\xi$, and thus for all $k$ there exists $f \in F$ such that $\displaystyle W_k = \sum_{w\in G} f(w)\xi_k(w)$.
\end{proof}
We now know that $W_{2k}$ will be one of finitely many expressions. The next lemma shows that each of these expressions is approximately exponential.
\begin{lemma} \label{lem: 1-lip are exponential}
For any Guvab $\mathcal{G}$ for which $W=0$ and any 1-Lipschitz function $f$, there exists some $0 < \lambda_f < 1$ and some constant $c_f$ such that $$\sum_{w\in G} f(w)\xi_{2k}(w) \sim c_f \cdot \lambda_f^{2k}$$ unless there exists some $N$ such that for all $k > N$ we have that $\sum_{w\in G} f(w)\xi_{2k}(w) = 0$.
\end{lemma}
\begin{proof}
Assume that there does not exist any $N$ such that for all $k > N$ we have that $\sum_{w\in G} f(w)\xi_{2k}(w) = 0$.
We know by Lemma \ref{lem: eileen the eigval sum fairy} that for all vertices $w$, there exist some constants $c^w_i$ such that for all $k \geq 1$, $$\xi_{2k}(w) = \sum_{i = 1}^m c^w_i \lambda_i^{2k} = \sum_{i = 1}^m c^w_i (\lambda_i^2)^k = \sum_{i = 1}^n c^w_i (\lambda_i^2)^k$$ where in the last sum the $\lambda_i^2$ are all distinct positive constants (by combining like terms in the sum with $m$ terms to get a sum with $n$ terms). Then $$\sum_{w\in G} f(w)\xi_{2k}(w) = \sum_{w\in G} f(w)\sum_{i = 1}^n c^w_i (\lambda_i^2)^k.$$ Thus, there exist constants $c_f^1, \ldots c_f^n$ such that $\sum_{w\in G} f(w)\xi_{2k}(w) = \sum_{i=1}^n c_f^i (\lambda_i^2)^k.$ Let $\lambda_f^2 = \max_{i, c_f^i \neq 0} \lambda_i^2$ (this is well-defined since if it wasn't well defined we would have $\sum_{w\in G} f(w)\xi_{2k}(w) = 0$ for all $k \geq 1$). Let $c_f$ be the constant corresponding to this $\lambda_f^2$. Then $$\frac{\sum_{w\in G} f(w)\xi_{2k}(w)}{c_f\cdot \lambda_f^{2k}} = \frac{\sum_{i=1}^n c_f^i (\lambda_i^2)^k}{c_f\cdot \lambda_f^{2k}} = 1 + O(c^{2k}),$$ where $0 < c < 1$. Thus we have that $$\sum_{w\in G} f(w)\xi_{2k}(w) \sim c_f \cdot \lambda_f^{2k}.$$
\end{proof}
We now have all the pieces we need to show that $W_{2k}$ is approximately some exponential. The following theorem finishes off the proof.
\begin{theorem} \label{thm: Simba the sim-an-exponential when W=0 fairy}
For any Guvab $\mathcal{G}$ for which $W=0$ and $\{W_{2k}\}$ is not eventually constant, we have that there exists some $0 < \lambda_{\emph{\textrm{even}}} < 1$ and some $c > 0$, such that $W_{2k} \sim c\cdot \lambda_{\emph{\textrm{even}}}^{2k}$.
\end{theorem}
\begin{proof}
By Lemma \ref{lem: finite function set}, there exists some set $F = \{f_1,\ldots f_n\}$ of 1-Lipschitz functions $f_i: V(G) \to \mathbb{R}$ such that for all $k$ there exists $f \in F$ such that $\displaystyle W_{2k} = \sum_{w\in G} f(w)\xi_{2k}(w)$. Furthermore, by Lemma \ref{lem: 1-lip are exponential}, for each of these $f \in F$ there exists some $\lambda_f$ and some positive constant $c_f$ such that $\sum_{w\in G} f(w)\xi_{2k}(w) \sim c_f \cdot \lambda_f^{2k}$, unless there exists some $N$ such that for all $k > N$ we have that $\sum_{w\in G} f(w)\xi_{2k}(w) = 0$. If for all $f \in F$, there exists some $N$ such that for all $k > N$ we have $\sum_{w\in G} f(w)\xi_{2k}(w) = 0$, then we have that $\{W_{2k}\}$ is eventually constant at 0. Otherwise, let $\Tilde{F}$ be the set of functions $f$ for which $\lambda_f$ is well-defined. Then let $\lambda_{\textrm{even}}$ be $\max_{f \in \Tilde{F}} \lambda_f$. Let $F' \subset \Tilde{F}$ be the set of $f$ such that $\lambda_f = \lambda_{\textrm{even}}$, and let $c$ be $\max_{f \in F'} c_f$. Finally, let $\mathcal{F} \subset \Tilde{F}$ be the set of $f \in \Tilde{F}$ such that $\lambda_f = \lambda_{\textrm{even}}$ and $c_f = c$. Then for all $f \in F$ such that $f \notin \mathcal{F}$, there exists some $N$ such that for all $k \geq N$ we have $$\sum_{w\in G} f(w)\xi_{2k}(w) < \max_{f \in \mathcal{F}} \sum_{w\in G} f(w)\xi_{2k}(w) \leq W_{2k}.$$ Thus, since $W_{2k}$ must be the output of some 1-Lipschitz function, there exists some $N$ such that for all $k \geq N$ we have $W_{2k} = \sum_{w\in G} f(w)\xi_{2k}(w)$ for some $f \in \mathcal{F}$, since it cannot be the output of any 1-Lipschitz function $f \notin \mathcal{F}$. However, for all $f \in \mathcal{F}$, we have that $\sum_{w\in G} f(w)\xi_{2k}(w) \sim c\cdot \lambda_{\textrm{even}}^{2k}$. Thus for all $k \geq N$, we have that $W_{2k} \sim c \cdot \lambda_{\textrm{even}}^{2k}$. Hence $W_{2k} \sim c \cdot \lambda_{\textrm{even}}^{2k}$.
\end{proof}
\begin{remark} \label{rem: odd simba}
Analogously, for any Guvab $\mathcal{G}$ for which $W=0$ and $\{W_{2k+1}\}$ is not eventually constant, we have that there exists some $0 < \lambda_{\emph{\textrm{odd}}} < 1$ and some $c > 0$, such that $W_{2k+1} \sim c\cdot \lambda_{\emph{\textrm{odd}}}^{2k+1}$.
\end{remark}
We now seek to explicitly characterize all the cases where $W = 0$ and $W_k$ is eventually constant. We start by understanding why we only need to consider the first few terms of $\{W_k\}$ to characterize all of these cases.
\begin{lemma} \label{lem: Mustard the Must-be-constant-after-1-step fairy} When $\displaystyle\lim_{k\to\infty}W_k = 0$, if there exists some $N \geq 0$ such that $\{W(\mu_k,\nu_k)\}_{k\geq N}$ is a constant sequence, then $\displaystyle\{W(\mu_k,\nu_k)\}_{k\geq 1}$ is also a constant sequence.\end{lemma}
\begin{proof} By Lemma \ref{lem: eileen the eigval sum fairy}, if we let the distinct eigenvalues of the transition matrices be $\lambda_1,\ldots, \lambda_n$, then for any vertex $w$ and for any $k \geq 1$ we can write $(\mu_k - \nu_k)_w = \sum_{i=1}^n c^w_i\lambda_i^k$ for some constants $c^w_1,\ldots, c^w_n$. Note that if $\{W(\mu_k,\nu_k)\}_{k\geq N}$ is a constant sequence, $0 = \lim_{k\to\infty}W_k = W(\mu_k,\nu_k)$ for all $k \geq N$. Thus for any vertex $w$, we will have $\sum_{i=1}^n c^w_i\lambda_i^k = 0$ for all $k \geq N$.
Suppose that for some $i$, we have that $c^w_i$ and $\lambda_i$ are nonzero. Then let $\Lambda$ be the set of all $\lambda_i$ for which $c^w_i$ and $\lambda_i$ are nonzero. Then let $\lambda_m = \max_{\lambda \in \Lambda} |\lambda|$. If there is only one $\lambda_i \in \Lambda$ such that $|\lambda_i| = \lambda_m$, then for some $N$, for all $k > N$ we will have that $|c^w_i\lambda_i^k| > \sum_{j\neq i} |c^w_j\lambda_j^k|$ so the left-hand-side term will dominate and $\sum_{i=1}^n c^w_i\lambda_i^k$ will be nonzero. Then $0 \neq \lim_{k\to\infty}W_k$. If there is more than one $\lambda \in \Lambda$ such that $|\lambda| = \lambda_m$, then those two $\lambda$s will be $\lambda_m$ and $\lambda_{m'} = -\lambda_m$, since those are the only two numbers with absolute value $\lambda_m$. We know that $c^w_m\lambda_m^k$ will stay the same sign regardless of $k$, while $c^w_{m'}\lambda_{m'}^k$ will switch sign with parity. Thus, for one of the parities, $c^w_m\lambda_m^k$ and $c^w_{m'}\lambda_{m'}^k$ will have the same sign. Thus, for some $N$, either for all even $k > N$ or for all odd $k > N$, we will have that $|c^w_m\lambda_m^k + c^w_{m'}\lambda_{m'}^k| > \sum_{j\neq m,m'} |c^w_j\lambda_j^k|$, so the left-hand-side term will dominate and $\sum_{i=1}^n c^w_i\lambda_i^k$ will be nonzero. Then $0 \neq \lim_{k\to\infty}W_k$. Thus, we must have for all $1 \leq i \leq n$ that either $c^w_i$ or $\lambda_i$ is 0.
This means that for all $1 \leq i \leq n$, either $c_i^w$ or $\lambda_i$ is 0. Thus, for all $k \geq 1$, we have that $c_i^w\lambda_i^k = 0$. Therefore $(\mu_k - \nu_k)_w = \sum_{i=1}^n c^w_i\lambda_i^k = 0$, so $\mu_k-\nu_k$ will be 0 at all vertices, so $W(\mu_k, \nu_k) = 0$ for all $k \geq 1$.
\end{proof}
With this lemma established, we proceed to characterize all the cases when the Wasserstein distance is eventually constant in the case where $W = 0$.
\begin{theorem} \label{thm: Constance the constant distance when W = 0 fairy}
When $\lim_{k\to\infty}W_k = 0$, we have that $W_k$ is eventually constant if and only if one of the following holds:
\begin{itemize}
\item $\alpha = \beta = 0$ and $N(u)=N(v)$,
\item $\displaystyle \alpha = \beta = \frac{1}{\deg u + 1}$, the edge $\{u,v\}\in E(G)$, and if the edge $\{u,v\}$ were removed from $E(G)$ then $u,v$ would have $N(u)=N(v)$,
\item $\alpha = \beta$ and $u = v$.
\end{itemize}
\end{theorem}
\begin{proof}
We know by Lemma \ref{lem: Mustard the Must-be-constant-after-1-step fairy} that if $\lim_{k\to\infty}W_k = 0$ and $W_k$ is eventually always 0, then $\mu_1 = \nu_1$ and $\mu_2 = \nu_2$. Let $\mu_1 = \nu_1$ be $\phi$. Recall that $P_\alpha$ is the transition matrix for $X$ and $P_\beta$ is the transition matrix for $Y$. Further recall that $P_\alpha = \alpha I + (1-\alpha) P$ and $P_\beta = \beta I + (1-\beta) P$. Then we have $\phi P_\alpha = \phi P_\beta$, so $\phi(\alpha I + (1-\alpha) P) = \phi(\beta I + (1-\beta) P)$. Then $\phi((\beta - \alpha)I + (\alpha - \beta)P) = 0$.
If $\alpha \neq \beta$ then dividing out by $(\alpha-\beta)$, we get $\phi P = \phi$. If such a $\phi$ exists, it must be the stationary distribution $\pi$. Then $\phi = \mu_0 P = \mathbbm{1}_u P = \pi$. However, $(\mathbbm{1}_u P)_u = P_{u,u} = 0$ by definition of $P$, and $\pi_u = \frac{\deg(u)}{2|E(G)|} > 0$. Thus, we cannot have $\mathbbm{1}_u P = \pi$, so we cannot have $\alpha \neq \beta$.
This means we have that $\alpha = \beta$.
We also know that, given that $\lim_{k\to\infty}W_k = 0$, if $\mu_1 = \nu_1$ and $\alpha = \beta$, then $\mu_k = \mu_1(P_\alpha)^{k-1} = \nu_1(P_\alpha)^{k-1} = \nu_k$ for all $k \geq 1$ so $W_k$ is eventually always 0.
It therefore suffices to characterize the cases where $\lim_{k\to\infty}W_k = 0$ and $\mu_1 = \nu_1$ and $\alpha = \beta$. We first note that if $u = v$ and $\alpha = \beta$, we are done. Otherwise, we assume that $\alpha = \beta$ and casework on the values of $\alpha$ to determine which cases yield $\lim_{k\to\infty}W_k = 0$ and $\mu_1 = \nu_1$.
If $\alpha = 0$, we need that $u$ and $v$ have the same neighbor set, since if $u$ had some neighbor $n$ that was not adjacent to $v$ then $\mu_1$ would have nonzero mass at $n$ and $\nu_1$ would not. We will also show that this is a sufficient condition. If $u$ and $v$ have the same neighbor set then $\deg(u) = \deg(v)$. For each neighbor $n$ of $u$ and $v$, we have that $\mu_1(n) = \frac{1-\alpha}{\deg(u)} = \frac{1-\beta}{\deg(v)} = \nu_1(n)$ and for all other vertices $w$, we have that $\mu_1(w)= \nu_1(w) = 0$. Thus $\mu_1 = \nu_1$. We also know that $\lim_{k\to\infty}W_k = 0$ by Theorem \ref{thm: 0-convergence} since $\alpha = \beta = 0$ and for any neighbor $n$ of $u$, the path $u \to n \to v$ has an even number of steps.
If $0 < \alpha < 1$, we first note that we need $u$ and $v$ to be adjacent, since $\mu_1(u) = \alpha > 0$ and $\nu_1(u) = 0$ if $u$ and $v$ are not adjacent. When $u$ and $v$ are adjacent we have that $\nu_1(u) = \frac{1-\alpha}{\deg(u)}$, so since $\nu_1(u) = \mu_1(u)$ we have $\frac{1-\alpha}{\deg(u)} = \alpha$, which yields $\alpha = \frac{1}{\deg u + 1}$. We also note that, similarly to before, aside from the edge $\{u,v\}$ we have that $u$ and $v$ need to have the same set of neighbors because if there was some vertex $n\neq u, v$ such that $n\sim u$ and $n\not \sim v$, then $\mu_1$ would have nonzero mass at $n$ and $\nu_1$ would not. We will finish by showing that if $\alpha,\beta,u,v$ satisfy these conditions, then $\mu_1 = \nu_1$ and $\lim_{k\to\infty}W_k = 0$.
Suppose that the conditions are satisfied. We know that $\deg(u) = \deg(v)$, so $\mu_1(u) = \alpha = \frac{1-\alpha}{\deg(u)} = \nu_1(u)$ and similarly $\mu_1(v) = \nu_1(v)$. We also know that for all $n\neq u,v$ such that $n\sim u$ and $n\sim v$, we have that $\mu_1(n) = \frac{1-\alpha}{\deg(u)} = \frac{1-\beta}{\deg(v)} = \nu_1(n)$ and for all other vertices $w$, we have that $\mu_1(w)= \nu_1(w) = 0$. Thus $\mu_1 = \nu_1$. Also, $\lim_{k\to\infty}W_k = 0$ by Theorem \ref{thm: 0-convergence} since $0 < \alpha \leq \beta < 1$.
\end{proof}
\section{Convergence when $\beta = 1$}
We next consider the case of Guvabs where $\beta = 1$. Similarly to the $W = 0$ case, we show that the rate of convergence is exponential unless the distance is eventually constant. Furthermore, when the Wasserstein distance is eventually constant, it is constant after exactly 1 step.
We first show that the rate of convergence is exponential unless the distance is eventually constant.
\begin{lemma} \label{lem: betty the beta=1 theorem}
Consider a Guvab where $\beta = 1$. Either $\{W_{2k}\}$ is eventually constant, or for some $c_e$ and some $\lambda_e$, we have that $|W_{2k} - \lim_{k\to\infty} W_{2k}| \sim c_e\cdot \lambda_e^{2k}$. Also, either $\{W_{2k+1}\}$ is eventually constant, or for some $c_o$ and some $\lambda_o$, we have that $|W_{2k+1} - \lim_{k\to\infty} W_{2k+1}| \sim c_o\cdot \lambda_o^{2k+1}$.
\end{lemma}
\begin{proof}
When $\beta = 1$, we know that $W_k = \sum_{w \in G} \mu_k(w)\textrm{d}(w,v) = \sum c_i \cdot \lambda_i^k$ for some constants $c_i$ and $\lambda_i$. Using the same reasoning as in the proof of Lemma \ref{lem: 1-lip are exponential}, we know that (unless $\sum c_i \cdot \lambda_i^{2k}$ is eventually constant) $\sum c_i \cdot \lambda_i^{2k} \sim c_e\cdot \lambda_e^{2k}$ for some $c_e, \lambda_e$. We also know that (unless $\sum c_i \cdot \lambda_i^{2k+1}$ is eventually constant) $\sum c_i \cdot \lambda_i^{2k+1} \sim c_o\cdot \lambda_o^{2k+1}$ for some $c_o, \lambda_o$. Thus, we attain the desired result.
\end{proof}
We now show that if the distance is eventually constant, it is constant after 1 step.
\begin{lemma} \label{lem: when beta is 1 we must be constant after 1 step}
When $\beta = 1$, if there exists some $N \geq 0$ such that $\{W(\mu_n,\nu_n)\}_{n\geq N}$ is a constant sequence, then $\{W(\mu_n,\nu_n)\}_{n\geq 1}$ is also a constant sequence.\end{lemma}
\begin{proof}
When $\beta = 1$, we have that $W_k$ is $\sum_{w \in G} \mu_k(w)\textrm{d}(w,v) = \sum c_i \cdot \lambda_i^k$ for some constants $c_i$ and $\lambda_i$. Thus, for similar reasons as in the proof of Lemma \ref{lem: Mustard the Must-be-constant-after-1-step fairy}, all the $c_i$ for $\lambda_i \neq 0,1$ are 0 so $\{W(\mu_n,\nu_n)\}_{n\geq 1}$ is constant.
\end{proof}
When $W = 0$, using a lemma similar to Lemma \ref{lem: when beta is 1 we must be constant after 1 step} we were able to explicitly characterize exactly when $W_k$ was eventually constant. Lemma \ref{lem: when beta is 1 we must be constant after 1 step} provides an important step towards making a similar characterization when $\beta=1$. To exemplify how a characterization could be made when $\beta=1$, we provide a family of examples of Guvabs where $W_k$ is eventually constant.
\begin{definition}
We define a \textbf{Gluvab} $\mathcal{J}$ to be a Guvab that satisfies all of the following conditions:
\begin{itemize}
\item $\beta = 1$,
\item $2\textrm{d}(u,v) = \max_{w\in G} \textrm{d}(w,v)$,
\item if $\textrm{d}(x,v) = \max_{w\in G} \textrm{d}(w,v)$, then for all $n \sim x$ we have that $\textrm{d}(n,v) < \textrm{d}(x,v)$,
\item if $0 < \textrm{d}(x,v) < \max_{w\in G} \textrm{d}(w,v)$, then for exactly half of the neighbors $n\sim x$ we have that $\textrm{d}(n,v) < \textrm{d}(x,v)$, and for exactly the other half we have that $\textrm{d}(n,v) > \textrm{d}(x,v)$.
\end{itemize}
\end{definition}
\begin{example}
Consider a Guvab with $G=P_3$ (where $P_3$ is the path graph with $3$ vertices), $v$ is the vertex of $P_3$ with degree 2, $u$ is either of the other two vertices, $\alpha=\frac{1}{3}$, and $\beta=1$. One can check that this Guvab is a Gluvab.
\end{example}
\begin{lemma}\label{lem: Garry the Gluvab Fairy}
Any Gluvab $\mathcal{J}$ satisfies $W_0 = W_1 = \cdots$.
\end{lemma}
\begin{proof}
We aim to prove this lemma by essentially reducing each Gluvab to a random walk on a path graph. In particular, each vertex $m_i$ in the path corresponds to the set of vertices $\{w\in G : d(w,v)= i\}$ at a given distance $i$ from $v$. After this, the desired result follows without much difficulty.
Construct the Markov chain $M$ that is simply a random walk with laziness $\alpha$ on a path of length $2\textrm{d}(u,v)$ with vertices $m_0, m_1, \ldots, m_{\textrm{d}(u,v)}, \ldots, m_{2\textrm{d}(u,v)}$. We let the starting point of this Markov chain be $m_{\textrm{d}(u,v)}$. It suffices to show that for all $i$, we have that $\displaystyle \sum_{w \in G, \, \textrm{d}(w,v) = i} \mu_k(w) = M_k(m_i)$, because that would mean that the distribution is always symmetric about $u$ so $\mu_k$ always has the same average distance $\textrm{d}(u,v)$.
We will show by induction on $k$ that for all $i$, $$ \sum_{w \in G, \, \textrm{d}(w,v) = i} \mu_k(w) = M_k(m_i).$$
Base case: At $k = 0$, we have that $\mu_k$ is only nonzero at $u$ and that $M_k$ is only nonzero at $m_{\textrm{d}(u,v)}$, so the claim holds.
Inductive step: We suppose that this claim holds for $k$. We will show that it holds for $k+1$. We know the following facts about $M$:
\begin{itemize}
\item $M_{k+1}(m_0) = \frac{1-\alpha}{2}M_k(m_1) + \alpha M_k(m_0)$,
\item $M_{k+1}(m_{2\textrm{d}(u,v)}) = \frac{1-\alpha}{2}M_k(m_{2\textrm{d}(u,v) - 1}) + \alpha M_k(m_{2\textrm{d}(u,v)})$,
\item $M_{k+1}(m_1) = \frac{1-\alpha}{2}M_k(m_2) + \alpha M_k(m_1) + (1-\alpha) M_k(m_0)$,
\item $M_{k+1}(m_{2\textrm{d}(u,v) - 1}) = \frac{1-\alpha}{2}M_k(m_{2\textrm{d}(u,v) - 2}) + \alpha M_k(m_{2\textrm{d}(u,v) - 1}) + (1-\alpha) M_k(m_{2\textrm{d}(u,v)})$,
\item for $1 < i < 2\textrm{d}(u,v)-1$, we have that $M_{k+1}(m_i) = \alpha M_k(m_i) + \frac{1-\alpha}{2}(M_k(m_{i-1}) + M_k(m_{i+1})).$
\end{itemize}
We now examine $\mu_{k+1}$, and in particular the amount of mass of $\mu_{k+1}$ at each level. We let $S_k(i)$ denote the mass of $\mu_k$ at the $i$th level; in other words, $$S_k(i) = \displaystyle \sum_{w \in G, \, d(w,v)=i} \mu_k(w).$$ For all $i$, we can calculate $S_{k+1}(i)$ by considering the $i$th level and considering how much mass from each level from $S_k$ goes to the $i$th level. This is possible because all vertices at the same level will have indistinguishable behavior with respect to their contribution to the $i$th level. By calculating the contribution of each different level to the $i$th level, we can check that
\begin{itemize}
\item $S_{k+1}(0) = \frac{1-\alpha}{2}S_k(1) + \alpha S_k(0)$,
\item $S_{k+1}(2\textrm{d}(u,v)) = \frac{1-\alpha}{2}S_k(2\textrm{d}(u,v) - 1) + \alpha S_k(2\textrm{d}(u,v))$,
\item $S_{k+1}(1) = \frac{1-\alpha}{2}S_k(2) + \alpha S_k(1) + (1-\alpha) S_k(0)$,
\item $S_{k+1}(2\textrm{d}(u,v) - 1) = \frac{1-\alpha}{2}S_k(2\textrm{d}(u,v) - 2) + \alpha S_k(2\textrm{d}(u,v) - 1) + (1-\alpha) S_k(2\textrm{d}(u,v))$,
\item for $1 < i < 2\textrm{d}(u,v)-1$, we have that $S_{k+1}(i) = \alpha S_k(i) + \frac{1-\alpha}{2}(S_k(i-1) + S_k(i+1)).$
\end{itemize}
This lines up exactly with our characterization of $M_{k+1}$, so for all $i$ we have $$\displaystyle \sum_{w \in G, \textrm{d}(w,v) = i} \mu_{k+1}(w) = M_{k+1}(m_i).$$
\end{proof}
\section{Main Convergence Theorems}
Since we have shown that all Guvabs have $W = 1$ or $W = \frac{1}{2}$ or $W = 0$ or $\beta = 1$, and we have some understanding of the rate of convergence of the Wasserstein distance in each of these cases, we make some general statements about convergence that apply to all Guvabs. The following theorems sum up the general convergence results obtained from considering the each of the cases $W = 1$, $W = \frac{1}{2}$, $W = 0$ and $\beta = 1$ in the previous sections.
The first theorem states that the rate of convergence of $\{W_{2k}\}$ and $\{W_{2k+1}\}$ is exponential unless it is eventually constant.
\begin{theorem} \label{thm: Guvab Convergence Theorem}
For any Guvab, we have that
\begin{itemize}
\item either $\{W_{2k}\}$ is eventually constant, or there exists a constant $\lambda_{\emph{\textrm{even}}} \in (-1,1)$ and a positive constant $c_{\emph{\textrm{even}}} > 0$ such that $|W_{2k} - \lim_{k\to\infty}W_{2k}| \sim c_{\emph{\textrm{even}}} \cdot |\lambda_{\emph{\textrm{even}}}|^{2k}$,
\item either $\{W_{2k+1}\}$ is eventually constant, or there exists a constant $\lambda_{\emph{\textrm{odd}}} \in (-1,1)$ and a positive constant $c_{\emph{\textrm{odd}}} > 0$ such that $|W_{2k+1} - \lim_{k\to\infty}W_{2k+1}| \sim c_{\emph{\textrm{odd}}} \cdot |\lambda_{\emph{\textrm{odd}}}|^{2k+1}$
\end{itemize}
\end{theorem}
\begin{proof}
To begin, note that when $\beta<1$, we have that $W_k$ converges and $W\in\{1,\frac{1}{2},0\}$ by Corollary \ref{cor: four guvab nations}. Further, when $\beta=1$, Lemma \ref{lem: betty the beta=1 theorem} implies exactly that the desired result holds. Thus, it suffices to consider each of these cases $W=1$, $W=\frac{1}{2}$, and $W=0$ separately.
First, when $W=1,$ Theorem \ref{thm:winnie the when it's constant for W = 1 fairy} implies $\{W_k\}$ is eventually constant (and hence, we have the same for $\{W_{2k}\}$ and $\{W_{2k+1}\}$). This gives the desired result in the case $W=1$.
When $W=\frac{1}{2},$ Theorem \ref{thm: washington the w=1/2 convergence theorem} implies that either $\beta=\frac{1}{2}$ and $\{W_k\}$ is eventually constant or else $|W_{k} - \lim_{k\to\infty}W_{k}| \sim 0.5 \cdot |1-2\beta|^{2k}$ (and hence, we have the same for $\{W_{2k}\}$ and $\{W_{2k+1}\}$). This gives the desired result for $W=\frac{1}{2}.$
Finally, we note that when $W=0$, Theorem \ref{thm: Simba the sim-an-exponential when W=0 fairy} (and Remark \ref{rem: odd simba}) gives exactly the desired result. Thus, having checked each case, we conclude the proof.
\end{proof}
The second theorem provides a characterization of when $\{W_k\}$ is eventually constant when $\beta < 1$.
\begin{theorem} \label{thm: Characterization of Constancy}
When $\beta < 1$, we have that $\{W_k\}$ is eventually constant if and only if one of the following holds:
\begin{itemize}
\item $\alpha = \beta = 0$, the graph $G$ is bipartite, and $\textrm{d}(u,v)$ is odd,
\item $\alpha = 0$ and $\beta = \frac{1}{2}$, and $G$ is bipartite,
\item $\alpha = \beta = 0$ and $N(u)=N(v)$,
\item $\displaystyle \alpha = \beta = \frac{1}{\deg u + 1}$, the edge $\{u,v\}\in E(G)$, and if the edge $\{u,v\}$ were removed from $E(G)$ then $u,v$ would have $N(u)=N(v)$,
\item $\alpha = \beta$ and $u = v$.
\end{itemize}
\end{theorem}
\begin{proof}
To begin, note that when $\beta<1$, we have that $W_k$ converges and $W\in\{1,\frac{1}{2},0\}$ by Corollary \ref{cor: four guvab nations}. Thus, it suffices to consider each of these cases where $W=1$, $W=\frac{1}{2}$ and $W=0$ separately.
First, we look at the case where $W=1$. Note that, in this case, Theorem \ref{thm:winnie the when it's constant for W = 1 fairy} implies that $\{W_k\}$ is always eventually constant. Further, by Theorem \ref{thm: convergence values}, we see this case is equivalent to $\alpha=\beta=0$ and $W \neq 0$. Further, by Theorem \ref{thm: 0-convergence}, this case occurs exactly when $\alpha = \beta = 0$, the graph $G$ is bipartite, and $\textrm{d}(u,v)$ is odd (i.e., the first item of the theorem statement).
Next, when $W=\frac{1}{2}$, Corollary \ref{cor: wendy the when it's constant for W = 1/2 fairy} implies $\{W_k\}$ is eventually constant exactly when $\beta=\frac{1}{2}$. By Theorem \ref{thm: convergence values}, this case occurs exactly when $\alpha = 0$ and $\beta = \frac{1}{2}$, and $G$ is bipartite (i.e., the second item of the theorem statement).
Finally, when $W=0$, we see that Theorem \ref{thm: Constance the constant distance when W = 0 fairy} implies $\{W_k\}$ is eventually constant exactly when one of the following holds:
\begin{itemize}
\item $\alpha = \beta = 0$ and $N(u)=N(v)$,
\item $\displaystyle \alpha = \beta = \frac{1}{\deg u + 1}$, the edge $\{u,v\}\in E(G)$, and if the edge $\{u,v\}$ were removed from $E(G)$ then $u,v$ would have $N(u)=N(v)$,
\item $\alpha = \beta$ and $u = v$.
\end{itemize}
Note that, each of these cases is indeed a case where $W=0$ by Theorem \ref{thm: 0-convergence}, so this case is equivalent to the final three items of the theorem statement.
Thus, considering each of these cases together, we obtain the desired result.
\end{proof}
\section{Open Problems}
The theorems presented in this paper open up several new questions and directions for further research, which the reader is invited to consider. Specifically, given Theorem~\ref{thm: Guvab Convergence Theorem}, the remaining questions regarding the behavior of Guvabs can be broken into three main categories: 1) determining when $\{W_{2k}\}$ and $\{W_{2k+1}\}$ are eventually constant, 2) in cases $\{W_{2k}\}$ and $\{W_{2k+1}\}$ are eventually constant, determining how long they take to become constant, and 3) determining $c$ and $\lambda$ when $\{W_{2k}\}$ and $\{W_{2k+1}\}$ are not eventually constant. In this section, we break down what we have shown and what is left to be done regarding each of these questions.
By Theorem \ref{thm: Characterization of Constancy}, we have characterized the cases where $\{W_k\}$ is constant in all cases where $\beta < 1$. Furthermore, in the cases of $W = 1$ and $W = \frac{1}{2}$, we know that $\{W_{2k}\}$ is eventually constant if and only if $\{W_k\}$ is eventually constant, and similarly $\{W_{2k+1}\}$ is eventually constant if and only if $\{W_k\}$ is eventually constant. In the case of $W = 0$, it remains to characterize the cases where either $\{W_{2k}\}$ or $\{W_{2k+1}\}$ individually are eventually constant, but $\{W_k\}$ is not. Further, in the $\beta = 1$ case we lack a complete characterization of when $\{W_k\}$ is eventually constant.
Question $2)$ remains largely unanswered and is a promising direction for future work. The progress so far in this paper is restricted to fairly weak upper and lower bounds when $W = 1$, and characterizations of when $\{W_k\}$ is eventually constant when $W = 0$ and $\beta = 1$. One interesting problem is that of tighter bounds for the case where $W = 1$, and similar bounds for the case when $W = \frac{1}{2}$ and $W$ is eventually constant. Also, depending on the answers to Question 1, there may be Guvabs where only one of $\{W_{2k}\}$ and $\{W_{2k+1}\}$ is eventually constant. If we find a specific Guvab that satisfies these criteria, it will be interesting to determine how long this Guvab takes to have either $\{W_{2k}\}$ or $\{W_{2k+1}\}$ be eventually constant.
Answering question $3)$ will require specific knowledge of eigenvectors and eigenvalues. In full generality, this is difficult, so a potential direction for future work would be addressing it in specific examples.
\section{Acknowledgements}
We would like to thank our mentor, Pakawut Jiradilok, for providing us with important knowledge, guidance, and assistance throughout our project. We would also like to thank Supanat Kamtue for the problem idea and helpful thoughts and guidance. Finally, we would like to thank the PRIMES-USA program for making this project possible.
\newpage
\bibliographystyle{alpha}
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\section{Introduction\label{section:introduction}}
Extensive experimental evidence \cite{Book,AMGE,MM} has shown that a complex dynamical 3-space
underlies reality. The evidence involves the repeated detection of the motion of the earth
relative to that 3-space using Michelson interferometers operating in gas mode \cite{MM},
particularly the experiment by Miller \cite{Miller} in 1925/26 at Mt.Wilson, and the coaxial cable RF travel
time measurements by Torr and Kolen in Utah in 1981, and the DeWitte experiment in 1991 in Brussels
\cite{MM}. All such 7 experiments are consistent with respect to speed and direction. It has
been shown that
effects caused by motion relative to this 3-space can mimic the formalism of
spacetime, but that it is the 3-space that is `real', simply because it is directly
observable \cite{Book}.
The 3-space is in differential motion, that is one part has a velocity relative
to other parts, and so involves a velocity field ${\bf v}({\bf r},t)$ description. To be
specific this velocity field must be described relative to a frame of observers, but the
formalism is such that the dynamical equations for this velocity field must transform
covariantly under a change of observer. As shown herein the experimental data from the DeWitte experiment
shows that ${\bf v}({\bf r},t)$ has a fractal structure. This arises because, in the absence of
matter, the dynamical equations for ${\bf v}({\bf r},t)$ have no scale. This implies that the
differential motion of 3-space manifests at all scales. This fractal differential motion of
3-space is missing from all the fundamental equations of physics, and so these equations
require a generalisation. Here we report on the necessary generalisation of the
Schr\"{o}dinger equation, and which results in some remarkable results: (i) the equivalence
principle emerges, as well as (ii) the effects of vorticity of this velocity field. These two effects are thus
seen to be quantum-theoretic effects, i.e. consequences of the wave nature of matter. The equivalence principle,
as originally formulated by Galileo and then Newton, asserts that the gravitational acceleration of an object is
independent of its composition and speed. However we shall see that via the vorticity effect, the velocity of
the object does affect the acceleration by causing rotations.
It has been shown \cite{Book,DMtrends} that the phenomenon of gravity is a consequence of the time-dependence
and inhomogeneities of ${\bf v}({\bf r},t)$. So the dynamical equations for ${\bf v}({\bf
r},t)$ give rise to a new theory of gravity, when combined with the generalised Schr\"{o}dinger equation, and
the generalised Maxwell and Dirac equations. The equations for ${\bf v}({\bf
r},t)$ involve the Newtonian gravitational constant $G$ and a dimensionless constant that determines the strength
of a new spatial self-interaction effect, which is missing from both Newtonian Gravity and General Relativity.
Experimental data has revealed
\cite{Book, DMtrends} the remarkable discovery that this constant is the fine structure constant $\alpha
\approx 1/137$. This dynamics then explains numerous gravitational anomalies, such as the bore hole $g$
anomaly, the so-called `dark matter' anomaly in the rotation speeds of spiral galaxies, and that the effective
mass of the necessary black holes at the centre of spherical matter systems, such as globular
clusters and spherical galaxies, is $\alpha/2$ times the total mass of these systems. This
prediction has been confirmed by astronomical observations \cite{BH}.
The occurrence of $\alpha$ suggests that space is itself a quantum system undergoing on-going
classicalisation. Just such a proposal has arisen in {\it Process Physics} \cite{Book} which is
an information-theoretic modelling of reality. There quantum space and matter arise in terms of
the Quantum Homotopic Field Theory (QHFT) which, in turn, may be related to the standard model of
matter. In the QHFT space at this quantum level is best described as a `quantum foam'. So
we interpret the observed fractal 3-space as a classical approximation
to this `quantum foam'.
While here we investigate the properties of the generalised Schr\"{o}dinger
equation, analogous generalisations of the Maxwell and Dirac equations, and in turn the
corresponding generalisations to the quantum field theories for such systems, may also be made.
In the case of the Maxwell equations we obtain the light bending effects, including in particular gravitational
lensing, caused by the 3-space
differential and time-dependent flow.
\section{The Physics of 3-Space\label{section:space}}
Because of the dominance of the spacetime ontology, which has been the
foundation of physics over the last century, the existence of a 3-space as an observable
phenomenon has been overlooked, despite extensive experimental detection over that period, and
earlier. This spacetime ontology is distinct from the role of spacetime as a mathematical
formalism implicitly incorporating some real dynamical effects, though this distinction is rarely made.
Consequently the existence of 3-space has been denied, and so there has never been a dynamical theory
for 3-space. In recent years this situation has dramatically changed. We briefly
summarise the key aspects to the dynamics of 3-space.
Relative to some observer 3-space is described by a velocity field ${\bf v}({\bf r},t)$.
It is important to note that the coordinate ${\bf r}$ is not itself 3-space, rather it
is merely a label for an element of 3-space that has velocity ${\bf v}$, relative to some
observer. This will become more evident when we consider the necessary generalisation of the
Schr\"{o}dinger equation. Also it is important to appreciate that this `moving' 3-space is not
itself embedded in a `space'; the 3-space is all there is, although as noted above its deeper
structure is that of a `quantum foam'.
In the case of zero vorticity $\nabla\times{\bf v}={\bf 0}$ the 3-space dynamics is given by, in
the non-relativistic limit,
\begin{equation}
\nabla.\left(\frac{\partial {\bf v} }{\partial t}+({\bf v}.{\bf \nabla}){\bf v}\right)
+\frac{\alpha}{8}\left((tr D)^2 - tr(D^2)\right)=
-4\pi G\rho,
\label{eqn:E1}\end{equation}
where $\rho$ is the matter density, and where
\begin{equation} D_{ij}=\frac{1}{2}\left(\frac{\partial v_i}{\partial x_j}+
\frac{\partial v_j}{\partial x_i}\right).
\label{eqn:E2}\end{equation}
The acceleration of an element of space is given by the Euler form
\begin{eqnarray}
{\bf g}({\bf r},t)&\equiv&\lim_{\Delta t \rightarrow 0}\frac{{\bf v}({\bf r}+{\bf v}({\bf r},t)\Delta t,t+\Delta
t)-{\bf v}({\bf r},t)}{\Delta t} \nonumber \\
&=&\frac{\partial {\bf v}}{\partial t}+({\bf v}.\nabla ){\bf v}
\label{eqn:E3}\end{eqnarray}
These forms are mandated by Galilean covariance under change of observer\footnote{However this does
not exclude so-called relativistic effects, such as the length contraction of moving rods or the time
dilations of moving clocks.}. This non-relativistic modelling of the dynamics for the
velocity field gives a direct account of the various phenomena noted above. A generalisation to include
vorticity and relativistic effects of the motion of matter through this 3-space is given in \cite{Book}.
From (\ref{eqn:E1}) and (\ref{eqn:E2}) we obtain that
\begin{equation}
\nabla.{\bf g}=-4\pi G\rho-4\pi G \rho_{DM},
\label{eqn:E4}\end{equation}
where
\begin{equation}
\rho_{DM}({\bf r})=\frac{\alpha}{32\pi G}( (tr D)^2-tr(D^2)).
\label{eqn:E5}\end{equation}
In this form we see that if $\alpha\rightarrow 0$, then the acceleration of the 3-space elements is given
by Newton's Law of Gravitation, in differential form. But for a non-zero $\alpha$ we see that the
3-space acceleration has an additional effect, the $\rho_{DM}$ term, which is an effective `matter density' that
mimics the new self-interaction dynamics. This has been shown to be the origin of the so-called `dark
matter' effect in spiral galaxies. It is important to note that (\ref{eqn:E4}) does not determine ${\bf
g}$ directly; rather the velocity dynamics in (\ref{eqn:E1}) must be solved, and then with ${\bf g}$
subsequently determined from (\ref{eqn:E3}). Eqn.(\ref{eqn:E4}) merely indicates that the resultant
non-Newtonian aspects to ${\bf g}$ could be mistaken as being the result of a new form of matter, whose density
is given by $\rho_{DM}$. Of course the saga of `dark matter' shows that this actually happened, and that there
has been a misguided and fruitless search for such `matter'.
The numerous experimental confirmations of (\ref{eqn:E1}) imply that Newtonian gravity is not universal
at all. Rather a key aspect to gravity was missed by Newton because it so happens that the
3-space self-interaction dynamics does not necessarily explicitly manifest outside of spherical matter systems,
such as the sun. To see this it is only necessary to see that the velocity field
\begin{equation}
{\bf v}({\bf r})=-\sqrt{\frac{2GM'}{r}}\hat{\bf r},
\label{eqn:E6}\end{equation}
is a solution to (\ref{eqn:E1}) external to a spherical mass $M$, where $M'=(1+\frac{\alpha}{2})M+..$.
Then (\ref{eqn:E6}) gives, using (\ref{eqn:E3}), the resultant external `inverse square law' acceleration
\begin{equation}
{\bf g}({\bf r})=-\frac{GM'}{r^2}\hat{\bf r}.
\label{eqn:E7}\end{equation}
Hence in this special case the 3-space dynamics predicts an inverse square law form for ${\bf g}$, as
confirmed in the non-relativistic regime by Kepler's laws for planetary motion, with only a modified value for
the effective mass $M'$. So for this reason we see how easy it was for Newton to have overlooked a velocity
formalism for gravity, and so missed the self-interaction dynamics in (\ref{eqn:E1}). Inside a spherical matter
system Newtonian gravity and the new gravity theory differ, and it was this difference that explained the bore
hole $g$ anomaly data
\cite{DMtrends}, namely that $g$ does not decrease down a bore hole as rapidly as Newtonian gravity predicts. It
was this anomaly that lead to the discovery that $\alpha$ was in fact the fine structure constant, up to
experimental errors.
As well the 3-space dynamics in (\ref{eqn:E1}) has `gravitational wave' solutions \cite{QFGGW}. Then there
are regions where the velocity differs slightly from the enveloping region. In the absence of matter these
waves will be in general fractal because there is no dimensioned constant, and so no natural scale. These waves
were seen by Miller, Torr and Kolen, and by DeWitte \cite{Book,QFGGW} as shown in Fig.\ref{fig:fractal}.
However an assumption made in previous analyses was that the acceleration of the 3-space itself, in
(\ref{eqn:E3}), was also the acceleration of matter located in that 3-space. The key result herein is to
derive this result by using the generalised Schr\"{o}dinger equation. In doing so we discover the additional
effect that vorticity in the velocity field causes quantum states to be rotated, as discussed in
Sect.\ref{section:GPB}.
\section{Newtonian Gravity and the Schr\"{o}dinger Equation\label{section:newtonian}}
Let us consider what might be regarded as the conventional `Newtonian' approach to including gravity in the
Schr\"{o}dinger equation \cite{Schrod}. There gravity is described by the Newtonian potential energy field
$\Phi({\bf r},t)$, such that ${\bf g}=-\nabla \Phi$, and we have for a `free-falling' quantum system, with mass
$m$,
\begin{equation}
i\hbar\frac{\partial \psi({\bf r},t)}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi({\bf r},t)+
m\Phi({\bf r},t)\psi({\bf r},t)\equiv H(t)\Psi,
\label{eqn:equiv1}\end{equation}
where the hamiltonian is in general now time dependent, because the masses producing the gravitational acceleration
may be moving. Then the classical-limit trajectory is obtained via the usual Ehrenfest method \cite{Ehrenfest}:
we first compute the time rate of change of the so-called position `expectation value'
\begin{eqnarray}
\frac{d\!\!<\!\!{\bf r}\!\!> }{dt} &\equiv& \frac{d }{dt}(\psi,{\bf
r}\psi)=\frac{i}{\hbar}(H\psi,{\bf r}\psi)-\frac{i}{\hbar}(\psi,{\bf r}H\psi)\nonumber
\\&=&\frac{i}{\hbar}(\psi,[H,{\bf r}]\psi),
\label{eqn:equiv2}\end{eqnarray}
which is valid for a normalised state $\psi$. The norm is time invariant when $H$ is hermitian
($H^\dagger=H$) even if $H$ itself is time dependent,
\begin{eqnarray}
\frac{d}{dt}(\psi,\psi)&=&\frac{i}{\hbar}(H\psi,\psi)-\frac{i}{\hbar}(\psi,H\psi)\nonumber \\&=&
\frac{i}{\hbar}(\psi,H^\dagger\psi)-\frac{i}{\hbar}(\psi,H\psi)=0.
\label{eqn:equiv3}\end{eqnarray}
Next we compute the matter `acceleration' from (\ref{eqn:equiv2}).
\begin{eqnarray}
\frac{d^2\!\!<\!\!{\bf r}\!\!> }{dt^2} &=& \frac{i}{\hbar}\frac{d }{dt}(\psi,[H,{\bf r}]\psi),\nonumber\\
&=&\left(\frac{i}{\hbar}\right)^2(\psi,[H,[H,{\bf r}]]\psi)+\frac{i}{\hbar}(\psi,[\frac{\partial H(t)
}{\partial t},{\bf r}]\psi),\nonumber\\
&=&-(\psi,\nabla \Phi\psi)=(\psi,{\bf g}({\bf r},t)\psi)=<\!{\bf g}({\bf r},t)\!>.
\label{eqn:equiv4}\end{eqnarray}
where for the commutator
\begin{equation}\left[\frac{\partial H(t)}{\partial t},{\bf r}\right]=\left[m\frac{\partial \Phi({\bf r}, t)}{\partial
t},{\bf r}\right]=0.\end{equation}
In the classical limit
$\psi$ has the form of a wavepacket where the spatial extent of $\psi$ is much smaller than the spatial region over
which ${\bf g}({\bf r},t)$ varies appreciably. Then we have the approximation $<\!{\bf g}({\bf r},t)\!>\approx {\bf
g}(<\!{\bf r}\!>,t)$, and finally we arrive at the Newtonian 2nd-law equation of motion for the wavepacket,
\begin{equation}
\frac{d^2\!\!<\!\!{\bf r}\!\!> }{dt^2}\approx {\bf g}(<\!{\bf r}\!>, t).
\label{eqn:equiv5}\end{equation}
In this classical
limit we obtain the equivalence principle, namely that the acceleration is independent of the mass $m$ and of
the velocity of that mass. But of course that followed by construction, as the equivalence principle is built
into (\ref{eqn:equiv1}) by having
$m$ as the coefficient of $\Phi$. In Newtonian gravity there is no explanation for the origin of $\Phi$ or
${\bf g}$. In the new theory gravity is explained in terms of a velocity field, which in turn has a deeper
explanation within {\it Process Physics}.
\section{Dynamical 3-Space and the Generalised Schr\"{o}dinger
Equation\label{section:schrodinger}}
The key insight is that conventional physics has neglected the interaction of various systems with the dynamical
3-space. Here we generalise the Schr\"{o}dinger equation to take account of this new physics. Now gravity is a
dynamical effect arising from the time-dependence and spatial inhomogeneities of the 3-space velocity field ${\bf
v}({\bf r},t)$, and for a `free-falling' quantum system with mass
$m$ the Schr\"{o}dinger equation now has the generalised form
\begin{equation}
i\hbar\left(\frac{\partial}{\partial t} +{\bf
v}.\nabla+\frac{1}{2}\nabla.{\bf v}\right) \psi({\bf r},t)=-\frac{\hbar^2}{2m}\nabla^2\psi({\bf r},t),
\label{eqn:equiv6}\end{equation}
which we write as
\begin{equation}
i\hbar\frac{\partial \psi({\bf r},t)}{\partial t}=H(t)\psi({\bf r},t),
\label{eqn:equiv7}\end{equation}
where now
\begin{equation}
H(t)=-i\hbar\left({\bf
v}.\nabla+\frac{1}{2}\nabla.{\bf v}\right)-\frac{\hbar^2}{2m}\nabla^2
\label{eqn:equiv8}\end{equation}
This form for $H$ specifies how the quantum system must couple to the velocity field, and it uniquely
follows from two considerations: (i) the generalised Schr\"{o}dinger equation must remain form invariant under a
change of observer, i.e. with $t
\rightarrow t$, and ${\bf r}
\rightarrow {\bf r}+{\bf V}t$, where ${\bf V}$ is the relative velocity of the two observers. Then we compute
that
$\displaystyle{\frac{\partial}{\partial t} +{\bf v}.\nabla +\frac{1}{2}\nabla.{\bf v} \rightarrow} $ $
\displaystyle{\frac{\partial}{\partial t} +{\bf v}.\nabla}+\frac{1}{2}\nabla.{\bf v}$, i.e. that it is an
invariant operator, and (ii) requiring that
$H(t)$ be hermitian, so that the wavefunction norm is an invariant of the time evolution. This implies that the
$\frac{1}{2}\nabla.{\bf v}$ term must be included, as ${\bf v}.\nabla$ by itself is not hermitian for an
inhomogeneous ${\bf v}({\bf r},t)$. Then the consequences for the motion of wavepackets are uniquely determined;
they are fixed by these two quantum-theoretic requirements\footnote{For two or more `particles' we have by the
same arguments $H(t)=\sum_j-i\hbar\left({\bf
v}.\nabla_j+\frac{1}{2}\nabla_j.{\bf v}\right)-\frac{\hbar^2}{2m_j}\nabla^2_j$}.
Then again the classical-limit trajectory is obtained via the position `expectation value', first with
\begin{eqnarray}
{\bf v}_O\equiv\frac{d\!\!<\!\!{\bf r}\!\!> }{dt} &=& \frac{d }{dt}(\psi,{\bf
r}\psi)=\frac{i}{\hbar}(\psi,[H,{\bf r}]\psi),\nonumber \\ &=&(\psi,({\bf v}({\bf r},
t)-\frac{i\hbar}{m}\nabla)\psi)\nonumber \\ &=&<\!\!{\bf v}({\bf r}, t)\!\!>-\frac{i\hbar}{m}<\!\!\nabla\!\!>,
\label{eqn:equiv9}\end{eqnarray}
on evaluating the commutator using $H(t)$ in (\ref{eqn:equiv8}), and which is again valid for a normalised state
$\psi$.
Then for the `acceleration' we obtain from (\ref{eqn:equiv9}) that\footnote{Care is needed to indicate the
range of the various $\nabla$'s. Extra parentheses $($ ... $)$ are used to limit the range when required.}
\begin{eqnarray}
\lefteqn{\frac{d^2\!\!<\!\!{\bf r}\!\!> }{dt^2} = \frac{d }{dt}(\psi,({\bf v}
-\frac{i\hbar}{m}\nabla)\psi)}\nonumber\\
& & =(\psi,\left(\frac{\partial {\bf v}({\bf r},t) }{\partial
t}+\frac{i}{\hbar}[H,({\bf v} -\frac{i\hbar}{m}\nabla)]\right)\psi),\nonumber\\
& &=(\psi,\frac{\partial {\bf v}({\bf r},t) }{\partial t}\psi)+
(\psi,
\left({\bf
v}.\nabla+\frac{1}{2}\nabla.{\bf v}-\frac{i\hbar}{2m}\nabla^2\right)\left({\bf v}
-\frac{i\hbar}{m}\nabla\right)\psi)-\nonumber\\
& &\mbox{\ \ \ }(\psi,\left.\left({\bf v}
-\frac{i\hbar}{m}\nabla\right)\left({\bf v}.\nabla+\frac{1}{2}\nabla.{\bf
v}-\frac{i\hbar}{2m}\nabla^2\right)\right)\psi), \nonumber\\
& & =(\psi,\left(\frac{\partial {\bf
v}({\bf r},t) }{\partial t}+(({\bf v}.\nabla){\bf v}) -\frac{i\hbar}{m}(\nabla\times{\bf
v})\times {\bf \nabla}\right)\psi)+(\psi,\frac{i\hbar}{2m}(\nabla\times(\nabla\times {\bf v}))\psi),\nonumber \\
& &\approx\frac{\partial{\bf v}}{\partial t}+({\bf v}.\nabla){\bf v}+(\nabla\times{\bf
v})\times\left(\frac{d\!\!<\!\!{\bf r}\!\!> }{dt}-{\bf v}\right)+\frac{i\hbar}{2m}(\nabla\times(\nabla\times{\bf
v})),\nonumber \\ & &=\frac{\partial{\bf v}}{\partial t}+({\bf v}.\nabla){\bf v}+(\nabla\times{\bf
v})\times\left(\frac{d\!\!<\!\!{\bf r}\!\!> }{dt}-{\bf v}\right)\nonumber \\
& &=\frac{\partial{\bf v}}{\partial t}+({\bf v}.\nabla){\bf v}+
(\nabla\times{\bf v})\times{\bf v}_R
\label{eqn:equiv10}\end{eqnarray}
where in arriving at the 3rd last line we have invoked the small-wavepacket approximation, and
also used (\ref{eqn:equiv9}) to identify
\begin{equation} {\bf v}_R \equiv -\frac{i\hbar}{m}<\!\!\nabla\!\!>={\bf v}_O-{\bf v},
\label{eqn:equiv11}\end{equation}
where ${\bf v}_O$ is the velocity of the wavepacket or object `O' relative to the observer, so then ${\bf v}_R$
is the velocity of the wavepacket relative to the local 3-space.
Then all
velocity field terms are now evaluated at the location of the wavepacket.
Note that the operator
\begin{equation}
-\frac{i\hbar}{m}(\nabla\times{\bf v})\times\nabla+\frac{i\hbar}{2m}(\nabla\times(\nabla\times{\bf v}))
\end{equation} is hermitian, but that separately neither of these two operators is hermitian. Then in general
the scalar product in (\ref{eqn:equiv10}) is real. But then in arriving at the last line in (\ref{eqn:equiv10})
by means of the small-wavepacket approximation, we must then self-consistently use that
$\nabla\times(\nabla\times{\bf v})={\bf 0}$, otherwise the acceleration acquires a spurious imaginary part.
This is consistent with (\ref{eqn:CG4b}) outside of any matter which contributes to the generation of the
velocity field, for there
$\rho=0$. These observations point to a deep connection between quantum theory and the velocity field dynamics,
as already argued in \cite{Book}.
We see that the
test `particle' acquires the acceleration of the velocity field, as in (\ref{eqn:E3}), and as well an
additional vorticity induced acceleration which is the analogue of the Helmholtz acceleration in fluid mechanics.
Then $\vec{\omega}/2$ is the instantaneous angular velocity of the local 3-space, relative to a distant
observer. Hence we find that the equivalence principle arises from the unique generalised Schr\"{o}dinger
equation and with the additional vorticity effect. This vorticity effect depends on the absolute velocity ${\bf
v}_R$ of the object relative to the local space, and so requires a change in the Galilean or Newtonian form of
the equivalence principle.
The vorticity acceleration effect is the origin of the
Lense-Thirring so-called `frame-dragging' \footnote{In the spacetime formalism it is mistakenly argued that it
is `spacetime' that is `dragged'.} effect
\cite{LT} discussed in Sect.\ref{section:GPB}. While the generation of the vorticity is a relativistic effect,
as in (\ref{eqn:CG4b}), the response of the test particle to that vorticity is a non-relativistic effect, and
follows from the generalised Schr\"{o}dinger equation, and which is not present in the standard Schr\"{o}dinger
equation with coupling to the Newtonian gravitational potential, as in (\ref{eqn:equiv1}). Hence the generalised
Schr\"{o}dinger equation with the new coupling to the velocity field is more fundamental. The Helmholtz term in
(\ref{eqn:equiv10}) is being explored by the Gravity Probe B gyroscope precession experiment, however the
vorticity caused by the motion of the earth is extremely small, as discussed in Sect.\ref{section:GPB}.
An important insight emerges from the form of (\ref{eqn:equiv7}) and (\ref{eqn:equiv8}): here the
generalised Schr\"{o}dinger equation involves two fields ${\bf v}({\bf r},t)$ and
$\psi({\bf r},t)$, where the coordinate ${\bf r}$ is merely a label to relate the two fields, and is
not itself the 3-space. In particular while ${\bf r}$ may have the form of a Euclidean 3-geometry,
the space itself has time-dependence and inhomogeneities, and as well in the more general case will
exhibit vorticity $\omega=\nabla\times{\bf v}$. Only in the unphysical case does the description
of the 3-space become identified with the coordinate system ${\bf r}$, and that is when the velocity
field ${\bf v}({\bf r},t)$ becomes uniform and time independent. Then by a suitable choice of
observer we may put ${\bf v}({\bf r},t)={\bf 0}$, and the generalised Schr\"{o}dinger equation
reduces to the usual `free' Schr\"{o}dinger equation. As we discuss
later the experimental evidence is that ${\bf v}({\bf r},t)$ is fractal and so cannot be removed by
a change to a preferred observer. Hence the generalised Schr\"{o}dinger equation in
(\ref{eqn:equiv7})-(\ref{eqn:equiv8}) is a major development for fundamental physics. Of course in general other
non-3-space potential energy terms may be added to the RHS of (\ref{eqn:equiv8}). A prediction of this
new quantum theory, which also extends to a generalised Dirac equation, is that the fractal structure to space
implies that even at the scale of atoms etc there will be time-dependencies and inhomogeneities, and that these
will affect transition rates of quantum systems. These effects are probably
those known as the Shnoll effects \cite{Shnoll1}.
\section{Free-Fall Minimum Proper-Time Trajectories\label{section:geodesic}}
The acceleration in (\ref{eqn:equiv10}) also arises from the following argument, which is the
analogue of the Fermat least-time formalism. Consider the elapsed time for a comoving clock
travelling with the test particle. Then taking account of the Lamour time-dilation effect that
time is given by
\begin{equation}
\tau[{\bf r}_0]=\int dt \left(1-\frac{{\bf v}_R^2}{c^2}\right)^{1/2}
\label{eqn:G1}
\end{equation}
with ${\bf v}_R$ given by (\ref{eqn:equiv11}) in terms of ${\bf v}_O$ and ${\bf v}$. Then this time effect
relates to the speed of the clock relative to the local 3-space, and that $c$ is the speed of light relative to
that local 3-space. We are using a relativistic treatment in (\ref{eqn:G1}) to demonstrate the generality of the
results\footnote{ A non-relativistic analysis may be alternatively pursued by first expanding (\ref{eqn:G1}) in
powers of $1/c^2$.}. Under a deformation of the trajectory
\begin{equation}{\bf r}_0(t)
\rightarrow {\bf r}_0(t) +\delta{\bf r}_0(t),
\mbox{\ \ }{\bf v}_0(t) \rightarrow {\bf v}_0(t) +\displaystyle\frac{d\delta{\bf r}_0(t)}{dt},\end{equation}
and then
\begin{equation}\label{eqn:G2}
{\bf v}({\bf r}_0(t)+\delta{\bf r}_0(t),t) ={\bf v}({\bf r}_0(t),t)+(\delta{\bf
r}_0(t).{\bf \nabla}) {\bf v}({\bf r}_0(t),t)+...
\end{equation}
Evaluating the change in proper travel time to lowest order
\begin{eqnarray*}\label{eqn:G3}
\delta\tau&=&\tau[{\bf r}_0+\delta{\bf r}_0]-\tau[{\bf r}_0] +... \nonumber\\
&=&-\int dt \:\frac{1}{c^2}{\bf v}_R. \delta{\bf v}_R\left(1-\displaystyle{\frac{{\bf
v}_R^2}{c^2}}\right)^{-1/2}+...\nonumber\\
&=&\int dt\frac{1}{c^2}\displaystyle{\frac{{\bf
v}_R.(\delta{\bf r}_0.{\bf \nabla}){\bf v}-{\bf v}_R.\displaystyle{\frac{d(\delta{\bf
r}_0)}{dt}}}{\sqrt{1-\displaystyle{\frac{{\bf v}_R^2}{c^2}}}}}+...\nonumber\\
&=&\int dt \frac{1}{c^2}\left(\frac{{\bf v}_R.(\delta{\bf r}_0.{\bf \nabla}){\bf v}}{
\sqrt{1-\displaystyle{\frac{{\bf
v}_R^2}{c^2}}}} +\delta{\bf r}_0.\frac{d}{dt}
\frac{{\bf v}_R}{\sqrt{1-\displaystyle{\frac{{\bf
v}_R^2}{c^2}}}}\right)+...\nonumber\\
&=&\int dt\: \frac{1}{c^2}\delta{\bf r}_0\:.\left(\frac{({\bf v}_R.{\bf \nabla}){\bf v}+{\bf v}_R\times({\bf
\nabla}\times{\bf v})}{
\sqrt{1-\displaystyle{\frac{{\bf v}_R^2}{c^2}}}}+\frac{d}{dt}
\frac{{\bf v}_R}{\sqrt{1-\displaystyle{\frac{{\bf
v}_R^2}{c^2}}}}\right)+...\nonumber \\
\end{eqnarray*}
Hence a
trajectory ${\bf r}_0(t)$ determined by $\delta \tau=0$ to $O(\delta{\bf r}_0(t)^2)$
satisfies
\begin{equation}\label{eqn:G4}
\frac{d}{dt}
\frac{{\bf v}_R}{\sqrt{1-\displaystyle{\frac{{\bf v}_R^2}{c^2}}}}=-\frac{({\bf
v}_R.{\bf \nabla}){\bf v}+{\bf v}_R\times({\bf
\nabla}\times{\bf v})}{
\sqrt{1-\displaystyle{\frac{{\bf v}_R^2}{c^2}}}}.
\label{eqn:vReqn}\end{equation}
Substituting ${\bf
v}_R(t)={\bf v}_0(t)-{\bf v}({\bf r}_0(t),t)$ and using
\begin{equation}\label{eqn:G5}
\frac{d{\bf v}({\bf r}_0(t),t)}{dt}=\frac{\partial {\bf v}}{\partial t}+({\bf v}_0.{\bf \nabla}){\bf
v},
\end{equation}
we obtain
\begin{equation}\label{eqn:CG6}
\frac{d {\bf v}_0}{dt}=\displaystyle{\frac{\partial {\bf v}}{\partial t}}+({\bf v}.{\bf \nabla}){\bf
v}+({\bf \nabla}\times{\bf v})\times{\bf v}_R-\frac{{\bf
v}_R}{1-\displaystyle{\frac{{\bf v}_R^2}{c^2}}}
\frac{1}{2}\frac{d}{dt}\left(\frac{{\bf v}_R^2}{c^2}\right).
\end{equation}
Then in the low speed limit $v_R \ll c $ we may neglect the last term, and we obtain
(\ref{eqn:equiv10}). Hence we see a close relationship between the geodesic equation, known first from General
Relativity, and the 3-space generalisation of the Schr\"{o}dinger equation, at least in the non-relativistic limit.
So in the classical limit, i.e when the wavepacket approximation is valid, the wavepacket trajectory
is specified by the least propertime geodesic.
The relativistic term in (\ref{eqn:CG6}) is responsible for the precession of
elliptical orbits and also for the event horizon effect. Hence the trajectory in (\ref{eqn:equiv10})
is a non-relativistic minimum travel-time trajectory, which is Fermat's Principle. The relativistic
term in (\ref{eqn:CG6}) will arise from a generalised Dirac equation which would then include the
dynamics of 3-space.
\begin{figure}[t]
\hspace{5mm}\includegraphics[scale=0.41]{CahillDeWitte8.eps}
\caption{\small{ Variations in twice the one-way travel time, in ns, for an RF signal to travel 1.5
km through a buried coaxial cable between Rue du Marais and Rue de
la Paille, Brussels. An offset has been used such that the average is zero. The cable has a
North-South orientation, and the data is $\pm$ difference of the travel times for NS and SN
propagation. The sidereal time for maximum effect of $\sim\!\!5$hr (or $\sim\!\!17$hr) (indicated
by vertical lines) agrees with the direction found by Miller \cite{Miller}. Plot shows
data over 3 sidereal days and is plotted against sidereal time. The main effect is caused by the rotation of the
earth. The superimposed fluctuations are evidence of turbulence i.e gravitational
waves. Removing the earth induced rotation effect we obtain the first experimental data of the fractal
structure of space, and is shown in Fig.\ref{fig:fractal}. DeWitte performed this experiment over 178 days, and
demonstrated that the effect tracked sidereal time and not solar time \cite{Book}. }
\label{fig:DeWittetimes}}\end{figure}
\begin{figure}[t]
\hspace{5mm}\includegraphics[scale=0.4]{CahillTurb8.eps}
\caption{\small{Shows the velocity fluctuations, essentially `gravitational waves' observed by DeWitte in 1991
from the measurement of variations in the RF coaxial-cable one-way travel times. This data is obtained from that
in Fig.\ref{fig:DeWittetimes} after removal of the dominant effect caused by the rotation of the earth.
Ideally the velocity fluctuations are three-dimensional, but the DeWitte experiment had only one
arm. This plot is suggestive of a fractal structure to the velocity field. This is confirmed by the
power law analysis shown in Fig.\ref{fig:powerlaw}.}
\label{fig:fractal}}\end{figure}
\section{Fractal 3-Space and the DeWitte Experimental Data\label{section:fractal}}
In 1991 Roland DeWitte working within Belgacom,
the Belgium telecommunications company, accidently made yet another detection of absolute motion, and one
which was 1st-order in $v/c$. 5MHz radio frequency (RF) signals were sent in both
directions through two buried coaxial cables linking the two clusters of cesium atomic clocks.
Changes in propagation times were observed and eventually observations over 178 days were recorded.
A sample of the data, plotted against sidereal time for just three days, is shown
in Fig.\ref{fig:DeWittetimes}. The DeWitte data was clear evidence of absolute motion with
the Right Ascension for minimum/maximum propagation time agreeing almost exactly with Miller's direction
\footnote{This velocity arises after removing the effects of the earth's orbital speed about the sun, 30km/s,
and the gravitational in-flow past the earth towards the sun, 42km/s, as in (\ref{eqn:E6}). } ($\alpha=5.2^{hr},
\delta=-67^0$)\footnote{The opposite direction is not easily excluded due to errors within the data, and so
should also be considered as possible. A new experiment will be capable of more accurately determining the
speed and direction, as well as the fractal structure of 3-space. The author is constructung a more compact
version of the Torr-Kolen - DeWitte coaxial cable RF travel-time experiment. New experimental techniques have
been developed to increase atomic-clock based timing accuracy and stability, so that shorter cables can be used,
which will permit 3-arm devices.}, and with speed
$420\pm 30$km/s. This local absolute motion is different from the CMB motion, in the direction
($\alpha=11.20^{hr},
\delta=-7.22^0$) with speed of
$369$km/s, for that would have given the data a totally different sidereal time signature, namely the times for
maximum/ minimum would have been shifted by 6hrs. The CMB velocity is motion relative to the distant early
universe, whereas the velocity measured in the DeWitte and related experiments is the velocity relative to the
local space. The declination of the velocity observed in this DeWitte experiment cannot be determined from the
data as only three days of data are available. However assuming exactly the same declination as Miller the
speed observed by DeWitte appears to be also in excellent agreement with the Miller speed. The dominant
effect in Fig.\ref{fig:DeWittetimes} is caused by the rotation of the earth, namely that the orientation of the
coaxial cable with respect to the direction of the flow past the earth changes as the earth rotates. This effect
may be approximately unfolded from the data, leaving the gravitational waves shown in Fig.\ref{fig:fractal}.
This is the first evidence that the velocity field describing 3-space has a complex structure, and is indeed
fractal.
The fractal structure, i.e. that there is an
intrinsic lack of scale, to these speed fluctuations is demonstrated by binning the absolute speeds $|v|$
and counting the number of speeds $p(|v|)$ within each bin. A least squares fit of the Log-Log plot to a
straightline was then made. Plotting Log$[p(|v|)]$ vs
Log$[|v|]$, as shown in Fig.\ref{fig:powerlaw}, we see that the fit gives
$p(v) \propto |v|^{-2.6}$. With the new experiment considerably more data will become available.
\begin{figure}[t]
\hspace{20mm}\includegraphics[scale=0.31]{Cahillpowerlawcut8.eps}
\caption{\small{Shows that the velocity fluctuations in Fig.\ref{fig:fractal} are scale free, as the
probability distribution from binning the speeds has the form $p(v) \propto |v|^{-2.6}$. This plot shows
Log$[p(v)]$ vs Log$[|v|]$. This shows that
the velocity field has a fractal structure, and so requiring the generalisation of the Schr\"{o}dinger
equation, as discussed herein, and also the Maxwell and Dirac equations (to be discussed elsewhere).}
\label{fig:powerlaw}}\end{figure}
\begin{figure}[t]
\hspace{7mm}\includegraphics[scale=1.6]{Cahill_GPB_3.eps}
\caption{\small{ Predicted variation of the precession angle $\Delta \Theta=|\Delta {{\bf S}}(t)|/|{\bf S}(0)|$, in
arcsec, over one 97 minute GP-B orbit, from the vorticity induced by the translation of the earth, as given by
(\ref{eqn:precession}). Predictions are for the months of April, August, September and February, labeled by
increasing dash length. The GP-B expected angle measurement accuracy is 0.0005 arcsec. }
\label{fig:GPB}}\end{figure}
\section{Observing 3-Space Vorticity \label{section:GPB}}
The vorticity effect in (\ref{eqn:equiv10}) can be studied experimentally in the Gravity Probe B (GP-B)
gyroscope satellite experiment in which
the precession of four on-board gyroscopes has been measured to unprecedented
accuracy \cite{GPB,Schiff}. In a generalisation of (\ref{eqn:E1}) \cite{Book} the vorticity
$\nabla\times{\bf v}$ is generated by matter in motion through the 3-space, where here ${\bf v}_R$ is the
absolute velocity of the matter relative to the local 3-space.
\begin{equation}\nabla \times(\nabla\times {\bf v}) =\frac{8\pi G\rho}{c^2}{\bf v}_R,
\label{eqn:CG4b}\end{equation}
We then obtain from
(\ref{eqn:CG4b}) the vorticity (ignoring homogeneous vortex solutions)
\begin{equation}
\vec{\omega}({\bf r},t)
=\frac{2G}{c^2}\int d^3 r^\prime \frac{\rho({\bf r}^\prime,t)}
{|{\bf r}-{\bf r}^\prime|^3}{\bf v}_R({\bf r}^\prime,t)\times({\bf r}-{\bf r}^\prime).
\label{eqn:omega}\end{equation}
For the smaller earth-rotation induced vorticity effect
${\bf v}_R({\bf r})={\bf w}\times{\bf r}$ in (\ref{eqn:omega}), where ${\bf w}$ is the angular
velocity of the earth, giving
\begin{equation}
\vec{\omega}({\bf r})_{rot}=4\frac{G}{c^2}\frac{3({\bf r}.{\bf L}){\bf r}-r^2{\bf L}}{2 r^5},
\label{eqn:rotation}\end{equation}
where ${\bf L}$ is the \index{angular momentum - earth} angular momentum of the earth, and ${\bf
r}$ is the distance from the centre.
In general the vorticity term in
(\ref{eqn:equiv10}) leads to an apparent `torque', according to a distant observer, acting on the angular
momentum
${\bf S}$ of the gyroscope,
\begin{equation}
\vec{\tau}= \int d^3 r \rho({\bf r})\; {\bf r}\times(\vec{\omega}({\bf r}) \times{\bf v}_R({\bf r})),
\label{eqn:torque1}\end{equation}
where $\rho$ is its density, and where now
${\bf v}_R$ is used here to describe the motion of the matter forming the gyroscope relative to the local
3-space. Then
$d{\bf S}=\vec{\tau}dt$ is the change in
${\bf S}$ over the time interval $dt$. For a gyroscope
${\bf v}_R({\bf r})={\bf s}\times{\bf r}$, where ${\bf s}$ is the angular velocity of the gyroscope.
This gives
\begin{equation}
\vec{\tau}=\frac{1}{2}\vec{\omega}\times{\bf S}
\label{eqn:torque2}\end{equation}
and so $\vec{\omega}/2$ is the instantaneous angular velocity of precession of the gyroscope, which is thus
equal to the instantaneous angular velocity of 3-space, also relative to a distant observer. The component of
the vorticity in (\ref{eqn:rotation}) has been determined from the laser-ranged satellites LAGEOS(NASA) and
LAGEOS 2(NASA-ASI)
\cite{Ciufolini}, and the data implies the indicated coefficient on the RHS of (\ref{eqn:CG4b}) to $\pm10\%$.
For GP-B the direction of ${\bf S}$ has been chosen so that this precession is cumulative and, on averaging
over an orbit, corresponds to some $7.7\times 10^{-6}$ arcsec per orbit, or 0.042 arcsec per year. GP-B has
been superbly engineered so that measurements to a precision of 0.0005 arcsec are possible.
However for the earth-translation induced precession if we use $v_R = 430$ km/s (in the
direction $\mbox{RA} =5.2^{hr}$, $\mbox{Dec} =-67^0$), (\ref{eqn:omega}) gives
\begin{equation}
\vec{\omega}({\bf r})_{trans}=\frac{2GM}{c^2}\frac{{\bf v}_R\times{\bf r}}{r^3},
\label{eqn:AMomega}\end{equation}
and then the total vorticity is $\vec{\omega}=\vec{\omega}_{rot}+\vec{\omega}_{trans}$.
The maximum magnitude of the speed of this precession component is $\omega_{trans}/2=gv_C/c^2=8
\times10^{-6}$arcsec/s,
where here
$g$ is the usual gravitational acceleration at the altitude of the satellite. This precession has a different
signature: it is not cumulative, and is detectable by its variation over each single orbit, as its orbital
average is zero, to first approximation. Fig.\ref{fig:GPB} shows $\Delta \Theta=|\Delta {{\bf
S}}(t)|/|{\bf S}(0)|$ over
one orbit, where,
\begin{equation}\Delta {{\bf S}}(t) =
\int_0^t dt^\prime \frac{1}{2}\vec{\omega}({\bf r}(t'))_{trans} \times {\bf S}(t^\prime)
\approx \left(\int_0^t dt^\prime \frac{1}{2}\vec{\omega}({\bf r}(t'))_{trans}\right) \times
{\bf S}(0).
\label{eqn:precession}\end{equation}
Here $\Delta {{\bf S}}(t)$ is the integrated change in spin, and where
the approximation arises because the change in
${\bf S}(t^\prime)$ on the RHS of (\ref{eqn:precession}) is negligible. The plot in Fig.\ref{fig:GPB} shows
this effect to be some 30$\times$ larger than the expected GP-B errors, and so easily detectable, if it exists as
predicted herein.
Essentially then these spin precessions are caused by the rotation of the `wavepackets' describing the matter
forming the gyroscopes, and caused in turn by the vorticity of 3-space. The above analysis shows that
the rotation is exactly the same as the rotation of the 3-space itself, just as the acceleration of
`matter' was exactly the same as the acceleration of the 3-space. We this obtain a much clearer insight
into the nature of motion, and which was not possible in the spacetime formalism.
\section{Conclusions\label{section:conclusions}}
We have seen herein that the new theory of 3-space has resulted in a number of fundamental developments, namely that a
complex `quantum foam' dynamical 3-space exists and has a fractal `flow' structure, as revealed most clearly by the
extraordinary DeWitte coaxial-cable experiment. This fractal structure requires that the fundamental equations of
physics be generalised to take account of, for the first time, the physics of this 3-space and, in
particular, here the inclusion of that dynamics within the dynamics of quantum systems. We saw that the
generalisation of the Schr\"{o}dinger equation is unique, and that from an Ehrenfest wavepacket analysis we
obtained the equivalence principle, with the acceleration of `matter' being shown to be identical to the
acceleration of the 3-space; which while not unexpected, is derived here for the first time. This result shows
that the equivalence principle is really a quantum-theoretic effect. As well we obtained by that same
analysis that any vorticity in the 3-space velocity field will result in a corresponding rotation of
wavepackets, and just such an effect is being studied in the GP-B gyroscope experiment. So for the first time
we see that the original Schr\"{o}dinger equation actually lacked a key dynamical ingredient. As well because
the 3-space is fractal the generalised Schr\"{o}dinger equation now contains a genuine element of stochasticity.
This research is supported by an Australian Research Council Discovery Grant.
\newpage
|
hep-th/0510178
|
\section{Introduction}
We will construct partition functions of conformal field theories with central
charge which is a multiple of $24$. Our construction is based on a unique modular function,
the so called $j$-invariant (or hauptmodule). The properties of this function guarantee (at least at modular level)
that any integer power of it will be again a modular function and this is of key importance in our construction.\\
Moreover, we report on a possible extension of a Monster moonshine, that relates
coefficients in a $q$-expansion of $j$-invariant function and dimensions of the irreducible
representations of the Monster sporadic group.
We are going to use lattices, or more precisely their $\Theta$-functions to describe a given CFT.
The $q$-expansion of a $\Theta$-function of a lattice $\Lambda$ is given as
\begin{eqnarray} Z_{\Lambda}=\sum_{x\in\Lambda}N(m)q^{m}\,,\end{eqnarray}
where we sum over all vectors $x$, in the lattice $\Lambda$, with a length $m=x\cdot x$. $N(m)$
is the number of vectors of norm $m$ and $q\equiv e^{i\pi\tau}$ in terms of the modular parameter $\tau$.
The spectra of meromorphic conformal field theories can be expressed in terms of partition functions of even self-dual lattices.
This means that the exponent $m$ in the $q$-expansion will be necessarily an even number.
Both self-duality and evenness of a lattice correspond to invariance of a partition function $\mathcal{Z}$,
closely related to $Z_{\Lambda}$, under the generators $S$ and $T$ of a modular group $SL(2,\mathds{Z})$.\\
Formally, $Z_{\Lambda}$ of a $d$ dimensional lattice $\Lambda$ is a modular form of weight $d/2$. A
partition function $\mathcal{Z}$ of a lattice is defined as follows
\begin{eqnarray}\label{partdef}\mathcal{Z}=Z_{\Lambda}/\eta^{d/2}\,,\end{eqnarray}
where $\eta(q)=q^{1/12}\prod_{m=1}^{\infty}(1-q^{2m})$
is Dedekind $\eta$-function which is a modular form of weight $1/2$.
Partition functions of a all the 24 dimensional even self-dual lattices (the Niemeier lattices)
can be written as
\begin{eqnarray}\label{part1}\mathcal{Z}=\left[J+24(h+1)\right]\eta^{24}\,,\end{eqnarray}
where $h$ is the Coxeter number of a given lattice. For example $h=0$ corresponds to the famous
Leech lattice, $h=30$ to the Niemeier lattice based on a root system of $E_{8}^{3}$, etc.
Physically $24(h+1)$ corresponds to a number of massless states in a given theory.\\
Using a technique presented in \cite{Jankiewicz:2005rx} one can choose any lattice $\Lambda_{1}$
to generate the $\Theta$-function of another lattice $\Lambda_{2}$. In the same paper it was shown
that it is possible to generate a partition function of any extremal lattice (by extremal,
we mean the lattice that has the tightest packing in a given dimension\footnote{If such a lattice exists.})
by taking the $k^{th}$ power of (\ref{part1}) and treating $x_{i}=24(h_{i}+1)$ (where $i=1,...,k$) as a free parameter.
\section{CFTs with $c=24k$: Systematic Approach}
Different choices of constant parameters $x_{i}s$ correspond to
different $\Theta$-functions. In principle one can use the same technique to find corresponding extremal
partition functions, that are related by (\ref{partdef}). One can write them as $q$-expansions of the form
\begin{eqnarray}\prod_{i=1}^{k}(J+24+x_{i})=\frac{1}{q^{2k}}\left[1+\sum_{m=(k-1)}^{\infty}f_{2m}(x_{1},...,x_{k})q^{2m-2k}\right]\end{eqnarray}
Here we want to focus on two choices of these parameters that lead to interesting families of partition functions,
that are motivated by both physics and mathematics.\\
The first choice is reminescent of the one introduced in the examples presented in the previous section,
namely the choice of $k-1$ parameters $x_{i}$s such that the lattice has densest possible packing in a given dimension.
More precisely, thanks to this parametrization, one eliminates coefficients of terms with negative powers in the
$q$-expansion that correspond (in a field theoretic language) to tachyonic states. In this setup we are left with only one free parameter $x_{k}$.
Different choices of $x_{k}$ would correspond to different partition functions of candidates for conformal field
theories with $c=24\cdot k$. Here we list the first three cases:
\begin{subequations}
\begin{align}
&\mathcal{G}_{1}(x_{1})\!=\!\frac{1}{q^{2}}+(24+x_{1})+196884q^{2}+...\\
&\mathcal{G}_{2}(x_{2})\!=\!\frac{1}{q^{4}}+(393192-48x_{2}-x_{2}^{2})+42987520q^{2}+...\\ \label{tach}
&\mathcal{G}_{3}(x_{3})\!=\!\frac{1}{q^{6}}+(50319456-588924x_{3}+72x_{3}^{2}+x_{3}^{3})+2592899910q^{2}+...
\end{align}\end{subequations}
Notice that in each case all of the tachyonic states (except the lowest one) are absent. Since the allowed
values \cite{Harvey:1988ur} of the coefficient of $q^{0}$ are integers that run from zero to the value of the
$q^{2}$ coefficient, one can easily find the number of ``allowed'' partition functions in $24\cdot k$ dimensions.\\
There exists an interesting alternative $x$-parametrization \cite{Apostol} of the partition functions
\begin{subequations}
\begin{align}
&\mathcal{H}_{1}=\frac{1}{q^{2}}+196884q^{2}+...\\
&\mathcal{H}_{2}=\frac{1}{q^{4}}+1+42987520q^{2}+...\\
&\mathcal{H}_{3}=\frac{1}{q^{6}}+\frac{1}{q^{2}}+1+2593096794q^{2}+...
\end{align}
\end{subequations}
Here we fix the tachyon levels, i.e., the levels with $q^{m}$ where $m<0$, and the massless level, i.e., $q^{0}$,
by appropriate choices of the $x$s, so that each level except the $(k-1)$th one contains a single state.
This parameterization is interesting since the first nontrivial coefficient corresponds to the characters of the
extremal vertex operator algebra of rank $24\cdot k$.
\section{Monster Moonshine and its Extension}
The extremal 24 dimensional case has been shown to be related to the Fischer-Griess monster group.
In mathematics this fact is known as Monster moonshine (\cite{Dolan:1989kf} and \cite{Borch}).
One can evaluate $\mathcal{G}_{1}$ at $x_{1}=-24$ which corresponds to the $j$-invariant to find
\begin{eqnarray}j=\frac{1}{q^{2}}+196884q^{2}+ 21493760q^{4} + 864299970q^{6} + 20245856256q^{8}+...\,.\end{eqnarray}
\noindent
The coefficients of this expansion decompose into dimensions of the irreducible representations of the
Monster\footnote{for explicit realization of the Monster moonshine see \cite{Jankiewicz:2005rx}.}, where we use the notation $j=\frac{1}{q^{2}}+j_{2}q^{2}+j_{4}q^{4}+...\,$.
Following this interpretation of the Monster moonshine theorem, one can easily generalize it to
higher dimensional cases, i.e., one can express coefficients of any partition functions,
for example $\mathcal{G}_{k}(x_{k})$ or $\mathcal{H}_{k}$, for any choice of $k$, in terms of the dimensions of
irreducible representations of the Monster group. We present the results in Table-\ref{tab-mon3},
where coefficients of both $G_{k}(x_{k})$ and $\mathcal{H}_{k}$ are expressed in terms of the coefficients of the invariant function $j$,
that (via the original Monster moonshine) are related to the Monster.
We notice that the coefficients the $g_{2n}$ and $h_{2n}$ fall into patterns with period $k!$.
We conjecture that this periodicity also continues to hold for all $k$.
The polynomial conditions to be satisfied to find the
extremal partition functions for large $k$ become increasingly more
difficult to solve with increasing $k$, so we do not have results
for $k>6$.\\
Table-\ref{tab-mon3} give the general periodicity in coefficients $g_{2n}$ and $h_{2n}$ of, respectively, $\mathcal{G}_{k}(x_{k})$ and $\mathcal{H}_{k}$.
\begin{table}\label{tab-mon3}
{\scriptsize\begin{tabular}{|l|l||l|l|}
\hline
$k=2$ & $k=2$ & $k=2$ & $k=2$ \\ \hline
$g_{4i+2}$ & $2j_{2(4i+2)}$ & $h_{4i+2}$& $2j_{2(4i+2)}$ \\
$g_{4i+4}$ & $2j_{2(4i+4)}+j_{2(2i+2)}$ & $h_{4i+4}$& $2j_{2(4i+4)}+j_{2(2i+2)}$ \\ \hline
$k=3$ & $k=3$ & $k=3$ & $k=3$ \\ \hline
$g_{6i+2}$ & $3j_{3(6i+2)}$ & $h_{6i+2}$& $3j_{3(6i+2)}+j_{6i+2}$ \\
$g_{6i+4}$ & $3j_{3(6i+4)}$ & $h_{6i+4}$& $3j_{3(6i+4)}+j_{6i+4}$ \\
$g_{6i+6}$ & $3j_{3(6i+6)}+j_{2i+2}$ & $h_{6i+6}$& $3j_{3(6i+6)}+j_{2i+2}+j_{6i+6}$ \\ \hline
$k=4$ & $k=4$ & $k=4$ & $k=4$ \\ \hline
$g_{8i+2}$ & $4j_{4(8i+2)}$ & $h_{8i+2}$ & $4j_{4(8i+2)}+2j_{2(8i+2)}+j_{8i+2}$ \\
$g_{8i+4}$ & $4j_{4(8i+4)}+2j_{2(2i+4)}$ & $h_{8i+4}$ & $4j_{4(8i+4)}+2j_{2(2i+4)}+2j_{2(8i+4)}$ \\
& & & $+j_{8i+4}+j_{4i+2}$ \\
$g_{8i+6}$ & $4j_{4(8i+6)}$ & $h_{8i+6}$ & $4j_{4(8i+6)}+2j_{2(8i+6)}+j_{8i+6}$ \\
$g_{8i+8}$ & $4j_{4(8i+8)}+2j_{(8i+8)}$ & $h_{8i+8}$ & $4j_{4(8i+8)}+2j_{(8i+8)}+j_{2i+2}$ \\
& $+j_{2i+2}$ & & $+2j_{2(8i+8)}+j_{8i+8}+j_{4i+4}$ \\ \hline
$k=5$ & $k=5$ & $k=5$ & $k=5$ \\ \hline
$g_{10i+2}$ & $5j_{5(10i+2)}$ & $h_{12i+2}$ & $g_{12i+2}+3j_{3(12i+2)}+2j_{2(12i+2)}+j_{12i+2}$ \\
$g_{10i+4}$ & $5j_{5(10i+4)}$ & $h_{12i+4}$ & $g_{12i+4}+3j_{3(12i+4)}+2j_{2(12i+4)}$ \\
& & & $+j_{12i+4}+j_{6i+2}$ \\
$g_{10i+6}$ & $5j_{5(10i+6)}$ & $h_{12i+6}$ & $g_{12i+6}+3j_{3(12i+6)}+2j_{2(12i+6)}$ \\
& & & $+j_{12i+6}+j_{4i+2}$ \\
$g_{10i+8}$ & $5j_{5(10i+8)}$ & $h_{12i+8}$ & $g_{12i+8}+3j_{3(12i+8)}+2j_{2(12i+8)}$ \\
& & & $+j_{12i+8}+j_{6i+4}$ \\
$g_{10i+10}$ & $5j_{5(10i+10)}+j_{2i+2}$ & $h_{12i+10}$& $g_{12i+10}+3j_{3(12i+10)}+2j_{2(12i+10)}+j_{12i+10}$ \\
& & $h_{12i+12}$& $g_{10i+12}+3j_{3(12i+12)}+2j_{2(12i+12)}+j_{12i+12}$ \\
& & & $+j_{6i+6}+j_{4i+4}$\\
\hline
\end{tabular}}\caption{Periodicity of the coefficients $g_{n}$ for $c=24\cdot k$ extremal partition functions $\mathcal{G}_{k}$,
and for $h_{n}$ coefficients of characters of the extremal vertex operator algebras $\mathcal{H}_{k}$
in terms of coefficients the $j_{2n}$ of the modular function $j$}\label{tab-mon3}\end{table}
These results are somewhat reminiscent of Bott periodicity for the
stable homotopy of the classical groups. Here we are dealing with (the
equivalent of) increasing level algebras.
To summarize, when $k=1$ it is known via standard Monster
Moonshine that the coefficients of $j$ decompose into Monster representations \cite{Borch}.
The fact that all the higher $k$ coefficients also decompose into Monster representations indicates
that they have large symmetries containing the Monster and the fact that they have these symmetries may
indicate that they are related to $24k$ dimensional lattices.
\section{Conclusions}
Using the techniques presented in \cite{Jankiewicz:2005rx}, one can construct a large class of conformal
field theories with central charge that is a multiple of 24. We have demonstrated (or at least conjecture)
the possibility of a new realization of Monster moonshine. This is realized as a periodicity
in a pattern of coefficients in $q$-expansions of the extremal partition functions.
\section*{Acknowledgments}
MJ thanks NSF and the QTS4 organizers for travel support.
This work was supported in part by U.S. DoE grant \#~DE-FG05-85ER40226.
|
math/0510044
|
\section{Introduction}\label{wp-intro}
The enumeration of permutation classes, whose ancestry can be traced back to at least 1915 (MacMahon~\cite{m:ca}), has frequently been accomplished by beautiful arguments utilizing such diverse objects as Young tableaux, Dyck paths, and planar maps, to name only a few. Our concern herein is not with attractive proofs, but rather with systematic methods for solving the enumeration problem. We adopt a strict definition of systematic, insisting that the computations can be performed without any human interaction whatsoever. For the definition of enumeration, we follow Wilf~\cite{wilf:formula} and insist only on a polynomial time (in $n$) algorithm to compute the number of length $n$ permutations in the class. We refer to such an algorithm as a {\it Wilfian formula\/}. To date, four techniques with wide applicability have been introduced which satisfy these goals:
\begin{itemize}
\item generating trees,
\item enumeration schemes,
\item substitution decompositions,
\item the insertion encoding.
\end{itemize}
The major aim of this paper, carried out in Section~\ref{wp-wp}, is to extend the method of enumeration schemes so that it can enumerate a wider variety of permutation classes and describe the Maple package {\sc WilfPlus}, which can rigorously and automatically find these extended schemes. Before that, we briefly examine the other methods in Sections~\ref{wp-gt}--\ref{wp-simple} and review enumeration schemes in Section~\ref{wp-wilf}. Section~\ref{wp-nonex} contains examples of classes which lie beyond the reach of even our more powerful enumeration schemes, while Section~\ref{wp-ex} gives numerous examples which can be handled. First we describe permutation classes.
Two sequences of natural numbers are said to be {\it order isomorphic\/} if they have the same pairwise comparisons, so $9,1,6,7,2$ is order isomorphic to $5,1,3,4,2$. Every sequence $w$ of natural numbers without repetition is order isomorphic to a unique permutation that we denote by $\operatorname{st}(w)$, so $\operatorname{st}(9,1,6,7,2)=5,1,3,4,2$, which we shorten to $51342$. We say that $\operatorname{st}(w)$ is the {\it standardization\/} of $w$. We further say that the permutation $\pi$ {\it contains\/} the permutation $\beta$ if $\pi$ contains a subsequence that is order isomorphic to $\beta$, and in this case we write $\beta\le\pi$. For example, $391867452$ contains $51342$, as can be seen by considering the subsequence $91672$. A permutation is said to {\it avoid\/} another if it does not contain it.
A {\it permutation class\/} is a lower order ideal in the containment ordering, meaning that if $\pi$ is contained in a permutation in the class, then $\pi$ itself lies in the class. Permutation classes can be specified in terms of the minimal permutations not lying in the class, which we call the {\it basis\/} of the class. By this minimality condition, bases are necessarily {\it antichains\/}, meaning that no element of a basis is contained in another. Although there are infinite antichains of permutations (see Atkinson, Murphy, and Ru\v{s}kuc~\cite{amr:pwocsop} for constructions and references to earlier work), we restrict our attention to finitely based classes. Given a set of permutations $B$, we define $\operatorname{Av}(B)$ to be the set of permutations that avoid all of the permutations in $B$. Thus if $\mathcal{C}$ is a closed class with basis $B$ then $\mathcal{C}=\operatorname{Av}(B)$, and for this reason the elements of a permutation class are often referred to as {\it restricted permutations\/}. We let $s_n(B)$ denote the number of permutations of length $n$ in $\operatorname{Av}(B)$, and refer to $\sum_n s_n(B)x^n$ as the generating function of $\operatorname{Av}(B)$. For more information on permutation classes, the reader is referred to B\'ona's text~\cite{bona:book}.
Each of the four systematic approaches for permutation class enumeration has a natural notion of a ``state,'' and in each case if the class is such that only finitely states are needed then --- at least in principle --- these methods give a Wilfian formula for the number of length $n$ permutations in the class. For generating trees, the states are the labels of the isomorphic generating tree. The classes possessing a generating tree with only finitely many labels are characterized in Vatter~\cite{finlabel}; this characterization appears here as Theorem~\ref{finlabel}. For the insertion encoding, which associates a language to the permutation class, the natural notion of ``state'' is a state in the accepting automaton for the associated language. The classes that require only finitely many states (or in other words, the classes that correspond to regular languages) were characterized by Albert, Linton, and Ru\v{s}kuc~\cite{insertion}; their result appears here as Theorem~\ref{insertion}. For enumeration schemes the translation of ``state'' is ``ES$^+$-irreducible permutation'' (or, for Zeilberger's original schemes, ``ES-irreducible permutation''). Should a class contain only finitely many such permutations then {\sc WilfPlus} can automatically enumerate it. No characterization of these classes is known\footnote{Zeilberger~\cite{z:wilf} dismisses this by stating ``if we know beforehand that we are guaranteed to succeed, then it is not research, but doing chores.''}. Moreover, unlike the other methods, there are subclasses of classes with finite enumeration schemes which do not themselves have finite enumeration schemes\footnote{In fact, the set of all permutations, $\operatorname{Av}(\emptyset)$, has a finite enumeration scheme (shown in Figure~\ref{all-perms-fig} on page~\pageref{all-perms-fig}), while several examples of classes without finite enumeration schemes are given in Section~\ref{wp-nonex}.}, indicating that such a characterization may be too much to hope for. For substitution decompositions, simple permutations play the role of states. As with enumeration schemes, there is no known characterization of the classes that contain only finitely many simple permutations.
\begin{figure}[t]
\begin{center}
\begin{psmatrix}[colsep=20pt,rowsep=10pt]
\psframebox[linearc=10pt,cornersize=absolute]{
\parbox{1in}{
\begin{center}
\begin{footnotesize}
\begin{tabular}{c}finite\\ enumeration\\ scheme\end{tabular}
\end{footnotesize}
\end{center}
}
}
&
\psframebox[linearc=10pt,cornersize=absolute]{
\parbox{1in}{
\begin{center}
\begin{footnotesize}
\begin{tabular}{c}regular\\ insertion\\encoding\end{tabular}
\end{footnotesize}
\end{center}
}
}
&
\psframebox[linearc=10pt,cornersize=absolute]{
\parbox{1in}{
\begin{center}
\begin{footnotesize}
\begin{tabular}{c}finitely\\ many simple\\ permutations\end{tabular}
\end{footnotesize}
\end{center}
}
}
\\
\multispan{2}
\begin{psmatrix}[colsep=0pt,rowsep=10pt]
\rput{30}{$\bigcup$}&&\rput{330}{$\bigcup$}\\
%
&
\psframebox[linearc=10pt,cornersize=absolute]{
\parbox{1in}{
\begin{center}
\begin{footnotesize}
\begin{tabular}{c}finitely labeled\\generating tree\end{tabular}
\end{footnotesize}
\end{center}
}
}
&
\end{psmatrix}
\end{psmatrix}
\end{center}
\caption{A depiction of the applicability of the four systematic enumeration techniques}\label{hasse-enum}
\end{figure}
The classes that these techniques can automatically enumerate are related as shown in Figure~\ref{hasse-enum}, which is to say, they are not very closely related at all (this is established via a series of examples in Section~\ref{wp-nonex} and remarks in Sections~\ref{wp-gt} and \ref{wp-ie}). Care should be taken when reading one symbol in this diagram; while the inclusion from finitely labeled generating trees to finite enumeration schemes indicates an increase in the number of classes that can be counted, there is a corresponding decrease in information. Finitely labeled generating trees and regular insertion encodings show that a class has a rational generating function, while classes with only finitely many simple permutations have algebraic generating functions. It is not yet known what types of generating functions can arise from finite enumeration schemes, but they need not be algebraic. For example, $\operatorname{Av}(1234)$, which has a holonomic\footnote{%
A generating function is said to be {\it holonomic\/} (or synonymously in the univariate case, {\it $D$-finite\/}) if its derivatives span a finite dimensional subspace over $\mathbb{C}(x)$. This is equivalent to the corresponding sequence $s_n$ being holonomic (again synonymously in the univariate case, {\it $P$-recursive\/}), which means that there are polynomials $p_0,p_1,\dots p_k$ so that
$
p_k(n)s_{n+k}+p_{k-1}(n)s_{n+k-1}+\cdots +p_0(n)s_n=0.
$
} but non-algebraic generating function (see Gessel~\cite{gessel}), has a finite enumeration scheme\footnote{%
A more trivial example would be the class of all permutations.%
} (shown in Figure~\ref{1234-fig} on page~\pageref{1234-fig}).
It is natural to hope that finite enumeration schemes produce only holonomic sequences, but this hope remains unproven.
Perhaps the greatest loss of information occurs with Wilf-equivalence. Two classes are said to be Wilf-equivalent if they are equinumerous. Clearly taking the reverse of a class yields a Wilf-equivalent class, as does taking the inverse, and these two operations generate the dihedral group with eight elements\footnote{%
With the exception of substitution decompositions, these techniques are not invariant under the eight permutation class symmetries. To be precise, there are classes that cannot be handled with these methods, while their inverses can be handled easily. Thus the comment of Albert, Linton, and Ru\v{s}kuc~\cite{insertion} that ``this apparent asymmetry does represent a possible flaw of the insertion encoding in general'' applies equally well to enumeration schemes and generating trees.%
}.
However, many examples of non-trivial Wilf-equivalences have been observed, ranging from the fact every class defined by avoiding a single pattern of length three is Wilf-equivalent\footnote{%
The classical bijective proof of this result is due to Simion and Schmidt~\cite{ss:rp}. Zeilberger~\cite{z:snappy} gives a proof using a technique quite like enumeration schemes that generalizes to permutations of a multiset.%
}
to the theorem of Atkinson, Murphy, and Ru\v{s}kuc~\cite{amr:twostacks} that $\operatorname{Av}(1342)$ is Wilf-equivalent to the infinitely based class
$$
\operatorname{Av}(\{2(2m-1)416385\cdots (2m)(2m-3) : m=2,3,\dots\}).
$$
If two classes both have finitely labeled generating trees, regular insertion encodings, or finitely many simple permutations then, since we can compute their generating functions from this information, we can decide whether or not they are Wilf-equivalent. For enumeration schemes this issue is not so clear. Occasionally, as with the enumeration schemes pictured in Figure~\ref{chow-west-fig} (a) and (b) on page~\pageref{chow-west-fig}, the Wilf-equivalence of two classes can be easily deduced from their enumeration schemes, but we present several examples in Section~\ref{wp-ex} where such deductions do not readily present themselves.
\section{Generating trees}\label{wp-gt}
Generating trees were introduced by Chung, Graham, Hoggatt, and Kleiman~\cite{cghk:baxter} and became quite popular after a pair of articles by West~\cite{west:cat, west:trees}. The closely related {\it ECO (enumerating combinatorial objects) method\/} (see Barcucci, Del Lungo, Pergola, and Pinzani~\cite{eco:survey} for a survey) extends the notion of generating trees to other settings.
We say that the permutation $\sigma$ of length $n$ is a {\it child\/} of $\pi\in S_{n-1}$ if $\sigma$ can be obtained by inserting $n$ into $\pi$. This defines a rooted tree $T$ on the set of all permutations. The {\it pattern-avoidance tree\/} of $\operatorname{Av}(B)$, denoted by $T(B)$, is then the subtree of $T$ with nodes $\operatorname{Av}(B)$. For example, the first four levels of $T(132,231)$ are shown in Figure~\ref{F-132-231}.
\begin{figure}[t]
\begin{footnotesize}
\begin{center}
\psset{xunit=0.03in, yunit=0.02in}
\psset{linewidth=0.25\psxunit}
\begin{pspicture}(-5,5)(155,78)
\pscircle*(10,10){1\psxunit}
\pscircle*(30,10){1\psxunit}
\pscircle*(50,10){1\psxunit}
\pscircle*(70,10){1\psxunit}
\pscircle*(90,10){1\psxunit}
\pscircle*(110,10){1\psxunit}
\pscircle*(130,10){1\psxunit}
\pscircle*(150,10){1\psxunit}
\rput[c](10,5){$1234$}
\rput[c](30,5){$4123$}
\rput[c](50,5){$3124$}
\rput[c](70,5){$4312$}
\rput[c](90,5){$2134$}
\rput[c](110,5){$4213$}
\rput[c](130,5){$3214$}
\rput[c](150,5){$4321$}
\pscircle*(20,30){1\psxunit}
\pscircle*(60,30){1\psxunit}
\pscircle*(100,30){1\psxunit}
\pscircle*(140,30){1\psxunit}
\rput[r](17,30){$123$}
\rput[l](63,30){$312$}
\rput[r](97,30){$213$}
\rput[l](143,30){$321$}
\psline(10,10)(20,30)
\psline(30,10)(20,30)
\psline(50,10)(60,30)
\psline(70,10)(60,30)
\psline(90,10)(100,30)
\psline(110,10)(100,30)
\psline(130,10)(140,30)
\psline(150,10)(140,30)
\pscircle*(40,50){1\psxunit}
\pscircle*(120,50){1\psxunit}
\rput[c](34,54){$12$}
\rput[c](126,54){$21$}
\psline(20,30)(40,50)
\psline(60,30)(40,50)
\psline(100,30)(120,50)
\psline(140,30)(120,50)
\pscircle*(80,70){1\psxunit}
\rput[c](80,76){$1$}
\psline(40,50)(80,70)
\psline(120,50)(80,70)
\end{pspicture}
\end{center}
\end{footnotesize}
\caption{The first four levels of the pattern-avoidance tree $T(132,231)$}\label{F-132-231}
\end{figure}
A generating tree, on the other hand, is a rooted, labeled tree such that the labels of the children of each node are determined by the label of that node. Sometimes the labels of the tree are taken to be natural numbers, but this is not necessary and frequently inconvenient. One specifies a generating tree by supplying the label of the {\it root\/} (also sometimes called the {\it axiom\/}) and a set of {\it succession rules\/} (also referred to as {\it inductive steps\/}). For example, the complete binary tree may be given by
$$
\begin{array}{llcl}
\mbox{Root:} & (2)&&\\
\mbox{Rule:} & (2)&\leadsto &(2)(2).
\end{array}
$$
In order to enumerate the permutation class $\operatorname{Av}(B)$, we want to find a generating tree isomorphic (as a rooted tree) to $T(B)$. For example, consider $T(132,231)$. We may obtain a permutation in $\operatorname{Av}_n(132,231)$ by inserting $n$ either at the beginning or the end of any $\pi\in\operatorname{Av}_{n-1}(132,231)$, but nowhere in between, so $T(132,231)$ is isomorphic to the complete binary tree and thus to the generating tree given above. For a more complicated example we turn to $T(1234)$, first described by West~\cite{west:cat}. This tree is isomorphic to generating tree defined by
$$
\begin{array}{llcl}
\mbox{Root:} & (2,2)&&\\
\mbox{Rule:} & (s,t)&\leadsto &(2,t+1)(3,t+1)\cdots (s,t+1)(s,s+1)(s,s+2)\cdots (s,t)(s+1,t+t).
\end{array}
$$
While verifying this isomorphism is not difficult (consider the lexicographically first ascent and the lexicographically first occurrence of $123$ in the permutation), it is much harder to obtain the generating function for $\operatorname{Av}(1234)$ from this tree; for the details of this see Bousquet-M\'elou~\cite{bm:four}.
Let $T(B;\pi)$ denote the subtree of $T(B)$ that is rooted at $\pi$ and contains all descendants of $\pi$. In an isomorphism between $T(B)$ and a generating tree, every permutation of $\operatorname{Av}(B)$ is assigned a label. Clearly two permutations $\pi$ and $\sigma$ may be assigned the same label if and only if $T(B;\pi)$ and $T(B;\sigma)$ are isomorphic (again, as rooted trees). Thus each pattern-avoidance tree $T(B)$ is isomorphic to a canonical generating tree whose labels correspond exactly to the isomorphism classes of $\{T(B;\pi) : \pi\in\operatorname{Av}(B)\}$.
In particular, $T(B)$ is isomorphic to a finitely labeled generating tree if and only if the set of all principal subtrees $\{T(B;\pi) : \pi\in\operatorname{Av}(B)\}$ contains only finitely many isomorphism classes. When this occurs, $\operatorname{Av}(B)$ has a rational generating function which may be routinely computed using the transfer matrix method (see Stanley's text~\cite[Section 4.7]{stanley:ec1} for details). The finitely based classes for which this is possible are characterized by the following theorem.
\begin{theorem}[Vatter~\cite{finlabel}]\label{finlabel}
Let $\mathcal{C}$ be a finitely based permutation class. The pattern-avoidance tree of $\mathcal{C}$ is isomorphic to a finitely labeled generating tree if and only if $\mathcal{C}$ omits both a child of an increasing permutation and a child of a decreasing permutation.
\end{theorem}
For example, $T(132,231)$ satisfies the hypotheses of Theorem~\ref{finlabel} because it omits both $132$ (a child of the increasing permutation $12$) and $231$ (a child of $21$). A less trivial example is given by $T(123,3214,2143,15432)$, which arose in Klazar~\cite{k:growth}. The Maple package {\sc FinLabel} (described in \cite{finlabel} and available at \url{http://math.rutgers.edu/~vatter/}) can find the generating functions for classes satisfying Theorem~\ref{finlabel} completely automatically.
It is easy to see that the hypotheses of Theorem~\ref{finlabel} are necessary%
\footnote{Suppose, without loss, that $\mathcal{C}$ contains all children of every increasing permutation. Then for each $n$, $12\cdots n$ has $n+1$ children in the pattern-avoidance tree of $\mathcal{C}$, and thus no two of these nodes may share the same label.}%
. The other direction is proved by showing that every sufficiently long permutation is ``GT-reducible.'' Since GT-reducibility is a stronger condition than the ES-reducibility of enumeration schemes, every class with a finitely labeled generating tree has a finite enumeration scheme.
\section{The insertion encoding}\label{wp-ie}
The insertion encoding, recently introduced by Albert, Linton, and Ru\v{s}kuc~\cite{insertion}, is a correspondence between permutation classes and languages. With it, one may attack the enumeration problem with all the tools of formal language theory. Roughly, this correspondence associates to each permutation a word describing how that permutation evolved. At each stage until the desired permutation has been constructed, at least one open {\it slot\/} (represented by a $\diamond$) exists in the intermediate {\it configuration\/}, and to proceed to the next configuration we insert a new maximal entry into one of these slots. This insertion can occur in four possible ways:
\begin{itemize}
\item the slot can be filled (replacing a $\diamond$ by $n$),
\item the new entry can be inserted to the left of the slot (replacing a $\diamond$ by $n\,\diamond$),
\item the new entry can be inserted to the right (replacing a $\diamond$ by $\diamond\,n$), or
\item the slot can be divided into two slots with the new entry in between (replacing a $\diamond$ by $\diamond\,n\,\diamond$).
\end{itemize}
These operations are denoted by the symbols $\ie{f}$, $\ie{l}$, $\ie{r}$, and $\ie{m}$, respectively. Since each of these operations can be performed on any open slot at any stage, we subscript their symbols with the number of the slot they were applied to (read from left to right). For example, the permutation $31254$ has the insertion encoding $\ie{m}_1\ie{l}_2\ie{f}_1\ie{r}_1\ie{f}_1$ because its evolution is
$$
\begin{array}{c}
\diamond\\
\diamond\,1\,\diamond\\
\diamond\,12\,\diamond\\
312\,\diamond\\
312\,\diamond\,4\\
31254
\end{array}
$$
Let $\sb(k)$ denote the permutation class whose basis consists of all length $2k+1$ permutations of the form $babab\cdots bab$ where the $a$'s represent the elements $\{1,2,\dots,k\}$ and the $b$'s represent the elements $\{k+1,k+2,\dots,2k+1\}$. These classes are called {\it slot bounded\/} because in the evolution of a permutation in $\sb(k)$ there are never more than $k$ open slots.
\begin{theorem}[Albert, Linton, and Ru\v{s}kuc~\cite{insertion}]\label{insertion}
The insertion encoding of a finitely based class is regular if and only if the class is a subclass of $\sb(k)$ for some $k$.
\end{theorem}
One can show using the Erd\H{o}s-Szekeres theorem~\cite{es:acpig} (or one can refer to the proof in \cite{insertion}) that Theorem~\ref{insertion} includes all of the classes identified by Theorem~\ref{finlabel} as having finitely labeled generating trees.
Even when the insertion encoding of a class is not regular, useful information can still be obtained by this correspondence. For example, Albert, Elder, Rechnitzer, Westcott, Zabrocki~\cite{1324} used regular approximations to the insertion encoding of $\operatorname{Av}(1324)$ to establish that $s_n(1324)>9.35^n$ for sufficiently large $n$, thereby disproving a conjecture of Arratia~\cite{arratia}. Additionally, Albert, Linton, and Ru\v{s}kuc~\cite{insertion} consider several classes with context-free insertion encodings and are able to obtain their (algebraic) generating functions from these languages. However, the derivation of insertion encodings is only automatic for subclasses of $\sb(k)$, and thus we choose to limit our focus to this case.
\section{Substitution decompositions}\label{wp-simple}
Substitution decompositions (also known as modular decompositions, disjunctive decompositions, and $X$-joins) have proven to be a useful technique in a wide range of settings, ranging from game theory to combinatorial optimization (see M\"ohring~\cite{m:aas} or M\"ohring and Radermacher~\cite{mr:sdd} for extensive references). Permutation class enumeration is no exception.
An {\it interval\/} (also called a {\it block\/}, or in other contexts, {\it factor\/}, {\it clan\/}, or even {\it convex subset\/}) in the permutation $\pi$ is an interval of indices $I=[a,b]$ such that the set of values $\{\pi(i) : i\in I\}$ also forms an interval. Clearly every permutation of length $n$ has $n$ trivial intervals of length one and one trivial interval of length $n$. A permutation that has no non-trivial intervals is called {\it simple\/} (the analogous term in other contexts is often {\it prime\/} or {\it primitive\/}).
Simple permutations first appear in the work of Atkinson and Stitt~\cite{as:wreath}, which is followed up by Albert and Atkinson~\cite{aa:simple}. Although in other contexts substitution decompositions are most often applied to algorithmic problems, they also have powerful enumerative applications. A class with only finitely many simple permutations has a recursive structure in which long permutations are built up from smaller permutations (their intervals). Thus it is natural to expect these classes to have algebraic generating functions, and this intuition is borne out by the following theorem.
\begin{theorem}[Albert and Atkinson~\cite{aa:simple}]\label{simple}
A permutation class with only finitely many simple permutations has an algebraic generating function.
\end{theorem}
The canonical example of a class with only finitely many simple permutations is $\operatorname{Av}(132)$. By considering the entries to the left and to the right of the $n$ in a permutation in $\operatorname{Av}_n(132)$ one simultaneously derives a decomposition of these permutations that leads immediately to the Catalan numbers and sees that this class contains no simple permutations of length three\footnote{Actually, there are no simple permutations of length three, $132$-avoiding or otherwise.} or longer.
Another example of a class with only finitely many simple permutations is the class of {\it separable permutations\/}. This class, first introduced by Bose, Buss, and Lubiw~\cite{bose:matching}, is essentially the permutation analogue of series-parallel posets (see Stanley~\cite{s:epg,stanley:ec1}) and complement reducible graphs (see Corneil, Lerchs, and Burlingham~\cite{clb:crg}). To define separable permutations we first need two binary operations on permutations. Given two permutations $\pi\in S_m$ and $\sigma\in S_n$ we define their {\it direct sum\/}, written $\pi\oplus\sigma$, by
$$
(\pi\oplus\sigma)(i)
=
\left\{
\begin{array}{ll}
\pi(i)&\mbox{if $i\in [m]$,}\\
\sigma(i-m)+m&\mbox{if $i\in[m+n]\setminus[m]$.}
\end{array}
\right.
$$
Similarly, we define their {\it skew sum\/}, $\pi\ominus\sigma$, by
$$
(\pi\ominus\sigma)(i)
=
\left\{
\begin{array}{ll}
\pi(i)+n&\mbox{if $i\in [m]$,}\\
\sigma(i-n)&\mbox{if $i\in[m+n]\setminus[m]$.}
\end{array}
\right.
$$
Given a class $\mathcal{C}$, we denote by $\operatorname{sc}(\mathcal{C})$ the {\it strong completion\/} of $\mathcal{C}$, which is the smallest class containing $\mathcal{C}$ such that both $\pi\oplus\sigma$ and $\pi\ominus\sigma$ lie in $\operatorname{sc}(\mathcal{C})$ for every $\pi,\sigma\in\operatorname{sc}(\mathcal{C})$.
The separable permutations are the strong completion of $\{1\}$. As was shown by Bose, Buss, and Lubiw~\cite{bose:matching}, this class can also be described as $\operatorname{Av}(2413,3142)$. The enumeration of this class (which can now be seen to follow routinely from Theorem~\ref{simple} and the fact that the only simple separable permutations are $1$, $12$, and $21$) was first undertaken by West~\cite{west:cat}. He used generating trees to show that the separable permutations are counted by the large Schr\"oder numbers. Later, Ehrenfeucht, Harju, ten Pas, and Rozenberg~\cite{ehpr:schroeder} (who also gave another proof that the basis of this class is $\{2413,3142\}$) presented a bijection between separable permutations and parenthesis words, the objects Schr\"oder was originally interested in counting.
One of the notable features of Theorem~\ref{simple} is that it does not seem to require the class to be finitely based. However, this is merely an illusion:
\begin{theorem}[Albert and Atkinson~\cite{aa:simple}, Murphy~\cite{maximillian}]\label{simplepwo}
A permutation class with only finitely many simple permutations is both finitely based and partially well-ordered\,\footnote{A partially ordered set is said to be partially well-ordered if contains neither an infinite strictly decreasing subsequence (which is never possible for a permutation class) nor an infinite antichain.}.
\end{theorem}
There is a semi-algorithm for establishing that a class contains only finitely many simple permutations. This semi-algorithm stems from the following theorem of Schmerl and Trotter~\cite{st:simple}, who proved it in the more general context of binary relational systems. Versions of the theorem for $2$-structures and $k$-structures are given by Ehrenfeucht and Rozenberg~\cite{er:ph2s} and Ehrenfeucht and McConnell~\cite{em:kgt}, repectively, and a proof for the special case of permutations can be found in Murphy's thesis~\cite{maximillian}.
\begin{theorem}[Schmerl and Trotter~\cite{st:simple}]\label{simple:hered}
Every simple permutation of length $n>2$ contains a simple permutation of length $n-1$ or $n-2$.
\end{theorem}
If the class $\mathcal{C}$ contains only finitely many simple permutations, then clearly there is an integer $n$ so that $\mathcal{C}$ does not contain any simple permutations of lengths $n-1$ or $n-2$. In the other direction, Theorem~\ref{simple:hered} shows that if we have found such an integer $n$ then $\mathcal{C}$ contains no simple permutations of length $n-2$ or longer. Therefore, when a class happens to contain only finitely many simple permutations, this fact can be verified automatically.
It remains an interesting open question if it is decidable whether a class contains only finitely many simple permutations.
\section{Zeilberger's original enumeration schemes}\label{wp-wilf}
Zeilberger~\cite{z:wilf} developed the notion of {\it enumeration schemes\/} and wrote the Maple package {\sc Wilf} to automate their discovery. Roughly, enumeration schemes are a divide and conquer technique which aims to partition the class into smaller pieces from which recurrences can be derived.
Take $\pi\in S_k$, suppose that $n\ge k$, and let $1\le i_1<i_2<\cdots<i_k\le n$. In Zeilberger's original formalization of enumeration schemes, we divide $\operatorname{Av}_n(B)$ into the sets
$$
A_\pi(n;B;i_1,i_2,\dots, i_k)=\{p\in\operatorname{Av}_n(B) : p(1)=i_{\pi(1)},\dots,p(k)=i_{\pi(k)}\}.
$$
In words, $A_\pi(n;B;i_1,i_2,\dots, i_k)$ is the set of $B$-avoiding length $n$ permutations that begin with the entries $i_1,i_2,\dots,i_k$, in the order specified by $\pi$. For example,
\begin{equation}
A_{312}(9;B;2,3,7)=\{723x_4x_5x_6x_7x_8x_9\in\operatorname{Av}_n(B)\}.
\label{z-set-example}
\end{equation}
In order to make enumeration schemes more closely resemble generating trees and the insertion encoding, we consider a symmetry of his approach. Everywhere Zeilberger mentions a permutation we consider its inverse. Thus we should specify the set of restrictions, $B$, a set of small entries of some length, $\pi$, and the positions in which the entries of $\pi$ occur. But instead of specifying the positions, we specify the gaps between the entries with a {\it gap vector\/}, ${\bf g}$. After performing these transformations, our version of \eqref{z-set-example} is
$$
Z(B;231;(1,0,3,2))=\{x_123x_4x_5x_61x_8x_9\in\operatorname{Av}_9(B)\},
$$
and in general we are concerned with the sets
$$
Z(B;\pi;{\bf g})=\{p\in\operatorname{Av}_{k+\|{\bf g}\|}(B) : p(g_1+1)=\pi(1),\dots,p(g_1+\cdots+g_{k}+k)=\pi(k)\},
$$
where $k$ is the length of $\pi$ and $\|{\bf g}\|$ denotes the sum of the components of ${\bf g}$.
Thus $Z(B;\pi;{\bf g})$ is the set of all $B$-avoiding permutations of length $k+\|{\bf g}\|$ whose least $k$ elements occur in the positions $g_1+1,g_1+g_2+2,\dots,g_1+g_2+\dots+g_k+k$ and form a $\pi$-subsequence.
Not all pairs $(\pi,{\bf g})$ result in a nonempty $Z$-set. Following Zeilberger, for a length $k$ permutation $\pi$ we define
$$
\mathcal{J}(\pi)=\{j\in[k+1] : Z(B;\pi;{\bf g})=\emptyset\mbox{ for all ${\bf g}$ with $g_j>0$}\}.
$$
Thus $Z(B;\pi;{\bf g})$ is guaranteed to be empty if ${\bf g}$ does not ``obey'' $\mathcal{J}(\pi)$, meaning that $g_j\neq 0$ for some $j\in\mathcal{J}(\pi)$.
For example, consider the case $B=\{132\}$. Then $2\in \mathcal{J}(12)$ because if $g_2>0$ then there is some entry between $1$ and $2$ in every permutation in $Z(B;\pi;{\bf g})$, and this gives a $132$-pattern. In order to check that $\mathcal{J}(12)=\{2\}$ we need merely observe that $312$ and $123$ avoid $132$. Our approach in this example can easily be generalized to compute $\mathcal{J}(\pi)$ for any $\pi$ and $B$.
\begin{proposition}\label{J-computable}
For any permutation $\pi$ and basis $B$, $\mathcal{J}(\pi)$ can be computed by inspecting the $B$-avoiding children of $\pi$.
\end{proposition}
\begin{proof}
Consider the vector ${\bf h}$ for which $h_i=0$ for all $i\neq j$ and $h_j=1$. If $Z(B;\pi;{\bf h})=\emptyset$ then $Z(B;\pi;{\bf g})=\emptyset$ for all ${\bf g}$ with $g_j>0$, so $j\in\mathcal{J}(\pi)$. If instead $Z(B;\pi;h)\neq\emptyset$ then $j\notin \mathcal{J}(\pi)$.
\end{proof}
For any $r\in[k]$, the set $Z(B;\pi;(g_1,\dots,g_{k+1}))$ embeds naturally (remove the entry $\pi(r)$ and standardize) into
\begin{equation}\label{embedding}
Z(B;\operatorname{st}(\pi-\pi(r));(g_1,\dots,g_{r-1},g_r+g_{r+1},g_{r+2},\dots,g_{k+1})),
\end{equation}
where $\pi-\pi(r)$ denotes the word obtained from $\pi$ by omitting the entry $\pi(r)$, so, for example, $51342-1=5342$. To make \eqref{embedding} easier to state, we define $\d_r(\pi)$ to be $\operatorname{st}(\pi-\pi(r))$ and let
$$
\d_r((g_1,\dots,g_{k+1}))=(g_1,\dots,g_{r-1},g_r+g_{r+1},g_{r+2},\dots,g_{k+1}).
$$
Sometimes the embedding of $Z(B;\pi;{\bf g})$ into $Z(B;\d_r(\pi);\d_r({\bf g}))$ is a bijection. If this is true for all gap vectors ${\bf g}$ that obey $\mathcal{J}(\pi)$, that is, that have $g_j=0$ for all $j\in\mathcal{J}(\pi)$, then we say that $\pi(r)$ is {\it enumeration-scheme-reducible for $\pi$ with respect to $B$\/}, or, for short, ES-reducible. (Zeilberger~\cite{z:wilf} refers to such entries as {\it reversely deleteable\/}.) We also say that a permutation with an ES-reducible entry is itself ES-reducible, and a permutation without an ES-reducible entry is ES-irreducible.
For example, suppose again that $B=\{132\}$ and consider the permutation $12$. We have already observed that $\mathcal{J}(12)=\{2\}$. Now we claim that the entry $1$ is ES-reducible. The gap vectors that obey $\mathcal{J}(12)$ are those of the form $(g_1,0,g_3)$, and thus we would like to verify that the embedding of $Z(\{132\};12;(g_1,0,g_3))$ into $Z(\{132\};1;(g_1,g_3))$ is a bijection.
Take $p\in Z(\{132\};1;(g_1,g_3))$ and consider inverting this embedding. In this case, that amounts to inserting the element $1$ into position $g_1+1$ and increasing all other entries of $p$ by $1$. Label the resulting permutation $p'$. For example, from ${\bf g}=(3,0,1)$ and $p=52314$ we obtain $p'=634125$.
We would like to show that $p'$ avoids $132$. To show this we consider all possible ways in which the new element $1$ could participate in a $132$-pattern. Clearly this entry must be the first entry in such a pattern. Now note that the $2$ in $p'$ cannot participate in this $132$-pattern, because the $1$ and $2$ are adjacent. But then there is a $132$-pattern in $p'$ which uses the $2$ instead of the $1$, and thus $p$ contains a $132$-pattern, a contradiction.
Thus we have shown that
$$
|Z_n(\{132\};12;(g_1,g_2,g_3))|
=
\case{0}{if $g_2>0$,}
{|Z_{n-1}(\{132\};1;(g_1,g_3))|}{if $g_2=0$.}
$$
Although this example did not demonstrate it, detecting and verifying ES-reducibility by hand can be enormously tedious. Fortunately, it is also unnecessary. By adapting the approach used in \cite{finlabel}, we arrive at the following test for ES-reducibility that can be routinely checked by computer%
\footnote{Zeilberger's approach in \cite{z:wilf} used what he referred to as ``logical reasoning,'' and while it is no less rigorous than this approach, Proposition~\ref{test-rd} has the advantage of being very explicit.}%
. In it we let $\|B\|_\infty$ denote the length of the longest permutation in $B$.
\begin{proposition}\label{test-rd}
The entry $\pi(r)$ of the permutation $\pi$ is ES-reducible if and only if
$$
|Z(B;\pi;{\bf g})|=|Z(B;\d_r(\pi);\d_r({\bf g}))|
$$
for all gap vectors ${\bf g}$ of the appropriate length that obey $J(\pi)$ and satisfy $\|{\bf g}\|\le\|B\|_\infty-1$.
\end{proposition}
\begin{proof}
If $\pi(r)$ is ES-reducible then the claim follows by definition. To establish the other direction, suppose that $\pi(r)$ is not ES-reducible, and choose ${\bf g}$ and $p\in Z(B;\d_r(\pi);\d_r({\bf g}))$ so that ${\bf g}$ obeys $\mathcal{J}(\pi)$ but $p$ cannot be obtained from a permutation in $Z(B;\pi;{\bf g})$ by removing $\pi(r)$ and standardizing.
First form the ($B$-containing) permutation $p'$ by incrementing each entry of $p$ that is at least $\pi(r)$ by $1$ and inserting $\pi(r)$ into position $g_1+\cdots+g_r+r$. Thus $p'$ is the permutation that would have mapped to $p$, except that $p'$ contains a pattern from $B$ and thus does not lie in $Z(B;\pi;{\bf g})$.
Now pick some $\beta\in B$ that is contained in $p'$, and choose a specific occurrence of $\beta$ in $p'$. Note that since $p=\operatorname{st}(p'-\pi(r))$ avoids $B$, this occurrence of $\beta$ must include the entry $\pi(r)$. Let $p''$ denote the standardization of the subsequence of $p'$ formed by all entries that are either in the chosen occurrence of $\beta$ or in $\pi$ (or in both), so $p''$ contains a $\beta$-pattern and lies in $Z(\emptyset;\pi;{\bf h})$ for some $\mathcal{J}(\pi)$-obeying ${\bf h}$ with $\|{\bf h}\|\le\|B\|_\infty-1$. On the other hand, $\operatorname{st}(p''-\pi(r))$ avoids $B$, which implies that $|Z(B;\d_r(\pi);{\bf h})|>|Z(B;\pi;{\bf h})|$, as desired.
\end{proof}
For example, consider the basis $B=\{132\}$ again. In order to show that the $1$ in $\pi=12$ is ES-reducible using this proposition, we first find that $\mathcal{J}(12)=\{2\}$ and then perform the following 10 computations:
\begin{footnotesize}
$$
\begin{array}{ccc}
{\bf g}&|Z(\{132\};12;{\bf g})|&|Z(\{132\};1;d_1({\bf g}))|\\\hline
(0,0,0)&1&1\\
(0,0,1)&1&1\\
(1,0,0)&1&1\\
(0,0,2)&1&1\\
(1,0,1)&2&2\\
(2,0,0)&2&2\\
(0,0,3)&1&1\\
(1,0,2)&3&3\\
(2,0,1)&5&5\\
(3,0,0)&5&5
\end{array}
$$
\end{footnotesize}
In a similar manner one can verify that the $1$ in $21$ is ES-reducible and that $\mathcal{J}(21)=\emptyset$. This gives the following enumeration scheme for $\operatorname{Av}(132)$:
\begin{eqnarray*}
s_n(132)
&=&
|Z(\{132\}; \emptyset; (n))|,
\\
|Z(\{132\}; \emptyset; (g_1))|
&=&
\sum_{i=0}^{g_1-1} |Z(\{132\}; 1; (i, g_1-i-1))|,
\\
|Z(\{132\}; 1; (g_1,g_2))|
&=&
\sum_{i=0}^{g_1-1} |Z(\{132\}; 21; (i, g_1-i-1, g_2))|
\\
&&
\quad+\sum_{i=0}^{g_2-1} |Z(\{132\}; 12; (g_1,i,g_2-i-1))|,
\\
&=&
\sum_{i=0}^{g_1} |Z(\{132\}; 1; (i, g_1+g_2-i-1))|.
\end{eqnarray*}
\section{Extending enumeration schemes}\label{wp-wp}
We will replace the sets $\mathcal{J}(\pi)$ in this section, giving us a more powerful version of enumeration schemes that can be found automatically with the Maple package {\sc WilfPlus}.
In order to motivate this change, we first consider a shortcoming of $\mathcal{J}(\pi)$. Let $B=\{1342,1432\}$. It can be shown easily, even by hand, that $12$ is ES-irreducible. To do so, first note that $\mathcal{J}(12)=\emptyset$, as witnessed by the permutations $312$, $132$, and $123$. Now consider the set $Z(B;12;(0,2,0))$. This set is empty, but removing $1$ gives the nonempty set $Z(B;1;(2,0))$ while removing $2$ gives the nonempty set $Z(B;1;(0,2))$. Indeed, this reasoning generalizes to show that all permutations of the form $\ominus^m 12$ are ES-irreducible, so $\operatorname{Av}(1342,1432)$ does not have a finite enumeration scheme, at least in Zeilberger's original sense.
Zeilberger's enumeration schemes fail in the previous example for a very simple reason: $\mathcal{J}(12)$ is too coarse to capture the fact that $(0,2,0)$ is not a valid gap vector for a $B$-avoiding descendant of $12$. We remedy this problem with the following definition.
\begin{definition}
The entry $\pi(r)$ of the length $k$ permutation $\pi$ is said to be {\it ES$^+$-reducible} if
\begin{eqnarray*}\label{wrd-def}
|Z(B;\pi;{\bf g})|=|Z(B;\d_r(\pi);\d_r({\bf g}))|
\end{eqnarray*}
whenever $Z_n(B;\pi;{\bf g})$ is nonempty. Further, we say that the permutation $\pi$ is ES$^+$-reducible if it contains an ES$^+$-reducible entry, and ES$^+$-irreducible otherwise.
\end{definition}
We then replace (for now) the set $\mathcal{J}$ by
$$
\mathcal{G}(\pi) = \{{\bf g} : Z(B;\pi;{\bf g})\neq\emptyset\}.
$$
Recall that for $\pi(r)$ to be ES-reducible, it had to satisfy (\ref{wrd-def}) for all ${\bf g}$ that contained $0$'s in the positions specified by $\mathcal{J}(\pi)$. Clearly if ${\bf g}\in\mathcal{G}(\pi)$ then ${\bf g}$ obeys $\mathcal{J}(\pi)$, but there can be gap vectors that obey $\mathcal{J}(\pi)$ and do not lie in $\mathcal{G}(\pi)$, as in our previous example with $B=\{1342,1432\}$. Thus we have obtained a weaker condition by requiring the satisfaction of (\ref{wrd-def}) less often.
The proof of Proposition~\ref{test-rd} carries over to this context to give the analogous result on testing for ES$^+$-reducibility.
\begin{proposition}\label{test-wrd}
The entry $\pi(r)$ of the permutation $\pi$ is ES$^+$-reducible if and only if
$$
|Z(B;\pi;{\bf g})|=|Z(B;\d_r(\pi);\d_r({\bf g}))|
$$
for all ${\bf g}\in\mathcal{G}(\pi)$ with $\|{\bf g}\|\le\|B\|_\infty-1$.
\end{proposition}
One can view $\mathcal{G}(\pi)$ as an lower order ideal\footnote{This means that ${\bf x}\in\mathcal{G}(\pi)$ whenever ${\bf x}\le {\bf y}$ for some ${\bf y}\in\mathcal{G}(\pi)$.} of $\mathbb{N}^{|\pi|+1}$ under the product order, where $(x_1,\dots,x_k)\le (y_1,\dots,y_k)$ if and only if $x_i\le y_i$ for all $i\in[k]$. Therefore we carry our definitions about permutation classes over to this context. In particular, we say that the {\it basis\/} of $\mathcal{G}(\pi)$ is the set of minimal vectors not in $\mathcal{G}(\pi)$, and if $B$ is a set of vectors then we write $\operatorname{Av}(B)$ to denote the set $\{{\bf g} : {\bf g}\not\ge\b\mbox{ for all }\b\in B\}$.
For example, let us compute the basis of $\mathcal{G}(12)$ when $B=\{1342,1432\}$. As already observed, $(0,2,0)$ does not lie in $\mathcal{G}(12)$. This gap vector is minimal in $\mathbb{N}^3\setminus \mathcal{G}(12)$ because $Z(B;12;(0,1,0))$ is nonempty. To show that the basis of $\mathcal{G}(12)$ is precisely $(0,2,0)$, it suffices to note that the permutation
$$
3\ 4\ \cdots\ (g_1+2)\ 1\ (g_1+3)\ 2\ (g_1+4)\ (g_1+5)\ \cdots\ (g_1+g_2+3)
$$
avoids $B$, so $(g_1,1,g_2)\in\mathcal{G}(12)$ for all $g_1,g_2\in\mathbb{N}$. Thus we have shown that $\mathcal{G}(12)=\operatorname{Av}((0,2,0))$.
Now that we have computed $\mathcal{G}(12)$, it is not hard to check that $2$ is ES$^+$-reducible for $12$. In order to do so we need to show that the embedding in question is a bijection for all gap vectors $(g_1,g_2,g_3)$ with $g_2\le 1$.
Suppose to the contrary that the embedding is not onto and take $p\in Z(B;1;(g_1,g_2+g_3))$ that is not mapped to. In other words, the permutation $p'$ obtained from $p$ by inserting $2$ into position $g_1+g_2+2$ and incrementing all the entries of $p$ of value at least $2$ contains a $1342$ or $1432$-pattern. Since $p$ avoids $\{1342,1432\}$, this pattern must involve the entry $2$. First, the $2$ cannot play the role of the ``$2$'' in such a pattern, because then the $1$ would be forced to play the role of the ``$1$,'' and there can be at most one entry between the $1$ and $2$ since $g_2\le 1$. The only other possible role for the $2$ is as the ``$1$,'' but in this case we could substitute the $1$, thereby finding a $B$-pattern in $p$, a contradiction.
Of course, we can replace $\mathcal{J}(\pi)$ by $\mathcal{G}(\pi)$ only if we are able to work with $\mathcal{G}(\pi)$. Because $\mathbb{N}^{|\pi|+1}$ is partially well-ordered by the product ordering\footnote{This fact, which is not difficult to prove, can be found many places, for example Nash-Williams~\cite{nw:wqo}.}, the basis of $\mathcal{G}(\pi)$, which is by definition an antichain, must be finite. However, this basis may be quite large, or may contain vectors with large components. For example, another way to state the Erd\H{o}s-Szekeres theorem~\cite{es:acpig} is that
$$
\mathcal{G}(\emptyset)=\operatorname{Av}(((j-1)(k-1)+1))
$$
when $B=\{12\cdots j, k\cdots 21\}$. While this does not preclude effective computation of $\mathcal{G}(\pi)$, it does suggest that such computations could be time consuming.
It happens that we can circumvent this problem by replacing $\mathcal{G}(\pi)$ by a different set of gap vectors. First we return to the way in which $\mathcal{G}(\pi)$ is used. By our definitions, if $\pi(r)$ is ES$^+$-reducible for $\pi$ then
\begin{eqnarray*}
|Z(B;\pi;{\bf g})|=
\case{0}{if ${\bf g}\notin\mathcal{G}(\pi)$,}
{|Z(B;\d_r(\pi);\d_r({\bf g}))|}{otherwise.}
\end{eqnarray*}
This equality shows that when enumerating the $B$-avoiding descendants of $\pi$ with gap vector ${\bf g}$, we first check to see if ${\bf g}$ lies in $\mathcal{G}(\pi)$. If ${\bf g}\notin\mathcal{G}(\pi)$ then we can be sure that no such descendants exist. If ${\bf g}\in\mathcal{G}(\pi)$, then $Z(B;\pi;{\bf g})$ is in one-to-one correspondence with $Z(B;\d_r(\pi);\d_r({\bf g}))$. Note that if $Z(B;\pi;{\bf g})$ and $Z(B;\d_r(\pi);\d_r({\bf g}))$ are both empty then they are trivially in one-to-one correspondence, so we could instead use the recurrence
\begin{eqnarray*}
|Z(B;\pi;{\bf g})|=
\case{0}{if ${\bf g}\notin\mathcal{G}(\pi)$ and $\d_r({\bf g})\in\mathcal{G}(\d_r(\pi))$,}
{|Z(B;\d_r(\pi);\d_r({\bf g}))|}{otherwise.}
\end{eqnarray*}
This equality shows that instead of considering $\mathcal{G}(\pi)$, we can look at the larger set of gap vectors for which either $Z(B;\pi;{\bf g})\neq\emptyset$ or $Z(B;d_r(\pi);d_r({\bf g}))=\emptyset$. Unfortunately, this set need not be an ideal%
\footnote{An example of this occurs with $B=\{231,k\cdots 21\}$. In Proposition~\ref{prop-chow-west} we observe that the set of ${\bf g}\in\mathbb{N}^3$ for which $Z(B;21;{\bf g})\neq\emptyset$ is $\operatorname{Av}((0,1,0),(k-2,0,0))$ while the set of ${\bf g}\in\mathbb{N}^2$ for which $Z(B;1;{\bf g})=\emptyset$ is $\operatorname{Av}((k-1,0))$, so the set of ${\bf g}\in\mathbb{N}^3$ for which either $Z(B;21;{\bf g})\neq\emptyset$ or $Z(B;1;\d_1({\bf g}))=\emptyset$ is $\operatorname{Av}((0,1,0))\setminus\{(k-2,0,0)\}$.},
so we consider the largest lower order ideal of $\mathbb{N}^{|\pi|+1}$ for which these conditions hold:
$$
\mathcal{G}_r(\pi)=\{{\bf g}\in\mathbb{N}^{|\pi|+1} : Z(B;\pi;{\bf h})\neq\emptyset\mbox{ or }Z(B;d_r(\pi);d_r({\bf h}))=\emptyset\mbox{ for all }{\bf h}\le{\bf g}\}.
$$
Note that $\mathcal{G}(\pi)\subseteq\mathcal{G}_r(\pi)$: if ${\bf g}\in\mathcal{G}(\pi)$ then $Z(B;\pi;{\bf g})\neq\emptyset$, so $Z(B;\pi;{\bf h})\neq\emptyset$ for all ${\bf h}\le{\bf g}$, so ${\bf g}\in\mathcal{G}_r(\pi)$. With this observation we have
\begin{eqnarray*}
|Z(B;\pi;{\bf g})|=
\case{0}{if ${\bf g}\notin\mathcal{G}_r(\pi)$,}
{|Z(B;\d_r(\pi);\d_r({\bf g}))|}{otherwise,}
\end{eqnarray*}
if $\pi(r)$ is ES$^+$-reducible for $\pi$ and $B$.
This new set has several advantages over $\mathcal{G}(\pi)$. For one, it may be considerably smaller, thus simplifying the scheme. More importantly, the following bound on basis elements implies that $\mathcal{G}_r(\pi)$ can be found automatically.
\begin{proposition}\label{Hbasis}
Suppose that $\pi(r)$ is an ES$^+$-reducible entry in $\pi$ (with respect to the permutation class with basis $B$). Then each basis vector $\b$ of $\mathcal{G}_r(\pi)$ satisfies $\|\b\|\le\|B\|_\infty-1$.
\end{proposition}
\begin{proof}
Suppose to the contrary that $\mathcal{G}_r(\pi)$ has a basis vector $\b$ with $\|\b\|\ge\|B\|_\infty$. Then $Z(B;\pi;\a)=\emptyset$ and $Z(B;\d_r(\pi);\d_r(\a))\neq\emptyset$ for some $\a\le\b$ since $\b\notin \mathcal{G}_r(\pi)$. However, because $\b$ is a basis vector for $\mathcal{G}_r(\pi)$, every $\a<\b$ lies in $\mathcal{G}_r(\pi)$, so
$Z(B;\pi;\a)\neq\emptyset$ or $Z(B;\d_r(\pi);\d_r(\a))=\emptyset$ for these vectors. If $Z(B;\d_r(\pi);\d_r(\a))=\emptyset$ for one of these vectors then $Z(B;\d_r(\pi);\d_r(\b))$ must also be empty, but this contradicts the fact that $\b\notin\mathcal{G}_r(\pi)$, so $Z(B;\pi;\a)\neq\emptyset$ for all $\a<\b$. Therefore $Z(B;\pi;\b)=\emptyset$ and $Z(B;\d_r(\pi);\d_r(\b))\neq\emptyset$.
Now, since $\pi(r)$ is ES$^+$-reducible and $Z(B;\pi;\a)\neq\emptyset$ for all $\a<\b$, we know that there is a bijection between $Z(B;\pi;\a)$ and $Z(B;\d_r(\pi);\d_r(\a))$ for all $\a<\b$. Thus we have the following diagram.
$$
\begin{psmatrix}[colsep=1.5cm,rowsep=1.5cm]
Z(B;\pi;\b)=\emptyset&Z(B;d_r(\pi);d_r(\b))\neq\emptyset\\
Z(B;\pi;\a)\neq\emptyset&Z(B;d_r(\pi);d_r(\a))\neq\emptyset
\psset{arrows=->,labelsep=3pt,nodesep=3pt}
\ncline{1,1}{2,1}
\ncline{1,2}{2,2}
\psset{arrows=<->}
\ncline{2,1}{2,2}
\end{psmatrix}
$$
The rest of the proof is similar to the proof of Proposition~\ref{test-rd}. Choose a permutation $p\in Z(B;\d_r(\pi);\d_r(\b))$ and form $p'$ by incrementing each entry of $p$ that is at least $\pi(r)$ by $1$ and inserting $\pi(r)$ into position $g_1+\cdots+g_r+r$. Choose a specific occurrence of some $\beta\in B$ in $p'$. Since $p$ avoids $B$, this occurrence of $\beta$ must involve the entry $\pi(r)$. Let $p''$ denote the standardization of the subsequence of $p'$ given by the entries from $\pi$ together with the entries from the chosen occurrence of $\beta$. Therefore $p''$ contains a $\beta$-pattern and lies in $Z(\emptyset;\pi;\a)$ for some $\a<\b$ with $\|\a\|\le\|B\|_\infty-1$. However, this is a contradiction because $\d_r(p'')$ avoids $B$ and thus lies in $Z(B;\d_r(\pi);\d_r(\a))$, and we have assumed that $\d_r$ is a bijection between $Z(B;\pi;\a)$ and $Z(B;\d_r(\pi);\d_r(\a))$.
\end{proof}
We conclude this section by writing out the enumeration scheme that we have derived for $\operatorname{Av}(1342,1432)$.
\begin{eqnarray*}
s_n(1342,1432)
&=&
|Z(\{1342,1432\}; \emptyset; (n))|,
\\
|Z(\{1342,1432\}; \emptyset; (g_1))|
&=&
\sum_{i=0}^{g_1-1} |Z(\{1342,1432\}; 1; (i, g_1-i-1))|,
\\
|Z(\{1342,1432\}; 1; (g_1,g_2))|
&=&
\sum_{i=0}^{g_1-1} |Z(\{1342,1432\}; 21; (i, g_1-i-1, g_2))|
\\
&&
\quad+\sum_{i=0}^{g_2-1} |Z(\{1342,1432\}; 12; (g_1,i,g_2-i-1))|,
\\
&=&
\sum_{i=0}^{g_1-1} |Z(\{1342,1432\}; 1; (i, g_1+g_2-i-1))|
\\
&&
\quad+2|Z(\{1342,1432\}; 1; (g_1,g_2-1))|.
\end{eqnarray*}
\section{A collection of failures}\label{wp-nonex}
In this section we collect numerous negative results needed to justify the lack of inclusions in Figure~\ref{hasse-enum}. The class $\operatorname{Av}(123)$ shows immediately that classes can have finite enumeration schemes without having regular insertion encodings. An example of a class with a regular insertion encoding but without a finite enumeration scheme is given by $\operatorname{Av}(1234,4231)$. It can be computed that this classes lies in $\sb(4)$, but the following proposition shows that it does not have a finite enumeration scheme.
\begin{proposition}\label{1234-4231-bad}
For all $k$, the permutation $k\cdots 21$ is ES$^+$-irreducible for $\operatorname{Av}(1234,4231)$.
\end{proposition}
\begin{proof}
Let $\pi=k\cdots 21$. First we show that the entries $\pi(r)=k-r+1$ for $r\in[k-1]$ are ES$^+$-irreducible. Let ${\bf g}$ denote the vector in $\mathbb{N}^{k+1}$ which is identically zero except for $g_{k-r+1}$ and $g_{k-r+2}$, which are both $1$ (these two components correspond to the gaps on either side of $\pi(k-r+1)$). Now observe that there are two permutations in $Z(\{1234,4231\};\d_r(\pi);\d_r({\bf g}))$:
$$
\begin{array}{ll}
\operatorname{st}(k(k-1)\cdots (k-r+2)(k+1)(k+2)(k-r)\cdots 21),&\mbox{and}\\
\operatorname{st}(k(k-1)\cdots (k-r+2)(k+2)(k+1)(k-r)\cdots 21),
\end{array}
$$
while $Z(\{1234,4231\};\pi;{\bf g})$ contains only one permutation,
$$
k(k-1)\cdots (k-r+2)(k+1)(k-r+1)(k+2)(k-r)\cdots 21.
$$
Thus ${\bf g}\in\mathcal{G}_r(\pi)$ but $Z(\{1234,4231\};\pi;{\bf g})$ and $Z(\{1234,4231\};\d_r(\pi);\d_r({\bf g}))$ are not in one-to-one correspondence, so $\pi(r)$ is not ES$^+$-reducible for any $r\in[k-1]$.
In order to show that $\pi(k)=1$ is not ES$^+$-reducible, consider the gap vector ${\bf g}=(0,0,\dots,0,3,0)$. Then $|Z(\{1234,4231\};\pi;{\bf g})|<|Z(\{1234,4231\};\d_k(\pi);\d_k({\bf g}))|$, finishing the proof.
\end{proof}
The substitution decomposition approach appears, at least on the surface, completely independent from the other three methods. The class $\operatorname{Av}(321,2341,3412,4123)$ has a finitely labeled generating tree by Theorem~\ref{finlabel} (and thus it also has a finite enumeration scheme and a regular insertion encoding), but it contains infinitely many simple permutations\footnote{These simple permutations can be defined as the standardizations of even-length initial segments of the sequence $4,1,6,3,8,5,\dots,2k+2,2k-1,\dots$.}.
Moreover, the separable permutations defined in Section~\ref{wp-simple}, which contain only three simple permutations, possess neither a finite enumeration scheme nor a regular insertion encoding. To see that they do not possess a regular insertion encoding, we need only note that they have an algebraic generating function (so, for this purpose, $\operatorname{Av}(132)$ would work just as well).
To establish that this class does not have a finite enumeration scheme, thereby completing our list of negative examples, we show that every permutation which consists of an increasing sequence followed by a decreasing sequence is ES$^+$-irreducible for $\operatorname{Av}(2413,3142)$. (This set forms a permutation class itself, with basis $\{213,312\}$.) Therefore, not only do the separable permutations not have a finite enumeration scheme, but for each $n$ there are $2^{n-1}$ ES$^+$-irreducible permutations. Moreover, one can observe (either from the definition or the basis) that this class is invariant under the eight permutation class symmetries, so none of these offer any simplification.
\begin{proposition}\label{wp:serparable:bad}
Every $\pi\in \operatorname{Av}_k(213,312)$ is ES$^+$-irreducible for $\operatorname{Av}(2413,3142)$.
\end{proposition}
\begin{proof}
Take $\pi\in\operatorname{Av}_k(213,312)$. First we show that $\pi(r)$ is not ES$^+$-reducible for any $2\le r\le k$. If $\pi(r-1)>\pi(r)$, consider the gap vector ${\bf g}$ with all components $0$ except for $g_{r}=g_{r+1}=1$. Then $Z(\{2413,3142\};\pi;{\bf g})$ contains at most one permutation, because to avoid $2413$, the smaller of these new entries must be to the left of the larger one. However, $Z(\{2413,3142\};d_r(\pi);d_r({\bf g}))$ contains $2$ permutations (the most possible). The case where $\pi(r-1)<\pi(r)$ can be handled similarly with the gap vector that is $0$ except for $g_{r-1}=g_r=1$. Finally, $\pi(1)$ is not ES$^+$-reducible because $Z(\{2413,3142\};\pi;(2,1,0,\dots,0))$ contains $5$ permutations -- every ordering of the new entries is allowed except $132$, because that would give rise to a $2413$-pattern -- while $Z(\{2413,3142\};d_1(\pi);(3,0,\dots,0))$ contains $6$ permutations.
\end{proof}
\section{An assortment of enumeration schemes successes}\label{wp-ex}
Here we present several examples of finite enumeration schemes. In these presentations, we adopt the following pictorial representation. If $\pi$ is ES$^+$-irreducible, so $|Z(B;\pi;{\bf g})|$ is computed by summing over the $B$-avoiding children of $\pi$, then we draw a solid arrow from $\pi$ to each of its children. If the entry $\pi(r)$ if ES$^+$-reducible in $\pi$ then we draw a dashed arrow from $\pi$ to $d_r(\pi)$, label this arrow with $d_r$, and indicate the basis of $G_r(\pi)$ beneath $\pi$. For example, the enumeration scheme for $\operatorname{Av}(1342,1432)$ is shown in Figure~\ref{schroeder-fig}. We also define the {\it depth\/} of an enumeration scheme to be the least integer $k$ so that every permutation of length at least $k$ is ES$^+$-reducible.
\begin{figure}[ht!]
\begin{footnotesize}
\begin{center}
\pstree[nodesep=3pt,treefit=loose,treesep=10pt,levelsep=50pt]{\TR[name=0]{$\emptyset$}}{
\pstree{\TR[name=1,edge={\ncline{->}}]{$1$}}{
\TR[name=12,edge={\ncline{->}}]{\basisnodeone{12}{(0,2,0)}}
\TR[name=21,edge={\ncline{->}}]{$21$}
\nccurve[linestyle=dashed,angleA=135,angleB=180]{->}{12}{1}
\ncput*{$d_2$}
\nccurve[linestyle=dashed,angleA=45,angleB=0]{->}{21}{1}
\ncput*{$d_2$}}}
\end{center}
\end{footnotesize}
\caption{The enumeration scheme for $\operatorname{Av}(1342,1432)$}\label{schroeder-fig}
\end{figure}
\begin{figure}[ht!]
\begin{footnotesize}
\begin{center}
\pstree[nodesep=3pt,treefit=loose,treesep=10pt,levelsep=50pt]{\TR[name=0]{$\emptyset$}}{
\TR[name=1,edge={\ncline{->}}]{$1$}
\nccurve[linestyle=dashed,angleA=45,angleB=315]{->}{1}{0}
\ncput*{$d_1$}}
\end{center}
\end{footnotesize}
\caption{The enumeration scheme for the set of all permutations, $\operatorname{Av}(\emptyset)$}\label{all-perms-fig}
\end{figure}
\begin{figure}[ht!]
\begin{footnotesize}
\begin{center}
\pstree[nodesep=3pt,treefit=loose,treesep=10pt,levelsep=50pt]{\TR[name=0]{$\emptyset$}}{
\TR[name=1,edge={\ncline{->}}]{\basisnodeone{1}{(a,b)}}
\nccurve[linestyle=dashed,angleA=45,angleB=315]{->}{1}{0}
\ncput*{$d_1$}}
\end{center}
\end{footnotesize}
\caption{The enumeration scheme for $\operatorname{Av}(M_{a,b})$}\label{mansour-1-fig}
\end{figure}
As can be seen in Figure~\ref{schroeder-fig}, we include the empty permutation, $\emptyset$, in our diagrams. Although this rarely has no more effect than making our diagrams consume more vertically space, there are a few exceptions. One is the set of all permutations, $\operatorname{Av}(\emptyset)$, which has the enumeration shown in Figure~\ref{all-perms-fig}. Another exception is $\operatorname{Av}(M_{a,b})$ where $M_{a,b}=\{\beta\in S_{a+b+1} : \beta(a+1)=1\}$. This class was first counted by Mansour~\cite{m:cont:and:avoid}. Its enumeration scheme is shown in Figure~\ref{mansour-1-fig}.
The scheme for $\operatorname{Av}(1234)$, which generates sequence \OEISlink{A005802} in the \href{http://www.research.att.com/\~njas/sequences/}{OEIS}~\cite{OEIS}, is shown in Figure~\ref{1234-fig}.
The $321$, hexagon-avoiding permutations, which can be defined as
$$
\operatorname{Av}(321,46718235,46781235,56718234, 56781234),
$$
were first introduced by Billey and Warrington~\cite{bw:321hex}, who showed how to compute the Kazhdan-Lusztig polynomials for them. Stankova and West~\cite{sw:321hex} proved that the number, $s_n$, of these permutations of length $n$ satisfies
$$
s_n=6s_{n-1}-11s_{n-2}+9s_{n-3}-4s_{n-4}-4s_{n-5}+s_{n-6}
$$
for all $n\ge 7$, which gives sequence \OEISlink{A058094} in the \href{http://www.research.att.com/\~njas/sequences/}{OEIS}~\cite{OEIS}. (So this complicated scheme produces only a sequence with a rational generating function.) Later, Mansour and Stankova~\cite{ms:321hex} counted $321$, $2k$-gon-avoiding permutations for all $k$.
\begin{figure}[t]
\begin{footnotesize}
\pstree[nodesep=3pt,treefit=loose,treesep=10pt,levelsep=50pt]{\TR[name=0]{$\emptyset$}}{
\pstree{\TR[name=1,edge={\ncline{->}}]{$1$}}{
\pstree{\TR[name=12,edge={\ncline{->}}]{$12$}}{
\TR[name=123,edge={\ncline{->}}]{\basisnodeone{123}{(0,0,0,1)}}
\TR[name=132,edge={\ncline{->}}]{$132$}
\pstree{\TR[name=312,edge={\ncline{->}}]{$312$}}
{%
\TR[name=3124,edge={\ncline{->}}]{\basisnodeone{3124}{(0,0,0,0,1)}}
\TR[name=4312,edge={\ncline{->}}]{$4312$}
\TR[name=3142,edge={\ncline{->}}]{$3142$}
\TR[name=3412,edge={\ncline{->}}]{$3412$}
}
}
{%
\pstree{\TR[name=21,edge={\ncline{->}}]{$21$}}
{
\TR[name=231,edge={\ncline[linestyle=none]}]{$231$}
}
}
\nccurve[linestyle=dashed,angleA=90,angleB=0]{->}{21}{1}
\ncput*{$d_2$}
\nccurve[linestyle=dashed,angleA=90,angleB=180]{->}{123}{12}
\ncput*{$d_3$}
\nccurve[linestyle=dashed,angleA=0,angleB=270]{->}{132}{12}
\ncput*{$d_3$}
\nccurve[linestyle=dashed,angleA=90,angleB=270]{->}{3412}{231}
\ncput*{$d_3$}
\nccurve[linestyle=dashed,angleA=90,angleB=180]{->}{3124}{312}
\ncput*{$d_4$}
\nccurve[linestyle=dashed,angleA=90,angleB=180]{->}{3142}{231}
\ncput*{$d_2$}
\nccurve[linestyle=dashed,angleA=150,angleB=255]{->}{4312}{312}
\ncput*{$d_2$}
\nccurve[linestyle=dashed,angleA=150,angleB=0]{->}{231}{12}
\ncput*{$d_3$}
}}
\end{footnotesize}
\caption{The enumeration scheme for $\operatorname{Av}(1234)$}\label{1234-fig}
\end{figure}
\begin{figure}[t]
\begin{footnotesize}
\pstree[nodesep=3pt,treefit=loose,treesep=5pt,levelsep=50pt]{\TR[name=0]{$\emptyset$}}{
\pstree{\TR[name=1,edge={\ncline{->}}]{$1$}}{
\pstree{\TR[name=12,edge={\ncline{->}}]{$12$}}
{%
\pstree{\TR[name=123,edge={\ncline{->}}]{$123$}}
{%
\TR[name=1234,edge={\ncline{->}}]{\basisnodetwo{1234}{(3,1,0,0,0)}{(4,0,0,0,0)}}
\TR[name=1243,edge={\ncline{->}}]{$1243$}
\TR[name=1423,edge={\ncline{->}}]{$1423$}
\pstree{\TR[name=4123,edge={\ncline{->}}]{$4123$}}
{%
\TR[name=41235,edge={\ncline{->}}]{%
\basisnodethree{41235}{(1,0,0,0,0,0)}{(0,2,1,0,0,0)}{(0,3,0,0,0,0)}}
\TR[name=41253,edge={\ncline{->}}]{$41253$}
\TR[name=41523,edge={\ncline{->}}]{$41523$}
\TR[name=45123,edge={\ncline{->}}]{$45123$}
}
}
\TR[name=132,edge={\ncline{->}}]{$132$}
\TR[name=312,edge={\ncline{->}}]{\basisnodeone{312}{(1,0,0,0)}}
}
\TR[name=21,edge={\ncline{->}}]{\basisnodeone{21}{(1,0,0)}}
\nccurve[linestyle=dashed,angleA=90,angleB=0]{->}{21}{1}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=45,angleB=135]{->}{132}{21}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=135,angleB=0]{->}{312}{12}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=90,angleB=180]{->}{1234}{123}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=30,angleB=180]{->}{1243}{132}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=45,angleB=225]{->}{1423}{312}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=135,angleB=315]{->}{41235}{1234}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=120,angleB=315]{->}{41253}{1243}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=135,angleB=0]{->}{41523}{1423}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=90,angleB=0]{->}{45123}{4123}
\ncput*{$d_1$}
}}
\end{footnotesize}
\caption{The enumeration scheme for the $321$, hexagon-avoiding permutations}\label{321-hex-fig}
\end{figure}
\begin{figure}[t]
\begin{footnotesize}
\pstree[nodesep=3pt,treefit=loose,treesep=10pt,levelsep=50pt]{\TR[name=0]{$\emptyset$}}{
\pstree{\TR[name=1,edge={\ncline{->}}]{$1$}}{
\pstree{\TR[name=12,edge={\ncline{->}}]{$12$}}{
\TR[name=123,edge={\ncline{->}}]{$123$}
\TR[name=132,edge={\ncline{->}}]{$132$}
\TR[name=312,edge={\ncline{->}}]{\basisnodeone{312}{(1,0,0,0)}}
}
{\pstree{\TR[name=21,edge={\ncline{->}}]{$21$}}{
\TR[name=213,edge={\ncline{->}}]{\basisnodeone{213}{(2,0,0,0)}}
\TR[name=231,edge={\ncline{->}}]{\basisnodeone{231}{(1,0,0,0)}}
\TR[name=321,edge={\ncline{->}}]{\basisnodetwo{321}{(1,0,0,0)}{(0,1,0,0)}}
}}}
\nccurve[linestyle=dashed,angleA=90,angleB=0]{->}{321}{21}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=155,angleB=225]{->}{231}{21}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=150,angleB=0]{->}{213}{12}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=60,angleB=330]{->}{312}{12}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=15,angleB=180]{->}{132}{21}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=90,angleB=180]{->}{123}{12}
\ncput*{$d_1$}
}
\end{footnotesize}
\caption{The enumeration scheme for the freely braided permutations, $\operatorname{Av}(3421,4231,4312,4321)$}
\label{freely-braided-fig}
\end{figure}
The {\it freely braided permutations\/} are the class
$$
\operatorname{Av}(3421,4231,4312,4321).
$$
This class was introduced by Green and Losonczy~\cite{gl:fb} and also arises in the work of Tenner~\cite{ten:rdp}. Mansour~\cite{mansour:fb} found that the freely braided permutations have the generating function
$$
\frac{1-3x-2x^2+(1+x)\sqrt{1-4x}}{1-4x-x^2+(1-x^2)\sqrt{1-4x}}.
$$
\OEIS{A108600} They also have an enumeration scheme of depth $3$, shown in Figure~\ref{freely-braided-fig}.
\begin{figure}[t]
\begin{tabular}{cc}
\subfigure[$\operatorname{Av}(321,23\cdots k1)$]{
\parbox{2in}{
\begin{footnotesize}
\pstree[nodesep=3pt,treefit=loose,treesep=10pt,levelsep=50pt]{
\TR[name=0]{$\emptyset$}}{
\pstree{\TR[name=1,edge={\ncline{->}}]{$1$}}{
\TR[name=12,edge={\ncline{->}}]{$12$}
\TR[name=21,edge={\ncline{->}}]{\basisnodetwo{21}{(1,0,0)}{(0,k-2,0)}}
\nccurve[linestyle=dashed,angleA=135,angleB=180]{->}{12}{1}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=45,angleB=0]{->}{21}{1}
\ncput*{$d_1$}}}
\end{footnotesize}
}
}
&
\subfigure[$\operatorname{Av}(231,k\cdots 21)$]{
\parbox{2in}{
\begin{footnotesize}
\pstree[nodesep=3pt,treefit=loose,treesep=10pt,levelsep=50pt]{
\TR[name=0]{$\emptyset$}}{
\pstree{\TR[name=1,edge={\ncline{->}}]{$1$}}{
\TR[name=12,edge={\ncline{->}}]{12}
\TR[name=21,edge={\ncline{->}}]{\basisnodetwo{21}{(0,1,0)}{(k-2,0,0)}}
\nccurve[linestyle=dashed,angleA=135,angleB=180]{->}{12}{1}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=45,angleB=0]{->}{21}{1}
\ncput*{$d_1$}}}
\end{footnotesize}
}
}
\\
\multicolumn{2}{c}{
\subfigure[$\operatorname{Av}(231,1k\cdots 32)$]{
\begin{footnotesize}
\pstree[nodesep=3pt,treefit=loose,treesep=10pt,levelsep=50pt]{
\TR[name=0]{$\emptyset$}}{
\pstree{\TR[name=1,edge={\ncline{->}}]{$1$}}{
\pstree{\TR[name=12,edge={\ncline{->}}]{$12$}}{
\TR[name=123,edge={\ncline{->}}]{123}
\TR[name=132,edge={\ncline{->}}]{\basisnodetwo{132}{(0,0,1,0)}{(0,k-3,0,0)}}
\TR[name=312,edge={\ncline{->}}]{\basisnodetwo{312}{(0,0,1,0)}{(0,1,0,0)}}
}
\TR[name=21,edge={\ncline{->}}]{\basisnodeone{21}{(0,1,0)}}
\nccurve[linestyle=dashed,angleA=90,angleB=0]{->}{21}{1}
\ncput*{$d_1$}
\nccurve[linestyle=dashed,angleA=90,angleB=180]{->}{123}{12}
\ncput*{$d_2$}
\nccurve[linestyle=dashed,angleA=30,angleB=285]{->}{132}{12}
\ncput*{$d_2$}
\nccurve[linestyle=dashed,angleA=45,angleB=0]{->}{312}{12}
\ncput*{$d_1$}}}
\end{footnotesize}
}}
\end{tabular}
\caption{Enumeration schemes for three Wilf-equivalent classes}\label{chow-west-fig}
\end{figure}
Chow and West~\cite{cw:cheby} showed using generating trees that the classes in Figure~\ref{chow-west-fig} are Wilf-equivalent, that their generating functions are rational\footnote{This fact can be verified quite quickly: by inverting the permutations in $\operatorname{Av}(321,23\cdots k1)$ one obtains the class $\operatorname{Av}(321,k12\cdots (k-1))$, which satisfies the hypotheses of Theorem~\ref{finlabel}, and thus also of Theorem~\ref{insertion}. Note, however, that while symmetries of these classes always have finitely labeled generating trees, and thus also regular insertion encodings, the complexity of their generating trees and insertion encodings increases with $k$, while the complexity of their enumeration schemes stay fixed.}, and that these generating functions can be expressed as quotients of Chebyshev polynomials. Krattenthaler~\cite{kratt:cheby} gave another proof via a bijection to Dyck paths (in fact, he found the bivariate generating functions for $132$-avoiding permutations by length and number of copies of $12\cdots k$ and for $123$-avoiding permutations by length and number of copies of $(k-1)\cdots 21k$). Around the same time as that work, several other authors studied these and similar classes: Jani and Rieper~\cite{jr:catalan}, Mansour and Vainshtein~\cite{mv:cheby1, mv:cheby2}, and Robertson, Wilf, and Zeilberger~\cite{rwz:fractions}. Deutsch, Hildebrand, and Wilf~\cite{dhw:lis} used these results in their study of the longest increasing subsequence problem for $132$-avoiding permutations.
For any fixed $k$, {\sc WilfPlus} can automatically and rigorously derive enumeration schemes for these classes. This makes it easy to make conjectures for the general form of these enumeration schemes, although they must be verified by hand. We carry this out for one of these classes below.
\begin{proposition}\label{prop-chow-west}
The enumeration scheme for $\operatorname{Av}(231,k\cdots 21)$ is as shown in Figure~\ref{chow-west-fig} (b).
\end{proposition}
\begin{proof}
Let $B=\{231,k\cdots 21\}$. First we verify the computations of $G_r(\pi)$ given in the diagram. For $12$, the diagram shows that $\mathcal{G}_1(12)=\mathbb{N}^3$. In any permutation in $Z(B;12;(g_1,g_2,g_3))$, the entries before the $2$ must be in decreasing order from left to right as any ascent before the $2$ would give rise to a $231$-pattern. Then, since these permutations must avoid $k\cdots 21$, there can be at most $k-2$ new entries before the $2$, so $(g_1,g_2,g_3)\notin \mathcal{G}(12)$ whenever $g_1+g_2\ge k-1$. The $B$-avoiding permutations
$$
g_1+g_2+2,\dots,g_2+4,g_2+3,1,g_2+2,\dots,4,3,2,g_1+g_2+3,g_1+g_2+4,\dots,g_1+g_2+g_3+2
$$
then show that $\mathcal{G}(12)$ is $\operatorname{Av}(\{(g_1,g_2,g_3) : g_1+g_2\ge k-1\})$. It can be shown similarly that $\mathcal{G}(1)=\operatorname{Av}((k-1,0))$. This shows that $\mathcal{G}_1(12)=\mathbb{N}^3$ because $Z(B;1;\d_1({\bf g}))$ is empty whenever $Z(B;12;{\bf g})$ is empty.
The computation for $21$ is similar. One first checks that
$$
\mathcal{G}(21)=\operatorname{Av}((0,1,0),(k-2,0,0)).
$$
Since $\mathcal{G}(1)=\operatorname{Av}((k-1,0))$, the set of gap vectors ${\bf g}$ for which either $Z(B;21;{\bf g})\neq\emptyset$ or $Z(B;1;\d_1({\bf g}))=\emptyset$ is $\operatorname{Av}((0,1,0))\setminus\{(k-2,0,0)\}$. This implies that $\mathcal{G}_1(21)=\operatorname{Av}((0,1,0),(k-2,0,0))$.
Now we must verify the ES$^+$-reducible entries. The $12$ case is clear because the only way $1$ can participate in a $231$ or $k\cdots 21$-pattern is as the minimal entry, and in either case the $2$ could play the same role. For $21$, first note that the $2$ also cannot participate in any $231$ or $k\cdots 21$-pattern if ${\bf g}\in\mathcal{G}_1(21)$: it cannot be the ``$2$'' in a $231$ because there can be nothing between the $2$ and the $1$, and it cannot be the ``$2$'' in a $k\cdots 21$-pattern because our restrictions on ${\bf g}$ prevent there from being enough entries to the left of the $2$ to accommodate such a pattern. This shows that the $2$ can only possibly be the minimal entry in a forbidden pattern, but then the $1$ could play the same role.
\end{proof}
In addition to the examples already mentioned, there are several other interesting classes avoiding a pair of permutations of length four\footnote{The characterization of the Wilf-equivalences between these classes has recently been completed by Le~\cite{le:length4}.}:
\begin{enumerate}
\item Kremer and Shiu~\cite{ks:len4} proved that there are four classes of this form counted by $(4^{n-1}+2)/3$. These all have finite enumeration schemes: $\operatorname{Av}(1234,2143)$ and $\operatorname{Av}(1432,2341)$ have schemes of depth 3, $\operatorname{Av}(2341,4321)$ has a scheme of depth 4, and $\operatorname{Av}(2143,4123)$ has a scheme of depth 6.
\item Kremer~\cite{k:sn} (see also Stanley~\cite[Exercise 6.39.l]{stanley:ec2}) completed the characterization of the classes defined by avoiding two permutations of length four that are counted by the large Schr\"oder numbers. Up to symmetry, there are ten such classes, seven of which have finite enumeration schemes:
\medskip
\begin{footnotesize}
\begin{center}
\begin{tabular}{l|l}
Class&Finite enumeration scheme?\\
\hline\hline
$\operatorname{Av}(1342,2341)$&Yes, of depth 3\\
$\operatorname{Av}(1342,1432)$&
\begin{minipage}[t]{2.2in}
Yes, of depth 2, shown in Figure~\ref{schroeder-fig}
\end{minipage}\\
$\operatorname{Av}(2341,2413)$&No\\
$\operatorname{Av}(2413,3142)$&
\begin{minipage}[t]{2.2in}
No, these are the separable permutations considered in Proposition~\ref{wp:serparable:bad}
\end{minipage}
\\
$\operatorname{Av}(2431,3241)$&No\\
$\operatorname{Av}(3241,3421)$&Yes, of depth 4\\
$\operatorname{Av}(3241,4231)$&
\begin{minipage}[t]{2.2in}
Yes, of depth 2 (Knuth~\cite{knuth1} proved that these are precisely the permutations that can be sorted by an input-restricted deque)
\end{minipage}
\\
$\operatorname{Av}(3412,3421)$&Yes, of depth 3\\
$\operatorname{Av}(3421,4321)$&Yes, of depth 2\\
$\operatorname{Av}(3421,4231)$&Yes, of depth 4
\end{tabular}
\end{center}
\end{footnotesize}
\item A permutation is said to be {\it skew-merged\/} if it is the union of an increasing subsequence and a decreasing subsequence. Stankova~\cite{stankova:fs} was the first to prove that the skew-merged permutations are given by $\operatorname{Av}(2143,3412)$. Later K\'ezdy, Snevily, and Wang~\cite{ksw:incdec} gave another proof of this result using F\"oldes and Hammer's characterization of split graphs~\cite{fh:sg}. Atkinson~\cite{a:skewmerged} showed that the generating function for this class is
$$
\frac{1-3x}{(1-2x)\sqrt{1-4x}},
$$
(sequence \OEISlink{A029759} in the \href{http://www.research.att.com/\~njas/sequences/}{OEIS}~\cite{OEIS}). This class has an enumeration scheme of depth four.
\end{enumerate}
\section{Conclusion}\label{wp-conclusion}
We have developed an extension of Zeilberger's enumeration schemes and provided the mechanisms for their automatic generation, a task implemented in the accompanying Maple package {\sc WilfPlus}, which is available at \url{http://math.rutgers.edu/~vatter/}. As the examples in Section~\ref{wp-nonex} demonstrate, there remain considerable differences in the applicability of enumeration schemes, substitution decompositions, and the insertion encoding. This obviously suggests the following question.
\begin{question}\label{systematic-gen}
Is there a systematic method of permutation class enumeration which is applicable to all classes with finite enumeration schemes, all classes with regular insertion encodings, and all classes with only finitely many simple permutations?
\end{question}
Additionally, one would like such a method to be invariant under the eight permutation class symmetries, but Question~\ref{systematic-gen} is probably demanding enough in the form above.
We conclude by collecting three of the questions raised earlier.
\begin{question}\label{prec}
Is every sequence produced by a finite enumeration scheme holonomic?
\end{question}
As demonstrated by the examples in Section~\ref{wp-ex}, another interesting question is the equivalence problem:
\begin{question}
Is it decidable whether two finite enumeration schemes produce the same sequence?
\end{question}
Brlek, Duchi, Pergola, and Rinaldi~\cite{bdpr:eps} consider the equivalence problem for generating trees with infinitely many labels.
Although only tangentially related to the main thrust of this article, the following question nevertheless seems intriguing enough to warrant its inclusion.
\begin{question}
Is it decidable whether a class contains only finitely many simple permutations?
\end{question}
\bigskip
\bibliographystyle{acm}
|
hep-lat/0510073
|
\section{Introduction}
Chiral gauge theory of weak interactions forms an important ingredient
of the standard model, but so far chiral gauge theories have defied a
definition beyond perturbation theory. These in fact form the only
class of perturbatively defined field theories with no
non-perturbative formulation. This problem is gaining practical import
since a number of proposed extensions to the standard model invoke
strongly interacting chiral dynamics, and a quantitative analysis
requires the evaluation of non-perturbative condensates in these
theories. The only known non-perturbative regulator in four-dimensions
is the lattice formulation and we investigate it here.
A fundamental problem in realizing a chiral gauge theory on the
lattice is encapsulated in the Nielsen Ninomiya
theorem~\cite{NielsenNinomiya}. According to this theorem, under mild
conditions of locality and analyticity of the propagator, a
translationally invariant fermion theory that preserves the continuum
chiral symmetry exactly on the lattice has paired left and right
chiral modes in the continuum limit. Since these modes are related by
boosts in the discretized theory, they cannot be differently charged
under any symmetry preserved by the discretization, and any na\"\i{}ve
attempt at obtaining an unbroken chiral gauge theory must fail. This
result is also expected based on our knowledge that chiral gauge
theories are undefined if anomalies do not cancel. The anomaly
cancellations can occur, as in the standard model, between fermion
species that interact only through the gauge fields. This cancellation
is difficult to preserve in any na\"\i{}ve discretization.
A way out of this conundrum was suggested by Kaplan~\cite{Kaplan} by
extending fermionic gauge theories to five dimensions. Theories on a
five dimensional interval typically have \emph{edge} states that are
chiral under the four dimensional Euclidean group acting on the edge
of the region. These states still come in chiral pairs,
but one can arrange the parameters to locate the left
and right chiral modes on the different edges, and hence separate them in
the fifth dimension. Since the fifth dimension is unphysical, one is
free to add interactions that are not translationally invariant along
this direction. Previous attempts to realize a chiral gauge theory by
confining the gauge interactions to only a part of the five
dimensional space were, however, unsuccessful.~\cite{waveguide}.
In this paper we consider an alternate formulation with a truly five
dimensional gauge field~\cite{us}. The translational invariance of this gauge
interaction is broken by invoking a Higgs' mechanism using extra
matter fields propagating only along one edge. This Higgs' phenomenon
makes the fermion states at that edge heavy, but has negligible effect
on states located elsewhere in the five dimensional world. A
classical analysis reveals that in the presence of background
five-dimensional space-time curvature we can take a limit in which a
four-dimensional gauge boson and one chiral fermion remain massless,
whereas all the other states in the theory become infinitely heavy and
decouple. If the resulting gauge theory has an uncancelled triangle
anomaly then this construction of a chiral gauge theory is easily seen
to fail at the quantum level. A complete quantum analysis remains
beyond our reach because even perturbation theory is not valid in some
regions in this five dimensional space.
\section{The flat space analogue}
\label{sec:flat}
Before constructing the full model, it is instructive to study an
analogue construction in flat five-dimensional space. We consider
this world bounded by flat Euclidean slices separated by distance
\(R'-R\). A free massive fermion in this spacetime is described
by the action
\begin{eqnarray}
S &=& \int d^4\,x \int_R^{R'} d\,z \big\{
-i\,\bar\psi\partial_\mu\bar\sigma^\mu\psi
-i\,\chi\partial_\mu\sigma^\mu\bar\chi
+\psi\partial_z\chi-\bar\chi\partial_z\bar\psi \nonumber\\
&&\qquad\qquad\qquad\qquad {} + M\,\psi\chi + M\,\bar\chi\bar\psi \big\}\,,
\nonumber
\end{eqnarray}
where \(\psi\) and \(\chi\) are two-component fermion fields of opposite
chirality, \(\sigma^\mu\) are the Pauli matrices (or identity for the
time direction), \(M\) is the five dimensional mass, and the overbar
represents conjugation. A Kaluza-Klein decomposition gives modes
\(\chi_n\) and \(\psi_n\) that satisfy
\[
(\partial_z - M)\bar\chi_n = m_n\bar\chi_n \qquad\qquad\qquad
-(\partial_z + M)\psi_n = m_n\psi_n\,.
\]
The edge states are the states with \(m_0=0\) and are exponentially
localized at the left and right walls (Fig.~\ref{fig:ferm}). The
mass scaleof the rest of the states in the theory, \(\pi/2(R'-R)\),
is set by the length of the fifth dimension.
\begin{figure}
\begin{center}
\leavevmode\vbox{\hbox{%
\includegraphics[width=2in]{profile-0.mps}%
\llap{\includegraphics[width=2in]{profile-1.mps}}%
\llap{\includegraphics[width=2in]{profile-2.mps}}%
\llap{\vbox to 1.1in{\hbox to 1.6in{%
\color{black}$\psi_0$\hspace*{0.8in} $\chi_0$\hss}\vss}}}
}
\end{center}
\caption{The exponentially localized edge states of the fermions,
\(\psi_0 \propto \exp{-Mz}\) and \(\bar\chi_0 \propto \exp{Mz}\).}
\label{fig:ferm}
\end{figure}
A gauge theory is obtained by changing the derivatives
above to gauge covariant ones and adding a kinetic term for the gauge
bosons
\[
\int d^4\,x \int_R^{R'} d\,z \frac1{4g_5^2}
\left( F_{\mu\nu} F^{\mu\nu} + 2 F_{\mu5}F^{\mu5} \right)\,.
\]
In such a theory the left and right chiral modes, \(\psi_0\) and
\(\chi_0\), have equal charge, and one needs to decouple one of them
(by making it heavy), say \(\chi_0\) on the right wall, to obtain a
chiral gauge theory. To this end, we introduce an Higgs' mechanism
localized on the right wall, and use the Higgs' field \(H\) to mix
\(\chi\) with a neutral fermions \(S_L\) with large Majorana mass,
\(m\):\footnote{The addition of a neutral right handed fermion \(S_R\)
at \(R'\) along with its Yukawa coupling to the bulk field
\(\bar\psi\) has, however, very little impact on the zero mode
\(\psi_0\) exponentially localized at \(R\). Such an addition is
necessary in the actual construction but is dropped in this exposition
for brevity.}
\[
y\,\chi H S_L + m S_L S_L + \mbox{h.c.}\,
\]
where \(y\) is the Yukawa coupling. This provides a mass of order \(y
\langle H\rangle\) to \(\chi_0\). Since Yukawa couplings are
infrared free it is difficult to make \(y\) much larger than the
gauge coupling \(g\). This is problematic since for small values of
\(\langle H\rangle\) the gauge boson also acquires a mass of order
\(g \langle H\rangle\), and it appears one cannot take a limit where
the unwanted mode \(\chi_0\) decouples, but the gauge boson stays in
the spectrum.
The situation is qualitatively different when the vacuum expectation
value is large. The odd Kaluza-Klein modes of the gauge boson, which
vanish at the boundary, do not feel the effect of the Higgs' mechanism
at the classical level and retain a mass, \({\tilde m}_1\), of the order of \(\pi/2(R'-R)\).
The even Kaluza-Klein modes pick up large masses by this Higgs'
mechanism. So, if the length of the fifth dimension is large, we do
find light gauge bosons and a massless chiral mode, \(\psi_0\) in the
spectrum\footnote{The masses of the Kaluza-Klein modes \(\chi_n\),
controlled by the scale \(M\) and not by the size of the extra
dimension, stay heavy.} as required to construct a chiral gauge
theory.
This construction, however, fails since \emph{all and not only the
lightest} of the odd Kaluza-Klein modes of the gauge field are
controlled by the same scale \(\pi/2(R'-R)\). Thus in the limit that
the extra dimension becomes large, the theory reverts to being five
dimensional.
\section{The Model in \(AdS_5\)}
\label{sec:model}
It has been known for a while~\cite{pheno} that if we consider an
interval in a curved five dimensional space, the gravitational
acceleration towards one wall can give rise to an effective four
dimensional theory \emph{even when the length of the extra dimension
is large.} Accordingly, we consider a Euclidean version of a slice of
\(AdS_5\) bounded by four-dimensional flat Minkowski slices separated
by a proper distance denoted by \(\ln R'/R\). \(AdS_5\) is a
homogeneous five-dimensional space with a constant negative radius of
curvature, which is also denoted by \(-R\). The comparison with our
flat space analysis is easiest in the metric
\[
ds^2 = \left(\frac Rz \right)^2
\left(\eta_{\mu\nu} dx^\mu dx^\nu - dz^2\right)\,,
\]
with the chosen interval being \((R,R')\). In this background, the
odd Kaluza-Klein modes of the gauge boson have masses given by
\[
{\tilde m}_1^2 = O\left(\frac1{R'^2\ln\left(R'/R\right)}\right)\,,\qquad\qquad\qquad
{\tilde m}_n^2 \approx O\left(\frac1{{R'}^2}\right)\,,
\]
whereas the fermion modes are not significantly affected for large
\(M\). This lets us take the limit
\[
R'\to 0\, \qquad\qquad\qquad\qquad\qquad R'^2 \ln (R'/R)\to\infty\,
\]
which decouples all the Kaluza-Klein modes leaving behind a massless
gauge boson interacting with the chiral fermion \(\psi_0\).
\section{Deconstruction}
\label{sec:deconstruction}
Five dimensional theories may not be renormalizable, and it may not
be possible to interpret the Lagrangian parameters as running
couplings at a certain scale. We, therefore, need to be careful in drawing
conclusions based on the tree level constructions provided in the
previous sections. To study this systematically we choose to
deconstruct~\cite{deconstruction} the five dimensional theory into a
stack of four dimensional continuum slices placed at discrete
positions along the fifth dimension. To maintain as many of the space time
symmetries of the \(AdS_5\) space as possible, we choose the positions of the
slices to be
\[
z_i = \left(1 + a\right)^{i-1} R, \qquad i = 1\ldots N,\qquad
a = \exp\left(\frac1{N-1}\ln \frac R{R'}\right)-1
\]
where we refer to the small dimensionless number \(a\) as the ``lattice
spacing''. The modes at each of these four dimensional slices
can be reinterpreted as those of fields transforming under a separate four
dimensional gauge group. It is easy to check that the gauge coupling
of the \(i^{\mbox{th}}\) gauge group is given by
\[
\frac1{g_i^2} = \frac{aR}{g_5(z_i)^2}\,,
\]
where we have generalized the model to allow for a variation of
\(g_5\) along the fifth dimension.
The link fields in the fifth dimension, \({\cal P}
\exp\int_{z_i}^{z_{i+1}} i\,A_5\,dz\) can then be interpreted as
bifundamental scalars (transforming according to the fundamental
(anti-fundamental) representation under the gauge theory at \(z_i\)
(\(z_{i+1}\))). With this identification, the original theory, before
we add the Higgs mechanism at \(R'\), can be seen to be invariant
under the product of the gauge groups at each site but realized in the
Higgs phase with the bifundamentals acquiring a vacuum expectation
value related to the lattice spacing. This view of the
five-dimensional theory is called deconstruction. The lowest gauge
boson mode is, not surprisingly, the discretization of the lowest
Kaluza-Klein mode: the equal superposition of the gauge fields on
all the slices, and its effective coupling is given by
\[
\frac1{g_4^2} \approx \sum_{i=1}^N \frac1{g_i^2}\,.
\]
The continuum \(AdS_5\) symmetry requires all lengths to scale in
proportion to the \(z\) coordinate as we move in the fifth dimension.
To maintain a discrete version of this we can choose the renormalized
couplings to satisfy
\[
g_i^R(\mu_i) = constant \qquad\qquad\mbox{if}\qquad\qquad
\mu_i \propto 1/z_i\,.
\]
A one loop calculation, assuming $\mu \ll 1/R'$, then yields
\[
\frac1{g_4^2(\mu)} \approx \frac N{g_1^2(1/aR)} +
\frac {\beta_0}{8\pi^2}\ln aR\mu\,
\]
whence we can obtain the \(\Lambda\) parameter of the theory associated
with the lowest mode to be
\[
aR\Lambda = \exp \frac{-8\pi^2N}{\beta_0 g_1^2(1/aR)}\,.
\]
To obtain a chiral gauge theory we need to take the limit
\[
\frac{m_{KK}}\Lambda \to \infty\,, \qquad\qquad\qquad\qquad
\frac{m_1}\Lambda \to 0\,,
\]
where \(m_{KK}\) is the typical Kaluza-Klein state and \(m_1\) is the
lightest gauge mode. Using
\[
m_{KK}^2 R^2 \sim (1 + a)^{2N}\,, \qquad\qquad\qquad\qquad
\frac {m_1^2}{m_{KK}^2} \sim \frac1{N\log(1+a)}\,,
\]
one finds that the limit can be realized with the choice
\[
N\to\infty \qquad\qquad\qquad\qquad
g_1^2(\frac1{aR}) \sim \frac{8\pi^2}{\beta_0 a}\,,
\]
with $a$ held fixed.
The arguments used in deriving these results, however, relied on the
wavefunctions on the lattice being a discretization of the continuum
tree-level wave functions. This property holds only if the lattice
spacing \(a\) is much smaller than unity, and this in turn implies we
need to choose the lattice couplings to be large. As a result, the
one-loop results presented here can only be treated as
suggestive.\footnote{A more detailed argument shows that it is
indeed the coupling at the scale \(1/aR\) which controls the validity
of the required one-loop calculation.}
\section{Potential problem}
\label{sec:problem}
It is well known that the massless limit of a vector field theory is
singular, and often involves strong couplings. It is instructive to
note that even in our construction, the `longitudinal' mode of the
gauge field becomes strongly interacting with an effective Yukawa like
interaction with the fermions. This interaction, of strength
\[
y(z) \propto \frac {z\ln(z/R)} {R'\ln(R'/R)}\,,
\]
pushes the unwanted chirality fermion away from the wall with the
Higgs' mechanism, and can potentially make it light. Explicit
evaluation in one-loop perturbation theory, however, shows that the
fermion decouples even in the presence of this interaction.
Unfortunately, the large gauge coupling precludes a pertubative
resolution of this question. To settle this issue a non-perturbative
calculation is required.
|
math/0510403
|
\section{INTRODUCTION}
A (nontrivial) trigonometric series\begin{equation}
\sum c(n)e^{int}\quad t\in \mathbb{T}=\mathbb{R}/2\pi \mathbb{Z}\label{eq:defser}\end{equation}
is called a null series if it converges to zero almost everywhere
(a.e.). The existence of such a series was discovered by D. E. Menshov
in 1916 (see \cite[chap.\ XIV]{B64}). He constructed a singular compactly
supported finite Borel measure on $\mathbb{T}$ with Fourier transform
vanishing at infinity. The Riemannian theory implies that the Fourier
series of this measure converges to zero at every point outside of
the support. This famous example of Menshov was the origin of the
modern uniqueness theory in Fourier Analysis, see \cite{B64,KS94,KL87}.
Clearly a null series can not belong to $L^{2}$. A less trivial observation
is that it can not be {}``analytic'' that is involve positive frequencies
only. This follows from the Abel summation and Privalov {}``angular
limit'' theorems. It turns out however that the {}``non-analytic''
part of a null series may belong to $L^{2}$.
\begin{thm}
\label{thm:Mensh}There is a null series (\ref{eq:defser}) such that\[
\sum _{n<0}|c(n)|^{2}<\infty \quad .\]
\end{thm}
An equivalent formulation of the result :
\begin{thm}
\label{thm:complx}There is a power series\[
F(z)=\sum c(n)z^{n}\]
converging a.e.~on the circle $|z|=1$ to some function $f\in L^{2}(\mathbb{T})$,
and $f$ does not belong to the Hardy space $H^{2}$.
\end{thm}
When we say that a function $f$ on the circle is in $H^{2}$ we mean
that it is a boundary limit of an $H^{2}$ function on the disk, or,
equivalently, that $f \in L^2$ and $\widehat{f}(-n)=0$ for $n=1,2,\dotsc $
To see that theorem \ref{thm:Mensh} implies \ref{thm:complx}, use
Carleson's convergence theorem \cite{C66} to get that the analytic
part of the sum, $\sum _{n\geq 0}c(n)z^{n}$ converges a.e.~on the
circle $|z|=1$ and then use, as above, Abel and Privalov's
theorems to get that the resulting $f$ is not in $H^{2}$.
Reversing these arguments one may derive theorem \ref{thm:Mensh}
from theorem \ref{thm:complx}.
It should be mentioned that usually if representation by harmonics
is unique then it is the Fourier series. Compare for instance the
classical Cantor and du Bois-Reymond \cite[pp.\ 193, 201]{B64} theorems
on pointwise convergence everywhere. Our result shows that this principle
is not universal. Indeed, any $f$ may have \emph{at most one representation}
by an a.e.~convergent series \begin{equation}
\sum _{n\geq 0}c(n)e^{int}\label{eq:sumanal}\end{equation}
however, even if $f\in L^{2}$, the coefficients, in general, can
not be recovered by Fourier's formula.
\section{PROOF}
\subsection{\label{sub:defFf}}
Our main goal is to construct an {}``analytic pseudofunction'',
that is\begin{align}
F(z) & =\sum _{n\geq 0}c(n)z^{n}\label{eq:defF}\\
c(n) & =o(1)\label{eq:cno1}
\end{align}
with the following properties:
\begin{enumerate}
\item \label{enu:FnotinH2}$F\not \in H^{2}$.
\item There is a compact $K\subset \mathbb{T}$ of Lebesgue measure zero
such that $F$ has boundary values on $K^{c}$\[
f(t):=\lim _{z\rightarrow e^{it}}F(z)\quad \forall t\not \in K.\]
\item $f\in L^{\infty }(\mathbb{T})$.
\item \label{enu:limunif}The limit is uniform on any closed arc $J\subset K^{c}$.
\end{enumerate}
Having a function $F$ with all the properties above one can get the
result easily. Indeed, the series (\ref{eq:defF}) on the boundary
represents a distribution\[
\bar{F}:=\sum \widehat{F}(n)e^{int},\]
which is the limit (in distributional sense) of $F_{r}:=F(re^{it})$
as $r \rightarrow 1$.
On the other hand $F(re^{it})\rightarrow f(t)$ uniformly on any closed arc
$J\subset K^{c}$, so the distribution $\bar{F}-f$ is supported on
$K$. Hence the condition (\ref{eq:cno1}) implies uniform convergence
of the Fourier series of $\bar{F}-f$ to zero on any such $J$, see
\cite[p.\ 54]{KS94}. Theorem \ref{thm:complx} (and hence theorem
\ref{thm:Mensh}) will follow.
The function $F$ will be obtained as $1/G$ where $G$ is a singular
inner function, so\begin{equation}
F(z)=\exp \Big (\int _{\mathbb{T}}\frac{e^{it}+z}{e^{it}-z}d\mu (t)\Big ).\label{eq:poisson}\end{equation}
This construction will ensure (\ref{enu:FnotinH2})-(\ref{enu:limunif}),
if $\mu $ is a positive measure supported on $K$, so our task in
sections \ref{sub:lnKnun}-\ref{sub:defmu} will be to construct a
singular $\mu $ such that (\ref{eq:cno1}) will be satisfied.
\subsection{\label{sub:lnKnun}}
Denoting $g(x):=xe^{2/x}+1-x$ we fix a sequence\[
l(1)>l(2)>...\rightarrow 0,\]
such that\[
g(l(n))-g(l(n-1))=o(1).\]
Proceed with the induction as follows. Let $K_{0}=\mathbb{T}$. Suppose
we already have a compact $K_{n-1}\subset \mathbb{T}$ which is a
finite union of segments of equal lengths. Divide each of them to
$q(n)$ equal subsegments $I$ and replace each $I$ by the concentric
segment $I'$,\[
|I'|=\frac{l(n)}{l(n-1)}|I|\]
(here and below by $|E|$ we denote the normalized Lebesgue measure
of a set $E\subset \mathbb{T}$). Set $K_{n}:=\cup I'$, so $|K_{n}|=l(n)$,
and\[
u_{n}:=\frac{1}{l(n)}\mathbf{1}_{K_{n}}.\]
\begin{claim}
If the number $q=q(n)$ is sufficiently large then the function $u_{n}-u_{n-1}$
is {}``almost orthogonal'' to any pre-given finite dimensional subspace
in $L^{2}(\mathbb{T})$. More precisely: for any $\epsilon >0$, $N\in \mathbb{N}$
there is a $Q\in \mathbb{N}$ such that $\forall q(n)>Q$,\[
|\widehat{u_{n}-u_{n-1}}(k)|<\epsilon \quad \forall k,|k|<N\]
(here and below the sign $\widehat{\cdot }$ stands for the Fourier
transform on $\mathbb{T}$. The term {}``sufficiently large'' means
that the minimal allowed value may depend on everything that happened
in previous stages of the induction).
\end{claim}
To prove the claim it is enough to mention that $u_{n}-u_{n-1}$ is
supported on the union of the segments $I$, the length of each $I$
is arbitrary small as $q$ gets large and the average of $u_{n}-u_{n-1}$
on $I$ equals to zero.\qed
Obviously the same inequality holds for the conjugate function $\widetilde{u_{n}-u_{n-1}}$
so we obtain
\begin{claim}
\label{cla:ununtilde}Given $\epsilon >0$, $N\in \mathbb{N}$ and
sufficiently large $q$ the function\[
h_{n}:=(u_{n}-u_{n-1})+i(\widetilde{u_{n}-u_{n-1}})\]
satisfies \[
|\widehat{h_{n}}(k)|<\epsilon \quad \forall k,0\leq k<N\]
\end{claim}
Now denote: $f_{n}=e^{u_{n}+i\widetilde{u_{n}}}$.
\begin{claim}
\label{cla:fnfn1korth}Given $\epsilon >0$, $N\in \mathbb{N}$ and
$q$ sufficiently large we have :\[
|\widehat{f_{n}-f_{n-1}}(k)|<\epsilon \quad \forall k,0\leq k<N\]
\end{claim}
Indeed, \[
f_{n}-f_{n-1}=f_{n-1}(e^{h_{n}}-1).\]
Clearly the fact that $h_{n}$ are analytic gives that they may be
exponentiated formally (e.g.~by extending to the disk $\mathbb{D}$
and using $\widehat{h}(k)=h^{(k)}(0)$) which gives that $\widehat{e^{h_{n}}-1}(k)$
is a polynomial with no constant term in $\widehat{h_{n}}(1),\dotsc ,\linebreak [0]\widehat{h_{n}}(k)$.
Since $f_{n-1}$ is also analytic, $\widehat{f_{n}-f_{n-1}}(k)$ is
a finite combination of $\widehat{f_{n-1}}(j)$ and $\widehat{e^{h_{n}}-1}(k-j)$
and the claim is a consequence of claim \ref{cla:ununtilde}.
As an immediate corollary we obtain :
\begin{claim}
\label{cla:fnfn1fn1orth}For any $\epsilon >0$ and sufficiently large
$q$\[
|\langle f_{n}-f_{n-1},f_{n-1}\rangle |<\epsilon .\]
\end{claim}
We mean here the usual inner product, $\langle f,g\rangle =\int _{\mathbb{T}}f\bar{g}$.
\subsection{}
Proceeding with the induction above we get
\begin{claim}
If the numbers $q(n)$ grow sufficiently fast then there are numbers
$N_{1}<N_{2}<\dotsm $ such that the functions $\{f_{n}\}$ satisfy,
for any $n$, the conditions:
\end{claim}
\begin{enumerate}
\item \label{enu:defNn}$|\widehat{f_{n-1}}(k)|<\frac{1}{n}$, for all $k$
such that $k>N_{n}$.
\item $|\widehat{f_{n}-f_{n-1}}(k)|<2^{-n}$, for all $k$ such that $0\leq k\leq N_{n}$.\label{enu:fnfn1k}
\item $|\langle f_{n}-f_{n-1},f_{n}\rangle |<\frac{1}{n}$.\label{enu:fnorth}
\end{enumerate}
It is enough on the $n$th step of the induction to choose $N_{n}$
so that (\ref{enu:defNn}) is fulfilled and then to use claims \ref{cla:fnfn1korth}
and \ref{cla:fnfn1fn1orth} to ensure (\ref{enu:fnfn1k}) and (\ref{enu:fnorth}).
Let the sequence $\{q(n)\}$ above be fixed. The {}``almost orthogonality''
condition (\ref{enu:fnorth}) implies the {}``almost Pythagorean''
equality:\[
||f_{n-1}||^{2}+||f_{n}-f_{n-1}||^{2}=||f_{n}||^{2}+o(1).\]
From section \ref{sub:lnKnun} we have $||f_{n}||^{2}=g(l(n))$, so
$||f_{n}-f_{n-1}||^{2}=o(1)$. Together with (\ref{enu:defNn}) and
(\ref{enu:fnfn1k}) this easily gives\begin{equation}
||\widehat{f_{n}-f_{m}}||_{\infty }\rightarrow 0\textrm{ as }n,m\rightarrow \infty \label{eq:fnfm20}\end{equation}
\subsection{\label{sub:defmu}}
Let $\mu $ be the weak limit of the measures $\mu _{n}(dt):=u_{n}(t)dt$.
Clearly it is a positive measure supported on $K:=\cap K_{n}$ and
$|K|=0$. Define $F$ by (\ref{eq:poisson}) and $F_{n}$ by the same
formula with $\mu $ replaced by $\mu _{n}$. Then $F_{n}\rightarrow F$
uniformly on compacts inside the unit disc $\mathbb{D}$. Therefore
each coefficient $c(k)$ of the expansion (\ref{eq:defF}) may be
obtained as the limit of the corresponding coefficients $c_{n}(k)$
which are just $\widehat{f_{n}}(k)$, so (\ref{eq:fnfm20}) implies
(\ref{eq:cno1}) which finishes the proof.
\begin{rem*}
It should be noted that our use of Carleson's convergence theorem
to prove the equivalence of theorems \ref{thm:Mensh} and \ref{thm:complx}
is unnecessary, since theorem \ref{thm:Mensh} may be proved directly
using the fact (which is easy to see) that the function $f$ defined
in section \ref{sub:defFf} is smooth on any closed arc $J\subset K^{c}$.
\end{rem*}
\section{REMARKS}
\subsection{}
In contrast to Menshov's original example, the series (\ref{eq:defser})
in theorem \ref{thm:Mensh} can not be the Fourier series of a measure.
Indeed , if \[
\mu \sim \sum c(n)e^{int}\quad \sum _{n<0}|c(n)|^{2}<\infty ,\]
then $\mu $ must be absolutely continuous, and cannot generate a
null-series. See, for example, \cite[sect. VIII.12]{B64}.
\subsection{}
Another contrast with the {}``non-analytic'' situation appears when
one considers the size of the exceptional set. It is well known that
a null series (\ref{eq:defser}) may converge to zero outside a {}``thin''
compact (of zero Hausdorff dimension). On the other hand the following
proposition is true\begin{ialt}If a series (\ref{eq:sumanal}) converges
to $f\in L^{1}(\mathbb{T})$ everywhere on $\mathbb{T}$ outside some
set of dimension $<1$ then it is the Fourier series of $f$.\end{ialt}
This follows from a Phragm\'en-Lindel\"of type theorem for analytic
functions in $\mathbb{D}$ of slow growth, see \cite[theorem 5]{B92}.
\subsection{}
Let $\mathcal{P}$ be the class of functions in $L^{2}$ which can be represented
by an a.e.\ converging sum (\ref{eq:sumanal}).
Theorem \ref{thm:complx}
shows us that $\mathcal{P}\setminus H^{2}$ is non-trivial. Further,
the proof actually gives a little more: $(\mathcal{P}\setminus H^{2})\cap L^{\infty }\neq \emptyset $.
The class $\mathcal{P}$ has some interesting properties. We plan
to analyze it in a separate paper.
|
hep-ph/0510371
|
\section{Introduction}\label{intro}
For low energies of the incoming beam in a hadron-nucleus collision,
the successive elastic rescatterings of the initial hadron on the
various nuclei of the nucleus are well described within the
probabilistic Glauber model \cite{glauber}. At higher energies,
corresponding to $E_{crit} \sim m_{\scriptstyle{N}} \mu R_A$, the
hadronic fluctuation length can become of the order of nuclear
radius and there will be coherent interaction of constituents of the
hadron with several nucleons of the nucleus. Within the Gribov
approach \cite{gribov1}, this corresponds to summing up
contributions of inelastic intermediate states, and leads to a
reduction of the total cross section of the reaction, i.~e. to nuclear
shadowing.
We calculate the total amount of gluon shadowing for low values
of the Bjorken variable $x$ for heavy ions, ignoring for
the time being the contribution from the quarks. The most recent data on
diffractive structure functions are used and much stronger shadowing
effects than previously expected are found. These effects will lead to
a strong multiplicity reduction in A+A collisions at RHIC
and LHC energies.
\section{The Model}
The diffractive $\gamma^* N$ scatterings are described by Pomeron
exchange. The scattering amplitude of an incoming photon with
virtuality $Q^2$ on a nuclear target, consisting of A nucleons, can
then be written as \cite{cap98}
\begin{eqnarray}
\label{eq:sum}
\sigma_A \;=\; A\sigma_N \,+\, \sigma_A^{(2)} \,+\, ...\;.
\end{eqnarray}
The second term in (\ref{eq:sum}) is negative and is related to
diffractive DIS through the AGK cutting rules \cite{agk}. Higher order
rescatterings in (\ref{eq:sum}) are model dependent. The Schwimmer
unitarization \cite{schwimmer} for the $\gamma^* A$ cross
section is used to obtain the shadowing ratio
\begin{eqnarray}
\label{eq:sch}
R^{Sch}_{\scriptscriptstyle{A/N}} (x) \,\equiv \,
\frac{\sigma_{\gamma* A}^{Sch}}{A \,\sigma_{\gamma^*
\scriptscriptstyle{N}}} \;=\; \int
\mbox{d}^2b \; \frac{T_A (b)}{1 \,+\, (A-1) f(x, Q^2) T_A (b) } \;,
\end{eqnarray}
where $f(x,Q^2)$ is the effective shadowing function, $T_A(b)$ is the
nuclear density profile normalized to unity and standard DIS
variables are used. Following \cite{cap01,H1abs} in choice of
parameters and kinematics, one can get the shadowing function as
\begin{eqnarray}
\label{eq:fsimpl}
f(x, Q^2) \;&=&\; 4\pi\; \int_x^{x_{\hspace{ -0.1em}I\hspace{-0.25em}P}^{max}} \mbox{d}x_{\hspace{ -0.1em}I\hspace{-0.25em}P} \,B(x_{\hspace{ -0.1em}I\hspace{-0.25em}P})\,
\frac{F_{2 {\cal D}}^{(3)}(x_{\hspace{ -0.1em}I\hspace{-0.25em}P}, Q^2, \beta)}{F_2 (x, Q^2)} \; F_A^2
(t_{min.}) \;.
\end{eqnarray}
Here $B(x_{\hspace{ -0.1em}I\hspace{-0.25em}P}) = 0.184 - 0.02 \ln \left(x_{\hspace{ -0.1em}I\hspace{-0.25em}P} \right)\,\mbox{fm}^2$, and $F_A$ is
the form factor of the nucleus.
Calculations are made both for $x_{\hspace{ -0.1em}I\hspace{-0.25em}P}^{max}
= 0.1$ as in \cite{cap98} and for $x_{\hspace{ -0.1em}I\hspace{-0.25em}P}^{max} = 0.03$ as in
\cite{fgs03}.
The structure functions $F_2 (x,Q^2)$ and
$F_{2 {\cal D}}^{(3)} (x_{\hspace{ -0.1em}I\hspace{-0.25em}P}, Q^2, \beta)$ are determined from
experiment. At small $x$, gluon shadowing is found to be dominant.
Quark contribution to the structure functions is not considered in
what follows.
Shadowing due to quarks, obtained wihtin the same approach, was discussed in
\cite{cap98}.
The gluon parton distribution functions (PDF) for nucleon and Pomeron were
measured at intermediate $Q^2$ at the HERA experiments ZEUS and H1,
correspondingly. The next to leading order (NLO)
ZEUS-S QCD fit for the gluon PDF of the nucleon \cite{chekanov03} at
$Q^2 = 7 \mbox{ GeV}^2$, and
the gluon PDF for the Pomeron (diffractive structure function)
\cite{H1abs} at $Q^2 = 6.5 \mbox{ GeV}^2$ were both parametrized by
\begin{eqnarray}
x \, g(x, Q^2) \;=\; A x^{-\delta} \left(1-x \right)^\gamma \;,
\label{eq:fitfunc}
\end{eqnarray}
where the fitting parameters $\left\{A,\delta,\gamma \right\}= \left\{
1.9,0.19,6.7 \right\}$ were obtained for the nucleon and
$\{0.38,0.28,0.17 \}$ for the Pomeron case, respectively. The $Q^2$-dependence of the fitting parameters is weak for moderate $Q^2$, and so we neglect it for the sake of simplicity.
\section{Numerical results}\label{result}
Gluon shadowing for various heavy ions (Ca, Pd and Pb) from (\ref{eq:sch})
is presented in Fig.~\ref{fig:AN}. The gluon shadowing is very strong
at small $x$, and disappearing at $x = x_{\hspace{ -0.1em}I\hspace{-0.25em}P}^{max}$. This is a
consequence of the coherence effect in the form factor,
and the vanishing integration domain in (\ref{eq:fsimpl}). Gluon
shadowing is as low as 0.2 for the Pb/nucleon ratio.
A comparison of our results for Pb/nucleon ratio at
$Q^2 = 6.5 \mbox{ GeV}^2$ with $x_{\hspace{ -0.1em}I\hspace{-0.25em}P}^{max} = 0.03$, with those of
others, calculated at $Q^2 =
5 \mbox{ GeV}^2$ is presented in Fig.~\ref{fig:comp}. For $x \leq
10^{-3}$ our model predicts the stronger gluon shadowing compared to
\cite{armesto02} (dashed-dotted line) and \cite{fgs03}
(dotted line), while \cite{hijing} (dashed line)
predicts the strongest effect down to $x \sim 10^{-4}$.
\begin{figure}[t]
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=\linewidth]{AN_results_alt.eps}
\caption{Gluon shadowing for heavy ions. Closed (open) symbols are for
$x_{\hspace{ -0.1em}I\hspace{-0.25em}P}^{max} = 0.1$ ($0.03$).}
\label{fig:AN}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=\linewidth,height=1.81in]{comparison3_alt.eps}
\caption{Comparison of theoretical predictions for the Pb/nucleon
ratio, at fixed $Q^2$.}
\label{fig:comp}
\end{minipage}
\end{figure}
\section{Shadowing effects in d+Au collisions}
The model is now employed to study multiplicity reduction in d+Au collisions
at ultra-relativistic energies. Deuteron is treated as a point-like
particle in impact parameter space, but with the shadowing
found from (\ref{eq:sch}). The collision is described by two-jet
kinematics through $ x_{p (t)} \;=\; c\,
p_T \, e^{\pm \eta}/ \sqrt{s} $, where $p_T$ is the transverse
momentum of the particle, and fixed at $Q^2$. We assume that most of the
high-$p_T$ particles come from jets $c$ times more energetic than the
measured one. The theoretical prediction is given by \cite{cap99}
\begin{eqnarray}
\label{eq:shadow}
R^{theo}_{d+Au} \;&=&\; R^{Sch}_d (x_{p}) R^{Sch}_{Au} (x_{t}) \;.
\end{eqnarray}
From here one obtains the multiplicity reduction due to shadowing
compared to the Glauber model. Then the model predictions for nuclear
modification factor (NMF) at forward rapidities are compared to BRAHMS
data at $\sqrt{s_{\scriptstyle{NN}}} = 200$ GeV \cite{brahms}.
We exclude the gluon shadowing effects in the BRAHMS NMF at $\eta =
0$ by defining $R^{\mathit{norm}}_{d Au} =
\left[R_{dAu}^{\mathit{exp}} / R_{d-Au}^{\mathit{theo}} \right]_{\eta
= 0}$. The multiplicity reduction due to shadowing effects will
appear when we compare the NMF at forward rapidities, $\eta = 1,
\,2.2, \,3.2$ to $R^{\mathit{norm}}_{d Au}$. The ratio $\tilde{R}
\;=\; \left[R^{exp}_{d Au}\right]_\eta / R^{\mathit{norm}}_{d Au}$ is
plotted in Fig. \ref{fig:res_c} together with the predictions of
(\ref{eq:shadow}) for two different values of the parameter $c$.
Statistical errors are denoted by the thick solid line, while the
systematic and statistical errors added up are denoted
by the dashed line. Cronin effect is assumed to be rapidity independent and is
cancelled out in the ratio.
\begin{figure}[t]
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=1.1\linewidth]{result_c3_alt_wsys.eps}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=1.1\linewidth]{result_c5_alt_wsys.eps}
\end{minipage}
\caption{NMF ratio for (a) $c = 3$ and (b) $c = 5$. See text for
details.}
\label{fig:res_c}
\end{figure}
The choice of $c$ does not affect the
result. Within the presented framework, one can conclude that suppression of the nuclear modification
factor at forward rapidities is mostly due to gluon shadowing in the
nuclei.
\linebreak
\linebreak
{\it{Acknowledgements:}} The authors acknowledge support from NFR and
\linebreak QUOTA program.
|
math/0510281
|
\section{Introduction}
In \cite{Gahler:1992}, G\"ahler gave a presentation of the category $\mathbf{Top}$ of topological spaces as a category, denoted here by $\mathbf{KlAlg}(\mathsf{F})$, of structured objects in the Kleisli category of the filter monad~$\mathsf{F}$:
\[
\mathbf{Top}\cong\mathbf{KlAlg}(\mathsf{F})\ .
\]
This result lead to a natural definition of fuzzy topological spaces by extending the previous monad to a fuzzy filter monad. Lax algebras on the other hand (see \cite{Barr:1970}, \cite{Clementino/all:2004}, \cite{Hofmann/Tholen:200?} and \cite{Seal:2005}) provide a setting for the presentation of topological spaces as structured objects in the category of sets and relations:
\[
\mathbf{Top}\cong\mathbf{Alg}(\mathsf{F},\mathbf{2})\ .
\]
The category $\mathbf{Alg}(\mathsf{F},\mathbf{2})$ of these structured objects depends on the filter monad $\mathsf{F}$ and the two-element ordered chain $\mathbf{2}$. In this context, a notion of fuzziness may be introduced by replacing the ordered chain $\mathbf{2}$ by a larger unital quantale $\mathbf{V}$.
Although the previous descriptions of topological spaces are both based on the filter monad, the first approach does not require the existence of a lax extension of the monad functor, an extension which is crucial for the second. This remark is at the origin of the present work, as it suggests that the information pertaining to the construction of a lax extension of an arbitrary functor $T$ may be extracted from the objects of the category $\mathbf{KlAlg}(\mathsf{T})$ associated with a monad $\mathsf{T}=(T,e,m)$. Not only is this the case, but the resulting category $\mathbf{Alg}(\mathsf{T},\mathbf{2})$ of lax algebras is isomorphic to $\mathbf{KlAlg}(\mathsf{T})$, thus generalizing the previous correspondence obtained for the filter monad. By modifying Zhang's tower extension construction \cite{Zhang:2000} to include unital quantales, we can moreover define a category $\mathbf{KlAlg}(\mathsf{T},\mathbf{V})$ such that
\[
\mathbf{KlAlg}(\mathsf{T},\mathbf{V})\cong\mathbf{Alg}(\mathsf{T},\mathbf{V})\ ,
\]
and for which $\mathbf{KlAlg}(\mathsf{T})$ is the $\mathbf{V}=\mathbf{2}$ instance: $\mathbf{KlAlg}(\mathsf{T},\mathbf{2})\cong\mathbf{KlAlg}(\mathsf{T})$.
The present paper is organized as follows. After establishing certain definitions in Section~\ref{intro}, we present a preliminary result that puts forth conditions a lax extension should satisfy in order to generalize the isomorphism between $\mathbf{KlAlg}(\mathsf{F})$ and $\mathbf{Alg}(\mathsf{F},\mathbf{2})$. These conditions lead in Section~\ref{laxextension} to the actual construction of a lax extension $T_{\!_M}$ of $T$ from quite a different perspective than in \cite{Barr:1970}, \cite{Clementino/Hofmann:2004}, \cite{Schubert:2006} or \cite{Seal:2005}. Such a lax extension yields a category $\mathbf{Alg}(\mathsf{T},\mathbf{V})$ of $(\mathsf{T},\mathbf{V})$-algebras, where $\mathbf{V}$ may be any unital quantale rather than just $\mathbf{2}$. In turn, this leads in Section~\ref{main} to the definition of the category $\mathbf{KlAlg}(\mathsf{T},\mathbf{V})$ of Kleisli $(\mathsf{T},\mathbf{V})$-algebras, which is a generalization of the category $\mathbf{KlAlg}(\mathsf{T})$. Our main result then conveniently states that the two categories $\mathbf{KlAlg}(\mathsf{T},\mathbf{V})$ and $\mathbf{Alg}(\mathsf{T},\mathbf{V})$ are isomorphic, thus allowing for an ``extension-free'' description of certain categories of lax algebras. In fact, if we refer to the original example, the Kleisli construction defines lax algebras via their ``neighborhood systems''. This is illustrated in Section \ref{clos}, in which the category $\mathbf{Cls}$ of closure spaces is presented by way of such structures (this also provides a new description of $\mathbf{Cls}$ as a category of lax algebras). Finally, Section \ref{fuz} is a brief incursion into the realm of fuzzy topology, in which fuzzy topological spaces (as defined for example in \cite{Hohle:2001}, or \cite{Gahler:1992}) are shown to be particular instances of lax algebras. This example is simply the original isomorphism $\mathbf{KlAlg}(\mathsf{F})\cong\mathbf{Alg}(\mathsf{F},\mathbf{2})$ in which the filter monad is replaced by a fuzzy filter monad.
It seems relevant now to mention another example arising in the context of lax algebras. Indeed, recall that the category $\mathbf{App}$ of approach spaces is isomorphic to $\mathbf{Alg}(\mathsf{F},\overline{\mathbf{R}}_+)$, where $\overline{\mathbf{R}}_+$ is the extended real line. Although the categories of fuzzy topological and approach spaces are both generalizations of $\mathbf{Top}\cong\mathbf{Alg}(\mathsf{F},\mathbf{2})$, the first is obtained by extending the filter monad $\mathsf{F}$, while the second by extending the underlying quantale $\mathbf{2}$. These examples clearly illustrate the difference between the two perspectives mentioned in the opening paragraph.
\section{Motivating result} \label{intro}
Before stating our preliminary result, we present a number of definitions, and recall some useful properties of the structures we will be using. For more details on lax algebras, we refer to the articles mentioned in the Introduction.
\begin{nr}\label{coherent}
\textbf{Monads factoring through a category.} Let $\mathbf{C}$ be a subcategory of the category $\mathbf{Ord}$ of preordered sets. A $\mathbf{Set}$-monad $\mathsf{T}=(T,e,m)$ \emph{factors through} $\mathbf{C}$ if there is a functor $S:\mathbf{Set}\to\mathbf{C}$ that composes with the forgetful functor to yield $T$, and such that $m_X:T^2X\to TX$ is the image of a morphism $m_X:STX\to SX$ of $\mathbf{C}$. To simplify notations, we will not distinguish between $SX$ and $TX$; for example, if $\mathsf{T}$ factors through the category $\mathbf{Sup}$ of complete lattices and sup-preserving maps, $\mathbf{Set}$-maps $Tf:TX\to TY$, as well as $m_X:T^2X\to TX$, will be considered as a sup-preserving maps between complete lattices.
The monad $\mathsf{T}$ factors \emph{coherently} through $\mathbf{C}$ if for any $f,g\in\mathbf{Set}(X,TY)$, we have
\begin{equation}\tag{$\ast$}
f\le g\implies m_Y\cdot Tf\le m_Y\cdot Tg\ ,
\end{equation}
where $\mathbf{Set}(X,TY)$ is equipped with the preorder induced by $TY$:
\begin{equation}\tag{$\ast\ast$}
f\le g\iff\text{ for all }x\in X,\text{ we have }f(x)\le g(x)\ .
\end{equation}
The notion of a monad factoring coherently though $\mathbf{Sup}$ is similar in spirit to the ordered monads of \cite{Gahler:1992}.
\end{nr}
\begin{nr}
\textbf{Kleisli $\mathsf{T}$-algebras.} Let $\mathsf{T}=(T,e,m)$ be a $\mathbf{Set}$-monad factoring coherently through $\mathbf{Ord}$, and denote by $\mathbf{Kl}(\mathsf{T})$ the associated Kleisli category. Recall that the \emph{Kleisli composition} $\beta\circ\alpha:X\to TZ$ of $\alpha:X\to TY$ and $\beta:Y\to TZ$ is given by $m_Z\cdot T\beta\cdot\alpha$, and the identity morphism in $\mathbf{Kl}(\mathsf{T})$ is $e_X:X\to TX$. By composing with $e_Y:Y\to TY$, a $\mathbf{Set}$-map $f:X\to Y$ becomes an element of $\mathbf{Set}(X,TY)=\mathbf{Kl}(\mathsf{T})(X,Y)$, and we write $f^\sharp:=e_Y\cdot f$. Remark that the condition ($\ast$) of \ref{coherent} is equivalent to preservation of the preorder on $\mathbf{Set}(X,TY)$ in the first variable of the Kleisli composition, while preservation of this preorder in the second variable simply follows from ($\ast\ast$).
The category $\mathbf{KlAlg}(\mathsf{T})$ of \emph{Kleisli $\mathsf{T}$-algebras}, has as objects pairs $(X,\alpha)$ with $X$ a set and
$\alpha:X\to TX$ a \emph{structure map} that is \emph{extensive} and \emph{idempotent}:
\begin{enumerate}[$(K_1)$]
\item $e_X\le\alpha$ ,
\item $\alpha\circ\alpha\le\alpha$ .
\end{enumerate}
Of course, in presence of the extensivity condition, idempotency may be expressed as an equality. Morphisms $f:(X,\alpha)\to(Y,\beta)$ are $\mathbf{Set}$-maps $f:X\to Y$ satisfying:
\begin{enumerate}[$(K_1)$]
\setcounter{enumi}{2}
\item $f^\sharp\circ\alpha\le\beta\circ f^\sharp$ ,
\end{enumerate}
and composing as in $\mathbf{Set}$.
\end{nr}
\begin{nr}
\textbf{Complete distributivity.} In this work, $\mathbf{V}$ will always denote a unital quantale with two-sided unit $k$, and we will assume that $\mathbf{V}$ is \emph{non-trivial}, that is, $\bot\ne k$. It will often be useful to suppose that $\mathbf{V}$ is \emph{completely distributive}, \textit{i.e.} that any $b\in\mathbf{V}$ may be obtained as
\[
b=\bigvee\{a\in\mathbf{V}\,|\,a\prec b\}\ ,
\]
where $a\prec b$ means that for any subset $S\subseteq\mathbf{V}$ with $b\le\bigvee S$, there exists $s\in S$
satisfying $a\le s$. The following properties follow from the definition of $\prec$:
\begin{enumerate}[i)]
\item $a\prec b$ implies $a\le b$ ;
\item $a\le a'\prec b'\le b$ implies $a\prec b$ ;
\item $a\prec\bigvee S$ implies there exists $s\in S$ with $a\prec s$ .
\end{enumerate}
\end{nr}
\begin{nr}
\textbf{Lax extensions.} Let $\mathbf{V}$ be a unital quantale, and denote by $\mathbf{Mat}(\mathbf{V})$ the category of \emph{$\mathbf{V}$-matrices} (or \emph{$\mathbf{V}$-relations}). Recall that the objects of $\mathbf{Mat}(\mathbf{V})$ are sets, morphisms $r:X\nrightarrow Y$ are maps $r:X\times Y\to\mathbf{V}$, and the transpose $r^\circ:Y\nrightarrow X$ of $r:X\nrightarrow Y$ is defined by $r^\circ(y,x)=r(x,y)$ for all $x\in X$, $y\in Y$. Composition of $r:X\nrightarrow Y$ and $s:Y\nrightarrow Z$ is given by
\[
s\cdot r(x,z)=\bigvee_{y\in Y}r(x,y)\otimes s(y,z)\ ,
\]
and the identity $1_X:X\nrightarrow X$ is defined by $1_X(x,y)=k$ if $x=y$ and $1_X(x,y)=\bot$ otherwise. There is also an order on the hom-sets of $\mathbf{Mat}(\mathbf{V})$ induced by the order on $\mathbf{V}$. Finally, a $\mathbf{Set}$-map $f:X\to Y$ will be identified with the matrix $f:X\nrightarrow Y$ given by $f(x,y)=k$ if $f(x)=y$ and $f(x,y)=\bot$ otherwise. We point out that if $f:X\to Y$, $g:W\to Z$ are $\mathbf{Set}$-maps, and $s:Y\nrightarrow Z$ is a $\mathbf{V}$-matrix, then
\[
g^\circ\cdot s\cdot f(x,w)=s(f(x),g(w))\ ,
\]
for all $x\in X$, $w\in W$.
A \emph{lax extension} of a $\mathbf{Set}$-functor $T$ is a map
\[
T_{\!_M}:\mathbf{Mat}(\mathbf{V})\to\mathbf{Mat}(\mathbf{V})\quad,\qquad(r:X\nrightarrow Y)\mapsto(T_{\!_M} r:TX\nrightarrow TY)
\]
which preserves the order on the hom-sets and satisfies
\begin{enumerate}[$(T_1)$]
\item $Tf\leT_{\!_M} f$ and $(Tf)^\circ\leT_{\!_M} f^\circ$ ,
\item $T_{\!_M} s\cdotT_{\!_M} r\leT_{\!_M}(s\cdot r)$ ,
\end{enumerate}
for all $f:X\to Y$, $r:X\nrightarrow Y$ and $s:Y\nrightarrow Z$. An important consequence of these conditions is that if
$f:X\nrightarrow Y$ and $g:Y\nrightarrow Z$ come from $\mathbf{Set}$-maps, then
\[
T_{\!_M}(s\cdot f)=T_{\!_M} s\cdotT_{\!_M} f=T_{\!_M} s\cdot Tf\quad\text{ and }\quadT_{\!_M}(g^\circ\cdot r)=T_{\!_M} g^\circ\cdotT_{\!_M} r=(Tg)^\circ\cdotT_{\!_M} r\ .
\]
Finally, when $TX$ is an ordered set, we say that $T_{\!_M}$ is \emph{order-compatible} if for all $\mathfrak{x},\mathfrak{y}\in
TX$, we have
\[
\mathfrak{x}\le\mathfrak{y}\iff k\leT_{\!_M} 1_X(\mathfrak{x},\mathfrak{y})\ .
\]
\end{nr}
\begin{nr}
\textbf{Lax algebras.} Let $\mathsf{T}=(T,e,m)$ be a $\mathbf{Set}$-monad equipped with a lax extension $T_{\!_M}$ of $T$. The category $\mathbf{Alg}(\mathsf{T},\mathbf{V})$ of \emph{$(\mathsf{T},\mathbf{V})$-algebras}, also called \emph{lax algebras}, has as objects pairs $(X,r)$, where $X$ is a set, and $r:TX\nrightarrow X$ a \emph{structure $\mathbf{V}$-matrix} satisfying the \emph{reflexivity} and \emph{transitivity} laws:
\begin{enumerate}[$(L_1)$]
\item $1_X\le r\cdot e_X$ ,
\item $r\cdotT_{\!_M} r\le r\cdot m_X$ .
\end{enumerate}
Morphisms $f:(X,r)\to(Y,s)$ are $\mathbf{Set}$-maps $f:X\to Y$ satisfying:
\begin{enumerate}[$(L_1)$]
\setcounter{enumi}{2}
\item $r\le f^\circ\cdot s\cdot T f$ ,
\end{enumerate}
and composing as in $\mathbf{Set}$. In the case where the lax extension $T_{\!_M}$ is order-compatible, then it follows that the structure matrix of a lax algebra $(X,r)$ reverses the order on $TX$, \textit{i.e.} for $\mathfrak{x},\mathfrak{y}\in TX$ and $z\in X$, we have
\[
\mathfrak{x}\le\mathfrak{y}\implies r(\mathfrak{y},z)\le r(\mathfrak{x},z)\ .
\]
\end{nr}
\begin{rem}
In \cite{Seal:2005}, it was noted that a lax extension $T_{\!_M}$ of $T$ naturally defined an order on $TX$. The order described therein was the opposite of the order given above in the definition of order-compatibility, so the structure matrices of the associated lax algebras \emph{preserved} that order rather than reversing it. In both cases however, the order on $TX$ is chosen as the natural one (for instance if $\mathsf{T}$ is the filter monad $\mathsf{F}$, then $\mathfrak{f}\le\mathfrak{g}$ \emph{always} means that the filter $\mathfrak{f}$ is finer than $\mathfrak{g}$).
\end{rem}
\begin{nr}
\textbf{Continuous lax algebras.} Let $\mathsf{T}=(T,e,m)$ be a $\mathbf{Set}$-monad factoring through $\mathbf{Sup}$, provided with an order-compatible lax extension $T_{\!_M}$. The $(\mathsf{T},\mathbf{V})$-algebra $(X,r)$ is said to be \emph{continuous} if for all $y\in X$ and $\mathcal{A}\subseteq TX$, we have
\[
\bigwedge_{\mathfrak{x}\in\mathcal{A}}r(\mathfrak{x},y)=r(\bigvee\mathcal{A},y)\ .
\]
The full subcategory of $\mathbf{Alg}(\mathsf{T},\mathbf{V})$ whose objects are the continuous lax algebras is denoted by $\mathbf{Alg}_\mathbf{cont}(\mathsf{T},\mathbf{V})$.
For example, if $T_{\!_M}$ is the op-canonical extension of either the filter or the powerset monad, then any lax algebra is continuous (this is a particular case of Proposition \ref{prop3}).
\end{nr}
\begin{nr}
\textbf{Kleisli $\mathsf{T}$-algebras and $(\mathsf{T},\mathbf{2})$-algebras.} The correspondence between Kleisli $\mathsf{F}$-al\-ge\-bras and $(\mathsf{F},\mathbf{2})$-algebras is given as a ``functional description of lax algebras'' in \cite{Hofmann/Tholen:200?}. In fact, the case where $\mathsf{T}$ is the filter monad provides an ideal setting in which the relation between Kleisli $\mathsf{T}$-algebras and $(\mathsf{T},\mathbf{2})$-algebras may be described. Indeed, recall that a topological space may be defined by two conditions on its neighborhood filters (the Kleisli presentation), or by two conditions on the ``convergence'' relation between filters and points (the lax algebra presentation). Given the first, one can obtain the second by stating that every filter finer than the neighborhood filter of a point converges to that point. Similarly, if the relation between filters and point is given, one can obtain the neighborhood filter of a point by taking the coarsest among all the filters that converge to that point. This correspondence is concretized in the following result.
\end{nr}
\begin{prop} \label{prop1}
Let $\mathsf{T}=(T,e,m)$ be a $\mathbf{Set}$-monad factoring coherently through $\mathbf{Sup}$ provided with an order-compatible lax extension $T_{\!_M}$.
\begin{enumerate}[i)]
\item There is a concrete functor $F:\mathbf{Alg}_\mathbf{cont}(\mathsf{T},\mathbf{2})\to\mathbf{KlAlg}(\mathsf{T})$ that associates to a structure matrix $r:TX\nrightarrow X$ the structure map $\alpha_r:X\to TX$ given by
\[
\alpha_r(y):=\bigvee\{\mathfrak{x}\in TX\,|\,r(\mathfrak{x},y)=\top\}\ .
\]
\item Suppose that the extension $T_{\!_M}$ satisfies
\[
T_{\!_M} r(\mathfrak{X},\mathfrak{y})=\top\implies m_X(\mathfrak{X})\le m_X\cdot T\alpha_r(\mathfrak{y})\ ,
\]
for all continuous structure matrices $r:TX\nrightarrow X$, and elements $\mathfrak{X}\in T^2X$, $\mathfrak{y}\in TX$. Then the concrete functor $G:\mathbf{KlAlg}(\mathsf{T})\to\mathbf{Alg}_\mathbf{cont}(\mathsf{T},\mathbf{2})$ that associates to a structure map $\alpha:X\to TX$ the structure matrix $r_\alpha:TX\nrightarrow X$ given by
\[
r_\alpha(\mathfrak{x},y)=\top\iff\mathfrak{x}\le\alpha(y)\
\]
is inverse to $F$.
\end{enumerate}
\end{prop}
\begin{proof}
The proof of this statement is almost identical to the proof of Theorem \ref{thm2}.
\end{proof}
\begin{rem}
In order to obtain a better description of the lax extension $T_{\!_M}$, one might be tempted to replace the previous condition ``$\,T_{\!_M} r(\mathfrak{X},\mathfrak{y})=\top\implies m_X(\mathfrak{X})\le m_X\cdot T\alpha_r(\mathfrak{y})\,$'' by a more restrictive one such as ``$\,T_{\!_M} r(\mathfrak{X},\mathfrak{y})=\top\iff\mathfrak{X}\le T\alpha_r(\mathfrak{y})\,$''. Although the result would remain true, this last equivalence is unfortunately not satisfied by the usual lax extensions of the powerset and filter monads.
\end{rem}
\section{A lax extension of $T$} \label{laxextension}
\begin{nr}
\textbf{The Kleisli extension.} Let $\mathsf{T}=(T,e,m)$ be a $\mathbf{Set}$-monad factoring coherently through $\mathbf{Sup}$, and consider the powerset monad $\P=(P,d,n)$. The unique sup-preserving map $\eta_X:PX\to TX$ extending $e_X:X\to TX$ along $d_X:X\to PX$ is given by
\[
\eta_X(A)=\bigvee\{e_X(x)\,|\,x\in A\}\ ,
\]
and defines a natural transformation $\eta:P\to T$ satisfying $\eta\cdot d=e$. Moreover, since $T\eta\cdot eP=eT\cdot\eta$ by naturality of $e$, we have for any $\mathcal{A}\in P^2X$ that
\[
m_X\cdot T\eta_X\cdot\eta_{PX}(\mathcal{A})=\bigvee_{A\in\mathcal{A}}m_X\cdot T\eta_X\cdot e_{PX}(A)=\bigvee_{A\in\mathcal{A}}\eta_X(A)=\eta_X\cdot n_X(\mathcal{A})\ .
\]
Thus, $\eta:\P\to\mathsf{T}$ is in fact a monad morphism.
For a $\mathbf{V}$-matrix $r:X\nrightarrow Y$, let $\rho_r=(\rho_r^a:Y\to PX)_{a\in\mathbf{V}}$ be the family of maps given by
\[
\rho_r^a(y):=\{x\in X\,|\,a\le r(x,y)\}\ .
\]
The \emph{Kleisli extension of $T$} is $T_{\!_M}:\mathbf{Mat}(\mathbf{V})\to\mathbf{Mat}(\mathbf{V})$ defined by
\[
T_{\!_M} r(\mathfrak{x},\mathfrak{y}):=\bigvee\{a\in\mathbf{V}\,|\,\mathfrak{x}\le m_X\cdot T(\eta_X\cdot\rho_r^a)(\mathfrak{y})\}\ ,
\]
for all $\mathfrak{x}\in TX$, and $\mathfrak{y}\in TY$.
\end{nr}
\begin{prop}
The Kleisli extension $T_{\!_M}$ of $T$ is a lax extension of $T$.
\end{prop}
\begin{proof}
Remark first that $T_{\!_M}$ preserves the order on the hom-sets because $\mathsf{T}$ factors coherently through $\mathbf{Sup}$. For a map $f:X\to Y$, $a\in\mathbf{V}$, $x\in X$, and $y\in Y$, we have
\[
\rho_f^a(y)=\left\{\begin{array}{l@{\quad}l}
X & \text{if }a=\bot\\
f^{-1}\{y\} & \text{if }\bot\ne a\le k\\
\emptyset & \text{otherwise,}
\end{array}\right.
\qquad\text{and}\qquad \rho_{f^\circ}^a(x)=\left\{\begin{array}{l@{\quad}l}
Y & \text{if }a=\bot\\
\{f(x)\} & \text{if }\bot\ne a\le k\\
\emptyset & \text{otherwise.}
\end{array}\right.
\]
On one hand, for all $a\le k$ we have $\eta_X\cdot d_X\le\eta_X\cdot\rho_f^a\cdot f$, so $\mathfrak{x}=m_X\cdot T(\eta_X\cdot d_X)(\mathfrak{x})\le m_X\cdot T(\eta_X\cdot\rho_f^a)\cdot Tf(\mathfrak{x})$; therefore, $k\leT_{\!_M} f(\mathfrak{x},Tf(\mathfrak{x}))$, and $Tf\leT_{\!_M} f$. On the other hand, for all $a\le k$ we have $\eta_X\cdot d_X\cdot f\le\eta_X\cdot\rho_{f^\circ}^a$, and we may proceed as before to get $k\leT_{\!_M} f^\circ(Tf(\mathfrak{x}),\mathfrak{x})$, or $(Tf)^\circ\leT_{\!_M} f^\circ$.
To prove that $T_{\!_M}$ is a lax functor, let $r:X\nrightarrow Y$ and $s:Y\nrightarrow Z$ be two $\mathbf{V}$-matrices, $\mathfrak{x}\in TX$, $\mathfrak{y}\in
TY$, and $\mathfrak{z}\in TZ$. Let $a,b\in\mathbf{V}$ be such that $\mathfrak{x}\le m_X\cdot T(\eta_X\cdot\rho_r^a)(\mathfrak{y})$ and $\mathfrak{y}\le m_Y\cdot T(\eta_Y\cdot\rho_s^b)(\mathfrak{z})$, so
$\mathfrak{x}\le m_X\cdot T(\eta_X\cdot\rho_r^a)\cdot m_Y\cdot T(\eta_Y\cdot\rho_s^b)(\mathfrak{z})$. Note furthermore that
\begin{align*}
m_X\cdot T(\eta_X\cdot\rho_r^a)\cdot m_Y\cdot T(\eta_Y\cdot\rho_s^b)&=m_X\cdot m_{TX}\cdot T(T(\eta_X\cdot\rho_r^a)\cdot\eta_Y\cdot\rho_s^b)\\
&=m_X\cdot T(m_X\cdot\eta_{TX}\cdot P(\eta_X\cdot\rho_r^a)\cdot\rho_s^b)\\
&=m_X\cdot T(\eta_X\cdot n_X\cdot P\rho_r^a\cdot\rho_s^b)\ .
\end{align*}
Since
\[
n_X\cdot P\rho_r^a\cdot\rho_s^b(z)=\{x\in X\,|\,\exists y\in Y:a\le r(x,y),b\le
s(y,z)\}\subseteq\rho_{s\cdot r}^{a\otimes b}(z)\ ,
\]
we obtain $\mathfrak{x}\le m_X\cdot T(\eta_X\cdot\rho_{s\cdot r}^{a\otimes b})(\mathfrak{z})$. Finally, suprema are preserved by $\otimes$ in each variable, so that $T_{\!_M} r(\mathfrak{x},\mathfrak{y})\otimesT_{\!_M} s(\mathfrak{y},\mathfrak{z})\leT_{\!_M}(s\cdot r)(\mathfrak{x},\mathfrak{z})$, and $T_{\!_M}$ is a lax extension of $T$ as claimed.
\end{proof}
\begin{prop} \label{prop3}
If $T_{\!_M}$ is the Kleisli extension of $T$, then the following assertions hold.
\begin{enumerate} [i)]
\item If $\mathbf{V}$ is non-trivial, then $T_{\!_M}$ is order-compatible.
\item For any set $X$ and $a\in\mathbf{V}$, $(a_{TX}\wedge 1_{TX})\leT_{\!_M}(a_X\wedge 1_X)$, where $a_X:X\nrightarrow X$ is
the $\mathbf{V}$-matrix with constant value $a\in\mathbf{V}$.
\item Any $(\mathsf{T},\mathbf{V})$-algebra structure $r:TX\nrightarrow X$ is continuous, so that
\[
\mathbf{Alg}_\mathbf{cont}(\mathsf{T},\mathbf{V})=\mathbf{Alg}(\mathsf{T},\mathbf{V})\ .
\]
\end{enumerate}
\end{prop}
\begin{proof}
\textit{i)} Consider the identity $1_X:X\to X$. Then $\rho_{1_X}^a$ is the unit $d_X$ of the powerset monad whenever $a\in\mathbf{V}$ satisfies $\bot\ne a\le k$. Thus, on one hand $\mathfrak{x}\le\mathfrak{y}$ implies $k\leT_{\!_M} 1_X(\mathfrak{x},\mathfrak{y})$. On the other hand, since $\mathbf{V}$ is non-trivial, $k\leT_{\!_M} 1_X(\mathfrak{x},\mathfrak{y})$ implies there exists $a\in\mathbf{V}$ such that $\bot\ne a\le k$ and $\mathfrak{x}\le m_X\cdot(T\eta_X\cdot\rho_{1_X}^a)(\mathfrak{y})=\mathfrak{y}$.
\textit{ii)} Let $a\in\mathbf{V}\setminus\{\bot\}$ and consider the relation $r=a_X\wedge 1_X:X\nrightarrow X$. Then $\rho_r^b=d_X$ if in particular $b=a\wedge k$, so that $m_X\cdot(T\eta_X\cdot d_X)(\mathfrak{x})=\mathfrak{x}$ yields $a\wedge k\leT_{\!_M} r(\mathfrak{x},\mathfrak{x})$ as required.
\textit{iii)} Suppose now that $r:TX\nrightarrow X$ is a $(\mathsf{T},\mathbf{V})$-algebra structure, and let $\mathcal{A}\subseteq TX$. Then $\mathfrak{X}=\eta_{TX}(\mathcal{A})$ naturally satisfies $m_X(\mathfrak{X})=\bigvee\mathcal{A}$, so $T_{\!_M} r(\mathfrak{X},e_X(z))\le r(\bigvee\mathcal{A},z)$. By using naturality of $e$ and the definition of $\eta$, we observe that
\[
T_{\!_M} r(\mathfrak{X},e_X(z))=\bigvee\big\{a\in\mathbf{V}\,\big|\,\textstyle{\bigvee}\{e_{TX}(\mathfrak{x})\,|\,\mathfrak{x}\in\mathcal{A}\} \le\textstyle{\bigvee}\{e_{TX}(\mathfrak{x})\,|\,\mathfrak{x}\in\rho_r^a(z)\}\big\}\ .
\]
Thus, for any $a$ such that $\mathcal{A}\subseteq\rho_r^a(z)$, we have $a\le T_{\!_M} r(\mathfrak{X},e_X(z))$. This is the case in particular for $a=\bigwedge_{\mathfrak{x}\in\mathcal{A}}r(\mathfrak{x},z)$, so that $\bigwedge_{\mathfrak{x}\in\mathcal{A}}r(\mathfrak{x},z)\le r(\bigvee\mathcal{A},z)$. We can conclude that this last inequality is in fact an equality since $r$ reverses the order in the first variable. \end{proof}
\begin{ex}
If $\mathbf{V}$ is a completely distributive lattice, then the Kleisli extensions of the powerset and the filter functors are given by
\begin{align*}
P_{\!_M} r(A,B)&=\bigwedge_{x\in A}\bigvee_{y\in B}r(x,y)\ ,\qquad\text{ and }\\
F_{\!_M} r(\mathfrak{f},\mathfrak{g})&=\bigwedge_{B\in\mathfrak{g}}\,\bigvee_{A\in\mathfrak{f}}\,\bigwedge_{x\in A}\,\bigvee_{y\in B}r(x,y)\
\end{align*}
respectively, where $r:X\nrightarrow Y$ is a $\mathbf{V}$-matrix, $A,B\in PX$, and $\mathfrak{f},\mathfrak{g}\in FX$. Note that these are also the op-canonical extensions of the corresponding functors.
\end{ex}
\section{Towers of Kleisli algebras} \label{main}
In Proposition \ref{prop1}, it has been shown how Kleisli $\mathsf{T}$-algebras may be related to $(\mathsf{T},\mathbf{2})$-algebras. In order to extend this correspondence to other quantales than $\mathbf{2}$, we introduce the following definition, which is based on Zhang's tower extensions \cite{Zhang:2000}.
\begin{nr}
\textbf{Kleisli $(\mathsf{T},\mathbf{V})$-algebras.} Let $\mathsf{T}=(T,e,m)$ be a $\mathbf{Set}$-monad factoring coherently through $\mathbf{Sup}$, $\mathbf{V}$ a unital quantale, and notice that $\mathbf{2}$ embeds into $\mathbf{V}$ via
\[
\bot\mapsto\bot\ ,\quad\top\mapsto k\ .
\]
The \emph{tower extension} of $\mathbf{Kl}(\mathsf{T})$ along $\mathbf{2}\to\mathbf{V}$ is the category $\mathbf{KlAlg}(\mathsf{T},\mathbf{V})$ (also called the category of \emph{Kleisli $(\mathsf{T},\mathbf{V})$-algebras}) whose objects are pairs $(X,\alpha)$, with $\alpha$ a $\mathbf{V}$-indexed family of morphisms $\alpha=(\alpha^a:X\to TX)_{a\in\mathbf{V}}$ satisfying the following conditions:
\begin{enumerate}
\item[$(K_0)$] $\alpha^{\bigvee\mathcal{A}}=\bigwedge_{a\in\mathcal{A}}\alpha^a$ ,
\item[$(K_1)$] $e_X\le\alpha^k$ ,
\item[$(K_2)$] $\alpha^a\circ\alpha^b\le\alpha^{a\otimes b}$ ,
\end{enumerate}
for all $\mathcal{A}\subseteq\mathbf{V}$, and $a,b\in\mathbf{V}$. When the monad $\mathsf{T}$ is clearly determined by the context, such a structure $\alpha$ will be called a \emph{$\mathbf{V}$-tower} on $X$. Morphisms $f:(X,\alpha)\to(Y,\beta)$ are maps $f:X\to Y$ such that
\begin{enumerate}
\item[$(K_3)$] $f^\sharp\circ\alpha^a\le\beta^a\circ f^\sharp$ ,
\end{enumerate}
for all $a\in\mathbf{V}$, and composing as in $\mathbf{Set}$. Since $(K_0)$ yields in particular that $\alpha^\bot(x)=\top$ for all $x\in X$, and tower $\alpha$, the category $\mathbf{KlAlg}(\mathsf{T},\mathbf{2})$ is concretely isomorphic to $\mathbf{KlAlg}(\mathsf{T})$. Moreover, if $\mathbf{V}$ is completely distributive, then $(K_0)$ is equivalent to
\begin{enumerate}
\item[$(K_0')$] $\alpha^a=\bigwedge_{b\prec a}\alpha^b$ ,
\end{enumerate}
for all $a\in\mathbf{V}$. Notice that a $\mathbf{V}$-tower $\alpha$ is in fact a sup-preserving map $\alpha:\mathbf{V}\to\mathbf{Set}(X,TX)^\mathrm{op}$ that forms an op-lax functor with respect to the multiplicative structures. The previous presentation via families of Kleisli endomorphisms appears however to be more practical for our purpose.
\end{nr}
\begin{rem}
The original definitions of tower extensions in \cite{Brock/Kent:1997} and \cite{Zhang:2000} only considered the indexing set $\mathbf{V}$ as a complete lattice, rather than a quantale. This explains in part why approach spaces---which explicitly make use of the addition of $\overline{\mathbf{R}}_+$ in their definition---were not directly described as tower extensions of topological spaces.
\end{rem}
\begin{thm} \label{thm2}
Let $(\mathsf{T},\mathbf{V})$ be a $\mathbf{Set}$-monad factoring coherently through $\mathbf{Sup}$, and suppose that $\mathbf{V}$ is completely
distributive. If $\mathbf{Alg}(\mathsf{T},\mathbf{V})$ denotes the category of lax algebras associated to the Kleisli extension $T_{\!_M}$,
then $\mathbf{Alg}(\mathsf{T},\mathbf{V})$ and $\mathbf{KlAlg}(\mathsf{T},\mathbf{V})$ are concretely isomorphic.
More precisely, to a $(\mathsf{T},\mathbf{V})$-algebra structure $r:TX\nrightarrow X$ can be associated a $\mathbf{V}$-tower $\alpha_r=(\alpha_r^a:X\to TX)_{a\in\mathbf{V}}$ defined by
\[
\alpha_r^a(y):=\bigvee\{\mathfrak{x}\,|\,a\le r(\mathfrak{x},y)\}\ ,
\]
and to a $\mathbf{V}$-tower $\alpha=(\alpha^a:X\to TX)_{a\in\mathbf{V}}$ can be associated a $(\mathsf{T},\mathbf{V})$-algebra structure $r_\alpha:TX\nrightarrow X$ given by
\[
r_\alpha(\mathfrak{x},y):=\bigvee\{a\in\mathbf{V}\,|\,\mathfrak{x}\le\alpha^a(y)\}\ .
\]
This correspondence yields two concrete functors
\[
F:\mathbf{Alg}(\mathsf{T},\mathbf{V})\to\mathbf{KlAlg}(\mathsf{T},\mathbf{V})\quad\text{ and }\quad G:\mathbf{KlAlg}(\mathsf{T},\mathbf{V})\to\mathbf{Alg}(\mathsf{T},\mathbf{V})
\]
that are inverses of each other.
\end{thm}
\begin{proof}
To prove that $F$ is well-defined, consider a $(\mathsf{T},\mathbf{V})$-algebra structure $r:TX\nrightarrow X$. In order to verify $(K_0)$ for $\alpha_r$, let $y\in X$, and recall that $\rho_r^b(y)=\{\mathfrak{x}\in TX\,|\,b\le r(\mathfrak{x},y)\}$ (where $b\in\mathbf{V}$), so we have $\alpha_r^b(y)=\bigvee\rho_r^b(y)$. If $\mathcal{A}\subseteq\mathbf{V}$, then by continuity of $r$, any $c\in\mathcal{A}$ satisfies
\[
c\le r(\textstyle{\bigvee}\rho_r^c(y),y)\le r(\textstyle{\bigwedge}_{b\in\mathcal{A}}\textstyle{\bigvee}\rho_r^b(y),y)\ ,
\]
so that $a=\bigvee_{c\in\mathcal{A}}c\le r(\bigwedge_{b\in\mathcal{A}}\alpha_r^b(y),y)$, and $\bigwedge_{b\in\mathcal{A}}\alpha_r^b(y)\le\alpha_r^a(y)$; equality follows, since $a\le b$ yields $\alpha_r^b\le\alpha_r^a$ for all $a,b\in\mathbf{V}$.
Reflexivity of $r$ immediately implies $(K_1)$. To verify $(K_2)$ it suffices to show that $a\otimes b\le r(\alpha_r^a\circ\alpha_r^b(y),y)$ for all $a,b\in\mathbf{V}$. For this, remark first that $r\cdot\alpha_r^b\ge(b_X\wedge 1_X)$, so $T_{\!_M}(r\cdot\alpha_r^a)\ge(a_{TX}\wedge 1_{TX})$ by Proposition \ref{prop3}. Therefore,
\begin{align*}
r(m_X\cdot T\alpha_r^a\cdot\alpha_r^b(y),y)
&\geT_{\!_M} r(T\alpha_r^a\cdot\alpha_r^b(y),\alpha_r^b(y))\otimes r(\alpha_r^b(y),y)\\
&=T_{\!_M}(r\cdot\alpha_r^a)(\alpha_r^b(y),\alpha_r^b(y))\otimes(r\cdot\alpha_r^b)(y,y)\ge a\otimes b
\end{align*}
by transitivity of $r$. Finally, if $f:(X,r)\to(Y,s)$ is a morphism of lax algebras, then $r(\mathfrak{x},y)\le s(Tf(\mathfrak{x}),f(y))$ implies that $Tf\cdot\alpha_r^a(y)\le\alpha_s^a\cdot f(y)$ for all $a\in\mathbf{V}$.
Consider now a $\mathbf{V}$-tower $\alpha=(\alpha^a:X\to TX)_{a\in\mathbf{V}}$. To verify that $G$ is well-defined, we first need to prove the equality $\alpha_{r_\alpha}=\alpha$. For this, let $a\in\mathbf{V}$, $y\in X$, and set $\mathcal{A}=\{\mathfrak{x}\in TX\,|\,a\le\bigvee\mathcal{B}_\mathfrak{x}\}$, where $\mathcal{B}_\mathfrak{x}=\{b\in\mathbf{V}\,|\,\mathfrak{x}\le\alpha^b(y)\}$, so that $\alpha_{r_\alpha}^a(y)=\bigvee\mathcal{A}$. On one hand, $\mathfrak{x}=\alpha^a(y)$ is in $\mathcal{A}$ (since $a$ is in $\mathcal{B}_\mathfrak{x}$), so $\alpha\le\alpha_{r_\alpha}$. On the other hand, if $c\in\mathbf{V}$ is such that $c\prec a$, and $\mathfrak{x}\in\mathcal{A}$, then by complete distributivity of $\mathbf{V}$ there exists $b\in\mathcal{B}_\mathfrak{x}$ with $c\le b$. Therefore, for any $\mathfrak{x}\in\mathcal{A}$ we have $\mathfrak{x}\le\alpha^c(y)$, so $\mathfrak{x}\le\bigwedge_{c\prec a}\alpha^c(y)=\alpha^a(y)$ by $(K_0')$. This implies $\alpha_{r_\alpha}\le\alpha$, as required.
Reflexivity of $r_\alpha$ is an immediate consequence of $(K_1)$. For transitivity, note that if $b\in\mathbf{V}$, then
\[
m_X\cdot m_{TX}\cdot T(\eta_{TX}\cdot\rho_{r_\alpha}^a)=m_X\cdot T(m_X\cdot\eta_{TX}\cdot\rho_{r_\alpha}^a)=m_X\cdot T\alpha_{r_\alpha}^a\ ,
\]
so that $T_{\!_M} r_\alpha(\mathfrak{X},\mathfrak{y})\le\bigvee\{a\in\mathbf{V},|\,m_X(\mathfrak{X})\le m_X\cdot T\alpha^a(\mathfrak{y})\}$ because $\alpha_{r_\alpha}\le\alpha$. By definition of $r_\alpha(\mathfrak{y},z)$, we have
\[
T_{\!_M} r_\alpha(\mathfrak{X},\mathfrak{y})\otimes r_\alpha(\mathfrak{y},z)\le\bigvee\{a\otimes b\,|\,m_X(\mathfrak{X})\le\alpha^a\circ\alpha^b(z)\}\le
r_\alpha(m_X(\mathfrak{X}),z)
\]
by $(K_2)$. Now, let $f:(X,\alpha)\to(Y,\beta)$ be a morphism, and suppose that $Tf\cdot\alpha^a(y)\le\beta^a\cdot f(y)$ for all $a\in\mathbf{V}$; this implies that $\{a\in\mathbf{V}\,|\,\mathfrak{x}\le\alpha^a(y)\}\subseteq\{a\in\mathbf{V}\,|\,Tf(\mathfrak{x})\le\beta^a\cdot f(y)\}$, and consequently
$r_\alpha(\mathfrak{x},y)\le r_\beta(Tf(\mathfrak{x}),f(y))$.
The proof that $r_{\alpha_r}=r$ is quite similar to that of $\alpha_{r_\alpha}=\alpha$. Indeed, let $\mathfrak{x}\in TX$, and $y\in X$. Consider $\mathcal{A}=\{a\in\mathbf{V}\,|\,\mathfrak{x}\le\bigvee\mathcal{B}_a\}$ where $\mathcal{B}_a=\{\mathfrak{y}\in TX\,|\,a\le r(\mathfrak{y},y)\}$, so that $r_{\alpha_r}(\mathfrak{x},y)=\bigvee\mathcal{A}$. On one hand, we observe that $a=r(\mathfrak{x},y)$ is in $\mathcal{A}$ (since $\mathfrak{x}$ is in $\mathcal{B}_a$), so $r\le r_{\alpha_r}$. On the other hand, if $a\in\mathbf{V}$ is such that $\mathfrak{x}\le\bigvee\mathcal{B}_a$, then by continuity of $r$ we have that $a\le r(\bigvee\mathcal{B}_a,y)\le r(\mathfrak{x},y)$, so $r_{\alpha_r}\le r$. This shows that $GF=\mathrm{Id}$. Since $\alpha_{r_\alpha}=\alpha$ implies that $FG=\mathrm{Id}$, we conclude that $G$ is an isomorphism with inverse $F$.
\end{proof}
\begin{rem}
The previous theorem yields another presentation of the category $\mathbf{App}\cong\mathbf{Alg}(\mathsf{F},\overline{\mathbf{R}}_+)$ of approach spaces \cite{Lowen:1997}, and suggests a new notion of ``approach system of neighborhoods''.
\end{rem}
\section{$\mathbf{V}$-valued closure spaces} \label{clos}
In \cite{Seal:2005}, it was shown that the category $\mathbf{Cls}$ of closure spaces could be seen as a category of $(\P,\mathbf{2})$-algebras. However, since $\mathbf{Top}\cong\mathbf{Alg}(\mathsf{F},\mathbf{2})$ is a full subcategory of $\mathbf{Cls}$, it would be useful to describe $\mathbf{Cls}$ as a lax algebra of the form $\mathbf{Alg}(\mathsf{D},\mathbf{2})$, with the filter monad $\mathsf{F}$ appearing as a submonad of $\mathsf{D}$. The aim of this Section is to provide such a description.
\begin{nr}
\textbf{The up-set monad.} An \emph{up-set} $\mathfrak{x}$ on $X$ is a set of subsets of $X$ such that for any $A,B\subseteq X$, we have
\[
A\subseteq B\text{ and }A\in\mathfrak{x}\implies B\in\mathfrak{x}\ .
\]
The set $DX$ of up-sets on $X$ is equipped with the order given by reverse inclusion:
\[
\mathfrak{x}\le\mathfrak{y}\iff\mathfrak{x}\supseteq\mathfrak{y}\ .
\]
In fact, $DX$ is a complete lattice, with supremum obtained via intersection, and infimum via union. There is only one up-set containing the empty set, namely the set $PX$ of subsets of $X$, which is also the bottom element of $DX$. On the other hand, the empty set is the top element of $DX$.
The \emph{up-set functor} $D$ assigns to a set $X$ the set $DX$ of up-sets on $X$, and sends a map $f:X\to Y$ to $Df:DX\to DY$ defined by
\[
B\in Df(\mathfrak{x})\iff f^{-1}(B)\in\mathfrak{x}\ ,
\]
where $\mathfrak{x}\in DX$. The \emph{up-set monad} $\mathsf{D}$ is the triple $(D,e,m)$, where $e:\mathrm{Id}\to D$ and $m:D^2\to
D$ are the natural transformations whose components at $X$ are given by
\[
A\in e_X(x)\iff x\in A\qquad\text{ and }\qquad A\in m_X(\mathfrak{X})\iff A^\sharp\in\mathfrak{X}\ ,
\]
where $A^\sharp=\{\mathfrak{x}\in DX\,|\,A\in\mathfrak{x}\}$, $x\in X$ and $\mathfrak{X}\in D^2X$. It follows immediately from this definition that the filter monad is a submonad of $\mathsf{D}$.
In view of Theorem \ref{thm2}, we point out that $\mathsf{D}$ factors coherently through $\mathbf{Sup}$.
\end{nr}
\begin{rem}
If $\mathbf{V}$ is completely distributive, then Theorem \ref{thm2} allows us to describe the category of $(\mathsf{T},\mathbf{V})$-algebras associated to the Kleisli extension $T_{\!_M}$ of $T$, without having to actually compute $T_{\!_M}$. However, it is not difficult to verify that in the present case the Kleisli extension of $D$ is given by
\[
D_{\!_M} r(\mathfrak{x},\mathfrak{y})=\bigwedge_{B\in\mathfrak{y}}\bigvee_{A\in\mathfrak{x}}\bigwedge_{x\in A}\bigvee_{y\in B}r(x,y) ,
\]
for a $\mathbf{V}$-matrix $r:X\nrightarrow Y$, and $\mathfrak{x},\mathfrak{y}\in DX$.
\end{rem}
\begin{nr}
\textbf{$\mathbf{V}$-valued closure operators.} The objects of the category $\mathbf{Cls}(\mathbf{V})$ are pairs $(X,c)$, where $X$ is a set and $c:PX\times X\to\mathbf{V}$ is a \emph{$\mathbf{V}$-valued closure operator} (called a \emph{closeness operator} in \cite{Seal:2005}), \textit{i.e.} a map satisfying:
\begin{enumerate}[$(C_1)$]
\item $x\in A\implies k\le c(A,x)$ ,
\item $A\subseteq B\implies c(A,x)\le c(B,x)$ ,
\item $a\otimes c(c_a[A],x)\le c(A,x)$ ,
\end{enumerate}
where $x\in X$, $A,B\subseteq X$, $a\in\mathbf{V}$ and $c_a[A]:=\{x\in X\,|\,a\le c(A,x)\}$. The pair $(X,c)$ is then a \emph{$\mathbf{V}$-closure space}. A morphism of $\mathbf{V}$-closure spaces $f:(X,c)\to(Y,d)$ is a $\mathbf{Set}$-map $f:X\to Y$ satisfying $c(A,x)\le d(f(A),f(x))$. Recall that if $\mathbf{V}=\mathbf{2}$, then a closure operator $\gamma:PX\to PX$ may be defined via
\[
x\in\gamma(A)\iff c(A,x)=\top\ ;
\]
in fact, this equivalence yields a concrete isomorphism between $\mathbf{Cls}(\mathbf{2})$ and the category $\mathbf{Cls}$ of closure spaces.
\end{nr}
\begin{nr}
\textbf{$\mathbf{V}$-graded closure operators.} Suppose that $\mathbf{V}$ is completely distributive. A \emph{$\mathbf{V}$-graded closure operator} on $X$ is a family $\gamma=(\gamma^a:PX\to PX)_{a\in\mathbf{V}}$ of operators satisfying:
\begin{enumerate}[$(\Gamma_1)$]
\item $A\subseteq\gamma^a(A)$ for all $a\le k$ ,
\item $A\subseteq B$ implies $\gamma^a(A)\subseteq \gamma^a(B)$ ,
\item $\gamma^b\cdot\gamma^a(A)\subseteq\gamma^{a\otimes b}(A)$ ,
\item $\gamma^a(A)=\bigcap_{b\prec a}\gamma^b(A)$ ,
\end{enumerate}
for all $A,B\subseteq X$, and $a,b\in\mathbf{V}$. A morphism $f:(X,(\gamma^a)_{a\in\mathbf{V}})\to(Y,(\delta^a)_{a\in\mathbf{V}})$ is a $\mathbf{Set}$-map $f:X\to Y$ satisfying $f(\gamma^a(A))\subseteq\delta^a(f(A))$ for all $a\in \mathbf{V}$. For convenience, the pair $(X,(\gamma^a)_{a\in\mathbf{V}})$ is also called a \emph{$\mathbf{V}$-closure space}, and the corresponding category denoted by $\mathbf{Cls}(\mathbf{V})$ (this abuse is justified by the following proposition).
\end{nr}
\begin{prop}
If $\mathbf{V}$ is completely distributive, then the category of $\mathbf{V}$-closure spaces given by $\mathbf{V}$-valued closure operators is concretely isomorphic to the category of $\mathbf{V}$-closure spaces given by $\mathbf{V}$-graded closure operators.
\end{prop}
\begin{proof}
Suppose first that $(X,c)$ satisfies $(C_1)$ to $(C_3)$, and set $\gamma^a(A):=c_a[A]$ for $A\in PX$. Then $(C_1)$ clearly implies $(\Gamma_1)$, $(C_2)$ implies $(\Gamma_2)$, and it is not hard to see that $(C_3)$ implies $(\Gamma_3)$. For $(\Gamma_4)$, observe on one hand that $b\prec a$ implies $\gamma^a(A)\subseteq \gamma^b(A)$, so $\gamma^a(A)\subseteq\bigcap_{b\prec a}\gamma^b(A)$. On the other hand, if $x\in\bigcap_{b\prec a}\gamma^b(A)$, then $b\le c(A,x)$ for all $b\prec a$, so that $a\le c(A,x)$, as required. If $f:(X,c)\to(Y,d)$ satisfies $c(A,x)\le d(f(A),f(x))$ for all $A\subseteq X$, and $x\in X$, then $x\in\gamma^a(A)$ implies $f(x)\in\delta^a(f(A))$, where $\delta^a(B):=d_a[B]$.
For a pair $(X,(\gamma^a)_{a\in\mathbf{V}})$ satisfying $(\Gamma_1)$ to $(\Gamma_4)$, set $c(A,x):=\bigvee\{a\in\mathbf{V}\,|\,x\in\gamma^a(A)\}$. Then $(\Gamma_1)$ immediately implies $(C_1)$, and $(\Gamma_2)$ implies $(C_2)$. Let $a\in\mathbf{V}$, and remark that $c_a[A]=\gamma^a(A)$ by using complete distributivity of $\mathbf{V}$ and $(\Gamma_4)$. Thus, $a\otimes c(c_a[A],x)=\bigvee\{a\otimes b\,|\,x\in \gamma^b\cdot\gamma^a(A)\}\le c(A,x)$ by $(\Gamma_3)$. Finally, if $f:(X,(\gamma^a)_{a\in\mathbf{V}})\to(Y,(\delta^a)_{a\in\mathbf{V}})$ satisfies $f(\gamma^a(A))\subseteq \delta^a(f(A))$ for all $a\in \mathbf{V}$, then it follows that $c(A,x)\le d(f(A),f(x))$, where $d$ is the $\mathbf{V}$-valued closure operator assigned to $(\delta^a)_{a\in\mathbf{V}}$.
The fact that these correspondences are inverses of one another has been proved partially in the previous paragraph, and the remaining part is clear.
\end{proof}
\begin{prop} \label{prop56}
If $\mathbf{V}$ is completely distributive, then the category of Kleisli $(\mathsf{D},\mathbf{V})$-algebras is concretely isomorphic to the category $\mathbf{Cls}(\mathbf{V})$ of $\mathbf{V}$-closure spaces.
More precisely, a $\mathbf{V}$-tower $\alpha=(\alpha^a: X\to DX)_{a\in\mathbf{V}}$ and a $\mathbf{V}$-graded closure operator $\gamma=(\gamma^a:PX\to PX)_{a\in\mathbf{V}}$ determine each other via
\[
x\in\gamma^a(A)\iff A^\c\notin\alpha^a(x)\ ,
\]
where $A\subseteq X$, and $A^\c$ denotes the complement of $A$ in $X$.
\end{prop}
\begin{proof}
Let $\alpha=(\alpha^a:X\to DX)_{a\in\mathbf{V}}$ be a $\mathbf{V}$-tower, and define $\gamma_\alpha=(\gamma_\alpha^a:PX\to PX)_{a\in\mathbf{V}}$ by $\gamma_\alpha^a(A):=\{x\in X\,|\,A^\c\notin\alpha^a(x)\}$, where $a\in\mathbf{V}$, and $A\subseteq X$. Since $\alpha$ satisfies $(K_0)$ and $(K_1)$, we have that $x\in A$ whenever $A\in\alpha^a(x)$ and $a\le k$. Thus, if $x\in A$, then $x\notin A^\c$, so $A^\c\notin\alpha^a(x)$ if $a\le k$, which proves $(\Gamma_1)$. If $A\subseteq B$ and $a\in\mathbf{V}$, $x\in X$ are such that $A^\c\notin\alpha^a(x)$, then $B^\c\subseteq A^\c$ implies that $B^\c\notin\alpha^a(x)$ because $\alpha^a(x)$ is an up-set, so we have $(\Gamma_2)$. To prove $(\Gamma_3)$, notice that for $a,b\in\mathbf{V}$ and $x\in X$,
\begin{align*}
A\in\alpha^a\circ\alpha^b(x)\iff A^\sharp\in D\alpha^a\cdot\alpha^b(x)\iff(\alpha^a)^{-1}(A^\sharp)\in\alpha^b(x)\ .
\end{align*}
Suppose now that $(\gamma_\alpha^a(A))^\c\notin\alpha^b(x)$. Since $(\gamma_\alpha^a(A))^\c=\{y\in X\,|\,A^\c\in\alpha^a(y)\}=(\alpha^a)^{-1}((A^\c)^\sharp)$, we have that $A^\c\notin\alpha^{a\otimes b}(x)$, as required. For $(\Gamma_4)$, we first note that if $b\prec a$, then $\alpha^a\le\alpha^b$, which implies that $\gamma_\alpha^a(A)\subseteq \gamma_\alpha^b(A)$ for all $A\subseteq X$. Thus, on one hand, we have $\gamma_\alpha^a(A)\subseteq\bigcap_{b\prec a}\gamma_\alpha^b(A)$. On the other hand, if $x\in\bigcap_{b\prec a}\gamma_\alpha^b(A)$, then $A^\c\notin\alpha^b(x)$ for all $b\prec a$. By $(K_0')$, we have $A^\c\notin\bigwedge_{b\prec a}\alpha^b(x)=\alpha^a(x)$, which proves $(\Gamma_4)$. If $f:(X,\alpha)\to(Y,\beta)$ satisfies $Df\cdot\alpha^a\le\beta^a\cdot f$ for all $a\in\mathbf{V}$, then $x\in \gamma_\alpha^a(A)\subseteq \gamma_\alpha^a(f^{-1}(f(A)))$, implies that $(f^{-1}(f(A)))^\c=f^{-1}(f(A)^\c)\notin\alpha^a(x)$, so $f(A)^\c\notin\beta^a(f(x))$, which shows that $f$ is a morphism of $\mathbf{V}$-closure spaces.
Suppose now that $\gamma=(\gamma^a:PX\to PX)_{a\in\mathbf{V}}$ is a $\mathbf{V}$-graded closure operator, and set $\alpha_\gamma^a(x):=\{A\in PX\,|\,x\notin \gamma^a(A^\c)\}$. To prove $(K_1)$, let $a\in\mathbf{V}$ be such that $a\le k$, and suppose that $A\in\alpha_\gamma^a(x)$. This implies that $x\notin \gamma^a(A^\c)$, so $x\notin A^\c$,
and $e_X(x)\le \alpha_\gamma^a(x)$. To verify $(K_2)$, recall from the previous paragraph that for $a,b\in\mathbf{V}$, we have $A\in\alpha_\gamma^a\circ\alpha_\gamma^b(x)$ if and only if $\{y\in X\,|\,A\in \alpha_\gamma^a(y)\}\in \alpha_\gamma^b(x)$. This last condition is equivalent to $\gamma^a(A^\c)^\c\in\alpha_\gamma^b(x)$, or $x\notin \gamma^b\cdot\gamma^a(A^\c)$, which yields $A\in\alpha_\gamma^{a\otimes b}(x)$, as required. Let now $f:(X,(\gamma^a)_{a\in\mathbf{V}})\to(Y,(\delta^a)_{a\in\mathbf{V}})$ be a morphism of $\mathbf{V}$-closure spaces. Then by using that $f(\gamma^a(f^{-1}(A^\c)))\subseteq\delta^a(f(f^{-1}(A^\c)))\subseteq\delta^a(A^\c)$, we observe that $A\in\alpha_\delta^a\cdot f(x)$ implies $f(x)\notin\delta^a(A^\c)$, so $x\notin \gamma^a(f^{-1}(A^\c))=\gamma^a(f^{-1}(A)^\c)$, and $f^{-1}(A)\in(\alpha_\gamma)^a(x)$. Therefore, $A\in\alpha_\delta^a(f(x))$ yields $A\in Df(\alpha_\gamma)^a(x)$, and $f$ is a morphism of Kleisli $(\mathsf{T},\mathbf{V})$-algebras.
To show that the previous correspondences yield an isomorphism, note that
\[
\gamma_{\alpha_\gamma}^a(A)=\{x\in X\,|\,x\in \gamma^a(A)\}=\gamma^a(A)\ ,
\]
for all $a\in\mathbf{V}$, and $A\subseteq X$. Furthermore,
\[
\alpha_{\gamma_\alpha}^a(x)=\{A\in PX\,|\,A\in\alpha^a(x)\}=\alpha^a(x)\ ,
\]
for all $a\in\mathbf{V}$, and $x\in X$, so we are done.
\end{proof}
\begin{rem}
In the case $\mathbf{V}=\mathbf{2}$, the Kleisli algebras present closure spaces by way of their ``neighborhood systems", where the neighborhood of a point $x\in X$ with respect to a closure operator $\gamma:PX\to PX$ is given by $\mathcal{N}(x)=\{A\subseteq X\,|\,x\in \gamma(A^\c)^\c\}$. Of course, this immediately leads us to the definition of an interior operator on a set $X$, and it is well known that the category of closure spaces may also be described by such operators.
\end{rem}
\begin{cor}
If $\mathbf{V}$ is completely distributive, then
\[
\mathbf{Alg}(\mathsf{D},\mathbf{V})\cong\mathbf{Cls}(\mathbf{V})\ ,
\]
where $\mathbf{Alg}(\mathsf{D},\mathbf{V})$ is the category of $(\mathsf{D},\mathbf{V})$-algebras associated to the Kleisli extension of $D$.
\end{cor}
\begin{proof}
This is a direct consequence of the previous proposition and Theorem \ref{thm2}. Note that the complete distributivity condition on $\mathbf{V}$ allows us to treat a number of isomorphisms in one stroke, but it is not necessarily the most efficient hypothesis for each one. For example, Schubert proved a similar result (\cite{Schubert:2006}, 4.4.2) by only assuming that $k=\top$.
\end{proof}
\begin{rem}
In the case $\mathbf{V}=\mathbf{2}$, the corollary suggests that up-sets might play the role of filters in a convergence theory for closure spaces. However, other candidates appear in the literature, such as the $p$-stacks of \cite{Kent/Min:2002}, or the rasters of \cite{Giuli/Slapal:2005}. Nonetheless, in the present context all these concepts are very similar. In the case $\mathbf{V}=\mathbf{2}$ for example, whenever $\alpha:X\to DX$ satisfies $e_X\le\alpha$, then the intersection of all elements of $\alpha(x)$ is non-empty, so that $\alpha(x)$ is both a $p$-stack and a raster (as long as the empty set is also considered to be such a structure), and the structures of the Kleisli $\mathsf{D}$-algebras restrict accordingly.
\end{rem}
\section{Many-valued topologies} \label{fuz}
\begin{nr}
\textbf{The $\L$-valued filter monad.} (See \cite{Hohle:2001}) Let $\L$ be a complete lattice provided with a binary operation $\ast$ that is monotone in both variables (in particular, any quantale $\mathbf{V}$ is such a lattice with its binary operation given by $\otimes$; better yet, any complete lattice has a binary operation given by infimum). There is an induced order on the set $\L^X$ of maps from $X$ to $\L$ defined by
\[
A\le B\iff\text{ for all }x\in X,\text{ we have }A(x)\le B(x)\ ,
\]
where $A,B\in\L^X$. The top and bottom elements of $\L$ are denoted by $\top$ and $\bot$, and those of $\L^X$ by $\top^X$ and $\bot^X$ respectively. An $\L$-valued filter on $X$ is a map $\mathfrak{f}:\L^X\to\L$ satisfying the following conditions for $A,B\in\L^X$:
\begin{enumerate}[$(F_1)$]
\item $\mathfrak{f}(\top^X)=\top$ ,
\item $A\le B$ implies $\mathfrak{f}(A)\le\mathfrak{f}(B)$ ,
\item $\mathfrak{f}(A)\ast\mathfrak{f}(B)\le\mathfrak{f}(A\ast B)$ .
\end{enumerate}
The set of all $\L$-valued filters on $X$ is denoted by $F_{\!_\L} X$. Of course, in the case $\L=\mathbf{2}$, we get the usual definition of filters, so that $F_{\!_\mathbf{2}}X=FX$.
The \emph{$\L$-valued filter functor} $F_{\!_\L}$ assigns to a set $X$ the set $F_{\!_\L} X$, and sends a map $f:X\to Y$ to $F_{\!_\L} f:F_{\!_\L} X\to F_{\!_\L} Y$ defined by
\[
[F_{\!_\L} f(\mathfrak{f})](A)=\mathfrak{f}(A\cdot f) ,
\]
where $A\in\L^Y$, and $\mathfrak{f}\inF_{\!_\L} X$. The \emph{$\L$-valued filter monad} $\mathsf{F}_{\!_\L}$ is the triple $(F_{\!_\L},e,m)$, where $e:\mathrm{Id}\toF_{\!_\L}$ and $m:F_{\!_\L}^2\toF_{\!_\L}$ are the natural transformations whose components at $X$ are obtained via
\[
[e_X(x)](A)=A(x)\qquad\text{ and }\qquad [m_X(\mathfrak{F})](A)=\mathfrak{F}(\mathrm{ev}_A)\ ,
\]
where $A\in\L^X$, $\mathfrak{F}\inF_{\!_\L}^2 X$, and $\mathrm{ev}_A:F_{\!_\L} X\to\L$ is given by $\mathrm{ev}_A(\mathfrak{f})=\mathfrak{f}(A)$. The order on $F_{\!_\L} X$ is defined by
\[
\mathfrak{f}\le\mathfrak{g}\iff\mathfrak{g}(A)\le\mathfrak{f}(A)\text{ for all }A\in\L^X\ .
\]
With this order, $F_{\!_\L}$ is a complete lattice, and it is easily checked that $F_{\!_\L}$ factors coherently through $\mathbf{Sup}$ (see also \cite{Hohle:2001}, Proposition 2.4.2.3).
The category of Kleisli $F_{\!_\L}$-algebras is called the category of \emph{$\L$-valued topological spaces} (or \emph{fuzzy topological spaces}, see \cite{Gahler:1995b}), and is denoted by $\mathbf{Top}(\L)$. Furthermore, if $(X,\alpha)$ is a Kleisli $F_{\!_\L}$-algebra, then for each $x\in X$, the $\L$-valued filter $\alpha(x)$ is called the \emph{$\L$-valued neighborhood system} of $x$.
\end{nr}
\begin{rem}
The previous definition of the $\L$-valued filter monad differs in two points with the one given in \cite{Hohle:2001}. First, we do not ask that filters $\mathfrak{f}\inF_{\!_\L} X$ satisfy $\mathfrak{f}(\bot^X)=\bot$. This will not be a problem in using results of \textit{op.cit.}, since $\L$-valued topologies are defined via neighborhoods, and a neighborhood $\mathfrak{f}$ of $x$ must satisfy $\mathfrak{f}(A)\le A(x)$ for all $A\in\L^X$; in particular $\mathfrak{f}(\bot^X)=\bot$, so the resulting $\L$-valued neighborhood systems are the same. Second, the order on $F_{\!_\L} X$ is chosen as opposite to the one in the cited reference. In particular, with the present definition, $F_{\!_\L} X$ is a complete lattice rather than an ``almost complete join-semilattice''.
\end{rem}
\begin{prop} \label{propfuz}
The category of $(\mathsf{F}_{\!_\L},\mathbf{V})$-algebras associated to the Kleisli extension of $F_{\!_\L}$ is isomorphic to the category of $\L$-valued topologies:
\[
\mathbf{Alg}(\mathsf{F}_{\!_\L},\mathbf{2})\cong\mathbf{Top}(\L)\ .
\]
\end{prop}
\begin{proof}
This is a direct consequence of Propositions \ref{prop1} and \ref{prop3}.
\end{proof}
\textbf{Acknowledgements.} The author is particularly indebted to Christoph Schubert for pointing out in the first place that the Kleisli presentation of topological spaces did not require a lax extension of the monad functor, for mentioning Brock and Kent's limit tower spaces \cite{Brock/Kent:1997}, and in general for a number of insightful discussions and pertinent remarks. He also wishes to thank Walter Tholen for his suggestions towards improving the presentation of this paper.
\bibliographystyle{plain}
|
hep-ph/0510195
|
\section{INTRODUCTION}
Precise measurements of top quark pair production
\begin{eqnarray}
\label{eett}
e^+e^- \rightarrow t \bar{t}
\end{eqnarray}
at the threshold and in the continuum region will
belong to the basic
physics program of the future International Linear Collider (ILC) \cite{ILC}.
In order to fully profit from these high precision measurements one has to
bring theoretical predictions to at least the same, or preferably
better, precision, which obviously requires taking into account radiative
corrections. The latter should be calculated not only for the on-shell
production process (\ref{eett}). Due to their large widths
the $t$- and $\bar{t}$-quark of reaction (\ref{eett}) almost immediately
decay into $bW^+$ and $\bar{b}W^-$, respectively, and the $W$-bosons
subsequently into 2 fermions each, thus constituting six-fermion reactions of
the form
\begin{equation}
\label{ee6f}
e^+e^-\;\; \rightarrow \;\; bf_1\bar{f'_1} \bar{b}f_2 \bar{f'_2},
\end{equation}
where $f_1, f'_2 =\nu_{e}, \nu_{\mu}, \nu_{\tau}, u, c$ and
$f'_1, f_2 = e^-, \mu^-, \tau^-, d, s$. Typical lowest order Feynman diagrams
of reaction (\ref{ee6f}) are shown in Fig.~1.
\begin{figure}[htb]
\vspace{140pt}
\special{psfile=diags0.epsi angle=0 hscale=100 vscale=100 hoffset=-57
voffset=-660}
\caption{Examples of Feynman diagrams of reaction (\ref{ee6f}):
(a) `signal', (b) and (c) `background'
diagrams.}
\label{fig:diags0}
\end{figure}
As decays of the top and antitop take place before toponium resonances
can form, the Standard Model (SM) predictions for reaction (\ref{eett})
can be obtained with the perturbative method. The QCD predictions for reaction
(\ref{eett}) in the threshold region were obtained in \cite{topQCD}
and then improved by calculation of the next-to-next-to-leading order QCD
corrections \cite{topNNLO}, and by including the effects of initial state
radiation and beamstrahlung \cite{topIR}.
The $\mathcal{O}(\alpha\alpha_s)$ \cite{topdec1,topdDS,topdD} and
$\mathcal{O}(\alpha\alpha_s^2)$ \cite{topdec2} corrections to the subsequent
top decay into a $W$ boson and a $b$ quark are also known.
In the continuum above the threshold, the QCD predictions for reaction
(\ref{eett}) are known to order $\alpha_s^2$ \cite{eettQCD} and the
electroweak (EW) corrections to one--loop order \cite{eettEW,topfit1,topfit3},
including the hard bremsstrahlung
corrections \cite{eetthb,topfit1}. The QCD and EW corrections are
large, typically of $\cal{O}$(10\%). Order $\alpha_s$ \cite{eettQCD1}
and $\alpha_s^2$ QCD, and EW corrections have been combined in \cite{eettcomb}.
Quite recently the EW radiative corrections to (\ref{eett}) have been
recalculated with a program {\tt topfit}
\cite{topfit1,topfit3} and thoroughly compared with results
of other calculations, with hard bremsstrahlung \cite{Fleischer:2002nn} and
without it \cite{Hahn:2003ab}.
Finally, the radiative corrections to $W$ decays into fermion pairs, which
have to be taken into account too, are also
known \cite{Bardin:1986fi,FJ2,Denner:1990tx}.
At tree level, reactions (\ref{ee6f}) can be studied with a
Monte Carlo (MC) program {\tt eett6f} \cite{eett6f,eett6f1} or with any other
MC program dedicated to the six fermion reactions, such as
{\sc Sixphact}~\cite{Sixphact}, {\sc Sixfap}~\cite{Sixfap},
{\sc Lusifer} \cite{Lusifer}, or with any of
multi-purpose generators, such as {\sc Amegic} \cite{Amegic},
{\sc Grace} \cite{Grace}/{\sc Bases} \cite{Bases},
{\sc Madgraph} \cite{Madgraph}/{\sc Madevent} \cite{Madevent},
{\sc Phegas} \cite{Phegas}/{\sc Helac} \cite{Helac},
or {\sc Whizard} \cite{Whizard}/{\sc Comphep}~\cite{Comphep},
{\sc Madgraph}~\cite{Madgraph}, or {\sc O'mega}~\cite{Omega}.
Thorough comparison of the lowest order predictions for several
different channels of (\ref{ee6f})
obtained with {\sc Amegic++}, {\tt eett6f}, {\sc Lusifer}, {\sc Phegas},
{\sc Sixfap} and {\sc Whizard} have been performed in the framework of
the Monte Carlo Generators group of the ECFA/DESY workshop \cite{didi}.
A survey of SM cross sections of all six fermion reactions
with up to four quarks
in the limit of
massless fermions (but the top quark), has been done
in \cite{Lusifer}. The latter contains also a fine tuned comparison of both
the lowest order and lowest order plus ISR results, obtained in the
structure function approach, between {\sc Lusifer} and {\sc Whizard}.
Concerning radiative corrections to the six-fermion reactions (\ref{ee6f}),
the situation is less advanced.
Already at the tree level, any of the reactions receives
contributions from typically several hundred Feynman diagrams, {\em e.g.}
in the unitary gauge, with neglect of the Higgs boson couplings to
fermions lighter than the $b$ quark, reactions $e^+ e^- \ra b \nu_{\mu} \mu^+ \bar{b} d \bar{u}$, $\eebnmbmn$,
and $e^+ e^- \ra b u \bar{d} \bar{b} d \bar{u}$ get contributions from 264, 452, and 1484 Feynman diagrams,
respectively. Hence, the calculation
of the full $\cal{O}(\alpha)$ radiative corrections to any of reactions
(\ref{ee6f}) seems not to be feasible at present.
Therefore, in the present note we will make a step
towards improving precision of the lowest order predictions for
(\ref{ee6f}) by including leading radiative effects, such as
initial state radiation (ISR) and factorizable EW radiative
corrections to the process of the on-shell top quark pair production
(\ref{eett}), to the decay of the $t$ ($\bar t$) into $bW^+$ ($\bar{b}W^-$) and
to the subsequent decays of the $W$-bosons.
We will illustrate an effect of of these corrections by showing numerical
results for the two selected six-fermion reactions
\begin{equation}
\label{nmmn}
\eebnmbmn
\end{equation}
and
\begin{equation}
\label{nmud}
e^+ e^- \ra b \nu_{\mu} \mu^+ \bar{b} d \bar{u}.
\end{equation}
\section{CALCULATIONAL SCHEME}
We calculate the ISR and the factorizable SM corrections
for the reaction
\begin{equation}
\label{ee6fmom}
e^+(p_1,\sigma_1)\;e^-(p_2,\sigma_2)\;\; \rightarrow \;\; b(p_3,\sigma_3)\;
f_1(p_4,\sigma_4)\; \bar{f'_1}(p_5,\sigma_5)\; \bar{b}(p_6,\sigma_6) \;
f_2(p_7,\sigma_7) \; \bar{f'_2}(p_8,\sigma_8),
\end{equation}
where the particle momenta and helicities have been indicated in the
parentheses, according
to the following formula:
\begin{eqnarray}
\label{LL}
{\rm d} \sigma=
\int_0^1 {\rm d} x_1 \int_0^1 {\rm d} x_2 \,
\Gamma_{ee}^{LL}\left(x_1,Q^2\right)
\Gamma_{ee}^{LL}\left(x_2,Q^2\right)
{\rm d}\sigma_{\rm Born+FEWC}\left(x_1 p_1,x_2 p_2\right),
\end{eqnarray}
where $x_1p_1$ ($x_2p_2$) is the four momentum of the positron
(electron) after emission of a collinear photon.
The structure function $\Gamma_{ee}^{LL}\left(x,Q^2\right)$ is given
by Eq.~(67) of \cite{Beenakker}, with {\tt `BETA'} choice for non-leading
terms. The splitting scale $Q^2$, which is not fixed in the LL approximation
is chosen to be $s=(p_1+p_2)^2$.
By ${\rm d}\sigma_{\rm Born+FEWC}$ we denote the cross section including
the factorizable EW $\cal{O}(\alpha)$ corrections
\begin{eqnarray}
\label{bpfewc}
{\rm d}\sigma_{\rm Born+FEWC}
=\frac{1}{2s}\left\{\overline{\left|M_{\rm Born}\right|^2}\;
+ 2\;{\rm Re}\overline{\left(M_{t\bar{t}}^{*}
\;\delta M_{t\bar{t},{\rm FEW}}\right)}\right\}{\rm d}\Phi_{6f},
\end{eqnarray}
where $M_{\rm Born}$ is the matrix element of reaction (\ref{ee6fmom})
obtained with the complete set of the lowest order Feynman diagrams,
$M_{t\bar{t}}$ and $\delta M_{t\bar{t},{\rm FEW}}$ is, respectively,
the lowest order amplitude of the `signal' Feynman diagram of
Fig.~\ref{fig:diags0}a and the corresponding factorizable EW
$\cal{O}(\alpha)$ correction, both in the pole approximation.
The overlines in (\ref{bpfewc}) denote, as usual, an initial state particle
spin average and a sum over final state particle polarizations,
and ${\rm d}\Phi_{6f}$ is the Lorentz invariant six-particle
phase space element. The basic phase space parametrizations which are used
in the program are given by Eqs.~(7)--(9) of \cite{eett6f}.
The corrections
that we take into account in $\delta M_{t\bar{t},{\rm FEW}}$
are illustrated diagramatically in Fig.~2.
\begin{figure}[htb]
\vspace{240pt}
\special{psfile=diags.epsi angle=0 hscale=90 vscale=90 hoffset=-57
voffset=-480}
\caption{Factorizable EW corrections to reaction (\ref{ee6f}).}
\label{fig:diags}
\end{figure}
In the pole approximation, the polarized lowest order amplitude
$M_{t\bar{t}}$ and the one--loop correction $\delta M_{t\bar{t},{\rm FEW}}$
of Eq.~(\ref{bpfewc}) can be expressed analytically as follows:
\begin{eqnarray}
M_{t\bar{t}}^{\sigma_1\sigma_2;\sigma_3\ldots\;\sigma_8}=
\frac{1}{D_t\left(p_{345}\right)D_t\left(p_{678}\right)}
\sum_{\sigma_t, \sigma_{\bar t}}& &\hspace*{-0.5cm}
M_{e^+e^-\rightarrow t\bar{t}}^{\sigma_1\sigma_2;\sigma_t \sigma_{\bar t}}\;
M_{t\rightarrow b f_1 f'_1}^{\sigma_t;\sigma_3\sigma_4\sigma_5}\;
M_{\bar{t}\rightarrow \bar{b} f_2f'_2}^{\sigma_{\bar{t}};\sigma_6\sigma_7\sigma_8}
\label{mtt}\\
\delta M_{t\bar{t}}^{\sigma_1\sigma_2;\sigma_3\ldots\;\sigma_8}=
\frac{1}{D_t\left(p_{345}\right)D_t\left(p_{678}\right)}
\sum_{\sigma_t, \sigma_{\bar t}}& &\hspace*{-0.5cm}\left[
\delta M_{e^+e^-\rightarrow t\bar{t}}^{\sigma_1\sigma_2\sigma_t, \sigma_{\bar t}}
M_{t\rightarrow b f_1 f'_1}^{\sigma_t;\sigma_3\sigma_4\sigma_5}\;
M_{\bar{t}\rightarrow \bar{b} f_2f'_2}^{\sigma_{\bar{t}};\sigma_6\sigma_7\sigma_8}
\right.\nonumber\\
&+&\hspace*{-0.3cm}\left.
M_{e^+e^-\rightarrow t\bar{t}}^{\sigma_1\sigma_2;\sigma_t\sigma_{\bar t}}\;
\delta M_{t\rightarrow b f_1 f'_1}^{\sigma_t\sigma_3\sigma_4\sigma_5}\;
M_{\bar{t}\rightarrow \bar{b} f_2f'_2}^{\sigma_{\bar{t}};\sigma_6\sigma_7\sigma_8}
\right.\label{dmtt}\\
&+&\hspace*{-0.3cm} \left.
M_{e^+e^-\rightarrow t\bar{t}}^{\sigma_1\sigma_2;\sigma_t\sigma_{\bar t}}\;
M_{t\rightarrow b f_1 f'_1}^{\sigma_t;\sigma_3\sigma_4\sigma_5}\;
\delta M_{\bar{t}\rightarrow \bar{b} f_2f'_2}^{\sigma_{\bar{t}};
\sigma_6\sigma_7\sigma_8}\right],\nonumber
\end{eqnarray}
where the lowest order $t$ and $\bar t$ decay amplitudes and the corresponding
one--loop corrections read
\begin{eqnarray}
M_{t\rightarrow b f_1 f'_1}^{\sigma_t\sigma_3\sigma_4\sigma_5}&=&
\frac{1}{D_W\left(p_{45}\right)}\sum_{\lambda_{W^+}}
M_{t\rightarrow b W^+}^{\sigma_t\sigma_3\lambda_{W^+}}
M_{W^+\rightarrow f_1 f'_1}^{\lambda_{W^+}\sigma_4\sigma_5},
\label{mtbff}\\
M_{\bar{t}\rightarrow \bar{b} f_2f'_2}^{\sigma_{\bar{t}}\sigma_6\sigma_7\sigma_8}
&=&\frac{1}{D_W\left(p_{78}\right)}\sum_{\lambda_{W^-}}
M_{\bar{t}\rightarrow \bar{b} W^-}^{\sigma_{\bar{t}}\sigma_6\lambda_{W^-}}
M_{W^-\rightarrow f_2 f'_2}^{\lambda_{W^-}\sigma_7\sigma_8},
\label{mtbbff}\\
\delta M_{t\rightarrow b f_1 f'_1}^{\sigma_t\sigma_3\sigma_4\sigma_5}&=&
\frac{1}{D_W\left(p_{45}\right)}\sum_{\lambda_{W^+}}\left[
\delta M_{t\rightarrow b W^+}^{\sigma_t\sigma_3\lambda_{W^+}}
M_{W^+\rightarrow f_1 f'_1}^{\lambda_{W^+}\sigma_4\sigma_5}
+M_{t\rightarrow b W^+}^{\sigma_t\sigma_3\lambda_{W^+}}
\delta M_{W^+\rightarrow f_1 f'_1}^{\lambda_{W^+}\sigma_4\sigma_5}\right],
\label{dmtbff}\\
\delta M_{\bar{t}\rightarrow \bar{b} f_2f'_2}^{\sigma_{\bar{t}}
\sigma_6\sigma_7\sigma_8}&=&\frac{1}{D_W\left(p_{78}\right)}
\sum_{\lambda_{W^-}}\left[
\delta M_{\bar{t}\rightarrow \bar{b} W^-}^{\sigma_{\bar{t}}\sigma_6\lambda_{W^-}}
M_{W^-\rightarrow f_2 f'_2}^{\lambda_{W^-}\sigma_7\sigma_8}
+M_{\bar{t}\rightarrow \bar{b} W^-}^{\sigma_{\bar{t}}\sigma_6\lambda_{W^-}}
\delta M_{W^-\rightarrow f_2 f'_2}^{\lambda_{W^-}\sigma_7\sigma_8}\right].
\label{dmtbbff}
\end{eqnarray}
In (\ref{mtt}--\ref{dmtbbff}),
$\sigma_t$, $\sigma_{\bar t}$ and $\lambda_{W^+}$, $\lambda_{W^-}$
denote polarizations of the intermediate top quarks and $W$ bosons
which are treated as on-shell particles, except for keeping their actual
off-shell momenta
\begin{equation}
p_{345}=p_3+p_4+p_5, \quad p_{678}=p_6+p_7+p_8, \qquad p_{78}=p_7+p_8,
\quad p_{45}=p_4+p_5
\end{equation}
in the denominators $D_t\left(p\right)$ and $D_W\left(p\right)$ of their
propagators
\begin{equation}
\label{props}
D_t\left(p\right)=p^2 - m_t^2 + i m_t\Gamma_t, \qquad
D_W\left(p\right)=p^2 - m_W^2 + i m_W\Gamma_W.
\end{equation}
The fixed widths $\Gamma_t$ and $\Gamma_W$ of (\ref{props}) are calculated in
the program for a given set of initial parameters. They are set to
their SM lowest order values, $\Gamma_t^{(0)}$ and $\Gamma_W^{(0)}$,
for the Born cross sections, or they
include radiative corrections of the same kind as those included in the
numerators of (\ref{dmtt}), (\ref{dmtbff}) and (\ref{dmtbbff}) for the
radiatively corrected cross sections.
While explaining further the notation of Eqs. (\ref{mtt}--\ref{dmtbbff}) we
will suppress the polarization indices.
$M_{e^+e^-\rightarrow t\bar{t}}$ and $\delta M_{e^+e^-\rightarrow t\bar{t}}$
are the lowest order and the EW one--loop amplitudes
of the on-shell top quark pair production process (\ref{eett}). They can be
decomposed in a basis composed of the following invariant amplitudes
\begin{eqnarray}
\label{amps}
{\cal M}_{1,\,{ab}} & =& \ensuremath { \bar{v}(p_1) \,\, } \gamma^{\mu}\, g_a \, \ensuremath { u(p_2) \,\, } \; \bar{u}(k_t)
\gamma_{\mu}\, g_b \, \bar{v}(k_{\bar t}),
\quad g_a, g_b={\rm 1\mskip-4.25mu l}, \gamma_5,\nonumber\\
{\cal M}_{3,{11}} & = &-\ensuremath { \bar{v}(p_1) \,\, } k\hspace{-0.43em}/}%\hspace{0.1em}_t \, \ensuremath { u(p_2) \,\, } \;\bar{u}(k_t)
\bar{v}(k_{\bar t}),\\
{\cal M}_{3,{51}} & = &-\ensuremath { \bar{v}(p_1) \,\, } k\hspace{-0.43em}/}%\hspace{0.1em}_t\, \gamma_5 \ensuremath { u(p_2) \,\, } \;
\bar{u}(k_t) \bar{v}(k_{\bar t}).\nonumber
\end{eqnarray}
The projected four momenta $k_t, k_{\bar t}$ of the on-shell
top- and antitop-quark of (\ref{amps}), as well as the four momenta
$k_{W^+}, k_{W^-}$ of the on-shell $W$-bosons and the four momenta
$k_3,\ldots,k_8$ of the decay fermions, which are used later,
have been obtained
from the four momenta of the final state fermions $p_3,\ldots,p_8$ of reaction
(\ref{ee6f}) with
the projection procedure described in Appendix A.
In terms of invariant amplitudes (\ref{amps}), the lowest order amplitude
of (\ref{eett}) reads
\begin{eqnarray}
\label{Bornamp}
\, M_{e^+e^-\rightarrow t\bar{t}} =
\sum_{a,b=1,5} {\rm F}_{1B}^{\,ab} \,\, {\cal M}_{1,{ab}},
\end{eqnarray}
where the 4 Born form factors ${\rm F}_{1B}^{\,ab}$ are given by
\\
\parbox{8cm}{
\begin{eqnarray*}
{\rm F}_{1B}^{\,11}&=&\frac{e_W^2\left(\chi_Z v_e v_t+Q_e Q_t\right)}{s},\\
{\rm F}_{1B}^{\,15}&=&-\frac{e_W^2\chi_Z v_e a_t}{s},
\end{eqnarray*}}
\hfill
\parbox{6cm}{
\begin{eqnarray*}
{\rm F}_{1B}^{\,51}&=&-\frac{e_W^2\chi_Z v_t a_e}{s},\\
{\rm F}_{1B}^{\,55}&=&\frac{e_W^2\chi_Z a_e a_t}{s}.
\end{eqnarray*}}
\hfill
\parbox{2cm}{\begin{eqnarray} \label{fborn} \end{eqnarray}}\\
In (\ref{fborn}), $e_W$ is the effective electric charge,
$e_W=\sqrt{4\pi\alpha_W}$, with
\begin{equation}
\label{sw2}
\alpha_W=\frac{\sqrt{2}G_{\mu}m_W^2 \sin^2\theta_W}{\pi}
\qquad {\rm and} \qquad \sin^2\theta_W=1-\frac{m_W^2}{m_Z^2},
\end{equation}
the $Z$-boson propagator is contained in the factor
\begin{equation}
\label{chiz}
\chi_Z=\frac{1}{4 \sin^2\theta_W \cos^2\theta_W}\;
\frac{s}{s-m_Z^2+im_Z\Gamma_Z}
\end{equation}
and we have used the following conventions for couplings of the electron
and top quark to a photon and $Z$-boson
\begin{equation}
Q_e=-1, \quad Q_t=\frac{2}{3}, \quad a_e=-a_t=-\frac{1}{2}, \quad
v_f=a_f\left(1-4\left|Q_f\right|\sin^2\theta_W\right), \quad f=e,t.
\end{equation}
We have introduced a constant $Z$-boson width $\Gamma_Z$ in (\ref{chiz}),
in a similar way as $\Gamma_t$ and $\Gamma_W$ have been introduced in
(\ref{props}), although the $Z$-boson propagator in the $e^+e^-$ annihilation
channel never becomes resonant in the CMS energy range above the
$t\bar t$-pair production threshold. Generally speaking,
the constant width
$\Gamma$ of an unstable particle is introduced into the lowest order
matrix elements by replacing its mass with the complex mass parameter
\begin{equation}
\label{m2}
m^2 \rightarrow m^2-im\Gamma
\end{equation}
in the corresponding propagator, both in the $s$- and $t$-channel one,
while keeping the electroweak mixing parameter $\sin^2\theta_W$ of
(\ref{sw2}) real.
This approach is usually referred to in the literature as the fixed width
scheme (FWS). The approach, in which $m_W^2$ and $m_Z^2$ are replaced with
their complex counterparts according to (\ref{m2}) also in $\sin^2\theta_W$ of
(\ref{sw2}) is on the other hand referred to as the complex mass scheme
\cite{Racoon}. The latter has the advantage that it
preserves Ward identities. Let us note, that in
Eqs.~(\ref{mtt}--\ref{dmtbbff}), substitution (\ref{m2}) is done
only in the denominators of the top-quark and $W$-boson propagators
and not in the one--loop amplitudes. Also the sums over the top-quark
and $W$-boson polarizations result in the numerators of the corresponding
propagators with real masses. However,
this does not violate the substitution rule of (\ref{m2}),
as the amplitudes of Eqs.~(\ref{mtt}--\ref{dmtbbff}) constitute
the factorizable one--loop correction term in (\ref{bpfewc}).
The EW one--loop amplitude of (\ref{eett}) reads
\begin{eqnarray}
\label{damp}
\delta M_{e^+e^-\rightarrow t\bar{t}} =
\sum_{a,b=1,5} \fh{1}{ab} \,\, {\cal M}_{1,{ab}} +
\fh{3}{11} \,\, {\cal M}_{3,{11}} +
\fh{3}{51} \,\, {\cal M}_{3,{51}},
\end{eqnarray}
with the six independent form factors: $\fh{1}{ab}, \; a,b=1,5$,
$\fh{3}{11}$ and $\fh{3}{51}$ which are calculated numerically with
a program {\tt topfit} \cite{topfit1,topfit3} that is tailored
to a subroutine of a new version of
{\tt eett6f}. Note that a factor $i$ has been omitted
on the left hand side of (\ref{amps}) compared to \cite{topfit1}.
Keeping it would result in an extra minus sign on the right hand
side of (\ref{mtt}) and (\ref{dmtt}), as we
neglect the $i$ factor in every vertex and propagator
and consequently the resulting common $+i$ factor for every Feynman
diagram in the present work.
The flags in {\tt topfit} switch off all photonic corrections there,
including the running of the electromagnetic coupling. This means that only
the genuine weak corrections will contribute.
In order to fix normalization we give the formula for the EW one--loop
corrected cross section ${\rm d}\sigma_{e^+e^-\rightarrow\; t \bar t}$
of the on-shell top production (\ref{eett})
\begin{eqnarray}
\label{cseett}
{\rm d}\sigma_{e^+e^-\rightarrow t\bar{t}}
=\frac{1}{2s}\left\{\left|M_{e^+e^-\rightarrow t\bar{t}}\right|^2\;
+ 2\;{\rm Re}\left(M_{e^+e^-\rightarrow t\bar{t}}^{*}
\;\delta M_{e^+e^-\rightarrow t\bar{t}}\right)\right\}{\rm d}\Phi_{2f},
\end{eqnarray}
where the matrix elements $M_{e^+e^-\rightarrow t\bar{t}}$ and
$\delta M_{e^+e^-\rightarrow t\bar{t}}$ are given by (\ref{Bornamp}) and
(\ref{damp}) and ${\rm d}\Phi_{2f}$ is the Lorentz invariant two-particle
phase space element
\begin{equation}
\label{dps2}
{\rm d}\Phi_{2f} = \frac{|\vec{p_t}|}{4\sqrt{s}} {\rm d} \Omega_t,
\end{equation}
with $\vec{p_t}$ being the momentum and $\Omega_t$ the solid angle
of the $t$-quark.
The $t$- and $\bar t$-quark decay amplitudes $M_{t \rightarrow b W^+}$
and $M_{\bar t \rightarrow \bar b W^-}$,
and the corresponding one--loop corrections $\delta M_{t \rightarrow b W^+}$
and $\delta M_{\bar t \rightarrow \bar b W^-}$ can be decomposed in terms
of the invariant amplitudes\\
\parbox{8cm}{
\begin{eqnarray*}
{\cal M}_{t,1}^{(\sigma)} &=& \bar{u}(k_3) \varepsilon\!\!\!/(k_{W^+})
P_{\sigma}u(k_t),\\
{\cal M}_{t,2}^{(\sigma)} &=& k_t\cdot \varepsilon(k_{W^+})\;
\bar{u}(k_3) P_{\sigma}u(k_t).
\end{eqnarray*}}
\hfill
\parbox{6cm}{
\begin{eqnarray*}
{\cal M}_{\bar t,1}^{(\sigma)} &=&
\bar{v}(k_{\bar t}) \varepsilon\!\!\!/(k_{W^-})P_{\sigma}v(k_6), \\
{\cal M}_{\bar t,2}^{(\sigma)} &=&-k_{\bar t}\cdot \varepsilon(k_{W^-})\;
\bar{v}(k_{\bar t}) P_{\sigma}v(k_6).
\end{eqnarray*}}
\hfill
\parbox{2cm}{\begin{eqnarray} \label{invtop} \end{eqnarray}},\\
where $P_{\sigma}=(1+\sigma\gamma_5)/2$, $\sigma=\pm 1$, are the chirality
projectors and we have used real polarization vectors for $W$ bosons.
The decomposition reads\\
\parbox{8cm}{
\begin{eqnarray*}
M_{t \rightarrow b W^+}&=&g_{Wff}\;{\cal M}_{t,1}^{(-)},\\
\delta M_{t \rightarrow b W^+}&=&g_{Wff}\;
\sum_{i=1,2\atop\sigma=\pm 1} F_{t,i}^{(\sigma)}{\cal M}_{t,i}^{(\sigma)},
\end{eqnarray*}}
\hfill
\parbox{6cm}{
\begin{eqnarray*}
M_{\bar t \rightarrow \bar b W^-}&=&g_{Wff}\;{\cal M}_{\bar t,1}^{(-)},\\
\delta M_{\bar t \rightarrow \bar b W^-}&=&g_{Wff}\;
\sum_{i=1,2\atop\sigma=\pm 1} F_{\bar t,i}^{(\sigma)}
{\cal M}_{\bar t,i}^{(\sigma)}.
\end{eqnarray*}}
\hfill
\parbox{2cm}{\begin{eqnarray} \label{amptop} \end{eqnarray}}\\
In (\ref{amptop}), $g_{Wff}$ is the SM $W$ boson coupling to
fermions which, similarly to the Born form factors of (\ref{fborn}), is
defined in terms of the effective electric charge $e_W$
\begin{equation}
g_{Wff}=-\frac{e_W}{\sqrt{2}\sin\theta_W},
\end{equation}
$F_{t,i}^{(\sigma)}$ and $F_{\bar t,i}^{(\sigma)}$ are the EW one--loop
form factors of the top- and antitop-quark decay, respectively. The form
factors $F_{t,i}^{(\sigma)}$ are calculated numerically with a newly written
dedicated subroutine that reproduces results of \cite{topdDS,topdD}.
The one--loop form factors of the antitop decay are then obtained assuming
$\cal{C}\cal{P}$ conservation which lead to the following relations
\begin{eqnarray}
F_{\bar t,1}^{(\sigma)}=F_{t,1}^{(\sigma)^*},\qquad
F_{\bar t,2}^{(\sigma)}=F_{t,2}^{(-\sigma)^*}.
\end{eqnarray}
Note that the imaginary parts of the form factors do not contribute
at the one-loop order.
Similarly the $W^+$- and $W^-$-boson decay amplitudes
$M_{W^+ \rightarrow f_1 \bar{f'}_1}$ and $M_{W^- \rightarrow f_2 \bar{f'}_2}$,
and the corresponding one--loop corrections
$\delta M_{W^+ \rightarrow f_1 \bar{f'}_1}$ and $\delta M_{W^- \rightarrow f_2 \bar{f'}_2}$
are given by\\
\parbox{8cm}{
\begin{eqnarray*}
M_{W^+ \rightarrow f_1 \bar{f'}_1}&=&g_{Wff}\;{\cal M}_{W^+,1}^{(-)},\\
\delta M_{W^+ \rightarrow f_1 \bar{f'}_1}&=&g_{Wff}\;
\sum_{i=1,2\atop\sigma=\pm 1} F_{W^+,i}^{(\sigma)}{\cal M}_{W^+,i}^{(\sigma)},
\end{eqnarray*}}
\hfill
\parbox{6cm}{
\begin{eqnarray*}
M_{W^- \rightarrow f_2 \bar{f'}_2}&=&g_{Wff}\;{\cal M}_{\bar W^-,1}^{(-)},\\
\delta M_{W^- \rightarrow f_2 \bar{f'}_2}&=&g_{Wff}\;
\sum_{i=1,2\atop\sigma=\pm 1} F_{W^-,i}^{(\sigma)}
{\cal M}_{W^-,i}^{(\sigma)},
\end{eqnarray*}}
\hfill
\parbox{2cm}{\begin{eqnarray} \label{ampw} \end{eqnarray}}\\
with the invariant amplitudes\\
\parbox{8cm}{
\begin{eqnarray*}
{\cal M}_{W^+,1}^{(\sigma)} &=& \bar{u}(k_4) \varepsilon\!\!\!/(k_{W^+})
P_{\sigma}v(k_5),\\
{\cal M}_{W^+,2}^{(\sigma)} &=& k_4\cdot \varepsilon(k_{W^+})\;
\bar{u}(k_4) P_{\sigma}v(k_5).
\end{eqnarray*}}
\hfill
\parbox{6cm}{
\begin{eqnarray*}
{\cal M}_{W^-,1}^{(\sigma)} &=& \bar{u}(k_7) \varepsilon\!\!\!/(k_{W^-})
P_{\sigma}v(k_8), \\
{\cal M}_{W^-,2}^{(\sigma)} &=&-k_8\cdot \varepsilon(k_{W^-})\;
\bar{u}(k_7) P_{\pm}v(k_8)
\end{eqnarray*}}
\hfill
\parbox{2cm}{\begin{eqnarray} \label{invw} \end{eqnarray}}\\
and the EW one--loop form factors of the $W$-boson decays
$F_{W^{\pm},i}^{(\sigma)}$ being calculated numerically, this time with
a new subroutine that reproduces results of \cite{Denner:1990tx,topdD}
for the EW corrected $W$-boson width. Again, the imaginary parts of
the form factors do not contribute
at the one-loop order.
The calculation of the EW factorizable corrections to reaction (\ref{ee6f})
in the pole approximation makes sense only if the invariant masses
\begin{eqnarray}
\label{mt}
m_{345}=\sqrt{\left(p_3+p_4+p_5\right)^2}, \qquad
m_{678}=\sqrt{\left(p_6+p_7+p_8\right)^2}
\end{eqnarray}
of the $bf_1\bar{f'_1}$ and $\bar{b}f_2 \bar{f'_2}$
are close to $m_t$ each and if
\begin{eqnarray}
\label{mw}
m_{45}=\sqrt{\left(p_4+p_5\right)^2}, \qquad
m_{78}=\sqrt{\left(p_7+p_8\right)^2}
\end{eqnarray}
of the $f_1\bar{f'_1}$ and $f_2 \bar{f'_2}$ do not depart too much from $m_W$.
Otherwise the signal diagrams of Fig.~\ref{fig:diags0}(a) stop to dominate
the cross section and the
association of the reduced phase space point, at which the EW factorizable
$\cal{O}(\alpha)$ corrections depicted in Fig.~\ref{fig:diags} are calculated,
with the phase space point of the full six particle phase space of (\ref{ee6f})
may lead to unnecessary distortion of the off resonance background
contributions.
Therefore in the following we will impose kinematical cuts on the quantities
\begin{eqnarray}
\label{deltas}
\delta_t=m_{345}/m_t-1, \quad \delta_{\bar t}=m_{678}/m_t-1, \quad
\delta_{W^+}=m_{45}/m_W-1,\quad \delta_{W^-}=m_{78}/m_W-1,
\end{eqnarray}
which describe the relative departures of the invariant masses
of (\ref{mt}) and (\ref{mw}) from $m_t$ and $m_W$, respectively.
\section{NUMERICAL RESULTS}
In this section, we will illustrate the effect of the factorizable EW
$\cal{O}(\alpha)$ corrections described in Section 2 on the SM predictions
for six fermion reactions relevant for detection of
the top quark pair production and decay at the ILC (\ref{ee6f})
by showing results for total cross sections of its two specific channels
(\ref{nmmn}) and (\ref{nmud}).
We choose the $Z$ boson mass, Fermi
coupling and fine structure constant in the Thomson limit
as the EW SM input parameters
\begin{eqnarray}
\label{params1}
m_Z=91.1876\; {\rm GeV},\qquad
G_{\mu}=1.16637 \times 10^{-5}\;{\rm GeV}^{-2}, \qquad
\alpha_0=1/137.0359895.
\end{eqnarray}
The external fermion masses of reaction (\ref{nmmn}) and the top quark mass
are the following:
\begin{eqnarray}
\label{params3}
m_e=0.51099907\;{\rm MeV},\quad m_{\mu}=105.658389\;{\rm MeV},
\quad m_b=4.7\;{\rm GeV}, \quad m_t=178\;{\rm GeV}.
\end{eqnarray}
For definiteness, we give also values of the other fermion masses
\begin{eqnarray}
\label{params4}
m_{\tau}=1.77705\;{\rm GeV}, \quad m_u=75\;{\rm MeV},
\quad m_d=75\;{\rm MeV},\quad m_s\!=250\;{\rm MeV},
\quad m_c\!=1.5\;{\rm GeV}
\end{eqnarray}
and the value of a strong coupling $\alpha_s(m_Z^2)=0.117$.
Assuming a value of the Higgs boson mass,
the $W$ boson mass and the $Z$ boson width are determined with
{\tt ZFITTER} \cite{ZFITTER}, while the SM Higgs boson width is calculated
with {\tt HDECAY} \cite{HDECAY}. We obtain the following values of these
parameters for $m_H=120$~GeV:
\begin{eqnarray}
\label{params5}
m_W = 80.38509\; {\rm GeV}, \qquad \Gamma_Z = 2.495270\;{\rm GeV},
\qquad \Gamma_H=3.2780~{\rm MeV}.
\end{eqnarray}
The actual values of the $Z$ and Higgs boson widths are not very relevant
in the context of the top quark pair production as they enter the calculation
through the off resonance background contributions.
The EW corrected top quark and $W$ boson widths, which on the
other hand play
an essential role for the calculation, are calculated with a newly written
dedicated subroutine that reproduces results of \cite{Denner:1990tx,topdD}.
We obtain the following values for them for
the parameters specified in (\ref{params1}--\ref{params4})
\begin{eqnarray}
\label{params6}
\Gamma_W = 2.03777\; {\rm GeV}, \qquad \Gamma_t=1.67432 {\rm GeV}.
\end{eqnarray}
We have neglected the QCD correction to the widths $\Gamma_W$ and
$\Gamma_t$, as no QCD corrections have been included in the one--loop
corrections to the $t\bar t$-pair production process.
The EW corrected widths of (\ref{params6}) are used in the calculation
of the cross sections that include the EW factorizable corrections. For the
calculation of the lowest order cross sections of (\ref{nmmn}) and
(\ref{nmud}) the corresponding lowest order SM values of the top quark and
$W$-boson widths are used.
Results for the total cross sections of reactions (\ref{nmmn}) and
(\ref{nmud}) at three different centre of mass (CMS) energies in the presence
of the following cuts on quantities $\delta_t, \delta_{\bar t},\delta_{W^+},
\delta_{W^-}$, defined in (\ref{deltas}),
\begin{eqnarray}
\label{cuts}
\delta_t < 0.1, \quad \delta_{\bar t} < 0.1, \qquad
\delta_{W^+} < 0.1, \quad \delta_{W^-} < 0.1,
\end{eqnarray}
are shown in Table 1. The second column shows the Born cross sections
calculated with the complete set of the lowest order Feynman diagrams.
The third column shows the Born `signal' cross section, {\em i.e.}
the cross section obtained with the two lowest order signal diagrams of
Fig.~\ref{fig:diags0}a only. We see that imposing the invariant mass cuts
(\ref{cuts}) efficiently reduces the off resonance background, which
becomes quite sizeable if the cuts are not imposed \cite{eett6f1,eett6f2}.
The fourth and fifth
columns show the cross sections including the ISR and factorizable EW
corrections separately and the sixth column shows the results including both
the ISR and EW factorizable corrections. Note that the cross sections
of (\ref{nmud}) are almost exactly 3 times larger than the cross sections of
(\ref{nmmn}), in agreement with the naive counting of the colour degrees
of freedom. This is because the neutral current off resonance background
contributions that make
reaction (\ref{nmmn}) differ from (\ref{nmud}) are almost completely suppressed
in the presence of cuts (\ref{cuts}).
\begin{table}
{\small Table~1: Total cross sections of reactions (\ref{nmmn}) and
(\ref{nmud}) in fb at three different CMS energies in the presence of
cuts (\ref{cuts}).
The numbers in parenthesis show the uncertainty of the last decimals.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\hline
\multicolumn{6}{|c|}{\rule{0mm}{7mm} $\eebnmbmn$} \\[1.5mm]
\hline
\hline
\rule{0mm}{7mm} $\sqrt{s}$ (GeV) & $\sigma_{\rm Born}$
& $\sigma_{\rm Born}^{t^*\bar{t}^*}$
& $\sigma_{\rm Born+ISR}$ & $\sigma_{\rm Born+FEWC}$ &
$\sigma_{\rm Born+ISR+FEWC}$ \\[2mm]
\hline
\rule{0mm}{7mm}
430 & 5.9117(54) & 5.8642(45) & 5.2919(91)& 5.6884(55) & 5.0978(53) \\[1.5mm]
500 & 5.3094(50) & 5.2849(43) & 5.0997(51)& 4.9909(49) & 4.8085(48) \\[1.5mm]
1000 & 1.6387(16) & 1.6369(15) & 1.8320(18)& 1.4243(14) & 1.6110(16) \\[1.5mm]
\hline
\hline
\multicolumn{6}{|c|}{\rule{0mm}{7mm} $e^+ e^- \ra b \nu_{\mu} \mu^+ \bar{b} d \bar{u}$} \\[1.5mm]
\hline
\hline
\rule{0mm}{7mm} $\sqrt{s}$ (GeV) & $\sigma_{\rm Born}$
& $\sigma_{\rm Born}^{t^*\bar{t}^*}$
& $\sigma_{\rm Born+ISR}$ & $\sigma_{\rm Born+FEWC}$ &
$\sigma_{\rm Born+ISR+FEWC}$ \\[2mm]
\hline
\rule{0mm}{7mm}
430 & 17.727(16) & 17.592(13) & 15.857(20) & 17.052(16)
& 15.283(16) \\[1.5mm]
500 & 15.950(15) & 15.855(13) & 15.311(15) & 14.977(16)
& 14.438(14) \\[1.5mm]
1000 & 4.9134(48) & 4.9106(46) & 5.4949(55) & 4.2697(40)
& 4.8287(47) \\[1.5mm]
\hline
\end{tabular}
\end{center}
\end{table}
How the radiative corrections for the six fermion reactions (\ref{ee6f})
depend on the CMS energy is illustrated in Fig.~\ref{fig3},
where, on the left hand side, we plot the total cross sections of reaction
(\ref{nmud}) as a function of the CMS energy, both in the lowest order
and including different classes of corrections.
The dashed-dotted line shows the Born cross section, the dotted line
is the cross section including the ISR correction, the dashed line shows
an effect of the factorizable EW correction while the solid line shows
an effect of the combined ISR and factorizable EW correction.
The plots on the right hand side of Fig.~\ref{fig3} show the corresponding
relative corrections
\begin{eqnarray}
\label{relcor}
\delta_{\rm cor.}&=&\frac{\sigma_{\rm Born+cor.}
-\sigma_{\rm Born}}{\sigma_{\rm Born}},
\qquad {\rm cor. = FEW, ISR, ISR+FEW.}
\end{eqnarray}
The dashed line shows the relative factorizable EW correction. The correction
is small and positive a few GeV above the $t\bar t$-pair production threshold,
but already about 20 GeV above the threshold it becomes negative
and it falls down logarithmically towards more and more negative values, due
to large logarithmic terms $\sim \left[\ln \left(m_W^2/s\right)\right]^2$
and $\sim \ln \left(m_W^2/s\right)$, reaching 20\% at $\sqrt{s}=2$~TeV.
The dotted line shows the relative ISR correction, which on the other
hand is dominated by large collinear logarithms
$\left[\ln \left(s/m_e^2\right)\right]^2$
and $\ln \left(s/m_e^2\right)$. It starts from about $-25$\% at energies
close to the threshold and grows to almost $+25$\% at $\sqrt{s}=2$~TeV.
Finally, the solid line shows the combined ISR and factorizable EW correction.
The net relative correction is dominated by the ISR: it is large and negative
for energies not far above the threshold and it becomes
positive at high energies, reaching 1.4\% at at $\sqrt{s}=2$~TeV.
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(35,35)(60,-50)
\rput(5.3,-6){\scalebox{0.6 0.6}{\epsfbox{nmud.tt.epsi}}}
\end{picture}
\begin{picture}(35,35)(10,-50)
\rput(5.3,-6){\scalebox{0.6 0.6}{\epsfbox{rel.tt.epsi}}}
\end{picture}
\end{center}
\vspace*{4.0cm}
\caption{Total cross sections of (\ref{nmud}) including different classes of
the SM radiative corrections (left) and corresponding relative corrections
(\ref{relcor})(right) as functions of the CMS energy.}
\label{fig3}
\end{figure}
\section{SUMMARY AND OUTLOOK}
We have calculated the SM predictions for top quark pair
production and decay into six fermions at a linear $e^+e^-$ collider.
We have included the factorizable EW $\cal{O}(\alpha)$ corrections in the
pole approximation and QED corrections due to the initial state radiation
in the structure
function approach into SM predictions for the top quark pair
production and decay into six fermions at the ILC.
We have illustrated an effect of the radiative corrections on the predictions
by showing numerical results for two selected six-fermion reactions
(\ref{nmmn}) and (\ref{nmud}).
The ISR and factorizable EW radiative corrections are sizeable and therefore
should be included
in the analysis of future precision data on the top quark pair production
and decay from the ILC.
In order to obtain a complete EW next to leading order result for six fermion
reactions (\ref{ee6f}) in the pole approximation one should include
the nonfactorizable virtual photonic corrections
corresponding to an exchange of a virtual photon between the electrically charged
lines of the signal diagrams of Fig.~\ref{fig:diags0}(a) which has not been
included in the shaded blobs of Fig.~\ref{fig:diags}. For example, an exchange of
a photon between the initial state electron and any of the final state fermions or
intermediate $W$ bosons, or between the $b$ and $\bar t$ quark, or its decay products
should be taken into account.
This would allow for inclusion of the real photon emission from the external legs in an
exclusive way. Taking into account the QCD coreections
would also be higly desirable.
{\Large \bf Acknowledgements}
K.K. is grateful to the Alexander von Humboldt Foundation for supporting
his stay at DESY, Zeuthen, where this work has been partly done and to
the Theory Group of DESY, Zeuthen for kind hospitality.
\begin{appendix}
\section{PROJECTION OF MOMENTA}
In this appendix, we describe the projection procedure that has been
used in order to associate each phase space point of the full 6-particle
phase space of reaction (\ref{ee6f}) with a point of the
reduced phase space of the on-shell top pair production
(\ref{eett}) and subsequent decay. The on-shell momenta $k_t$ and $k_{\bar t}$
of the $t$-quar and $\bar{t}$-antiquark, $k_{W^{\pm}}$ of the decay
$W^{\pm}$-bosons,
and $k_i$, $i=3,...,8$, of the decay fermions of reaction
(\ref{ee6f}) are constructed from the four momenta $p_i$, $i=3,...,8$,
of the final state fermions of reaction (\ref{ee6f}) with
the following projection procedure.
First the on-shell four momenta of $t$ and $\bar{t}$ in the
centre of mass system (CMS) are found in the following way
\parbox{8cm}{
\begin{eqnarray*}
\left|\vec{k}_{t}\right|&=&
\frac{\lambda^{\frac{1}{2}}\left(s,m_t^2,m_t^2\right)}{2s^{\frac{1}{2}}},\\
k^0_{t}&=&\left(\vec{k}_{t}^2 + m_t^2\right)^{\frac{1}{2}},
\end{eqnarray*}}
\hfill
\parbox{6cm}{
\begin{eqnarray*}
\vec{k}_{t}&=&\left|\vec{k}_t\right|
\frac{\vec{p}_3+\vec{p}_4+\vec{p}_5}
{\left|\vec{p}_3+\vec{p}_4+\vec{p}_5\right|}, \\
\vec{k}_{\bar{t}}&=&-\vec{k}_t, \quad
k^0_{\bar{t}}=\sqrt{s}-k^0_t.
\end{eqnarray*}}
\hfill
\parbox{2cm}{\begin{eqnarray} \label{momtt} \end{eqnarray}}
Then the four momenta $p_3$, $p_4$ and $p_5$ ($p_6$, $p_7$ and $p_8$)
are boosted to the rest frame of the $b f_1 \bar{f'_1}$
($\bar{b} f_2 \bar{f'_2}$) subsystem of reaction (\ref{ee6f}), where they
are denoted $p'_3$, $p'_4$ and $p'_5$
($p'_6$, $p'_7$ and $p'_8$). The projected four momentum
$k'_3$ of $b$ ($k'_6$ of $\bar b$) is determined in the rest frame
of $b f_1 \bar{f'_1}$ ($\bar{b} f_2 \bar{f'_2}$) according to
\begin{eqnarray}
\left|\vec{k'}_{i}\right|=
\frac{\lambda^{\frac{1}{2}}\left(m_t^2,m_i^2,m_W^2\right)}{2m_t},
\qquad \vec{k'}_{i}=
\left|\vec{k'}_{i}\right|\frac{\vec{p'}_i}{\left|\vec{p'}_{i}\right|},
\qquad
{k'}_i^0=\left(\vec{k'}_{i}^2 + m_i^2\right)^{\frac{1}{2}}, \quad
i=3,6,
\label{k3k6}
\end{eqnarray}
which means that the directions of the $b$ and $\bar b$ momenta are kept unchanged
while their lengths are being altered.
The four momenta $p'_4$ and $p'_5$ ($p'_7$ and $p'_8$) are further boosted
to the rest frame of $f_1 \bar{f'_1}$ ($f_2 \bar{f'_2}$), where they
are denoted $p''_4$ and $p''_5$ ($p''_7$ and $p''_8$).
The projected four momenta $k''_4$ and $k''_5$ of $f_1$ and
$\bar{f'_1}$ ($k''_7$ and $k''_8$ of $f_2$ and $\bar{f'_2}$)
are in this frame determined according to
\begin{eqnarray}
\left|\vec{k'}_{4}\right|\!\!\!&=&\!\!\!
\frac{\lambda^{\frac{1}{2}}\left(m_W^2,m_4^2,m_5^2\right)}{2m_W},
\qquad
\left|\vec{k'}_{7}\right|=
\frac{\lambda^{\frac{1}{2}}\left(m_W^2,m_7^2,m_8^2\right)}{2m_W}, \qquad
\vec{k''}_{i}=
\left|\vec{k''}_{i}\right|\frac{\vec{p''}_i}{\left|\vec{p''}_{i}\right|},
\quad i=4,7, \nonumber\\
\vec{k''}_{5}\!\!\!&=&\!\!\!-\vec{k''}_{4},
\qquad \vec{k''}_{8}=-\vec{k''}_{7}, \qquad\qquad
{k''}^0_{j}=\left(\vec{k''}_{j}^2 + m_j^2\right)^{\frac{1}{2}},
\quad j=4,5,7,8.
\label{k4578}
\end{eqnarray}
This again means that the directions of momenta of $f_1$, $\bar{f'_1}$,
$f_2$ and $\bar{f'_2}$ are kept unchanged while their lengths
are being altered.
The four momenta $k''_4$ and $k''_5$ ($k''_7$ and $k''_8$)
are now boosted to the rest frame of the on-shell $t$ ($\bar t$)
and, finally, $k'_3$, $k'_4$ and $k'_5$ ($k'_6$, $k'_7$ and $k'_8$)
are boosted from the $t$ ($\bar t$) rest frame to the CMS giving
the desired projected four momenta $k_i$, $i=3,...,8$.
As one can easily see from Eqs.~(\ref{momtt}--\ref{k4578}),
the projected momenta, except for satisfying the necessary on-shell relations
$k_i^2=m_i^2$, $i=3,...,8$,
fulfil also other required on-shell relations
\begin{eqnarray}
\left(k_3+k_4+k_5\right)^2=\left(k_6+k_7+k_8\right)^2=
m_t^2, \qquad \left(k_4+k_5\right)^2=\left(k_7+k_8\right)^2=
m_W^2.
\end{eqnarray}
The described projection procedure is not unique. Moreover, it
strongly depends on the departures (\ref{deltas}) of invariant masses
$m_{345}$, $m_{678}$ of (\ref{mt}) from $m_t$, and of the invariant masses
$m_{45}$, $m_{78}$ of (\ref{mw}) from $m_W$.
How it works in practice is illustrated in Table~\ref{tabmom},
where two randomly selected sets of four momenta
$p_i$, $i=3,...,8$, $p_3+p_4+p_5$, $p_6+p_7+p_8$ , $p_4+p_5$,
$p_7+p_8$, and their projections $k_i$, $i=3,...,8$ , $k_t$,
$k_{\bar t}$ , $k_{W^+}$, $k_{W^-}$, respectively, are compared.
Momenta $p_i$ have been generated according to the Breit--Wigner
distribution in such a way that the invariant masses
of the $b f_1 \bar{f'_1}$, $\bar{b} f_2 \bar{f'_2}$,
$f_1 \bar{f'_1}$ and $f_2 \bar{f'_2}$ subsystems of reaction (\ref{ee6f})
fall into the vicinity of the masses of the corresponding primary on-shell
particles: $t$-quark, $\bar t$-antiquark, $W^+$- and $W^-$-boson, respectively.
\begin{table}
\label{tabmom}
{\small Table~A: A comparison of two randomly selected sets of the four momenta
$p_i$, $i=3,...,8$, $p_3+p_4+p_5$, $p_6+p_7+p_8$ , $p_4+p_5$,
$p_7+p_8$ and their projections $k_i$, $i=3,...,8$ , $k_t$,
$k_{\bar t}$ , $k_{W^+}$, $k_{W^-}$, respectively. Quantities
$\delta_t=m_{345}/m_t-1$, $\delta_{\bar t}=m_{678}/m_t-1$,
$\delta_{W^+}=m_{45}/m_W-1$ and $\delta_{W^-}=m_{78}/m_W-1$
describe relative departures of the corresponding final state
particle subsystems from a mass-shell of the $t$,
$\bar t$, $W^+$ and $W^-$, respectively.}
{\small
\begin{center}
\begin{tabular}{|c|rrrr|rrrr|}
\hline
& \multicolumn{4}{|c|}{\rule{0mm}{7mm}
$\delta_t=0.03\%$, $\delta_{\bar t}=0.19\%$,} &
\multicolumn{4}{c|}{$\delta_t=0.06\%$, $\delta_{\bar t}=0.17\%$,}\\
GeV & \multicolumn{4}{|c|}{$\delta_{W^+}=0.26\%$, $\delta_{W^-}=0.85\%$} &
\multicolumn{4}{c|}{$\delta_{W^+}=0.78\%$, $\delta_{W^-}=3.23\%$}\\[2mm]
\hline
\hline
\rule{0mm}{7mm}
$\!p_3$ & 154.0 & 141.4 & --28.1 & 53.8
& 116.5 & 89.3 & 13.1 & 73.6\\
$k_3$ & 153.8 & 141.3 & --28.0 & 53.8
& 116.9 & 89.6 & 13.1 & 73.8\\[1mm]
$p_4$ & 22.9 & --15.1 & --7.7 & --15.4
& 117.6 & 92.8 & --20.6 & --69.3\\
$k_4$ & 22.9 & --15.0 & --7.8 & --15.4
& 117.4 & 92.5 & --20.6 & --69.2\\[1mm]
$p_5$ & 73.1 & 24.6 & 35.7 & 58.8
& 16.0 & --3.2 & 7.5 & 13.8\\
$k_5$ & 73.3 & 24.8 & 35.8 & 59.0
& 15.7 & --3.3 & 7.5 & 13.5\\[1mm]
$p_6$ & 64.5 & --10.0 & 57.5 & --27.0
& 109.8 & --78.5 & --12.1 & --75.7\\
$k_6$ & 64.0 & --10.0 & 57.1 & --26.9
& 108.1 & --77.2 & --11.9 & --74.6\\[1mm]
$p_7$ & 112.1 & --109.2 & --20.7 & --15.1
& 106.1 & --95.4 & 33.7 & 32.0\\
$k_7$ & 112.4 & --109.5 & --20.4 & --15.1
& 107.8 & --97.2 & 34.5 & 31.4\\[1mm]
$p_8$ & 73.5 & --31.7 & --36.8 & --55.1
& 33.9 & --4.9 & --21.6 & 25.7\\
$k_8$ & 73.6 & --31.5 & --36.7 & --55.5
& 34.1 & --4.4 & --22.6 & 25.1\\[1mm]
$p_3+p_4+p_5$ & 249.9 & 150.9 & 0.0 & 97.3
& 250.1 & 178.9 & 0.0 & 18.0\\
$k_t$ & 250.0 & 151.0 & 0.0 & 97.4
& 250.0 & 178.8 & 0.0 & 18.0\\[1mm]
$p_6+p_7+p_8$ & 250.1 & --150.9 & 0.0 & --97.3
& 249.9 & --178.9 & 0.0 & --18.0\\
$k_{\bar t}$ & 250.0 & --151.0 & 0.0 & --97.4
& 250.0 & --178.8 & 0.0 & --18.0\\[1mm]
$p_4+p_5$ & 95.9 & 9.5 & 28.1 & 43.5
& 133.6 & 89.6 & --13.1 & --55.5\\
$k_{W^+}$ & 96.2 & 9.8 & 28.0 & 43.6
& 133.1 & 89.2 & --13.1 & --55.8\\[1mm]
$p_7+p_8$ & 185.6 & --140.9 & --57.5 & --70.3
& 140.0 & --100.4 & 12.1 & 57.7\\
$k_{W^-}$ & 186.0 & --141.0 & --57.1 & --70.5
& 141.9 & --101.6 & 11.9 & 56.6\\[1mm]
\hline
\end{tabular}
\end{center}}
\end{table}
\end{appendix}
\newpage
|
math/0510292
|
\section{Introduction}
Let $(M,g)$ be a compact Riemannian manifold without boundary, denote
by $\Delta_{g}$ its Laplace-Beltrami operator, and consider the
nonlinear Klein-Gordon equation
\be \label{KG}
(\partial_{t}^2-\Delta_{g}+V+m^2)v=-\partial_{2}f(x,v)
\ee
where $m$ is a strictly positive constant, $V$ is a smooth nonnegative
potential on $M$ and $f\in C^\infty(M\times \R)$ vanishes at least at
order 3 in $v$, $\partial_{2}f$ being the derivative with respect to
the second variable. In this work we prove that, for a special class
of manifolds and for almost every value of $m>0$, this
\textit{Hamiltonian} partial differential equation admits a Birkhoff
normal form at any order. The principal dynamical consequence is the
almost global existence of small amplitude solutions for such a
nonlinear Klein-Gordon equation.
More precisely, if $M$ is a Zoll manifold (i.e. a compact manifold
whose geodesic flow is periodic, e.g. a sphere), for almost every
value of $m>0$ and for any $N\in \Nn$, we prove that there is $s\gg 1$
such that, if the initial data
$(v\arrowvert_{t=0},\partial_{t}v\arrowvert_{t=0})$ are of size $\epsilon
\ll 1$ in $H^s \times H^{s-1}$, \eqref{KG} has a solution defined on a
time interval of length $C_{N}\ \epsilon^{-N}$. As far as we know,
this is the first result of that type when the dimension of the
manifold is larger or equal to 2.
\medskip
Let us recall some known results for the similar problem on $\R^d$,
when the Cauchy data are smooth, compactly supported, of size
$\epsilon\ll 1$. In this case, linear solutions decay in $L^\infty$
like $t^{-d/2}$ when $t\to\infty$. This allows one to get global
solutions including quasi-linear versions of (\ref{KG}), when
$d\geq2$ (see Klainerman \cite{K1} and Shatah \cite{Sh} if $d\geq3$
and Ozawa, Tsutaya and Tsutsumi \cite{OTT} if $d=2$). When $d=1$
Moriyama, Tonegawa and Tsutsumi \cite{MTT} proved that
solutions exist over intervals of time of exponential length
$e^{c/\epsilon^2}$. This result is in general optimal (see references
in \cite{D}), but global existence for small $\epsilon>0$ was proved
in \cite{D} when the nonlinearity satisfies a special condition (a ``null
condition'' in the terminology introduced by Klainerman in the case
of the wave equation in 3--space dimensions \cite{K}).
For the problem we are studying here, since we have no dispersion on a
compact manifold, we cannot hope to exploit any time decay of the
solutions of the linear equation. Instead we shall use a normal form
method. Remark that if in (\ref{KG}) the nonlinearity vanishes at
order $p\geq2$ at $v=0$, local existence theory gives a solution
defined on an interval of length $c\epsilon^{-p+1}$. Recently, in
\cite{DS1}, \cite{DS2} Delort and Szeftel proved that the solution of
the same equation exists, for almost all $m>0$, over a time interval
of length $c\epsilon^{-q+1}$, where $q$ is an explicit number strictly
larger than $p$ (typically $q=2p-1$). Actually these papers concern
more general nonlinearities than the one in \eqref{KG}, namely a
suitable class of non Hamiltonian nonlinearities depending on time and
space derivatives of $v$.
One of the ideas developed by Delort-Szeftel consists in reducing, by
normal form procedure, \eqref{KG} to a new system in which the
nonlinearity vanishes at order $q>p$ at the origin. In \cite{DS2} an
explicit computation showed that the first order normal form (which
leads to a nonlinearity of degree $q$) conserves also the $H^s$ norm
for any large $s$, whence the result cited above.
\medskip
On the other hand in \cite{BG} Bambusi and Gr{\'e}bert proved an
abstract Birkhoff normal form theorem for Hamiltonian PDEs. Although
that theorem remains valid in all dimensions, it supposes that the
nonlinearity satisfies a ``tame modulus" property. In \cite{BG} this
property was only verified for a quite general class of $1-d$ PDEs and
for a particular NLS equation on the torus $\T^d$ with arbitrary $d$.
Actually in that paper, the tame modulus property was verified by the
use of the property of ``well localization with respect to the
exponentials" established by Craig and Wayne \cite{CW}, a property which
has no equivalent in higher dimensions.
It turns out that in \cite{DS2} Delort and Szeftel proved an estimate
concerning multilinear forms defined on $M$ that implies a weaker form
of the tame modulus property assumed in \cite{BG}.
The present paper is the result of the combination of the arguments
of \cite{DS1}, \cite{DS2} and of \cite{BG}.
\medskip
We recall that some other partial normal form results for PDEs have been
previously obtained by Kuksin and P{\"o}schel \cite{KP96}, by Bourgain
\cite{Bo96,Bo04} and, for perturbations of completely resonant systems, by
Bambusi and Nekhoroshev \cite{BN98}. For a more precise discussion we
refer to the introduction of \cite{BG}.
\medskip
Let us conclude this introduction mentioning several open questions.
The first concerns the possibility of proving almost global existence
for more general nonlinearities than the Hamiltonian ones we consider
here. Of course, one cannot expect to be able to do so for any
nonlinearity depending on $v$ and its first order derivatives: in
\cite{D1} an example is given on the circle ${\mathbb{S}}^1$ of a
nonlinearity for which the solution does not exist over a time
interval of length larger than the one given by local existence theory
(Remark that this example holds true for any value of $m>0$). On the
other hand, Delort and Szeftel constructed in \cite{DS3} almost global
solutions of equations of type \eqref{KG} on manifolds of revolution,
for radial data, with a nonlinearity $f$ depending on
$(v,\partial_{t}v)$ and even in $\partial_{t}v$. We thus ask the
question of finding a ``null condition" (in the spirit of Klainerman
\cite{K}) for semi-linear nonlinearities $f(v,\partial_{t}v, \nabla
v)$, which would allow almost global existence of small $H^s$
solutions for almost every $m>0$.
The second question we would like to mention concerns the
exceptional values of $m$ which are excluded of our result. The
conservation of the Hamiltonian of equation \eqref{KG} allows one to
control the $H^1$-norm of small solutions. This implies global
existence of small $H^1$ solutions in one or two space dimensions. The
results we establish in the present paper show that for almost every
$m>0$, the $H^s$-norms of these solutions remain small over long time
intervals if they are so at $t=0$. What happens when $m$ is in the
exceptional set? In \cite{Bo96b} Bourgain constructed, in one space
dimension and for a convenient perturbation of $-\Delta$, an example
of a solution whose $H^s$-norm grows with time. Nothing seems to be
known in larger dimensions. In particular, if $d\geq 3$, one does not
even know if for all $m>0$ a solution exists almost globally,
eventually without staying small in $H^s$ ($s\gg 1$).
\section{Statement of main results}
We begin, in section \ref{subsec1.1}, by a precise exposition of our
result concerning the almost globality. The Birkhoff normal form
theorem for equation \eqref{KG} that implies the almost globality
result will be presented in section \ref{subsec1.3}, after the
introduction of the Hamiltonian formalism in section \ref{subsec1.2}.
\subsection{Almost global solution}\label{subsec1.1}
Let $(M,g)$ be a compact Riemannian manifold without boundary of
dimension $d\geq 1$. Denote by $\Delta_{g}$ its Laplace-Beltrami
operator. Let $V$ be a smooth nonnegative potential on $M$ and $m\in
(0,\infty)$. Let $f\in C^\infty(M\times \R)$ be such that $f$ vanishes at
least at order 3 in $v$. We consider the following Cauchy problem for
the nonlinear Klein-Gordon equation
\begin{align}
\begin{split} \label{KGCauchy}
(\partial_{t}^2-\Delta_{g}+V+m^2)v&=-\partial_{2}f(x,v)
\\
v\arrowvert_{t=0}&=\epsilon v_{0}
\\
\partial_{t}v\arrowvert_{t=0}&=\epsilon v_{1}
\end{split}
\end{align}
where $v_{0}\in H^s(M,\R)$, $v_{1}\in H^{s-1}(M,\R)$ are real valued
given data and $\epsilon>0$. We shall prove
that the above problem has almost global solutions for almost every
$m$ when $\epsilon>0$ is small enough and $s$ is large enough, under
the following geometric assumption on $M$:
\definition One says that $(M,g)$ is a Zoll manifold if and only if
the geodesic flow is periodic on the cosphere bundle of $M$.
\medskip
Our main dynamical result is the following:
\begin{theorem} \label{thm1}
Let $(M,g)$ be a Zoll manifold and let $V:M\to \R$ be a smooth
nonnegative potential. Let $r\in \Nn$ be an arbitrary integer. There
is a zero measure subset $\mathcal N$ of $(0,+\infty)$, and for any
$m\in (0,+\infty) \setminus \mathcal N$, there is $s_{0}\in\Nn$ such that
for any $s\geq s_{0}$, for any real valued $f\in C^\infty(M\times \R)$
vanishing at least at order 3 at $v=0$, there are $\epsilon_{0}>0$,
$c>0$, such that for any pair $(v_{0},v_{1})$ of real valued functions
belonging to the unit ball of $H^s(M,\R)\times H^{s-1}(M, \R)$, any
$\epsilon \in (0,\epsilon_{0})$, the Cauchy problem \eqref{KGCauchy}
has a unique solution
$$ v\in C^0((-T_{\epsilon},T_{\epsilon}),H^s(M,\R))\cap
C^1((-T_{\epsilon},T_{\epsilon}),H^{s-1}(M,\R))$$ with
$T_{\epsilon}\geq c\epsilon^{-r}$. Moreover there is $C>0$ such that,
for any $t\in (-T_{\epsilon},T_{\epsilon})$, one has
\be
\label{estimHs} \Vert{v(t,\cdot )}\Vert_{H^s}+\Vert\partial_{t}
v(t,\cdot )\Vert_{H^{s-1}}\leq C\epsilon \ .
\ee
\end{theorem}
\noindent
{\bf Comments}
The above theorem provides Sobolev bounded almost global solutions for
equation \eqref{KGCauchy} with small smooth Cauchy data on a
convenient class of compact manifolds. To our knowledge this is the
first result of this kind on compact manifolds of dimension larger or equal
to 2. In the case of one dimensional compact manifolds, similar
statements have been obtained by Bourgain \cite{Bo96,Bo04} (with a loss on
the number of derivatives of the solution with respect to those of the
data), by Bambusi \cite{Bam03} and by Bambusi-Gr{\'e}bert
\cite{BG}. Remark that in this case, because of the conservation of
the Hamiltonian of the equation, one controls uniformly the $H^1$-norm
of small solutions, which implies global existence of such
solutions. The results of the preceding authors allow to control
$H^s$-norms of these solutions for very long times. In the case of
compact manifolds of revolution and for convenient radial data,
Delort and Szeftel got in \cite{DS3} Sobolev bounded almost global
solutions (remark that this result is morally one-dimensional).
\medskip
The assumption that $M$ is a Zoll manifold will be used in the proof
through distribution properties of the eigenvalues of the Laplacian
of $M$. Actually we shall prove theorem \ref{thm1} for any
compact manifold without boundary ($M,g)$ such that if
\be \label{P}
P=\sqrt{-\Delta_{g} +V},
\ee
the spectrum $\sigma(P)$ of $P$ satisfies the following condition:
there are constants $\tau >0$, $\alpha \in \R$, $c_0 >0$, $\delta
>0$, $C_0 >0$, $D\geq 0$, and a family of disjoint compact
intervals $(K_{n})_{n\geq 1}$, with $K_{1}$ at the left of $K_{2}$
and for $n\geq 2$
\be \label{Kn}
K_{n}=\left[ \frac{2\pi}{\tau}n+\alpha -\frac{c_{0}}{n^\delta},
\frac{2\pi}{\tau}n+\alpha +\frac{c_{0}}{n^\delta}\right],
\ee
such that
\begin{align}\begin{split}\label{115}
\sigma(P)&\subset \bigcup_{n\geq 1}K_{n}\\
\#(\sigma(P)\cap K_{n})&\leq C_{0}n^D\ .
\end{split}\end{align}
If $M$ is a Zoll manifold, and if $\tau >0$ is the minimal period of
the geodesic flow on $M$, the results of Colin de Verdi{\`e}re
\cite{CV} (see also Guillemin \cite{G} and Weinstein \cite{W}) show
that the large eigenvalues of $P$ are contained inside the union of
the intervals
$$
\left[ \frac{2\pi}{\tau}n+\alpha -\frac{C}{n},
\frac{2\pi}{\tau}n+\alpha +\frac{C}{n}\right]
$$
for $n$ large enough and for some constant $C>0$. Making a translation
in $n$ and $\alpha$, and changing the definition of the constants, one
sees that this implies conditions \eqref{Kn}, \eqref{115} for any
$\delta \in (0,1)$ (remark that the second condition in \eqref{115}
holds true with $D=d-1$ because of Weyl law).
On the other hand conditions \eqref{Kn}, \eqref{115} are not more
general than the assumption that $M$ is a Zoll manifold, since by
theorem 3.2 in Duistermaat and Guillemin~ \cite{DG}, they imply
that the geodesic flow is periodic.
\subsection{Hamiltonian formalism}\label{subsec1.2}
We introduce here (see e.g. \cite{CheMa}) the Hamiltonian
formalism we shall use to solve the equation. We denote by
\begin{equation}
\label{1.2.1}
\langle f_1,f_2\rangle
\end{equation}
the bilinear pairing between complex valued distributions and test
functions on $M$. We shall use the same notation for vector valued
$f_1,f_2$.
If $F$ is a $C^\infty$ function on an open subset $\U$ of the Sobolev
space of real valued functions $\hsi$, $\sig\geq 0$, we define for
$p\in\U$, the $L^2$ gradient $\nabla F(p)$ by
\begin{equation}
\label{1.2.2}
\partial F(p)h= \langle\nabla F(p), h \rangle\ ,\quad \forall h \in \hsi,
\end{equation}
$\partial F$ denoting the differential. In that way $\nabla F(p) $ is
an element of $H^{-\sig}(M,\R)$. When we consider real valued
$C^\infty$ functions defined on an open subset of
$\hsi\times \hsi\equiv \hsi^2$, $(p,q)\mapsto F(p,q)$ we write
\begin{eqnarray*}
\partial F(p,q)&=& (\partial_p F(p,q),\partial_q F(p,q) )
\\
\nabla F(p,q)&=& (\nabla_p F(p,q),\nabla_q F(p,q) )\in
H^{-\sig}(M,\R)\times H^{-\sig}(M,\R).
\end{eqnarray*}
Endow $\hsi^2$ with the weak symplectic structure
\begin{equation}
\label{1.2.4}
\Omega\left((p,q),(p',q')\right):= \langle q,p'
\rangle - \langle q',p\rangle = \langle J^{-1}(p,q),(p',q') \rangle
\end{equation}
where $J$ is given by
\begin{equation}
\label{1.2.5}
J=
\left[
\begin{matrix}
0 & -\uno
\\
\uno &0
\end{matrix}
\right].
\end{equation}
If $\U$ is an open subset of $\hsi^2$ and $F\in C^\infty (\U,\R)$, then,
for $(p,q)\in\U$, we define its Hamiltonian vector field by
\begin{equation}
\label{1.2.6}
X_F(p,q)=J\nabla F(p,q)=(-\nabla_qF(p,q),\nabla_pF(p,q))
\end{equation}
which is characterized by
\begin{equation}
\label{1.2.7}
\Omega\left( X_F,(h_p,h_q)\right)=\partial F (h_p,h_q)= \partial_p F
h_p+\partial_q F h_q
\end{equation}
for any $(h_p,h_q)\in\hsi^2$.
A special role is played by the functions whose Hamiltonian vector
field is an $\hsi^2$ valued function. Thus we give the following
\begin{definition}
\label{d.1.2.1}
If $\U$ is an open subset of $\hsi^2$, we denote by $\csi$
(resp. $C^\infty_\sig(\U,\C)$) the space of real (resp. complex)
valued $C^\infty$ functions defined on $\U$ such that
\begin{equation}
\label{1.2.8}
X_F\in C^\infty(\U,\hsi^2 )\quad (\text{or}\quad \nabla F\in
C^\infty(\U,\hsi^2 ) ),
\end{equation}
resp.
\begin{equation}
\label{1.2.8bis}
X_F\in C^\infty(\U,\hsi^2 \otimes\C)\quad (\text{or}\quad \nabla F\in
C^\infty(\U,\hsi^2 \otimes\C) ).
\end{equation}
\end{definition}
We shall use complex coordinates in $\hsi^2$ identifying this space
with $\hsic$, through $(p,q)\mapsto u=(p+\im q)/\sqrt2$. We set
\begin{eqnarray}
\label{1.2.9}
&\partial_u=\frac{1}{\sqrt2}(\partial_p-\im\partial_q)
&\partial_{\bar u}=\frac{1}{\sqrt2}(\partial_p+\im\partial_q)
\\
\label{1.2.9.1}
&\nabla_u=\frac{1}{\sqrt2}(\nabla_p-\im\nabla_q)
&\nabla_{\bar u}=\frac{1}{\sqrt2}(\nabla_p+\im\nabla_q)
\end{eqnarray}
so that, if $F$ is a $C^1$ real valued function, we have an identification
\begin{equation}
\label{1.2.10}
X_F(u,\bar u)=\im \nabla_{\bar u}F(u,\bar u)\ .
\end{equation}
If $F\in\csi$, then clearly $X_F\in C^\infty(\U,\hsic)$.
For $m \in (0,+\infty)$ let us define
\begin{equation}\label{1.2.3}
\Lambda_m = \sqrt{-\Delta_g+V+m^2}.
\end{equation}
Let $\sig>(d-1)/2$. We shall write equation \eqref{KG} as a
Hamiltonian system for $p=\lm^{-1/2}\partial_t v $ and $q=\lm^{1/2}v$
on $\hsi^2$. Define
\begin{equation}
\label{1.2.11}
G_2(p,q)=\frac{1}{2}\int_M
\bigl(\bigl|\lm^{1/2}p\bigr|^2+\bigl|\lm^{1/2}q\bigr|^2\bigr) dx\
,\quad \tilde G(p,q)=\int_M f(x,\lm^{-1/2}q) dx
\end{equation}
where $dx$ is the Riemannian volume on $M$, and set
\begin{equation}
\label{1.2.12}
G=G_2+\tilde G.
\end{equation}
Then by (\ref{1.2.6})
\begin{eqnarray}
\label{1.2.13}
X_{G_2}(p,q)=(-\lm q,\lm p)\ ,\quad X_{\tilde G}(p,q)= (-\lm^{-1/2}
\partial_2f(x,\lm^{-1/2} q),0)
\end{eqnarray}
where $\partial_2f$ is the derivative with respect to the second
argument. Then one
has that $\tilde G\in \csi$ with $\U=\hsi^2$ (actually $X_{\tilde G}$ takes
values in $H^{\sig+1}(M,\R)^2$).
It follows also that equation (\ref{KG}) can be written as
\begin{equation}
\label{1.2.15}
(\dot p,\dot q)=X_G(p,q)
\end{equation}
or, using (\ref{1.2.10})
\begin{equation}
\label{1.2.16}
\dot u=\im \nabla_{\bar u}G(u,\bar u).
\end{equation}
In the rest of this section we shall give a few technical results that
we shall need for the proofs of theorems \ref{thm1}, \ref{thm2}.
\begin{definition}
\label{d.1.2.2}
Let $\U$ be an open subset of $\hsi^2$ and $F_j\in\csi$,
$j=1,2$. Then their Poisson bracket is defined by
\begin{equation}
\label{1.2.17}
\left\{F_1,F_2\right\}=\partial F_2
\cdot X_{F_1}=\Omega(X_{F_2},X_{F_1})
\end{equation}
and one has $\{F_1,F_2\}\in\csi $.
One extends the definition to complex valued functions by linearity of
the bracket relatively of each of its arguments.
\end{definition}
The fact that (\ref{1.2.17}) has a smooth vector field follows from
the well known formula
\begin{equation}
\label{1.2.17.1}
X_{\{ F_1,F_2\}}=[X_{F_1},X_{F_2}]=\partial X_{F_2}\cdot X_{F_1}-\partial
X_{F_1}\cdot X_{F_2}\ ,
\end{equation}
with the square bracket denoting the Lie bracket of vector fields (for
a proof of this formula in the case of weak symplectic manifolds see
\cite{Bam93}). In case either $F_1$ or $F_2$ do not have a smooth
vector field, one can also define their Poisson brackets by formula
(\ref{1.2.17}) but one has to check that it is a well defined
function, using the fact that we may write
\begin{equation}
\begin{split}
\label{1.2.18}
\left\{F_1,F_2\right\} &= - (\partial_p F_2)(\nabla_q F_1) +
(\partial_q F_2)(\nabla_p F_1)\\ &= - \langle\nabla_p F_2,\nabla_q F_1
\rangle + \langle\nabla_q F_2,\nabla_p F_1 \rangle \\ &= \im
(\partial_u F_2)(\nabla_{\bar u} F_1) -\im (\partial_{\bar u}
F_2)(\nabla_u F_1).
\end{split}
\end{equation}
Let us recall also the rule of transformation of vector fields and
Poisson brackets under symplectomorphism. Let $\U$ and $\V$ be open
subsets of $\hsi^2$, and $\chi:\U\to\V$ be a smooth symplectic
diffeomorphism. We have by definition for any $u \in \U$
\begin{equation}\label{1.2.19}
(\partial\chi(u))^{-1} = J{}^t(\partial\chi(u))J^{-1}.
\end{equation}
For $F\in C^{\infty}_\sig(\V,\R)$ one has
\begin{equation}
\label{1.2.20}
X_{F\circ \chi}(u)=(\partial \chi(u))^{-1}X_F(\chi(u))
\end{equation}
and therefore $F\circ\chi\in C^{\infty}_\sig(\U,\R) $ (actually
(\ref{1.2.20}) holds in the more general context where $\nabla F$ has
a domain which is left invariant by $\chi$). We also remark that for
any $C^1$ real-valued function $F_1$ on $\V$ and for any $F_2$ in
$C^{\infty}_\sig(\V,\R)$ one has
\begin{equation}
\label{1.2.21}
\left\{F_1\circ\chi,F_2\circ\chi\right\}=
\left\{F_1,F_2\right\}\circ\chi .
\end{equation}
To conclude this subsection let us state as a lemma the well known
formula that is the root of the Birkhoff normal form method as
developed using Lie transform.
\begin{lemma}
\label{l.1.2.3}
Let $F,G$ be two real valued functions defined on
$\U\subset\hsi^2$. Assume that $F\in C^\infty_\sig(\U,\R)$ and
$G\in C^\infty(\U,\R)$. Denote by
$(Ad\, F)\, h=\left\{F,h\right\}$. Then $(Ad\,F)G$ is well
defined, and if we assume that for some $n\geq 1$
\begin{equation}
\label{1.2.22}
F_n:= (Ad\,F)^n G
\end{equation}
is well defined and belongs to $C^\infty_\sig(\U,\R)$, then $F_{n+1}$
is also well defined.
Let $\V$ be such that $\overline{\V}\subset \U$. There
exists a positive $T$ such that the flow $\V\ni(p,q)\mapsto
\Phi^t(p,q)\in\U$ of $X_F$ is well defined and smooth for
$|t|<T$. Moreover, for $|t|<T$ and $(p,q)\in\V$, one has for any $r\in
\Nn$ the formula
\begin{equation}
\label{1.2.23}
G(\Phi^t(p,q))=\sum_{n=0}^{r}\frac{t^n}{n!}F_n(p,q)+\frac{1}{r!}\int
_0^t (t-s)^{r}F_{r+1}(\Phi^s(p,q))ds.
\end{equation}
\end{lemma}
\proof Remark first that $(Ad\,F)G$ is well defined by \eqref{1.2.18},
and that under our assumptions, for $n\geq 2$, $F_n$ is well defined
by definition~\ref{d.1.2.2}. Since $X_F$ is smooth on $\U$ the flow
$\Phi^t(.)$ is a smooth symplectic diffeomorphism on $\V$. For fixed
$(p,q)$ put $\phi(t)=G(\Phi^t(p,q))$. Formula (\ref{1.2.23}) follows
from Taylor formula since $\phi(t)$ is $C^\infty$. We thus have
$\phi'(t)=[(Ad\, F)\, G ](\Phi^t(p,q))=F_1(\Phi^t(p,q))$. Using
(\ref{1.2.22}) one proves by induction that
$\phi^{(n)}(t)=F_n(\Phi^t(p,q))$ and the conclusion follows. \qed
\subsection{Birkhoff Normal Form}
\label{subsec1.3}
Using the notation of section \ref{subsec1.1}, we define for $n\geq 1$
spectral projectors
\be \label{pin}
\Pi_{n}=\mathbf{1}_{K_{n}}(P)\ .
\ee
Then, for $(p,q)\in H^{s}(M,\R)^2$ we introduce the quantities
\be \label{Jn}
J_{n}(p,q)=\frac{1}{2}\left( \Vert\Pi_{n}p\Vert^2_{L^2}
+\Vert\Pi_{n}q\Vert^2_{L^2}\right)\ .
\ee
For $(p,q)\in H^{s}(M,\R)^2$ we denote
$$
\norma{(p,q)}_{s}^2:= \Vert p\Vert^2_{H^s}+\Vert q\Vert^2_{H^s}\
$$
We can now state our Birkhoff normal form result for the nonlinear
Klein-Gordon equation on Zoll manifolds:
\begin{theorem}\label{thm2}
Let $G$ be the Hamiltonian given by \eqref{1.2.11},
\eqref{1.2.12}. Then for any $r\geq 1$, there exists a zero
measure subset $\mathcal N$ of $(0,+\infty)$, and for any $m\in
(0,+\infty) \setminus \mathcal N$, there exists a large $s_0$ with
the following properties: For any $s\geq s_0$, there exist two
neighborhoods of the origin $\U$, $\V$, and a bijective canonical
transformation $\Tr:\V\to\U$
which puts the Hamiltonian in the form \begin{equation}
\label{for.nor} G\circ\Tr= G_{2}+\Ze+\resto \end{equation} where
$\Ze$ is a real valued continuous polynomial of degree at most
$r+2$ satisfying
\begin{equation}
\label{stime1}
\left\{J_n,\Ze \right\}=0\ ,\quad \forall n\geq 1
\end{equation}
and $\resto\in C^\infty_s(\V,\R)$ has a zero of order $r+3$ at
the origin. Precisely its vector field fulfills the estimate
\begin{equation}
\label{stime2}
\norma{X_{\resto}(p,q)}_s\leq
{C_s} \norma{(p,q)}_s^{r+2}\ ,\quad (p,q)\in\V.
\end{equation}
Finally the canonical transformation satisfies
\begin{equation}
\label{stime}
\norma{(p,q)-\Tr(p,q)}_s\leq C_s
\norma{(p,q)}_s^2\ ,\quad (p,q)\in\V.
\end{equation}
Exactly the same estimate is fulfilled on $\U$ by the inverse canonical
transformation.
\end{theorem}
From \eqref{stime} it follows $\Tr(0)=0$ and $\partial\Tr(0)=\uno $.
Theorem \ref{thm2} implies theorem \ref{thm1} (see the proof of theorem
\ref{thm1} in section \ref{sec:proofthm1}) but it says more:
namely, the $J_{n}$ are almost conserved
quantities for the equation \eqref{KG}. More
precisely, with the notation of theorems \ref{thm1} and \ref{thm2},
for any $n\geq 1$
\be \label{estimJn} \vert J_{n}(p(t),q(t))-
J_{n}(p(0),q(0))\vert\leq \frac C{n^{2s}}\epsilon^3
\quad \mbox{ for } |t|\leq \epsilon^{-r}
\ee
where $p(t)=\lm^{-1/2}\partial_t v(t) $ and $q(t)=\lm^{1/2}v(t)$ (for
the proof see the end of section \ref{sec:proofthm1}).
Roughly speaking, the last property means that energy transfers are
allowed only between modes corresponding to frequencies in the same
spectral interval $K_{n}$.
\section{Proof of the main results}
\label{proof}
In this section we prove theorem \ref{thm2} and then deduce theorem
\ref{thm1}. The proof uses a Birkhoff procedure described in subsection
\ref{birk}.
Formally this procedure is very close to the classical Birkhoff scheme
in finite dimension. Nevertheless, in infinite dimension, we need to
define a convenient framework in order to justify the formal
constructions. This framework, first introduced in \cite{DS2}, is
presented, and adapted to our context, in the next subsection.
\subsection{Multilinear Forms}
\label{multi}
Let us introduce some notations. If $n_1,\ldots,n_{k+1}$ are in $\Nn^*$,
we denote the second and third largest elements of this family by
\begin{equation}
\begin{split}
\label{2.1.1}
\max\!{}_2(n_1,\ldots,n_{k+1})&=\max\left(\left\{n_1,\ldots,n_{k+1}\right\}
-\left\{n_{i_0}
\right\}\right)
\\
\mu (n_1,\ldots,n_{k+1})&=\max\left(\left\{n_1,\ldots,n_{k+1}\right\}
-\left\{n_{i_0},n_{i_1}
\right\}\right)
\end{split}
\end{equation}
where $i_0$ and $i_1$ are the indices such that
$$ n_{i_0}=\max\!{}(n_1,\ldots,n_{k+1})\ ,\quad
n_{i_1}=\max\!{}_2(n_1,\ldots,n_{k+1})
$$
and where by convention, when $k=1$,
$\mu (n_1,n_2)=1$.
We define then
\begin{equation}
\begin{split}
\label{2.1.2}
S(n_1,\ldots,n_{k+1})&=\sum_{\ell=1}^{k+1}[n_\ell-\sum_{j\not=\ell}n_j]_++
\mu (n_1,\ldots,n_{k+1})
\end{split}
\end{equation}
where $[a]_+=\max(a,0)$. If $n_k$ and $n_{k+1}$ are the largest two among
$n_1,\ldots,n_{k+1}$, we have
\begin{equation}
\begin{split}
\label{2.1.3}
\mu (n_1,\ldots,n_{k+1})&\sim n_1+\cdots+n_{k-1} +1
\\
S(n_1,\ldots,n_{k+1})&\sim \left|n_k-n_{k+1}\right|+n_1+\cdots+n_{k-1}+1\ .
\end{split}
\end{equation}
We shall denote by $\E$ the algebraic direct sum of the ranges of the
$\Pi_n$'s defined by (\ref{pin}).
\begin{definition}
\label{d.2.1.1}
Let $k\in\Nn^*$, $\nu\in[0,+\infty)$, $N\in\Nn$.
\begin{itemize}
\item[i)] We denote by $\EL\nu N{k+1}$ the space of $(k+1)$--linear
forms $L:\E\times \cdots\times \E\to\C$ for which there exists $C>0$ such
that for any $u_1,\ldots,u_{k+1}\in\E$, any $n_1,\ldots,n_{k+1}$ in $\Nn^*$
\begin{equation}
\label{2.1.4}
\left|L(\Pi_{n_1}u_1,\ldots,\Pi_{n_{k+1}}u_{k+1}) \right|\leq
C\frac{\mu (n_1,\ldots,n_{k+1})^{\nu+N}}
{S(n_1,\ldots,n_{k+1})^N}\prod_{j=1}^{k+1} \norma{u_j}_{L^2}.
\end{equation}
\item[ii)] We denote by $\EM\nu N{k}$ the space of $k$--linear maps
$M:\E\times \cdots \times \E\to L^2(M,\C)$ for which there exists
$C>0$ such that for any $u_1,\ldots,u_{k}\in\E$ any
$n_1,\ldots,n_{k+1}$ in $\Nn^*$
\begin{equation}
\label{2.1.5}
\left\Vert \Pi_{n_{k+1}} M(\Pi_{n_1}u_1,\ldots,\Pi_{n_{k}}u_{k})
\right\Vert_{L^2}
\leq C\frac{\mu (n_1,\ldots,n_{k+1})^{\nu+N}}
{S(n_1,\ldots,n_{k+1})^N}\prod_{j=1}^{k} \norma{u_j}_{L^2}.
\end{equation}
\end{itemize}
The best constant $C$ in (\ref{2.1.4}), (\ref{2.1.5}) defines a norm
on the above spaces. We set also $\EL\nu {+\infty}{k+1} =
\bigcap_{N\in\Nn} \EL\nu N{k+1}$.
\end{definition}
Consider $L\in \EL \nu N{k+1}$ with $N>1$ and fix an integer
$j\in\{1,\ldots,k+1\}$ and elements $u_\ell\in\E$ for
$\ell\in\left\{1,\ldots,k+1\right\} -\left\{j\right\}$. Then by
\eqref{2.1.3}, \eqref{2.1.4}
$$
\sum_{n_j}L(u_1,\ldots,u_{j-1},\Pi_{n_j}u_j,
u_{j+1},\ldots,u_{k+1})
$$
converges for any $u_j\in L^2(M,\C)$, so $u_j\mapsto
L(u_1,\ldots,u_{k+1})$ extends as a continuous linear form on
$L^2(M,\C)$. Consequently, there is a unique element
$M_{L,j}(u_1,\ldots,\widehat {u_j},\ldots,u_{k+1})$ of $L^2(M,\C)$ with
\begin{equation}
\label{2.1.6}
L(u_1,\ldots,u_{k+1})=\left\langle u_j, M_{L,j}(u_1,\ldots,\widehat
{u_j},\ldots,u_{k+1}) \right\rangle
\end{equation}
for all $u_1,\ldots, u_{k+1}\in\E$. By \eqref{2.1.4}, $M_{L,j}$ satisfies
\eqref{2.1.5}, i.e. defines an element of $\EM\nu Nk$. Conversely, if
we are given an element of $\EM\nu Nk$, we define a multilinear form
belonging to $\EL\nu N{k+1}$ by a formula of type \eqref{2.1.6}.
The
basic example satisfying definition \ref{d.2.1.1} is provided by the
following result proved in \cite{DS2} (proposition 1.2.1).
\begin{proposition}
\label{p.2.1.2}
Let $k\in \Nn^*$. Denote by $dx$ any measure on $M$ with a $C^\infty$
density with respect to the Riemannian volume. There is
$\nu\in(0,+\infty)$ such that the map
\begin{equation}
\label{2.1.7}
(u_1,\ldots,u_{k+1})\mapsto \int_M u_1\cdots u_{k+1} dx
\end{equation}
defines an element of $\EL\nu{+\infty}{k+1}$.
\end{proposition}
\begin{remark}
\nonumber Up to now we did not use the spectral assumption
\eqref{115} on the manifold $M$. Actually proposition 1.2.1 of
\cite{DS2} is proved on any compact manifold without boundary,
replacing in \eqref{2.1.4} the spectral projectors $\Pi_n$ defined in
\eqref{pin} by spectral projectors $\Pi_\lambda$ associated to
arbitrary intervals of center $\lambda$ and length $O(1)$.
\end{remark}
We now use the fundamental example given by the previous proposition
to verify that the nonlinearity $\tilde G$ defined in \eqref{1.2.11}
is in a good class of Hamiltonian functions. If $L$ is a
$(k+1)$-linear map, and if $a\in \Nn$ satisfies $0\leq a\leq k+1$, we
set for $u,\bar u \in \mathcal E$
\be \label{2.1.10}
\underline L^a(u,\bar u)= L(u,\ldots,u,\bar u,\ldots,\bar u)
\ee
where in the right hand side one has $a$ times $u$ and $(k+1-a)$--times
$\bar u$. We then define the following class of Hamiltonian functions:
\begin{definition}
\label{multi.d}
For $k \in \Nn$ and $s, \nu \in\R$ with $s>\nu + \frac{3}{2}$, we define
$\mathcal{H}_{k+1}^s(\nu)$ as the
space of all real valued smooth functions defined on $H^s(M,\C)$,
$(u,\bar{u}) \to Q(u,\bar{u})$, such that there are for $\ell = 0,\ldots,k+1$
multilinear forms $L_\ell \in \mathcal{L}_{k+1}^{\nu,+\infty}$ with
\[Q(u,\bar{u}) = \sum_{\ell =0}^{k+1} \underline{L}_\ell^\ell(u,\bar{u}).\]
\end{definition}
This definition is obtain by adapting to our context the usual
definition of polynomial used for example in the theory of analytic
functions on Banach spaces (see for example \cite{Muj} or
\cite{Nik86}).
As a consequence of proposition \ref{p.2.1.2} one gets:
\begin{lemma}\label{lem:G}
Let $P$ be the Taylor's polynomial of $\tilde G$ at degree $k$.
Then there exists
$\nu\in(0,+\infty)$ such that $P$ can be decomposed as
$$P=\sum_{j=3}^{k} P_{j}$$
where $P_{j} \in \mathcal{H}_{j}^s(\nu)$.
\end{lemma}
Let us recall the main properties for $\EM\nu N{k}$ established
in proposition 2.1.3 and theorem 2.1.4 of \cite{DS2}.
\begin{proposition}
\label{p.2.1.3.bis}
\begin{itemize}
\item[i)] Let $\nu\in[0,+\infty)$, $s\in\R$, $s>\nu+3/2$, $N\in\Nn$,
$N>s+1$. Then, any element
$M\in\EM\nu N{k}$ extends as a bounded operator from
${\hcs s}^{k}$ to $\hcs s$.
Moreover, for any $s_0\in(\nu+3/2,s]$, there is $C>0$ such that for any
$u_\ell\in\hcs s$, $\ell\in\{1,\ldots,k\}$
\begin{equation}
\label{2.1.8.bis}
\norma{M(u_1,\ldots,u_{k})}_{H^s}\leq C \norma{M}_{\EM\nu
N{k}}\Big(\sum_{{1\leq \ell\leq
k}}\norma{u_\ell}_{H^s}\prod_{{\ell'\not=\ell}}
\norma{u_{\ell'}}_{H^{s_0}}\Big).
\end{equation}
\item[ii)] Let $k_1,k_2\in\Nn^*$, $\nu_1,\nu_2\in[0,+\infty)$, $1\leq
\ell\leq k_2$. For $M_1\in\EM{\nu_1}N{k_1}$, $M_2\in\EM{\nu_2}N{k_2}$
with $N> 1+\max(\nu_1,\nu_2)$, define a $(k_1+k_2-1)$--linear operator on
$\E^{k_1+k_2-1}$
$$
(u_1,\ldots,u_{k_1+k_2-1})\to M(u_1,\ldots,u_{k_1+k_2-1})
$$
by
\begin{equation}
\label{2.1.9.bis}
\begin{split}
M(u_1,\ldots,u_{k_1+k_2-1})= \makebox[7cm]{}\\
M_2(u_1,\ldots,u_{\ell-1},M_1(u_\ell,\ldots,u_{\ell+k_1-1}),
u_{\ell+k_1},\ldots,u_{k_1+k_2-1})\ .
\end{split}
\end{equation}
Then $M$ belongs to $\EM{\nu_1+\nu_2+1}{N-\max(\nu_1,\nu_2)-1}{k_1+k_2-1}$
and the map $(M_1,M_2)\mapsto M$ is bounded from
$\EM{\nu_1}N{k_1}\times \EM{\nu_2}N{k_2}$ to the preceding space.
\end{itemize}
\end{proposition}
Using the duality formula \eqref{2.1.6}, proposition \ref{p.2.1.3.bis}
immediately implies the corresponding properties for the multilinear forms
of $\EL\nu N{k+1}$.
\begin{proposition}
\label{p.2.1.3}
\begin{itemize}
\item[i)] Let $\nu\in[0,+\infty)$, $s\in\R$, $s>\nu+3/2$, $N\in\Nn$,
$N>s+1$. Then for any $j\in\{1,\ldots,k+1\}$, any multilinear form
$L\in\EL\nu N{k+1}$ extends as a continuous multilinear form
$(u_1,\ldots,u_{j},\ldots,u_{k+1})\mapsto L
(u_1,\ldots,u_{j},\ldots,u_{k+1}) $ on
$$ \hcs s\times \cdots\times \hcs s\times \hcs{-s}\times \hcs s\times
\cdots\times \hcs s.
$$
Moreover for any $s_0\in(\nu+3/2,s]$, there is $C>0$ such that for any
$u_\ell\in\hcs s$, $\ell\in\{1,\ldots,k+1\}-\{j\}$, any $u_j\in\hcs{-s}$
\begin{equation}
\label{2.1.8}
\left|L(u_1,\ldots,u_{k+1})\right|\leq C \norma{L}_{\EL\nu
N{k+1}}\norma{u_j}_{H^{-s}} \Big(\sum_{{1\leq\ell\leq k+1\atop \ell\not=j
}}\norma{u_\ell}_{H^s}\prod_{{\ell'\not=\ell\atop \ell'\not=j}}
\norma{u_{\ell'}}_{H^{s_0}}\Big).
\end{equation}
\item[ii)] Let $k_1,k_2\in\Nn^*$, $\nu_1,\nu_2\in[0,+\infty)$, $1\leq
\ell\leq k_2+1$. For $M\in\EM{\nu_1}N{k_1}$, $L\in\EL{\nu_2}N{k_2+1}$
with $N> 1+\max(\nu_1,\nu_2)$ define a $(k_1+k_2)$--linear form on
$\E^{k_1+k_2}$
$$
(u_1,\ldots,u_{k_1+k_2})\to\tilde L(u_1,\ldots,u_{k_1+k_2})
$$
by
\begin{equation}
\label{2.1.9}
\tilde L(u_1,\ldots,u_{k_1+k_2})=
L(u_1,\ldots,u_{\ell-1},M(u_\ell,\ldots,u_{\ell+k_1-1}),
u_{\ell+k_1},\ldots,u_{k_1+k_2}).
\end{equation}
Then $\tilde L\in\EL{\nu_1+\nu_2+1}{N-\max(\nu_1,\nu_2)-1}{k_1+k_2}$
and the map $(M,L)\mapsto \tilde L$ is bounded from
$\EM{\nu_1}N{k_1}\times \EL{\nu_2}N{k_2+1}$ to the preceding space.
\end{itemize}
\end{proposition}
We shall denote, for any $N,\nu$ by
\begin{equation}
\label{2.1.14}
\Sigma:\EL\nu N{k+1}\to \EM\nu N k
\end{equation}
the map given, using notation (\ref{2.1.6}), by
$\Sigma(L)=M_{L,k+1}$. This is an isomorphism.
In order to apply a Birkhoff procedure, it is necessary to verify that
our framework is stable by Poisson brackets.
\begin{proposition}\label{prop2.1.4}
Let $k_1,k_2\in\Nn^*$, $\nu_1,\nu_2\in[0,+\infty)$,
$N>\frac{5}{2}+\max(\nu_1,\nu_2)$. Let $L_1\in\EL{\nu_1}N{k_1+1}$,
$L_2\in\EL{\nu_2}N{k_2+1}$, $\ell_1\in \{0,\ldots,k_1+1\}$, $\ell_2\in
\{0,\ldots,k_2+1\}$. Then
$\bigl\{\underline{L}\null_1^{\ell_1},\underline{L}\null_2^{\ell_2}
\bigr\}$ may be written
\begin{equation}
\label{2.1.11}
\bigl\{\underline{L}\null_1^{\ell_1},\underline{L}\null_2^{\ell_2}
\bigr\}(u,\bar u) = \underline{L}\null_3^{\ell_1+\ell_2-1}(u,\bar
u)
\end{equation}
for a multilinear form $L_3\in
\EL{\nu_1+\nu_2+1}{N-\max(\nu_1,\nu_2)-1}{k_1+k_2}$.
\end{proposition}
\proof We can choose $s$ with
$N-1>s>\frac{3}{2}+\max(\nu_1,\nu_2)$. By i) of proposition
\ref{p.2.1.3}, $\underline{L}\null_i^{\ell_i}(u,\bar u)$ $i=1,2$ is then a
smooth function on $\hcs s $. Using \eqref{2.1.6} we may write for
any $h\in\E$, $i=1,2$
\begin{equation*}
\partial_u \underline{L}\null_i^{\ell_i}\cdot h
=
\sum_{j=1}^{\ell_i}L_i(u,\ldots,h,\ldots,u,\bar u,\ldots,\bar u)
=\sum_{j=1}^{\ell_i} \bigl\langle h,
\underline{M}_{L_i,j}^{\ell_i-1}(u,\bar u)\bigr\rangle
\end{equation*}
where in the first sum $h$ stands at the $j$-th place. We have a
similar formula for $\partial_{\bar u} \underline{L}\null_i^{\ell_i}. h
$. In other words, we may write
\begin{equation}
\begin{split}
\label{2.1.12}
\nabla_u \underline{L}\null_i^{\ell_i}(u,\bar u)&= \sum_{j=1}^{\ell_i}
\underline{M}_{L_i,j}^{\ell_i-1}(u,\bar u)
\\
\nabla_{\bar u} \underline{L}\null_i^{\ell_i}(u,\bar u)&=
\sum_{j=\ell_i+1}^{k_i+1}
\underline{M}_{L_i,j}^{\ell_i}(u,\bar u).
\end{split}
\end{equation}
By i) of proposition \ref{p.2.1.3.bis} these quantities are smooth
functions of $u$ with values in $\hcs s $,
i.e. $\underline{L}\null_i^{\ell_i}\in C_{s}^\infty(\hcs s,\C)$. We may thus
apply definition \ref{d.1.2.2} and \eqref{1.2.18} to write
\begin{equation}
\begin{split}
\label{2.1.12'}
\bigl\{\underline{L}\null_1^{\ell_1},\underline{L}\null_2^{\ell_2}
\bigr\}(u,\bar u) = \im &\Big[
\sum_{j_2=1}^{\ell_2}\sum_{j_1=\ell_1+1}^{k_1+1}
L_2(u,\ldots,\underline{M}_{L_1,j_1}^{\ell_1}(u,\bar u),\ldots, u,\bar
u,\ldots,\bar u ) \\ &-
\sum_{j_2=\ell_2+1}^{k_2+1}\sum_{j_1=1}^{\ell_1} L_2(u,\ldots,u,\bar
u,\ldots,\underline{M}_{L_1,j_1}^{\ell_1-1}(u,\bar u),\ldots,\bar u )
\Big]
\end{split}
\end{equation}
where the $M$--term in the argument of $L_2$ stays at the $j_2$-th
place. Since $M_{L_1,j_1}$ belongs to $\EM{\nu_1}{N}{k_1}$ we just
have to apply (ii) of proposition \ref{p.2.1.3} to write this last
expression in terms of a new multilinear form $L_3$.
\qed
\medskip
In order to prove our main theorem we have to decompose the
multilinear forms of $\EL\nu N{k+1}$ in the sum of a resonant
and of a non-resonant part.
\begin{definition}
\label{d.2.1.4}
(Non-resonant multilinear form) Fix $k\in \Nn$ and let $1\leq \ell\leq
k+1$ be a fixed integer.
\begin{itemize}
\item If $2\ell\not =k+1$ we set $ \ELt\nu N{k+1}= \EL\nu N{k+1}$,
$\EMt\nu N{k}= \EM\nu N{k}$ .
\item If $2\ell=k+1$ we define $\ELt\nu N{k+1}$ (resp. $\EMt\nu N{k}$)
as the subspace of those $L\in\EL\nu N{k+1} $ (resp. $M\in\EM\nu
N{k}$) such that respectively
\begin{equation}
\label{2.1.13}
L(\Pi_{n_1}u_1,\ldots,\Pi_{n_{k+1}} u_{k+1})\equiv 0\ ,\quad
\Pi_{n_{k+1}}M(\Pi_{n_1}u_1,\ldots,\Pi_{n_{k}} u_{k})\equiv 0
\end{equation}
for any $u_1,\ldots,u_{k+1}\in\E$ and any
$(n_1,\ldots,n_{k+1})\in(\Nn^*)^{k+1}$ such that
$$
\left\{n_1,\ldots,n_{\ell}\right\}=\left\{n_{\ell+1},\ldots,n_{k+1}\right\}.
$$
\end{itemize}
\end{definition}
Remark that the map $\Sigma$ given by (\ref{2.1.14}) induces an isomorphism
between $\ELt\nu N{k+1}$ and $\EMt \nu N{k}$.
\begin{definition}
\label{d.2.1.4.1} (Resonant multilinear form)
Fix $k\in \Nn$ and let $1\leq\ell\leq k+1$. We define the space of
$\ell$--resonant multilinear forms
$\widehat{\mathcal{L}}_{k+1,\ell}^{\nu,N}$ as the subspace of those
$L\in\EL \nu N{k+1}$ verifying
\begin{equation}
\label{2.1.13.1}
L(\Pi_{n_1}u_1,\ldots,\Pi_{n_{k+1}} u_{k+1})\equiv 0\ ,
\end{equation}
for any $u_1,\ldots,u_{k+1}\in\E $ and any
$(n_1,\ldots,n_{k+1})\in(\Nn^*)^{k+1}$ such that
$$ \left\{n_1,\ldots,n_{\ell}\right\}\not=\left\{n_{\ell+1},\ldots,
n_{k+1}\right\}.
$$
\end{definition}
Remark that $\widehat{\mathcal{L}}_{k+1,\ell}^{\nu,N} = 0$ if $k$ is even
or $k$ is odd and $\ell \neq \frac{k+1}{2}$. If $k$ is odd and
$\ell = \frac{k+1}{2}$, one gets a direct sum decomposition
\begin{equation}
\label{decomp}
\EL \nu N{k+1} = \widehat{\mathcal{L}}_{k+1,\ell}^{\nu,N} \oplus
\tilde{\mathcal{L}}_{k+1,\ell}^{\nu,N}.
\end{equation}
The main feature of the above definitions is captured by the following
proposition:
\begin{proposition}
\label{p.jein}
Assume that $L\in\EL\nu N{k+1} $ is $\ell$ resonant. Then for any
$a\in\Nn$, $a\geq 1$ one has
$$
\big\{ \underline L^\ell,J_a \big\}\equiv 0.$$
\end{proposition}
\proof Remark first that one has $2\ell=k+1$ and that $J_a(u,\bar
u)=\langle \Pi_a u,\Pi_a\bar u\rangle$, from which, using
\eqref{2.1.12'}, one gets
$$
\big\{ \underline L^\ell,J_a \big\}=-\im\underline{\tilde L}^\ell
$$
with
\begin{displaymath}
\begin{array}{l}
\displaystyle\tilde L(u_1,\ldots,u_{k+1})=\\
\displaystyle\Big[\sum_{j=1}^{\ell}L(u_1,\ldots,{\Pi_a} u_j
,\ldots,u_{k+1}) -\sum_{j=\ell+1}^{k+1}
L(u_1,\ldots,{\Pi_a} u_j,\ldots,u_{k+1}) \Big].
\end{array}
\end{displaymath}
Then the above expression is equal to
\begin{equation*}
\begin{split}
\sum_{{n_1,\ldots,n_{k+1}}}\Big[ \sum_{j=1}^{\ell}
L(\Pi_{n_1}u_1,\ldots,\Pi_a\Pi_{n_j}u_j,\ldots,\Pi_{n_{k+1}}u_{k+1})
\makebox[2.5cm]{} \\ -\sum_{j=\ell+1}^{2\ell}
L(\Pi_{n_1}u_1,\ldots,\Pi_a\Pi_{n_j}u_j,\ldots,\Pi_{n_{k+1}}u_{k+1})\Big]
\\ =\sum_{n_1,\ldots,n_{k+1}} \Big[\sum_{j=1}^{\ell} \delta_{n_j,a}-
\sum_{j=\ell+1}^{2\ell}\delta_{n_j,a}\Big] L(\Pi_{n_1}u_1,\dots,
\Pi_{n_{2\ell}}u_{2\ell}).
\end{split}
\end{equation*}
Since for an $\ell$ resonant form
$$
\left\{n_1,\ldots,n_{\ell}\right\} =\left\{n_{\ell+1},
\ldots,n_{2\ell}\right\} ,
$$
the quantity $\sum_{j=1}^{\ell}
\delta_{n_j,a}- \sum_{j=\ell+1}^{2\ell}\delta_{n_j,a}$ always vanishes.\qed
\begin{definition}
\label{2.1.d}
For given integers $\ell, k$ satisfying $1\leq \ell\leq k+1$,
we define an operator $\psi_\ell$ acting on $\EL\nu
N{k+1}$ by
\begin{equation}
\begin{split}
\label{2.1.10.a}
\psi_\ell(L)(u_1,\ldots,u_{k+1}) \makebox[9cm]{}
\\
=
\Bigl[\sum_{j=1}^{\ell}L(u_1,\ldots,{\Lambda_m u}_j
,\ldots,u_{k+1}) -\sum_{j=\ell+1}^{k+1}L(u_1,\ldots,{\Lambda_m
u}_j,\ldots,u_{k+1}) \Bigr].
\end{split}
\end{equation}
\end{definition}
Remark that writing $G_{2}(u,\bar u) = \langle \Lambda_{m}u, \bar
u\rangle$, and using \eqref{1.2.18} one gets
\be \label{Gpsi}
\big\{ \underline L^\ell,G_2\big\}(u,\bar u)=
-\im \psi_\ell(L)(u,\ldots,u,\bar u,\ldots,\bar u)
\ee
where in the right hand side one has $\ell$ times $u$ and $k+1-\ell$
times $\bar u$.
\begin{proposition}
\label{prop3.5}
There is a zero measure subset $\N$ of $(0,+\infty)$ such that for any
$k\in\Nn^*$, any $m\in (0,+\infty)-\N$, any $0\leq \ell\leq k+1$, there is
a $\bar \nu\in\R_+$, and for any $(\nu,N)\in\R_+\times \Nn, N>2$, there is
an operator
\begin{equation}
\label{2.1.18}
\psi_\ell^{-1}:\ELt\nu N{k+1}\to \ELt{\nu+\bar\nu} N{k+1}
\end{equation}
such that for any $L\in \ELt\nu N{k+1}$, $\psi_\ell(\psi_\ell^{-1}(L))=L$.
Moreover there exists $C>0$ such that
\begin{equation}
\label{2.1.18.1}
\norma{\psi_\ell^{-1}(L)}_{\EL{\nu+\bar\nu}N{k+1}} \leq C
\norma{L}_{\EL{\nu}N{k+1}}.
\end{equation}
\end{proposition}
\proof We reduce the proof to proposition 2.2.4 of
\cite{DS2}. Let
$\rho:\left\{1,\ldots,k+1\right\}\to\left\{-1,1\right\}$ be the map given
by $\rho(j)=1$ if $j=1,\ldots,\ell$ and $\rho(j)=-1$ if
$j=\ell+1,\ldots,k+1$, and for $M\in\EM \nu N k$ define
\begin{equation}
\label{2.1.18.b}
\begin{array}{l}
\displaystyle\tilde\psi_\ell(M)(u_1,\ldots,u_{k})=\\
\displaystyle\sum_{j=1}^{k}\rho(j)
M(u_1,\ldots,\Lambda_mu_j,\ldots,u_{k})+\rho(k+1)\Lambda_mM
(u_1,\ldots,u_{k}).
\end{array}
\end{equation}
One has, if $\Sigma$ is the map defined in \eqref{2.1.14},
\begin{equation}
\label{2.1.18.a}
\Sigma^{-1}\circ\tilde\psi_\ell(M)=\psi_\ell\circ\Sigma^{-1}(M)
\end{equation}
for any $M \in \mathcal{M}^{\nu,N}_k$ such that $\tilde\psi_\ell(M)$
belongs to $\mathcal{M}^{\nu',N}_k$ for some $\nu'\geq 0$.
By proposition 2.2.4 in \cite{DS2}, there are $\bar\nu\in\R_+$ and an
operator $\tilde \psi_\ell^{-1}:\EMt\nu Nk\to \EMt{\nu+\bar\nu} Nk$
such that for any $M\in \EMt\nu Nk$,
$\tilde\psi_\ell(\tilde\psi_\ell^{-1}(M))=M$ and such that the
equivalent for $M$ of the estimate (\ref{2.1.18.1}) holds true. We
just set $\psi_\ell^{-1}=\Sigma^{-1}\circ \tilde \psi_\ell^{-1}
\circ\Sigma$, and the conclusion follows from equation
(\ref{2.1.18.a}). \qed
\medskip
The construction of the operator $\tilde \psi_\ell^{-1}$ in \cite{DS2}
relies in an essential way on the spectral assumption (\ref{Kn}) and
(\ref{115}), i.e. on the fact that $M$ is a Zoll manifold. For the
reader's convenience, we give a direct proof of
proposition \ref{prop3.5} in the case where $M={\mathbb{S}}^d$ and $V=0$.
In this case, the eigenvalues $\lambda_n$ of $P$ and
$\omega_n$ of $\Lambda_m$ are respectively given by
\begin{equation}
\label{A.3}
\lambda_n=\sqrt{n(n+d-1)}\ ,\quad \omega_n=\sqrt{\lambda_n^2+m^2}\ ,
\end{equation}
and moreover $P\Pi_n=\lambda_n\Pi_n$,
$\Lambda_m\Pi_n=\omega_n\Pi_n$. Thus, from equation (\ref{2.1.10.a}) one
has
\begin{equation}
\begin{split}
\label{A.1}
&\psi_\ell(L)(u_1,\ldots,u_{k+1}) \\ &= \sum_{n_1,\ldots,n_{k+1}}
(\omega_{n_1}+\cdots+\omega_{n_\ell}-\omega_{n_{\ell+1}}-\cdots
-\omega_{n_{k+1}} ) L(\Pi_{n_1}u_1,\ldots,\Pi_{n_{k+1}}u_{k+1}).
\end{split}
\end{equation}
Remark also that, if $L\in\ELt \nu N{k+1}$, then the sum is restricted
to those $(n_1,\ldots,n_{k+1})$ such that
$$ \left\{n_1,\ldots,n_{\ell}\right\}\not=
\left\{n_{\ell+1},\ldots,n_{k+1}\right\}\ .
$$
The following proposition was proved in \cite{DS1} (see Proposition
4.8) and is also a minor variant of theorem 3.12 of \cite{BG}.
\begin{proposition}
\label{A.4}
There is a zero measure subset $\N$ of $(0,+\infty)$ such that for any
$m\in (0,+\infty)-\N$ and any $k\in\Nn^*$, there are $ c>0$ and
$\bar \nu\in\R_+$ such that for any $0\leq \ell\leq k+1$, one has
\begin{equation}
\label{A.2}
\left|\omega_{n_1}+\cdots+\omega_{n_\ell}-\omega_{n_{\ell+1}}-\cdots
-\omega_{n_{k+1}} \right|\geq c\mu (n_1,\ldots,n_{k+1})^{-\bar\nu}
\end{equation}
for any choice of $(n_1,\ldots,n_{k+1})$ such that
$$ \left\{n_1,\ldots,n_{\ell}\right\}\not=
\left\{n_{\ell+1},\ldots,n_{k+1}\right\}\ .
$$
\end{proposition}
It is now immediate to obtain the
\noindent
{\bf Proof of Proposition \ref{prop3.5} in the case $M={\mathbb{S}}^d$,
$V\equiv 0$.} Given $L\in\ELt \nu N{k+1}$ define
\begin{equation}
\label{A.12}
\tilde L(u_1,\ldots,u_{k+1}) = \sum_{n_1,\ldots,n_{k+1}}
\frac{L(\Pi_{n_1}u_1,\ldots,\Pi_{n_{k+1}}u_{k+1})}{
(\omega_{n_1}+\cdots+\omega_{n_\ell}-\omega_{n_{\ell+1}}-\cdots
-\omega_{n_{k+1}} ) }.
\end{equation}
Then by (\ref{A.2}) one has $\tilde L\in\ELt {\nu+\bar\nu} N{k+1}$,
and by (\ref{A.12}) $\psi_\ell(\tilde L)=L$; finally also the estimate
(\ref{2.1.18.1}) immediately follows. On a general Zoll manifold, the
construction of the map $L \to \tilde{L}$ is made in~\cite{DS1} through an
approximation argument and a suitable use of Neumann series. \qed
\medskip
Finally we end this subsection with two lemmas that will be useful to
verify that certain Hamiltonian functions are real valued.
\begin{lemma}
\label{lemma3.13}
Assume $m\in (0,+\infty)-\N$ and let $L\in \ELt\nu N{k+1}$.\\ i)
Assume that for any $u\in\E$, $\psi_\ell(L)(u,\ldots,u,\bar
u,\ldots,\bar u)=0$ (where one has $\ell$ times $u$ and $k+1-\ell$
times $\bar u$). Then $\underline L^\ell(u,\bar u)=0$.\\ ii) Assume
$\bigl\{\Im \underline{L}^\ell,G_2\bigr\} \equiv 0$. Then
$\Im\underline{L}^\ell(u,\bar{u}) \equiv 0$.
\end{lemma}
\proof i) Let $\mathfrak{S}_{\ell,k}$ be the product of the group of
permutations of $\{1,\ldots,\ell\}$ by the group of permutations of
$\{\ell+1,\ldots,k+1\}$. For
$(\sigma,\sigma')\in\mathfrak{S}_{\ell,k}$ define
\[((\sigma,\sigma')\cdot L)(u_1,\ldots,u_{k+1})=L(u_{\sigma(1)},
\ldots,u_{\sigma(\ell)},u_{\sigma'(\ell+1)},
\ldots,u_{\sigma'(k+1)}).\]
Replacing $L$ by
\[\frac{1}{\ell!(k+1-\ell)!} \sum_{a\in\mathfrak{S}_{\ell,k}}(a\cdot L)
(u_1,\ldots,u_{k+1})\]
does no affect the hypotheses nor the conclusion (since $\psi_\ell$
commutes to the $\mathfrak{S}_{\ell,k}$-action), so we can assume that
$L$ -- and thus $\psi_\ell(L)$ -- is
$\mathfrak{S}_{\ell,k}$-invariant. Write the assumption
$\psi_\ell(L)(u,\ldots,u,\bar u,\ldots,\bar u)=0$ with
\[u=u_1+\cdots+u_\ell+\overline{u_{\ell+1}}+\cdots+\overline{u_{k+1}}\]
for arbitrary $u_j$'s belonging
to $\mathcal{E}$. If one expands this expression by multilinearity,
sorts the different contributions according to their homogeneity
degree in $u_j,\bar{u}_j$, and uses the
$\mathfrak{S}_{\ell,k}$-invariance, one gets
\begin{equation}\label{inj1}
\psi_\ell(L)(u_1,\ldots,u_{k+1})=0
\end{equation}
for any $u_1,\ldots,u_{k+1}$ in $\mathcal{E}$. Take a family of
positive integers $(n_1,\ldots,n_{k+1})$ such that
$\{n_1,\ldots,n_\ell\}\neq\{n_{\ell+1},\ldots,n_{k+1}\}$ if
$2\ell=k+1$. We apply \eqref{inj1} taking for all $u_j$ an
eigenfunction associated to an eigenvalue $\lambda_{n_j}\in K_{n_j}$
$j=1,\ldots,k+1$ so that $\Lambda_mu_j=\omega_{n_j}u_j$,
$\omega_{n_j}= \sqrt{m^2+\lambda_{n_j}^2}$.
By \eqref{2.1.10.a} we obtain
\[\Big(\sum_{j=1}^\ell
\omega_{n_j}-
\sum_{\ell+1}^{k+1}\omega_{n_j}\Big)L(u_1,\ldots,
u_{k+1})=0.\] By proposition 2.2.1 and formula (2.2.3) of \cite{DS2}
(see also proposition \ref{A.4} of the present paper in the case of
the sphere), the first factor is nonzero for $m\in (0,+\infty)-\N$, so
$L(u_1,\ldots,u_{k+1})=0$ for any family $(u_1,\ldots,u_{k+1})$ of the
preceding form. The definition of $\ELt\nu N{k+1}$ implies that
$\underline L^\ell(u,\bar u)=0$.
ii) We may write when $\ell \neq \frac{k+1}{2}$
$\Im\underline{L}^\ell(u,\bar{u}) =
\underline{\Gamma}_1^\ell(u,\bar{u}) +
\underline{\Gamma}_2^{k+1-\ell}(u,\bar{u})$ for $\Gamma_1 \in \ELt\nu
N{k+1}$, $\Gamma_2 \in \tilde{\mathcal{L}}^{\nu,N}_{k+1,k+1-\ell}$. By
homogeneity, $\bigl\{G_2,\underline{\Gamma}_1^\ell +
\underline{\Gamma}_2^{k+1-\ell}\bigr\} \equiv 0$ implies that
$\bigl\{G_2,\underline{\Gamma}_1^\ell\bigr\} =$
\mbox{$\bigl\{G_2,\underline{\Gamma}_2^{k+1-\ell}\bigr\} \equiv 0$},
whence $\underline{\Gamma}_1^\ell = \underline{\Gamma}_2^{k+1-\ell}
=0$ by \eqref{Gpsi} and assertion i). If $\ell = \frac{k+1}{2}$, we
have $\Im\underline{L}^\ell(u,\bar{u}) =
\underline{\Gamma}^\ell(u,\bar{u})$ for a $\Gamma \in
\widetilde{\mathcal{L}}^{\nu,N}_{k+1}$, and the result follows again
from i). \qed
\begin{lemma}
\label{lemma3.13bis}
Assume $m\in (0,+\infty)-\N$ and $k$ odd. Set $\ell = \frac{k+1}{2}$
and consider $L_1\in \ELt\nu N{k+1}$ and $L_2 \in
\widehat{\mathcal{L}}_{k+1,\ell}^{\nu,N}$. Set $L = L_1+L_2$ and
assume that for any $u\in\E$, $\underline L^\ell(u,\bar u)$ is real
valued. Then $\underline {L_1}^\ell(u,\bar u)$ and $\underline
{L_2}^\ell(u,\bar u)$ are real valued.
\end{lemma}
\proof Since $\underline L^\ell(u,\bar u)$ is real valued,
$\{\Im\underline L^\ell,G_2\}(u,\bar u)=0$. As \[\{\underline
L^\ell,G_2\}(u,\bar u) =\{\underline {L_1}^\ell,G_2\}(u,\bar u)\] by
proposition \ref{p.jein}, this yields $\{\Im\underline
{L_1}^\ell,G_2\}(u,\bar u)=0$. Now, ii) of lemma \ref{lemma3.13}
implies $\Im\underline {L_1}^\ell(u,\bar u)=0$. Therefore,
$\underline{L_1}^\ell(u,\bar u)$ and $\underline {L_2}^\ell(u,\bar u)$
are real valued. \qed
\subsection{Proof of theorem \ref{thm2}.}\label{birk}
We use a Birkhoff scheme to put the Hamiltonian system with the
Hamiltonian $G$ of \eqref{1.2.12} in normal form. Having fixed some
$r_0\geq 1$, the idea is to construct iteratively for $r =
0,\ldots,r_0$, $\U_r$ a neighborhood of $0$ in $H^s(M,\C)$ for $s\gg
1$, a canonical transformation $\Tr_r$, defined on $\U_r$, an
increasing sequence $(\nu_r)_{r=1,\ldots,r_0}$ of positive numbers,
and functions $\Ze^{(r)}, P^{(r)}, \resto^{(r)}$ such that
\begin{equation}
\label{b.1}
G^{(r)}:=G\circ\Tr_r=G_2+\Ze^{(r)}+P^{(r)}+\resto^{(r)}.
\end{equation}
Moreover, these functions will decompose as
\begin{eqnarray}
\label{b.2}
\Ze^{(r)}&=&\sum_{j=1}^r \Ye_j
\\
\label{b.3}
P^{(r)}&=&\sum_{j=r+1}^{r_0} Q^{(r)}_j
\end{eqnarray}
where $\Ye_j$ is in $\mathcal{H}_{j+2}^s(\nu_j)$ and Poisson commutes
with $J_n$ for any $n$, $Q^{(r)}_j$ is in
$\mathcal{H}_{j+2}^s(\nu_r)$, by convention $P^{(r_{0})}=0$, and
$\resto^{(r)}\in C_s^\infty(\U_r,\R)$ has a zero of order $r_0+3$ at the
origin.
First remark that the Hamiltonian (\ref{1.2.12}) has the form
(\ref{b.1}), (\ref{b.2}), (\ref{b.3}) with $r=0$ and $\Tr_r=I$,
$P^{(0)}$ being the Taylor's polynomial of $\tilde{G}$ at degree
$r_{0}$ (see lemma \ref{lem:G}). We
show now how to pass from $r$ to $r+1$ provided one is able to solve
the homological equation below.
\begin{lemma}
\label{lemma3.14}
Assume we are given $0<\nu_r$ and functions $\Ze^{(r)}, P^{(r)},
\resto^{(r)}$ satisfying the above conditions. Assume that there are
$\nu'_r>\nu_r$ and a function $F^{(r+1)}$ of $(u,\bar u)$
with the properties that
\begin{eqnarray}
\label{b.4}
F^{(r+1)}&\in& \mathcal{H}_{r+3}^s(\nu'_r)
\\
\label{b.5}
\{F^{(r+1)},G_2\}&\in& \mathcal{H}_{r+3}^s(\nu'_r).
\end{eqnarray}
Assume moreover one is able to choose $F^{(r+1)}$ with the further
property that $\Ye_{r+1}$ defined by
\begin{equation}
\label{b.10}
\Ye_{r+1}=\bigl\{F^{(r+1)},G_2\bigr\}+Q^{(r)}_{r+1}
\end{equation}
Poisson commutes with $J_n$ for any $n$. Denote by $\Phi^t_{r+1}$ the
flow generated by $X_{F^{(r+1)}}$. Then, there are $\nu_{r+1}>\nu'_r$
and, for large enough $s$, a sufficiently small neighborhood
$\U_{r+1}$ of the origin of $H^s(M,\C)$, such that $G^{(r+1)}=
G^{(r)}\circ\Phi^1_{r+1}$ has the same structure as $G^{(r)}$ but with
$r$ replaced by $r+1$ and $\U_r$ replaced by $\U_{r+1}$.
\end{lemma}
\proof If
$\U_{r+1}$ is a sufficiently small neighborhood of the origin of
$H^s(M,\C)$, then $\Phi_{r+1}^1:\U_{r+1}\to\U_{r}$ is well defined.
We decompose $G^{(r)}\circ \Phi^1_{r+1}$ as follows
\begin{eqnarray}
\label{b.6}
G^{(r)}\circ \Phi^1_{r+1}&=& G_2+\bigl\{F^{(r+1)},G_2\bigr\}+Q^{(r)}_{r+1}+
\Ze^{(r)} + \resto^{(r)}\circ\Phi^1_{r+1}
\\
\label{b.7}
&+& P^{(r)}\circ\Phi^1_{r+1}-Q^{(r)}_{r+1}
\\
\label{b.8}
&+& \Ze^{(r)}\circ\Phi^1_{r+1}-\Ze^{(r)}
\\
\label{b.9}
&+& G_2\circ\Phi^1_{r+1}-G_2-\bigl\{F^{(r+1)},G_2\bigr\}.
\end{eqnarray}
Using the fact that $\Ye_{r+1}$ Poisson commutes with $J_n$ for any
$n$ and belongs to $\mathcal{H}_{r+3}^s(\nu'_r) \subset
\mathcal{H}_{r+3}^s(\nu_{r+1})$ by \eqref{b.5}, we may define
$\Ze^{(r+1)}:=\Ze^{(r)}+\Ye_{r+1}$.
If $s$ is large enough, \eqref{b.4} implies that $F^{(r+1)} \in
C^\infty_s(\U_r,\R)$, and we may apply lemma \ref{l.1.2.3} with
$F=F^{(r+1)}$ to $P^{(r)}\circ\Phi^1_{r+1}$ and
$\Ze^{(r)}\circ\Phi^1_{r+1}$. Using proposition~\ref{prop2.1.4} to
write the iterated Poisson brackets of the right hand side
of~\eqref{1.2.23} in terms of multilinear forms, we thus see that
\eqref{b.7}, \eqref{b.8} may be decomposed in a sum of elements of
$\mathcal{H}^s_{j+2}(\nu_{r+1})$ for $s, \nu_{r+1}$ large enough and
$j= r+2,\ldots,r_0$. Consequently these two terms will contribute to
$P^{(r+1)}, \resto^{(r+1)}$ in \eqref{b.1} written with $r$ replaced
by $r+1$. In the same way, lemma~\ref{l.1.2.3} applied to
$G_2\circ\Phi_{r+1}^1$ shows that, for large enough $r$ and
$\nu_{r+1}$, \eqref{b.9} gives a contribution to $P^{(r+1)}+
\resto^{(r+1)}$ in \eqref{b.1} at step $r+1$. The conclusion follows.
\qed
\medskip
Let us remark that the above lemma implies theorem 2.6. Actually, if
we are able to apply lemma 3.17 up to step $r_0 -1$, we get (3.31)
with $r=r_0$, which is the conclusion of the theorem. Our remaining
task is thus to solve the homological equation (3.36). This will be
achieved in the following lemma.
\begin{lemma}
\label{hom.true}
Let $2\leq r\leq r_0+2, \nu\in \R_+^*$, and assume $m\in
(0,+\infty)-\N$. For any $Q\in \Hs{r+1}(\nu)$ there are $\nu'>\nu$,
$F\in\Hs{r+1}(\nu')$ and $\Ye\in\Hs{r+1} (\nu)$, with $\Ye$ which
Poisson commutes with $J_n$ for any $n\geq1$, such that
\begin{equation}
\label{hom.1}
\left\{F,G_2\right\}+Q=\Ye\, .
\end{equation}
As a consequence one also has $\{F,G_2\}\in\Hs{r+1}(\nu)$.
\end{lemma}
\proof If $r+1$ is odd then we define $\Ye=0$. As $Q$ is
in $\mathcal{H}_{r+1}^s(\nu)$, it decomposes in the form
\begin{equation}
\label{F.11}
Q=\sum_{\ell=0}^{r+1} \underline{L_{\ell}}\null^\ell
\end{equation}
where $L_{\ell}$ are multilinear forms in $\EL{\nu} {+\infty}{r+1}$.
We remark that, since $r+1$ is odd, the $L_{\ell}$ are all
non-resonant, i.e. $L_{\ell} \in \ELt{\nu} {+\infty}{r+1}$. Therefore
by proposition \ref{prop3.5}, we can define
$F_{\ell}\in \ELt{\nu+\bar\nu} {+\infty}{r+1}$ by
\be\label{F1}
F_{\ell}=-\im \Psi_{\ell}^{-1}(L_{\ell})
\ee
and in view of \eqref{Gpsi}, the Hamiltonian function
\be\label{F2}
F=\sum_{\ell=0}^{r+1} \underline{F_{\ell}}\null^\ell
\ee
satisfies the homological equation \eqref{hom.1}.
If $r+1$ is even, set $\tilde{L}_{\ell} = L_{\ell}
$ if $\ell \neq \frac{r+1}{2}$. When $\ell=\frac{r+1}{2}$, write
\[ L_{\frac{r+1}{2}} = Y +
\tilde{L}_{\frac{r+1}{2}} \in
\widehat{\mathcal{L}}^{\nu,+\infty}_{r+1,\frac{r+1}{2}} \oplus
\widetilde{\mathcal{L}}^{\nu,+\infty}_{r+1,\frac{r+1}{2}}\]
using decomposition \eqref{decomp}. Then if
$\Ye:=\underline{Y}^{\frac{r+1}{2}}$,
\[Q-\Ye=\sum_{\ell=0}^{r+1}
\underline{(\tilde{L}_{\ell})}\null^\ell \]
and if we define $F$ by
\eqref{F2} with $F_{\ell} =
-\im\psi_\ell^{-1}(\tilde{L}_{\ell})$, we still obtain
that equation \eqref{hom.1} is satisfied.
It remains to show that $F$ is real valued.
As $Q$ is real, using \eqref{F.11} yields for any
$\ell\in\{0,\ldots,r+1\}$
\begin{equation}
\label{F.13}
\overline{\underline{L_{\ell}}\null^\ell}
=\underline{L_{r+1-\ell}}\null^{r+1-\ell}
\end{equation}
by homogeneity. If $r+1$ is even, \eqref{F.13} implies that
$\underline{L_{\frac{r+1}{2}}}\null^\frac{r+1}{2}$ is real
valued. Using lemma \ref{lemma3.13bis}, we obtain that $\Ye$ is real
valued (remark that if $r+1$ is odd, $\Ye=0$ is also real
valued). Therefore, $\left\{F,G_2\right\}$ is real valued by
\eqref{hom.1}. So $\left\{\Im F,G_2\right\}=0$ which implies by
homogeneity that $\bigl\{\Im
\underline{F_{\ell}}\null^\ell,G_2\bigr\}=0$ for any $\ell$. We may now use
lemma
\ref{lemma3.13} to obtain that $\Im
\underline{F_{\ell}}\null^\ell=0$ for any
$\ell\in\{0,\ldots,r+3\}$. Therefore, $F$ is real valued.\qed
\subsection{Proof of theorem \ref{thm1}.}\label{sec:proofthm1}
Let $\Tr$ be the canonical transformation defined in theorem \ref{thm2}.
Define on $\U=\Tr(\V)$ the function
$$
E(u,\bar u):=\sum_{n\geq 1}n^{2s}J_n\circ\Tr^{-1}(u,\bar u).
$$ We shall control $E(u,\bar{u})$ along long time intervals. To take
into account the loss of derivatives coming from the linear part of
the equation, we proceed by regularization. Fix $\sigma=s+1$ and take
the Cauchy data such that $u_0=\epsilon (\Lambda_m^{-1/2}v_{1} + \im
\Lambda_m^{1/2}v_{0})/\sqrt{2}$ is in $H^\sigma(M,\C)\cap\U$. Let
$u(t)\equiv u(t,.)$ be the corresponding solution of $\dot
u=X_G(u)\equiv \im \nabla_{\bar{u}} G(u,\bar{u})$. Since $X_G$ is semilinear and
$H^\sigma$ is its domain, as far as $\norma{u(t)}_{H^s}<\infty$ one
has $u(t)\in H^\sigma$. Thus, as far as $u(t)\in\U$
\begin{equation}\label{F.17}
\frac{dE}{dt}=\partial E\cdot X_G=\left\{G,E\right\}
\end{equation}
which is well defined since $E\in C^\infty(\U,\R)$, with $\U\subset
H^s$ and $X_G(u)\in H^s$ for $u\in H^\sigma$. So we may write
\[\left\{G,E\right\}(u,\bar u) =\displaystyle\sum_{n\geq
1}n^{2s}\left\{G,J_n\circ\Tr^{-1}\right\}.\]
If we use \eqref{1.2.21}, \eqref{for.nor} and \eqref{stime1} we get then
\begin{equation}\label{F.18}
\begin{array}{ll}
\left\{G,E\right\} &
=\displaystyle\sum_{n\geq
1}n^{2s}\left\{G\circ\Tr,J_n\right\}\circ\Tr^{-1}\\ &
=\displaystyle\sum_{n\geq
1}n^{2s}\left\{G_2+\Ze+\resto,J_n\right\}\circ\Tr^{-1}\\ &
=\left\{\resto\circ\Tr^{-1},E\right\}.
\end{array}
\end{equation}
Thus
\[\frac{dE}{dt}=\left\{\resto\circ\Tr^{-1},E\right\}\]
which, by taking an approximating sequence is seen to hold also for
initial data which are not in $H^\sigma$, but only in $\U$.
Using \eqref{stime2} one has
\begin{equation}\label{F.14}
\left|\frac{dE(t)}{dt}\right|\leq C\Vert{u(t,\cdot )}\Vert_{H^s}^{r+3}.
\end{equation}
Remark that by definition of $E(u,\bar{u})$ and because $\Tr(0) = 0$,
as long as $u$ stays in a small enough neighborhood of 0, we have
\begin{equation}
\label{la.e}
\frac{1}{2}E(u,\bar u)\leq \norma{u}^2_{H^s}\leq 2E(u,\bar u).
\end{equation}
We deduce then by integration of \eqref{F.14} the estimate
\begin{equation}\label{F.16}
\Vert{u(t,\cdot )}\Vert_{H^s}^2 \leq C'\left(\Vert{u_0}\Vert_{H^s}^2
+\left|\int_0^t\Vert{u(\tau,\cdot )}\Vert_{H^s}^{r+3}d\tau\right|\right)
\end{equation}
which holds true as long as $u$ remains in a small enough neighborhood
of 0. It is classical to deduce from this inequality that there are
$C>0, c>0, \epsilon_0>0$ such that, if the
Cauchy data $u_0$ is in the $H^s$ ball of center 0 and radius $\epsilon
<\epsilon_0$, the solution exists over an interval of length at least
$c\epsilon^{-r-1}$, and for any $t$ in that interval $\Vert
u(t,\cdot )\Vert_{H^s} \leq C\epsilon$. This concludes the proof.
\qed
\begin{remark}
The proof of \eqref{estimJn} is similar. As in \eqref{F.17} and
\eqref{F.18}, we see that
\[\frac{dJ_n\circ \Tr^{-1}(u,\bar u)}{dt}=\left\{\resto\circ\Tr^{-1},
J_n\right\}(u,\bar u)\]
which together with the bound $\Vert{u(t,\cdot
)}\Vert_{H^s}\leq C_1\epsilon$ yields
\begin{equation}\label{F.19}
|J_n\circ\Tr^{-1}(u(t),\bar{u}(t))-J_n\circ\Tr^{-1}(u_0,\bar{u}_0)|\leq
\frac{C\epsilon^3}{n^{2s}}
\end{equation}
for times $|t|\leq\epsilon^{-r}$. Finally, using \eqref{F.19},
\eqref{stime} and the inequality
\begin{displaymath}
\begin{array}{l}
|J_n(u(t),\bar{u}(t))-J_n(u_0,\bar{u}_0)|\leq
|J_n(u(t),\bar{u}(t))-J_n\circ\Tr^{-1}(u(t),\bar{u}(t))|\\
\hspace{0.4cm}+|J_n\circ\Tr^{-1}(u(t),\bar{u}(t))-J_n \circ\Tr^{-1}(u_0,
\bar{u}_0)|+|J_n(u_0,\bar{u}_0)-J_n\circ\Tr^{-1}(u_0,\bar{u}_0)|
\end{array}
\end{displaymath}
implies \eqref{estimJn}.
\end{remark}
|
1911.04386
|
\section{Introduction}\label{sec:intro}
In industrial manufacturing processes, a {\em fault} is defined as any abnormal deviation from the normal operating conditions (NOC). Faults are a concern because even small faults in a complex industrial system can initiate a series of events that result in loss of efficiency and reliability. As a result, there is a need for techniques to improve the process's reliability and up-time. Effective fault detection and identification (FDI) is important for monitoring components for making appropriate maintenance decisions. First, fault detection determines whether a fault has occurred in the system (also characterized as anomaly detection in other applications). Then fault identification determines which observation variables are most relevant to diagnosing the fault detected, thereby helping operators to focus on specific subsystems. Systems that can accurately and promptly detect and identify faults can more effectively inform operators and engineers and significantly reduce the effort and time to recover the system.
A number of FDI methods have been proposed in the literature. Since analytical and knowledge-based methods are impractical in most large-scale modern industrial processes, data-driven methods have dominated the literature for the past decade and have been effective in practice, taking advantage of increasing levels of instrumentation and widespread availability of sensor data \citep{Qin2009,Ge2013,Yin2014,Chiang2000}. The choice of model used to characterize the NOC and deviations thereof is still a crucial aspect in these methods because the limitations of the model lead to decreased detection rates or increased occurrence of fault alarms.
A number of data-driven methods including principal component analysis (PCA) \citep{Jolliffe2011}, partial least squares \citep{Kourti1996}, Fisher discriminant analysis \citep{Fisher1936,Chiang2000}, and support vector machines \citep{Chiang2004},
have been applied for fault detection and identification in industry with varying degrees of success. The most widely used method is PCA which models the correlations between the process variables. PCA can detect faults effectively when the sensor measurements are highly correlated, which is often the case. For many processes, the temporal dynamics also need to be taken into consideration, especially when fast sampling rates are used, because the dynamics provide additional information through which to detect deviations from the NOC. To that end, DPCA has been proposed to handle serially correlated multivariate observations \citep{Ku1995}. DPCA can be viewed as a multivariate autoregressive model with exogenous inputs (ARX). PCA and DPCA are both limited by the linear model structure and correlations in the process' dynamics. Methods for extending PCA to nonlinearities such as kernel PCA and neural network-based PCA \citep{Lee20041,Hsieh2007} have been well studied only for static systems. As such, the development of approaches that can effectively model nonlinear system structure and dynamics has been an active research field.
Neural network (NN) based methods have also received significant attention because of their capability and flexibility for modeling complex structure and temporal dynamics. NN models have been used for fault detection in three general frameworks: (1)~as a fault classification tool between normal and known faulty conditions \citep{Zarei2014,Chine2016,Ince2016,Jia2016,Wu2018,Hu2018,Lee2017,Li2014,Zhang2017}, (2)~as a model of the input-output variable relationships during NOC \citep{Malhotra2015,Patan2008,Moustapha2008,Talebi2008,Wang2017,Nie2018}, or (3) as a generalization of the basic fault detection methods such as NN-based PCA \citep{Kramer1992,Dong1996} using statistical indices to monitor the process. The first two approaches are dominant for NN-based fault detection. The first approach can be highly effective due to the up-front knowledge of specific fault conditions to detect. It can also be set up to classify each fault which directly enables fault diagnosis. On the other hand, training these NNs requires substantial amounts of data under fault conditions but these data are usually quite limited for chemical manufacturing processes compared to NOC data. Moreover, it is hard to assess the performance of these methods for fault conditions other than for which the classifier is explicitly trained for.
In the second approach, NNs are used to model the process by capturing the nonlinear, multivariate, and temporal dependencies from inputs to outputs. In this approach, the NN models are typically trained on NOC data to predict the system outputs. This NN is then used for fault detection during runtime by comparing the predictions of the NN with the actual system output measurements, and a fault is detected if the difference is significantly large. This approach has the advantage that the NNs are trained using only NOC data, which is usually abundant, and that the NNs are not constrained by the type of fault because detection is marked from any significant deviation from the NOC. On the other hand, the model must accurately characterize these complex and potentially nonlinear structures between inputs and outputs in the process, or its fault detection will perform poorly as a result. Moreover, only faults that break the input-output relationships are considered, meaning that faults due to input disturbances will likely not be detected.
In the third approach, NNs are used to account for nonlinearities in the process but, like other PCA-based methods, fail to appropriately model the process dynamics.
There are other challenges regarding both approaches, which have limited the application of NNs in industrial process monitoring. First, fault identification has not yet been properly addressed. Once a fault is detected, it is typically difficult to identify the input variables most relevant to the fault from a complex NN model. Even if an NN is trained to directly classify the fault, the underlying cause may still be unclear if there are multiple explanations for the observed fault type. Secondly, standard NNs are deterministic models which lack an estimate of the uncertainty in the model outputs. However, uncertainty and probabilistic estimates are important to assess the confidence level associated with the decision of detecting a fault and for fault identification. Lastly, NNs are prone to overfitting, meaning that they `memorize' the particular characteristics of the training data that are not relevant for new data. This overfitting must be addressed to ensure good generalization to the full space of operating conditions of a complex industrial process.
For fault identification, contribution plots \citep{Miller1998} are one of the most popular techniques for providing information on the variables that are most strongly related to the faults. In the context of PCA-based methods, contribution plots are obtained by quantifying the contribution of each process variable to the individual scores of the PCA representation \citep{Westerhuis2000}. Methods based on the contribution of each process variable in the residual space have also been developed \citep{Wise1989}. However, the aforementioned limitations of PCA-based approaches will also be reflected in the identification procedure and those methods require extra processing steps after fault detection. Moreover, those methods only provide the relative contribution value of each variable which is not very useful. A more valuable and precise measure to aid operators in diagnosis would be the probable severity of each affected variable. On the other hand, it has been hard to extend contributions plots to NNs due to the complex and nonlinear relationships between predictions and model inputs.
This article proposes a novel end-to-end FDI framework, which adopts a recently developed Bayesian recurrent neural network (BRNN) architecture \citep{Gal2016theoretically}. The proposed FDI framework is fundamentally different from the two types of frameworks that have been previously used in the NN-based fault detection literature. The proposed framework uses BRNNs to model the joint distribution and dynamics between all process variables. This framework provides estimates of the prediction uncertainty, which capture both model uncertainty and the inherent noise in the data. The BRNN is realized using the variational dropout approach proposed in \citep{Gal2016dropout,Gal2016theoretically} due to its simplicity, regularization capability, strong generalization ability, and scalability.
To the best of our knowledge, this work is the first time that Bayesian spatio-temporal models, and BRNNs in particular, have been successfully applied to FDI in the chemical manufacturing industry. The proposed approach tackles three key challenges typical of manufacturing systems: (1)~nonlinearity, (2)~non-Gaussian distributed variables, and (3)~high degree of spatio-temporal correlations (i.e., temporal and sensor correlations). Furthermore, the probabilistic framework provided by BRNNs enables more sensitive and robust FDI. Fault identification through the proposed BRNN-based approach provides easily interpretable visualizations to the plant operators, for quick fault type categorization, analysis of the possible fault propagation path, and root cause determination using engineering judgment.
The remainder of this paper is organized as follows. Section~\ref{section:BRNN2} provides a brief introduction to RNNs and BRNNs, and describes the variational dropout approach used in this paper for inference in BRNNs. Section~\ref{sec:methodology-for-fault-detection-and-identification} presents the proposed BRNN-based FDI methodology. In Section~\ref{sec:case-studies}, the effectiveness of the proposed approach is demonstrated and compared to (D)PCA-based methods in the Tennessee Eastman process and a real chemical manufacturing process, followed by the conclusion in Section~\ref{sec:conclusion}.
\section{Background}\label{section:BRNN2}
\subsection{Recurrent Neural Networks}\label{sec:rnns}
RNNs were developed in the 1980s~\citep{Rumelhart1986}. Since then, RNNs have been shown to achieve state-of-the-art performance on a wide range of sequential data modeling tasks, including language modeling, speech recognition, image captioning, and music composition \citep{Wu2016,Jozefowicz2016,Merity2016}. Generally speaking, an RNN comprises an input layer, one or more hidden recurrent layers, and an output layer. The input layer corresponds directly to the input data, and hidden recurrent layers capture the state with the response of its nodes being added to the inputs on the next time step. At each time $t$, denote the input to the network as $\bm{x}_t\in\ensuremath{\mathbb{R}}^{m_x}$, the state (i.e., output of the hidden layer) as $\bm{s}_t\in\ensuremath{\mathbb{R}}^{m_s}$, and the RNN output as $\hat{\bm{y}}_t\in\ensuremath{\mathbb{R}}^{m_y}$. They are represented as row vectors in the equations.
Accordingly, the state and output layers have the general form
\begin{equation}\label{eq:rnn}
\begin{split}
\bm{s}_t &= f_s\!\left(\bm{x}_t, \bm{s}_{t-1} \right | \theta_s) \\
\hat{\bm{y}}_t &= \bm{W}_y\bm{s}_t + \bm{b}_y \\
\end{split}
\end{equation}
where the subscript $s=1,\dots,m_s$ is the index over hidden layer nodes, $\theta_s$ and $f_s$ denote the corresponding hidden layer parameter/weights and nonlinear operator for each node, and $\bm{W}_y\in\ensuremath{\mathbb{R}}^{m_y \times m_s}$ and $\bm{b}_y\in\ensuremath{\mathbb{R}}^{m_y}$ are the output layer parameters. The new state of the network depends on its value at the previous time step, emblematic of recurrent architectures. This dependency, and the unfolding through time, is depicted in Figure~\ref{arm:BRNN1}. A linear output layer is commonly used for regression tasks.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{010_rnn_unfolding}
\caption{A simple RNN structure with one recurrent layer and showing the unfolding in time of the sequence of its forward computation. The RNN includes the input variable $\bm{x}_t$, state variable $\bm{s}_t$ and outputs $\hat{\bm{y}}_t$. The state variable $\bm{s}_t$ is calculated based on the previous state $\bm{s}_{t-1}$ and the current input $\bm{x}_t$. The RNN output $\hat{\bm{y}}_t$ is then calculated based on the current state. In this way, the input sequence $\bm{x}_t$ is mapped to output sequence $\hat{\bm{y}}_t$ with each $\hat{\bm{y}}_t$ depending on all previous inputs. The model parameters $\omega = \{\bm{W}_s, \bm{U}_s, \bm{W}_s, \bm{b}_s, \bm{b}_y\}$ are shared at each time step.}
\label{arm:BRNN1}
\end{figure}
In the simpler form of nodes, the state is computed as \citep{Elman1990}
\begin{equation}\label{eq:rnn:state.vanilla}
\bm{s}_t = \phi\!\left(\bm{W}_s\bm{x}_t + \bm{U}_s\bm{s}_{t-1} + \bm{b}_s \right)
\end{equation}
where $\bm{W}_s\in\ensuremath{\mathbb{R}}^{m_s \times m_x}$, $\bm{U}_s\in\ensuremath{\mathbb{R}}^{m_s \times m_s}$, and $\bm{b}_s\in\ensuremath{\mathbb{R}}^{m_s}$ are the hidden layer parameters (denoted $\theta$ above), and $\phi$ is an element-wise activation function such as the logistic, hyperbolic tangent, or a rectifier linear function.
As can be explicitly observed from the mathematical formulation in Equation~\ref{eq:rnn}, RNNs are essentially state-space models capable of modeling nonlinear dependencies. RNNs can capture complex nonlinear dynamics of a system in the state. Also, by appropriately training the parameters, RNNs can adapt to the right level of temporal depth. Thus, RNN models are considerably more powerful for modeling complex industrial processes in comparison to traditional statistical methods.
It is worth noting that different RNN architectures have been proposed \citep{Jordan1997}, with the formulation in Equation~\ref{eq:rnn} corresponding to Elman's architecture \citep{Elman1990}, which has been widely used in the recent deep learning RNN implementations and applications.
In order to learn the parameters of the RNN, an optimization problem is defined with regard to an appropriate loss function. For regression tasks, the loss function is typically chosen to be the mean squared loss,
\begin{equation}
J(\Theta) = \frac{1}{N}\sum_{t = 1}^{N}\Vert\bm{y}_t - \hat{\bm{y}}_t\Vert^2_2,
\end{equation}
or the cross-entropy loss for classification purposes,
\begin{equation}
J(\Theta) = - \sum_{t=1}^{N}\bm{y}_t\log \hat{\bm{y}}_t
\end{equation}
where $\Theta$ denotes the collection of all RNN model parameters, and $\bm{y}_t$ is the desired output at time step $t$. In addition, $L_2$ regularization terms are often added to help prevent overfitting, resulting in the overall minimization objective
\begin{equation}
L(\Theta) = J(\Theta) + \lambda\left( \Vert{\bm{W}_s}\Vert^2 + \Vert\bm{W}_y\Vert^2 + \Vert{\bm{U}_s}\Vert^2 \right)
\end{equation}
where $\lambda$ is the regularization (aka weight decay) parameter.
Because the recurrence introduces dependencies between time steps, training RNNs involves backpropagation through time~(BPTT) to compute the gradient update of the model parameters that minimizes the loss function~\citep{Werbos1974,Werbos1990}. BPTT corresponds to an unfolding of the network over a number of time steps, as depicted in Figure~\ref{arm:BRNN1}. For BPTT, the difference between network outputs and target values is first calculated and stored for each time step in a forward pass, and then the weight gradient updates are calculated as the network is ``rolled back''. However, simple RNNs trained with BPTT can have difficulties learning long-range time dependencies due to the vanishing gradient problem \citep{Bengio1994}. To alleviate the vanishing gradient problem, recurrent node gating mechanisms have been recently developed. These gating mechanisms allow information and the gradients to flow through the unrolled network with minimal attenuation if determined to be necessary by BPTT. These gating mechanisms resulted in two popular variations on RNN hidden units: LSTM units \citep{Hochreiter1997} and GRUs \citep{Cho2014}. RNNs with LSTM and GRU units have been reported to show salient performance \citep{Graves2013,Cakir2015}.
\subsection{BRNNs}\label{sec:brnns}
BRNNs combine statistical modeling of RNN parameters to obtain a probabilistic model of input-output mapping. As such, instead of point estimates, BRNNs can effectively perform Bayesian inference which provides probabilistic distributions over the outputs.
To realize that capability, BRNNs view the model parameters $\omega = \{\bm{W}_s,\bm{W}_y,\bm{U}_s,\bm{b}_s,\bm{b}_y \}$ as random variables from a prior distribution $p(\omega)$. Expressing the functional dependence in Equation~\ref{eq:rnn} as $\bm{s}_t = \bm{f}_s^{\omega}(\bm{x}_t,\bm{s}_{t - 1})$ and $\hat{\bm{y}}_t = \bm{f}_y^{\omega}\left( \bm{s}_t \right)$, the likelihood of the output for each data point is
\begin{equation}
p(\bm{y}_t | \omega, \bm{x}_t, \bm{s}_t, \tau)
= N\!\left(\bm{y}_t | \bm{f}_{y}^{\omega}\left(\bm{f}_{s}^{\omega}\left( \bm{x}_t, \bm{s}_{t-1} \right),
\tau^{-1}\bm{I}_{D} \right) \right)
\label{eq:training.posterior}
\end{equation}
where $\tau$ is the precision parameter that reflects the intrinsic noise in the data, and the likelihood function is assumed to have a normal distribution for simplicity. Note how the likelihood function is evaluated with respect to forward passes through the NN.
Then, given a training dataset comprising $\bm{X}$ and $\bm{Y}$, learning entails estimating the posterior distribution $p(\omega|\bm{X},\bm{Y})$ over the space of parameters. With the updated distribution, the distribution of a predicted output $\bm{y}^*$ can be obtained by integration
\begin{equation}
p(\bm{y}^* | \bm{x}^*, \bm{X}, \bm{Y})
= \int p(\bm{y}^* | \bm{x}^*, \omega) p(\omega|\bm{X},\bm{Y}) d\omega
\label{eq:output.posterior}
\end{equation}
where the dependency on the precision parameter, state, and past inputs are not shown to simplify the expression.
For the prior distribution, standard zero-mean Gaussian priors over the weight matrices $p(\bm{W})$ and $p(\bm{U})$ are typically chosen, with point estimates for the bias vectors assumed for simplicity.
The uncertainty in the prediction will be directly reflected in the posterior distribution $p(\bm{y}^* | \bm{x}^*, \bm{X}, \bm{Y})$.
In complex models such as NNs, however, the exact inference of the posterior is not possible. Moreover, traditional algorithms for approximating the Bayesian inference are generally not applicable to train RNNs having a large number of parameters or complex architectures.
To overcome this limitation, several approximation inference methods have been proposed, including variational dropout \citep{Gal2016theoretically,Gal2016dropout}, Bayes by BackProp \citep{Pawlowski2017, Fortunato2017}, multiplicative normalizing flows \citep{Louizos2017}, and probabilistic backpropagation \citep{Hernandez2015}. Among all those techniques, the variational dropout technique proposed by \citet{Gal2016theoretically} is adopted in this paper because of its simplicity and generalization capability.
In particular, variational dropout can be applied to any RNN architecture without modification on the underlying NN structure, and only concurrent runs of the trained model are needed for online application. Details of this algorithm are reviewed in the next section.
\subsection{Variational Dropout as Bayesian Approximation}\label{sec:variational-dropout-as-bayesian-approximation}
\citet{Gal2016dropout} showed how dropout could be used as a general variational approximation to the posterior of Bayesian neural networks (BNNs), which can be applied directly to a variety of NN architectures. The main advantage of this `variational dropout' approach is that it does not require significant modifications to the model architecture and training method, unlike other probabilistic approximation methods. Moreover, the uncertainty estimation incurs only the computation cost due to multiple stochastic forward passes through the network to generate samples of the posterior distribution.
Therefore, variational dropout is used here as a variational inference approach for BNNs. Variational inference is a technique used to approximate an intractable posterior distribution $p(\omega|\bm{X}, \bm{Y})$ with a simpler parameterized distribution $q(\omega)$. Then, the integration in Equation \ref{eq:output.posterior} can be approximated simply by MC integration using $q(\omega)$. Specifically, the approximation distribution is factorized over the weight matrices in $\omega$. For each row $\bm{w}_k$, variational dropout imposes a variation distribution comprising a mixture of two Gaussian distributions with small variances,
\begin{equation}
q(\bm{w}_k)
= p N(\bm{w}_k | 0, \sigma^2\bm{I})
+ (1-p)N(\bm{w}_k | \bm{m}_k,\sigma^2\bm{I}),
\end{equation}
where $p$ is the predefined dropout probability, $\sigma^2$ is a small precision parameter, and $\bm{m}_k$ is a variational parameter. The learning problem is then casted into an optimization problem by minimizing the KLD between $q(\omega)$ and $p(\omega | \bm{X}, \bm{Y})$. It can be shown that optimizing the loss function using dropout is equivalent to minimizing KL$( q(\omega) \| p(\omega | \bm{X}, \bm{Y})))$ \citep{Gal2016dropout}, which updates the variational parameter. Although variational inference is a biased approximation, it has been shown to work well in practice.
\begin{figure}
\centering
\includegraphics[width=11cm]{020_rnn_dropout}
\caption{Illustration of the variational dropout technique (right) compared to standard dropout technique (left) for a simple RNN. Each graph shows units unfolded over time, with the lower level for inputs, middle level for state units, and upper level for output units. Vertical arrows represent the connections from inputs to outputs while horizontal arrows represent recurrent connections. The arrows with dashed grey lines represent the standard connection without dropout. Colored lines represent dropout connections with different colors for different dropout masks. (Left) In the standard dropout technique, no dropout is applied for the recurrent layers, while other connections have different dropout masks at different time steps. (Right) For the variational dropout approach proposed in \citep{Gal2016theoretically}, dropout is applied to both input, recurrent, and output layers with the same dropout mask at different time steps. Variational dropout is applied during both training and testing.}
\label{fig:variationalRNN}
\end{figure}
Variational dropout requires caution when applied in the context of RNNs, however. Because of the recurrence, na\"ively applying standard dropout \citep{Srivastava2014} with different masks at each time step of an RNN can lead to model instabilities and disrupt an RNN's capability to model a sequence \citep{Pham2014,Pachitariu2013}. We use the approach in \citep{Gal2016theoretically} to resolve these issues. Under these circumstances, variational dropout has been shown to also act as an effective regularization method for reducing overfitting by preventing co-adaptions in RNNs \citep{Gal2016theoretically}.
The implementation of BRNNs with variational dropout is relatively straightforward. During both training and testing, the variational approximation involves sampling the model distribution with regard to the variational distribution over the weights, which is implemented by dropping out (i.e., forcing to zero) randomly chosen inputs, outputs, and hidden states. This step results in multiple random realizations of the RNN model, each obtained by implicitly removing a portion of the inputs, outputs, or hidden states. However, as detailed in \citep{Gal2016theoretically}, it is crucial for RNNs that the dropout mask used for each model realization be kept fixed between time steps. In other words, the dropout mask of which elements are zeroed out is sampled and frozen for each time sequence sample. This sampling characteristic is contrasted to standard dropout in Figure \ref{fig:variationalRNN}.
Variational dropout applied during testing can be viewed as an approximation to MC samples from the posterior predictive distribution, $p\left( \omega | \bm{X},\bm{Y} \right)$. Given a new observation $\bm{x}^*$, by forward passing it $N$ times, $N$ samples $\left\{ {\hat{\bm{y}}}^*(i) \right\}_{i = 1,\dots,N}$ are collected of the approximate predictive posterior. The corresponding empirical estimators for the posterior predictive mean, standard deviation, and covariance are
\begin{gather}
E(\hat{\bm{y}}^*) \approx \frac{1}{N}\sum_{i = 1}^{N}\hat{\bm{y}}^*(i)
\\
\text{std}(\hat{y}^*) \approx \sqrt{\tau^{-1} + \frac{1}{N}\sum_{i=1}^{N} \left(\hat{y}^*(i)\right)^2 - E(\hat{y}^*)^2}
\\
\text{cov}( \hat{\bm{y}}^*) \approx \tau^{-1}\bm{I}_{D} + \frac{1}{N}\sum_{i=1}^{N}{{\hat{\bm{y}}^*(i)}^\top \hat{\bm{y}}^*(i)} - E(\hat{\bm{y}}^*)^\top E(\hat{\bm{y}}^*)
\end{gather}
where $\tau$ can be estimated as $\tau = \frac{pl^2}{2N\lambda}$ given a predefined regularization/weight-decay parameter $\lambda$, and prior length scale $l$ \citep{Gal2016dropout}. Higher order statistics can also be estimated using the samples by moment-matching.
Since the forward passes involve simply a number of independent and fixed realizations of the RNN model distribution, they can be done concurrently, thus making variational dropout a good candidate for online monitoring. In the next section, the proposed novel FDI scheme is explained in detail. While this methodology is described here in the context of chemical process monitoring, it can be observed that it could be readily extended to other manufacturing industries.
\section{Methodology for FDI}\label{sec:methodology-for-fault-detection-and-identification}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{030_fdi_framework}
\caption{General procedure for process monitoring system development (left) versus procedure for developing BRNN-based FDI system (right). The general framework to establish a monitoring system begins with a model to characterize NOC behavior, such as using the BRNN model to learn the NOC pattern from the training data. Then, the method to measure the deviation of a particular observation to the NOC region is chosen. In our case, the process observations are compared to the BRNN posterior predictive distributions. Finally, the decision will involve determining whether the acquired observation is from the NOC or not (i.e., compare deviations of observations for fault detection and assess which variables significantly deviate from the NOC for identification).}
\label{fig:detection.process}
\end{figure}
The design of a FDI system generally begins with the development of a model to characterize the normal operating characteristics of a process. Historical data collected during the NOC are used to build the model, which means this learning problem is unsupervised. Then, an approach must be established to characterize the magnitude of the deviation from the NOC based on the developed model and to determine when deviations are considered to be outside of the NOC. For example, the $T^2$ and $Q$ statistics are commonly used to measure the distance of the observation to the NOC region in PCA-based models and thresholds thereon \citep{Chiang2000}. Finally, given a new observation $\bm{x}^*$, these measures are calculated to determine whether $\bm{x}^*$ deviates substantially from the NOC (fault detection) and, if that is the case, which variables are significantly affected (fault identification), thereby assisting in locating and troubleshooting the fault.
Specifically, this paper proposes using a BRNN with variational dropout to build the probabilistic model, denoted as $f^{\omega}(\cdot)$, and characterize the NOC and its intrinsic variability. As discussed earlier, BRNNs are capable of extracting the nonlinear spatial and temporal signatures in the data that are critical for characterizing complex chemical processes. Moreover, BRNNs provide probabilistic estimates of the likelihood of the observations with regard to its inferred posterior distribution of the variable values. These likelihood estimates lend themselves to be used to assess the current deviation level from the NOC region. Accordingly, observations are detected as faults whenever their deviation is above a threshold, determined such that the number of false alarms under the NOC does not exceed a predefined level. Fault identification then involves determining which process attributes are deviating significantly. This general framework for BRNN-based FDI is summarized in Figure \ref{fig:detection.process} and described in detail in the next sections.
\subsection{Fault Detection}\label{fault-detection}
The first step toward fault detection is to learn a model to characterizing the NOC. This step involves training a BRNN with variational dropout to model the dynamics in time, correlations between sensors, and the prediction uncertainty resulting from model mismatch and inherent system variability/noise.
\begin{figure}
\centering
\includegraphics[width=0.44\textwidth]{040_brnn_MCdropout}
\caption{Depiction of BRNN model using variational dropout (left) for
FDI. The BRNN model uses the current
observation and state to predict the next system observation. The BRNN
model is unrolled in two dimensions (right): the time of the computation
involved in its forward computation and the stochastic repetition by
variational dropout. At each time step, stochastic variational dropout
is applied $N$ times and the corresponding MC prediction samples
$\left\{ \hat{\bm{x}}_t(i) \right\}_{i = 1,\dots,N}$
are used to approximate the posterior predictive distribution for
that time step. For the next time step, the same procedure is repeated
and MC samples
$\left\{ \hat{\bm{x}}_{t + 1}(i) \right\}_{i = 1,\dots,N}$
are collected and used to approximate the distribution.}
\label{fig:brnn.MCdropout}
\end{figure}
The BRNN model is trained directly on historical NOC data. Specifically, this step involves setting a training problem wherein the BRNN model uses the past context (as captured by its state) and current observation to predict the next observation, as depicted in Figure \ref{fig:brnn.MCdropout}. During training, BPTT is applied to batches of time subsequences with one variational dropout mask sampled per sequence, as explained in Section \ref{sec:variational-dropout-as-bayesian-approximation}.
After training, the model output $\hat{\bm{x}}_{t+1}$ from the BRNN is sampled from the posterior predictive distribution for next observation $\bm{x}_{t+1}$ via variational dropout model realizations. That is, at each time step $t$, the stochastic forward pass is repeated $N$ times, each with a different dropout mask, and the predictive distribution of the output for $t+1$ is approximated based on the MC samples of the BRNN model, $\left\{ \hat{\bm{x}}_{t+1}(i) \right\}_{i = 1,\dots,N}$. Then, when the true observation $\bm{x}_{t+1}$ is available, it is compared to the predictive distribution and deemed as abnormal if it significantly deviates from the predictive distribution. Finally, the true observation is fed into the BRNN model and the procedure is repeated for the next time step.
Notice that the predictive distribution is evolving, which provides an adaptive decision boundary for the next measurement. This adaptive decision boundary is calculated based on all the useful past system information, which takes into consideration both the spatial and temporal correlations in the data. Further combined with the potential ability to model nonlinear correlations, this property increases both the detection sensitivity and robustness because of the increasing accuracy in modeling NOC pattern.
Depending on the complexity of the system and observed properties of the predictive distribution from the BRNN model, below is a description of two methodologies to quantify the deviation magnitude of each observation to its corresponding predictive distribution. The first method is faster and simpler to implement, but is limited to Gaussian predictive posterior distributions. The second method approximates the posterior distribution non-parametrically and is much more flexible, but requires tuning an additional density estimation parameter.
\subsubsection{Method 1: Squared Mahalanobis distance for Gaussian predictive distributions}
If the predictive distribution is Gaussian, or well approximated as such, the squared Mahalanobis distance can be used to characterize the magnitude of the deviation. First, the MC samples at time $t$ of the predictive distribution, $\left\{\hat{\bm{x}}_t(i) \right\}_{i = 1,\dots,N}$, are used to approximate the sample mean $\bm{\mu}_t$ and covariance $\bm{S}_t$:
\begin{gather}
\bm{\mu}_t \approx \frac{1}{N}\sum_{i = 1}^{N}\hat{\bm{x}}_{t(i)}
\\
\bm{S}_t \approx \tau^{-1}\bm{I}_D
+ \frac{1}{N}\sum_{i = 1}^{N}{{\hat{\bm{x}}_t(i)}^\top {\hat{\bm{x}}}_t(i)}
- \bm{\mu}_t^\top \bm{\mu}_t.
\end{gather}
Then, when the true observation $\bm{x}_t$ is available, the squared Mahalanobis distance is calculated as
\begin{equation}
M^2 = (\bm{x}_t - \bm{\mu}_t)^\top \bm{S}_t^{-1} (\bm{x}_t - \bm{\mu}_t).
\end{equation}
A larger value of $M^2$ indicates that the observation $\bm{x}_t$ is far away from the predicted mean and there is a higher likelihood that it corresponds to a fault. The detection threshold $M_{\text{th}}^2$ is determined with regard to a chosen maximum FAR $\alpha$ on a validation dataset. That is, the threshold is the $100(1-\alpha)^{\mathrm{th}}$ percentile of the $M^2$ statistic in the MC samples of the validation dataset. Therefore, any data point with $M^2$ exceeding the threshold ($M^2 > M_{\text{th}}^2)$ should be detected as a fault.
\subsubsection{Method 2: Local density ratio (LDR) for non-Gaussian predictive distributions}
If the predictive distribution is not well characterized by a Gaussian distribution (e.g., is multi-modal), then non-parametric methods are necessary to quantify the abnormality of each observation. For those cases, a LDR method is proposed, which is closely related to the so-called local outlier factor \citep{Breunig2000}. The LDR statistic quantifies the abnormality of each new observation with respect to its predictive distribution using an estimate of the density around the observation based on its $k$NN.
The $k$NN local density estimate $\hat{f}\left( \bm{x} \right)$ can be calculated as \citep{Duda2001}
\begin{equation}
\hat{f}(\bm{x}) = \frac{k}{\sum_{p \in N_k(\bm{x})} d(p,\bm{x})}
\end{equation}
where $N_k(\bm{x})$ denotes the set of $k$NN of $\bm{x}$ in $\left\{ {\hat{\bm{x}}}_t(i) \right\}_{i=1,\dots,N}$ and $d(p,\bm{x})$ is the Euclidean distance between $\bm{x}$ and a point $p \in N_k(\bm{x})$. Intuitively, the points close to its $k$NN will have high local density values, whereas points in more sparsely sampled or spread out areas will have low density.
Then, the LDR for an observation $\bm{x}_t$ is defined as
\begin{equation}
\text{LDR}(\bm{x}_t) = \frac{\frac{1}{k}\sum_{p \in N_k(\bm{x}_t)} \hat{f}(p)}{\hat{f}(\bm{x}_t)}
\end{equation}
which is the ratio of the averaged local density of the $k$NN of $\bm{x}_t$ in $\left\{ {\hat{\bm{x}}}_t(i) \right\}_{i = 1,\dots,N}$ to the local density of $\bm{x}_t$. A larger value of the LDR$(\bm{x}_t)$ means that the observed point is far away from the samples of the prediction posterior and thus indicates higher likelihood that the observation $\bm{x}_t$ is abnormal.
The number of $k$NN specifies the smallest number of data points in a cluster that will be considered as abnormal and is crucial for the algorithm to perform properly. In general, this number defines a tradeoff, because a small value of $k$ will result in large fluctuations, whereas a very large value of $k$ will reduce the detection sensitivity. As recommended in \citep{Breunig2000}, a minimum and maximum $k$ can be chosen and, for each observation, the final value can be set equal to the maximum of LDR over $k$. The detection threshold $\text{LDR}_{\text{th}}$ is obtained similarly to $M_{\text{th}}^2$.
\subsection{Fault Identification}\label{fault-identification}
Once a fault is detected, the next goal is to identify the main variables associated with the fault. Without using labeled fault examples, this step involves determining the observation variables with the abnormal deviations, which are most relevant to locate and troubleshoot the fault.
BRNN fault identification is obtained by applying the fault detection approach but independently for each variable. To determine which variables deviate abnormally, each observation variable is compared to its corresponding predicted marginal posterior distribution estimated from the BRNN samples. More specifically, the observation $x_t^j$, corresponding to the $j^{\mathrm{th}}$ system variable at time $t$, is compared to the predictive posterior distribution characterized by the samples $\{\hat{x}_t^j(i)\}_{i = 1,\dots,N}$. This variable-wise comparison allows the identification of variables with values in low probability areas and thus more likely to be relevant for diagnosing the fault.
The marginal posterior distributions used for identification still take into consideration the spatial and temporal correlations in past data observations. Hence, the marginal distribution for each variable also evolves with dynamics that depend on other variables and past observations. This analysis sacrifices some information with regard to the complete joint distribution, as considered during fault detection, but is necessary to obtain variable specificity.
As with fault detection, two methodologies to quantify the fault identification deviation are described here, depending on the properties or assumptions placed on the predictive distribution. The same considerations apply to these methodologies.
\subsubsection{Method 1: Standard deviation for Gaussian predictive distributions}
Assuming that the predictive distribution can be approximated by a Gaussian distribution, the number of standard deviations of each variable to its predictive mean can be used to measure the deviation. Using the same MC samples of the posterior predictive distribution generated for fault detection at time $t$, $\left\{ \hat{\bm{x}}_t(i) = \{\hat{{x}}_t^l(i)\}_{l = 1,\ldots,m_x} \right\}_{i = 1,\dots,N}$, the mean $\mu_t^l$ and standard deviation $\sigma_t^l$ of each variable ${\hat{x}}_t^l$ can be estimated by
\begin{gather}
\mu_t^l \approx \frac{1}{N} \sum_{i = 1}^{N} \hat{x}_t^l(i)
\\
\sigma_t^l \approx \sqrt{\tau^{-1} + \frac{1}{N}\sum_{i = 1}^{N} \left(\hat{x}_t^l(i)\right)^2 - (\mu_t^l)^2}
\end{gather}
The deviation for each variable $D^l$, $l\in\{1,\dots,m_x\}$, is then calculated from
\begin{equation}
D^l = \frac{x_t^l - \mu_t^l}{\sigma_t^l}
\label{eq:fault.id:Dl.stat}
\end{equation}
The $D^l$ can be either negative or positive, unlike $M^2$ which can only be positive. Under a Gaussian approximation, variables are identified as significantly affected by the disturbance based only on whether $D^l$ has a large magnitude (i.e., absolute value). Still, the sign of the deviation (positive or negative) can be helpful to operators because the sign explicitly indicates whether the variable is significantly higher or lower than expected. For a predefined significance level, the NOC validation dataset can be used to determine thresholds $\{D_{\text{th}}^l\}_{i = 1,\dots,N}$ such that variables with ($D^l >$ $D_{\text{th}}^l$) are explicitly highlighted as abnormal.
\subsubsection{Method 2: LDR for non-Gaussian predictive distributions}
For more general distributions, and similarly to the fault detection procedure, the LDR can be used element-wise for fault identification by considering each variable separately in the calculation of the LDR. Given the true measurement $\bm{x}_t = \left\{ x_t^l \right\}_{l = 1,\dots,m_x}$ and MC samples from predictive distribution $\left\{ {\{\hat{x}_t^l(i)\}}_{l = 1,\dots,m_x} \right\}_{i = 1,\dots,N}$, the LDR for variable $l$ can be calculated by
\begin{gather}
\hat{f}(x_t^l) = \frac{k}{\sum_{p^l \in N_k(x_t^l)} d(p^l, x_t^l)}
\\
\text{LDR}^l(x_t^l) = \frac{\frac{1}{k}\sum_{p^l \in N_k(x_t^l)} \hat{f}(p^l)}{\hat{f}(x_t^l)}
\end{gather}
where $p^l$ is one of the $k$NN of $x_t^l$ and $d(p^l, x_t^l)$ is the Euclidean distance between the $p^l$ and $x_t^l$ sample. The same rule for selecting the number of nearest neighbors $k$ discussed with regard to fault detection can be used here.
The variables associated with a large value of $\text{LDR}^l$ can be explicitly selected as significantly affected by the fault. This is done similarly as for fault detection using the NOC validation dataset to determine a threshold $\text{LDR}_{\text{th}}^l$ above which variables are considered abnormal. Alternatively, the variables can simply be sorted from the largest to the smallest such to emphasize the system variables that deviate the most.
\subsubsection{Fault identification plots}\label{sec:fault.idplots}
The fault identification statistics of each variable can be visualized by plotting their values over time. The resulting plots are visually similar to contribution plots \citep{Miller1998,Zhu2014}. Their interpretation and analysis, however, are fundamentally different and are referred to here as {\em identification plots}. The main distinction is that the statistics in identification plots are specific to the current status of each variable and its dynamics, rather than as a relative component of a global statistic. These plots provide greater specificity in the analysis and allows the interpretation of the status of each variable directly.
\subsection{FDI Scheme}
\label{sec:fault-detection-and-identification-scheme}
\begin{figure*}
\centering
\includegraphics[width=0.78\textwidth]{050_methodology_diagram}
\caption{Flowchart of the BRNN-based FDI methodology. The offline training stage (left) and the online monitoring stage (right) are shown in the figure. The procedure starts with offline training, and then the offline-trained model is used during online monitoring. The choice of statistics for detection and identification is made at design time.}
\label{fig:flowchart}
\end{figure*}
For completeness, the overall methodology is summarized in Figure \ref{fig:flowchart}.
Although the figure explicitly shows the two methods for detecting and identifying faults, this decision of which method to use is actually done at the design stage rather than during operations.
In either case, the BRNN model with variational dropout is crucial to the methodology by providing samples that characterize the uncertainty and directly enable both FDI. The $M^2$ or LDR statistics are used to detect the fault in the system, while the $D^l$ or $\text{LDR}^l$ statistics are used to identify the impacted variables useful for locating the fault and possible root cause analysis. Minimal computation is needed for fault identification, having to calculate only some additional statistics on the same samples.
\section{Case Studies}\label{sec:case-studies}
In this section, the effectiveness of the proposed BRNN-based FDI method is demonstrated in two case studies: the benchmark Tennessee Eastman process synthetic dataset and a real dataset from a chemical plant.
For comparison, results are also shown for PCA \citep{Jackson1979,Kourti1996} and DPCA \citep{Ku1995} FDI methods.
For each method, both models with and without dimension reduction are considered, and identified by prefix `r-' or `f-', respectively. For the models with reduced dimension, parallel analysis \citep{Downs1993} is used to determine the number of PCs $a$ to retain in the model.
These (D)PCA-based methods are commonly accepted benchmark methods for algorithm comparison in the FDI community \citep{Chiang2000,Yin2014,De2015, Venkatasubramanian20032}. DPCA in particular provides an interesting contrast to the proposed BRNN method because DPCA also models both spatial and temporal correlations, albeit in a limited form. As mentioned in the introduction, DPCA is limited to linear dynamics and correlations and scale poorly with increased temporal memory depth. The proposed BRNN method does not have these limitations.
In both case studies, the BRNN model constructions were implemented in TensorFlow \citep{TensorFlow}, and a number of BRNN model configurations and hyperparameters were tested. The choices included different recurrent node types (regular RNN, GRU, and LSTM cells), activation functions (i.e., linear, sigmoid, hyperbolic tangent, and rectifier linear), number of recurrent nodes/states $m_s$, number of recurrent layers, regularization hyperparameter values $\lambda$, dropout probabilities $p_{d}$, and RNN training parameters (e.g., learning rate). BRNN models were trained for each variation of these configurations and hyperparameters. The final model configuration and hyperparameters were selected as the ones that gave the maximum likelihood on the validation dataset. Only the results for the final BRNN model are shown in the next sections.
\subsection{Tennessee Eastman Process}\label{sec:tep}
\begin{figure}
\centering
\includegraphics[width=0.74\textwidth]{060_tep_process_diagram}
\caption{A process flowsheet for the TEP with the second control structure in \citep{Downs1993}.}
\label{fig:TEP.flowsheet}
\end{figure}
The Tennessee Eastman process (TEP) is a well-known benchmark by the Eastman Chemical Company for process monitoring and control studies. It is based on a realistic industrial process with properly modified components, kinetics, and operating conditions~\citep{Downs1993}. In this study, the second plant-wide control strategy was utilized, with the process flowsheet as shown in \figref{fig:TEP.flowsheet}. The process contains eight components (A, B, C, D, E, F, G, and H) and five major units (a reactor, condenser, compressor, separator and stripper).
In this case study, $m_x = 52$ variables are used to construct the monitoring system, of which 41 are sensor measurements (XMEAS(1)--XMEAS (41)) and 11 are manipulated variables (XMV(1)--XMV(11)). During the NOC, the system is operating under one production mode and the sampling period is set to 3 min. The training data contains 480 samples and the validation data contains 960 samples. The TEP simulation contains 21 preprogrammed faults with different disturbance types and locations. Once a fault is introduced in the system, the system will either behave normally if the control system is effective in controlling the disturbance, or it will evolve outside the NOC region. For each set of data with a fault condition, the simulator first runs for 160 time points in the normal state, and then the corresponding fault disturbance is introduced with the simulator continuing to run for another 800 samples. The dataset used in this case study can be downloaded from the website of Prof.\ Richard Braatz \citep{TEPBraatz}. For further details about the TEP dataset, the reader is referred to \citet{Downs1993} or \citet{Chiang2000}.
Although a number of BRNN model architecture variations were tried as previously mentioned, the final BRNN model used in this case study contains one recurrent layer with regular RNN cell and linear activation function. A linear dense layer is used for the output layer, as is commonly done for regression tasks. This structure means that, in this case, the BRNN model implements a probabilistic linear state-space model. Although this architecture is simpler, it is also easier to train and achieved better performance than more complex structures and neuron types. The results are likely due to the fact that the inherent correlations and dynamics in the NOC data of TEP are well modeled as linear \citep{WeikeThesis}. The final model has $m_s = 80$ hidden nodes in the recurrent layer, and is trained with regularization parameter $\lambda = 10^{-4}$ and dropout rate $p_{d} = 0.1$. Given the linear structure of the model and by the central limit theorem, the predictive distribution by the final BRNN model is well approximated by the Gaussian distribution in this case. Thus, the $M^2$ and $D^l$ statistics are used for FDI. In any event, the results are quite similar for a number of configurations of the hyperparameters within a reasonable tuning range.
For the reduced dimensionality (D)PCA models used in the comparison, the number of PCs determined by parallel analysis is $a = 12$ for r-PCA and $a = 25$ for r-DPCA (with $\text{lag}=1$). The fault detection procedure for r-PCA and r-DPCA use both the $T^2$ and $Q$ statistics, whereas f-PCA and f-DPCA use only the $T^2$ statistic for fault detection, which plays the same role as the $M^2$ statistic. Contribution plots are used for (D)PCA-based fault identification. For implementation details on the (D)PCA methods, the reader is referred to \citep{Russell2000}, \citep{Zhu2014}, or \citep{Chiang2000}.
The false alarm rate~(FAR) and fault detection rate~(FDR) are used to evaluate the fault detection performance of different algorithms:
\begin{align}
\text{FAR} &= \frac{\mbox{\# of samples with alarm during NOC\xspace}}{\mbox{total \# of samples during NOC\xspace}} \\
\text{FDR} &= \frac{\mbox{\# of samples with alarm after the fault is introduced in the system}}{\mbox{total \# of samples after the fault is introduced in the system}}
\end{align}
In words, the FAR corresponds to the frequency of spurious detection of faults under NOC, and FDR is the sample frequency of a fault being detected when a fault situation is present.
The dataset contains three types of faults: controllable faults, back-to-control faults, and uncontrollable faults. Controllable faults are disturbances that can be well compensated by the control system, and therefore the disturbance does not significantly affect the process state. In these situations, since the operator is not required to intervene, the FDR should ideally be as low as the FAR to avoid distracting the operators. Back-to-control faults are disturbances that are large enough to cause the system to initially deviate from the NOC, but for which the control system is able to compensate at least some aspects of the disturbance after some time. The process measurements return to the normal region after some time, but certain manipulated or input variables remain outside the normal regime. These represent sub-optimal or off-spec conditions that ought to be handled by an operator and to be detected accordingly. Moreover, the FDI result should accurately reflect the system state, such that its evolution back to control is apparent. Finally, uncontrollable faults are faults that cannot be handled adequately by the control system and require operator intervention. For both back-to-control and uncontrollable faults, the fault detection algorithm should ideally yield high FDR to notify the operator that the system has been disturbed outside the original NOC. It is worth noting that this ``classification'' is based on prior knowledge of the faults and used here only to facilitate the interpretation of the results; it was not used anywhere in the model training.
For fault identification, the proposed BRNN-based identification plots are compared with the (D)PCA-based contribution plots, which are shown for a representative fault of each of the aforementioned types. Note that, ideally, fault identification should accurately pinpoint the variables that are affected by the fault to provide the operators with specific information for them to analyze the situation and quickly diagnose the underlying root cause.
Accordingly, it should be verified that no variable should be identified as abnormal for controllable faults. For back-to-control faults, the abnormal variables should first be identified, and only the corresponding tuned manipulated variables should remain identified once the system is back to control. For uncontrollable faults, the identification procedure should correctly locate the abnormal variables as soon as they are outside the NOC regime.
\subsubsection{Training and validation results on the NOC data}
\label{sec:tep:noc-data}
\begin{figure*}
\centering
\includegraphics[width=0.96\textwidth]{070_tep_noc_training}
\caption{BRNN model outputs for TEP NOC training data. The plot shows all 52 variables in TEP. The dark blue lines are the TEP measurements and the light blue lines correspond to the BRNN model predictive distribution outputs for the NOC data. For measurements under the NOC, the dark blue lines should lie within the predictive distribution.}
\label{fig:TEP.training}
\end{figure*}
The training and validation results are first shown to demonstrate how the posterior predictions of the BRNN model characterize the NOC.
The training results of the total 52 variables with centered and normalized values are shown in Figure \ref{fig:TEP.training}. The dark blue lines are the real data and the light blue lines are the posterior prediction samples by the BRNN model with $N = 400$ model samples by variational dropout. The results do not differ significantly for $N>100$.
When the real measurements are within the predictive distribution (in dark and light blue in the figures, respectively), the system is considered normal. This condition can be observed in Figure \ref{fig:TEP.training}, which indicates that the trained BRNN model accurately captures the NOC pattern.
\begin{figure*}
\centering
\includegraphics[width=0.96\textwidth]{080_tep_noc_validation}
\caption{BRNN model outputs for TEP NOC validation data.}
\label{fig:TEP.validation}
\end{figure*}
Then, to validate the model, the trained BRNN model is applied to a separate NOC validation dataset. These results are shown in Figure \ref{fig:TEP.validation}. As observed in the training results, the real validation measurements lie within the predictive distribution. This result indicates that the model generalizes well, meaning that it is able to capture the normal pattern without overfitting to the training data, which is crucial to avoiding high FARs.
\subsubsection{Fault detection results}\label{sec:tep:fault.detection}
\begin{table}[h]
\centering
\caption{TEP fault detection percentage results. The FAR is shown for the NOC (in the first row), and the FDR is given for the 21 fault conditions.}%
\label{tab:tep:detection.results}
\resizebox{\columnwidth}{!}{\begin{tabular}{ccccccc}
\hline
\multicolumn{1}{c}{Type} & \multicolumn{1}{c}{Fault ID} & \multicolumn{1}{c}{BRNN} & \multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}r-PCA\\ ($a=12$)\end{tabular}} & \multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}f-PCA\\ ($a=52$)\end{tabular}} & \multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}r-DPCA\\ ($a=25$)\end{tabular}} & \multicolumn{1}{c}{\begin{tabular}[c]{@{}c@{}}f-DPCA\\ ($a=104$)\end{tabular}} \\ \hline
NOC & -- & 4.75 & 5.00 & 4.88 & 5.00 & 5.00 \\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}controllable\\ faults\end{tabular}} & IDV(3) & 5.00 & 7.00 & 19.75 & 6.12 & 22.25 \\
& IDV(9) & 5.00 & 7.88 & 15.25 & 8.87 & 21.37 \\
& IDV(15) & 7.12 & 10.62 & 26.87 & 11.13 & 36.63 \\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}back to control\\ faults\end{tabular}} & IDV(4) & 100.00 & 98.88 & 100.00 & 100.00 & 100.00 \\
& IDV(5) & 100.00 & 32.62 & 100.00 & 34.50 & 100.00 \\
& IDV(7) & 100.00 & 100.00 & 100.00 & 100.00 & 100.00 \\ \hline
\multirow{15}{*}{\begin{tabular}[c]{@{}c@{}}uncontrollable\\ faults\end{tabular}} & IDV(1) & 99.75 & 99.75 & 100.00 & 99.75 & 99.25 \\
& IDV(2) & 99.00 & 98.75 & 99.12 & 98.62 & 99.12 \\
& IDV(6) & 100.00 & 100.00 & 100.00 & 100.00 & 100.00 \\
& IDV(8) & 98.12 & 98.00 & 98.25 & 97.75 & 98.38 \\
& IDV(10) & 87.38 & 54.13 & 93.50 & 55.75 & 94.63 \\
& IDV(11) & 74.75 & 74.25 & 87.25 & 80.75 & 92.75 \\
& IDV(12) & 99.75 & 99.00 & 100.00 & 99.25 & 100.00 \\
& IDV(13) & 95.75 & 95.50 & 95.75 & 95.50 & 96.25 \\
& IDV(14) & 100.00 & 100.00 & 100.00 & 100.00 & 100.00 \\
& IDV(16) & 90.38 & 46.50 & 95.50 & 48.50 & 97.00 \\
& IDV(17) & 96.13 & 93.13 & 97.75 & 94.78 & 98.12 \\
& IDV(18) & 90.63 & 90.38 & 91.50 & 90.50 & 92.87 \\
& IDV(19) & 88.25 & 25.12 & 96.00 & 34.00 & 99.50 \\
& IDV(20) & 78.63 & 58.25 & 92.13 & 61.75 & 92.37 \\
& IDV(21) & 48.00 & 48.50 & 61.62 & 47.88 & 59.38 \\ \hline
\end{tabular}}
\end{table}
The fault detection results for the 21 predefined faults are shown in Table \ref{tab:tep:detection.results}. The results are grouped according to one of the above-mentioned three types of faults. For all algorithms, the FDRs are estimated with regard to the threshold estimated for a FAR of 5\%, and validated on NOC data as shown on the first row of the table.
As shown in Table \ref{tab:tep:detection.results}, the proposed BRNN-based method yields close to 5\% FDR on controllable faults, which is almost as low as the pre-determined FAR level. On the other hand, (D)PCA-based methods are overly sensitive in these cases, especially the models without model reduction, f-PCA and f-DPCA. These results show that (D)PCA-based methods cannot accurately differentiate the controllable faults from the other cases, because they do not appropriately characterize the dynamics of NOC, such as to determine if the situation is ultimately controllable.
As previously explained, controllable faults should not trigger an alert because they are handled directly by the control system. The ability of the fault detection approach to differentiate between these situations is of crucial practical importance because alerts due to these situations will often be perceived as false alarms and can erode an operator's confidence in the method and the significance of its alerts. The BRNN method is observed to be more robust to controllable fault than the (D)PCA methods.
For both back-to-control and uncontrollable faults, the BRNN method reliably detected faults with high FDRs.
Full PCA and DPCA models with the squared Mahalanobis distance were also able to detect the back-to-control and uncontrollable faults with high FDR. However, the (D)PCA models were overly sensitive for general fault detection purposes because they overreacted to controllable faults. Compared to the BRNN method, (D)PCA models emphasized higher sensitivity to disturbances at the expense of an increased likelihood of unwarranted alerts.
The reduced dimensionality (D)PCA models (i.e., r-PCA and r-DPCA), with the number of PCs determined by parallel analysis, responded more reasonably to controllable faults but also yield much worse performance compared to the BRNN method. In fact, they fail to reliably detect several back-to-control and uncontrollable faults (Faults 5, 16, and 19, for example).
It is insightful to consider how the temporal dynamics interact with the detection approach to lead to the measured FDR results.
If a disturbance is such that the measurements oscillate around the NOC region, there will be moments in time that are momentarily indistinguishable from those in the NOC region. Since the BRNN is trained such that its state characterizes the NOC distribution in state space, it is understandable that some of these time points may not be detected as faulty. This observation explains the slightly lower FDR of the BRNN method for those cases.
Unlike the BRNN, the (D)PCA methods do not model internal system dynamics under the NOC. Hence, since the internal dynamics are not considered when explaining the observed data, these models do not have this detection ambiguity. This lack of ambiguity is achieved at the expense of the inability by (D)PCA to assess whether a fault is controllable.
In summary, the fault detection results indicate the BRNN has high detection accuracy and is able to more robustly detect faults when operator intervention is truly necessary.
Specific FDI results are presented and discussed in detail below. Faults~1, 3, and 5 are presented because they are representative of each type. There was no significant differences between fault within the same type. The use of the BRNN contribution plot results for fault propagation analysis and one example is given for Fault 6.
\subsubsection{Controllable fault: Fault 3}%
\label{sec:tep:fault3}
Fault 3 is considered first as a demonstrative controllable fault.
For Fault 3, the D feed temperature in Stream 2 has a step change at the $160^{\mathrm{th}}$ time point. Since this change in feed temperature is handled immediately and directly by the control system, the process is not driven outside its normal operating state. In this case, a data-driven fault detection algorithm should not trigger the alarm (beyond the chosen FAR).
\begin{figure*}
\centering
\includegraphics[width=0.96\textwidth]{090_tep_fault3_outputs}
\caption{BRNN model outputs for TEP Fault 3.}
\label{fig:TEP:fault3.outputs}
\end{figure*}
The prediction results by the BRNN model are shown in Figure \ref{fig:TEP:fault3.outputs}. Similarly to the NOC case, the dark blue lines (i.e., real measurements) are within the distribution high-likelihood area characterized by the light blue lines, which indicates that the system is operating under the NOC.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{100_tep_fault3_id_brnn}
\caption{Fault identification plot of the BRNN-$D^l$ statistic for Fault 3. The $\left\{ D^l \right\}_{l = 1,\dots,52}$ values for the 960 timesteps are color coded in the identification plot. Variables with dark blues have high values of $D^l$, meaning that the variable has positively deviated from the NOC region. Conversely, variables with dark red have low values of $D^l$ and have negatively deviated from the NOC region. A light color means the variable is not significantly affected by the disturbance. As expected, no variable significantly deviates from the NOC.}
\label{fig:TEP:fault3.idplot}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.96\textwidth]{110_tep_fault3_id_pca}
\caption{Contribution plots for Fault 3 from (a) r-PCA, (b) f-PCA, (c) r-DPCA, and (d) f-DPCA. The plot shows the contribution factor and with the darkness of the blue color indicating the amount of deviation of the variable from the NOC region.}
\label{fig:TEP:fault3.contribplot}
\end{figure*}
Fault identification results by the BRNN and (D)PCA methods are shown in Figures \ref{fig:TEP:fault3.idplot} and \ref{fig:TEP:fault3.contribplot}, respectively. The color indicates the deviation from the NOC over time for each of the 52 variables. The $D^l$ statistic (c.f.\ Equation \ref{eq:fault.id:Dl.stat}) is used in the BRNN identification plot.
As shown in Figure \ref{fig:TEP:fault3.idplot}, the BRNN model identifies that no variable has its normal operating dynamics significantly affected by Fault 3, as is expected. In contrast, the contribution plots in Figure \ref{fig:TEP:fault3.contribplot}bd, by the full PCA and DPCA models, incorrectly identify several variables as being affected by the disturbance even before the introduction of the disturbance (at the $160^{\mathrm{th}}$ sample). This further demonstrates the oversensitiveness of those models. The r-PCA and r-DPCA models have identification results that are similar to those of the BRNN model and are somewhat robust to controllable faults, but at the expense of robustness in fault detection.
In summary, for controllable faults, BRNN-based FDI is robust and successfully characterizes those disturbances as corresponding to NOC. r-PCA and r-DPCA methods gave similar fault identification results but have lower detection rates (c.f.\ Table \ref{tab:tep:detection.results}). The f-PCA and f-DPCA models were clearly oversensitive, have high FARs, and incorrectly characterized the controllable faults in the contribution plots.
\subsubsection{Back-to-control fault: Fault 5}%
\label{sec:tep:fault5}
Fault 5 is a representative example of a back-to-control fault.
This fault involves a step change in condenser cooling water inlet temperature. This step change requires a step change in the condenser cooling water flow rate XMV(11) by the control system. While the fault is ultimately controllable, the fault causes the system at first to operate off-spec, or at least sub-optimally. In this particular, immediately after the fault occurs, the system oscillates with about 32 variables exhibiting this similar transient oscillation behavior. The process returns to control after about 10~hours, at which point the sensor measurements XMEAS(1)--XMEAS(41) are back to their pre-disturbance set-points, and only the manipulated variable XMV(11) remains outside the NOC regime, tuned so as to compensate the step change in condenser cooling water inlet temperature.
\begin{figure*}
\centering
\includegraphics[width=0.96\textwidth]{120_tep_fault5_outputs}
\caption{BRNN model outputs for TEP Fault 5.}
\label{fig:TEP:fault5.outputs}
\end{figure*}
The BRNN results for all of the variables are shown in Figure \ref{fig:TEP:fault5.outputs}. As expected, when the fault is introduced, several measurements in dark blue lines deviate from the posterior predictive distribution under the NOC shown in light blue. After about 200 data points the system returns to control, verified by the fact that all the system measurements (XMEAS(1)--XMEAS(41)) are back within the predictive NOC region while only the manipulated variable XMV(11) maintains a systematic deviation off-the-center of the BRNN model prediction distribution.
These results show that the BRNN model is able to correctly identify the NOC pattern and how the deviation from the predictive distribution accurately locates the faulty variable under disturbance.
The BRNN model is also able to better assess the state of the system, distinguishing the back to control faults from the uncontrollable faults by showing the transient deviation of the process variables and their return to the NOC region.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{130_tep_fault5_id_brnn}
\caption{Fault identification plot by BRNN-$D^l$ for Fault 5. The
switch between dark blue and red colors shows that the system is
undergoing large fluctuation.}
\label{fig:TEP:fault5.idplot}
\end{figure}
The fault identification plot by BRNN is shown in Figure \ref{fig:TEP:fault5.idplot}. This example showcases the typical pattern of back to control faults, with several measurements outside the predictive region after the fault is introduced and only the manipulated variables deviating once the system is back to steady state. In this case, about 32 variables are affected once the disturbance is introduced to the system, and the color switches between blue and red, indicating the system is oscillating. The plot also clearly shows how, after the $360^{\mathrm{th}}$ time point, all system variables except XMV(11) are undoubtedly back to normal. XMV(11) remains consistently above the predictive mean after the fault as that is forced by the controller to compensate for the fault. However, the magnitude of the deviation of XMV(11) is relatively small, indicating that the disturbance is no longer critical.
The BRNN identification plot also contains crucial information for locating the likely root cause of the fault. The plot clearly shows that XMEAS(22) is the first variable positively deviated from the predictive distribution, which indicates the higher than normal separator cooling water outlet temperature. Combined with the fact that the condenser cooling water flow rate is increased to compensate for the disturbance to the system, one would reason that the root cause is the increase in the condenser cooling water temperature. After the condenser cooling water temperature increases, the outlet stream from the condenser to the separator also increases the temperature, resulting in an increase in the temperature in the separator, which finally results in the increase in separator cooling water outlet temperature.
\begin{figure*}
\centering
\includegraphics[width=0.96\textwidth]{140_tep_fault5_id_pca}
\caption{Contribution plots for Fault 5 from (a) r-PCA, (b) f-PCA, (c) r-DPCA, and (d) f-DPCA.}
\label{fig:TEP:fault5.contribplot}
\end{figure*}
For comparison purposes, the contribution plots by PCA and DPCA methods are shown in Figure \ref{fig:TEP:fault5.contribplot}. The r-PCA and r-DPCA model results in Figures \ref{fig:TEP:fault5.contribplot}ac fail to identify the consistent deviation in XMV(11), which clearly explains their detection results for this fault. These results demonstrate again that the r-PCA and r-DPCA models can exhibit much lower detection and identification sensitivity than the BRNN method. The results of the f-PCA and f-DPCA models in Figures \ref{fig:TEP:fault5.contribplot}bd clearly identify the deviations in several variables. However,the f-PCA and f-DPCA are oversensitive and identify variables in an unspecific manner, which prevents those statistics from being used, at least directly, by operators for diagnosing the root cause of the fault.
For the back-to-control fault, BRNN FDI had high accuracy and robustness. Moreover, this method yielded more specific information for evaluating the state of the system. By inspecting the identification plot, operators have a clear view about which variables are affected by the disturbance and are able to assess the type of the fault occurring and the current stage of the system.
\subsubsection{Uncontrollable fault: Fault~1}%
\label{sec:tep:fault1}
An uncontrollable fault is now considered.
Fault 1 involves a step change in the A/C feed ratio in Stream 4, which results in an increase in the C feed and a decrease in the A feed. This subsequently leads to a decrease in feed A in the recycle Stream 5 and the controller reacts by increasing the A feed flow in Stream 1. These two effects conflict with each other, thereby shifting the system to an uncontrollable operating situation.
\begin{figure*}
\centering
\includegraphics[width=15cm]{150_tep_fault1_outputs}
\caption{BRNN model outputs for TEP Fault 1.}
\label{fig:TEP:fault1.outputs}
\end{figure*}
The BRNN model output results are shown in Figure \ref{fig:TEP:fault1.outputs}. After the fault is introduced to the system, more than half of the variables are observed to deviate significantly from the BRNN predictive NOC region. All of the (D)PCA methods are also capable of detecting this fault.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{160_tep_fault1_id_brnn}
\caption{Fault identification plot by BRNN-$D^l$ for Fault 1.
The root cause for this uncontrollable fault can be assessed by looking
at the variables that are persistently off the NOC region.}
\label{fig:TEP:fault1.idplot}
\end{figure}
The corresponding BRNN fault identification statistics are shown in Figure \ref{fig:TEP:fault1.idplot}. Since the system is seriously affected by the disturbance and several variables associated with material balances (e.g., composition, pressure) change significantly, this fault is easily detected. The long-term and uncontrollable nature of the fault on these measurements and manipulated variables can also be observed in the identification plot, making the fault easy to diagnose based on those variables.
\begin{figure*}
\centering
\includegraphics[width=0.96\textwidth]{170_tep_fault1_id_pca}
\caption{Contribution plots for Fault~1 from (a) r-PCA, (b) f-PCA, (c) r-DPCA, and (d) f-DPCA.}
\label{fig:TEP:fault1.contribplot}
\end{figure*}
As before, the contribution plots by (D)PCA methods are shown in Figure \ref{fig:TEP:fault1.contribplot}. The r-PCA and r-DPCA models, in Figure \ref{fig:TEP:fault1.contribplot}ac, both give somewhat results similar to those of the BRNN model in Figure \ref{fig:TEP:fault1.idplot}. However, both of them fail to identify the continued deviation in XMV(4) (shown between XMV(3) and XMV(5) in Figure \ref{fig:TEP:fault1.contribplot}) for instance, which is the manipulated variable for total feed flow in Stream 4 and clearly plays a central role in the fault. In contrast, it is clearly identifiable from the BRNN results in Figure \ref{fig:TEP:fault1.idplot} that XMV(4) has negatively deviated from the NOC region.
The contribution plots of f-PCA and f-PCA in Figure \ref{fig:TEP:fault1.contribplot}bd show the identification of the involved variables, but again highlight several other variables that are unrelated to the fault and operating within their normal pattern (such as XMV(11)). As previously observed, this again shows that the f-PCA and f-DPCA models are overly sensitive and their identification results require substantial additional processing such that operators cannot directly use them to diagnose the fault.
In summary, the BRNN model is able to accurately and robustly detect and identify uncontrollable faults. Perhaps most crucially, BRNN identification plots provide clear information that is directly useful for root cause analysis. While several (D)PCA models are also able to detect uncontrollable faults, their identification results are less accurate and precise than for the BRNN model.
\subsubsection{Fault propagation path analysis: Fault 6}%
\label{sec:tep:fault-propagation-fault6}
This section shows how the accuracy and specificity of the BRNN identification statistics can be used for fault propagation path analysis. The key observation is that the chronological sequence of events of when each variable deviates significantly from its NOC is useful information to understand the start and evolution of the disturbance through the process \citep{Chiang2003}. The BRNN method can extract this information with a high degree of temporal precision. This information can then be combined with expert knowledge of the process to examine the propagation of the fault through the system.
This approach is exemplified here using Fault 6, which is an uncontrollable fault induced by a loss of feed A in Stream 1. The loss of component A thus causes the control system to increase the manipulated variable XMV(3) in order to increase A in the system and attempt to compensate for the disturbance. However, since there is no component A in Stream 1, the control system fails to take the system back to NOC. Due to the severity of this fault, a large portion of system variables is affected.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{180_tep_fault6_id_brnn_sorted}
\caption{Sorted fault identification plot according to the detected deviation occurrence time of BRNN-$D^l$ for Fault 6.}
\label{fig:TEP:fault6.idplot}
\end{figure}
The temporal sequence of the fault through the process is achieved by sorting the identification plot according to the time when each variable significantly deviates from the NOC. For this approach, one needs to estimate the threshold used to determine when the deviation is significant. In our case, this is estimated using the NOC validation set and determined to be $D_{\text{th}}^l = 4.8$. Then, once a fault is detected, the process variables are sorted according to the time index at which its $D^l$ statistic first exceeds the threshold, yielding the sorted identification plot shown in Figure \ref{fig:TEP:fault6.idplot}. The $y$-axis numbers 1--52 correspond to $[\text{XMEAS}(1), \dots, \text{XMEAS}(41), \text{XMV}(1), \dots, \text{XMV}(11)]$.
\begin{figure}
\centering
\includegraphics[width=0.74\textwidth]{190_tep_fault6_brnn_propag_path}
\caption{Fault~6 propagation path at the $180^{\mathrm{th}}$ data point (1 hour after the fault occurs). Colored nodes indicate that the corresponding variable has been detected as deviating significantly from the NOC.}
\label{fig:TEP:fault6.propagpath}
\end{figure}
A diagram of the fault propagation path is then obtained by combining the timing results of the sorted BRNN identification plot with the knowledge of the process, as demonstrated in Figure \ref{fig:TEP:fault6.propagpath} for Fault 6. When the fault is introduced in the system, XMEAS(1) and XMV(3) are affected and deviates from the NOC first. Then, after a few minutes, the reactor pressure measurement XMEAS(7) is affected. Then the reactor cooling water system is also affected due to the change in the mass inside the reactor and both XMEAS(21) and XMV(10) deviate from the NOC. The diagram in Figure \ref{fig:TEP:fault6.propagpath} highlights that, after 1 hour, the fault has already propagated to the final product and the concentration of A and C have been affected, thus clearly showing the impact of the fault in the system at that point in time.
The approach outlined here shows how the properties of the BRNN method can be used to easily determine and visualize the fault propagation path. This information is crucial to operators to accurately diagnose the fault and determine which parts of the process have been affected.
\subsection{Real Industrial Dataset}%
\label{sec:real-data}
The next case study further demonstrates the efficiency of the proposed BRNN method on a real dataset from a chemical manufacturing process. The use of this method for real-time FDI is a promising application for the next generation of process monitoring systems in chemical plants. The complex nature of real chemical manufacturing processes and their intricate control system dynamics, make BRNN the best-suited tool to extract and recognize these patterns from data in comparison to traditional methods.
The dataset pertains to the operation of an amine tower. The column experienced foaming issues resulting in faults that decrease the efficiency of the process. There are a total of 20 sensor measurements with a sampling time of $t = 1$ min. A total of two months of data are available. Two events has been recorded by operators as a result of the foaming issue in the tower. However, it is also possible that additional disturbances are encountered during the two-month operating window that have been previously missed.
The final BRNN model uses standard RNN cells with the sigmoid activation function. There is one hidden recurrent layer with 40 units (i.e., `state variables'). The dropout probability is set to $p_d = 0.1$ and the regularization parameter to $\lambda = 10^{-5}$. The BRNN model using LSTM or GRU cells yield similar performance in spite of the higher complexity and thus those results are omitted. Similar to the TEP data, the results are similar for a number of configurations of the hyperparameters within a reasonable tuning range.
For comparison, the PCA and DPCA models, with the number of PCs determined by parallel analysis and full models without dimension reduction, are also applied. The number of PCs from parallel analysis is determined to be $a=6$ for r-PCA and $a=9$ for r-DPCA.
Due to the sensitivity of the data, the actual measurement values and the BRNN model predictions are omitted and only the detection and identification results are shown. The posterior predictive distribution is observed to be multi-modal and thus the LDR statistics are used for FDI. The number of $k$NN is set to the range of 10 to 20. The dataset is divided into a 35-day training dataset, a 17-day validation dataset, and two testing datasets. The first testing dataset spanned 7 days containing Fault 1, and the second testing dataset spanned 14 days containing Fault 2.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{200_realdata_fault1_brnn}
\caption{FDI by the BRNN model on the Fault 1 testing data. The red box indicates the period with the foaming event recorded by the operator.}
\label{fig:fault1:brnn}
\end{figure}
\subsubsection{Fault 1 results}%
\label{sec:real-data:fault1}
The BRNN FDI results for Fault 1 are shown in Figure \ref{fig:fault1:brnn}. The BRNN model successfully detects the documented event, marked by the red box in the figure. The model also accurately pinpoints the variables that are most affected by the foaming issue, X1.PV and X6.PV. Moreover, it also highlights several points after the $8000^{\mathrm{th}}$ time point that may have been originally missed. During these later periods, the X1.MV sensor measurement is identified by the BRNN method. This is subsequently verified to have been the result of large unexplained fluctuations in that variable and that the BRNN has performed as expected.
\begin{figure}
\centering
\includegraphics[width=0.96\textwidth]{210_realdata_fault1_pca}
\caption{FDI by (D)PCA methods for Fault 1: (a) r-PCA, (b) f-PCA, (c) r-DPCA, and (d) f-DPCA.
The red box indicates the period with the foaming event recorded by the operator.}
\label{fig:fault1:pca}
\end{figure}
For comparison, the corresponding FDI results by (D)PCA methods are shown in Figure \ref{fig:fault1:pca}. The r-PCA and r-DPCA models simply fail to detect the fault. In the contribution plots, the r-PCA and r-DPCA models also fail to identify any variable that is noticeably affected by the foaming event. While (D)PCA models with reduced dimensionality determined by parallel analysis have been widely applied \citep{Chiang2000,Yin2014,De2015,Valle1999}, they are incapable of accurately detecting the main fault in this case.
For f-PCA and f-DPCA models, the $T^2$ statistic is able to detect the documented fault. The contribution plots also identify X1.PV and X6.PV as being associated with the fault. However, X1.MV and X6.MV are also identified as abnormal and as more significantly than X1.PV and X6.PV. While those variables are likely affected by the fault, they are operating normally with respect to the control system dynamics and thus should not have been identified. Furthermore, some of these variables continue to be highlighted well after the issue is resolved. The (D)PCA models also only scantly detect the deviations in the later time that are correctly highlighted by the BRNN.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{220_realdata_fault2_brnn}
\caption{FDI by BRNN for Fault 2.
The red box indicates the period with the foaming issue as recorded by the operator.}
\label{fig:fault2:brnn}
\end{figure}
\subsubsection{Fault 2 results}%
\label{sec:real-data:fault2}
The BRNN FDI results for Fault 2 are shown in Figure \ref{fig:fault2:brnn}. The proposed method successfully detects the documented fault and identifying the related variables, as marked by time interval with the red box. For the earlier period around the $6000^{\mathrm{th}}$ time point, the BRNN model detects a disturbance and identifies the deviation in X14.PV, X15.PV, and X9.PV. The relative magnitude of the deviation during those periods is not as significant as that during the documented fault period. This assessment was then verified to be fully warranted by inspection of the recorded sensor measurements. Analysis of the period around the $15000^{\mathrm{th}}$ data point yield similar results.
\begin{figure}
\centering
\includegraphics[width=0.96\textwidth]{230_realdata_fault2_pca}
\caption{FDI by (D)PCA methods for Fault 2: (a) r-PCA, (b) f-PCA, (c) r-DPCA, and (d) f-DPCA.
The red box indicates the period with the foaming issue as recorded by the operator.}
\label{fig:fault2:pca}
\end{figure}
As before, the (D)PCA FDI methods are also applied. Their results are shown in Figure \ref{fig:fault2:pca}. As observed for Fault 1, the r-PCA and r-DPCA models are not as sensitive to the fault and only partially detect the documented fault period, and they did not detect the earlier event highlighted by the BRNN method. The identification plots by the r-PCA and r-DPCA models in Figures \ref{fig:fault2:pca}ac also only identify a limited number of faulty variables. The f-PCA and f-DPCA models are again overly sensitive for both FDI, flagging much of the data period. After the foaming issue occurred and the operator intervention, the control system is able to compensate for the disturbance after a while. However, f-PCA and f-DPCA models incorrectly continue to assess the system as in an abnormal state even though the foaming issue has been fully resolved. This can also be observed from the contribution plots in Figures \ref{fig:fault2:pca}bd, wherein variables X1.PV and X6.MV are identified as problematic during and long after the resolution of the fault.
To summarize, this case study on real data from a chemical process demonstrates the higher accuracy, specificity, and robustness in FDI of the BRNN-based method over (D)PCA-based methods. The proposed method is also shown to provide precise and easily interpretable results for prompt diagnosis and mitigation of fault events in real manufacturing processes.
\section{Conclusion}%
\label{sec:conclusion}
This article proposes a novel BRNN-based FDI method for manufacturing processes.
The proposed method simultaneously tackles three key challenges in modeling real process data: (1) concurrent spatio-temporal correlations, (2) nonlinearity, and (3) incomplete characterization of the uncertainty in process noise and dynamics.
The BRNN model addresses these challenges because of its probabilistic framework built on RNN models. And, for implementation efficiency, the inference is made using variational dropout, which both regularizes the NN during training and efficiently estimates the uncertainty as it evolves through time.
The uncertainty estimates of the BRNN model play a crucial role in FDI. By continuously estimating the uncertainty, the BRNN model provides adaptive confidence intervals that fully characterize the system dynamics based on the current and past information. As demonstrated here, the BRNN framework therefore enables:
\begin{enumerate}[(1)]
\item fault detection in processes with nonlinear dynamics, and
\item direct fault identification with easily interpreted identification plots and fault
propagation path analysis.
\end{enumerate}
The effectiveness of the proposed BRNN method is demonstrated in two case studies: (1) the benchmark TEP dataset and (2) a real chemical manufacturing dataset. The proposed method is compared to the widely applied PCA and DPCA methods, using either full and reduced dimension models. The comparisons show that the BRNN model provides results that are accurate and more specific and directly relevant for fault identification. Furthermore, based on its results, one can distinguish the nature of the faults, between controllable, back to control, or uncontrollable faults.
More broadly, the application of Bayesian methods to fault detection is not a widely explored field. To that end, this paper demonstrates a novel framework involving the systematic application of spatio-temporal models with Bayesian estimation such that the posterior inference results are directly relevant for detection and identification. In this case, a BRNN is used, but the strategy could be adapted for other spatio-temporal models such as dynamic process models.
The proposed BRNN-based FDI framework can be directly applied to any manufacturing process with historical NOC measurements without significant modifications. Moreover, the easy implementation of variational dropout to any model architecture and concurrent online calculating capability make BRNN feasible for large-scale industrial applications.
Some considerations for future work might include:
\begin{enumerate}[(1)]
\item The online adaptation of the BRNN model for changing NOC. In real chemical processes, the process conditions evolve and it is unlikely that the training data can cover all of the NOC modes. Thus, online adaptation is crucial for reducing false alarms and maintenance costs.
\item While proposed here for FDI, the BRNN model framework also has broad potential applications in industrial manufacturing processes related to time series analysis. The variational dropout can be applied to any deep learning model without modification of the model architecture, which makes it a preferable probabilistic model as compared to other recent advanced techniques.
\end{enumerate}
\include{refs}
\end{document}
|
1108.2265
|
\section{Introduction}
An important emerging issue in contemporary cosmology is to search for
signatures of primordial non-Gaussianities in the Cosmic Microwave
Background (CMB). A single field inflation scenario with standard
kinetic term, slow roll potential, and standard vacuum initial
condition produces a nearly scale-invariant and nearly Gaussian
distribution of fluctuations in the CMB, while generalizations of the
simplest model, as well as alternatives to inflation, can lead to
non-Gaussianities with different sizes and shapes. Thus, a detection of
non-Gaussianity would be a significant discovery, providing strong
hints about the nature of inflationary or alternative models.
Among different types of non-Gaussianities, the most natural and well
studied one is the local shape non-Gaussianity,
\begin{equation}\label{local}
\Phi(x)=\Phi_g(x)+f_{NL}\left(\Phi_g(x)^2-\langle\Phi_g(x)^2\rangle\right)
+g_{NL}\Phi_g^3(x)+\cdots\ , \end{equation}
where $\Phi_g(x)$ is a field of Gaussian fluctuations. In this note,
we focus on the lowest order of
this local form non-Gaussianity, parameterized by
$f_{NL}$.
Currently there have been a number of methods to search for
non-Gaussian signatures, for example, bispectrum analysis
\cite{Komatsu:2001rj}, Minkowski functionals \cite{0067-0049-141-1-1},
and mode decomposition \cite{Fergusson:2006pr}. When applied to WMAP
\cite{Komatsu:2010fb} data with resolution of 0.21 degrees, current
constraints of non-Gaussianity are of order $-10<f_{NL}<74$
(95\% CL) from the bispectrum method. Current CMB experiments such as
Planck \cite{2010A&A...520A...1T} with resolution of 5 arcminutes, the
Atacama Cosmology Telescope \cite{Kosowsky:2004sw}
with resolution of 0.9 arcminutes, and the
South Pole Telescope \cite{Ruhl:2004kv} with
resolution of 0.25 arcminutes will result in improved constraints with
their expanded range of multipole moments and increased sensitivity
and resolution.
We will explore whether edge detection algorithms are efficient at
distinguishing non-Gaussian from Gaussian CMB skies. In recent years,
there has emerged an interest in applying the Canny algorithm
\cite{Canny:1986:ACA}, an edge detection algorithm which searches for
steep gradients in images, to cosmological data
\cite{2008JCAP...04..015A, 2009JCAP...02..009S,2010IJMPD..19..183D,
2010JCAP...02..033D,2010arXiv1012.3667F}. When applied to CMB
temperature maps, the Canny algorithm selects the steep gradients in
temperature and stores them as edges. For example, an edge map of a
temperature map in which the background possesses one temperature and
the area inside a circle possesses a different temperature would appear
as just the outline of the circle, since the edge of the circle is a
region with a steep gradient.
Since the temperature fluctuations of Gaussian and non-Gaussian maps
have different probability distributions, locally maximal gradients
occur at different locations in the maps. More concretely,
\cite{Pogosyan:2009rg,Pogosyan:2011qq}
have shown that the gradients of Gaussian and non-Gaussian maps have
different probability distributions; as edges are a subset of
gradients, we expect that the edge distribution will differ between
Gaussian and non-Gaussian maps. Heuristically,
\cite{Pogosyan:2009rg,Pogosyan:2011qq} also
demonstrated that the number of local temperature maxima increases (or
decreases) in a non-Gaussian map as compared to a Gaussian map, along
with a corresponding decrease (or increase) in the number of local
minima. Since one place edges will appear is along the steep
gradient between a local maximum and a local minimum, we have another
reason to expect a difference in edge statistics between Gaussian and
non-Gaussian maps. In this paper, we take an empirical approach to
ask how sensitive the Canny algorithm is to local-form
non-Gaussianity.\footnote{It is well known that the single
point probability distribution is not efficient enough to detect
non-Gaussianity. However, here by considering edges, spatial
correlation of data points are considered thus the information
contained in edge detection is more than that in the single point
probability distribution function.}
To study this
question, we have used the publicly available full sky maps provided by
Elsner and Wandelt \cite{Elsner:2009md}.
In this letter, we report preliminary results testing the sensitivity
of the Canny algorithm to non-Gaussianities of the local shape in the
CMB. We begin with a brief discussion of our simulated skies, then
review the Canny algorithm and our rough optimization of it, continue
with a discussion of our statistics and results, and conclude with
predictions related to current experiments and future simulations.
\section{Simulations}
A challenge in developing new methods for detecting CMB
non-Gaussianity is the production of simulated non-Gaussian CMB sky
maps evolved to the time of recombination with the appropriate
transfer function; this is a computationally intensive process.
Fortunately, the authors of \cite{Elsner:2009md} have provided the
spherical harmonic coefficients $a_{lm}$ for a thousand realizations
of full sky maps for both the linear and nonlinear components of the CMB
with a local form of non-Gaussianity. That is, they provide the set
of $a_{lm}$ for both the $\Phi_g$ and $\Phi_g^2$ terms in
(\ref{local}). These simulations include up to mulitpole moments of
$l=1024$; the spectrum of $C_l$ used in the simulations is available with
their simulations.
With these simulated maps, the work of constructing local shape
non-Gaussian maps reduces to a superposition between the Gaussian and
the non-Gaussian maps, with coefficient $f_{NL}$ in front of the
non-Gaussian maps. We anticipate future studies when simulations with
higher multipole moments and non-trivial trispectra are available.
Since our current implementation of the Canny algorithm requires the flat
sky approximation, we cut approximately 57 degree by 57 degree
Cartesian windows from these simulations. These window sizes are
compatible with the flat sky approximation. Any distortion due to the
Cartesian slicing and flat sky approximation applies equally to both the
Gaussian and non-Gaussian maps and hence does not affect our results.
As a check that our algorithm is not sensitive to differences between
independent Gaussian maps, we have compared two sets of 120 Gaussian
maps using the statistics described below. We find that there is a 96\%
chance that two sets of maps drawn from the same distribution would
have greater statistical differences, so they are
indistinguishable. In addition, we have checked that the non-Gaussian
maps and Gaussian maps have the same spectrum ($C_l$).
\begin{figure}
\center
\includegraphics[width=0.42\textwidth]{gaussianimage.pdf}
\caption{\label{fig:gaussianmap}
A sample of the Gaussian map, in a window of 57 degree by 57 degree
sky. }
\end{figure}
\begin{figure}
\center
\includegraphics[width=0.42\textwidth]{nongaussianfNL1000.pdf}
\caption{\label{fig:ngmap} A sample of the non-Gaussian map with
$f_{NL}=1000$, in a window of 57 degree by 57 degree sky. }
\end{figure}
\section{Review of Canny Algorithm}
We refer the reader to \cite{2010IJMPD..19..183D} for a complete
description of our implementation of the Canny algorithm. In brief,
we search for local maxima in the gradient
of the map along the direction of the gradient, in the following
steps:
\begin{enumerate}
\item Convert the temperature map to a gradient map,
recording magnitude and direction of the derivative at each pixel.
\item Scan the map along eight directions (vertically, horizontally,
and along both diagonals) retaining
only local maxima along the direction of the gradient.
\item Filter the remaining gradient map such that only
gradients with magnitudes between a lower threshold $t_l$ and a
cut-off threshold $t_c$ are kept.
\item Impose an upper threshold $t_u$. The remaining points
above this threshold are considered as belonging to an edge.
Points below this threshold are considered as belonging to an edge
only if they are connected to an edge in a direction perpendicular to their
gradient.
\item Count and store the numbers and lengths of edges to
perform statistical analysis.
\end{enumerate}
The thresholds are measured in units of a maximal gradient $G_m$, which
we define as the lesser of the averages of the maximum gradient magnitude
for all the Gaussian and non-Gaussian maps. Again, we refer the reader to
\cite{2010IJMPD..19..183D} for more details.
One significant change made for the above standard implementation
refers to Appendix A of \cite{2010IJMPD..19..183D}, which removes
doubles of locally maximal gradients. In choosing which doubled pixel
to discard, instead of choosing the pixel with lower temperature,
which artificially introduces more edges with lower temperature
fluctuations, we chose the pixel with the lower absolute value of the
temperature fluctuation.
In addition, we implemented a routine to optimize the thresholds used.
We sampled ten values of the threshold parameters and minimized the
quadratic fit to the probabilities for $f_{NL}=1000$ (see the
description of our statistics below). The final thresholds used were
$t_c=3.09414$, $t_u=0.257424$, and $t_l=0.104205$.
\section{Statistics and Results}
To differentiate statistically between sets of Gaussian and non-Gaussian
simulated images, we applied the same tests as described in
\cite{2010IJMPD..19..183D} for detection of cosmic strings. In brief, for
each CMB map (a window of 500 pixels on a side), we divided all the edges
into bins by edge length; all edges longer than a determined maximum length
$k$ were binned with that length. Then we found the distribution of all the
windows within each bin for both Gaussian and non-Gaussian maps.
At that point, we apply the Student t-test to the two distributions in
each bin. From the t-test we obtained
the $p$-values, which give probability information,
which we then combined using the Fisher combined probability test,
\begin{equation}
\chi_{2k}^2 = -2\sum_{l=1}^k \ln(p_l)\ ,
\end{equation}
to compute the total $\chi^2$ separating the two sets of maps, which follows
the $\chi^2$ distribution with $2k$ degrees of freedom.
We then find the probability that the two sets of CMB maps
could have that value of $\chi^2$ (or larger) if they were
drawn from the same larger distribution of maps. In
the following, we use an optimal value of $k=2$.
We applied the Canny algorithm and statistical analyses to 120 windows of
approximately 57 degrees (500 pixels) per side and an $f_{NL}=350$. Our
results indicate that there is a 0.1\% probability that the non-Gaussian
simulations and Gaussian simulations would have such a large value of
$\chi^2$ if drawn from the same distribution,
which would constitute a $3\sigma$ detection. We find similar statistical
significance distinguishing Gaussian simulations and simulations with
a negative non-Gaussianity parameter of $f_{NL}=-700$.
The statistics for the comparison of $f_{NL}=350$ and Gaussian simulations
are plotted in figure
\ref{fig:fnl350}, showing the distribution in each bin.
As a comparison, statistics for $f_{NL}=1000$ are also shown in figure
\ref{fig:fnl1000}. An astute reader will note that the Gaussian simulations
give different edge counts in the two figures; this is because they are
compared to simulations with different amounts of non-Gaussianity, resulting
in different values of the parameter $G_m$ defined above.
The reader will also note that our $3\sigma$ detection level is
asymmetric in $f_{NL}$; that is, it seems possible to distinguish smaller
$|f_{NL}|$ from Gaussian maps if the sign of $f_{NL}$ is positive. We
have checked that this is not due to a bias in the algorithm by reversing
the sign of all temperature fluctuations on a set of maps; these maps give
edge counts that are indistinguishable from the original set of maps.
It appears that the asymmetry is an intrinsic property of local-type
non-Gaussianity due to the nonlinearity of the fluctuations.
As the Canny algorithm is designed to detect edges, it may be possible to
optimize it for the study of primodial non-Gaussianity (for example,
by finding new statistics to differentiate Gaussian and non-Gaussian
skies). However, there are also prospects for improvement even with the
un-modified algorithm and statistics presented here. For example,
our method is
sensitive to the number of pixels in each window. A comparison of the same
non-Gaussian and Gaussian simulations in windows with 200 pixels per side
yields a probability of 16.3\%. Therefore, we expect that the better
resolutions offered by future simulations and experiments such as Planck,
ACT, and SPT should allow greater sensitivity.
In contrast, our results
were less sensitive to the number of windows considered.
When we applied the
Canny algorithm to 30 non-Gaussian realizations with $f_{NL}=350$ and
120 Gaussian realizations
(we have no upper constraints to the number of Gaussian realizations we can
produce), we obtained probabilities of 4\% that the maps were drawn from the
same distribution.
In addition, we found that reducing the number of Gaussian simulations
(to 30 Gaussian and 30 non-Gaussian simulations) resulted in
probabilities of 17\% for $f_{NL}=350$. Therefore, we conversely expect
that increasing the number of Gaussian simulations could improve our
results (concominant with improved resolution) due to the fact that the
standard error of the mean decreases with increasing sample size.
This offers another
potential avenue to improve the Canny algorithm's sensitivity to
local-type non-Gaussianity.
\begin{figure}
\center
\includegraphics[width=0.49\textwidth]{NGfNL350_lengthmax2_plot.pdf}
\caption{\label{fig:fnl350}
Edge statistics for $f_{NL}=350$. The blue (upper) dot represents the
non-Gaussian simulations, and the red (lower) dot represents the Gaussian
simulations. Error bars are the standard error of the mean (1$\sigma$).
}
\end{figure}
\begin{figure}
\center
\includegraphics[width=0.49\textwidth]{NGfNL1000_lengthmax2_plot.pdf}
\caption{\label{fig:fnl1000}
Edge statistics for $f_{NL}=1000$. The blue (upper) dot represents the
non-Gaussian simulations, and the red (lower) dot represents the Gaussian
simulations. Error bars are the standard error of the mean (1$\sigma$).
}
\end{figure}
\section{Conclusion}
We have shown that applying the Canny algorithm to segments of full
sky Gaussian and non-Gaussian maps can differentiate the two sets of
maps at the $3\sigma$ level down to $f_{NL}=350$ or $f_{NL}=-700$ (note
that current observational limits are quoted at the 95\% confidence level).
Since tests of our
application greatly improved for 500 pixels per window side compared
to 200 pixels per window side, we anticipate that implementation of
this algorithm on high resolution data should dramatically improve our
results. For example,
SPT \cite{Ruhl:2004kv} will provide
data with up to 2400 pixels per 10 degree side.
In particular, our tests
indicated that larger pixel numbers were more relevant to
substantially improved results than larger numbers of images. For
this reason, application of the Canny algorithm
to Planck \cite{2010A&A...520A...1T}, ACT \cite{Kosowsky:2004sw},
and SPT \cite{Ruhl:2004kv} data should also be very promising for
detection of primordial non-Gaussianities.
In the present note, we considered the local type bispectrum as the form
of non-Gaussianity of interest. However, our method is general and can be
performed on non-Gaussian maps with other types of non-Gaussianity,
namely bispectra with other shapes as well as a nontrivial trispectrum.
We especially
expect our estimator to be sensitive to the local shape
trispectrum, where the Gaussian distribution is deformed by kurtosis
instead of skewness. In this case the slope of the probability
distribution function is changed symmetrically, so more (less) edges
should be produced when the kurtosis is positive (negative) respectively.
On the other hand, it remains unclear whether we can distinguish
different types or shapes of non-Gaussianity or the contribution from
cosmic strings. We will leave the comparison between these signals to
future work.
In our current approach, the error bars in the figures are determined
numerically by simulations of Gaussian and non-Gaussian maps. It
remains interesting to see whether these error bars could also be
determined theoretically. In \cite{Pogosyan:2011qq}, the extrema
counts for CMB maps are calculated theoretically. It may be possible to
obtain analytical bounds on edge number statistics. The analytical extrema
counts may also suggest new types of statistical analysis to perform on
edge maps.
Another important issue that we have not addressed in the present note
is to add noise to the simulated maps. By adding noise, according to
the sensitivities of WMAP, Planck, SPT or ACT, we could tell what
value of $f_{NL}$ could be detected in the corresponding
experiments using the Canny algorithm. We hope to address this issue
in the future.
Finally, the Canny algorithm is originally developed to detect edges
instead of non-Gaussianity. Thus, although we have shown that the
algorithm is sensitive to non-Gaussianity, we expect there is considerable
potential to optimize the algorithm, such as through the identification of
a new statistic to distinguish Gaussian and non-Gaussian maps. Therefore,
we are optimistic about the potential of the Canny algorithm for development
as an estimator of non-Gaussianity in the CMB.
\vspace{.5cm}
\begin{acknowledgments}
We would like
to thank R.~Brandenberger, G.~Holder, L.~LeBlond, and A.~ van Engelen
for useful discussions. RJD thanks the University of Winnipeg Department
of Physics for hospitality during the completion of this work, as well
as K.~Dasgupta and M.~Venditti for their support.
\end{acknowledgments}
|
1108.2243
|
\section{Introduction}
The role of local regularity for nonconvex minimization problems or nonmonotone variational inequalities
is well-established. In broad terms, a generalized equation
is said to be ``regular" (or ``metrically regular") if the distance from a proposed
solution to an exact solution can be bounded by a constant multiple of the
model error of the proposed solution. A particular focus has been the
proximal point algorithm and alternating projections
\cite{Ara05, Ius03, Pen02, LewisLukeMalick08}.
It is often the case, however, that the problems in question are
{\em ill-posed}; in other words, there is no constant of proportionality
between the model error and the distance of an approximate solution
to the true solution. For some algorithms such an ill-posedness would not
prevent the iterates from converging to a {\em best approximate solution}, but
numerical performance will suffer. An example of such behavior
can be observed with the classical alternating projection algorithm of von
Neumann \cite{Neumann49} applied to a general {\em feasibility problem:} that is,
the problem of finding the intersection of sets. Ill-posedness for feasibility
problems can be characterized by problem {\em inconsistency}, that is,
the nonexistence of an intersection of the sets in question. More
generally, the feasibility problem will be ill-posed if the intersection
vanishes under arbitrarily small perturbations of the sets.
For the applications we have in mind, at least
one of the sets in question comes from a finite precision measurement or
calculation. It is quite reasonable to expect an inconsistency between the idealized
model and the measured data, which can be represented as a perturbation of the
idealized data set. When only two convex sets
are involved, alternating projections can be shown to converge to
{\em nearest points} \cite[Theorem 4]{CheneyGoldstein59},
however the {\em rate} of convergence will in general be
arbitrarily slow. For
other algorithms ill-posedness leads to instability in the sense that the iterates
do not converge to a fixed point. The Douglas Rachford
algorithm, for example, applied to inconsistent feasibility problems has no fixed points
\cite{LionsMercier79,BCL3, Luke08}.
Insofar as ill-posed problems can be {\em regularized}, the
theory cited above can be applied to numerical methods for the regularized problems. Our
focus here is on a particular regularization for ill-posed feasibility problems and efficient
{\em approximate} projection algorithms. The problem of nonconvex best approximation
was considered in \cite{Luke05a, Luke08} where the focus was on instability of the Douglas Rachford
algorithm resulting from problem inconsistency. A relaxation of this
algorithm was proposed that has fixed points for inconsistent problems and has been
successful in practice \cite{Marchesini07}. As is often the case for relaxed projection
algorithms, there is no systematic rule for choosing the relaxation parameter. It was shown
in \cite{Luke08} that the size of the relaxation parameter at the solution is related
to the optimal gap distance between the sets. This observation suggests a different
approach to algorithmic design that is based on regularization of the underlying
problem rather than stabilization of the algorithm as was the focus in \cite{Luke05a}.
We further develop this viewpoint here, where we study {\em local} regularization of the
underlying problem while retaining the character of the original problem. In
particular, we expand one of the sets in order to
create an intersection with all the desired regularity properties described in
\cite{LewisLukeMalick08}.
The strategy is a local regularization in the sense that indicator functions
are still used as the central penalty function, in
contrast to \cite{Luke08} where the indicator function was relaxed to a distance
function. One then can apply any number of algorithms for finding the intersection
of regularized sets. We are particularly interested in projection algorithms and specifically
the classical alternating projection algorithm.
We show in section \ref{s:problem} that, for the problems of interest to us, such a
regularization of the sets results in a significant increase in the complexity of computing the
corresponding projections. To address computational complexity of the regularized problem
we consider approximate alternating projections based on the projection
operators of the original, unregularized problem. An approximate algorithm
is stated in section \ref{s:inexact alternating projections}.
We prove local linear convergence of this
algorithm to a solution of the regularized problem under regularity
assumptions that are natural for regularized problems. In section
\ref{s:regularized feasibility} we apply a specific
approximation motivated in section \ref{s:problem} to the approximate
projection algorithm and prove that this
approximation is guaranteed to succeed under certain conditions.
We demonstrate the effectiveness of this approximation in section \ref{s:numerics} with
an example from diffraction imaging with real experimental data. We do not claim
that the approximate alternating projection algorithm is the best, or even a very good
strategy for solving this particular problem. However to our knowledge, our analysis
yields the first mathematically sound stopping criteria for alternating projections
applied to the phase retrieval problem. Our goal is to demonstrate the theory
and to motivate the adaptation of our proposed regularization and approximation
to more sophisticated projection algorithms.
\section{Notation, Definitions and Basic Theory}
\label{s: notaion}
We begin with basic theory and notation. For the most part, we present
only the results with pointers to the literature for interested readers.
The setting we consider is finite dimensional Euclidean space $\mathbb{E}$.
The closed unit ball centered at $x$ is denoted by ${\Bbb}(x)$; when it is centered at the origin, we
simply write ${\Bbb}$. We denote the open interval from $a$ to $b$ by $(a,b)$; the closed
interval is denoted as usual by $[a,b]$.
Given a set $C \subset \mathbb{E}$, we define the {\em distance function} and (multivalued) {\em projection} for $C$ by
\begin{eqnarray*}
d_C(x) & = & d(x,C) ~=~ \inf \{ \|z-x\| : z \in C \} \\
P_C(x) & = & \argmin \{ \|z-x\| : z \in C \}.
\end{eqnarray*}
If $C$ is closed, then the projection is nonempty.
Following \cite[Definition 1.6]{Mor06} we define the {\em normal cone} to a closed set
$C\subset \mathbb{E}$ as follows:
\begin{defn}[normal cone]
\label{d:normal cone}
A vector $v$ is {\em normal} to a closed set $C\subset\mathbb{E}$ at
${\overline{x}}$, written $v\in N_C({\overline{x}})$
if there are sequences $x^k\to{\overline{x}}$ and $v^k\to v$ with
\[
v^k\in\set{t(x^k-z)}{t\geq 0,~z\in P_C(x^k)}\quad\mbox{ for all }k\in\mathbb{N}.
\]
The vectors $v^k$ are {\em proximal normals} to $C$ at $z\in P_C(x^k)$ and
the cone of proximal normals at $z$ is denoted $N^P_C(z)$.
\end{defn}
It follows immediately from the definition that the normal cone is a closed multifunction:
for any sequence of points
$x^k \rightarrow {\overline{x}}$ in $C$,
any limit of a sequence of normals $v^k \in N_C(x^k)$ must lie in $N_C({\overline{x}})$.
The relation of the projection to the normal cone is also evident from the definition:
\begin{equation}\label{e:PC to NC}
z \in P_C(x) ~~\Rightarrow~~ x-z \in N_C(z).
\end{equation}
Notice too that $N_C(x)=\{0\}\iff x\in \intr C$.
\begin{defn}[basic set intersection qualification]
\label{d:strong regularity}
A family of closed sets $C_1$,$C_2,\ldots$ $C_m$ $\subset \mathbb{E}$
satisfies the basic set intersection qualification at a point ${\overline{x}} \in \cap_i C_i$,
if the only solution to
\[
\displaystyle{\sum_{i=1}^m} y_i = 0,\quad
y_i \in N_{C_i}({\overline{x}}) ~~ (i=1,2,\ldots,m)
\]
is $y_i = 0$ for $i=1,2,\ldots,m$. We say that the intersection is {\em strongly regular} at ${\overline{x}}$
if the basic set constraint qualification is satisfied there.
\end{defn}
In the case $m=2$, this condition can be written
\[
N_{C_1}(\bar x) \cap - N_{C_2}(\bar x) =\{0\}.
\]
The two set case is is called the {\em basic constraint qualification for sets} in \cite[Definition 3.2]{Mor06} and has its origins
in the the {\em generalized property of nonseparability} \cite{Mord84} which is the $n$-set case. It was later recovered as
a dual characterization of what is called {\em strong regularity} of the intersection in \cite[Proposition 2]{Kru06}.
This property was called {\em linear regularity} in \cite{LewisLukeMalick08}.
The case of two sets also yields the following simple quantitative characterization of strong regularity.
\begin{propn}[Theorem 5.16 of \cite{LewisLukeMalick08}]
\label{t:cbar}
Suppose that $C_1$ and $C_2$ are closed subsets of $\mathbb{E}$. The intersection
$C_1\cap C_2$ satisfies the basic set intersection qualification at ${\overline{x}}$ if and only if the constant
\begin{equation}\label{e:cbar}
{\overline{c}} ~\equiv~ \max \set{\ip{u}{v}}{u \in N_{C_1}({\overline{x}}) \cap {\Bbb},~ v \in -N_{C_2}({\overline{x}}) \cap {\Bbb}}<1.
\end{equation}
\end{propn}
\noindent
\begin{defn}[angle of regular intersections]
We say that the intersection $C_1\cap C_2$ is
{\em strongly regular at ${\overline{x}}$ with angle ${\overline{\theta}}\equiv\cos^{-1}({\overline{c}})>0$}
where ${\overline{c}}$ is given by \eqref{e:cbar}.
\end{defn}
In order to achieve linear rates of convergence of alternating projections to
the intersection of sets, we
require pointwise strong regularity of the intersection \cite{LewisLukeMalick08}.
In the absence of this property the above definitions suggest a general regularization
philosophy: {\em promote strong regularity}. This
is most obviously achieved by augmenting at least one of the sets
by some $\epsilon$ ball: $C_1(\epsilon)=C_1+\epsilon{\Bbb}$, for instance.
Similar ideas been used extensively in the development of proximally smooth
sets by Clarke, Stern and Wolenski \cite{ClarkeSternWolenski95}.
We pursue this idea in section \ref{s:problem} with the generalization that the
ball, or ``tube'' around the set of interest is with respect to a generic distance in the image
space of a continuous mapping, the tube having no relation to the native space in which
the projectors onto the sets are defined.
Somewhat stronger results are possible when the sets have additional
regularity. We call a set $C \subset \mathbb{E}$ is {\em prox-regular} at a point
${\overline{x}} \in C$ if the projection mapping $P_C$ is single-valued around ${\overline{x}}$ \cite{PolRockThib00}.
Convex sets, in particular, are prox-regular. More generally,
any set defined by $C^2$ equations and inequalities is prox-regular at any
point satisfying the Mangasarian-Fromovitz constraint qualification, for instance.
\begin{propn}[angle of normals of prox-regular set]
\label{t:prox angle}
Suppose the set $C \subset \mathbb{E}$ is prox-regular at the point ${\overline{x}} \in C$.
Then for any constant $\delta > 0$, any points $y,z \in C$ near ${\overline{x}}$
and any normal vector $v \in N_C(y)$ satisfy the inequality
\[
\ip{v}{z-y} \le \delta\|v\| \cdot \|z-y\|.
\]
\end{propn}
\begin{proof}
This is a special case of the same property for
{\em super regular sets} (\cite[Definition 4.3]{LewisLukeMalick08}
and \cite[Proposition 4.4]{LewisLukeMalick08}) since
by \cite[Proposition 4.9]{LewisLukeMalick08} prox-regularity implies
super regularity.
Alternatively, for prox-regular sets we can proceed directly
from \cite[Proposition 1.2]{PolRockThib00} which shows that,
for any sequences of
points $y^k,z^k \in C$ converging to ${\overline{x}}$ and any
corresponding sequence of normal vectors
$v^k \in N_C(y^k)$,
there exist
constants $\epsilon, \rho > 0$ such that
\[
\ip{\frac{\epsilon}{2\|v^k\|} v^k}{z^k - y^k}
\le \frac{\rho}{2} \|z^k - y^k\|^2
\]
for all large $k$. Since for any fixed $\delta>0$ we will eventually have
$\|z^k - y^k\| \le \frac{\delta \epsilon}{\rho}$, it
follows that
\[
\ip{v^k}{z^k-y^k} \leq \delta\|v^k\| \cdot \|z^k-y^k\|
\]
for $k$ large enough.
\end{proof}
The next result builds upon Proposition \ref{t:prox angle} and
provides bounds on the angle between sets in the neighborhood
of a point in a strongly regular intersection of a closed and a prox-regular set.
In \cite[Theorem 5.2]{LewisLukeMalick08} implications \eqref{e:cond1} and
\eqref{e:cond2} are used to characterize sets for which linear convergence of the alternating projections algorithm holds.
We do not seek such generality here
and are content with identifying classes of sets which satisfy these conditions,
namely prox-regular sets. The proof of the following assertion can be found
in the proof of Theorem 5.16 of \cite{LewisLukeMalick08}.
\begin{propn}
\label{t:bones}
Let $M,C \subset \mathbb{E}$ be closed. Suppose that $C$ is prox-regular
at a point ${\overline{x}} \in M \cap C$ and that $M$ and $C$ have strongly
regular intersection at ${\overline{x}}$ with angle ${\overline{\theta}}$. Define
${\overline{c}}\equiv\cos({\overline{\theta}})$ and
fix the constant $c'$ with ${\overline{c}}<c'<1$.
There exists a constant $\epsilon > 0$
such that
\begin{equation} \label{e:cond1}
\left.
\begin{array}{cc}
x \in M \cap ({\overline{x}} + \epsilon {\Bbb}), & u \in -N_M(x) \cap {\Bbb} \\
y \in C \cap ({\overline{x}} + \epsilon {\Bbb}), & v \in N_C(y) \cap {\Bbb}
\end{array}
\right\}
~~\implies~~ \ip{u}{v} \le c',
\end{equation}
and, for some constant $\delta \in [0,\frac{1-c'}{2})$,
\begin{equation}\label{e:cond2}
\left.
\begin{array}{rcl}
y,z & \in & C \cap ({\overline{x}} + \epsilon {\Bbb}) \\
v & \in & N_C(y) \cap {\Bbb}
\end{array}
\right\}
~~\implies~~ \ip{v}{z-y} \le \delta\|z-y\|.
\end{equation}
\end{propn}
In what follows, we define an approximate alternating projection algorithm in terms of
the distance of the normal cone associated with the approximate projection to the
``true'' normal cone. In order to guarantee that for our proposed approximation we
can get arbitrarily close to the true projection, we need the notion of convergence
of the associated normal cone mappings. Let $\mmap{S}{\mathbb{E}}{\mathbb{Y}}$ denote
a set-valued mapping where $\mathbb{Y}$ is another Euclidean space. We define the
domain of $S$ to be the set of points whose image is not empty, that is
\[
\dom S\equiv\set{x}{S(x)\neq\emptyset}.
\]
Following \cite[Definition 4.1]{VA} we define continuous set-valued mappings relative to
some subset $D$ as those which are both outer and inner semicontinuous
relative to $D$.
\begin{defn}[continuity of set-valued mappings]
\label{d:set continuity}
A set-valued mapping $\mmap{S}{\mathbb{E}}{\mathbb{Y}}$ is continuous at a point
${\overline{x}}\in D$ relative to $D\subset\mathbb{E}$ if
\begin{eqnarray*}
&&\!\!\!\!\!\!\!\!\!S({\overline{x}})\subset\\
&& \!\!\! \set{y}{\forall~x^k\attains{D} {\overline{x}}, ~\exists~ K>0 \mbox{ such that for }k>K,~ y^k\to y \mbox{ with } y^k\in S(x^k) }
\nonumber
\end{eqnarray*}
and
\[
\set{y}{\exists~x^k\attains{D} {\overline{x}},~\exists~ y^k\to y~\mbox{ with }y^k\in S(x^k)}
\subset S({\overline{x}})
\]
where $\attains{D}$ indicates that the sequence lies within $D$. We denote this as
$S(x)\to S({\overline{x}})$ for all sequences $x\attains{D}{\overline{x}}$.
\end{defn}
\section{The problem}
\label{s:problem}
In this section we formulate our abstract problem and
motivate the regularization and approximation strategies
that we propose.
Our initial, naive problem formulation involves finding points $x\in C\subset\mathbb{E}$,
a Euclidean space,
that explain some measurement $b\in \mathbb{Y}$ modeled as the image of
the continuous mapping $\map{g}{\mathbb{E}}{\mathbb{Y}}$, that is
\[
\mbox{Find }x\in C\cap M_0
\]
for
\[
M_0\equiv \set{x\in\mathbb{E}}{g(x)=b}.
\]
The set $C$ usually captures a qualitative feature of solutions, such
as nonnegativity, or a prescribed support.
If $b$ is a physical/empirical measurement, it is likely
that the intersection is empty, or that the solution consists only of
extremal points.
In the case of measurements with discrepancies modeled by statistical
noise, the noise could be Gaussian or
Poisson distributed (among still other possibilities). To accommodate
a variety of instances we consider
the following regularizations of the set $M_0$:
\begin{equation}\label{e:fatM}
M_\epsilon\equiv \set{x\in\mathbb{E}}{d_\phi(g(x),b)\leq \epsilon}
\end{equation}
where $\epsilon\geq 0$ and $d_\phi$ is a {\em Bregman distance}
defined by
\[
d_\phi(z,y)\equiv \phi(z)-\phi(y)-\phi'(y)(z-y)
\]
for $\map{\phi}{\mathbb{Y}}{(-\infty,+\infty]}$ strictly convex and differentiable on $\intr(\dom \phi)$ .
The Bregman distance with $\phi\equiv \frac12\|\cdot\|^2$ corresponds to the
Euclidean norm which is appropriate for Gaussian noise. If $\mathbb{Y}={\Rbb^m}$ and
\[
\phi(y) = \sum_{j=1}^m h(y_j)\quad\mbox{ for }
h(t)\equiv\begin{cases}t\log t - t &\mbox{ for }t>0\\
0& \mbox{ for }t=0\\
+\infty& \mbox{ for }t<0
\end{cases}
\]
then the Bregman distance leads to the {\em Kullback-Leibler divergence},
\begin{equation}\label{e:KL}
d_\phi(z,y)=KL(x,y)\equiv \sum_{j=1}^m z_j\log\frac{z_j}{y_j}+y_j-z_j.
\end{equation}
The Kullback-Leibler divergence is appropriate for Poisson noise.
\begin{remark}
The regularization \eqref{e:fatM} bears some resemblance to closed neighborhoods of the
type $X(\epsilon)\equiv\set{x}{d(x,X)\leq \epsilon}$ considered by Clarke, Stern and
Wolenski \cite{ClarkeSternWolenski95} in their development of proximally smooth
sets, except that the neighborhood around the set of interest is with respect to a
generic distance in the image space of a continuous mapping, the neighborhood having no relation
to the metric upon which the projectors onto the sets are defined.
Still, we will rely on prox-regularity of the regularized set for the
approximation strategy discussed in Section \ref{s:regularized feasibility}.
\endproof
\end{remark}
Regardless of the distance, the first algorithm we consider for finding this
intersection is the classical
alternating projection algorithm.
\begin{alg}[exact alternating projections]\label{alg:exact ap}
\begin{description}
\item Choose $x^0 \in C$.
For $k=1,2,3,\ldots$
generate the sequence $\{x^{2k}\}\subset C$ with
$
x^{2k} \in P_C(x^{2k-1})
$
where the sequence $\{x^{2k+1}\}$ consists of points
$x^{2k+1}\in P_{M_\epsilon}(x^{2k})$.
\end{description}
\end{alg}
\noindent We show next that the projection onto the fattened set $M_\epsilon$
could be considerably more costly to calculate than
for the unregularized set $M_0$. This motivates the approximate
projection algorithm studied in section \ref{s:inexact alternating projections}
We want to compute
\[
x^*\in P_{M_\epsilon}({\widehat{x}})\equiv \argmin_{x\in M_\epsilon}\tfrac12 \|x-{\widehat{x}}\|^2.
\]
Assume $d_\phi(g({\widehat{x}}),b)> \epsilon$, then we seek a solution on the
$\epsilon$-sphere
around $b$ with respect to $d_\phi$. This is an instance of a {\em trust region}
problem.
Suppose that ${\overline{x}}\in P_{M_\epsilon}({\widehat{x}})$
and that the standard constraint qualification holds, that is
\begin{equation}\label{e:Vanderslice}
-\nabla d_\phi(g({\overline{x}},b))^*\eta=0, \quad \eta\geq 0\quad\implies\quad \eta=0.
\end{equation}
Then
\begin{eqnarray}\label{e:KKT1}
({\overline{x}}-{\widehat{x}}) + \nabla d_\phi(g({\overline{x}}),b)^*{\overline{\eta}}&=&0\qquad({\overline{\eta}}\geq 0)\\
d_\phi(g({\overline{x}}),b)-\epsilon&=&0.
\label{e:KKT2}
\end{eqnarray}
These are the standard KKT conditions (see, for example \cite[Theorem 10.6]{VA}).
Numerical methods for computing the projection $P_{M_\epsilon}({\widehat{x}})$ involve
solving a possibly large-scale nonlinear system of equations with respect to $x$ and $\eta$;
this could well be as difficult to solve as the original problem.
\begin{eg}[affine subspaces]
\label{eg:linear}
Let $\mathbb{E}={\Rbb^n}$, $\mathbb{Y}={\Rbb^m}$ with $m<n$. Take $g$ to be the linear mapping
$\map{A}{{\Rbb^n}}{{\Rbb^m}}$ and $d_\phi(x,y)=\frac{1}{2}\|x-y\|^2$
(that is, $\phi(x)=\frac12\|x\|^2$).
The projection can then be written as the solution to a quadratically
constrained quadratic program:
\[
\cmin{\tfrac12 \|x-z\|^2}{x\in {\Rbb^n}}{\frac{1}{2}\|Ax-b\|^2\leq \epsilon.}
\]
For small problem sizes this can be efficiently solved via interior point methods.
Still, even the most efficient numerical methods cannot compare to computing the
projection onto the affine space $M_0\equiv \set{x\in \mathbb{E}}{Ax=b}$ which has the
trivial closed form
\[
P_{M_0}(z) = (I-A^T(AA^T)^{-1}A)z + A^T(AA^T)^{-1}b.
\]
This suggests an alternative strategy for computing the
projection onto the ``fattened'' set.
Indeed, we can efficiently compute the projection $P_{M_\epsilon}(z)$
as a convex combination of the
points $y=P_{M_0}(z)$ and $z$
\[
x^*= \lambda_\epsilon z + (1-\lambda_\epsilon)y
\]
where $\lambda_\epsilon\in [0,1)$ solves $\frac{1}{2}(1-\lambda)^2\|z-y\|^2=\epsilon$.
This also has a closed form: the quadratic formula. For general Bregman distances
such shortcuts are not available, but this forms the basis for our approximations.
$\Box$
\end{eg}
\begin{eg}[boxes]
Let $\mathbb{E}=\mathbb{Y}={\Rbb^n}$. Define $\map{g}{{\Rbb^n}}{{\Rbb^n}}$ by
\[
g(x)=\left(|x_1|^2,\dots,|x_n|^2\right)^T
\]
and, again, let the distance $d_\phi$ be the standard
normalized squared Euclidean distance to some point $b\in {\Rbb^n_+}$.
The projection can then be written as the solution to the nonconvex optimization
problem
\[
\cmin{\tfrac12 \|x-{\widehat{x}}\|^2}{x\in {\Rbb^n}}{\frac12\sum_{j=1}^n (|x_j|^2-b_j)^2=\epsilon.}
\]
Notice that the corresponding set
$M_\epsilon$ is not convex: the origin is projected in the positive and negative direction
in each component. Generally, nonconvex problems are hard to solve.
On the other hand, the projection onto the box with length $2b$,
$y = (y_1,\dots,y_n)^T\in P_{M_0}({\widehat{x}})$, is trivial and has the form
\[
y_j \begin{cases}=
b_j\frac{{\widehat{x}}_j}{|{\widehat{x}}_j|}& {\widehat{x}}_j \neq 0\\
\in\{-b_j,b_j\}&{\widehat{x}}_j=0.
\end{cases}
\]
See \cite{BurkeLuke03} for analysis of this projection in higher dimensional
product spaces.
For this example there is no shortcut to computing the projection $P_{M_\epsilon}$ for $\epsilon>0$,
but we show below that the convex combination of the projection of ${\widehat{x}}$ onto
$M_0$ and ${\widehat{x}}$ is a effective approximation that still yields linear rates of
convergence for the method of alternating projections for finding the intersection
of $M_\epsilon\cap C$.
$\Box$
\end{eg}
\begin{remark}\label{r:911}
We note that in both of the above examples the constraint
qualification \eqref{e:Vanderslice} is no longer satisfied in
the limit $\epsilon=0$ for the set $M_\epsilon$. This obviously
does not prevent us from calculating the projection onto the set
$M_0$. Indeed, as we showed, the projection sometimes even has
an explicit representation.
\endproof
\end{remark}
\section{Inexact alternating projections}
\label{s:inexact alternating projections}
There is more than one way to formulate inexact algorithms. One template
for this is to add summable error terms to the operators involved in the
exact algorithm. Another approach --
the one we take here -- is less general but
has a more geometric appeal. More to the point, it is appropriate for our
intended application.
Given two iterates $x^{2k-1} \in M$ and $x^{2k} \in C$, a necessary condition
for the new iterate $x^{2k+1}$ to be an exact projection on $M$, that is
$x^{2k+1} \in P_M(x^{2k})$, is
\[
\|x^{2k+1} - x^{2k}\| \le \|x^{2k} - x^{2k-1}\|
~~\mbox{and}~~
x^{2k} - x^{2k+1} \in N_M(x^{2k+1}).
\]
In a modification of \cite{LewisLukeMalick08} we assume only that we choose the odd
iterates $x^{2k+1}$
to satisfy a relaxed version of this condition, where we replace the second
part by the assumption that the distance of
the normalized direction of the current step to the normal cone to $M$ at the intersection
of the boundary of $M$ with the line
segment between $x^{2k+1}$ and $x^{2k}$ is small.
Consider the following inexact alternating projection iteration for
finding the intersection of two sets $M,C \subset \mathbb{E}$.
\begin{alg}[inexact alternating projections]\label{alg:inexact ap}
\begin{description}
\item Fix $\gamma>0$ and choose $x^0 \in C$ and $x^1 \in M$.
For $k=1,2,3,\ldots$
generate the sequence $\{x^{2k}\}\subset C$ with
$
x^{2k} \in P_C(x^{2k-1})
$
where the sequence $\{x^{2k+1}\}\subset M$ satisfies
\begin{subequations}\label{e:pooping}
\begin{eqnarray}
&& \|x^{2k+1} - x^{2k}\| \le \|x^{2k}-x^{2k-1}\|,\label{e:pooping_a}\\
&&x^{2k+1} = x^{2k}\quad\mbox{ if }~x_*^{2k+1}=x^{2k},\label{e:pooping_b} \\
\mbox{ and }&& d_{N_M(x_*^{2k+1})}({\widehat{z}}^k)\le \gamma\label{e:pooping_c}
\end{eqnarray}
\end{subequations}
for
\[
x_*^{2k+1}= P_{M\cap\{x^{2k}-\tau{\widehat{z}}^k,~\tau\ge 0\}}(x^{2k})
\]
and
\[
{\widehat{z}}^k\equiv \begin{cases}
\frac{x^{2k} - x^{2k+1}}{\|x^{2k} - x^{2k+1}\|} &\mbox{ if }~x_*^{2k+1}\neq x^{2k}\\
0&\mbox{ if }~ x_*^{2k+1}=x^{2k}.
\end{cases}
\]
\end{description}
\end{alg}
Note that the odd iterates $x^{2k+1}$ can lie on the interior of $M$.
This is the major difference between Algorithm \ref{alg:inexact ap} and the one specified in
\cite{LewisLukeMalick08} where all of the iterates are assumed to lie on the boundary
of $M$. We include this feature to allow for {\em extrapolated} iterates in the case
where $M$ has interior. Extrapolation, or over relaxation, is a common
technique for accelerating algorithms, though its basis is rather
heuristic. Empirical experience reported in the literature shows that extrapolation can be quite
effective (see \cite{Spingarn85, Combettes97b}). The algorithm given in Theorem \ref{t:implementation}
below explicitly includes extrapolation. Our numerical results at the end of this paper do
not contradict the conventional experience with extrapolation.
Lemma \ref{t:Elephant} and Theorem \ref{t:approx proj} below were
sketched in \cite[Theorem 6.1]{LewisLukeMalick08} for the variation of
Algorithm \ref{alg:inexact ap} just described.
\begin{lemma}\label{t:Elephant}
Let $M,C \subset \mathbb{E}$ be closed. Suppose that $C$ is prox-regular
at a point ${\overline{x}} \in M \cap C$ and that $M$ and $C$ have strongly
regular intersection at ${\overline{x}}$ with angle ${\overline{\theta}}$. Define
${\overline{c}}\equiv\cos({\overline{\theta}})$ and
fix the constants $c$ with ${\overline{c}}<c<1$
and $\gamma < \sqrt{1-c^2}$. Then there is an $\epsilon>0$ such that
the iterates of Algorithm
\ref{alg:inexact ap} satisfy
\begin{equation}\label{e:Sesamstrasse}
\left.\begin{array}{cc}
\|x^{2k+1}-{\overline{x}}\|&\leq\frac{\epsilon}{2}\\
\|x^{2k+1}-x^{2k}\|&\leq\frac{\epsilon}{2}
\end{array}\right\}~~\implies~~\|x^{2k+2}-x^{2k+1}\|\leq \eta
\|x^{2k+1}-x^{2k}\|
\end{equation}
for $\eta= c\sqrt{1-\gamma^2}+\gamma\sqrt{1-c^2}<1$.
\end{lemma}
\begin{proof}
Fix $c'$ with ${\overline{c}}< c'<c<1$ and define
$
\delta = \tfrac12\left(\eta - \eta'\right)
$
where $\eta' = c'\sqrt{1-\gamma^2}+\gamma\sqrt{1-c'^2}$.
($\delta>0$ since, as is easily verified,
$c\sqrt{1-\gamma^2}+\gamma\sqrt{1-c^2}$ increases monotonically
with respect to $c$ on $[0,1]$.) Since $C$ is prox-regular at ${\overline{x}}$ and
the intersection is strongly regular, by Proposition \ref{t:bones}
for this $\delta$ there is an $\epsilon>0$ such that
implications \eqref{e:cond1} and \eqref{e:cond2} hold. We apply this
result here.
The assumptions and the triangle inequality yield
\begin{equation}\label{e:playing}
\|x^{2k}-{\overline{x}}\|\leq \|x^{2k}-x^{2k+1}\|+\|{\overline{x}}-x^{2k+1}\|\leq \epsilon.
\end{equation}
By the definition of $x_*^{2k+1}$ we have $x_*^{2k+1} = (1-\lambda)x^{2k}+\lambda x^{2k+1}$
for some $\lambda\in [0,1]$ so that
\begin{eqnarray}
\|x^{2k+1}_* - {\overline{x}}\| &=& \|\lambda (x^{2k+1}-{\overline{x}})+(1-\lambda)(x^{2k}-{\overline{x}})\|\nonumber\\
&\leq&\lambda\|x^{2k+1}- {\overline{x}}\| + (1-\lambda)\|x^{2k}- {\overline{x}}\|\nonumber\\
&\leq&\lambda\frac\epsilon2 +(1-\lambda)\epsilon\leq\epsilon\quad(\lambda\in[0,1])
\label{e:at home}
\end{eqnarray}
where the last inequality combines the left hand side of\eqref{e:Sesamstrasse} and \eqref{e:playing}.
Next, by the triangle inequality and the definition of the projection
\begin{eqnarray}
\|x^{2k+2}-{\overline{x}}\|&\leq& \|x^{2k+2}-x^{2k+1}\| + \|{\overline{x}}-x^{2k+1}\|\nonumber\\
&\leq& \|x^{2k}-x^{2k+1}\|+ \|{\overline{x}}-x^{2k+1}\|\leq \epsilon.
\label{e:Rolling}
\end{eqnarray}
If $x^{2k+1}=x^{2k}$
then this is a fixed point of the algorithm and the result is trivial. Similarly, if
$x^{2k+1}=x^{2k+2}$, then $x^{2k+1}\in C\cap M$ and by the first condition in
\eqref{e:pooping} this is a fixed point of the algorithm. So we assume that
$x^{2k+1}\neq x^{2k}$ and define ${\widehat{w}}\in N_M(x^{2k+1}_*)$ with $\|{\widehat{w}}\|=1$ and
${\widehat{u}}\equiv\frac{x^{2k+2}-x^{2k+1}}{\| x^{2k+2}-x^{2k+1} \|}$.
Now applying Proposition \ref{t:bones} to $x^{2k+1}_*$ satisfying \eqref{e:at home}
with $-{\widehat{w}}\in -N_M(x^{2k+1}_*)\cap {\Bbb}$ and to $x^{2k+2}$
satisfying \eqref{e:Rolling} with $-{\widehat{u}}\in N_C(x^{2k+2})\cap {\Bbb}$ we have that
for $\epsilon $ small enough
\begin{equation}
\label{e:sleep}
\ip{{\widehat{w}}}{{\widehat{u}}} =\ip{-{\widehat{w}}}{-{\widehat{u}}}\leq c'.
\end{equation}
In other words the angular separation between the unit vectors ${\widehat{w}}$ and
${\widehat{u}}$ is bounded below by $\arccos c'$.
On the other hand,
define
\[
{\widehat{z}} \equiv \frac{x^{2k} - x^{2k+1}}{\|x^{2k} - x^{2k+1}\|}.
\]
Our goal is to obtain a lower bound the angle between ${\widehat{z}}$ and ${\widehat{u}}$.
If it were the case that $x^{2k+1} \in P_M(x^{2k})$ then ${\widehat{z}}={\widehat{w}}$ and
$c'$ would already be our bound. But since $x^{2k+1}$
only approximates the projection, we must work a little harder.
Since the iterates satisfy \eqref{e:pooping}, for some $w \in N_M(x^{2k+1}_*)$ we have
$\|w-{\widehat{z}}\| \le \gamma$.
There are two cases to consider. If ${\widehat{z}}=0$, then we are done. Otherwise
${\widehat{z}}$ has length one, and
\[
\frac{\|w\|^2 + 1- \gamma^2}{2\|w\|}\leq\ip{\frac{w}{\|w\|}}{{\widehat{z}}}.
\]
Maximizing the left hand side as a function of $\|w\|\in [1-\gamma,1+\gamma]$
yields the largest possible angular separation from ${\widehat{z}}$, that is
\begin{equation}\label{e:Ella}
\ip{{\widehat{w}}}{{\widehat{z}}}\geq \sqrt{1-\gamma^2}
\end{equation}
where ${\widehat{w}} = \frac{w}{\|w\|}$.
Note that
$\gamma<\sqrt{1-c^2}<\sqrt{1-c'^2}$ for $c'<c$ so that $c'<\sqrt{1-\gamma^2}$.
Thus, combining \eqref{e:sleep} and \eqref{e:Ella}, we have
\begin{eqnarray*}
\ip{{\widehat{w}}}{{\widehat{z}}}\geq\sqrt{1-\gamma^2}&>&c'\geq\ip{{\widehat{w}}}{{\widehat{u}}}\\
&\iff&\\
\arccos\ip{{\widehat{w}}}{{\widehat{z}}}\leq \arccos(\sqrt{1-\gamma^2})&<&\arccos c'\leq \arccos\ip{{\widehat{w}}}{{\widehat{u}}}.
\end{eqnarray*}
It follows immediately, then, that
\begin{eqnarray*}
0&<&\arccos c' - \arccos(\sqrt{1-\gamma^2})\\
&<& \arccos\ip{{\widehat{w}}}{{\widehat{u}}}-\arccos\ip{{\widehat{w}}}{{\widehat{z}}}
\leq \arccos\ip{{\widehat{u}}}{{\widehat{z}}}
\end{eqnarray*}
which is equivalent to
\[
\ip{{\widehat{z}}}{{\widehat{u}}} ~\le~ \cos \Big( \arccos c' - \arccos(\sqrt{1-\gamma^2}) \Big)
~=~
c'\sqrt{1-\gamma^2} + \gamma \sqrt{1-c'^2}<1.
\]
Letting $\eta' = c'\sqrt{1-\gamma^2} + \gamma \sqrt{1-c'^2}$ and
removing the normalization yields
\begin{equation}\label{e:Hermann}
\ip{x^{2k}-x^{2k+1}}{x^{2k+2}-x^{2k+1}}\leq \eta'\|x^{2k}-x^{2k+1}\|
\|x^{2k+2}-x^{2k+1}\|.
\end{equation}
Now by our choice of $\epsilon$, implication \eqref{e:cond2} holds
for $x^{2k}$ and $x^{2k+2}\in C\cap\{{\overline{x}}+\epsilon {\Bbb}\}$ with
$-{\widehat{u}}\in N_C(x^{2k+2})\cap{\Bbb}$, namely
\[
\ip{-{\widehat{u}}}{x^{2k}-x^{2k+2}}\leq \delta\|x^{2k+2}-x^{2k}\|
\]
which is equivalent to
\[
\ip{x^{2k+2}-x^{2k+1}}{x^{2k+2}-x^{2k}}\leq \delta\|x^{2k+2}-x^{2k}\|
\|x^{2k+2}-x^{2k+1}\|.
\]
By the triangle inequality and the definition of the projection
\[
\|x^{2k+2}-x^{2k}\|\leq \|x^{2k+2}-x^{2k+1}\|+
\|x^{2k+1}-x^{2k}\|\leq 2\|x^{2k+1}-x^{2k}\|
\]
so that
\begin{equation}\label{e:Foege}
\ip{x^{2k+2}-x^{2k+1}}{x^{2k+2}-x^{2k}}\leq 2\delta\|x^{2k+1}-x^{2k}\|
\|x^{2k+2}-x^{2k+1}\|.
\end{equation}
Adding \eqref{e:Hermann} and \eqref{e:Foege} yields
\[
\|x^{2k+2}-x^{2k+1}\|^2\leq \left( 2\delta +\eta'\right)
\|x^{2k+1}-x^{2k}\|
\|x^{2k+2}-x^{2k+1}\|,
\]
which by our construction of $\delta$ yields
\[
\|x^{2k+2}-x^{2k+1}\|\leq \eta \|x^{2k+1}-x^{2k}\|
\]
as claimed.
\end{proof}
\begin{lemma}\label{t:Orton}
With the same assumptions as Lemma \ref{t:Elephant}, choose $x^0$ and $x^1$ so that
\begin{equation}\label{e:sunny}
\|x^1-{\overline{x}}\|\leq \|x^0-{\overline{x}}\|=\beta<\frac{1-\eta}{4}\epsilon
\end{equation}
where
$\epsilon$ is chosen to satisfy \eqref{e:Sesamstrasse}.
Let $\eta= c\sqrt{1-\gamma^2}+\gamma\sqrt{1-c^2}$. Then for all $k\geq 0$
\begin{subequations}
\begin{eqnarray}
\|x^{2k+1}-{\overline{x}}\|&\leq & 2\beta\frac{1-\eta^{k+1}}{1-\eta}<\frac{\epsilon}{2},
\label{e:dining}\\
\|x^{2k+1}-x^{2k}\|&\leq & \beta\eta^k<\frac{\epsilon}{2}~~\mbox{ and }
\label{e:room}\\
\|x^{2k+2}-x^{2k+1}\|&\leq & \beta\eta^{k+1}.
\label{e:table}
\end{eqnarray}
\end{subequations}
If in addition $M$ is prox-regular at ${\overline{x}}$, then for all $k\geq 0$
\begin{subequations}
\begin{eqnarray}
\|x^{k+1}-{\overline{x}}\|&\leq & 2\beta\frac{1-\eta^{k+1}}{1-\eta}<\frac{\epsilon}{2},
\label{e:dining_prox}\\
\|x^{k+1}-x^{k}\|&\leq& \beta\eta^k<\frac{\epsilon}{2}~~\mbox{ and }
\label{e:room_prox}\\
\|x^{k+2}-x^{k+1}\|&\leq & \beta\eta^{k+1}.
\label{e:table_prox}
\end{eqnarray}
\end{subequations}
\end{lemma}
\begin{proof}
The proof is by induction. For the case $k=0$ inequality \eqref{e:dining} holds
trivially. Inequality \eqref{e:room} follows from the triangle
inequality and \eqref{e:sunny}. Inequality \eqref{e:table}
then follows from \eqref{e:dining}, \eqref{e:room} and
Lemma \ref{t:Elephant}. Since for the case $k=0$ inequalities
\eqref{e:dining}-\eqref{e:table} are equivalent to \eqref{e:dining_prox}-\eqref{e:table_prox}
this case is true whether $M$ is prox-regular or not.
To show that these relations hold for $k+1$ with $M$ {\em not} prox-regular, note
that by \eqref{e:pooping}
$\|x^{2k+3}-x^{2k+2}\|\leq \|x^{2k+2}-x^{2k+1}\|$. In light of
\eqref{e:table} this implies
\begin{equation}\label{e:room2}
\|x^{2k+3}-x^{2k+2}\|\leq \beta\eta^{k+1}<\frac{\epsilon}{2}.
\end{equation}
This together with \eqref{e:dining} and \eqref{e:table}, yields
\begin{eqnarray}
\|x^{2k+3}-{\overline{x}}\|&\leq& \|x^{2k+3}-x^{2k+2}\| + \|x^{2k+2}-x^{2k+1}\|+
\|x^{2k+1}-{\overline{x}}\|\nonumber\\
&\leq& \beta\eta^{k+1} + \beta\eta^{k+1}+
2\beta\frac{1-\eta^{k+1}}{1-\eta}\nonumber\\
&\leq &2\beta\eta^{k+1} +2\beta\frac{1-\eta^{k+1}}{1-\eta} =
2\beta\frac{1-\eta^{k+2}}{1-\eta}< \frac{\epsilon}{2}.
\label{e:dining2}
\end{eqnarray}
Now, Lemma \ref{t:Elephant} applied to \eqref{e:room2} and
\eqref{e:dining2} yields
\begin{equation}\label{e:table2}
\|x^{2k+4}-x^{2k+3}\|\leq \eta\|x^{2k+3}-x^{2k+2}\| \leq
\beta\eta^{k+2}.
\end{equation}
As \eqref{e:room2}-\eqref{e:table2} are just \eqref{e:dining}-\eqref{e:table}
with $k$ replaced by $k+1$, this completes the induction and the proof for the
case where $M$ is not prox-regular.
If we assume, in addition, that $M$ is prox-regular, then by \eqref{e:table_prox}
\begin{equation}\label{e:room_prox2}
\|x^{k+2}-x^{k+1}\|\leq \beta\eta^{k+1}<\frac{\epsilon}{2}.
\end{equation}
This together with \eqref{e:dining_prox} yields
\begin{eqnarray}
\|x^{k+2}-{\overline{x}}\|\leq \|x^{k+2}-x^{k+1}\| +\|x^{k+1}-{\overline{x}}\|\nonumber\\
&\leq& \beta\eta^{k+1}+\beta\frac{1-\eta^{k+1}}{1-\eta}\nonumber\\
&=&\beta\frac{1-\eta^{k+2}}{1-\eta}\leq \frac{\eta}{2}
\label{e:dining_prox2}
\end{eqnarray}
Now Lemma \ref{t:Elephant} with the rolls of $C$ and $M$ reversed, together
with \eqref{e:table_prox} yields
\begin{equation}\label{e:table_prox2}
\|x^{k+3}-x^{k+2}\|\leq \eta\|x^{k+2}-x^{k+1}\|\leq \beta\eta^{k+2}
\end{equation}
Again, since \eqref{e:room_prox2}-\eqref{e:table_prox2} are just \eqref{e:dining_prox}-\eqref{e:table_prox}
with $k$ replaced by $k+1$, this completes the induction and the proof.
\end{proof}
\begin{thm}[convergence of inexact alternating projections]
\label{t:approx proj}
Let $M,C \subset \mathbb{E}$ and suppose $C$ is prox-regular at a point
${\overline{x}} \in M \cap C$. Suppose furthermore that $M$ and $C$ have strongly
regular intersection at ${\overline{x}}$ with angle ${\overline{\theta}}$. Define
${\overline{c}}\equiv\cos({\overline{\theta}})<1$ and
fix the constants $c \in ({\overline{c}},1)$
and $\gamma < \sqrt{1-c^2}$.
For $x^0$ and $x^1$ close enough to ${\overline{x}}$, the iterates in Algorithm
\ref{alg:inexact ap} converge to a point in
$M \cap C$ with R-linear rate
\[
\sqrt{ c\sqrt{1-\gamma^2} + \gamma \sqrt{1-c^2} } ~<~ 1.
\]
If, in addition, $M$ is prox-regular at ${\overline{x}}$, then
the iterates converge with R-linear rate
\[
c\sqrt{1-\gamma^2} + \gamma \sqrt{1-c^2} ~<~ 1.
\]
\end{thm}
\noindent
\begin{proof}
We prove in detail the case where $M$ is {\em not} assumed to be prox-regular.
Choose $x^0$ and $x^1$ so that \eqref{e:sunny} holds with
$\epsilon$ is chosen as in Lemma \ref{t:Elephant}.
Let $\eta= c\sqrt{1-\gamma^2}+\gamma\sqrt{1-c^2}$.
To establish convergence of the sequence we
check that the iterates form a Cauchy sequence. To see this, note that for any
integer $k=0,1,2,\dots$ and any integer $j>2k$, by \eqref{e:room} and
\eqref{e:table} of Lemma \ref{t:Orton} we have
\begin{eqnarray*}
\|x^j-x^{2k}\|&\leq& \sum_{i=2k}^{j-1}\|x^{i+1}-x^i\|\\
&\leq&\beta\left(\eta^k +2\eta^{k+1} +2\eta^{k+2} +\dots\right)\\
&\leq&\beta\frac{1+\eta}{1-\eta}\eta^k
\end{eqnarray*}
Similarly, it can be shown that
\[
\|x^{j+1}-x^{2k+1}\|\leq \beta\frac{\eta^{k+1}}{1-\eta}
\]
So the sequence is a Cauchy sequence and converges to some ${\widehat{x}}\in\mathbb{E}$.
The fixed point of the sequence must belong to $M\cap C$ and satisfies
\[
\|{\widehat{x}}-x^0\|\leq \beta\frac{1+\eta}{1-\eta}.
\]
Moreover, for all $j=0,1,2,\dots$
\[
\|{\widehat{x}}-x^j\|\leq \beta\eta^{j/2}\frac{1+\eta}{1-\eta}.
\]
We conclude that convergence is R-linear with rate $\sqrt{\eta}$ as claimed.
The proof for the case where $M$ is also prox-regular at ${\overline{x}}$ proceeds
analogously using inequalities \eqref{e:dining_prox}-\eqref{e:table_prox}
of Lemma \ref{t:Elephant} instead.
\end{proof}
Note that the worse the approximation to the
projection, the slower the convergence. As we showed in the previous section,
the projection onto the unfattened set can be easier (sometimes {\em much} easier)
to compute than the projection onto the fattened set, so although the rate of convergence suffers
from taking only an approximate projection,
we gain in the per-iteration complexity of calculating the projections.
\section{Approximate alternating projections onto fattened sets}
\label{s:regularized feasibility}
\begin{thm}\label{t:implementation}
Let $\mathbb{E}$ and $\mathbb{Y}$ be Euclidean spaces, and $\map{\phi}{\mathbb{Y}}{(-\infty,+\infty]}$ be lsc,
strictly convex and differentiable on $\intr(\dom \phi)$.
Let $C\subset\mathbb{E}$ be closed and $M_\epsilon \cap C\neq \emptyset$ for all $\epsilon\geq 0$
where $M_\epsilon$ is defined by
\[
M_\epsilon\equiv \set{x\in\mathbb{E}}{f\equiv d_\phi(g(x),b)\leq \epsilon}
\]
for $d_\phi$ a Bregman distance to $b\in \mathbb{Y}$
and $\map{g}{\mathbb{E}}{\mathbb{Y}}$ continuous with $\range g\subset \dom\phi$ and
\[
\liminf_{|x|\to \infty}\frac{d_\phi(g(x),b)}{|x|}>0.
\]
Suppose that there is a ${\overline{\theta}}_0>0$ and, for all $\epsilon>0$ small enough,
a point ${\overline{x}}_0 \in M_0 \cap C$ with nearby points ${\overline{x}}_\epsilon$ at which the
intersection $M_\epsilon \cap C$ is strongly regular with
angle ${\overline{\theta}}_\epsilon\geq {\overline{\theta}}_0>0$.
Suppose further that $C$ and $M_\epsilon$
are prox-regular on a neighborhood of ${\overline{x}}_0$ and that $M_\epsilon$
has nonzero proximal normals at all boundary points within this neighborhood.
Define
${\overline{c}}_0\equiv\cos({\overline{\theta}}_0)<1$ and
fix the constants $c \in ({\overline{c}}_0,1)$
and $\gamma < \sqrt{1-c^2}$.
Compute the sequence $\{x^{k}\}$ via
Algorithm \ref{alg:inexact ap} with the odd iterates generated by
\begin{equation}\label{e:odd iterates}
x^{2k+1}=(1-\lambda_k)x^{2k} +\lambda_k x^{2k+1}_0
\end{equation}
for $x^{2k+1}_0\in P_{M_0}(x^{2k})$ and $\lambda_k>0$.
For $x^0$ and $x^1$ close enough to ${\overline{x}}_0$, there exist $\{\lambda_k\}>0$
and $\epsilon>0$ such that for all $k\in \mathbb{N}$ the iterates satisfy \eqref{e:pooping_a},
and \eqref{e:pooping_c} with $\{x^{2k+1}\}\in M_\epsilon$, and
the sequence of points converges to a point in $M_\epsilon \cap C$ with at least R-linear rate
\[
c\sqrt{1-\gamma^2} + \gamma \sqrt{1-c^2} ~<~ 1.
\]
\end{thm}
The odd iterates of the proposed algorithm do not necessarily lie on the surface of the regularized set
$M_\epsilon$, but could be on the interior of this set. Were we computing true projections, all the
odd iterates would lie on the boundary of $M_\epsilon$ -- instead we take {\em larger} steps than the
projections would indicate. In this sense, the algorithm defined in
Theorem \ref{t:implementation} is a regularized approximate alternating projection with
{\em extrapolation}. The theorem does not tell us what such extrapolation buys us, but at least it says
that we will not do any worse than without it.
We begin next developing the groundwork for the proof of Theorem \ref{t:implementation}.
\begin{lemma}[level-boundedness]
\label{t:level-bounded}
Let $\mathbb{E}$ and $\mathbb{Y}$ be Euclidean spaces, and $\map{\phi}{\mathbb{Y}}{(-\infty,+\infty]}$ be lsc,
strictly convex and differentiable on $\intr(\dom \phi)$.
Define the function $f\equiv d_\phi(g(\cdot),b)$ where $d_\phi(y,b)$ is
the Bregman distance of $y$ to the point $b\in \dom \phi$ and
the function $\map{g}{\mathbb{E}}{\mathbb{Y}}$ is continuous with $\range g\subset \dom\phi$ and
satisfies
\begin{equation}\label{e:level coercive}
\liminf_{|x|\to \infty}\frac{d_\phi(g(x),b)}{|x|}>0.
\end{equation}
Then the lower level sets of $f$, $\set{x\in\mathbb{E}}{f(x)\leq \alpha}$ for fixed $\alpha\in \mathbb{R}$, are
compact. In particular, the set $\argmin f$ is nonempty and compact and $\inf f=\min f\geq 0$.
\end{lemma}
\begin{proof}
For easy reference we recall the definition of the Bregman distance:
\[
f(x)\equiv d_\phi(g(x),b) = \phi(g(x))-\phi(b) - \ip{\phi'(b)}{g(x)-b}.
\]
Since $\range g\subset \dom\phi$ and $b\in \dom \phi$ there is an $x\in\mathbb{E}$ at which
$f(x)<\infty$. Moreover, since $\phi$ is convex, the Bregman distance is bounded
below by $0$, hence $\inf f\geq 0$ and $f$ is {\em proper}
(that is, not everywhere equal to infinity, and does not take the value $-\infty$ on $\mathbb{E}$).
Also $f$ is lsc as the composition of the
sum of a lsc function $\phi$ and a linear function $\ip{\phi'(b)}{\cdot}$ with a continuous
function $g$. The lower level sets of $f$ are therefore closed (see for instance \cite[Theorem 1.6]{VA}).
The coercivity condition \eqref{e:level coercive} then implies that the lower level-sets are {\em bounded}
\cite[Corollary 3.27]{VA}, thus the lower level sets are compact and $\argmin f$ is nonempty and compact.
\end{proof}
\begin{thm}[continuity of the level set mapping]
\label{t:continuity}
Let $f\equiv d_\phi(g(\cdot),b)$ with $\phi$, $g$, $b$ and $d_\phi$
as in Lemma \ref{t:level-bounded}. The corresponding
level-set mapping
\begin{equation}\label{e:fatM2}
M(\alpha)\equiv \set{x\in\mathbb{E}}{f(x)\leq \alpha}
\end{equation}
is continuous on
$[{\overline{\epsilon}},\infty)$ where
${\overline{\epsilon}}\equiv\min f$.
\end{thm}
\begin{proof}
By Lemma \ref{t:level-bounded}
$M(\cdot)$ is compact and $\dom M(\cdot) = [{\overline{\epsilon}},\infty)\subset[0,\infty)$.
Consequently the graph of $M(\cdot)$ is closed
(in fact, closed-valued) in $\mathbb{E}\times\mathbb{R}$ and satisfies
\begin{equation}\label{e:osc}
\set{y}{\exists~\alpha^k\to {\overline{\alpha}},~\exists~ y^k\to y~\mbox{ with }y^k\in M(\alpha^k)}
\subset M({\overline{\alpha}})\quad\mbox{ for all } {\overline{\alpha}}\in \mathbb{R}.
\end{equation}
On the other hand,
the inverse of the level-set mapping (the epigraphical profile mapping)
\[
M^{-1}(x)\equiv\set{\alpha\in\mathbb{R}}{\alpha\geq f(x)}
\]
maps open sets to open sets relative to $[{\overline{\epsilon}},\infty)$, that is
$M^{-1}(O)$ is open relative to $[{\overline{\epsilon}},\infty)$ for every open
set $O\subset\mathbb{E}$. Thus by \cite[Theorem 5.7]{VA}
the level set mapping satisfies
\begin{eqnarray}\label{e:isc}
&&\!\!\!\!\!\!\!\!\!M({\overline{\alpha}})\subset\\
&& \!\!\! \set{y}{\forall~\alpha^k\attains{{[{\overline{\epsilon}},\infty)}} {\overline{\alpha}}, ~\exists~ K>0 \mbox{ such that for }
k>K,~ y^k\to y \mbox{ with } y^k\in M(\alpha^k) }
\nonumber
\end{eqnarray}
for all ${\overline{\alpha}}\geq {\overline{\epsilon}}$.
Since the right hand side of \eqref{e:isc} is a subset of the left hand side of \eqref{e:osc}
we have equality of these limiting procedures, and thus continuity of $M(\cdot)$ on
$[{\overline{\epsilon}},\infty)$ according to Definition \ref{d:set continuity}.
\end{proof}
\begin{propn}\label{t:projection convergence}
Let $f\equiv d_\phi(g(\cdot),b)$ with $\phi$, $g$, $b$ and $d_\phi$
be as in Lemma \ref{t:level-bounded} and let $M(\alpha)$ be
defined by \eqref{e:fatM2}.
For $\{\alpha^k\}\subset [{\overline{\epsilon}},\infty)$ with
$\alpha^k\to {\overline{\alpha}}$ where
${\overline{\epsilon}}\equiv\min f$,
the corresponding sequence of projections onto $M(\alpha^k)$,
$P_{M(\alpha^k)}$, converges graphically to $P_{M({\overline{\alpha}})}$,
that is
\[
\gph P_{M(\alpha^k)}\to \gph P_{M({\overline{\alpha}})}.
\]
\end{propn}
\begin{proof}
Since $M(\alpha^k)\to M({\overline{\alpha}})$ by Theorem \ref{t:continuity}, graphical
convergence of the projection mapping follows from a minor
extension of \cite[Proposition 4.9]{VA} (see \cite[Example 5.35]{VA}).
\end{proof}
In light of the discussion in section \ref{s:problem}, our numerical strategy for approximating
the projection to the regularized set $M_\epsilon$ defined by \eqref{e:fatM} will be to
compute the intersection of the boundary of $M_\epsilon$ with line segment between the
current iterate and the projection onto the unregularized set $M_0$. Specifically,
for $x\notin M_\epsilon$ we
define $x_0=P_{M_0}(x)$ and calculate the point
\begin{equation}\label{e:approximate projection}
x_\epsilon\equiv (1-\tau_\epsilon) x+\tau_\epsilon x_0
\quad\mbox{ where } \quad\tau_\epsilon\equiv\min\{\tau>0~|~ (1-\tau) x+\tau x_0\in M_\epsilon\}.
\end{equation}
The next proposition shows that this approximation can achieve any specified
accuracy for sets with a certain regularity.
This will then be used to guarantee that the approximation to the projection given by
\eqref{e:approximate projection}
satisfies \eqref{e:pooping_c} on neighborhoods of a fixed point of Algorithm
\ref{alg:inexact ap}.
\begin{propn}[uniform normal cone approximation]\label{t:uniform approximation}
Let ${\overline{\epsilon}}>0$ and $M_\epsilon$ $(\epsilon\in[0,{\overline{\epsilon}}])$ be defined by
\eqref{e:fatM}.
Let $x_0\in M_0$, and $(x_0+\rho {\Bbb})\cap(\mathbb{E}\setminus M_{\overline{\epsilon}})\neq \emptyset$
for $\rho>0$ fixed.
In addition to the assumptions of Lemma \ref{t:level-bounded},
suppose that $M_\epsilon$ $(\epsilon\in[0,{\overline{\epsilon}}])$
is prox-regular at all points
$x\in (x_0+\rho {\Bbb})\cap M_\epsilon$ with nonzero proximal normals
at points $x\in \left[(x_0+\rho {\Bbb})\cap M_\epsilon\right]\setminus\intr(M_\epsilon)$.
Then given any $\gamma>0$
there exists an $\epsilon'\in(0,{\overline{\epsilon}}]$ such that for all
$\epsilon\in (0,\epsilon']$
\begin{equation}\label{e:pooping_cp}
d_{N_{M_\epsilon}(z_\epsilon)}\left(\frac{z-z_0}{\|z-z_0\|}\right)<\gamma
\end{equation}
holds where $z_0=P_{M_0}(z)$, $z_\epsilon$ is given by \eqref{e:approximate projection}
and $z$ is any point near
$(x_0+\rho {\Bbb})\cap M_\epsilon$.
\end{propn}
\begin{proof}
Since for all $\epsilon\in[0,{\overline{\epsilon}}]$ the sets $M_\epsilon$ are prox-regular on
$(x_0+\rho {\Bbb})\cap M_\epsilon$,
all nonzero proximal normals to $M_\epsilon$ can be realized by an $r$-ball
on open neighborhoods of points on $(x_0+\rho {\Bbb})\cap M_\epsilon$ for $r$
small enough \cite[Theorem 1.3.f]{PolRockThib00}. There is thus a ball
with radius $r_\epsilon>0$ on which the nonzero proximal normals to $M_\epsilon$ can
be realized uniformly on $(x_0+\rho {\Bbb})\cap M_\epsilon$. Also by assumption, the
proximal normal cones to all points on the boundary of
$(x_0+\rho {\Bbb})\cap M_\epsilon$ are nonzero. Thus, by Definition \ref{d:normal cone}
the normal cone to $M_\epsilon$ at all points on the boundary of
$(x_0+\rho {\Bbb})\cap M_\epsilon$ can be identified with the projection of points
$z$ in a $r_\epsilon$-neighborhood of this boundary. The result then follows from
Proposition \ref{t:projection convergence},
identifying the level set mapping $M(\epsilon)$ with
the parameterized set $M_\epsilon$.
\end{proof}
\begin{remark}
We conjecture that the assumption of prox-regularity and nontriviality of the proximal
normal can be relaxed. The assumptions of Lemma
\ref{t:level-bounded} are used to guarantee graphical convergence of the projection mappings; the
issue here is that the points on the boundary of the $M_\epsilon$ generated by
\eqref{e:approximate projection} do not have to correspond to projections.
Prox-regularity, and more restrictive still, the nontriviality of the proximal normals to $M_\epsilon$
on the boundary is used, in essence, locally to guarantee the reverse implication of \eqref{e:PC to NC}.
Definition \ref{d:normal cone} only relies on the {\em existence} of sequences of proximal normals whose
limits constitute the normal cone. Our approximation scheme \eqref{e:approximate projection},
in contrast, generates a specific
sequence of points, which could conceivably correspond only to zero proximal normals without further
assumptions on the regularity of $M_\epsilon$,
though we are unaware of a counterexample.
That prox-regularity alone is not enough to assure that the proximal normal cone is nonzero is
nicely illustrated by the set $M=\set{x\in {\Rbb^2}}{x_2\geq x_1^{3/5}}$ which is prox-regular at the origin,
but has only a zero proximal normal cone there (see \cite[Fig. 6-12.]{VA}).
Obviously, such regularity will depend on the distance $d_\phi$ and the mapping $g$ used in the
construction of $M_\epsilon$. \endproof
\end{remark}
{\em Proof of Theorem \ref{t:implementation}.}
We show first that there are $\lambda_k>0$ such that for any $\epsilon\geq 0$
the iterates $x^{2k+1}$ lie in $M_\epsilon$ and
satisfy \eqref{e:pooping_a}. Consider $\lambda_k=1$ for all $k$. Then
$x^{2k+1}=x_0^{2k+1}\in M_\epsilon$ for all $k$ and all $\epsilon\geq 0$
and by the definition of the projection
\[
\|x_0^{2k+1}-x^{2k}\|\leq \|x^{2k}-x_0^{2k-1}\|
\]
which suffices to prove the claim.
For existence of $\epsilon>0$ such that \eqref{e:pooping_c} is satisfied, note that
for all $\epsilon$ sufficiently small
${\overline{c}}_\epsilon \equiv \cos({\overline{\theta}}_\epsilon)\leq\cos({\overline{\theta}}_0) \equiv {\overline{c}}_0<1$ so that
the choice of $c\in ({\overline{c}}_0,1)$ satisfies $c\in({\overline{c}}_\epsilon,1)$ and consequently a fixed
$\gamma < \sqrt{1-c^2}$ suffices for all $\epsilon$ sufficiently small.
The result then follows immediately
from Proposition \ref{t:uniform approximation}. The assumptions of Theorem \ref{t:approx proj}
then apply to guarantee linear convergence, which completes the proof.
\endproof
\begin{remark}\label{r:slither}
The theorem above guarantees convergence of Algorithm \ref{alg:inexact ap} with
approximation strategy given by \eqref{e:odd iterates} for instances where the
intersection of the unregularized problem need not be strongly regular. When the unregularized
problem is inconsistent the strategy may fail. In particular,
suppose that $M_0\cap C=\emptyset$. Then for some ${\overline{\epsilon}}$ the intersection
$M_\epsilon\cap C=\emptyset$ for all $\epsilon<{\overline{\epsilon}}$. If $\gamma$ is such
that \eqref{e:pooping_c} is only satisfied for $\epsilon<{\overline{\epsilon}}$, then the
proposed approximation will fail.
To the degree that the coupling between the regularization parameter $\epsilon$ and
$\gamma$ is {\em weak}, we can still obtain positive results. One instance
where the coupling is very weak is if the fattened set has interior and ${\overline{x}}$ is
some point in this interior. In this case ${\overline{c}} =0$ in \eqref{e:cbar}, $\gamma$ can
be arbitrarily close to $1$ and the condition
\eqref{e:pooping_c} is almost trivial to satisfy. This is indeed the case for our
intended application. Of course, the closer $\gamma$ is to $1$, that is, the
worse our approximation of the true projection, the slower the convergence;
so the trade off between efficient computations and rates of convergence
must be balanced. The addition of
extrapolation to the approximate algorithm is meant to mitigate
any adverse effects of the approximation. The effectiveness of
extrapolation is illustrated in the following section.
\endproof
\end{remark}
\section{An example from diffraction imaging}
\label{s:numerics}
We present an application of the theory developed here to
image reconstruction from laser diffraction experiments produced
at the Institute for X-Ray Physics at the University of
G\"ottingen. Shown in Figure \ref{f:setup} is the observed diffraction
image produced by an object resembling a coffee cup that has been
placed in the path of a helium-neon laser. The imaging model is
\begin{equation}\label{e:diffraction}
|Fx|^2 = b
\end{equation}
where $b\in {\Rbb^n}$ is the observed image intensity, $F$ is a discrete
Fourier transform, $|\cdot|^2$ is the componentwise (pixelwise) modulus-squared,
and $x\in \mathbb{C}^n$ is the object to be found. The
image is corrupted by noise modeled by a Poisson distribution. In the
context of \eqref{e:fatM} the solution we seek lies in the fattened set
\begin{equation}\label{e:X-ray set}
M_\epsilon\equiv\set{x}{KL(|Fx|^2, b)\leq \epsilon}
\end{equation}
for $KL(x,y)$ the Kullback-Leibler divergence given by \eqref{e:KL}.
This set can be shown to be prox-regular everywhere with nonzero
proximal normals at all points on the boundary.
To this, we add the qualitative constraint that the object is nonnegative
(that is, real) and lies within a specified support: for a given index set
$\mathbb{J}\subset\{1,2,\dots,n\}$
\[
C\equiv\set{x\in{\Rbb^n}_{\!\!\!\!+}}{x_j=0 \mbox{ for }j\in\mathbb{J}} .
\]
This set is not only prox-regular, but in fact convex.
Despite the good features of these sets, the problem is inconsistent/ill-posed.
The set $C$ is a set of real vectors, but the observation $b$ is corrupted
by noise. If $b$ is not symmetric, as happens to be the case here,
then the image cannot come from a real-valued object. Sometimes
practitioners will ``preprocess'' the data by symmetrizing the raw data. If
this is done, then the corresponding feasibility problem is provably consistent,
and the results of Theorem \ref{t:implementation} can be applied.
In the numerical examples below, however, we choose to keep closer to the
true nature of the experiment and demonstrate the success of
Algorithm \ref{alg:inexact ap} as prescribed by Theorem \ref{t:approx proj} despite
the absence of guarantees that the condition \eqref{e:pooping_c} is satisfied.
\begin{figure}
\centering
\includegraphics[width=0.52\linewidth]{fig1_imaging_data}
\caption{\label{f:setup}
Diffraction image of real object.
}
\end{figure}
The state of the art for iterative methods for solving this problem can be found in
\cite{Marchesini07}. The main problem for these algorithms is the absence of
a stopping criterion. Often what is done in practice is one algorithm (often
the Douglas Rachford algorithm or variants \cite{BCL1, BCL2, Luke05a}) is used
to get close to a solution, and then alternating projections is used to refine the
image according to the ``eye-ball'' norm. In the application literature alternating
projections is often known as the ``Error Reduction'' algorithm.
Different communities have different
opinions as to what constitutes a stopping criteria, but in our reading of the application
literature, seldom do the proposed criteria involve iterates approaching a numerical fixed
point. Typical behavior of alternating projections onto the unregularized
problem, together with the corresponding reconstruction are shown in Figure
\ref{f:convention}. The true object was a coffee cup, which can be seen, upside down, in the lower
right hand corner of the reconstruction in Figure \ref{f:convention}, with the handle on
the left hand side. The reconstruction of the true object is only unique up to rotations, shifts and
reflections. This is why the reconstruction is upside down relative to the true object.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]%
{fig2_unregularized_CDI}
\caption{\label{f:convention}
Reconstruction and behavior of odd and even iterates of unregularized
(data set $M_0$ given by \eqref{e:X-ray set}) exact alternating
projections applied to the diffraction imaging problem. Only $500$
iterations are shown. The algorithm appeared to find a best approximation
pair after about $24,000$ iterations. ($x^{2k}\to x^{2k+2}$ but $\|x^{2k}- x^{2k+1}\|$ is
bounded above zero.)
}
\end{figure}
Next we apply Algorithm \ref{alg:inexact ap} with the approximate projection
computed as in Theorem \ref{t:implementation} for different regularization
parameters $\epsilon$ and different step-length strategies. Figure \ref{f:regularized}
shows the reconstruction and behavior of iterates for $\epsilon=1.9$ and $\lambda_k$
chosen so that the iterates remain on the surface of the $M_\epsilon$ set.
Figure \ref{f:regularized_extrapolated}
shows the reconstruction and behavior of iterates for $\epsilon=1.2$ and $\lambda_k=1$
for all $k$.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth]{fig3_regularized_CDI}
\end{center}
\caption{\label{f:regularized}
Reconstruction and behavior of odd and even iterates of regularized
(data set $M_\epsilon$ given by \eqref{e:X-ray set} with $\epsilon=1.9$) inexact
alternating projections with $\lambda_k$ chosen so that the iterates lie on the
surface of the $M_\epsilon$ set. }
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth]{fig4_regularized_extrapolated_CDI}
\end{center}
\caption{\label{f:regularized_extrapolated}
Reconstruction and behavior of odd and even iterates of regularized
(data set $M_\epsilon$ given by \eqref{e:X-ray set} with $\epsilon=1.2$) inexact
extrapolated alternating projections with $\lambda_k=1$ for all $k$. The algorithm
terminates at the 5th iterate which achieves condition \eqref{e:pooping_b} to numerical
precision.}
\end{figure}
Figure \ref{f:comparison}
shows the {\em apparent} convergence rates for different values of the relaxation
parameter $\epsilon$ in \eqref{e:X-ray set} and different settings for the
step-length parameters $\lambda_k$. The black solid line shows again the change between the
even iterates of the unregularized, exact alternating projection algorithm.
The blue and green dashed lines show the apparent
rate of convergence of the regularized problems without extrapolation,
that is, $\lambda_k$ is computed
so that the iterates lie on the surface of the set $M_\epsilon$ (to numerical precision).
As expected, the lower the value of
$\epsilon$, the poorer the (asymptotic) rate of convergence since the sets are
closer to ill-posedness for smaller regularization values. The red dashed-dotted line shows what can be
gained by extrapolation. Here the step-length parameter $\lambda_k=1$ for all $k$ and
the algorithm proceeds with a convergence rate
indicated by Theorem \ref{t:approx proj}, but then terminates finitely as it finds a
point on intersection interior to the regularized set $M_\epsilon$.
Note that the only difference between this
implementation and the unregularized exact alternating projections implementation
(the black solid line) is early termination of the algorithm. This is what is usually done
heuristically in practice. What this example shows is a mathematically sound
explanation of this practice in terms of regularization, extrapolation and approximate
alternating projections.
For this example it is not possible to compute an a priori rate of convergence as specified by
Theorem \ref{t:approx proj} since the set $M_\epsilon$ has no analytic form and we are
unable to compute the angle of the intersection. We observe a linear convergence rate, at least
to the limit of machine precision. To a certain extent, this is beside the point.
The value of the theory outlined above lies not with the computation of rates of convergence, but rather
with the provision of regularization strategies and corresponding stopping rules. We can, however,
verify numerically whether a point lies in the interior of the regularized set at the point of intersection with
the qualitative constraint set. For the extrapolated example shown in Figure \ref{f:regularized_extrapolated}
it was verified that this point lies in the interior of the set $M_\epsilon$ by perturbing the point slightly and
verifying that it still lies in the set $M_\epsilon$. Thus, even if the unregularized
problem is not consistent as required by Theorem \ref{t:implementation} to {\em guarantee} that the
approximate projection achieves a sufficient accuracy for linear convergence, since the fixed point of
the algorithm is an interior point, the required accuracy for the approximate projection is quite easy to
satisfy as discussed in Remark \ref{r:slither}.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{fig5_comparison_Poisson}
\caption{\label{f:comparison}
Comparison of implementations of Algorithm \ref{alg:inexact ap} with
the approximate projection computed as in Theorem \ref{t:implementation}
for different parameters $\epsilon$ and step-length strategies ($\lambda_k$)
for the fattened set $M_\epsilon$
given by \eqref{e:X-ray set}. The black line is the unregularized alternating
projection algorithm with exact projections. The blue and green lines are the
regularized approximate alternating projection algorithms with step lengths $\lambda_k$
computed so that the iterates lie on the surface of the $M_\epsilon$ set. The red line
is the extrapolated approximate alternating projection algorithm with $\lambda_k=1$
for all $k$.
}
\end{figure}
Finally, note that while the rate of convergence for the more regularized problems is better,
as indicated by comparing the reconstructions in Figures \ref{f:regularized} and
\ref{f:regularized_extrapolated}, the reconstruction can be poorer since this
reconstruction is apparently further
away from the ideal solution than the less regularized reconstructions.
\section{Conclusion}
The main achievement of this note is not
our algorithm. Indeed, the regularized extrapolated ($\lambda_k=1$ for all $k$)
inexact projection algorithm specified in Theorem \ref{t:implementation} in fact has been used successfully
for decades in diffraction imaging with heuristic stopping criteria and {\em early termination} effectively
serving as the regularization.
What the analysis here provides, for the first time, is a
regularization strategy that fits naturally with many ill-posed inverse problems, and a mathematically
sound stopping criterion. The conventional early termination applied in practice to the unregularized problem
can be justified fully in the framework of this regularization strategy together with approximate projections.
While all of the regularity
assumptions on the sets $M_\epsilon$ and $C$ are
satisfied for the finite dimensional phase problem discussed in Section \ref{s:numerics},
since the unregularized phase problem with noise
is still inconsistent, Theorem \ref{t:implementation} does not apply. If exact projections onto the
regularized sets were computed, then Theorem 5.16 of \cite{LewisLukeMalick08} would suffice
to prove convergence of exact alternating projections applied to the regularized problem. Proof of convergence of
the inexact algorithm with extrapolation strategy $\lambda_k=1$ for all $k$
for the regularized, inconsistent phase retrieval problem (what has in fact been applied in the application
literature for decades) hinges on verifying that condition \ref{e:pooping_c} of Algorithm \ref{alg:inexact ap} is satisfied locally
for all iterates. This is an open problem.
\section*{Acknowledgements} We thank Katharina Echternkamp, Ann-Kathrin G\"unther, Daja Herzog,
Dong-Du Mai, Jelena Panke, Aike Ruhlandt and Jan Thiart at the Institut f\"ur R\"ontgenphysik
at the University of G\"ottingen for the diffraction data used in our numerical experiments.
|
1705.07438
|
\section{Introduction and main results}
In this article, we are concerned with the global weighted Lorentz space estimates for gradients of very weak solutions to linear parabolic equations in divergence form:
\begin{equation}\label{5hh070120148}
\left\{
\begin{array}
[c]{l}%
{u_{t}}-\operatorname{div}(A(x,t)\nabla u)=\operatorname{
div}(F)~~\text{in }\Omega_T,\\
u=0~~~~~~~\text{on}~~
\partial_p(\Omega \times (0,T)),
\\
\end{array}
\right.
\end{equation} where $\Omega_T:=\Omega\times (0,T)$ is a bounded open subset of $\mathbb{R}^{N+1}$, $N\geq2$, $ \partial_p(\Omega \times (0,T))=(\partial\Omega\times(0,T))\cup (\Omega\times\{t=0\})$, $F\in L^p(\Omega_T,\mathbb{R}^N),~p>1$ is a given vector field and the matrix function $A:\mathbb{R}^N\times\mathbb{R}\times \mathbb{R}^N\to \mathbb{R}^N$ is a Carath\'eodory vector valued function, i.e. $A$ is measurable in $(x,t)$ and continuous with respect to $\nabla u$ for a.e. $(x,t)$.\\
We suppose in this paper that $A$ satisfies
\begin{align}
\label{5hhconda} \Lambda ^{-1}|\xi|^2\leq \langle A(x,t)\xi,\xi\rangle\le \Lambda |\xi|^2,
\end{align}
for every $\xi\in \mathbb{R}^N$ and a.e. $(x,t)\in \mathbb{R}^N\times \mathbb{R}$, where $\Lambda$ is a positive constant.
Our main result is that, for any $q>1$ and any $ w\in \mathbf{A}_q$ (the Muckenhoupt class for parabolic, see below), $F\in L^q_w(\Omega_T,\mathbb{R}^N)$, and under some additional conditions on the matrix $A$ and on the boundary of $\Omega$, there exists a unique very weak solution $u\in L^{q_0}(0,T,W_{0}^{1,q_0}(\Omega))$ for some $q_0>1$ of \eqref{5hh070120148} satisfying
\begin{align}\label{es1}
\int_{\Omega_T}|\nabla u|^q w dxdt\leq C \int_{\Omega_T}|F|^q w dxdt
\end{align}
In this paper, a very weak solution $u$ of \eqref{5hh070120148} is understood in the standard weak (distributional) sense, that is $u\in L^1(0,T,W_0^{1,1}(\Omega))$ is a very weak solution of \eqref{5hh070120148} if
\begin{align*}
-\int_{\Omega_T}u \varphi_tdxdt+\int_{\Omega_T}A(x,t)\nabla u\nabla \varphi dxdt= -\int_{\Omega_T}F\nabla\varphi dxdt
\end{align*}
for all $\varphi \in C_c^1([0,T)\times \Omega)$.
\\
Case $w\equiv1$, the result was obtained by Byun and Wang in \cite{55BW1,55BW4}. Moreover, $w\in A_{q/2}$ for $q\geq 2$ and $w^{3}\in \mathbf{A}_1$ for $1<p<2$ was proved by author in \cite[see Theorem 1.3]{55QH3}.
The result of this paper is inspired by \cite{AdiMenPhuc}, they have demonstrated for linear elliptic equation, their approach employs a local version of the sharp maximal function of Fefferman and Stein. Our approach in this paper is different from \cite{AdiMenPhuc}, we use Hardy-Littlewood maximal function.
It is worth mentioning that the result of this paper can imply results in \cite{AdiMenPhuc}, see Corollary \ref{101120143b}. Furthermore, the requirement $w\in \mathbf{A}_q$ in \eqref{es1} is optimal, this was discussed in \cite{AdiMenPhuc}.
For our purpose, we need to assume that $\Omega$ is a Lipschitz domain with small Lipschiptz constant. We say that $\Omega$ is a $(\delta,R_0)-$Lip domain for $\delta\in (0,1)$ and $R_0>0$ if for every $x\in\partial \Omega$, there exists a map $\Gamma:\mathbb{R}^{n-1}\to \mathbb{R}$ such that $||\nabla \Gamma||_{L^\infty(\mathbb{R}^{n-1})}\leq \delta$ and, upon rotating and relabeling of coordinates if necessary,
\begin{align*}
\Omega\cap B_{R_0}(x_0)=\{(x',x_n)\in B_{R_0}(x_0): x_n>\Gamma(x')\}.
\end{align*}
It is well-known that $\Omega$ is a $(\delta,R_0)-$Lip domain for $\delta\in (0,1)$ and $R_0>0$ then, $\Omega$ is also a $(\delta,R_0)-$Reifenberg flat domain, see \cite{55BW1,55BW4,55QH3}.
We also require that the matrix function $A$ satisfies a smallness condition of BMO type in the $x$-variable in the sense that $A(x,t)$ satisfies a $(\delta,R_0)$-BMO condition for some $\delta, R_0>0$ if
\begin{equation*}
[A]_{R_0}:=\mathop {\sup }\limits_{(y,s)\in \mathbb{R}^N\times\mathbb{R},0<r\leq R_0}\fint_{Q_r(y,s)}|A(x,t)-\overline{A}_{B_r(y)}(t)|dxdt\leq \delta,
\end{equation*}
where
$\overline{A}_{B_r(y)}(t)$ is denoted the average of $A(t,.)$ over the ball $B_r(y)$, i.e,
\begin{equation*}
\overline{A}_{B_r(y)}(t):=\fint_{B_r(y)}A(x,t)dx.
\end{equation*}
The above condition appeared in our previous paper \cite{55QH2}. It is easy to see that the $(\delta,R_0)-$BMO is satisfied when $A$ is continuous or has small jump discontinuities with respect to $x$.
We recall that a positive function $w\in L^1_{\text{loc}}(\mathbb{R}^{N+1})$ is called an $\mathbf{A}_{p}$ weight, $1\leq p<\infty$ if there holds
\begin{align*}
[w]_{\mathbf{A}_{p}}:= \mathop {\sup }\limits_{\tilde{Q}_\rho(x,t)\subset\mathbb{
R}^{N+1}}\left(\fint_{\tilde{Q}_\rho(x,t)}w(y,s)dyds
\right)\left(\fint_{\tilde{Q}_\rho(x,t)}w(y,s)^{-\frac{1}{p-1}}dyds
\right)^{p-1}<\infty~~\text{when }~p>1,
\end{align*}
The $[w]_{\mathbf{A}_p}$ is called the $\mathbf{A}_{p}$ constant of $w$. \\
A positive function $w\in L^1_{\text{loc}}(\mathbb{R}^{N+1})$ is called an $\mathbf{A}_{\infty}$ weight if there are two positive constants $C$ and $\nu$ such that
$$w(E)\le C \left(\frac{|E|}{|Q|}\right)^\nu w(Q),
$$
for all cylinder $Q=\tilde{Q}_\rho(x,t)$ and all measurable subsets $E$ of $Q$. The pair $(C,\nu) $ is called the $\mathbf{A}_\infty$ constant of $w$ and is denoted by $[w]_{\mathbf{A}_\infty}$.
It is well known that this class is the union of $\mathbf{A}_p$ for all $p\in (1,\infty)$, see \cite{55Gra}. Furthermore, if $w\in \mathbf{A}_p$ with $[w]_{\mathbf{A}_p}\leq M$ then there exists a constant $\varepsilon_0=\varepsilon(N,p,M)$, and a constant $M_0=M(N,p,M)$ such that $[w]_{\mathbf{A}_{p-\varepsilon_0}}\leq M_0$.
If $w$ is a weight function belonging to $w\in \mathbf{A}_{\infty}$ and $E\subset \mathbb{R}^{N+1}$ a Borel set, $0<q<\infty$, $0<p\leq\infty$, the weighted Lorentz space $L^{q,p}_w(E)$ is the set of measurable functions $g$ on $E$ such that
\begin{equation*}
||g||_{L^{q,p}_w(E)}:=\left\{ \begin{array}{l}
\left(q\int_{0}^{\infty}\left(\rho^qw\left(\{(x,t)\in E:|g(x,t)|>\rho\}\right)\right)^{\frac{p}{q}}\frac{d\rho}{\rho}\right)^{1/p}<\infty~\text{ if }~s<\infty, \\
\sup_{\rho>0}\rho \left( w\left(\{(x,t)\in E:|g(x,t)|>\rho\}\right)\right)^{1/q}<\infty~~\text{ if }~p=\infty. \\
\end{array} \right.
\end{equation*}
Here we write $w(O)=\int_{O}w(x,t)dxdt$ for a measurable set $O\subset \mathbb{R}^{N+1}$. Throughout the paper, we always denote $T_0=\text{diam}(\Omega)+T^{1/2}$ and $Q_\rho(x,t)=B_\rho(x)\times (t-\rho^2,t)$ $\tilde{Q}_\rho(x,t)=B_\rho(x)\times (t-\rho^2/2,t+\rho^2/2)$ for $(x,t)\in\mathbb{R}^{N+1}$ and $\rho>0$. Moreover, $\mathcal{M}$ denotes the parabolic Hardy-Littlewood maximal function defined for each locally integrable function $f$ in $\mathbb{R}^{N+1}$ by
\begin{equation*}
\mathcal{M}(f)(x,t)=\sup_{\rho>0}\fint_{\tilde{Q}_\rho(x,t)}|f(y,s)|dyds~~\forall (x,t)\in\mathbb{R}^{N+1}.
\end{equation*}
If $q>1$ and $w\in \mathbf{A}_q$ we verify that $\mathcal{M}$ is operator from $L^1(\mathbb{R}^{N+1})$ into $L^{1,\infty}(\mathbb{R}^{N+1})$ and $L^{q,p}_w(\mathbb{R}^{N+1})$ into itself for $0<p\leq \infty$, see \cite{55Stein2,55Stein3,55Tur}.
\\ We would like to mention that the use
of the Hardy-Littlewood maximal function in non-linear degenerate problems was started in the elliptic setting by T. Iwaniec in his fundamental paper \cite{Iwa}. \\
We now state the main result of the paper.
\begin{theorem} \label{101120143} For any $w\in \mathbf{A}_{q}$, $1< q<\infty$, $0<p\leq\infty$ we find $\delta=\delta(N,\Lambda, q,p, [w]_{\mathbf{A}_{q}})\in (0,1)$ such that if $\Omega$ is $(\delta,R_0)$-Lip domain $\Omega$ and $[A]_{R_0}\le \delta$ for some $R_0>0$ and $F\in L^{q,p}_w(\Omega_T)$, then there exists a unique weak solution $u\in L^{q_0}(0,T,W_{0}^{1,q_0}(\Omega))$ for some $q_0>1$ of
\eqref{5hh070120148} satisfying
\begin{equation}\label{101120144}
|||\nabla u|||_{L^{q,p}_w(\Omega_T)}\leq C ||F||_{L^{q,p}_w(\Omega_T)}.
\end{equation}
Here $C$ depends only on $N,\Lambda,q,p, [w]_{\mathbf{A}_q}$ and $T_0/R_0$.
\end{theorem}
As an immediate consequence of Theorem \ref{101120143}, we obtain a version of Theorem \ref{101120143} for the linear elliptic equations. This result was obtained in \cite{AdiMenPhuc}.
\begin{corollary}\label{101120143b} Assume that $ A(x)=A(x,t)$ for all $(x,t)\in \mathbb{R}^{N+1}$. For any $w(x)=w(x,t)\in \mathbf{A}_{q}$, $1< q<\infty$, $0<p\leq\infty$ we find $\delta=\delta(N,\Lambda, q_0,q,p, [w]_{\mathbf{A}_{q}})\in (0,1)$ such that if $\Omega$ is $(\delta,R_0)$-Lip domain $\Omega$ and $[A]_{R_0}\le \delta$ for some $R_0>0$ and $G\in L^{q,p}_w(\Omega)$ then there exists a unique very weak solution $u\in W_{0}^{1,q_0}(\Omega)$ for some $q_0>1$ of
\begin{equation}\label{pro-elliptic}
\left\{
\begin{array}
[c]{l}%
-\operatorname{div}(A(x)\nabla u)=\operatorname{
div}(G)~~\text{in }\Omega,\\
u=0~~~~~~~\text{on}~~
\partial \Omega.
\\
\end{array}
\right.
\end{equation} satisfying
\begin{equation}\label{101120144b}
|||\nabla u|||_{L^{q,p}_w(\Omega)}\leq C ||F||_{L^{q,p}_w(\Omega)}.
\end{equation}
Here $C$ depends only on $N,\Lambda,q,p, [w]_{\mathbf{A}_q}$ and $diam(\Omega)/R_0$.
\end{corollary}
\section{Interior estimates and boundary estimates for parabolic equations}
In this section we present various local interior and boundary estimates for weak solution $u$ of \eqref{5hh070120148}. They will be used for our global estimates later.
In \cite{55QH3}, author proved the following result.
\begin{theorem} \label{161120141} Let $q>1$ and $G\in L^{q}(\Omega_T,\mathbb{R}^N)$. We find a $\delta=\delta(N,\Lambda,q)\in (0,1)$ such that if $\Omega$ is a $(\delta,R_0)$-Lip domain and $[A]_{R_0}\le \delta$ for some $R_0>0$ there exists a very unique weak solution $v\in L^q(0,T,W_0^{1,q}(\Omega))$ of
\begin{equation}\label{161120143}
\left\{
\begin{array}
[c]{l}%
{v_{t}}-\operatorname{div}(A(x,t)\nabla u)=\operatorname{div}(G)~~\text{in }\Omega_T,\\
u=0~~~~~~~\text{on}~~
\partial_p(\Omega \times (0,T)),\\
\end{array}
\right.
\end{equation}
Furthermore, there holds
\begin{equation}\label{161120144} ||\nabla u||_{L^{q}(\Omega_T)}\leq C ||F||_{L^{q}(\Omega_T)}
\end{equation}
where $C$ depends only on $N,\Lambda,q$ and $(diam(\Omega)+T^{1/2})/R_0$.
\end{theorem}
Let $s>0$. We apply Theorem \eqref{161120141} to $G=F$and $q=s$, there is a constant $\delta_0=\delta_0(N,\Lambda,s)\in (0,1/4)$ such that if $\Omega$ is a $(\delta_0,R_0)$-Lip domain and $[A]_{R_0}\le \delta_0$ for some $R_0>0$ , then the problem \eqref{5hh070120148} has a unique very weak solution $u\in L^{s}(0,T,W_{0}^{1,s}(\Omega))$ satisfying
\begin{equation*} ||\nabla u||_{L^{s}(\Omega_T)}\leq C ||F||_{L^{s}(\Omega_T)}.
\end{equation*}
where $C=C(N,\Lambda,s,T_0/R_0)$.\\
In this section, we assume that $\Omega$ is a $(\delta_0,R_0)$-Lip domain and $[A]_{R_0}\le \delta_0$ for some $R_0>0$, where $\delta_0$ is as above. For some technical reasons, throughout this section, we always assume that $u\in L^{s}(-\infty,T;W^{1,s}_0(\Omega))$ is a very weak solution to equation \eqref{5hh070120148} in $\Omega\times (-\infty,T)$ with $F=0$ in $\Omega\times (-\infty,0)$. \\
\subsection{Interior Estimates}
Let $R \in (0,R_0)$, $B_{2R}=B_{2R}(x_0)\subset\subset\Omega$ and $t_0\in (0,T)$ . Set $Q_{2R}=B_{2R} \times (t_0-4R^2,t_0)$ and $\partial_{p}Q_{2R}= \left( {\partial B_{2R} \times (t_0-4R^2,t_0)} \right) \cup \left( {B_{2R} \times \left\{ {t = t_0-4R^2} \right\}} \right) $. Since $\Omega$ is a $(\delta_0,R)$-Lip domain and $[A]_{R}\le \delta_0$, thus, applying Theorem \ref{161120141} to $\Omega_T=Q_{2R}$ and $G=F$, the following equation
\begin{equation}
\label{111120146}\left\{ \begin{array}{l}
{W_t} - \operatorname{div}\left( {A(x,t)\nabla w} \right) = \operatorname{div}(F) \;in\;Q_{2R}, \\
W = 0\quad \quad on~~\partial_{p}Q_{2R}, \\
\end{array} \right.
\end{equation}
has a unique very weak solution $W\in L^s(t_0-4R^2,t_0;W_0^{1,s}(B_{2R}))$. Moreover, we have
\begin{align}\label{1111201410}
\fint_{Q_{2R}}|\nabla W|^{s}dxdt\leq C\fint_{Q_{2R}}|F|^{s}dxdt.
\end{align}
where $C$ depends only on $N,\Lambda,s$. Note that the constant $(diam(\Omega)+T^{1/2})/R_0$ in Theorem \ref{161120141} equals 6 in this case. \\
We now set $w=u-W$, so $w$ is a solution of
\begin{equation}
\label{111120146b}
{w_t} - \operatorname{div}\left( {A(x,t)\nabla w} \right) = 0 \;in\;Q_{2R}
\end{equation}
The following a variant of Gehring's lemma was proved in \cite{55Nau,55DuzaMing}.
\begin{lemma} \label{111120147} There exist a constant $C>0$ depending only on $N,\Lambda$ such that the following estimate
\begin{equation}\label{111120148}
\left(\fint_{Q_{\rho/2}(y,s)}|\nabla w|^{2} dxdt\right)^{\frac{1}{2}}\leq C\fint_{Q_{\rho}(y,s)}|\nabla w| dxdt,
\end{equation}holds
for all $Q_{\rho}(y,s)\subset \subset Q_{2R}$.
\end{lemma} \medskip
To continue, we denote by $v$ the unique solution
$v\in L^2(t_0-R^2,t_0;H^1(B_{R}))$
of the following equation
\begin{equation}\label{5hheq4}
\left\{ \begin{array}{l}
{v_t} - \operatorname{div}\left( {\overline{A}_{B_R}(t)\nabla v} \right) = 0 \;in\;Q_{R}, \\
v = w\quad \quad on~~\partial_{p}Q_{R}, \\
\end{array} \right.
\end{equation}
where $B_R=B_R(x_0)$, $Q_{R}=B_{R}\times(t_0-R^2,t_0)$ and $$\partial_{p}Q_{R}= \left( {\partial B_{R} \times (t_0-R^2,t_0)} \right) \cup \left( {B_{R} \times \left\{ {t = t_0-R^2} \right\}} \right).$$.
\begin{lemma}\label{5hh21101319} There exist constants $C_1=C_1(N,\Lambda)$ and $C_2=C_2(\Lambda)$ such that \begin{eqnarray}
\left( \fint_{Q_R}|\nabla (w-v)|^2dxdt\right)^{1/2}\leq C_1 [A]_{R_0} \fint_{Q_{2R}}|\nabla w|dxdt\, \label{5hh18094}
\end{eqnarray}
and \begin{equation}\label{5hh18091}
C_2^{-1} \int_{Q_R}|\nabla v|^2dxdt\leq \int_{Q_R}|\nabla w|^2dxdt\leq C_2\int_{Q_R}|\nabla v|^2dxdt.
\end{equation}
\end{lemma}
\begin{proof} The proof can be found in \cite[Lemma 7.3]{55QH2}.
\end{proof}
\begin{proposition}\label{5hh24092} There holds $$v\in L^2(t_0-R^2,t_0;H^1(B_{R}))\cap L^\infty(t_0-\frac{1}{4}R^2,t_0;W^{1,\infty}(B_{R/2})), $$
and
\begin{equation}\label{5hh18092}
||\nabla v||^{s}_{L^\infty(Q_{R/2})}\leq C \fint_{Q_{2R}}|\nabla u|^{s} dxdt +C\fint_{Q_{2R}}|F|^{s} dxdt,
\end{equation}
\begin{eqnarray}
\fint_{Q_R}|\nabla (u-v)|^{s}dxdt\leq C\left([A]_{R_0}\right)^{s}\fint_{Q_{2R}}|\nabla u|^{s}dxdt+ C\fint_{Q_{2R}}|F|^{s} dxdt,\label{5hh18093}
\end{eqnarray}
where $C$ depends only on $N,\Lambda,s.$\\
\end{proposition}
\begin{proof} By standard interior regularity and inequality \eqref{111120148} in Lemma \ref{111120147} and \eqref{5hh18091} in Lemma \ref{5hh21101319} we have\begin{align*}
||\nabla v||_{L^\infty(Q_{R/2})}&\leq C \left(\fint_{Q_R}|\nabla v|^2dxdt\right)^{1/2}\\&\leq C \left(\fint_{Q_R}|\nabla w|^2dxdt\right)^{1/2}\\&\leq C \fint_{Q_{2R}}|\nabla w|dxdt.
\end{align*}
Thus, from this and inequality \eqref{1111201410}, we get \eqref{5hh18092}. \\
On the other hand, applying \eqref{5hh18094} in Lemma \ref{5hh21101319} yields
\begin{align*}
\fint_{Q_R}|\nabla (u- v)|^{s}dxdt\leq C \fint_{Q_R}|\nabla (u- w)|^{s}dxdt+ C\left([A]_{R}\right)^{s}\fint_{Q_{2R}}|\nabla w|^{s} dxdt.
\end{align*}
Combining this with \eqref{1111201410}, we get \eqref{5hh18093}. The proof is complete.
\end{proof}
\subsection{Boundary Estimates}
In this subsection, we focus on the corresponding estimates near the boundary. \\
Throughout this subsection, $\Omega$ is a $(\delta/4,R_0)$-Lip domain and $[A]_{R_0}\le \delta/4$ for $\delta<\delta_0$. Let $x_0\in \partial\Omega$ be a boundary point and $0<R<R_0$ and $t_0\in (0,T)$.
Since, for any $\eta>0$, $B_{\frac{1}{8}}((0,...,\frac{1}{8}-\varepsilon))\cap B_1(0)$ is $(\eta,\eta_0)$-Lip domain for some $\varepsilon>0$ and $\eta_0>0$. Therefore, there a ball $B$ of radius $R/8$ and $\varepsilon_1,\varepsilon_2>0$ depending only on $N$ such that $B_{\varepsilon_1 R}(x)\subset B\subset B_{R}(x)$ and $B\cap \Omega$ is $(\delta,\varepsilon_2R)-$ Lip domain.
We set $\tilde{\Omega}_{R/8}=\tilde{\Omega}_{R/8}(x_0,t_0)=\left(\Omega\cap B \right)\times (t_0-(R/8)^2,t_0)$. Since $B\cap \Omega$ is $(\delta_0,\varepsilon_2R)-$ Lip domain and $[A]_{\varepsilon_2R}\le \delta_0$, we
apply Theorem \ref{161120141} to $\Omega_T= \tilde{\Omega}_{R/8}$, $G=F$ and $q=s$, there exists a unique very weak solution $W$ to
\begin{equation}\label{161120143b}
\left\{
\begin{array}
[c]{l}%
{W_{t}}-\operatorname{div}(A(x,t)\nabla W)=\operatorname{div}(F)~~\text{in }\tilde{\Omega}_{R/8},\\
W=0~~~~~~~\text{on}~~
\partial_p\tilde{\Omega}_{R/8},\\
\end{array}
\right.
\end{equation}
satisfying
\begin{equation}\label{141120145} ||\nabla W||_{L^{s}(\tilde{\Omega}_{R/8})}\leq C ||F||_{L^{s}(\tilde{\Omega}_{R/8})}
\end{equation}
where $C$ depends only on $N,\Lambda,s$. Note that the constant $(diam(\Omega)+T^{1/2})/R_0$ in Theorem \ref{161120141} equals $\frac{3}{8\varepsilon_2}$ in this case.
In what follows we extend $F$ by zero to $\left(\Omega\times (-\infty,T)\right)^c$, and $W$ by zero to $\mathbb{R}^{N+1}\backslash \tilde{\Omega}_{R/8}$.
We now set $w=u-W$, so $w$ is a solution of
\begin{equation}
\label{141120142}\left\{ \begin{array}{l}
{w_t} - \operatorname{div}\left( {A(x,t,\nabla w)} \right) = 0 \;in\;\tilde{\Omega}_{R/8}, \\
w = u\quad \quad on~~\partial_{p}\tilde{\Omega}_{R/8}. \\
\end{array} \right.
\end{equation}
\begin{lemma}\label{141120141}
There exist a constant $C>0$ depending only on $N,\Lambda$ such that the following estimate
\begin{equation}\label{141120143}
\left(\fint_{Q_{\rho/2}(y,s)}|\nabla w|^{2} dxdt\right)^{\frac{1}{2}}\leq C\fint_{Q_{3\rho}(y,s)}|\nabla w| dxdt,
\end{equation}
holds for all $Q_{3\rho}(z,s)\subset B\times (t_0-(R/8)^2,t_0) $.
\end{lemma}
Above lemma was proved in \cite[Theorem 7.5]{55QH2}.
Next, we set $\rho=\varepsilon_1R(1-\delta)/8$ so that $0<\rho/(1-\delta)<\varepsilon_1R_0/8$. By the definition of Lipschiptz domains and $B_{\varepsilon_1 R}(x_0)\subset B$, there exists a coordinate system $\{y_1,y_2,...,y_N\}$ with the
origin $0\in\Omega$ such that in this coordinate system $x_0=(0,...,0,-\frac{\rho\delta}{4(1-\delta)})$ and $B_\rho(0)\subset B$,
\begin{equation*}
B^+_\rho(0)\subset \Omega\cap B_\rho(0)\subset B_\rho(0)\cap \left\{y=(y_1,y_2,....,y_N):y_N>-\frac{\rho\delta}{2(1-\delta)}\right\}.
\end{equation*}
Since $\delta<1/4$, we have
\begin{equation}\label{5hh1610138}
B^+_\rho(0)\subset \Omega\cap B_\rho(0)\subset B_\rho(0)\cap \{y=(y_1,y_2,....,y_N):y_N>-\rho\delta\},
\end{equation}
where $B^+_\rho(0):=B_\rho(0)\cap\{y=(y_1,y_2,...,y_N):y_N>0\}$.\\
Furthermore we consider the unique solution
\begin{equation*}
v\in L^2(t_0-\rho^2,t_0;H^1(\Omega\cap B_\rho(0)))
\end{equation*}
to the following equation
\begin{equation}\label{5hh1610131}
\left\{ \begin{array}{l}
{v_t} - \operatorname{div}\left( {\overline{A}_{B_{\rho}(0)}(t)\nabla v} \right) = 0 \;in\;\tilde{\Omega}_\rho(0),\\
v = w\quad \quad on~~\partial_{p}\tilde{\Omega}_\rho(0), \\
\end{array} \right.
\end{equation}
where $\tilde{\Omega}_\rho(0)=\left(\Omega\cap B_{\rho}(0)\right)\times (t_0-\rho^2,t_0)$.
We put $v=w$ outside $\tilde{\Omega}_\rho(0)$. As Lemma \ref{5hh21101319} (see \cite[Lemma 2.8]{55QH3}) we have the following result.
\begin{lemma}\label{5hh1610139} There exist positive constants $C_1=C_1(N,\Lambda)$ and $C_2=C_2(\Lambda)$ such that \begin{eqnarray}
\fint_{Q_{\rho}(0,t_0)}|\nabla (w-v)|^2dxdt\leq C_1 \left([A]_{R}\right)^2 \fint_{Q_{\rho}(0,t_0)}|\nabla w|^2dxdt, \label{5hh1610132}
\end{eqnarray}
and \begin{equation}\label{5hh1610133}
C_2^{-1} \int_{Q_{\rho}(0,t_0)}|\nabla v|^2dxdt\leq \int_{Q_{\rho}(0,t_0)}|\nabla w|^2dxdt\leq C_2\int_{Q_{\rho}(0,t_0)}|\nabla v|^2dxdt.
\end{equation}
\end{lemma}
We can see that if the boundary of $\Omega$ is irregular enough, then the $L^\infty$-norm of $\nabla v$ up to $\partial\Omega\cap B_\rho(0)\times (t_0-\rho^2,t_0)$ may not exist. However, we have the following Lemma obtained in \cite[Lemma 7.12]{55QH2}.
\begin{lemma}\label{5hh21101314}
For any $\varepsilon>0$, there exists a small $\delta_1=\delta_1(N,\Lambda,\varepsilon)\in (0,\delta_0)$ such that if $\delta\in (0,\delta_1)$, there exists a function $V\in C(t_0-\rho^2,t_0;L^2( B_\rho^+(0)))\cap L^2(t_0-\rho^2,t_0;H^1( B_\rho^+(0)))$ satisfying
\begin{align*}
||\nabla V||^2_{L^\infty(Q_{\rho/4}(0,t_0))}\leq C \fint_{Q_{\rho}(0,t_0)} |\nabla v|^2dxdt,
\end{align*}
and
\begin{align*}
\fint_{Q_{\rho/8}(0,t_0)}|\nabla (v-V)|^2dxdt\leq \varepsilon^2\fint_{Q_{\rho}(0,t_0)} |\nabla v|^2dxdt,
\end{align*}
for some $C=C(N,\Lambda)>0$.
\end{lemma}
\begin{proposition}\label{5hh16101310}For any $\varepsilon>0$ there exists a small $\delta_1=\delta_1(N,\Lambda,s,q_0,\varepsilon)\in (0,\delta_0)$ such that the following holds. If $\Omega$ is a $(\delta/4,R_0)$-Lip domain with $\delta\in (0,\delta_1)$, there is a function $V\in L^2(t_0-(R/9)^2,t_0;H^1( B_{R/9}(x_0)))\cap L^\infty(t_0-(R/9)^2,t_0;W^{1,\infty}( B_{R/9}(x_0)))$ such that
\begin{equation}\label{5hh21101317}
||\nabla V||^s_{L^\infty(Q_{\varepsilon_1 R/500})}\leq C\fint_{Q_{R}}|\nabla u|^sdxdt+C \fint_{Q_{R}}|F|^sdxdt,
\end{equation}
and
\begin{align}
\fint_{Q_{\varepsilon_1 R/500}}|\nabla (u-V)|^sdxdt\leq C (\varepsilon^s+([A]_{R_0})^s)\fint_{Q_{R}}|\nabla u|^sdxdt+ C\fint_{Q_{R}}|F|^sdxdt,\label{5hh21101318}
\end{align}
for some $C=C(N,\Lambda,s)>0$. Here $Q_{\rho}=Q_{\rho}(x_0,t_0)$ for all $\rho>0$.
\end{proposition}
\begin{proof} We can assume that $\delta\in (0,1/100) $. So
\begin{equation}\label{5hh090520141}
Q_{\varepsilon_1 R/500}\subset Q_{\rho/8}(0,t_0)\subset Q_{6\rho}(0,t_0)\subset Q_{\varepsilon_1 R}\subset Q_{R}
\end{equation}
By Lemma \ref{5hh21101314} for any $\varepsilon>0$, we can find a small positive $\delta=\delta(N,\Lambda,s,q_0,\varepsilon)<1/100$ such that there is a function $V\in L^2(t_0-\rho^2,t_0;H^1( B_{\rho}(0)))\cap L^\infty(t_0-\rho^2,t_0;W^{1,\infty}( B_{\rho}(0)))$ satisfying
\begin{align*}
& ||\nabla V||^2_{L^\infty(Q_{\rho/4}(0,t_0))}\leq C \fint_{Q_{\rho}(0,t_0)} |\nabla v|^2dxdt,\\&
\fint_{Q_{\rho/8}(0,t_0)}|\nabla (v-V)|^2\leq \varepsilon^2\fint_{Q_{\rho}(0,t_0)} |\nabla v|^2dxdt.
\end{align*}
Then, by \eqref{5hh1610133} in Lemma \ref{5hh1610139} and \eqref{141120143} in Lemma \ref{141120141} and \eqref{5hh090520141}, we get
\begin{align}
\nonumber
||\nabla V||^s_{L^\infty(Q_{\varepsilon_1 R/500})} &\leq C \left(\fint_{Q_{\rho}(0,t_0)} |\nabla w|^2dxdt\right)^{\frac{s}{2}}\nonumber\\&\leq C\fint_{Q_{\varepsilon_1R}} |\nabla w|^sdxdt, \label{5hh21101316}
\end{align}
and
\begin{align}
\fint_{Q_{\varepsilon_1 R/500}}|\nabla (v-V)|^{s}dxdt\leq C \varepsilon^s\fint_{Q_{\varepsilon_1R}} |\nabla w|^sdxdt.\label{5hh21101320}
\end{align}
Therefore, from \eqref{141120145} and \eqref{5hh21101316} we get \eqref{5hh21101317}.\\
Next we prove \eqref{5hh21101318}. Since \eqref{5hh090520141},
\begin{align*}
&\fint_{Q_{\varepsilon_1 R/500}}|\nabla (u-V)|^sdxdt\leq C\fint_{Q_{\rho/8}(0,t_0)}|\nabla (u-V)|^sdxdt
\\&~~~~~~~~\leq C\fint_{Q_{\rho/8}(0,t_0)}|\nabla (u- w)|^sdxdt+C\fint_{Q_{\rho/8}(0,t_0)}|\nabla (w- v)|^sdxdt\\&~~~~~~~~~~+C\fint_{Q_{\rho/8}(0,t_0)}|\nabla (v- V)|^sdxdt.
\end{align*}
Using \eqref{141120145} and \eqref{5hh1610132}, \eqref{5hh1610133} in Lemma \ref{5hh1610139} and \eqref{5hh21101320} we find that \begin{align*}
&\fint_{Q_{\rho/8}(0,t_0)}|\nabla (u-w)|^sdxdt\leq C \fint_{Q_{R}}|F|^sdxdt,
\\&\fint_{Q_{\rho/8}(0,t_0)}|\nabla( v- w)|^sdxdt \leq C([A]_{R_0})^s \fint_{Q_{\varepsilon_1R}}|\nabla w|^sdxdt\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\leq C([A]_{R_0})^s\left(\fint_{Q_{ R}}|\nabla u|^sdxdt+\fint_{Q_{R}}|F|^sdxdt\right),
\end{align*}
and
\begin{align*}
\fint_{Q_{\rho/8}(0,t_0)}|\nabla (v-V)|^s dxdt \leq C\varepsilon^s\left(\fint_{Q_{ R}}|\nabla u|^sdxdt+\fint_{Q_{R}}|F|^sdxdt\right).
\end{align*}
Then we derive \eqref{5hh21101318}. This completes the proof.
\end{proof}
\section{Global integral gradient bounds for parabolic equations }
The following good-$\lambda$ type estimate will be essential for our global estimates later.
\begin{theorem}\label{5hh23101312} Let $w\in \mathbf{A}_\infty$, $F\in L^s(\Omega_T,\mathbb{R}^N), s>1$. For any $\varepsilon>0,R_0>0$ one finds $\delta_1=\delta_1(N,\Lambda,s,\varepsilon,[w]_{\mathbf{A}_\infty})\in (0,1/2)$ and $\delta_2=\delta_2(N,\Lambda,s,\varepsilon,[w]_{\mathbf{A}_\infty},T_0/R_0)\in (0,1)$ and $\Lambda_0=\Lambda_0(N,\Lambda,s)>0$ such that if $\Omega$ is a $(\delta_1,R_0)$-Lip domain and $[A]_{R_0}\le \delta_1$ then there exists a unique solution $u\in L^s(0,T;W^{1,s}_0(\Omega))$ to equation \eqref{5hh070120148} satisfying
\begin{equation}\label{5hh16101311}
w(\{\mathcal{M}(|\nabla u|^s)>\Lambda_0\lambda, \mathcal{M}(|F|^s)\le \delta_2\lambda \}\cap \Omega_T)\le C\varepsilon w(\{ \mathcal{M}(|\nabla u|^s)> \lambda\}\cap \Omega_T)
\end{equation}
for all $\lambda>0$,
where the constant $C$ depends only on $N,\Lambda,s, T_0/R_0, [w]_{\mathbf{A}_\infty}$.
\end{theorem}
To prove above estimate, we will use L. Caddarelli and I. Peral's technique in \cite{CaPe}. Namely, it is based on the following technical lemma whose proof is a consequence of Lebesgue Differentiation Theorem and the standard Vitali covering lemma, can be found in \cite{55BW4,55MePh2} with some modifications to fit the setting here.
\begin{lemma}\label{5hhvitali2} Let $\Omega$ be a $(\delta,R_0)$-Reifenberg flat domain with $\delta<1/4$ and let $w$ be an $\mathbf{A}_\infty$ weight. Suppose that the sequence of balls $\{B_r(y_i)\}_{i=1}^L$ with centers $y_i\in\overline{\Omega}$ and radius $r\leq R_0/4$ covers $\Omega$. Set $s_i=T-ir^2/2$ for all $i=0,1,...,[\frac{2T}{r^2}]$. Let $E\subset F\subset \Omega_T$ be measurable sets for which there exists $0<\varepsilon<1$ such that $w(E)<\varepsilon w(\tilde{Q}_r(y_i,s_j))$ for all $i=1,...,L$, $j=0,1,...,[\frac{2T}{r^2}]$; and for all $(x,t)\in \Omega_T$, $\rho\in (0,2r]$, we have
$\tilde{Q}_\rho(x,t)\cap \Omega_T\subset F$
if $w(E\cap \tilde{Q}_\rho(x,t))\geq \varepsilon w(\tilde{Q}_\rho(x,t))$. Then $
w(E)\leq \varepsilon Bw(F)$
for a constant $B$ depending only on $N$ and $[w]_{\mathbf{A}_\infty}$.
\end{lemma}
\begin{proof}[Proof of Theorem \ref{5hh23101312}] By Theorem \ref{161120141}, we find $\delta_0=\delta_0(N,\Lambda,s)$ then there exists a unique solution $u$ to solution $u\in L^s(0,T;W^{1,s}_0(\Omega))$ to equation \eqref{5hh070120148} satisfying
\begin{align}\label{es-U}
\int_{\Omega_T}|\nabla u|^s dxdt\leq C \int_{\Omega_T}|F|^s dxdt
\end{align}
provided that $\Omega$ is a $(\delta,R_0)$-Lip domain and $[A]_{R_0}\le \delta$ for $\delta<\delta_0=\delta_0(N,\Lambda,s)$ and $R_0>0$. Let $\varepsilon \in (0,1)$. Set $$E_{\lambda,\delta_2}=\{\mathcal{M}(|\nabla u|^s)>\Lambda_0\lambda, \mathcal{M}(|F|^s)\leq \delta_2\lambda\}\cap \Omega_T $$ and $$F_\lambda=\{\mathcal{M}(|\nabla u|^s)>\lambda \}\cap \Omega_T$$ for $\delta_2\in (0,1),\Lambda>0$ and $\lambda>0$.
Let $\{y_i\}_{i=1}^L\subset \Omega$ and a ball $B_0$ with radius $2T_0$ such that
$$
\Omega\subset \bigcup\limits_{i = 1}^L {{B_{r_0}}({y_i})} \subset {B_0},$$
where $r_0=\min\{R_0/1000,T_0\}$. Let
$s_j=T-jr_0^2/2$ for all $j=0,1,...,[\frac{2T}{r_0^2}]$ and $Q_{2T_0}=B_0\times (T-4T_0^2,T)$. So,
\begin{equation*}
\Omega_T\subset \bigcup\limits_{i,j} {{Q_{r_0}}({y_i,s_j})} \subset {Q_{2T_0}}.
\end{equation*}
We verify that
\begin{equation}\label{5hh2310131}
w(E_{\lambda,\delta_2})\leq \varepsilon w({\tilde{Q}_{r_0}}({y_i,s_j})) ~~\forall ~\lambda>0,
\end{equation}
for some $\delta_2$ small enough depending on $N,s,\epsilon,[w]_{\mathbf{A}_\infty},T_0/R_0$.\\
In fact, we can assume that $E_{\lambda,\delta_2}\not=\emptyset$, so $\int_{\Omega_T}|F|^sdxdt\leq C|Q_{2T_0}|\delta_2\lambda$. Since $\mathcal{M}$ is a bounded operator from $L^1(\mathbb{R}^{N+1})$ into $L^{1,\infty}(\mathbb{R}^{N+1})$ and \eqref{es-U} we get
\begin{align*}
|E_{\lambda,\delta_2}|& \leq \frac{C}{\Lambda\lambda}\int_{\Omega_T}|\nabla u|^sdxdt\\& \leq
\frac{C}{\Lambda\lambda}\int_{\Omega_T}|F|^sdxdt\\&
\leq C\delta_2 |Q_{2T_0}|,
\end{align*}
which implies
\begin{align*}
w(E_{\lambda,\delta_2})\leq c\left(\frac{|E_{\lambda,\delta_2}|}{|Q_{2T_0}|}\right)^\nu w(Q_{2T_0})\leq C\delta_2^\nu w(Q_{2T_0}),
\end{align*}
where $(c,\nu)=[w]_{\mathbf{A}_\infty}$. It is well-known that (see, e.g \cite{55Gra}) there exist $c_1=c_1(N,c,\nu)$ and $\nu_1=\nu_1(N,c,\nu)$ such that
\begin{align*}
\frac{w(Q_{2T_0})}{w({\tilde{Q}_{r_0}}({y_i,s_j}))}\leq c_1\left(\frac{|Q_{2T_0}|}{|{\tilde{Q}_{r_0}}({y_i,s_j})|}\right)^{\nu_1}~~\forall i,j.
\end{align*}
Therefore,
\begin{align*}
w(E_{\lambda,\delta_2})\leq C\delta_2^\nu c_1\left(\frac{|Q_{2T_0}|}{|{\tilde{Q}_{r_0}}({y_i,s_j})|}\right)^{\nu_1} w({\tilde{Q}_{r_0}}({y_i,s_j}))
< \varepsilon w({\tilde{Q}_{r_0}}({y_i,s_j}))~~\forall ~i,j,
\end{align*}
for $\delta_2$ small enough depending on $N,s,\epsilon,[w]_{\mathbf{A}_\infty},T_0/R_0$. Thus \eqref{5hh2310131} follows.\\
Next we verify that for all $(x,t)\in \Omega_T$, $r\in (0,2r_0]$ and $\lambda>0$ we have
$
\tilde{Q}_r(x,t)\cap \Omega_T\subset F_\lambda
$
provided $$
w(E_{\lambda,\delta_2}\cap \tilde{Q}_r(x,t))\geq \varepsilon w(\tilde{Q}_r(x,t)),
$$
for some $\delta_2$ small enough depending on $N,s,\epsilon,[w]_{\mathbf{A}_\infty},T_0/R_0$.
Indeed,
take $(x,t)\in \Omega_T$ and $0<r\leq 2r_0$, we set $$\tilde{Q}_\rho=\tilde{Q}_\rho(x,t)~~\forall \rho>0.$$
Now assume that $\tilde{Q}_r\cap \Omega_T\cap F^c_\lambda\not= \emptyset$ and $E_{\lambda,\delta_2}\cap \tilde{Q}_r\not = \emptyset$ i.e, there exist $(x_1,t_1),(x_2,t_2)\in \tilde{Q}_r\cap \Omega_T$ such that $\mathcal{M}(|\nabla u|^s)(x_1,t_1)\leq \lambda$ and $\mathcal{M}(|F|^s)(x_2,t_2)\le \delta_2 \lambda$.
We need to prove that
\begin{equation}\label{5hh2310133}
w(E_{\lambda,\delta_2}\cap \tilde{Q}_r)< \varepsilon w(\tilde{Q}_r).
\end{equation}
Using $\mathcal{M}(|\nabla u|^2)(x_1,t_1)\leq \lambda$, we can see that
\begin{equation*}
\mathcal{M}(|\nabla u|^s)(y,t')\leq \max\left\{\mathcal{M}\left(\chi_{\tilde{Q}_{2r}}|\nabla u|^s\right)(y,t'),3^{N+2}\lambda\right\}~~\forall (y,t')\in \tilde{Q}_r.
\end{equation*}
Therefore, for all $\lambda>0$ and $\Lambda_0\geq 3^{N+2}$,
\begin{eqnarray}\label{5hh2310134}E_{\lambda,\delta_2}\cap \tilde{Q}_r=\left\{\mathcal{M}\left(\chi_{\tilde{Q}_{2r}}|\nabla u|^s\right)>\Lambda_0\lambda, \mathcal{M}(|F|^s)\leq \delta_2\lambda\right\}\cap \Omega_T \cap \tilde{Q}_r.
\end{eqnarray}
In particular, $E_{\lambda,\delta_2}\cap \tilde{Q}_r=\emptyset$ if $\overline{B}_{8r}(x)\subset\subset \mathbb{R}^{N}\backslash \Omega$.
Thus, it is enough to consider the case $B_{8r}(x)\subset\subset\Omega$ and the case $B_{8r}(x)\cap\Omega\not=\emptyset$.\\
First assume $B_{8r}(x)\subset\subset\Omega$. Let $v$ be as in Theorem \ref{5hh24092} with $Q_{2R}=Q_{8r}(x,t_0)$ and $t_0=\min\{t+2r^2,T\}$. We have
\begin{equation}\label{5hh2310135}
||\nabla v||^s_{L^\infty(Q_{2r}(x,t_0))}\leq C \fint_{Q_{8r}(x,t_0)}|\nabla u|^s dxdt +C\fint_{Q_{8r}(x,t_0)}|F|^s dxdt,
\end{equation}
and
\begin{align*}
\fint_{Q_{4r}(x,t_0)}|\nabla (u- v)|^sdxdt\leq C\fint_{Q_{8r}(x,t_0)}|F|^s dxdt+ C([A]_{R_0})^s\fint_{Q_{8r}(x,t_0)}|\nabla u|^sdxdt.
\end{align*}
Here constants $C$ in above two depend only $N,\Lambda,s$. \\
Thanks to $\mathcal{M}(|\nabla u|^s)(x_1,t_1)\leq \lambda$ and $\mathcal{M}(|F|^s)(x_2,t_2)\le \delta_2 \lambda$ with $(x_1,t_1),(x_2,t_2)\in Q_r(x,t)$, we find $Q_{8r}(x,t_0)\subset\tilde{Q}_{17r}(x_1,t_1),\tilde{Q}_{17r}(x_2,t_2) $ and
\begin{align}\nonumber
||\nabla v||^s_{L^\infty(Q_{2r}(x,t_0))}&\leq C\fint_{\tilde{Q}_{17r}(x_1,t_1)}|\nabla u|^s dxdt +C\fint_{\tilde{Q}_{17r}(x_2,t_2)}|F|^s dxdt\\&\nonumber\leq
C(1+\delta_2)\lambda\\&\leq
C\lambda,\label{1411201410}
\end{align}
and
\begin{align}\nonumber
\fint_{Q_{4r}(x,t_0)}|\nabla (u- v)|^2dxdt&\leq C\delta_2\lambda+ C([A]_{R_0})^s\lambda\\&\leq
C(\delta_2+\delta_1^s)\lambda.\label{1411201411}
\end{align} Here we used $[A]_{R_0}\leq \delta_1$ in the last inequality. \\
In view of \eqref{1411201410}, we have that for $\Lambda_0\geq \max\{3^{N+2},2C\}$, $C$ is the constant in \eqref{1411201410}.
\begin{align*} |\{\mathcal{M}\left(\chi_{\tilde{Q}_{2r}}|\nabla v|^s\right)>\Lambda_0\lambda/4\}\cap \tilde{Q}_r|=0.
\end{align*}
It follows that
\begin{align*}
|E_{\lambda,\delta_2}\cap \tilde{Q}_r|&\leq |\{\mathcal{M}\left(\chi_{\tilde{Q}_{2r}}|\nabla (u- v)|^s\right)>\Lambda_0\lambda/4\}\cap \tilde{Q}_r|.
\end{align*}
Therefore, $\mathcal{M}$ is a bounded operator from $L^1(\mathbb{R}^{N+1})$ into $L^{1,\infty}(\mathbb{R}^{N+1})$ and \eqref{1411201411}, $\tilde{Q}_{2r}\subset Q_{4r}(x,t_0)$ we deduce
\begin{align*}
|E_{\lambda,\delta_2}\cap \tilde{Q}_r|&\leq \frac{C}{\lambda}\int_{\tilde{Q}_{2r}} |\nabla (u-v)|^sdxdt \\&\leq C \left(\delta_2+\delta_1^s\right)|\tilde{Q}_r|.
\end{align*}
Thus,
\begin{align*}
w(E_{\lambda,\delta_2}\cap \tilde{Q}_r)&\leq c\left(\frac{|E_{\lambda,\delta_2}\cap \tilde{Q}_r |}{|\tilde{Q}_r|}\right)^\nu w(\tilde{Q}_r)
\\&\leq C\left(\delta_2+\delta_1^2\right)^\nu w(\tilde{Q}_r)
\\&< \varepsilon w(\tilde{Q}_r).
\end{align*}
where $\delta_2,\delta_1\leq \delta(N,\Lambda,s,\varepsilon,[w]_{\mathbf{A}_\infty})$ and $(c,\nu)=[w]_{\mathbf{A}_\infty}$.\\
Next assume $B_{8r}(x)\cap\Omega\not=\emptyset$. Let $x_3\in\partial \Omega$ such that $|x_3-x|=\text{dist}(x,\partial\Omega)$. Set $t_0=\min\{t+2r^2,T\}$. We have
\begin{equation}\label{5hh2310138}
Q_{2r}(x,t_0)\subset Q_{10r}(x_3,t_0)\subset Q_{5000r/\varepsilon_1}(x_3,t_0) \subset \tilde{Q}_{10^4r/\varepsilon_1}(x_3,t)\subset \tilde{Q}_{10^5r}(x,t)\subset \tilde{Q}_{10^6r}(x_1,t_1),
\end{equation}
and
\begin{equation}\label{5hh2310139}
Q_{5000r/\varepsilon_1}(x_3,t_0) \subset \tilde{Q}_{10^4r/\varepsilon_1}(x_3,t)\subset \tilde{Q}_{10^5r}(x,t)\subset \tilde{Q}_{10^6r}(x_2,t_2)
\end{equation}
Applying Theorem
\ref{5hh16101310} with $Q_{R}=Q_{5000r/\varepsilon_1}(x_3,t_0)$ and $\varepsilon=\delta_3\in (0,1)$, there exists a constant $\delta_0'=\delta_0'(N,\Lambda,s,\delta_3)\in (0,\delta_0)$ such that if $\Omega$ is a $(\delta_0',R_0)$-Lip domain then
\begin{equation*}
||\nabla V||^s_{L^\infty(Q_{10r}(x_3,t_0))}\leq C\fint_{Q_{5000r/\varepsilon_1}(x_3,t_0)}|\nabla u|^sdxdt+C\fint_{Q_{5000r/\varepsilon_1}(x_3,t_0)}|F|^sdxdt,
\end{equation*}
and
\begin{align*}
\nonumber&\fint_{Q_{10r}(x_3,t_0)}|\nabla (u-V)|^sdxdt\\&~~~~\leq C (\delta_3^s+[A]_{R_0}^s)\fint_{Q_{5000r/\varepsilon_1}(x_3,t_0)}|\nabla u|^sdxdt+ C\fint_{Q_{5000r/\varepsilon_1}(x_3,t_0)}|F|^sdxdt.
\end{align*}
Since $\mathcal{M}(|\nabla u|^s)(x_1,t_1)\leq \lambda$, $\mathcal{M}(|F|^s)(x_2,t_2)\le \delta_2 \lambda$ and \eqref{5hh2310138}, \eqref{5hh2310139} we get
\begin{align}\nonumber
||\nabla V||^s_{L^\infty(Q_{10r}(x_3,t_0))}&\nonumber\leq C\fint_{\tilde{Q}_{10^6r}(x_1,t_1)}|\nabla u|^sdxdt+C \fint_{\tilde{Q}_{10^6r}(x_1,t_1)}|F|^sdxdt\\&\nonumber\leq
C(1+\delta_2)\lambda
\\&\leq C\lambda,\label{1411201412"}
\end{align}
and
\begin{align}
\nonumber&\fint_{Q_{10r}(x_3,t_0)}|\nabla (u-V)|^sdxdt\leq C \left(\delta_3^s+([A]_{R_0})^s+ \delta_2\right)\lambda\\&\leq C\left(\delta_3^s+\delta_1^s+\delta_2\right)\lambda.\label{1411201412}
\end{align}
Notice that we have used $[A]_{R_0}\leq \delta_1$ in the last inequality.\\
As above we also have that for $\Lambda_0\geq \max\{3^{N+2},4C\}$, the constant $C$ is in \eqref{1411201412"}.
\begin{align*}
|E_{\lambda,\delta_2}\cap \tilde{Q}_r|&\leq |\{\mathcal{M}\left(\chi_{\tilde{Q}_{2r}}|\nabla (u- V)|^s\right)>\Lambda_0\lambda/4\}\cap \tilde{Q}_r|.
\end{align*}
Note that the constant $\Lambda_0$ depends only on $N,\Lambda,s$. \\
Therefore using \eqref{1411201412} we obtain
\begin{align*}
|E_{\lambda,\delta_2}\cap \tilde{Q}_r|&\leq \frac{C}{\lambda}\int_{\tilde{Q}_{2r}} |\nabla (u- V)|^sdxdt \\&\leq C \left(\delta_3^s+\delta_1^s+ \delta_2\right)|\tilde{Q}_r|.
\end{align*}
Thus
\begin{align*}
w(E_{\lambda,\delta_2}\cap \tilde{Q}_r)&\leq c\left(\frac{|E_{\lambda,\delta_2}\cap \tilde{Q}_r|}{|\tilde{Q}_r|}\right)^\nu w(\tilde{Q}_r)
\\&\leq C\left(\delta_3^s+\delta_1^s+\delta_2\right)^\nu w(\tilde{Q}_r)
\\&< \varepsilon w(\tilde{Q}_r),
\end{align*}
where $\delta_1,\delta_2,\delta_3\leq \delta'(N,\Lambda,s,\varepsilon,[w]_{\mathbf{A}_\infty})$ and $(c,\nu)=[w]_{\mathbf{A}_\infty}$.\\
Therefore, for all $(x,t)\in \Omega_T$, $r\in (0,2r_0]$ and $\lambda>0$, if
$$w(E_{\lambda,\delta_2}\cap \tilde{Q}_r(x,t))\geq \varepsilon w(\tilde{Q}_r(x,t)),$$
then $$ \tilde{Q}_r(x,t)\cap \Omega_T\subset F_\lambda,$$
where $\Omega$ is a $(\delta_1,R_0)$-Lip domain and $[A]_{R_0}\le \delta_1$ with $\delta_1=\delta_1(N,\Lambda,s,\varepsilon,[w]_{\mathbf{A}_\infty})\in (0,\delta_0)$, $\delta_2=\delta_2(N,\Lambda,s,\varepsilon,[w]_{\mathbf{A}_\infty},T_0/R_0)\in (0,1)$. Hence, combining this with \eqref{5hh2310131}, we can apply Lemma \ref{5hhvitali2} to get the result.
\end{proof}\medskip\\
\begin{proof}[Proof of Theorem \ref{101120143}] Let $F\in L^{q,p}_w(\Omega_T)$. Since $L^{q,p}_w(\Omega_T)\subset L^{q_0}(\Omega_T)$ for some $q_0>1$. By Theorem \ref{161120141}, we find $\delta_0=\delta_0(N,\Lambda,q,p)$ then there exists a unique very weak solution $u$ to solution $u\in L^{q_0}(0,T;W^{1,q_0}_0(\Omega))$ to equation \eqref{5hh070120148} satisfying
\begin{align*}
\int_{\Omega_T}|\nabla u|^{q_0} dxdt\leq C \int_{\Omega_T}|F|^{q_0} dxdt
\end{align*}
provided that $\Omega$ is a $(\delta,R_0)$-Lip domain and $[A]_{R_0}\le \delta$ for $\delta<\delta_0=\delta_0(N,\Lambda,q,p)$.\\
By Theorem \ref{5hh23101312}, for any $s\in (0,q_0), \varepsilon>0,R_0>0$ one finds $\delta=\delta(N,\Lambda,\varepsilon,s,[w]_{\mathbf{A}_\infty})\in (0,\delta_0)$ and $\delta_2=\delta_2(N,\Lambda,\varepsilon,s,[w]_{\mathbf{A}_\infty},T_0/R_0)\in (0,1)$ and $\Lambda_0=\Lambda_0(N,\Lambda)>0$ such that if $\Omega$ is a $(\delta,R_0)$- Lip domain and $[A]_{R_0}\le \delta$ then
\begin{equation}\label{1411201413}
w(\{\mathcal{M}(|\nabla u|^s)>\Lambda_0\lambda, \mathcal{M}[|F|^s]\le \delta_2\lambda \}\cap \Omega_T)\le C\varepsilon w(\{ \mathcal{M}(|\nabla u|^s)> \lambda\}\cap \Omega_T),
\end{equation}
for all $\lambda>0$,
where the constant $C$ depends only on $N,\Lambda,s, T_0/R_0, [w]_{\mathbf{A}_\infty}$.
Thus, for $s<\infty,$
\begin{align*}
& ||\mathcal{M}(|\nabla u|^s)||_{L^{q_1,p_1}_w(\Omega_T)}^{p_1}
=q_1\Lambda_0^{p_1}\int_{0}^{\infty}\lambda^{p_1}\left(w(\{\mathcal{M}(|\nabla u|^s)>\Lambda_0\lambda \}\cap \Omega_T)\right)^{p_1/q_1}\frac{d\lambda}{\lambda} \\&~~~\leq
q_1\Lambda_0^{p_1}2^{p_1/q_1}(C\varepsilon)^{p_1/q_1}\int_{0}^{\infty}\lambda^{p_1}\left(w(\{\mathcal{M}(|\nabla u|^s)>\lambda \}\cap \Omega_T)\right)^{p_1/q_1}\frac{d\lambda}{\lambda}
\\&~~~~+ q_1\Lambda_0^{p_1}2^{p_1/q_1}\int_{0}^{\infty}\lambda^{p_1}\left(w(\{\mathcal{M}(|F|^s)>\delta_2\lambda \}\cap \Omega_T)\right)^{p_1/q_1}\frac{d\lambda}{\lambda}
\\&~~~ = \Lambda_0^{p_1}2^{p_1/q_1}(C\varepsilon)^{p_1/q_1}||\mathcal{M}(|\nabla u|^s)||_{L^{q_1,p_1}_w(\Omega_T)}^{p_1}+\Lambda_0^{p_1}2^{p_1/q_1}\delta_2^{-p_1}||\mathcal{M}(|F|^s)||_{L^{q_1,p_1}_w(\Omega_T)}^{p_1}.
\end{align*}
It implies
\begin{align*}
||\mathcal{M}(|\nabla u|^s)||_{L^{q_1,p_1}_w(\Omega_T)}&\leq 2^{1/p_1}\Lambda_02^{1/q_1}(C\varepsilon)^{1/q_1}||\mathcal{M}(|\nabla u|^2)||_{L^{q_1,p_1}_w(\Omega_T)}\\&~~~+2^{1/p_1}\Lambda_0 2^{1/q_1}\delta_2^{-1}||\mathcal{M}(|F|^s)||_{L^{q_1,p_1}_w(\Omega_T)}
\end{align*}
and this inequalities is also true when $p_1=\infty$. \\
We can choose $\varepsilon=\varepsilon(N,\Lambda,q_1,p_1,C)>0$ such that $2^{1/p_1}\Lambda2^{1/q_1}(C\varepsilon)^{1/q_1}\leq 1/2$, then we get
\begin{align}\label{es-U'}
||\mathcal{M}(|\nabla u|^s)||_{L^{q_1,p_1}_w(\Omega_T)}\leq C||\mathcal{M}(|F|^s)||_{L^{q_1,p_1}_w(\Omega_T)}
\end{align}
Let $w\in \mathbf{A}_q $, there exists $q_2=q_2(N,q,[w]_{\mathbf{A}_q})\in (1,q)$ such that $[w]_{\mathbf{A}_{q_2}}\leq C_0=C_0(N,q, [w]_{\mathbf{A}_q})$.Thus,
\begin{align*}
||\mathcal{M}(|F|^{q/q_2})||_{L^{q_2,pq_2}_w(\Omega_T)}\leq C ||F||_{L^{q,p}_w(\Omega_T)}.
\end{align*} Applying \eqref{es-U'} to $s=q/q_2$ and $q=q_2,p_1=pq_2$, we have
\begin{align*}
|||\nabla u|||_{L^{q,p}_w(\Omega_T)}&\leq ||\mathcal{M}(|\nabla u|^{q/q_2})||_{L^{q_2,pq_2}_w(\Omega_T)}\\&\leq C||\mathcal{M}(|F|^{q/q_2})||_{L^{q_2,pq_2}_w(\Omega_T)}
\\&\leq C ||F||_{L^{q,p}_w(\Omega_T)}.
\end{align*}
We get the result. The proof is complete.
\end{proof}
|
1705.07409
|
\section{Introduction}
Every finite, simple, and undirected graph has at least two vertices of equal degree,
and this lower bound on the number of repeated degrees can be improved for restricted graph classes \cite{cw}.
Caro, Shapira, and Yuster \cite{csy} proved the surprising result that,
for every positive integer $k$,
there is a constant $c_k$ such that,
for every graph $G$, there is a set $X$ of at most $c_k$ vertices such that
$G-X$ has at least $\min\{ k,n(G)-|X|\}$ many vertices of equal degree,
where $n(G)$ denotes the order of $G$.
In \cite{cy} Caro and Yuster considered an analogous problem for the maximum degree.
For an integer $k$ at least $2$, and a graph $G$,
let $f_k(G)$ be the minimum cardinality of a set $X$ of vertices of $G$
such that $G-X$ has either $k$ vertices of maximum degree or order less than $k$.
Caro and Yuster pose the following intriguing conjecture.
\begin{conjecture}[Caro and Yuster \cite{cy}]\label{conjecturecy}
For every integer $k$ at least $2$, there is a constant $c_k$ such that
$f_k(G)\leq c_k \sqrt{n(G)}$ for every graph $G$.
\end{conjecture}
They describe graphs $G$ with $f_2(G)\geq (1-o(1))\sqrt{n(G)}$
showing that the upper bound in Conjecture \ref{conjecturecy} has the best possible growth rate,
that is, forcing many vertices of maximum degree is considerably harder than forcing many vertices of equal degree.
Furthermore, they verify the conjecture for $k\in \{ 2,3\}$
by showing that $c_2=\sqrt{8}$ and $c_3=43$ have the desired properties.
They also prove the following result, which implies the conjecture for $C_4$-free graphs.
\begin{theorem}[Caro and Yuster \cite{cy}]\label{theoremcy}
Let $k$ and $t$ be positive integers at least $2$.
If $G$ is a $K_{2,t}$-free graph of order at least $t^2{k\choose 2}^2$,
then $f_k(F)\leq (3k-3)\sqrt{n(G)}$.
\end{theorem}
In \cite{clz} Caro, Lauri, and Zarb show that $\sqrt{2}$ is the best possible value for $c_2$,
and, for forests $F$, they improve the growth rate of the upper bound on $f_k(F)$ from the second to the third root of the order as follows.
\begin{theorem}[Caro, Lauri, and Zarb \cite{clz}]\label{theoremclz}
If $k$ is an integer at least $2$, and $F$ is a forest of order at least $(2k-1)^3$,
then $f_k(F)\leq (2k-1)n(F)^{\frac{1}{3}}$.
\end{theorem}
For $k=2$, they formulate a precise conjecture,
and construct graphs showing that their conjecture would be tight.
\begin{conjecture}[Caro, Lauri, and Zarb \cite{clz}]\label{conjecture1}
If $t$ is a positive integer, and $F$ is a forest of order at most $\frac{1}{6}\left(t^3+6t^2+17t+12\right)$,
then $f_2(F)\leq t$.
\end{conjecture}
In the present paper we show this conjecture.
Furthermore, we study $f_3(F)$ for forests $F$ in more detail obtaining almost tight results,
and we give improved upper bounds on $f_k(G)$ for graphs $G$ of girth at least $5$.
For graphs $G$ of girth more than $2p$, for $p$ at least $3$,
our results imply $f_k(G)=O\left(n(G)^{\frac{p+1}{3p}}\right)$,
and we obtain considerable improvements of Theorem \ref{theoremclz}.
Finally, we show that, for every fixed integer $k$ at least $2$,
and every given forest $F$,
the value of $f_k(F)$ can be determined in polynomial time.
The influence of degree multiplicities on graph parameters or
large sets of vertices of equal degree satisfying additional properties
have been studied in several papers such as \cite{ab,bs};
see \cite{clz} for further discussion.
Before we proceed to our results, we collect some notation.
Let $G$ be a graph.
The size of $G$ is denoted by $m(G)$.
For a vertex $u$ of $G$, the degree of $u$ is denoted by $d_G(u)$.
The maximum degree of $G$ is denoted by $\Delta(G)$.
For an integer $n$, let $[n]$ be the set of all positive integers at most $n$.
If $G$ has order $n$ and degree sequence $d_1\geq d_2 \geq \ldots \geq d_n$,
then let $\Delta_i(G)$ be $d_i$ for $i\in [n]$; in particular,
$\Delta_1(G)$ is the maximum degree of $G$,
and $G$ has at least $k$ vertices of maximum degree if and only if $\Delta_1(G)=\Delta_k(G)$.
\section{Upper bounds}
Our first goal is the proof of Conjecture \ref{conjecture1}.
The following result from \cite{clz} was the key insight needed to obtain the best possible value for $c_2$.
\begin{theorem}[Caro, Lauri, and Zarb \cite{clz}]\label{lemma1}
If $t$ is a positive integer, and $G$ is a graph with $\Delta(G)\leq {t+2\choose 2}$, then $f_2(G)\leq t$.
\end{theorem}
Since a forest has less edges than vertices,
the following result immediately implies Conjecture \ref{conjecture1}.
\begin{theorem}\label{theorem1}
If $t$ is a positive integer, and $F$ is a forest of size less than $\frac{1}{6}\left(t^3+6t^2+17t+12\right)$,
then $f_2(F)\leq t$.
\end{theorem}
{\it Proof:} For a positive integer $t$, let $n(t)=\frac{1}{6}\left(t^3+6t^2+17t+12\right)$.
The proof is by induction on $t$.
Let $\Delta_i=\Delta_i(F)$ and let $u_i$ be such that $d_F(u_i)=\Delta_i$ for $i\in [2]$,
where $u_1$ and $u_2$ are distinct.
For $t=1$, we have $m(F)\leq n(1)-1=5$.
Clearly, we may assume that $f_2(F)>0$, that is, $\Delta_1>\Delta_2\geq 1$.
If $\Delta_2=1$, then $F$ is the union of a star $K_{1,\Delta_1}$ and copies of $K_1$ and $K_2$,
and removing $u_1$ yields two vertices of maximum degree $0$ or $1$.
Hence, we may assume that $\Delta_2\geq 2$,
which, using $m(F)\leq 5$, implies that $3\leq \Delta_1\leq 4$.
If $\Delta_1=3$, then removing a neighbor of $u_1$ that does not lie in $N_F[u_2]$
yields two vertices of maximum degree $2$.
Note that such a neighbor exists, because $F$ is a forest.
Hence, we may assume that $\Delta_1=4$,
which implies that $F$ arises by subdividing one edge of a star $K_{1,4}$ once,
and removing $u_1$ yields two vertices of maximum degree $1$.
Now, let $t\geq 2$.
If $\Delta_1\leq {t+2\choose 2}$, then Theorem \ref{lemma1} implies $f_2(F)\leq t$.
Hence, we may assume that $\Delta_1\geq {t+2\choose 2}+1$.
If $F'=F-u_1$, then
\begin{eqnarray*}
m(F') & = & m(F)-\Delta_1\\
& < & \frac{1}{6}\Big(t^3+6t^2+17t+12\Big)-\Big(\frac{1}{2}t^2+\frac{3}{2}t+2\Big)\\
& = & \frac{1}{6}\Big((t-1)^3+6(t-1)^2+17(t-1)+12\Big)\\
& = & n(t-1).
\end{eqnarray*}
By induction, we obtain $f_2(F)\leq 1+f_2(F')\leq 1+(t-1)=t$,
which completes the proof. $\Box$
\medskip
\noindent In order to better understand $f_k(F)$ for forests $F$, we first consider the case $k=3$.
Our next result suitably generalizes Theorem \ref{lemma1}.
\begin{theorem}\label{theorem2}
If $t$ is an integer at least $2$, and $F$ is a forest with $\Delta_1(F)+2\Delta_2(F)\leq {t+2\choose 2}+2$,
then $f_3(F)\leq t$.
\end{theorem}
{\it Proof:} The proof is by induction on $t$.
Clearly, we may assume that $F$ has at least three vertices,
and that $\Delta_1(F)>\Delta_3(F)$.
Let $\Delta_i=\Delta_i(F)$ and let $u_i$ be such that $d_F(u_i)=\Delta_i$ for $i\in [3]$,
where $u_1$, $u_2$, and $u_3$ are distinct.
For $t=2$, we have $\Delta_1+2\Delta_2\leq {2+2\choose 2}+2=8$.
If $\Delta_1=1$, then $F$ is the union of copies of $K_1$ and $K_2$,
and removing one vertex of degree $1$
yields either three vertices of maximum degree $0$,
or a graph with less than $3$ vertices.
Hence, we may assume that $\Delta_1\geq 2$.
If $\Delta_2=1$, then $F$ is the union of a star $K_{1,\Delta_1}$,
copies of $K_1$, and $p$ copies of $K_2$.
If $p=0$ or $p\geq 2$, then let $X=\{ u_1\}$, and,
if $p=1$, then let $X$ contain $u_1$ and exactly one vertex from the unique $K_2$ component.
It is easy to check that $F-X$ has three vertices of maximum degree.
Hence, we may assume that $\Delta_2\geq 2$,
which, using the upper bound on $\Delta_1+2\Delta_2$,
implies $\Delta_1\in \{ 2,3,4\}$ and $\Delta_2=2$.
First, we assume that $\Delta_1=2$.
Clearly, we may assume that $\Delta_3=1$.
If $u_1$ and $u_2$ are non-adjacent, then $F$ contains two copies of $P_3$,
and removing one endvertex from each copy yields four vertices of maximum degree $1$.
If $u_1$ and $u_2$ are adjacent, let $F$ contain $p$ $K_2$ components.
If $p=0$, then let $X=\{ u_1,u_2\}$, and,
if $p\geq 1$, then let $X=\{ u_1\}$.
It is easy to check that either $n(F-X)<3$ or
$F-X$ has three vertices of maximum degree.
Hence, we may assume that $\Delta_1\geq 3$.
If $\Delta_3=2$, then removing $\Delta_1-2$ neighbors of $u_1$
that do not belong to $N_F[u_2]\cup N_F[u_3]$
yields three vertices of maximum degree $2$.
Hence, we may assume that $\Delta_3=1$.
If $F$ has no $K_2$ component, then removing $u_1$ and $u_2$ yields
three vertices of maximum degree $0$.
Hence, we may assume that $F$ has a $K_2$ component.
If $u_1$ is adjacent to $u_2$, then let $X=\{ u_1\}$, and,
if $u_1$ is non-adjacent to $u_2$, then let $X$ contain $u_1$ and exactly one neighbor of $u_2$.
It is easy to check that
$F-X$ has three vertices of maximum degree.
Now, let $t\geq 3$.
First, suppose that $\Delta_1+\Delta_2-2\Delta_3\leq t$.
Clearly, we may assume that $\Delta_3\geq 1$.
If $\Delta_3=1$, then
either removing $u_1$ and $u_2$
or removing $\Delta_1-1$ neighbors of $u_1$ that do not belong to $N_F[u_2]$
and $\Delta_2-1$ neighbors of $u_2$ that do not belong to $N_F[u_1]$
yields three vertices of maximum degree $0$ or $1$.
Hence, we may assume that $\Delta_3\geq 2$.
Now, removing
$\Delta_1-\Delta_3$ neighbors of $u_1$ that do not belong to $N_F[u_2]\cup N_F[u_3]$
and
$\Delta_2-\Delta_3$ neighbors of $u_2$ that do not belong to $N_F[u_1]\cup N_F[u_3]$
yields three vertices of maximum degree $\Delta_3$.
Again, all these vertices exist, because $F$ is a forest.
Hence, we may assume that $\Delta_1+\Delta_2-2\Delta_3\geq t+1$.
Let $F'=F-u_1$.
Clearly, $\Delta_1(F')\leq \Delta_2$ and $\Delta_2(F')\leq \Delta_3$.
If $\Delta_2+2\Delta_3\leq {t+1\choose 2}+2$,
then $\Delta_1(F')+2\Delta_2(F')\leq {t+1\choose 2}+2$, and, by induction,
$f_3(F)\leq 1+f_3(F')\leq 1+(t-1)=t$.
Hence, we may assume that $\Delta_2+2\Delta_3\geq {t+1\choose 2}+3$,
and, we obtain
\begin{eqnarray*}
\Delta_1+2\Delta_2 & = & \Big(\Delta_1+\Delta_2-2\Delta_3\Big)+\Big(\Delta_2+2\Delta_3\Big)\\
& \geq & \Big(t+1\Big)+\left({t+1\choose 2}+3\right)\\
& = & {t+2\choose 2}+3,
\end{eqnarray*}
which is a contradiction.
$\Box$
\medskip
\noindent Since $f_3(K_{1,5}\cup P_3\cup P_3)=3$,
the base case of the induction in the previous proof is best possible.
Note that $f_3(K_{1,3}\cup K_2)=2$
shows that Theorem \ref{theorem2} is not true for $t=1$.
By a simple inductive argument,
Theorem \ref{theorem2} implies a lower bound on the sum of the largest degrees in terms of $f_3(F)$.
\begin{corollary}\label{corollary1}
If $t$ is an integer at least $2$, and $F$ is a forest with $f_3(F)>t$, then
\begin{enumerate}[(i)]
\item $\Delta_t(F)\geq 2$,
\item $\Delta_{t+1-i}(F)+2\Delta_{t+2-i}(F)\geq {i+2\choose 2}+3$ for every $i\in [t]\setminus \{ 1\}$, and
\item $\Delta_1(F)+\Delta_2(F)+\cdots+\Delta_t(F)\geq \frac{1}{18}t^3 + \frac{1}{3}t^2 + \frac{29}{18}t$.
\end{enumerate}
\end{corollary}
{\it Proof:} Let $\Delta_i=\Delta_i(F)$ and let $d_F(u_i)=\Delta_i$ for $i\in [t]$,
where $u_1,\ldots,u_t$ are distinct vertices.
\medskip
\noindent (i) Suppose that $\Delta_t\leq 1$.
If every vertex of degree $1$ is in $N_F[u_1]\cup\cdots\cup N_F[u_{t-1}]$,
then removing $X=\{ u_1,\ldots,u_{t-1}\}$ yields
three vertices of maximum degree $0$
or a forest of order less than $3$.
Hence, we may assume that $u_t$ is not adjacent to any vertex in $X$.
Now, either removing $X$ yields three vertices of maximum degree $1$,
or removing $X\cup \{ u_t\}$ yields three vertices of maximum degree $0$
or a forest of order less than $3$.
Hence, $\Delta_t\geq 2$.
\medskip
\noindent (ii) Suppose that
$\Delta_{t+1-i}+2\Delta_{t+2-i}\leq {i+2\choose 2}+2$ for some $i\in [t]\setminus \{ 1\}$.
If $X=\{ u_1,\ldots,u_{t-i}\}$,
then
$\Delta_1(F-X)+2\Delta_2(F-X)
\leq \Delta_{t+1-i}+2\Delta_{t+2-i}\leq {i+2\choose 2}+2,$
and, Theorem \ref{theorem2} implies the contradiction
$f_3(F)\leq (t-i)+f_3(F-X)\leq (t-i)+i=t$,
which completes the proof of (ii).
\medskip
\noindent (iii) By (i) and (ii), we obtain
\begin{eqnarray*}
\Big(\Delta_1+2\Delta_2\Big)+\Big(\Delta_2+2\Delta_3\Big)+\cdots+\Big(\Delta_{t-1}+2\Delta_t\Big)+\Delta_t
& \geq & \sum\limits_{i=2}^t\left({i+2\choose 2}+3\right)+2\\
& = & \frac{1}{6}t^3 + t^2 + \frac{29}{6}t - 4.
\end{eqnarray*}
Since $\Delta_1\geq 2$, this implies
$3\Big(\Delta_1+\Delta_2+\cdots+\Delta_t\Big)\geq \frac{1}{6}t^3 + t^2 + \frac{29}{6}t$,
which implies (iii).
$\Box$
\medskip
\noindent We obtain a result similar to Theorem \ref{theorem1}.
\begin{corollary}\label{corollary2}
If $t$ is an integer at least $2$, and $F$ is a forest of size less than
$\frac{1}{18}t^3 + \frac{1}{3}t^2 + \frac{11}{18}t+1$,
then $f_3(F)\leq t$.
\end{corollary}
{\it Proof:} Clearly, we may assume that $F$ has at least $t$ vertices.
Since
$\Delta_1(F)+\Delta_2(F)+\cdots+\Delta_t(F)\leq m(F)+(t-1)$,
Corollary \ref{corollary1}(iii) implies $f_3(F)\leq t$. $\Box$
\medskip
\noindent In order to understand how tight Corollary \ref{corollary2} actually is,
we construct forests $F$ with few edges and a large value of $f_3(F)$.
Therefore, let $a_1=1$, $a_2=3$, and, for every integer $i$ at least $3$, let
\begin{eqnarray}\label{e1}
a_i&=&\max\Big\{ a_{i-1},i-a_{i-1}+2a_{i-2}\Big\}.
\end{eqnarray}
It is easy to verify by induction that $a_{2i+1}=a_{2i}=i^2+i+1$ for every positive integer $i$.
For a positive integer $t$, let $F_t=K_{1,a_1}\cup K_{1,a_2}\cup\cdots\cup K_{1,a_t}$.
\begin{lemma}\label{lemma2}
If $t$ is a positive integer, then
$f_3(F_t)=t$ and $m(F_t)=\frac{t^3}{12}+O(t^2)$;
more precisely
$$m(F_t)=
\begin{cases}
\frac{2}{3}k^3+2k^2 +\frac{10}{3}k+ 1 & \mbox{, if $t=2k+1$, and}\\
\frac{2}{3}k^3+k^2 +\frac{7}{3}k & \mbox{, if $t=2k$.}
\end{cases}$$
\end{lemma}
{\it Proof:} Since the statement about the size of $F_t$ follows from a straightforward calculation
using the closed formula for the $a_i$, we only give details for the proof of $f_3(F_t)=t$.
Clearly, removing the $t$ centers of the stars results in an edgeless forest,
which implies $f_3(F_t)\leq t$.
Now, let $X$ be a minimum set of vertices of $F_t$ such that
$F_t-X$ has at least three vertices of maximum degree.
Let $\Delta=\Delta_1(F_t-X)$,
and, let $d_{F_t-X}(v_i)=\Delta$ for $i\in [3]$,
where $v_1$, $v_2$, and $v_3$ are distinct.
If $\Delta=0$, then clearly $|X|\geq t$.
Since removing the $t-2$ vertices of largest degree and $2$ endvertices from $K_{1,a_2}$
yields three vertices of maximum degree $1$ in the most efficient way,
if $\Delta=1$, then $|X|\geq (t-2)+2=t$.
Hence, we may assume that $\Delta\geq 2$,
which implies that $v_1$, $v_2$, and $v_3$ are distinct centers
of some star components $K_{1,a_i}$ of $F_t$.
Let
$v_1$ be the center of the component $K_{1,a_p}$,
$v_2$ be the center of the component $K_{1,a_q}$, and
$v_3$ be the center of the component $K_{1,a_r}$, where $p<q<r$.
Clearly, $X$ contains
$a_r-a_p$ neighbors of $v_3$,
$a_q-a_p$ neighbors of $v_2$,
and at least one vertex from every star component $K_{1,a_i}$ with $q<i<r$ or $r<i\leq t$. Using the monotonicity of the $a_i$ and ($\ref{e1}$), this implies
\begin{eqnarray*}
|X|
& \geq & (a_r-a_p)+(a_q-a_p)+(r-q-1)+(t-r)\\
& = & a_r+a_q-2a_p+(t-q-1)\\
& \stackrel{mon.}{\geq} & a_{q+1}+a_q-2a_{q-1}+(t-q-1)\\
& \stackrel{(\ref{e1})}{\geq} & (q+1)+(t-q-1)\\
& = & t,
\end{eqnarray*}
which completes the proof. $\Box$
\medskip
\noindent Lemma \ref{lemma2} implies that in any version of Corollary \ref{corollary2},
the upper bound on the size is at most $\frac{t^3}{12}+O(t^2)$,
that is, the bound in Corollary \ref{corollary2} might be improved by an asymptotic factor of $3/2$.
The following lemma will be used to extend Theorem \ref{theorem2} to graphs of girth at least $5$ and larger values of $k$.
\begin{lemma}\label{lemma3}
Let $k$ and $t$ be integers with $k\geq 2$ and $t\geq (k-1)^2$.
If $G$ is a graph of girth at least $5$, and
$$\Delta_1(G)+\cdots+\Delta_{k-1}(G)-(k-1)\Delta_k(G)\leq t,$$
then $f_k(G)\leq t$.
\end{lemma}
{\it Proof:} Let $\Delta_i=\Delta_i(G)$ and let $d_G(u_i)=\Delta_i$ for $i\in [k]$,
where $u_1,\ldots,u_k$ are distinct vertices.
First, suppose that $\Delta_k<k-1$.
We remove $u_1,\ldots,u_{k-1}$,
and, as long as the current graph has order at least $k$
but less than $k$ vertices of maximum degree,
we iteratively remove all vertices of maximum degree from the current graph.
Therefore,
removing $u_1,\ldots,u_{k-1}$,
at most $(k-1)$ further vertices of degree $k-2$,
at most $(k-1)$ further vertices of degree $k-3$,
and so on,
until at most $(k-1)$ further vertices of degree $1$,
yields either a graph with $k$ vertices of maximum degree
or a graph with less than $k$ vertices.
Since we removed at most $(k-1)+(k-1)(k-2)=(k-1)^2\leq t$ vertices,
we obtain $f_k(G)\leq t$.
Hence, we may assume that $\Delta_k\geq k-1$.
Let $i\in [k-1]$.
By the girth condition,
$u_i$ has at most $k-1$ neighbors in
$$N_i=N_G[u_1]\cup \cdots \cup N_G[u_{i-1}]\cup N_G[u_{i+1}]\cup \cdots \cup N_G[u_k].$$
Therefore, there are $\Delta_i-\Delta_k\leq \Delta_i-(k-1)$ neighbors of $u_i$
outside of $N_i$ whose removal results in a graph in which $u_i$ has degree $\Delta_k$.
Doing this for every $i$ in $[k-1]$ yields $k$ vertices of maximum degree $\Delta_k$.
$\Box$
\medskip
\noindent We proceed to the extension of Theorem \ref{theorem2}.
\begin{theorem}\label{theorem3}
Let $k$ and $t$ be integers with $k\geq 2$ and $t\geq (k-1)^2$.
There is some integer $c_k$ such that,
if $G$ is a graph of girth at least $5$, and
$$\Delta_1(G)+2\Delta_2(G)+3\Delta_3(G)+\cdots+(k-1)\Delta_{k-1}(G)\leq {t+2\choose 2}+c_k,$$
then $f_k(G)\leq t$.
\end{theorem}
{\it Proof:} Clearly, we may assume that $G$ has at least $t+k$ vertices.
Let $\Delta_i=\Delta_i(G)$ and let $d_G(u_i)=\Delta_i$ for $i\in [k]$,
where $u_1,\ldots,u_k$ are distinct vertices.
The proof is by induction on $t$.
First, let $t=(k-1)^2$.
Let $c_k$ be such that ${(k-1)^2+2\choose 2}+c_k=k-1$.
We obtain that $\Delta_1\leq k-1$, and
removing
at most $(k-1)$ vertices of degree $k-1$,
at most $(k-1)$ further vertices of degree $k-2$,
and so on,
until at most $(k-1)$ further vertices of degree $1$,
yields either a graph with $k$ vertices of maximum degree
or a graph with less than $k$ vertices.
Since we removed at most $(k-1)^2=t$ vertices,
we obtain $f_k(G)\leq t$.
Next, let $t>(k-1)^2$.
By Lemma \ref{lemma3}, we may assume that
$\Delta_1(G)+\cdots+\Delta_{k-1}(G)-(k-1)\Delta_k(G)\geq t+1$.
Similarly as in the proof of Theorem \ref{theorem2},
we may assume, by induction, that
$$\Delta_2(G)+2\Delta_3(G)+3\Delta_4(G)+\cdots+(k-1)\Delta_k(G)\geq {t-1+2\choose 2}+c_k+1.$$
Adding these two inequalities implies a contradiction,
which completes the proof. $\Box$
\medskip
\noindent Theorem \ref{theorem3} has several interesting consequences.
\begin{corollary}\label{corollary3}
Let $k$ be a fixed integer at least $2$.
There is a function $g:\mathbb{N}\to \mathbb{Z}$ with $|g(t)|=O(t^2)$ such that,
if $t$ is some positive integer, and
$G$ is a graph of size at most $\frac{t^3}{6{k\choose 2}}+g(t)$ and girth at least $5$,
then $f_k(G)\leq t$.
\end{corollary}
{\it Proof:} Choosing $g(t)$ equal to $-\frac{t^3}{6{k\choose 2}}$ for $t<(k-1)^2$,
the statement becomes trivial for $t<(k-1)^2$.
Hence, we may assume that $t\geq (k-1)^2$.
Let the graph $G$ of girth at least $5$ be such that $f_k(G)>t$;
in particular, $G$ has at least $t+k$ vertices.
Let $\Delta_i=\Delta_i(G)$ for $i\in [t]$.
Arguing similarly as in the proof of Corollary \ref{corollary1} (ii), we obtain that
$$\Delta_{t+1-i}+2\Delta_{t+2-i}+\cdots+(k-1)\Delta_{t+k-1-i}\geq {i+2\choose 2}+c_k+1$$
for every $i\in [t]\setminus \Big[(k-1)^2-1\Big]$.
Adding all these inequalities, we obtain, using $1+2+\cdots+(k-1)={k\choose 2}$, that
\begin{eqnarray*}
{k\choose 2}\Big(\Delta_1+\cdots+\Delta_{t+k-1-(k-1)^2}\Big) & \geq &
\sum\limits_{i=(k-1)^2}^t\left({i+2\choose 2}+c_k+1\right)
=\frac{t^3}{6}+O(t^2),
\end{eqnarray*}
where the implicit constants depend on the fixed value of $k$.
If $H$ is the subgraph of $G$ induced by the $t+k-1-(k-1)^2<t$ vertices of the largest degrees,
then
$$m(G)\geq
\Big(\Delta_1+\cdots+\Delta_{t+k-1-(k-1)^2}\Big)-m(H)
\geq \frac{t^3}{6{k\choose 2}}+O(t^2),$$
which completes the proof. $\Box$
\medskip
\noindent It is a simple consequence of the Moore bound \cite{ms} that,
for every positive integer $p$,
we have $m(G)\leq 2n(G)^{\frac{p+1}{p}}$
for every graph $G$ of girth more than $2p$.
\begin{corollary}\label{corollary4}
Let $k$ and $p$ be fixed integers with $k\geq 2$ and $p\geq 3$.
If $G$ has girth more than $2p$, then
$$f_k(G)\leq \Big(1+o(1)\Big)\left(12{k\choose 2}\right)^{\frac{1}{3}}n(G)^{\frac{p+1}{3p}}.$$
\end{corollary}
{\it Proof:} Let $G$ be a graph of girth more than $2p$, and let $t=f_k(G)-1$.
By the above consequence of the Moore bound and Corollary \ref{corollary3}, we obtain
\begin{eqnarray*}
n(G) & \geq & \left(\frac{1}{2}m(G)\right)^{\frac{p}{p+1}}
> \left(\frac{1}{2}\left(\frac{1}{6{k\choose 2}}+o(1)\right)t^3\right)^{\frac{p}{p+1}}
= \left(\left(\frac{1}{12{k\choose 2}}+o(1)\right)t^3\right)^{\frac{p}{p+1}}.
\end{eqnarray*}
This implies
$t<\Big(1+o(1)\Big)\Big(12{k\choose 2}\Big)^{\frac{1}{3}}n(G)^{\frac{p+1}{3p}}$,
which completes the proof. $\Box$
\medskip
\noindent Arguing in a similar way for forests, we obtain the following considerable improvement of Theorem \ref{theoremclz}.
\begin{corollary}\label{corollary5}
Let $k$ be a fixed integer with $k\geq 2$.
If $F$ is a forest, then
$$f_k(G)\leq \Big(1+o(1)\Big)\left(6{k\choose 2}\right)^{\frac{1}{3}}n(G)^{\frac{1}{3}}.$$
\end{corollary}
\section{An algorithm for forests}
In this section we describe an efficient algorithm calculating $f_k(F)$ for a given forest $F$.
Let $k$ be an integer at least $2$.
Let $T$ be a tree of order more than $k$,
let $S$ be a set of $k$ distinct vertices of $T$, and,
let $\Delta$ be some non-negative integer at most $\Delta(T)$.
The vertices in $S$ are called {\it special}.
We root $T$ in some non-special vertex $r$,
and, for every vertex $u$ of $T$,
we denote by $T(u)$ the subtree of $T$ rooted in $u$
and containing $u$ as well as all descendants of $u$.
For a vertex $u$ of $T$, let $(n_1(u),n_2(u),n_3(u))$ be a triple of integers, where
\begin{enumerate}[(i)]
\item $n_1(u)$ is the maximum order of an induced subforest $T_1(u)$ of $T(u)$ such that
\begin{itemize}
\item $u\not\in V(T_1(u))$,
\item $S\cap V(T(u))\subseteq V(T_1(u))$,
\item $\Delta(T_1(u))\leq \Delta$, and
\item $d_{T_1(u)}(v)=\Delta$ for every vertex $v\in S\cap V(T(u))$.
\end{itemize}
Note that, if $u$ is special, then $n_1(u)=\max\emptyset$, which, by convention, is $-\infty$.
\item $n_2(u)$ is the maximum order of an induced subforest $T_2(u)$ of $T(u)$ such that
\begin{itemize}
\item $\{ u\} \cup \Big(S\cap V(T(u))\Big)\subseteq V(T_2(u))$,
\item $\Delta(T_2(u))\leq \Delta$, and
\item $d_{T_2(u)}(v)=\Delta$ for every vertex $v\in \{ u\}\cup \Big(S\cap V(T(u))\Big)$.
\end{itemize}
\item $n_3(u)$ is the maximum order of an induced subforest $T_3(u)$ of $T(u)$ such that
\begin{itemize}
\item $\{ u\} \cup \Big(S\cap V(T(u))\Big)\subseteq V(T_3(u))$,
\item $\Delta(T_3(u))\leq \Delta$,
\item $d_{T_3(u)}(v)=\Delta$ for every vertex $\Big(S\cap V(T(u))\Big)\setminus \{ u\}$, and
\item if $u$ is special, then $d_{T_3(u)}(u)=\Delta-1$,
and, if $u$ is non-special, then $d_{T_3(u)}(u)\leq \Delta-1$.
\end{itemize}
\end{enumerate}
If $u$ is a non-special leaf of $T$, then
$$(n_1(u),n_2(u),n_3(u))=
\begin{cases}
(0,1,-\infty) & \mbox{, if $\Delta=0$ and}\\
(0,-\infty,1) & \mbox{, if $\Delta\geq 1$,}
\end{cases}$$
and, if $u$ is a special leaf of $T$, then
$$(n_1(u),n_2(u),n_3(u))=
\begin{cases}
(-\infty,1,-\infty) & \mbox{, if $\Delta=0$,}\\
(-\infty,-\infty,1) & \mbox{, if $\Delta=1$, and}\\
(-\infty,-\infty,-\infty) & \mbox{, if $\Delta\geq 2$.}
\end{cases}$$
The following lemma gives recursions for non-leaf vertices of $T$.
\begin{lemma}\label{lemma4}
Let $u$ be a non-leaf vertex of $T$, where we use the notation introduced above.
Let $v_1,\ldots,v_p$ be the special children of $u$,
and, let $w_1,\ldots,w_q$ be the non-special children of $u$.
Let
$$n_3(w_1)-n_1(w_1)\geq n_3(w_2)-n_1(w_2)\geq \ldots \geq n_3(w_q)-n_1(w_q).$$
If $n_3(w_1)-n_1(w_1)<0$, let $q'=0$, and,
if $n_3(w_1)-n_1(w_1)\geq 0$,
let $q'\in [q]$ be maximum such that $n_3(w_{q'})-n_1(w_{q'})\geq 0$.
\begin{enumerate}[(i)]
\item If $u$ is non-special, then
$$n_1(u)=
\sum_{i=1}^p n_2(v_i)+\sum_{j=1}^q\max\Big\{ n_1(w_j),n_2(w_j),n_3(w_j)\Big\}.$$
\item If $p>\Delta$ or $p+q<\Delta$, then $n_2(u)=-\infty$, and, if $p\leq \Delta\leq p+q$, then
$$n_2(u)=
\sum_{i=1}^p n_3(v_i)
+\sum_{j=1}^{\Delta-p}n_3(w_j)
+\sum_{j=\Delta-p+1}^{q}n_1(w_j).$$
\item If $u$ is special, and $p>\Delta-1$ or $p+q<\Delta-1$, then $n_3(u)=-\infty$,
and, if $u$ is special, and $p\leq \Delta-1\leq p+q$, then
$$n_3(u)=
\sum_{i=1}^p n_3(v_i)
+\sum_{j=1}^{\Delta-1-p}n_3(w_j)
+\sum_{j=\Delta-p}^{q}n_1(w_j).$$
\item If $u$ is non-special and $p>\Delta-1$, then $n_3(u)=-\infty$,
and, if $u$ is non-special and $p\leq \Delta-1$, then
$$n_3(u)=
\sum_{i=1}^p n_3(v_i)
+\sum_{j=1}^{\min\{ q',\Delta-1-p\}}n_3(w_j)
+\sum_{j=\min\{ q',\Delta-1-p\}+1}^{q}n_1(w_j).$$
\end{enumerate}
\end{lemma}
{\it Proof:} (i) Since $u$ does not belong to $T_1(u)$,
for every special child $v_i$ of $u$,
the forest $T_1(u)\cap T(v_i)$
has at most as many vertices as $T_2(v_i)$,
and,
for every non-special child $w_j$ of $u$,
the forest $T_1(u)\cap T(w_j)$
has at most as many vertices as
the forest of largest order in $\{ T_1(w_j),T_2(w_j),T_3(w_j)\}$,
which implies that $n_1(u)$ is at most the specified value.
On the other hand,
combining the mentioned forests in the obvious way,
it follows that $n_1(u)$ is also at least the specified value,
which completes the proof of (i).
\medskip
\noindent (ii) If $p>\Delta$ or $p+q<\Delta$,
then no forest with the properties required for $T_2(u)$
exists, and, hence, $n_2(u)=-\infty$.
If $p\leq \Delta\leq p+q$,
then, since $u$ belongs to $T_2(u)$ and has degree exactly $\Delta$ in $T_2(u)$,
for every special child $v_i$ of $u$,
the forest $T_2(u)\cap T(v_i)$
has at most as many vertices as $T_3(v_i)$,
there are exactly $\Delta-p$ non-special children $w_j$ of $u$
that belong to $T_2(u)$,
and the forest $T_2(u)\cap T(w_j)$ for such a $w_j$
has at most as many vertices as $T_3(w_j)$,
and, for the remaining $q-(\Delta-p)$ non-special children $w_j$ of $u$
that do not belong to $T_2(u)$,
the forest $T_2(u)\cap T(w_j)$ for such a $w_j$
has at most as many vertices as $T_1(w_j)$.
In view of the ordering of the non-special children $w_j$ of $u$,
this implies that $n_2(u)$ has at most the specified value.
Again, on the other hand,
combining the mentioned forests in the obvious way,
it follows that $n_2(u)$ is also at least the specified value,
which completes the proof of (ii).
\medskip
\noindent Since the proof of (iii) is almost identical to the proof of (ii),
we proceed to the proof of (iv).
Since $u$ belongs to $T_3(u)$ and has degree at most $\Delta-1$,
some non-special child $w_j$ of $u$ may only belong to $T_3(u)$
if $n_3(w_j)-n_1(w_j)\geq 0$,
which easily implies (iv) arguing similarly as for the proof of (ii). $\Box$
\begin{theorem}\label{theorem4}
For a fixed integer $k$ at least $2$, and a given forest $F$, the value $f_k(F)$
can be determined in polynomial time.
\end{theorem}
{\it Proof:} Clearly, we may assume that $F$ has more than $k$ vertices.
Let $S$ be a set of $k$ distinct vertices of $F$, and let $\Delta$ be a non-negative integer at most $\Delta(F)$.
If $F$ is disconnected, then we add a vertex $r$ with a neighbor in each component of $F$,
and denote the resulting tree by $T$.
Otherwise, let $T=F$, and let $r$ be a vertex of $F$ that does not belong to $S$.
Using the recursions from Lemma \ref{lemma4},
we can determine, in polynomial time,
$(n_1(r),n_2(r), n_3(r))$ for $T$,
denoted by
$\left(n_1^{(S,\Delta)}(r),n_2^{(S,\Delta)}(r),n_3^{(S,\Delta)}(r)\right)$
for this specific choice of $S$ and $\Delta$.
If $F$ is connected, then $n(F)-f_k(F)$ equals
$$
\max\left\{
\max\left\{ n_1^{(S,\Delta)}(r),n_2^{(S,\Delta)}(r),n_3^{(S,\Delta)}(r)\right\}:
S\in{ V(F)\choose k}\mbox{ and }\Delta\in \{ 0\}\cup [\Delta(F)]\right\},$$
and,
if $F$ is not connected, then $n(F)-f_k(F)$ equals
$$
\max\left\{ n_1^{(S,\Delta)}(r):
S\in{ V(F)\choose k}\mbox{ and }\Delta\in \{ 0\}\cup [\Delta(F)]\right\}.$$
Since these maxima are taken over polynomially many values,
the desired statement follows.
$\Box$
\medskip
\noindent It seems possible yet challenging to extend this approach to graphs of bounded tree width.
|
1705.07608
|
\section{Introduction}
The realization of nanoscale, magnetic skyrmions in metallic multilayer films has generated a surge of research \cite{Romming2013, MoreauLuchaire2016, Woo2016,Boulle2016}.
Understanding the structure and behavior of these localized, two-dimensional (2D) spin-textures is fundamental \cite{Nagaosa2013, Soumyanarayanan2016a, Wiesendanger2016}, with implications for spintronics technologies.
The unique properties of skyrmions stem from their topologically non-trivial spin-configuration. The spin at the center of a skyrmion is opposite to the out-of-plane (OP) spin direction of the background [{Fig.~\ref{fig:skyrmionfield_ODE}(a-d)], and reverses over a length-scale defining the skyrmion radius ($\rsk$), which can vary from a few nanometers to microns \cite{Romming2015,Jiang2015}.
The in-plane (IP) spin component winds chirally with helicity $\gamma$ \cite{Nagaosa2013}, ranging from \Neel \cite{Wiesendanger2016} ($\gamma=0,\pi$) to Bloch \cite{Yu2010} ($\gamma=\pm\pi/2$) texture [Fig.~\ref{fig:skyrmionfield_ODE}(a-d)].
Skyrmions are generated by the anti-symmetric \DM\ interaction (DMI) found in chiral magnets \cite{Muhlbauer2009, Lee2009, Yu2010, Kiselev2011, Milde2013} and at ferromagnet/heavy-metal interfaces \cite{Fert1990, Bode2007}.
Efforts to realize interfacial DMI have rapidly shifted from epitaxial monolayers \cite{Boulle2016} to sputtered multilayer films that host columnar room-temperature (RT) skyrmions \cite{MoreauLuchaire2016, Boulle2016, Woo2016, Nandy2016, Soumyanarayanan2017}.
The properties of multilayer skyrmions show considerably more variation than their epitaxial counterparts.
First, $\rsk$ can be inhomogeneous, with up to $\times2$ variations over a $\mathrm{\mu m}$-range \cite{Woo2016}.
Next, the spin structure can evolve in all three dimensions with columnar skyrmions potentially consisting of inertial cores \cite{Woo2016, Boulle2016, Buttner2015}.
Finally, the granularity of magnetic interactions can result in varying skyrmion configurations \cite{Soumyanarayanan2017}, which affect stability, dynamics, and switching properties \cite{Legrand2017}.
Any effort to understand and exploit such skyrmions requires spatially resolved information about their static properties (e.g. size, helicity, robustness to perturbations), and an understanding of how these influence their dynamic behavior.
Here we use magnetic force microscopy (MFM) to investigate magnetic textures in a {[}Ir(1)/Fe(0.5)/ Co(0.5)/Pt(1){]}$_\mathrm{20}$ (in parenthesis -- thickness in nm) multilayer film sputtered on a $\mathrm{SiO_2}$ substrate \cite{SuM}. Such multilayers host RT skyrmions \cite{Soumyanarayanan2017}, which we find persist down to $T=5$~K.
MFM is an established technique for magnetic characterization on the nanoscale, with unique, yet-untapped advantages for investigating skyrmions.
First, MFM allows for high-resolution imaging of magnetic textures in films and devices on substrates, enabling direct comparisons with transport and thermodynamic techniques \cite{Soumyanarayanan2017}.
Next, while MFM has been used for direct/in-situ imaging of skyrmion dynamics \cite{Hrabec2017,Legrand2017}, using it in conjunction with a quantitative physical model enables determination of individual skyrmion properties across the disordered magnetic landscape.
Crucially, the magnetic MFM tip, when in close proximity to skyrmions, provides a unique window into the response of individual skyrmions to perturbations (c.f. vortices \cite{Auslaender2009}), which may facilitate experimentally-driven modeling of mobility and switching by charge and spin currents.
Motivated by the potential of quantitative MFM, we utilize it here to investigate the characteristics of individual skyrmions. To provide an accurate physical description of the MFM signal we develop a multipole expansion for the magnetic field from skyrmions (MEFS), and fit it to our model using only two free parameters per skyrmion.
Our fit results enable us to determine {(i)} their \Neel texture and helicity ($|\gamma|<\pi/2$), {(ii)} to quantify $\rsk$, {(iii)} to map the spatial variation of their properties, and {(iv)} to estimate the force required to move individual skyrmions.
In this work, skyrmions were stabilized at $5$~K by finite OP magnetic field, $\mu_0H$, after saturation at $-0.5$~T. MFM imaging was performed by rastering a magnetic tip above the planar ($x-y$) surface of the sample. The MFM signal arises from the variation of the tip-sample interaction force ($F_z$) induced by oscillating the height between $h$ and $h+2a$, which we track by measuring the change in resonant frequency ($\Delta f$) of the cantilever holding the tip \cite{Albrecht1991}. Such a response can be well-described provided that the cantilever motion is harmonic and that $\Delta f\ll f_{0}$, the free resonant frequency \cite{Giessibl1997}. Adapting to MFM raster scanning \cite{SuM}, the 2D Fourier transform (FT) of $\Delta f$ is related to the FT of $\partial F_{z}/\partial h$, by $\widehat{\Delta f}_{\mathbf{q}}(h)={\cal T}(qa)\widehat{\partial {F}_z/\partial h}$,
where $q= (q_x^2+q_y^2)^{1/2}$ and ${\cal T}(qa)=-k_0^{-1}f_0I_1(qa)\exp{(-qa)}/(qa)$. Here
$k_0\approx1$~N/m is the spring constant of the cantilever, $I_1(x)$ is a Bessel function, $f_0\approx75$~kHz, $a\approx 30$~nm, and $h\lesssim 50$~nm for sufficiently high resolution. As expected for $a\rightarrow0$, ${\cal T}(qa) \approx -f_0/2k_0$ \cite{Albrecht1991}.
Figure~\ref{fig:data}(a) is a typical MFM image acquired at $-0.3$~T. We identify the small round features as skyrmions \cite{Soumyanarayanan2017}.
As the tip and sample were polarized together, the uniformly magnetized background interacts weakly with the tip, with small variations indicating disorder \cite{Bacani2016}.
In contrast, the skyrmions, magnetized opposite to the background, display a much stronger interaction with the tip.
The skyrmions are randomly dispersed suggesting that disorder is more important than skyrmion-skyrmion interactions under these conditions.
The disorder reveals its role also in the zoom in Fig.~\ref{fig:data}(c), which shows that the skyrmions are not identical.
\begin{figure}
\centering
\includegraphics[width=3.4in]{"Yagil_et_al_Fig1"}
\caption
\textbf{(a)} MFM image for $\mu_0H=-0.3$~T with $h=50~\mathrm{nm}$. White frames show zoom areas for (c), (e).
\textbf{(b)} Result of the MEFS fit assuming each skyrmion is different.
\textbf{(c)} Zoom on visibly different skyrmions.
\textbf{(d)} Line-cuts through the the two skyrmions in (c)
\textbf{(e)} Zoom on an individual skyrmion. Arrows indicate the positions of line-cuts in (f). \textbf{Insets:} the difference between the data and the MEFS fit [left, detail from Fig.~\ref{fig:diff_fit}(b)], or a 2D Gaussian fit (right). [Scale bars: $200$~nm.]
\textbf{(f)} Line-cuts through the data in (d) offset for clarity (x's), and the fit (lines). }
\label{fig:data}
\end{figure}
We now focus on understanding the signal profile of individual skyrmions [cf. Fig.~\ref{fig:data}(c),(e)].
Previously their profile has been fit to a standard line-shape, e.g. an isotropic Gaussian [cf. Fig.~\ref{fig:data}(e), right inset]. Here we present an improved framework for describing the profile [Fig.~\ref{fig:data}(e), left inset], which is physically justified from a microscopic model, more accurate, and helps unveil useful skyrmion characteristics.
In particular, we have found that the sum of a dipolar field and a quadrupolar field describes the magnetic field of a skyrmion well [cf. fit in Fig.~\ref{fig:data}(b)].
Below we describe the motivation for this description, and examine the relationship between the dipole ($P_{i}$) and quadrupole ($Q_{ij}$) moments and the MFM signal.
\begin{figure}
\centering
\includegraphics[width=3.4in]{Yagil_et_al_Fig2}
\caption
\textbf{(a-d)} Schematic spin texture in \Neel (a,b) and Bloch (c,d) skyrmions with $m=1$ and $\gamma=0,\pi,\pm\pi/2$.
\textbf{(e)} Plot of $\theta(\rho)$ for $k=1.536$, $b=1.474$ \cite{magpar} from a numerical solution of Eq.~\ref{eq:ODE} (circles), and a fit to Eq.~\ref{eq:romming} (line), which gives: $\sigma=0.140$, $\rho_0=0.037$. The vertical lines show $\rho_0$ (solid) and $\rho_0+\sigma$ (dashed). $\rho=2\pi\rsk/L_D$ corresponds to $\theta/\pi=0.5$.
\textbf{(f)} Illustration of the bottom of the tip, length-scales that we use, and a cut through a $\gamma=0$ \Neel\ skyrmion.
\textbf{(g)} $\mu_0H_z$, in a cut through the center of a skyrmion in a $d=20$~nm film for various values of $w=h+d/2$, with the curves offset for clarity.
Shown are $\mu_0H_z$ obtained from $\theta(\rho)$ in (e) as well as from the dipole and quadrupole fields ($\equiv H^P_z$, $\equiv H^Q_z $) from Eqs.~\ref{eq:P},~\ref{eq:Q} with $\gamma=0$.
\textbf{(h)} $H^Q_z/H^P_z$ vs. $w$ calculated using Eqs.~\ref{eq:P},~\ref{eq:Q} for $\gamma=0,\pm \pi/2,\pi$ as a function of $w$ for $R=0$.
}
\label{fig:skyrmionfield_ODE}
\end{figure}
The magnetic field generated by the skyrmion magnetization ($\mathbf{M}$) determines its MFM signature.
For a uniformly magnetized thin film hosting an axially-symmetric skyrmion with vorticity $m$ magnetized along $\pm\hat{z}$, $\mathbf{M}$ is given by \cite{Nagaosa2013}:
$\mathbf{M}(\rho,z)/M_s(z)=\sin\theta(\rho)\cos\psi(\varphi)\hat{x}+\sin\theta(\rho)\sin\psi(\varphi)\hat{y}+
\left[\pm1+\cos\theta(\rho)\right]\hat{z}$.
Here $\psi(\varphi)=m\varphi+\gamma$, where $\varphi$ is the axial angle, $\theta$ is the polar angle, $\rho=2\pi r/L_{D}$ ($r$ is the distance from the skyrmion center, $L_{D}\equiv4\pi A/|D|$ is the domain wall thickness, $A$ is the exchange stiffness and $D$ is the micromagnetic DMI strength). Meanwhile $M_{s}(z)=M_{s}^{0}\left[\Theta({z+d/2})-\Theta({z-d/2})\right]$, where $M_{s}^{0}$ is the saturation magnetization, $d$ is the film thickness, and $\Theta(z)$ is the Heaviside function. $\theta(\rho)$ is a solution to well-known ordinary differential equations \cite{Leonov2016a} with appropriate boundary conditions. For $\theta(0)=\pi$, $\theta(\infty)=0$, and $m=1$ we have \cite{Nagaosa2013, Soumyanarayanan2016a}:
\begin{equation}\label{eq:ODE}
\theta'' + \frac{\theta'+2\sin^2\!\theta}{\rho}-\sin\!\theta\cos\!\theta\left({\rho}^{-2}+k\right) -b\sin\!\theta=0.
\end{equation}
Here $b=B/B_D$ ($B_D\equiv D^2/2AM_s^0$), and $k=4AK/D^2$, where $K$ is the effective anisotropy. Fig.~\ref{fig:skyrmionfield_ODE}(e) shows a solution for parameters typical to multilayers \cite{magpar}.
The magnetic field from a localized magnetic structure can be described by a multipole expansion \cite{Jackson1998}.
For this purpose we define a magnetic scalar potential $\mathbf{H}=-\nabla\Phi$ \cite{SuM}.
The first term of the resulting MEFS is proportional to $P_{i}\equiv-\int r_{i}{\mathbf{\nabla}\cdot\mathbf{M}}\,dv$, the second to $Q_{ij}\equiv-\int\left(3r_{i}r_{j}-\mathbf{r}^{2}\delta_{ij}\right){\mathbf{\nabla}\cdot\mathbf{M}}\,dv$. For axially symmetric skyrmions with $m=1$, $\mathbf{P}=P\hat{z}$ and $Q_{ij}$ is diagonal with $Q_{xx}=Q_{yy}=-Q_{zz}/2\equiv Q$. Thus
\begin{equation}\label{eq:multipole}
\Phi\left(r,h\right)\approx\frac{1}{4\pi}\left(\frac{Pw(h)}{\left[r^2+w(h)^2\right]^{3/2}}+\frac{Q}{2}\frac{r^2-2w(h)^2}{\left[r^2+w(h)^2\right]^{5/2}}\right),
\end{equation}
where $w(h)\equiv h+d/2$ and \cite{SuM}:
\begin{eqnarray}
\label{eq:P} P &=&
2\pi M_s^0d\left(\frac{L_D}{2\pi}\right)^2\int_0^\infty\!\!\! d\rho\rho \left\{\pm1+\cos\left[\theta(\rho)\right]\right\}, \\
\label{eq:Q} Q&=&2\pi M_s^0d\left(\frac{L_D}{2\pi}\right)^3\cos{\gamma}\int_0^\infty\!\!\!d\rho\rho^2\sin\left[\theta(\rho)\right].
\end{eqnarray}
The sign of $P$ corresponds to the OP magnetization of the skyrmion ($\pm\hat{z}$). The sign of $Q$ indicates whether the IP magnetization points away ($+$) or towards ($-$) the center, thus determining the helicity of \Neel skyrmions, which is difficult to extract from other techniques \cite{Pulecio2016}. Importantly, for Bloch skyrmions $Q=0$.
To estimate $P$ and $Q$, we approximate the solution of Eq.~\ref{eq:ODE} \cite{Romming2015}:
\begin{equation}\label{eq:romming}
\pi-\theta(\rho)\approx\sin^{-1}\left[\tanh\left(\eta_+\right)\right]+\sin^{-1}\left[\tanh\left(\eta_-\right)\right],
\end{equation}
where $\eta_\pm\equiv(\rho\pm\rho_0)/\sigma$, $\sigma$ parameterizes the domain wall thickness and $\rho_0$ the skyrmion radius. For $\rho_0/\sigma\ll1$ $Q/|P|\approx\rsk\cos\gamma$, where $M_z(r=\rsk)=0$ \cite{SuM}.
For Fig.~\ref{fig:skyrmionfield_ODE}(e) a fit to Eq.~\ref{eq:romming} gives $\sigma=0.140$, $\rho_0=0.037$.
The analytical model is substantiated by numerical calculations of the magnetic field from a $\gamma=0$ skyrmion in a film with $d=20$~nm, as illustrated in Fig.~\ref{fig:skyrmionfield_ODE}(f).
A comparison between the exact solution and MEFS, shown in Fig.~\ref{fig:skyrmionfield_ODE}(g), suggests that Eq.~\ref{eq:multipole} describes the stray field very well with the quadrupolar contribution increasing gradually as $w(h)$ is reduced.
Figure~\ref{fig:skyrmionfield_ODE}(h) shows the ratio between the quadrupole ($H_z^Q$) and the dipole ($H_z^P$) contributions to $H_z$, from Eqs.~\ref{eq:multipole}-\ref{eq:Q} with $\gamma=0,\pm\pi/2,\pi$.
As expected, for Bloch skyrmions ($\gamma=\pm\pi/2$) $Q=0$, and the sign of $Q$ is opposite for the two kinds of \Neel skyrmions ($\gamma=0,\pi$). Therefore $Q/P$ allows the direct determination of skyrmion helicity.
To fit the MFM data using Eq.~\ref{eq:multipole}, we model the tip as a thin shell with axial symmetry (\cite{SuM}), and illustrated in Fig.~\ref{fig:skyrmionfield_ODE})}: $M^t_z\!(\mathbf{R},z)=m_0 \delta\left[|\mathbf{R}|-g(z)\right].$
Here $m_{0}=M_{0}\,t$, where $M_{0}$ is the tip magnetization and $t$ is the thickness of the magnetic coating; $z$ is along the tip axis and $g(z)$ is the radius of the tip in a constant-$z$ cut. Given $g(z)$, Eq.~\ref{eq:multipole} implies \cite{SuM}:
\begin{eqnarray}\label{eq: wide tip int}
\frac{\partial F_z}{\partial h}=&&-C\!\!\int_0^\infty\!\!\!\!\!\!\! dq \!\! \int_0^\infty\!\!\!\!\!\!\! dz q^4\left(1-q\frac{Q}{2P}\right
J_0[g(z)q]g(z)J_0(rq)\nonumber\\ &&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times e^{-q\left[w(h)+z\right]},
\end{eqnarray}
where we have used the FT of $\Phi\left(r,h=0\right)$, $J_0(x)$ is a Bessel function, and $C$ is a constant proportional to $P$ that determines the scale of the skyrmion-tip interaction.
A fit using Eq.~\ref{eq: wide tip int} is computationally expensive. Therefore, we first determine the skyrmion positions, peak magnitude ($\Delta f_\mathrm{max}$) and full-width-at-half-maximum (FWHM) by fitting to a simplified model of the tip \cite{SuM}.
Next, we fit the signal from the skyrmions using a more accurate model for the tip: $g(z)=\alpha (z^4+\beta z)^{{1}/{4}}$, with $\alpha=0.24$ and $\beta=2.7\cdot10^{6}$~$\mathrm{nm}^{3}$ \cite{SuM}.
This fit includes only two free parameters per skyrmion ($C$ and $Q/P$), as the positions are set from the fit to the simplified tip model, $\alpha$ and $\beta$ are determined from scanning electron microscopy, and we measure $d$ and $h$.
Figure~\ref{fig:diff_fit}(a) shows the dependence of $\chi$ (root-mean-square of the error) on $Q/P$ for a representative skyrmion for three values of $w$, including the actual value for the data in Fig.\ref{fig:data}(a), $w(h)=82$~nm. This value includes $2$~nm for a capping-layer, and the $d=60$~nm magnetic part of the stack \cite{SuM}. For all skyrmions we find a single global minimum corresponding to the optimal $Q/P$, that becomes shallower for larger $w(h)$. As expected, $h$ and $d$ have a direct impact on how precisely $Q/P$ can be determined. Based on such analysis we conclude that $Q/P\gtrsim0$ for the skyrmions in our film, indicating \Neel texture
\begin{figure}
\centering
\includegraphics[width=3.4in]{"Yagil_et_al_Fig3"}
\caption
\textbf{(a)} Plot of $\chi(Q/P)$ for a representative skyrmion for the measured $w(h)=82$~nm. The other curves show the influence of $w$. For each $Q/P$ we chose $C$ that minimizes $\chi$ in the area marked by the dashed rectangle in (b) after subtracting the contributions of all other skyrmions, which were fit using a simpler model \cite{SuM}.
\textbf{(b)} The difference between data and fit in Fig.~\ref{fig:data}(a),(b). The dashed rectangle shows a $650$~nm square, where $\chi$ is calculated for a representative skyrmion [in (a)].
The location of representative skyrmions showing discontinuous (circles) or negligible (rectangles) residual are highlighted.
}
\label{fig:diff_fit}
\end{figure}
Figure~\ref{fig:data}(b) shows the fit and Fig.~\ref{fig:diff_fit}(b) the residual we obtain upon repeating this fitting procedure for all skyrmions in Fig.~\ref{fig:diff_fit}(a). This reveals several subtle features.
First, the nanoscale variations in the background that are typical of the inhomogeneous magnetic structure of sputtered multilayer films \cite{Bacani2016}.
Second, are discontinuities for some skyrmions [e.g. circles in Fig.~\ref{fig:diff_fit}(b)], likely due to MFM tip-induced skyrmion motion. Other explanations, such as irregular skyrmion shapes, cannot give such sharp fit residuals.
These observations, in conjunction with the variability in skyrmion properties, are direct consequences of inhomogeneous magnetic interactions \cite{Bacani2016}, and reinforce the need for individual fit parameters to accurately describe multilayer skyrmions.
Figure~\ref{fig:indi_fit_hist} shows histograms of the individual skyrmion parameters we obtain from the fit to Fig.~\ref{fig:data}(a).
Figure~\ref{fig:indi_fit_hist}(a) shows that FWHM, which includes tip effects, varies by $\sim20\%$ ($100-120$~nm). Its larger magnitude compared to RT values for similar films \cite{Soumyanarayanan2017} is likely due to the changed magnetic parameters at $5$~K.
Notably however, the uniform FWHM [Fig.~\ref{fig:indi_fit_hist}(a)] is in contrast to the considerable variability of $\Delta f_\mathrm{max}$ [Fig.~\ref{fig:indi_fit_hist}(b)], indicating a significant variation in the stray field strength of the skyrmions.
\begin{figure}
\centering
\includegraphics[width=3.4in]{"Yagil_et_al_Fig4"}
\caption{Histograms and plots of skyrmion parameters from the fit to Fig.~\ref{fig:data}(a). Dots: mean, horizontal bars: $70\%$ confidence intervals.
\textbf{(a,b)} FWHM and $\Delta f_\mathrm{max}$.
\textbf{(c)} Histograms of $Q/P$ and a plot of $C$ vs. $Q/P$. The large error bars reflect the proximity of the resolution limit. The narrow distribution in (c) is directly from the fit, while the wide distribution is derived by accounting for the large error bars by assuming they are normally distributed.
\textbf{(d)} $C$.
[The data in (a)-(d) was taken from $91$ out of $104$ skyrmions which did not exhibit residual discontinuities, cf. Fig.~\ref{fig:diff_fit}(b)].
}
\label{fig:indi_fit_hist}
\end{figure}
The model allows us to go beyond conventional MFM to extract a typical length scale ($Q/P$) that, unlike FWHM, is dissociated from both the tip shape and the effect of $h$, and can therefore be lower. This ability to extract information on true length scales indicates the power of MEFS.
Figure~\ref{fig:indi_fit_hist}(c) shows histograms for $Q/P$. The narrow histogram is for the actual fit results, but as the fit uncertainty is large [cf. $C$ vs. $Q/P$ and Fig.~\ref{fig:diff_fit}(a)], we generated the wider histogram. For this we assumed that each value of $Q/P$ is drawn from a normal distribution $\mathcal{N}(Q/P,\delta\frac{Q}{P})$ with the width $\delta\frac{Q}{P}$ given by the error shown in the $C(Q/P)$ plot.
We conclude that {$Q/P\approx34$~nm} (standard deviation $27$~nm).
This number represents the shape of skyrmions that do not exhibit discontinuities. By comparing to scans with larger $h$ we conclude that changes induced by the field from the tip are too subtle for us to observe. Integrating the wide histogram we find that $Q/P>0$ with probability $0.9$.
This likely rules out Bloch skyrmions and implies that our skyrmions have \Neel texture with helicity $|\gamma|<\pi/2$. While this is consistent with \Neel skyrmions with helicity $\gamma=0$ [Fig.~\ref{fig:skyrmionfield_ODE}(a),(f)], we cannot rule out the presence of a partial Bloch component \cite{Rowland2016}.
Figures~\ref{fig:indi_fit_hist}(c),(d) show no correlation between $C$ and $Q/P$, and that the relative spread of $C$ is smaller ($C=61\pm9~\mathrm{nN\cdot nm^2}$).
The contrasting spreads are likely due to the inherent sensitivity of $C$ to $P$, rather than to $Q/P$. $Q/P$ provides finer information on the skyrmion structure \cite{SuM}, and is therefore more sensitive to disorder, which in turn contributes to its spread. Crucially, a key utility of our model is the ability to calculate the force exerted by the tip on skyrmions. We estimate that a skyrmion with the mean $Q/P$ and $C$ experiences a lateral force of $F_{tip}\approx1$~pN as a result of interaction with the MFM tip \cite{SuM}. As this force was sufficient to move only some of the skyrmions, we estimate $F_\mathrm{pin}\approx F_\mathrm{tip}$, where $F_\mathrm{pin}$ is the typical force required to move a skyrmion. Using the Lorentz force to estimate a critical current for adiabatic manipulation of skyrmions, we obtain \cite{Lin2013} $J=(F_\mathrm{pin}/d)/(h/e)\approx4\cdot10^{9}\mathrm{A/m^2}$. This $5$~K value is smaller than RT values reported previously on similar samples \cite{Woo2016}, and indicates that accounting for non-adiabatic processes and interlayer interactions may provide an improved estimate for bottom-up predictive modeling of skyrmion dynamics \cite{Finocchio2016}.
In summary, we have shown that MFM images of skyrmions can be quantitatively reproduced by modeling the magnetic field from a skyrmion using a closed expression from a multipole expansion with two free parameters per skyrmion, with several conclusions.
First, based on $|Q/P|\gtrsim0$ we can rule out with $\approx90\%$ certainty the skyrmions in our Ir/Fe/Co/Pt multilayers as purely Bloch textured.
The sign of $Q/P$ independently establishes that these skyrmions are \Neel textured with helicity $|\gamma|<\pi/2$, consistent with micromagnetic calculations \cite{Soumyanarayanan2017}.
Second, the magnitude of $Q/P$ provides the estimate $\rsk=36\pm28$~nm for $\gamma=0$.
Third, the spread of $\rsk$ and the $\Delta f_\mathrm{max}$ can be directly used to estimate the corresponding inhomogeneity of magnetic interactions.
In particular, $\Delta f_\mathrm{max}$ [Fig.~\ref{fig:indi_fit_hist}(b)] is expected to be sensitive to variations in $D$ \cite{Bacani2016}, and $\rsk$ is expected to be sensitive to variations in $\Keff$ \cite{Kim2017}.
Fourth, we have estimated the pinning force skyrmions experience, and the critical current density for skyrmion motion.
Finally, the utility of the physical analysis we presented beyond MFM, the compatibility with device configurations, and the relative computational simplicity that allows to apply it easily to large arrays of skyrmions, all bode well for future use towards both applications and basic science.
\begin{acknowledgments}
We are grateful for input from D. Arovas, K. Kuchuk, Shi-Zeng Lin, D. Podolsky, Y. Shechner, I. Schlesinger, U. Sivan, and A. Turner. The work in Technion was supported by the Israel Science Foundation (Grant no. 1897/14). The work in Singapore was supported by the Ministry of Education (MoE) -- Academic Research Fund (Ref. No. MOE2014-T2-1-050), the National Research Foundation -- NRF Investigatorship (Reference No. NRF-NRFI2015-04), and the A{*}STAR Pharos Fund (1527400026). We would also like to thank the Micro Nano Fabrication Unit at the Technion.
\end{acknowledgments}
\end{document}
|
1704.06278
|
\section{Introduction}
Several observations highlight the presence of tiny, unresolved
structure in atomic gas across a wide range of astrophysical
environments. For instance, the wide, smooth emission lines in quasar
spectra suggest the atomic gas close to the black hole has both a
suprathermal velocity dispersion but low volume filling factor
\citep[e.\,g.][]{Rees1987,1997MNRAS.288.1015A}. Moreover, studies of
the diffuse gas in the halos of massive galaxies at redshifts
$z\sim{2-3}$ routinely find that these galaxies are filled with tiny
clouds of neutral gas, again with a high covering factor, but with a
low overall volume filling factor
\citep[e.\,g.][]{Rauch1999,Cantalupo2014,Hennawi2015}. Similar
evidence for tiny-scale structure in neutral gas may be found in
galactic winds and in high-velocity clouds in the Milky Way (see,
e.\,g. \citealt{McCourt2016} for a summary).
The physical origin of these clumps has been investigated recently by
\citet{McCourt2016}, who find that cooling gas clouds are prone to
rapid fragmentation akin to the Jeans instability. They suggest this
fragmentation rapidly ``shatters'' cold gas into tiny cloudlets of a
characteristic size $l \sim 0.1{\,\rm pc} (n / \,\ifmmode{{\rm cm}}\else cm\fi^{-3})^{-1}$, or
equivalently a column density $N_{\mathrm{cloudlet}} \sim
10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$. \citet{McCourt2016} argue that this length scale is
consistent with a number of observational upper limits, but
unfortunately such small scales are extremely difficult to probe
directly in distant objects. In this paper, we show that radiative
transfer of the resonant Ly$\alpha$ line can indeed probe sub-parsec
scales, even in distant galaxies.
There are several reasons why the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi emission line hydrogen is an ideal probe for tiny-scale structures.
As the most prominent transition line of the most abundant element, \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi is a sensitive probe of neutral gas enabling us to study even the most distant objects in the Universe.
Recently, instruments such as
\textit{MUSE} \citep{2010SPIE.7735E..08B} reveal the ubiquity of \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi emission throughout the observable space. In particular, \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi is used to study our galactic neighborhood \citep{Hayes2015}, galaxies at the peak of cosmic star formation \citep{Barnes2014}, and the later stages of reionization \citep{Dijkstra2014_review}.
Apart from this practical reason, the resonant nature of the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi
transition gives \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi observations a potentially great constraining
power in studying otherwise unresolvable structure. This is due to
the strong frequency dependence of the neutral hydrogen scattering cross
section which leads to many orders of magnitude of variation in the photon
mean free path. For instance, in a medium with one neutral hydrogen
atom per $\,\mathrm{cm}^{-3}$, a \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photon travels on average only $\sim 1\,$AU
if it is in the core of the line; however, this distance grows by
nearly five orders of magnitude to $\sim 0.5\,$pc if the frequency is
shifted merely five Doppler widths ($\sim{60}$\,km/s) away from line
center. The mean free is therefore also sensitive to gas motions on
the scale of $\sim(1-100)$\,km/s, providing powerful constraints on
the kinematic properties of galaxies and their surrounding environments.
In this paper, we revisit Ly$\alpha$ radiative transfer through a
simplified clumpy medium consisting of spherical ``clouds'' of neutral
hydrogen embedded in an ionized surrounding medium. While this setup
has been considered many times before
\citep[e.\,g.][]{Neufeld1991,Hansen2005,Laursen2012,Duval2013,Gronke2016a},
all of these previous studies have focused on a part of the parameter
space with a relatively low ($\lesssim 10$) number of clumps per
sightline. In light of the observations and the shattering scenario
discussed above, we now consider the limit with many more clouds per
sightline, exploring the full range from $\sim{1}$ to $\sim{1000}$s.
We show that this has tremendous influence on the propagation of
Ly$\alpha$, and we provide simple scaling relations which enable
simple, order-of-magnitude calculations for Ly$\alpha$ radiative
transfer through clumpy media.
Our paper is structured as follows. In Sec.~\ref{sec:analyt-cons-new}, we discuss the problem analytically and estimate the expected results.
In Sec.~\ref{sec:method} we describe briefly our \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer calculations, and introduce our model. We present the simulation results in Sec.~\ref{sec:results} with particular focus on the spectral shape and the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction as well as their connections to a corresponding homogeneous medium. We then discuss the results in Sec.~\ref{sec:discussion} before we conclude in Sec.~\ref{sec:conclusion}.
\section{Analytic results}
\label{sec:analyt-cons-new}
We find several distinct regimes for \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer in multiphase
media, which we summarize in figure~\ref{fig:sketch_regimes}. In this
section, we describe the physics relevant to each regime and provide
analytic estimates for the boundaries separating them. Since it will
prove essential for our analysis, we first review some general results
about \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape from a homogeneous slab
(\S~\ref{sec:radi-transf-homog}) before describing radiative transfer
in clumpy medium (\S~\ref{sec:radi-transf-clumpy}). We confirm these
analytic results numerically in section~\ref{sec:results} using
Monte Carlo radiative transfer simulations.
\input{content/table_static-regimes}
\subsection{Definitions \& notation conventions}
\label{sec:general-conventions}
The basics of \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer have been described in the literature \citep[e.g., recently in a review by][]{Dijkstra2014_review} and will not be repeated in detail here. Instead, we summarize the most relevant quantities for our present applications.
\begin{itemize}
\item We express the photon's frequency $\nu$ in terms of its Doppler parameter
\begin{equation}
x = \frac{\nu - \nu_0}{\Delta \nu}\;,
\end{equation}
where $\nu_0 \approx 2.47\times 10^{15}\,{\rm s}^{-1}$ is the frequency at line center, and $\Delta\nu_D = v_{\rm th} \nu_0 / c = \sqrt{2 k_B T/m_H} \nu_0 / c$ is the line width due to thermal motions of the atoms.
\item Temperature dependence is expressed through the Voigt parameter
\begin{equation}
a_v = \frac{\Delta\nu_L}{2 \Delta_D} \approx 4.7\times 10^{-4} \left(\frac{T}{10^4\,{\rm K}}\right)^{-1/2}\;.
\end{equation}
Here, $\Delta\nu_L= 9.939\times 10^7\,\mrm{s}^{-1}$ is the natural
(i.\,e., quantum mechanical) line broadening due to the finite lifetime of the transition.
\item The \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi cross section of neutral hydrogen is
\begin{align}
\sigma_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace}(x, T) =&
\sigma_0 H(a_v, x) \nonumber \\
=& \frac{\sigma_0 a_v}{\pi} \int\limits_{-\infty}^{\infty}\mathrm{d} y \frac{e^{-y^2}}{(y - x)^2 + a_v^2}
\end{align}
where $\sigma_0 \approx 5.895\times 10^{-14} (T / 10^4\,\mrm{K})^{-1/2}\,\ifmmode{{\rm cm}}\else cm\fi^{2}$ denotes its value at line center and $H(a_v, x)$
is the Voigt function which can be approximated as $H(a_v,
x)\sim{e^{-x^2}}$ in the core of the line and as $\sim
a_v/(\sqrt{\pi}x^2)$ in the wing of the line. The transition occurs
at a frequency $x_* \approx 3.26$ for $T = 10^4\,$K. The normalized Voigt distribution
$\phi(x) = H(a_v,x) / \sqrt{\pi}$ represents the probability of a photon in the frequency interval $[x\pm \mathrm{d} x / 2]$ to interact with an atom.
\item The optical depth per length $d$ is, hence,
\begin{equation}
\tau(x) = \int\limits_0^d\mathrm{d} s\, \sigma_{\rm HI}(x) n_{\rm HI}(s)
\end{equation}
where $n_{\rm HI}$ denotes the number density of neutral hydrogen atoms. Note that we did not include the contribution of dust in the above expression as its impact is modelled in post-process (see \S~\ref{sec:dust-within-clumps} for details).
\end{itemize}
\subsection{Radiative transfer in a homogeneous slab}
\label{sec:radi-transf-homog}
Since it is crucial for our analytic work below, we briefly review some classical solutions for \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer in a semi-infinite (that is, only one dimension is finite) slab with half-height $B$ and optical depth $\tau_0$.
\ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape can be seen as a random-walk in both real space and frequency space, as every scattering event (that is, absorption and quick re-emission from a neutral hydrogen atom) alters the frequency and direction of the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photon. However, due to the large value of $\sigma_{0}$, the mean free path of a photon close to line center is very small ($\lambda_{\mathrm{mfp}}\sim 5.5 \times 10^{-6} \left(n / \,\ifmmode{{\rm cm}}\else cm\fi^{-3}\right)^{-1}\,$pc for $T=10^4\,$K), and most scatterings are spatially close to each other.
Thus, the vast majority of scatterings do not contribute significantly towards the escape of the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photon (at least in optically thick media\footnote{For lower optical depths (when $\tau(x_*)\lesssim 1$), an escape in single flight is possible even in the Doppler core. Such escape occurs via rare scattering events when a photon near line center encounters a fast moving atom, with large velocities perpendicular to the photon's direction. When this photon is re-emitted, it will be far from line center, and if $\tau(x)\approx\tau_{0}{\rm e}^{-x^{2}} < 1$, it can escape (also see \S~\ref{sec:divis-betw-regim}).}). Instead, \citet{Adams1972} found that \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons escape in several consecutive wing-scatterings where the mean free path is significantly enhanced (for instance, $\lambda_{\mathrm{mfp}} \sim 0.48\,$pc at $x=5$ for the above setup).
The random-walk in frequency space is therefore crucial to the escape
of Ly$\alpha$.
These series of wing-scatterings are referred to as `excursion', and this is thought of as the common way \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons propagate in an astrophysical context.
To estimate the average displacement per `excursion,' one has to take into account its random walk in frequency. Specifically, a \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photon in the wing of the line at frequency $x$ has a slight tendency to return to the core of the line with mean frequency shift per scattering event of $-1/x$ \citep{Osterbrock1962}. This means it will scatter $\sim x^2$ times before returning to the core with a mean free path of $l = B \sigma_0 / (\sigma_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace}(x) \tau_0) = B / (H(x) \tau_0)$
between each scatter. Since an excursion itself can be seen as a random walk, \citet{Adams1972} obtained $d_{\mrm{exc}}= \sqrt{N_{\mrm{sct, exc}}} l = x B / (H(x) \tau_0)$ as mean distance per excursion. Furthermore, by using the wing-approximation $H(x)\sim a_v/(\sqrt{\pi} x^2)$ described above, and setting $d_{\mrm{exc}}= \sqrt{3} B$ \citep[where the factor $\sqrt{3}$ arises due to geometrical considerations,][]{Adams1975}, he obtained
\begin{equation}
x_{\mrm{esc}} = \left(\tau_0 a_v \sqrt{3/\pi}\right)^{1/3} \approx 6.5 \left(\frac{N_{\text{H\MakeUppercase{\romannumeral 1}}}\xspace}{10^{19}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}} \frac{10^4\,\mathrm{K}}{T} \right)^{1/3}
\label{eq:xesc_adams}
\end{equation}
as an expression for the most likely escape frequency.
\citet{Adams1972} continues to calculate the number of scatterings it takes for a photon to reach a frequency $|x| \ge x_{\mrm{esc}}$ which allows for escape. In an optically thick medium
photons undergo many scatterings and the frequency distribution $J(x)$ is roughly constant\footnote{Like any diffusive process, frequency diffusion can be represented by a Fokker-Planck equation, for which the steady state solution is $J(x)=\mathrm{const}$. This is independent of the form of the frequency diffusion coefficient (and thus independent of the redistribution function).}. Thus, the probability to find an arbitrary photon with frequency in the interval $[x\pm \mathrm{d} x / 2]$ is $\sim \phi(x) \mathrm{d} x$ (complete redistribution approximation\footnote{This approximation holds only for $|x|\lesssim x_{\mathrm{esc}}$, beyond which photons leave the system and $J(x)$ tends towards zero \citep[over the intervals $\pm \lbrack x_{\mathrm{esc}},\,2 x_{\mathrm{esc}}\rbrack$,][]{Adams1972}. Taking this into account only changes the pre-factors by order unity.}). However, a given photon will scatter $\sim x^2$ times at the frequency $\sim x$.
Consequently, $\sim x^2$ scattering events are not \textit{into} a frequency interval which allows for escape, and thus the probability to scatter \textit{into} $[x\pm \mathrm{d} x / 2]$ is $\sim \phi(x)/x^2 \mathrm{d} x$. This implies a cumulative escape probability
\begin{equation}
P_{\mrm{esc}} = 2 \int\limits_{x_{\mrm{esc}}}^{\infty}\mathrm{d} x \frac{\phi(x)}{x^2} = \frac{2 a_v}{3 \pi x_{\mrm{esc}}^3}\;,
\end{equation}
where in the last equality we have used the wing-approximation for $\phi(x)$. The number of scatterings required to escape is $1/P_{\mrm{esc}}$ and plugging in $x_{\mrm{esc}}$ from Eq.~\eqref{eq:xesc_adams} one obtains
\begin{equation}
N_{\mrm{sct}}^{\mrm{esc}} \approx 4.6 \tau_0 \approx 2.7\times 10^6 \left(\frac{N_{\text{H\MakeUppercase{\romannumeral 1}}}\xspace}{10^{19}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}}\right) \left(\frac{T}{10^4\,\mathrm{K}} \right)^{-1/2}\;.
\label{eq:N_sct_esc}
\end{equation}
This relation differs only by a factor of a few with the exact solution of \citet{1973MNRAS.162...43H} which has been backed up by numerical results \citep[e.g.,][]{1979ApJ...233..649B,2006ApJ...649...14D}.
In summary, \citet{Adams1972} found that a typical \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photon leaving
an optically-thick slab scatters a large number of times
essentially in-place (Eq.~\eqref{eq:N_sct_esc}), until reaching the
frequency $x_{\mrm{esc}}$ (Eq.~\eqref{eq:xesc_adams}), after which it
escapes undergoing $N_{\mrm{sct}}^{\mrm{exc}}\sim x^{2}_{\mrm{esc}}$
scattering interactions in the wing of the line.
\subsection{Radiative transfer in clumpy medium}
\label{sec:radi-transf-clumpy}
Resonant line transfer in a clumpy medium has fundamentally different
behavior than in a homogeneous medium, because much of the distance
can be traversed in the optically-thin medium between the clumps. As
we discussed in the previous section, due to its highly variable
interaction cross section, \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escapes through `excursion' in regimes
where the mean free path at the initial frequency is short. In a
multiphase medium, however, a significant fraction of the volume may
have no neutral hydrogen at all. The gas opacity thus varies
strongly with position, even at line center. This opens up an
alternate escape route in which \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons ``solve the maze'' by
scattering into the optically thin medium between clouds. This
possibility is essential to consider, since astrophysical systems such
as the ISM and CGM are thought to have a multiphase nature
\citep[e.g.,][]{McKee1977}.
\subsubsection{Model parameters}
In this section, we describe the expected propagation of \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photon in a clumpy medium, which we model using spherical clumps of radius $r_{\mathrm{cl}}$ and {\text{H\MakeUppercase{\romannumeral 1}}}\xspace number density $n_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}}$ placed in an otherwise empty, semi-infinite slab of height $2B$.
In what follows, we will use the clump column density $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}}=r_{\mathrm{cl}} n_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}}$ and optical depth $\tau_{\mathrm{cl}}(x, T) = N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}}\sigma_{\text{H\MakeUppercase{\romannumeral 1}}}\xspace(x, T)$ as convenient notation.
The most important parameter of a clumpy medium is the covering factor \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi which describes the average number of clumps per orthogonal sightline between the midplane and the surface of the slab. These sight-lines will intercept a column density of $N_{\rm HI, total} = \frac{4}{3} \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi N_{\rm HI, cl}$ where the factor $4/3$ is due to the spherical geometry of the clumps\footnote{The mean path length through a sphere of radius $r$ is $\mathrm{Volume} / \mathrm{Area} = 4/3 \pi r^3 / (\pi r^2) = 4/3 r$.}.
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/regimes_sketch_new.pdf}
\caption{Sketch of radiative transfer regimes in a static, clumpy medium discussed in \S~\ref{sec:radi-transf-clumpy}. The $x$-axis shows the total optical depth at line center and $y$-axis the covering factor \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi.}
\label{fig:sketch_regimes}
\end{figure}
\subsubsection{Escape regimes}
\label{sec:escape-regimes}
In a static, clumpy medium several regimes are possible for the escape
of a \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photon.
We introduced some of them in \citet{clumps1}, but will describe each
regime below. Furthermore, we will sketch \textit{(i)} under which
circumstances each regime is active, \textit{(ii)} on average, how
many clumps a photon encounters $N_{{\mathrm{cl}}}$, and \textit{(iii)} which emergent spectrum can be expected. Additionally, Fig.~\ref{fig:sketch_regimes} provides a visual overview of the regimes, and a similar overview for a non-static setup is given in Appendix~\ref{sec:regimes_moving}.
\begin{itemize}[itemsep=10pt]
\item \textit{Porous regime.} If a substantial number of sight-lines do not intercept any clumps, many photons will not scatter and simply escape at their intrinsic frequency.
The fraction of sight-lines without any clumps can be estimated
assuming the clumps are Poisson distributed with mean $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ yielding
$\exp(-\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi)$ \citep[cf][]{Gronke2016a,DijkstraLyaLyC2016}. This area of the
parameter space has been explored in previous work
\citep[][]{Hansen2005,Laursen2012,Gronke2016a} and is of interest as
the empty sight-lines allow for ionizing photon escape
\citep{2015A&A...578A...7V,DijkstraLyaLyC2016} and might allow for directionally dependent
photon escape \citep{Gronke2014a}. This is the regime suggested by
cosmological simulations of the CGM \citep[e.g.,][]{2015MNRAS.449..987F,2016MNRAS.458.1164L}, though we note
that may be a consequence of their limited resolution, which strongly
limits the number of clumps to be no more than $\sim$ a few.
\item \textit{Random walk regime.} If the clumps are optically thick
to the photons, i.e., $\tau_{{\mathrm{cl}}}\gtrsim 1$, the photons are
expected to scatter at every clump encounter. When $\tau_{{\mathrm{cl}}}\gg 1$
the photon scatters close to the surface of each clump and
effectively random-walks between the clumps, rather than through
them. This regime has been studied by \citet{Neufeld1991} and by \citet{Hansen2005}.
In this `random walk regime' the number of clumps a photon intercepts scales as $\propto \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2$. This is because after $N_{{\mathrm{cl}}}$ interactions a photon has travelled on average a distance $\sqrt{N_{{\mathrm{cl}}}} B / \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ away from the midplane. Thus, to escape this distance has to be $\sim B$ which yields $N_{{\mathrm{cl}}} = \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2$. \citet{Hansen2005} found the scaling to be
\begin{equation}
N_{{\mathrm{cl}}} \sim \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2 + \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi,
\label{eq:N_cl_generic}
\end{equation}
with pre-factors of order unity which depend somewhat on the geometry
(see \citealt{Hansen2005} for details).
\item \textit{Optically-thin regime.} If the clumps are, on the other hand, optically thin to the intrinsic radiation ($\tau_{0,{\mathrm{cl}}}\lesssim 1$), not every cloud interception will necessarily cause the photon to scatter. In particular, the probability of a scattering event to happen in this case is $1 - \exp(-\tau_{0,{\mathrm{cl}}}) \approx \tau_{0,{\mathrm{cl}}}$. This implies that in order to describe the expected scaling in this regime, we can replace in the above considerations $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ by $\tau_{0,{\mathrm{cl}}}\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$. As in this regime $\tau_{0,{\mathrm{cl}}}\lesssim 1$, this corresponds to effectively decrease the pre-factors in Eq.~\eqref{eq:N_cl_generic}. Specifically, if \textit{all clumps} in a given sightline are optically thin the intrinsic photons (i.e., at line center), that is, $\tau_{0,\mrm{total}}\equiv 4/3 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \tau_{0,{\mathrm{cl}}} \lesssim 1$ most \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons will not interact before escaping. This means they will simply stream through all the clumps (leading to $N_{\mathrm{cl}} \propto \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$) keeping their intrinsic frequency (i.e., a peak frequency of $x_{\mrm{p}}\sim 0$). We call this `optically-thin' regime.
\item \textit{Homogeneous regime.} Since $\tau_{{\mathrm{cl}}}$ depends strongly
on the frequency of the photon which changes throughout the
scattering process, \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi might also escape from clumpy media in a
frequency excursion as discussed in the homogeneous slab. In
particular, during the course of the $\sim\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2$ scatterings needed
to random-walk through the clumpy medium, the photon may scatter far
enough into the wing of the line to escape the medium in a single excursion as described in \S~\ref{sec:radi-transf-homog}. If this happens, most clumps become optically thin for the photon and one can generalize the argument made above when describing the `optically thin regime' by replacing $\tau_{\mrm{0,cl}}$ by $\tau_{{\mathrm{cl}}}(x)$.
Since this possibility becomes increasingly likely with every
scattering event, we anticipate that above some critical value of
$\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$, clumpy media behave like homogeneous slabs in the sense that
photos escape via frequency excursion.
\end{itemize}
Conclusively, the four regimes are different in their preferential
escape route of the photons which depends in the static case on the
covering factor \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi and on the optical depth of individual clumps. Each escape route implies that the photon experiences a clumpy medium differently which leaves a clear imprint on the emergent spectrum. One way to characterize these regimes is: the `optically thin' and `porous' regimes represent escape without significant interaction, while the `random walk' and `homogeneous' regimes represent escape primarily via spatial or frequency diffusion respectively.
Table~\ref{tab:regimes} provides a brief summary of the different regimes.
\subsubsection{Division between the regimes}
\label{sec:divis-betw-regim}
In the last section we introduced the four different routes for a \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi
photon to escape from a clumpy medium. We also briefly discussed the
physical conditions under which each escape route is favored. In this
section, we quantify these boundaries more precisely.
We denote the boundary between the homogeneous and the other regimes
with \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi, and specifically to the random walk regime with
$\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi^{\mrm{exc}}$. Physically, this value characterizes the
critical number of clumps per sightline when a excursion-like escape
becomes faster than a random-walk diffusion. In order to find this
boundary let us follow this argument and compute the criteria for when it is possible for the photons to stream through the clumps\footnote{This derivation of \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi in a static setup is complementary to the one presented in \citet{clumps1} where we used a time-scale argument.}.
As stated above, the characteristic escape frequency is given by Eq.~\eqref{eq:xesc_adams}, and in a clumpy medium the total line center optical depth is $\tau_0 = 4/3 f_c N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}} \sigma_0$. The transition occurs when photons can stream through individual clumps, that is when $4/3\tau_{{\mathrm{cl}}}(x_{\mathrm{esc}}) = 1$. Using the wing approximation for the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi cross section, this yields
\begin{equation}
\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi^{\mathrm{exc}} = \frac{2 \sqrt{a_v \tau_{0, {\mathrm{cl}}}}}{3
\pi^{1/4}}
\approx \left(\frac{N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}}}{10^{17}\,\text{cm}^{-2}}\right)^{1/2}\left(\frac{T}{10^{4}\,\text{K}}\right)^{-1}\;.
\label{eq:fccrit_exc}
\end{equation}
For optically thinner medium an escape through excursion is not possible as a frequency shift into the wings of the lines will lead to immediate escape. Specifically, this happens if the wings become optically thin, i.e., if $\sqrt{3}\tau(x_*) < 1$ which translates to $\sqrt{3}\tau_0 a_v < \sqrt{\pi} x_*^2 \approx 18.78$ (where we included factors of $\sqrt{3}$ due to the rectangular geometry). This transition happens for an homogeneous medium as well as a clumpy medium and sets a lower limit to \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi. However, if the individual clumps possess an optical depth at line center of $\tau_{0,{\mathrm{cl}}} \lesssim 1$, not every clump encounter leads necessarily to a scattering, and, thus introduces the additional factor of $1-\mathrm{e}^{-\tau_{0,{\mathrm{cl}}}}$ (as described for the \textit{optically-thin} regime in \S~\ref{sec:divis-betw-regim}).
In conclusion, we expect the transition to the homogeneous regime to occur if
\begin{equation}
\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi = \begin{dcases}
\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi^{\mrm{exc}} = \frac{2 \sqrt{a_v \tau_{0, {\mathrm{cl}}}}}{3 \pi^{1/4}} & \text{ for } \sqrt{3} a_v \tau_0 > \sqrt{\pi} x_*^2\\
\frac{\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi^{\mrm{sf}}}{1-\mathrm{e}^{-\tau_{0,{\mathrm{cl}}}}} & \text{ otherwise.}
\end{dcases}
\label{eq:fccrit}
\end{equation}
where $\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi^{\mrm{sf}} =2 x_*/ 3^{5/4} \approx 1.65$ is due to continuity of \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi at $\sqrt{3}a_v\tau_0 = \sqrt{\pi} x_*^2$, that is, at the transition from excursion to single-flight escape.
\subsubsection{Non-static case}
\label{sec:non-static-case}
Since the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi cross section depends sensitively on the frequency $x$, clump motions can dramatically influence radiative transfer.
If a clump moves with a velocity
$\gtrsim{x_*}v_{\text{th}}\sim{50}$\,km/s, a single clump interaction
will put the photon far enough into the wing of the line to
allow the photon to escape directly in a single excursion. This
possibility is important to consider because random velocities
$v\sim{100}$\,km/s may be typical for the CGM in galaxies and in
galactic winds, and velocities $v\sim{1000}$\,km/s may be typical in
the regions around black holes.
This means that, for large $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ the medium behaves as a slab with an increased temperature of
\begin{equation}
T_{\rm eff} = T + \frac{\sigma_{{\mathrm{cl}}}^2 m_H}{2 k_B}
\label{eq:Teff}
\end{equation}
where $\sigma_{\mathrm{cl}}$ is the 1$D$ velocity dispersion of the clumps.
For a lower number of clumps the overall velocity distribution is not well-sampled which leads to sight-lines with no clumps in the core of the line. In this case the photons escape without any clump interaction.
We can estimate this to happen if the mean separation of two clumps in velocity space becomes larger than the velocity range over which a clump can provide $\tau_{{\mathrm{cl}}}\gtrsim 1$. For a Gaussian velocity distribution with variance of $\sigma_{\mathrm{cl}}^2$ the average separation is approximately given by $\sigma_{\mathrm{cl}} / (\alpha\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi)$ where $\alpha$ is the fraction of clumps within core of the velocity distribution, i.e., in our case $\alpha\approx 0.68$. Consequently, the transition to the homogeneous regime for randomly moving clumps occurs at $4/3\tau_{\mathrm{cl}}(\sigma_{\mathrm{cl}} / (\alpha \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi v_{\mrm{th}})) = 1$ which -- using the core approximation and including geometrical factors -- can be written as a critical covering factor for the randomly moving case
\begin{equation}
\label{eq:fccrit_moving}
\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi =
\begin{dcases}
\frac{\sigma_{\mathrm{cl}}}{\alpha v_{\rm th} \sqrt{\ln(4/3 \tau_{0,{\mathrm{cl}}})}} & \text{if }\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi > 1/\alpha\\
\frac{1}{\alpha} & \text{otherwise.}
\end{dcases}
\end{equation}
Here, the lower boundary of $1/\alpha$ results simply from the requirement that at least one clump within the core of the velocity distribution function is necessary in order to sample the core of the velocity distribution. We expect for larger covering factors the system to behave as a homogeneous slab of temperature $T_{\mathrm{eff}}$. See also Appendix~\ref{sec:regimes_moving} for more details about the expected behavior in the case of uncorrelated clump motion.\\
In the case of clumps with a systematic velocity structure (for instance, outflowing clumps), the above requirement of a `well-sampled' velocity field is fulfilled if the adjacent clump is optically thick to the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photon, i.e., if $4/3\tau_{\mathrm{cl}}(x_{\rm next}) \gtrsim 1$ where $x_{\mathrm{next}}$ depends on the exact velocity profile. For a linearly scaled (Hubble-like) outflow from $0$ at
midplane to $|v_{\mathrm{max}}|$ at the boundaries of the slab, we have $x_{\rm next} = v_{\rm max} / (\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi v_{\rm th})$.
In addition, a photon might be artificially forced into the wing of the line if $x_{\rm next} > x_*$ due to the sampling of the velocity field. This does not occur in a homogeneous medium, and thus, for a Hubble-like outflow the criterion to be fulfilled in order the be in the homogeneous regime is
\begin{equation}
\label{eq:fccrit_outflow}
\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi =
\begin{dcases}
\frac{\sqrt{\pi} v_{\mathrm{max}}^2}{a_v N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, \mathrm{total}} v_{\mathrm{th}}^2 \sigma_0} & \text{if }v_{\mathrm{max}} > \hat v_{\mathrm{max}}\\
x_* v_{\mathrm{th}}/v_{\mathrm{max}} & \text{otherwise.}
\end{dcases}
\end{equation}
where $\hat v_{\mathrm{max}} = v_{\mathrm{th}}\left(a_v N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, \mathrm{total}} x_* \sigma_0/\sqrt{\pi}\right)^{1/3}$.
\section{Numerical Method}
\label{sec:method}
\subsection{\ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer}
\label{sec:lya-radi-transf}
Due to the complexity of the resonant line transfer, Monte-Carlo radiative transfer codes are commonly in use \citep[e.g.,][]{Auer1968,Ahn2001a,Zheng2002}. This algorithm works by following individual photon packages in a stochastic manner through real- and frequency-space until their escape.
In this work, we use the code \texttt{tlac} which has been used and described previously, e.g., in \citet{Gronke2014a}. In particular, we make use of \texttt{tlac}'s features \textit{(i)} to handle embedded spherical grids within a Cartesian grid, and \textit{(ii)} employ a dynamical core-skipping scheme \citep[as described in][]{Smith2014,Gronke2016a}. We also turned off the dynamical core-skipping for a few models and checked that the emergent spectra are identical.
We run most setups using $\sim 10^4$ photon packages but use occasionally more to obtain a higher-resolution spectrum.
\input{content/table_model-params}
\subsection{Model parameters}
\label{sec:params}
Analogous to \S~\ref{sec:radi-transf-clumpy}, our setup consists of a slab with half-height $B$ in which we distribute spherical clumps with radius $r_{{\mathrm{cl}}}$ randomly in the box until a fraction of the total volume $F_V$ is filled. This means the number density of clumps is $n_{\mathrm{cl}} = F_V / (4/3 \pi r_{\mathrm{cl}}^3)$ where $r_{\mathrm{cl}}$ is the clump radius.
The connection between the volume filling factor $F_V$ and the previously introduced covering factor \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi (which describes the average number of clumps a line orthogonal to the slab intercepts between the midplane and the boundary of the box) is given by the
integration along the finite axis of the slab, that is,
\begin{equation}
\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi = \int\limits_0^B \mathrm{d} r\; \pi n_{\mathrm{cl}} r_{\mathrm{cl}}^2 = \frac{3 F_V B}{4 r_{{\mathrm{cl}}}}\;.
\end{equation}
The clumps are filled with neutral hydrogen with a number density of $n_{\rm HI, cl}$ and temperature $T$, leading to a column density between the center of the clumps to their outskirts of $N_{\rm HI, cl} = r_{{\mathrm{cl}}} n_{\rm HI, cl}$. As described in \S~\ref{sec:radi-transf-clumpy}, this means on average the shortest path between the midplane and the boundary of the box will intercept a column density of
\begin{equation}
N_{\rm HI, total} = \frac{4}{3} \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi N_{\rm HI, cl} = F_V B n_{\rm HI, cl}\;.
\end{equation}
In general, we consider three cases: the static case with no motion, the randomly moving case, and an outflowing case.
In the randomly moving case, we assign each clump a random velocity by drawing each component from a Gaussian with standard deviation $\sigma_{{\mathrm{cl}}}$. This represents a ``white noise'' spectrum, with velocity differences which are statistically equally probable on all spatial scales.
For the outflow, we choose a simple linear velocity scaling from $0\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$ to $v_{\rm max}$ at the midplane and boundary of the slab, respectively.
We will investigate models with correlated turbulence, and different velocity profiles in future work.
Furthermore, we study two different emission sites for the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons. First, simply the midplane of the box, and secondly randomly chosen emission within the clumps. While the former is useful in order to study merely the radiative transfer processes through the clumpy medium from an external source such as a star-forming region, the latter case represents a physically motivated scenario in which \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi are produced via recombination events within the clumps. Both scenarios might be responsible, e.g., for the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi halos found around galaxies \citep[e.g.][]{Dijksta2012MNRAS.424.1672D,Lluis2016ApJ...822...84M}.
\section{Numerical Results}
\label{sec:results}
In this section we present the results from our numerical radiative transfer simulations. In particular, we focus on three quantities, namely the number of clumps encountered by the photons $N_{\mathrm{cl}}$, and the emergent \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra. $N_{\mathrm{cl}}$ is a useful diagnostic, since we expect $N_{\mathrm{cl}} \sim \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2$ for escape via random walk in position space, and $N_{\mathrm{cl}}\sim\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ for escape via excursion and single-flight as described in the previous section.
The section is approximately ordered by ascending complexity. In \S~\ref{sec:static-case}, we discuss the static case, in \S~\ref{sec:random-motion} and \S~\ref{sec:outflows} we introduce random clump motions and outflows, respectively. Moreover, in \S~\ref{sec:clump-emission} we change the emission site of the photons to be inside the clumps which resembles a case of fluorescent emission. Finally, we study the effect of dust inside the clumps on the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape in a clumpy medium (\S~\ref{sec:dust-within-clumps}) which we quantify through the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction.
\begin{figure}
\centering \includegraphics[width=.95\linewidth]{plots/Ncl_randomly_moving.pdf}
\caption{Number of clumps passed versus $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ for clumps with $N_{\rm HI, cl}=10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ and uncorrelated, random motion with various $\sigma_{{\mathrm{cl}}}$. The dashed lines show fits of Eq.~\eqref{eq:fit_N_cl} to the data points and the grey solid line shows the limit with $f_{\rm c, crit} > 10^3$.}
\label{fig:nclumps_moving}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/slabs_b_all.pdf}
\caption{The \textit{solid lines} show the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra for a constant clump column density $N_{\rm HI,cl}=10^{16}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$, and covering factor $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi\approx 1000$. The \textit{dashed lines} show as comparison the spectra obtained from slabs with $T_{\rm eff}(\sigma_{{\mathrm{cl}}})$.}
\label{fig:spectra_vs_slabs}
\end{figure}
\subsection{Static case}
\label{sec:static-case}
Fig.~\ref{fig:nclumps_static} shows the number of clumps a \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photon passed through before escaping the box versus the covering factor $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ which we vary over $\sim 3$ orders of magnitude. Each symbol and color represents different values of $N_{\rm HI, cl}$ and, thus, different clump optical depths at line center $\tau_{cl, 0}$ which we vary from $\sim 0.06$ (optically thin) to $\sim 6\times 10^8$ (optically thick). Note, that we also ran each combination of $(N_{\rm HI, cl},\,\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi)$ with two different cloud radii $r_{{\mathrm{cl}}}=\{10^{-2},\,10^{-3}\}\,$pc to confirm that this parameter is not important \citep{Hansen2005}.
The dashed lines in the corresponding color show curves following
\begin{equation}
N_{{\mathrm{cl}}} = \begin{cases}
a_1 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2 + b_1 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi & \text{ for } \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi < \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi\\
a_2 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2 + b_2 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi & \text{ for } \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \ge \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi
\end{cases}\;.
\label{eq:fit_N_cl}
\end{equation}
We fit the data points for $N_{\rm HI, cl}=10^{22}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \le 100$ to determine $(a_1\,b_1) = (3/2,\,2)$ (the best fit values are $(1.51,\,1.90)$ which -- given the uncertainty -- we rounded to the nearest convenient fraction for simplicity).
These coefficients represent geometrical factors in the surface scattering regime (where clouds are optically thick), and thus independent of $N_{\rm HI,cl}$. We have directly verified this numerically.
These values are then fixed for all $N_{\rm HI, cl}$ in order to fit each $N_{\rm HI, cl}$-curve for $(\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi,\,a_2)$ while $b_2$ is fixed by requiring continuity at \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi. Fig.~\ref{fig:nclumps_static} shows the resulting fits as well as the obtained values for $(\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi,\,a_2)$. The break in the scaling relation at \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi is clearly visible (for visual aid, Fig.~\ref{fig:nclumps_static} also shows the $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi = a_1\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2 + b_1\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ curve from which the scaling departures for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi > \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$).
We find that the obtained \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi for high column densities ($N_{\rm HI, cl}\gtrsim 10^{20}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$) matches the prediction from \S~\ref{sec:divis-betw-regim} reasonably well, this breaks down for lower optical depths.
Also, for $N_{\rm HI, cl}=10^{12}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$, i.e., when the clumps are always optically thin for \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons, we obtain $a_2\approx \tau_{0, cl}^2 \approx 0.010$ as discussed in Sec.~\ref{sec:analyt-cons-new}. With increasing $N_{\rm HI, cl}$ we find a decreasing $a_2$ to match the data. Thus, we can identify the escape regimes (characterized by the number of clumps encountered) described in the analytic model in Sec.~\ref{sec:analyt-cons-new}.
In summary, Fig.~\ref{fig:nclumps_static} shows that for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi < \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$, $N_{{\mathrm{cl}}} \propto \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2$ (as expected for a random walk), while for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi > \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$, $N_{\mathrm{cl}} \propto \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ (as expected for escape through excursion).
Fig.~\ref{fig:spectra_fc} shows the corresponding spectra for $N_{\rm HI, cl}=10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$. Note, that for this column density we found $\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi \sim 2$ which corresponds roughly to the boundary between single and double peaked spectra. In particular, we recover the spectral shape of \citet{Hansen2005} for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \ll \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$ while obtaining wide, double peaked spectra with zero flux at line center for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \gg \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$. This means, that the escape regimes do not only impact the photons' paths but modify the escape frequencies', and hence, leave a clear observational signature on the emergent spectra.
\begin{figure}
\centering \includegraphics[width=.95\linewidth]{plots/Ncl_outflowing.pdf}
\caption{Number of clumps passed versus $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ for clumps with $N_{\rm HI, cl}=10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ and outflowing motions with different maxima $v_{\rm max}$. The dashed lines show fits of Eq.~\eqref{eq:fit_N_cl} to the data points and the grey solid line shows the limit with $f_{\rm c, crit} > 10^3$.}
\label{fig:nclumps_outflow}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=.95\linewidth]{plots/general_cloud-outflow_multi.pdf}
\caption{\ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra using a setup of outflowing clumps with linear velocity profile for $N_{\rm HI, cl}=10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ and four different maximal velocities $v_{\rm max}$. The \textit{dashed lines} in corresponding colors show the spectra emergent from a slab with the same column density and velocity structure. Each sub-panel displays a case with different covering factor corresponding to increasing agreement with the homogeneous setup.}
\label{fig:outflows}
\end{figure*}
\subsection{Random motion}
\label{sec:random-motion}
Fig.~\ref{fig:nclumps_moving} shows the $N_{{\mathrm{cl}}}-\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ scaling relation in the case of random clump motion for a fixed clump's column density of $N_{\rm HI, cl}=10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$.
As conjectured in \S~\ref{sec:non-static-case}, compared to the static case the photons spend less time until escape and thus the number of clumps passed is smaller. This is due to the fact that in the case of fast moving clumps, the photons escape either through `holes' in velocity space (where, $N_{\mathrm{cl}} \propto \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$), or via single flight (in which case also $N_{\mathrm{cl}} \propto \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$). Departures from that are either due to convergence to the static case (for $\sigma_{\mathrm{cl}} \rightarrow 0$), or when escape via single flight involves multiple surface scatterings (for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi\ll\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$) when the interaction with another clump is non-negligible.
In Fig.~\ref{fig:spectra_vs_slabs} we show the emergent spectra from this setup. In particular, we focus on the case with $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi\approx 1000$ and $N_{\rm HI, cl}=10^{16}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ and four different values of clump velocity dispersion $\sigma_{{\mathrm{cl}}}$. Also in Fig.~\ref{fig:spectra_vs_slabs} we overlay spectra from homogeneous slabs with an effective temperature $T_{\rm eff}$ (see Eq.~\eqref{eq:Teff}) corresponding to the respective value of $\sigma_{{\mathrm{cl}}}$. Clearly, the spectra match quite well -- especially the peak separation. However, with increasing $T_{\rm eff}$ the matches become worse, which makes sense, since the wider velocity space is more poorly sampled.
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/outflows_asymmetry_cut.pdf}
\caption{Integrated blue over integrated red flux (minus one) versus covering factor for different combinations of $v_{\rm max}$ and $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, \mathrm{total}}$. With increasing \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi the spectra become more redshifted. See \S\ref{sec:outflows} for details}
\label{fig:outflow_asymmetry_cut}
\end{figure}
\subsection{Outflows}
\label{sec:outflows}
Fig.~\ref{fig:nclumps_outflow} shows the $N_{{\mathrm{cl}}}-\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ relation in the presence of linearly scaled outflows with maximum velocity $v_{\rm max}$ (as described in \S~\ref{sec:params}). We can see that a flattening of the curve still exists which we interpret again as the transition between the `random walk' and `homogeneous regime'.
As expected, with increasing outflow speed this threshold decreases.
Fig.~\ref{fig:outflows} illustrates the change in spectral shape when introducing outflows. In each subpanel, the solid lines show the emergent spectrum from the clumpy model and the dashed lines in matching color the ones from a homogeneously filled slab with the same total column density and velocity structure.
We focus on the case with constant clump column density $N_{\rm HI, cl}=10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ and show three cases $f_{\rm c}=\{1,\,100,\,1000\}$ which match increasingly well the slab case -- for all four values of $v_{\rm max}$.
This implies that the spectra become more asymmetric as they converge towards the homogeneous limit. The asymmetry develops because the outflow shifts the scattering cross section in the observers reference frame towards the blue. Thus, the optical depth for photons with frequency redward of line-center (e.g., `back-scattered' ones off the far-side of the system) is lowered allowing for easier escape.
We will discuss the result of higher asymmetry with increased number of clumps further in \S~\ref{sec:discussion}.
Fig.~\ref{fig:outflow_asymmetry_cut} shows this increase in asymmetry with greater \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi for fixed outflow velocities of $v_{\mathrm{max}} = \{100,\,1000\}\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$ and total column densities of $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, \mathrm{total}}=4/3\times\{10^{18},\,10^{19},\,10^{20}\}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$. We characterize the spectral asymmetry by the integrated flux ratio of the blue over the red part of the spectra (minus one), i.e., a value of $-1$ means that all photons escape redward of line center ($x \le 0$) and if this quantity is zero the spectrum is symmetric (around $x = 0$). While the transition from symmetric to dominantly red spectra for $v_{\mathrm{max}}=100\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$ is nearly independent of the column density at $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \sim 10$, this is not the case for $v_{\mathrm{max}}=1000\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$ where a larger total column density implies a shift at lower \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi. This is due to the dependence of \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi on $v_{\mathrm{max}}$ and $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, \mathrm{total}}$ described in \S~\ref{sec:non-static-case}. In particular, as seen in Eq.~(\ref{eq:fccrit_outflow}) \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi does not depend on the column density if $v_{\mathrm{max}}$ is small.
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/general_cloud-emission_spectra.pdf}
\caption{\ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra for a constant clump column density $N_{\rm HI,cl}=10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ and three values of $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$. The \textit{solid lines} mark the spectral shape with emission inside the clumps whereas the \textit{dashed lines} show as comparison the spectra obtained from midplane emission.}
\label{fig:spectra_clumpy-emission}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/general_cloud-emission_spectra_moving.pdf}
\caption{\ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra for a constant geometry with $N_{\rm HI,cl}=10^{16}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$, $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi=1000$ and three values of $\sigma_{{\mathrm{cl}}}$. The \textit{solid lines} mark the spectral shape with emission inside the clumps whereas the \textit{dashed lines} show as comparison the spectra obtained from midplane emission. Please note that for presentation purposes we rescaled the $x$-axis according to the value of $\sigma_{{\mathrm{cl}}}$. The \textit{black dotted line} shows the intrinsic spectrum which has the same width for all $\sigma_{{\mathrm{cl}}}$ due to the rescaling.}
\label{fig:spectra_clumpy-emission_moving}
\end{figure}
\subsection{Emission within the clumps}
\label{sec:clump-emission}
In this section we study the effect of emission originating from inside the clumps. This case resembles \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi production due to cooling in the inner parts of the clumps, or to recombination events in the outer layer of (self-shielding) clumps caused by an external ionizing source. The latter is sometimes referred to as fluorescence. In both cases, \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi are produced in the reference frame of the clumps, and experience an initial optical depth before entering the inter-clump medium -- both effects shape the `intrinsic' spectrum.
Fig.~\ref{fig:spectra_clumpy-emission} compares some spectra with starting position inside the clumps to the ones previously presented, i.e., with starting position at midplane. The clumps in this case possess a column density of $N_{\rm HI, cl}= 10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$, which means they are optically thick to \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiation. The effect of this can be seen best in the spectrum with $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi = 1$ (red curve in Fig.~\ref{fig:spectra_clumpy-emission}) which contrasts a double peak profile due to the escape from the clump to the single peaked profile from the random walk process between the clumps. For greater values of $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$, however, this `initial feature' gets washed out from the scatterings off subsequent clumps and the spectra are independent of the emission site.
In Fig.~\ref{fig:spectra_clumpy-emission_moving} we show a similar plot for moving clumps. Note that in this case the photons' frequencies are rescaled according to the value of $\sigma_{{\mathrm{cl}}}$ due to presentation purposes. This means that in the spectra for $\sigma_{{\mathrm{cl}}}=10^4\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$ (shown in purple) are with a full width at half maximum (FWHM) of $\Delta x\sim 5000$ the widest of the presented spectra.
As previously for the large $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi\gg 1$ cases, the spectra with the emission sites within the clumps resemble closely the ones with emission sites in the midplane. The only difference is that the latter are slightly wider and possess a smaller flux at line center which is simply due to the fact that a number of clumps are located at the boundary of the slab. This is encouraging as it shows that our results are quite general, that is, not dependent on the exact emission site. However, a small caveat is that for spherical geometries most clumps are located at large radii which might make this setup more sensitive to in-clump emission (on the other hand, the outermost clumps might emit less \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons as some are `shadowed' by clumps closer to the ionizing source).
\subsection{Dusty clumps}
\label{sec:dust-within-clumps}
When placing absorbing dust in the clumps -- which we characterize by the all-absorbing dust optical depth $\tau_{{\mathrm{cl}}, d}$ -- \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi can be destroyed leading to an escape fraction $f_{\rm esc}\le 1$. Interestingly, in clumpy medium the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction might be larger than the continuum one as predicted by \citet{Neufeld1991}.
This `Neufeld effect' is due to the fact that \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons may `surface scatter' off the neutral clumps, thus, effectively shielding the dust from them. Therefore, one expects the observed \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi equivalent widths to be potentially much larger than the intrinsic ones. \citet{Hansen2005} characterized this effect more systematically using \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer simulations for a wide range of parameters.
Building upon their work, \citet{Laursen2012} found, however, that in a part of the parameter space which they tried to constrain by observations the boosting vanishes. Specifically, out of their $4\times 10^3$ models only a few percent showed an equivalent width boost \citep[see also ][for a study of the `Neufeld effect' in clumpy shells]{Duval2013}. \citet{Laursen2012} thus concluded ``consider the Neufeld model to be an extremely unlikely reason for the observed high EWs''. All these studies focused on values of $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \sim 1$ and we want to re-visit \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape in clumpy medium with several orders of magnitude greater covering factors. Thus, it is not entirely clear from the literature whether radiative transfer effects from clumpy media can explain the extreme equivalent width measurements observed in some galaxies. However, \citet{Laursen2012} identified some criteria which have to be fulfilled such as relatively slowly moving clumps with high dust optical depths.
Instead of re-running the radiative transfer simulations for various dust contents, we use the information of the hydrogen column density `seen' by each photon package to compute the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction as in \citet{Gronke2015} which yields an escape fraction for each photon package that is
\begin{equation}
f_{{\rm esc,} i} = \exp\left[-\frac{\hat N_{{\rm HI}, i}}{N_{\rm HI, cl}} \tau_{\rm d, cl}\right]\;.
\end{equation}
Here, $\hat N_{{\rm HI}, i}$ is the column density experienced by photon package $i$. Given $f_{{\rm esc}, i}$ for a certain setup one can now obtain \textit{(a)} the overall \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction as the average of $f_{{\rm esc}, i}$, and \textit{(b)} the spectral shape altered through dust by simply assigning each photon package the weight $f_{{\rm esc}, i}$ when assembling the spectrum.
In Fig.~\ref{fig:fesc_static_simple} we plot the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction versus $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ for a constant total dust and hydrogen number content. For all three values of $\tau_{\mathrm{d, total}}$ displayed a similar trend is visible: with increasing \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi, first a approximately linear fall off in escape fraction before a flattening occurs, that is, $f_{\mrm{esc}}\sim \mrm{const.}$ for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \gtrsim 40$.
Interestingly, the position of this threshold is independent of $\tau_{\mrm{d,total}}$ which hints towards a origin in the nature of the radiative transfer.
The flattening occurs at the boundary between the boundary between the `free streaming' and `homogeneous' regime because in the former the probability of absorption is proportional to the number of clump interactions (and, thus, \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi) whereas in the latter the escape fraction is set by the total dust content only and does not grow further with \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi. We discuss this phenomenon in more detail in \S~\ref{sec:escape-fractions-ew}.
An implication of the respective escape fractions of the two regimes is visible in Fig.~\ref{fig:fesc_static}. Here we show several values of $N_{\rm HI, cl}$ for the static setup using $\tau_{\rm d, cl}=10^{-4}$ (empty symbols) and $\tau_{\rm d, cl}=1$ (filled symbols) which correspond to metallicities of $Z/Z_\odot = 0.63\left(\tau_d / 10^{-4}\right)\left(10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2} / N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}}\right)$ \citep{Pei1992,Laursen2009} which reaches clearly unrealistic values. However, as in this paper we're interested in the fundamental impact of the individual parameters we also study these extreme value.
Also shown in Fig.~\ref{fig:fesc_static} (with a black [grey] solid line for the low [high] dust content) is the proposed analytic solution for $f_{\rm esc}$ by \citet{Hansen2005}
\begin{equation}
f_{\rm esc}^{\rm HO06} = 1/{\rm cosh}(\sqrt{2 N_{{\mathrm{cl}}}\epsilon})\;,
\label{eq:fescHO06}
\end{equation}
where for $N_{{\mathrm{cl}}}$ we used Eq.~\eqref{eq:N_cl_generic} (with $(a_1,\,b_1) = (3/2,\,2)$ as found in \S~\ref{sec:static-case}) and for the clump albedo (i.e., the fraction of incoming photons which are reflected) $\epsilon$ we adopted a value of $c_1 (1 - e^{-\tau_{\rm d,cl}})$ with $c_1 = 1.6$ [$c_1=0.06$] to match the $N_{\rm HI, cl}=10^{22}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ data points for $\tau_{\rm d, cl}=10^{-4}$ [$\tau_{\rm d, cl}=1$].
Note, that the behaviour for the low and high dust contents is quite different. On the one hand, the escape fractions versus $N_{\rm HI,cl}$ scales for $\tau_{\rm d,cl}=1$ (filled symbols in Fig.~\ref{fig:fesc_static}) as predicted by \citet{Hansen2005} in their `surface scatter' approximation, that is, a larger clump hydrogen column density `shields' the dust better from the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons and thus prevents more efficiently their destruction. On the other hand, however, this is not the case for the low-dust scenario presented in Fig.~\ref{fig:fesc_static} (with unfilled symbols) where a larger value of $N_{\rm HI, cl}$ implies a lower $f_{\rm esc}$. This is due to the fact that here the dust optical depth through all the clumps (shown in the black dotted line in Fig.~\ref{fig:fesc_static}) is lower than the accumulated one through the subsequent random-walk clump encounters (black solid line), i.e, $\exp(-4/3 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \tau_{\rm d, cl}) \lesssim f_{\rm esc}^{\rm HO06}$. Consequently, configurations in the `free-streaming' regime can possess enhanced \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fractions compared to the `random walk' regime (see \S~\ref{sec:escape-fractions-ew} for a more detailed discussion).
Still, both cases possess (much) larger escape fractions than a homogeneous slab which is shown in Fig.~\ref{fig:fesc_static} with a black dashed line. Here, we use the derived escape fraction by \citet{Neufeld1990} with $N_{\rm HI}=4/3\times \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi 10^{22}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ and $\tau_{\rm d} = 4/3 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \tau_{\rm d, cl}$ -- i.e., with equal column densities as in the $N_{\rm HI, cl}=10^{22}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ case.
The same quantity, i.e., $f_{\rm esc}$ versus $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$, for the case of randomly moving clumps is plotted in Fig.~\ref{fig:fesc_moving}. As previously, the escape fraction departures from the curve given by Eq.~\eqref{eq:fescHO06} for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \gtrsim \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$. The lower number of clump encounters in this regime leads to a significantly higher escape fraction, e.g., $f_{\rm esc}\sim 10^{-1}$ for $\sigma_{{\mathrm{cl}}}=500\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$.
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/general_fesc_NHI-total_const.pdf}
\caption{\ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction versus $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ for a fixed total hydrogen column density $N_{\rm HI, total}$ and dust optical depth $\tau_{\mathrm{d, total}}$.}
\label{fig:fesc_static_simple}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/general_fesc_static.pdf}
\caption{\ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction versus $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ for various values of $N_{\rm HI, cl}$, $\tau_{\rm d, cl}=1$ and $\tau_{\rm d, cl}=10^{-4}$ (filled and unfilled symbols, respectively). The black [grey] curves show some analytic curves for $\tau_{\rm d, cl} = 10^{-4}$ [$\tau_{\rm d, cl} = 1$]. The solid curve shows the \citet{Hansen2005} formula as given by Eq.~\eqref{eq:fescHO06}, the dashed line is the escape fraction from a homogeneous slab with $N_{\rm HI} = 4/3 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi 10^{22}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ as given by \citet{Neufeld1990}, and the dotted line is simply $\exp(-4/3 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \tau_{\rm d, cl})$ symbolizing a continuum escape fraction.}
\label{fig:fesc_static}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/general_fesc_moving.pdf}
\caption{\ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction versus $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ for fixed clumps with $\tau_{\rm d, cl}=10^{-4}$, $N_{\rm HI, cl}=10^{20}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ and various values of $\sigma_{{\mathrm{cl}}}$. The curves are the same as in Fig.~\ref{fig:fesc_static} for comparison.}
\label{fig:fesc_moving}
\end{figure}
\section{Discussion}
\label{sec:discussion}
In this section, we will discuss our results in the light of the various escape regimes discussed in Sec.~\ref{sec:analyt-cons-new} (\S~\ref{sec:regimes-clumpy-model}). Furthermore, we analyze what implications our results have for `\ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi equivalent width boosting' (in \S~\ref{sec:escape-fractions-ew}), and make the connection to observational results (of `shell-model' fitting; \S~\ref{sec:clumpy-solid-conn}) as well as to radiative transfer results through hydrodynamical simulations (\S~\ref{sec:impl-lya-radi}).
\subsection{The regimes of the clumpy model}
\label{sec:regimes-clumpy-model}
Fig.~\ref{fig:result_overview} summarizes our findings for the static case. Here, color shows the flux at line center expressed in units of flux at the peak of the spectra $F(x=0)/F_{\rm peak}$. This measure is $\sim 1$ for a single peaked spectra and is less for double peaked spectra; a value of $\sim 0$ corresponds to an optically thick, `slab like' spectrum. We highlighted the dividing value of $F(x=0)/F_{\rm peak}=1/2$ specifically.
Also visible in Fig.~\ref{fig:result_overview} are the three regimes described Sec.~\ref{sec:analyt-cons-new}, along with our analytic estimates. They can be summarized as follows:
\begin{itemize}
\item \textit{Optically thin regime.} For an overall optical depth $\tau_{0, \mathrm{total}}=4/3 \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi\tau_{\rm 0, cl}\lesssim 1$ the $N_{{\mathrm{cl}}}-\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ scaling is shallow and the emergent spectra are single peaked. The dotted line in Fig.~\ref{fig:result_overview} mark this boundary.
\item \textit{Homogeneous regime.} If not in the `optically thin regime', for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \gtrsim f_{\rm c,crit}$ we found also a shallower $N_{{\mathrm{cl}}}-\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ scaling than \citet{Hansen2005}. This is due to the preferential escape in an optically thick medium through single excursion -- which causes broad, double peaked spectra. Fig.~\ref{fig:result_overview} shows $f_{\rm c,crit}$ as a function of $N_{\rm HI,cl}$ as the dashed line.
Note that above this line we find $F(x=0)/F_{\rm peak}\rightarrow 0$ denoting double peaked spectra as predicted. Similarly, below this line the numerical results show single-peaked spectra.
\item \textit{Random-walk regime.} For optically thick clumps and $f_{\rm c}\lesssim \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$ we recovered the results of \citet{Hansen2005}, i.e., $N_{{\mathrm{cl}}}\propto \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi^2$ and single peaked spectra due to a surface-scattering escape of the photons.
\end{itemize}
As noted in \S~\ref{sec:divis-betw-regim} these regimes break down for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi\lesssim 3$ -- an area of the parameter space which has previously by studied by \citet{Hansen2005,Laursen2012,Gronke2016a} -- where the probability of not finding a clump in a certain sightline is non-negligible \citep[this allows for non-zero ionizing photon escape fraction, see ][]{DijkstraLyaLyC2016}.
Fig.~\ref{fig:result_overview_moving} displays the transition from double- to single-peaked spectra for randomly moving clumps. The color coding shows in this case the peak position of the spectrum with white being $x_{\mathrm{peak}}\sim 3$, that is, when the peak position moves outside the core of the line\footnote{We used an alternative criterion because for larger $\sigma_{{\mathrm{cl}}}$ the spectra can be very broad, and thus, $F(x=0)/F_{\rm peak}$ becomes noise dominated. However, both measures can be used to distinguish between single- and double-peaked.}.
For faster clumps, this boundary moves to greater values of \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi, making it more likely to obtain a single-peaked spectrum (at line center).
The black dashed lines in Fig.~\ref{fig:result_overview_moving} denotes \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi from Eq.~\eqref{eq:fccrit_moving}\footnote{In fact, we used the exact functional form for $\sigma_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace}(x)$ and did not resort to the approximation as in Eq.~\eqref{eq:fccrit_moving} which yields a slightly better fit to the data..} -- in other words below this line the velocity space is not well sampled and allows photons at line center to escape.
The same line is marked also in Fig.~\ref{fig:overview_NHI17} where we focus on the clump column density $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, {\mathrm{cl}}}=10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ as predicted by `shattering' \citep{McCourt2016}. Here the peak position (in log scale) is color coded as a function of covering factor and clump velocity dispersion.
For large values of $\sigma_{{\mathrm{cl}}}$, the transition to double peaked spectra occurs at a larger covering fraction, since more clumps are required to sample the broader velocity distribution.
Below this threshold, we see a single-peaked spectrum from photons which escape through holes in velocity space.
Fig.~\ref{fig:overview_outflow_asymmetry} shows this increase in asymmetry with greater \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi (for fixed outflow velocity and total column density of $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, \mathrm{total}}=4/3\times 10^{19}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$). Here, the color corresponds to the asymmetry of the spectra which we define as in \S~\ref{sec:outflows} to be the ratio of the integrated blue over the red flux minus one. In Fig.~\ref{fig:overview_outflow_asymmetry} we also mark graphically the conditions for homogeneous escape discussed in \S~\ref{sec:non-static-case}, that is, that the adjacent clump is optically thick ($4/3 \tau_{\mathrm{cl}}(x_{\mathrm{next}}) \gtrsim 1$ with $x_{\mathrm{next}} \equiv v_{\mathrm{max}} / (\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi v_{\mathrm{th}})$), and that the initial scatterings occur in the core of the line ($x_{\mathrm{next}} < x_*$). If both conditions are fulfilled (and sufficient outflows are present, i.e., $v_{\mathrm{max}}\gtrsim 50\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$), the emergent spectrum is asymmetric towards the red side (as visible from the red region in Fig.~\ref{fig:overview_outflow_asymmetry}).
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/result_overview_static.pdf}
\caption{Overview of the different regimes for the static ($\sigma_{\mathrm{cl}}=v_{\mathrm{max}}=0$) setup. The color coding shows our (interpolated) numerical results in terms of the flux at line center divided by the peak flux of the spectrum, i.e., a value of $\sim 0$ [$\sim 1$] quantifies a double [single] peaked spectrum. Specifically this quantity is $1/2$ at the \textit{solid line}. The \textit{dashed} line marks the $\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$ (Eq.~\eqref{eq:fccrit}), and the \textit{dotted line} is the boundary to the low-density regime ($\tau_{0, \mathrm{total}}=1$).}
\label{fig:result_overview}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=.95\linewidth]{plots/general_moving_xpeak_overview_contour.pdf}
\caption{Overview of the $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$-$N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}}$-plane with moving clumps for two different values of $\sigma_{{\mathrm{cl}}}$. The color coding shows the spectral peak position $x_{\mathrm{peak}}$ (truncated at $x_{\mathrm{peak}}=6$).
The \textit{dashed lines} show \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi in the moving case (\S~\ref{sec:non-static-case}), i.e., below this line the velocity distribution of clumps is not sampled well.
}
\label{fig:result_overview_moving}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/overview_NHI17_fc-sigmacl.pdf}
\caption{Overview of the spectral shape for $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}} = 10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$, the clump column density predicted by \citet{McCourt2016}. The color coding denotes (the log of) the peak position $x_{\mathrm{peak}}$, i.e., low values (in black) represent a single peaked feature. The white dashed line is again the \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi boundary in the moving case.}
\label{fig:overview_NHI17}
\end{figure}
\subsection{Escape fractions and EW boosting}
\label{sec:escape-fractions-ew}
The different regimes for \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer through a multi-phase gas have different implications for the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction when dust in present within the clumps.
\begin{itemize}
\item Inside the `optically thin regime' ($\tau_{0, \mrm{total}}\lesssim 1$), the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction is equal to the continuum one as \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons stream through all the clumps and are affected by the dust content within them. Hence, $f_{\mrm{esc}}\approx \exp(-\tau_{\mrm{d, total}})$. This can be seen for the $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}}\lesssim 10^{14}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$, $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \lesssim\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$ data points in Fig.~\ref{fig:fesc_static}.
\item In the `random-walk regime', we confirm the escape fraction given by \citet{Hansen2005} (see Eq.~\eqref{eq:fescHO06}) (apart from geometrical pre-factors). In this regime the governing quantity for the escape fraction is $\epsilon$, i.e., the absorption probability per clump interaction and the number of clumps encountered $N_{\mathrm{cl}}$. In this regime, the latter is merely a function of \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi (see Eq.~\eqref{eq:N_cl_generic}) but $\epsilon$ depends non-trivially on $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, {\mathrm{cl}}}$ and the clump movement due to variations in how deep the photons penetrate into the clumps. This is why there is some scatter in $f_{\mrm{esc}}$ in this regime; visible, for instance, in Fig.~\ref{fig:fesc_static}. This is the only regime where \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons are shielded from dust, thus, allowing for `EW-boosting' \citep{Hansen2005}. That is, the ratio between the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi and UV escape fraction might be greater than unity.
\item Finally, in the `homogeneous regime', the behaviour is a combination of the above two behaviours. Initially, the photons will (on average) interact with $\sim N_{\mathrm{cl}}(\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi)$ clumps before diffusing to the line wings and escaping through free-streaming which leads to another $\sim \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ clump encounters (cf. Fig.~\ref{fig:nclumps_static}). Consequently, in this regime the escape fraction is approximately given by $f_{\mrm{esc}}\sim f_{\mrm{esc}}^{\mrm{HO06}}(\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi)e^{-\tau_{\mrm{d,total}}}$. This causes the flattening of $f_{\mrm{esc}}$ versus \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi in Fig.~\ref{fig:fesc_static_simple} as in this case $\tau_{\mrm{d, total}}$ is kept constant.
\end{itemize}
From the above considerations, one can see that the escape fraction
depends on several parameters and is therefore non-trivial to predict.
As a consequence, in \S~\ref{sec:dust-within-clumps} we demonstrated
that the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction may either increase \textit{or} decrease
with increasing metallicity which is $Z \propto \tau_d / n_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace}$ \citep{Pei1992,Laursen2009}, depending on the dust optical depth through an individual clump $\tau_{\mrm{d,cl}}$ (see the trends in the filled and unfilled symbols in
Figs.~\ref{fig:fesc_static} \& \ref{fig:fesc_moving}).
The controlling parameter is essentially the
ratio of absorption probability per surface scatter to the absorption
probability per clump passing.
Moreover, we have shown that \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fractions can be large, even
for large values of \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi. Thus, we find homogeneous, `slab-like'
spectra can be observable even in models with significant dust content
(as is realistic; see \S~\ref{sec:clumpy-solid-conn} \&
\S~\ref{sec:impl-lya-radi}).
Regarding the EW boosting we found, one necessary requirement for the `Neufeld effect' to be active is that \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons escape via surface scatterings off the clumps, i.e., in the `random walk' regime. This implies that the emergent spectrum is narrow, and single peaked at line center \citep[as already noted by ][]{Laursen2012} -- a clear observational signature for EW boosting to be active\footnote{Note however that if the line is narrow and concentrated on line center, that then the IGM can suppress the flux, as this is where we expect the IGM opacity to peak \citep[see, e.g.,][]{Laursen2011}.}.
\subsection{Connection to a homogeneous medium}
\label{sec:clumpy-solid-conn}
Observed \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra can often successfully modeled using a simple model called the `shell-model' \citep[see, for instance,][]{Hashimoto2015,Karman2016_dl}. This shell-model consists of a central \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi (and continuum) emitting region which is surrounded by a moving shell of hydrogen and dust \citep{Ahn2003MNRAS.340..863A,Verhamme2006A&A...460..397V}. It is somewhat surprising that this simple, six-parameter model can account for the likely radiative transfer effects happening in the complex, multiphase medium of a variety of galaxies and their environments.
Since the shell-model is clearly very idealized, it is unclear what the extracted shell-model parameters mean physically. In \citet{Gronke2016a} we found that a simple one-to-one mapping between the shell-model parameters and the ones from a clumpy medium is not possible -- for the most part, the shell-model cannot reproduce the spectra emergent from a multiphase medium. This failure mostly results from the high fluxes at line-center from the multiphase simulations which are hard to obtain through radiative transfer through a uniform gas distribution (such as a shell).
However, in \citet{Gronke2016a} we restricted our analysis to covering factors of $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \sim \mathcal{O}(1)$ (and $\sigma_{{\mathrm{cl}}}\lesssim 100\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$) -- i.e., the `random-walk' and `optically thin' regime. As we showed here, for (much) greater number of clouds the system approaches a `slab like' state which leads to, e.g., much lower fluxes at line center for the resulting \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectrum. Hence, these multiphase spectra might be closer to observed ones.
Whether or not the shell-model parameters correspond to physical parameters of a clumpy medium with large $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ is part of future work. However, our results show that for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi\gtrsim \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$ the spectra are similar to a slab with the same column density. Furthermore, we fitted shell-models to three spectra originating from a clumpy medium with $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, \mathrm{total}}=4/3\times 10^{19}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$, $v_{\mathrm{max}}=50\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$, and various covering factors. Prior to fitting, we smoothed the spectra using a Gaussian kernel with FWHM $W\sim 24\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$.
Fig.~\ref{fig:shell_model_fits} shows the three spectra as well as the best-fit shell model spectra. The resulting shell-model parameters are also displayed in the figure. While the fits for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi = 3$ and $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi = 10$ are rather poor and the recovered shell-model column densities are more than an order of magnitude off, the spectra for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi=50$ can be remarkably well recovered. Interestingly, here the shell column density is very close to the input value, and the recovered shell of $v_{\mathrm{exp}}\approx 25\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$ outflow velocity corresponds to the mass weighted mean of the used Hubble-like outflow. Also, the dust content and to some extent the temperature of the gas are recovered. On the other hand, as the photons are injected at line center, the recovered widths of the intrinsic spectra ($\sigma_i$) are too large.
This may be to compensate the narrow coverage of the shell in velocity space. Note that a similar discrepancy of the intrinsic profile width is also found in the literature \citep[e.g.,][by comparing $\sigma_{\mathrm{i}}$ with the width of the H$\alpha$ line]{Yang2015} where it might also originate from radiative transfer effects.
All these points suggest that at least some of the shell-model parameters might have a true physical meaning. In this work, we provide an equally simple but physically meaningful model which serves as theoretical justification for the shell-model.
The full mapping from shell-model to parameters of a multiphase medium with large \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi is part of future work. However, from our single example it is already apparent that if an observed \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra can be modeled using a simple, homogeneous shell, one can -- and we encourage the reader to do so -- think instead about a fog of droplets (with $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \gg \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$), which is more realistic given our knowledge about gas properties.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{plots/overview_moving_asymmetry.pdf}
\caption{Asymmetry of the spectra (color coded) as a function of outflow velocity $v_{\mathrm{max}}$ and covering factor \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi for a fixed total column density of $N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, \mathrm{total}}=4/3\times 10^{19}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$. As tracer of the asymmetry we chose to display the ratio of the integrated flux on the blue side ($x \ge 0$) of the line $L_{\mathrm{blue}}$ over the integrated red flux $L_{\mathrm{red}}$ minus one. This implies a value of $0$ (in white) corresponds to a symmetric spectrum whereas $-1$ (in dark red) to a spectrum where all flux is redward of line center. The contour lines highlight values of $(-0.75,\,-0.5,\,-0.25)$. Also shown are the \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi boundary (Eq.~\eqref{eq:fccrit_outflow}), and the more precise $4/3\tau_{{\mathrm{cl}}}(x_{\rm next})=1$ deviation. The relatively low values of \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi imply that large covering factors as predicted by \citet{McCourt2016} will lead to asymmetric spectra.}
\label{fig:overview_outflow_asymmetry}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=.95\textwidth]{plots/spectra_comparison_plot.pdf}
\caption{Fitting shell models to spectra of $\log N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace, \mathrm{total}} / cm^{-2} \approx 19.1$ and $v_{\mathrm{max}}=50\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$ and covering factors of $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi = \{3,\,10,\,50\}$ (left to right panel). The black points and red lines show the spectra of the multiphase media, and the best fit shell-model spectra. See \S\ref{sec:clumpy-solid-conn} for details.}
\label{fig:shell_model_fits}
\end{figure*}
\subsection{Implications for \textit{ab initio} \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer simulations}
\label{sec:impl-lya-radi}
Our findings suggest a possible cause for the mismatch between observed \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra and the ones computed using snapshots of hydrodynamical simulations as input -- which are sometimes referred to \textit{ab initio} \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer simulations.
Observed \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra from $z\sim 0$ to higher redshifts show several common features:
\begin{itemize}[itemsep=10pt]
\item A significant shift redwards of the main emitting peak. For instance, at $z\sim 2-3$ galaxies selected due to their strong \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi emission as well as dropout-selected galaxies (LAEs and LBGs, respectively) show shifts of several hundred \,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi\ \citep[e.g.,][]{Steidel2010a,Kulas2011,Erb2014,2014ApJ...791....3S,Trainor2015,Hashimoto2015}.
\item Asymmetric profiles with mostly stronger red than blue component. For instance, \citet{Erb2014} measured the median EW ratio $W_{\rm blue} / W_{\rm red}$ in their sample of $36$ LAEs at $z\sim 2-3$ to be $\sim 0.4$. This is consistent with the findings at $z\sim 0.2$ that also show a dominant red side (by a factor of a few) \citep{Henry2015,Yang2015,Yang2017}.
\item There are roughly as many single as double peaked spectra. For instance, in the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi selected galaxy sample presented by \citet{Trainor2015} of $318$ LAEs at $z\sim 2.5-3$, $41\%$ show a double peaked spectra. This fraction agrees well with the double peaked fraction of $45\%$ they found in the KBSS-MOSFIRE LBG sample \citep{2014ApJ...795..165S}, and the ones of other studies \cite[e.g.,][found ratios of $\sim 1/3$ and $1/2$, respectively]{Kulas2011,2012ApJ...751...29Y}. Note that only a small part of the double peaked profiles show a dominant blue peak which agrees with flux ratio discussed above. \citet{Trainor2015} quantifies these to be $\sim 10\%$ of the double peaked spectra.
\item For double peaked spectra, the flux in the `valley' between the peaks is small. Because of smoothing and resolution effects due to the observational aperture, measuring this quantity is challenging -- in particular for higher redshifts. However, at lower redshifts \citet{Yang2015} find in their sample of $12$ galaxies at $z\sim 0.2$ the flux ratio between the valley and the red peak to be $0.03^{+0.08}_{-0.02}$ and never greater than $0.27$.
Also the $14$ galaxies of the `Lyman-$\alpha$ reference sample' \citep[LARS; ][]{Ostlin2014} at $0.02 < z < 0.2$ have a flux ratio between the maximum and the minimum of $<0.1$, mostly even consistent with zero \citep{Rivera-thorsen2014}.
\end{itemize}
These findings seem to be in stark contrast to \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer simulations that use a snapshot of a (high-resolution) hydro-dynamical simulation of a galaxy as input geometry
\citep[e.g.,][]{2006ApJ...645..792T,Laursen2007ApJ...657L..69L,Zheng2009,2011MNRAS.416.1723B,Verhamme2012,Behrens2014a,Smith2014,Trebitsch2016}. Due to computational cost, and probable directional dependence of the emergent spectrum \citep{Verhamme2012,Behrens2014a}, no statistical compilation of simulation-based spectra has yet been assembled.
However, existing predicted spectra are generally too symmetric and / or possess a too high flux at line center. This is commonly attributed to
\textit{(i)} CGM in combination with instrumental effects \citep[as discussed in ][]{Gronke2016a}, \textit{(ii)} radiative transfer effects in close proximity to the origin of the photon, and / or
\textit{(iii)} IGM absorption \citep{Dijkstra2007,Laursen2011}.
All these arguments move the problem to different spatial location (in case of \textit{(ii)} even to a subgrid scale). However, the last solution cannot be universally invoked, especially at lower redshifts. For instance, \citet{Laursen2011} found that for $z\lesssim 3.5$ only $\lesssim 30\%$ of the sightlines show a full absorption feature which would lead to a low flux in the `valley'.
Furthermore, while the IGM opacity increases with redshift, we see that the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi escape fraction from star-forming galaxies also increases with redshift \citep{Hayes2011,2011ApJ...736...31B,2013MNRAS.435.3333D}.
Both arguments strongly suggest that IGM absorption cannot be the dominant mechanism regulating the visibility of \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi emission.
We have shown here that this discrepancy between observations and simulations can be understood easily.
Simulations with Lagrangian-type techniques such as adaptive-mesh-refinement or smooth-particle-hydrodynamics (AMR and SPH, respectively) reach
their highest resolution in the densest regions such as the mid-plane
of the galaxy disk. While future simulations will likely reach peak
resolutions approaching the $\sim0.1$\,pc scale we expect, we note that this \textit{still} won't capture clump formation
and evolution at large distances in the CGM, where the density and
resolution remains low.
This means the clumps are unresolved and, thus, the covering factor per resolution element is less than unity -- compared to potentially hundreds as suggested theoretically by \citet{McCourt2016}. This lower $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ (while keeping the column density and global structure unchanged) leads to a higher flux at line center (as shown in Fig.~\ref{fig:spectra_fc}), less asymmetric spectra (Fig.~\ref{fig:overview_outflow_asymmetry}), and in general more `unrealistic spectra' \citep[cf.][]{Gronke2016a}.
Therefore, small-scale structure in the CGM is crucial for modeling radiative transfer through the galaxy.
We expect that
direct simulation of the multiphase CGM will be essentially
impossible, precisely because it requires very high spatial
resolution, even in parts of the galaxy which are typically empty:
for example, a spatial resolution of $\sim0.1$\,pc in the outskirts of
a galaxy corresponds to a mass resolution of
$\sim{10}^{-10}$\,--\,$10^{-9}$ solar masses.
Instead, we propose to study \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative on the smallest scales, and then to use this knowledge as a sub-grid recipe.
\section{Conclusion}
\label{sec:conclusion}
Motivated by several observations and recent theoretical work by \citet{McCourt2016} we studied \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer in an extremely clumpy medium, i.e., with large number of clumps per sightline (up to $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \sim 1000$). Our main findings on \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer through clumpy media are:
\begin{itemize}[itemsep=10pt]
\item The behaviour of a multiphase medium depends strongly on the `clumpiness' of the system -- even when keeping the other parameters such as the total column density constant.
\item In particular, we identify a threshold above which \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi photons escape preferentially via frequency excursion, i.e above which multiphase media affect \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi as if they were homogeneous.
This transition depends on clump column density and can be estimated analytically.
We found the threshold for the static case to be:
\begin{equation}
\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi \approx \begin{dcases}
\frac{2 \sqrt{a_v \tau_{0, {\mathrm{cl}}}}}{3 \pi^{1/4}} & \text{ for } \sqrt{3} a_v \tau_0 \gtrsim 19\\
\frac{1.65}{1-\mathrm{e}^{-\tau_{0,{\mathrm{cl}}}}} & \text{ otherwise.}
\end{dcases}
\end{equation}
\item The value of this threshold between clumpy- and homogeneous nature further depends on the clump kinematics in a way that can also be estimated analytically.
If the clump motion is uncorrelated and Maxwellian, we find that the threshold is given by:
\begin{equation}
\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi \approx \mathrm{max}\left( \frac{3\sigma_{\mathrm{cl}}}{2 v_{\rm th} \sqrt{\ln(4/3 \tau_{0,{\mathrm{cl}}})}},\; 1.5 \right).
\end{equation}
This is valid for sufficiently large clump motion, i.e., $\sigma_{\mathrm{cl}} \gtrsim v_{\mathrm{th}}$. For smaller values, the system approaches the static case above. Furthermore, we expect for large-scale correlations in velocity the transition to happen between these extreme cases. We will investigate this in a future study.
\item A similar threshold was found for outflowing clumps (\S~\ref{sec:non-static-case}). We also showed that for outflowing clumps, increasing $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$ naturally leads to more asymmetric line profiles, in much better agreement with what has been observed in observations of galaxies.
\end{itemize}
These results suggest important implications for the interpretation of observed \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra, as a multiphase medium is physically more motivated than simplified homogeneous geometries such as the `shell-model'. Nevertheless since shell models successfully reproduce observed spectra, they are frequently used to model observations. Because a medium with sufficiently large covering factor behaves as a homogeneous medium, the success of shell models may indicate
large covering factors are typical in galaxies as predicted by \citet{McCourt2016}.
Specifically, we found typical values of $\relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi \sim 10-50$, much smaller than $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \gtrsim 1000$ predicted in their work. In this picture, it is easy to understand the convergence to the shell model.
Motivated by these results, we fitted shell models to spectra emerging from extremely clumpy outflows undergoing Hubble flow. We found that the column density from the shell closely matches that of the collection of clumps as a whole, and the shell expansion velocity appears to be the mass weighted average velocity. This result is very promising as it suggests that the shell model provides us with a fast method of extracting some physical properties of the interstellar and circumgalactic medium from the \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectral line shape.
In addition, the value of other shell parameters (e.g. intrinsic \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi line width prior to scattering) should not necessarily be interpreted literally as physical.
We will explore this systematically in future work.
Another implication concerns the mismatch between observed \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi spectra and the ones predicted by theoretical studies of \ifmmode{\mathrm{Ly}\alpha}\else Ly$\alpha$\xspace\fi radiative transfer utilizing hydrodynamical simulations for their input geometry. Our work suggests that this mismatch can be due to the existence of tiny clumps in the observed systems which cannot form even in the most modern hydrodynamical simulations of galaxies due to their limited resolution. Thus, setups of these simulations might yield effective covering factors which are too low causing the spectra to possess, e.g., a too large flux at line center.
We will use our results for radiative transfer on small scales to develop an effective theory which can be implemented as a sub-grid model in global simulations of galaxies.
\section{Regimes of a medium with uncorrelated clump motion}
\label{sec:regimes_moving}
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/regimes_sketch_moving.pdf}
\caption{Escape regimes of a medium with (uncorrelated) randomly moving clumps as discussed in Appendix~\ref{sec:regimes_moving}.}
\label{fig:regimes_sketch_moving}
\end{figure}
For randomly moving, optically thick clumps the photons either escape through holes in velocity space (if $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \lesssim \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$, Eq.~\eqref{eq:fccrit_moving}) or escape in single-flight (for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi\gtrsim \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$) -- as described in \S~\ref{sec:non-static-case}. In the former case, the emergent spectrum will be similar to the intrinsic one, that is, narrow, and singly peaked. If, however, $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi > \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$ the radiative transfer process will be similar to a slab with temperature $T_{\mathrm{eff}}$ (Eq.~\eqref{eq:Teff}) which means the photons will escape in a single flight after interaction with one (fast-moving) clump, and so will the emergent double peaked spectrum, i.e., a peak position of $x_{\rm p}\sim x_*$ or in (observed) velocity units
\begin{equation}
\label{eq:v_peak_moving}
v_{\mathrm{p}}\approx x_* v_{\mathrm{th}}(T_{\mathrm{eff}}) \approx 3.8\sigma_{{\mathrm{cl}}}\;.
\end{equation}
Fig.~\ref{fig:regimes_sketch_moving} shows a visual overview of these regimes. In this figure, we also marked that a smaller clump motion than the internal thermal motion of the atoms (for the parameters used in this work of $\sigma_{\mathrm{cl}} \lesssim 13\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi$) leads to a convergence back to the static case.
Another interesting part of the parameter space is between the two regimes, for $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi \sim \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$. Here, the velocity space is sufficiently sampled so that hardly any photons can escape without clump interaction. However, after an interaction with a (slowly moving) clump the probability of interacting with another clump is small -- even if the photon is still in the core of the line. This is because the velocity distribution is not \textit{that} well sampled to provide $\tau_0 \gg 1$.
As a result, the emergent spectrum will directly represent the clumps' velocity dispersion -- which means a single peaked spectrum a line center of width $\sim \sigma_{\mathrm{cl}}$.
To summarize: with increasing covering factor, a medium with uncorrelated clump motion can lead to a narrow or wide single peaked spectrum (of widths of the intrinsic spectrum or the clump velocity dispersion, respectively) at line center, or a wide double-peaked spectrum (if $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi\gtrsim \relax\ifmmode f_{\mathrm{c,\,crit}}\else $f_{\mathrm{c,\,crit}}$\xspace\fi$).
\section{Additional numerical results}
\label{sec:additional-results}
\begin{figure}
\centering
\includegraphics[width=.95\linewidth]{plots/transmission.pdf}
\caption{Transmission through a clumpy slab with column density $N_{\mathrm{HI,\,total}}=\frac{4}{3}\times 10^{19}\,\ifmmode{{\rm cm}}\else cm\fi^{-2}$ (as before, measured per half-height) versus covering factor $\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi$. As comparison we show the transmission through a (solid) slab with the same column density.}
\label{fig:transmission}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=.95\linewidth]{plots/traj_multiplot.png}
\caption{Examples of photon trajectories. The \textit{left panel} shows a photon escaping through random-walk from a static medium with $(N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}},\,\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi) = (10^{20}\,\ifmmode{{\rm cm}}\else cm\fi^{-2},\,100)$. In the \textit{central panel} the photon escapes in an excursion [$(N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}},\,\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi) = (10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2},\,100)$] after some random walk, and in the \textit{right panel} nearly directly through excursion / single flight due to movement of the clumps [$(\,N_{{\text{H\MakeUppercase{\romannumeral 1}}}\xspace,{\mathrm{cl}}},\,\relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi,\sigma_{\mathrm{cl}}) = (10^{17}\,\ifmmode{{\rm cm}}\else cm\fi^{-2},\,100,\,100\,\ifmmode{\mathrm{km}\,\mathrm{s}^{-1}}\else km\,s${}^{-1}$\fi)$]. In each panel, the escape frequency $x$ is displayed as well as the total number of scatterings $n$, and the distance travelled $d$. In addition, the color coding represents the frequency of the photon (truncated at $\pm 5$). An animated version of this figure is available at \url{http://bit.ly/a-in-a-fog}.
}
\label{fig:example_traj}
\end{figure*}
\subsection{Transmission through a clumpy slab}
\label{sec:transm-thro-clumpy}
Fig.~\ref{fig:transmission} shows the fraction of photons that passed through a clumpy medium when emitted at the boundary of the box for a fixed total column density but various number of clumps per sightline. The transmitted fraction of photons decreases with increasing covering factor and approaches the limit of a homogeneous slab. We attribute this dependence on \relax\ifmmode{f_{\mathrm{c}}}\else $f_{\mathrm{c}}$\xspace\fi as well as the differences to `surface effects', i.e., due to the roughness of the boundary it is easier for photons to get ``trapped'' in the slab.
\subsection{Examples of photon trajectories}
\label{sec:exampl-phot-traj}
Fig.~\ref{fig:example_traj} shows examples for the three different escape mechanism discussed in this work. The left panel shows a random walk in a static medium, the central panel escape through excursion, and the right panel the escape through single flight. An animated version of Fig.~\ref{fig:example_traj} is available online\footnote{\url{http://bit.ly/a-in-a-fog}}.
|
2302.13516
|
\section{Introduction}
\subsection{Tilings and Aperiodic Protosets}\label{sec:tiling}
A \emph{\textbf{tiling}} of the Euclidean plane $\mathbb{E}^2$ is a collection $\{T_i\}$ of distinct sets called \emph{\textbf{tiles}} (which are typically topological disks) such that $\cup T_i = \mathbb{E}^2$ and for all $i \neq j$, $\text{int}(T_i) \cap \text{int}(T_j) = \emptyset$. In other words, a tiling is a covering of $\mathbb{E}^2$ in which each set can only overlap at its boundary.
A \emph{\textbf{protoset}} for a tiling $\mathscr{T}$ is a collection of tiles $\mathcal{T}$ such that each tile of $\mathscr{T}$ is congruent to a tile in $\mathcal{T}$, in which case we say that $\mathcal{T}$ \emph{\textbf{admits}} the tiling $\mathscr{T}$. The \emph{\textbf{symmetry group}} of a tiling $\mathscr{T}$ is the set of rigid planar transformations $\mathcal{S}(\mathscr{T})$ such that $\sigma(\mathscr{T}) = \mathscr{T}$ for all $\sigma \in \mathcal{S}(\mathscr{T})$. If $\sigma(\mathscr{T})$ contains two nonparallel translations, we say that $\mathscr{T}$ is \emph{periodic}; otherwise, we say $\mathscr{T}$ is \emph\textbf{{nonperiodic}}. If every tiling admitted by a protoset $\mathcal{T}$ is \emph{\textbf{nonperiodic}}, we say that $\mathcal{T}$ is an \emph{\textbf{aperiodic protoset}}.
Aperiodic protosets are somewhat rare, and indeed, until the 1960s, no aperiodic protosets were known to exist. H. Wang famously conjectured that aperiodic protosets cannot exist \cite{Wang}; that is, Wang conjectured that any protoset that admits a tiling of the plane must admit at least one periodic tiling. Wang's conjecture was refuted in 1964 by his doctoral student, R. Berger, in his PhD thesis \cite{MR2939561,MR216954} where an aperiodic protoset consisting of over 20,000 edge-marked squares called Wang tiles was described. Wang tiles are a bit of a special case in the theory of tilings in that tilings formed from a protoset of Wang tiles consist only of translates of the prototiles; however, this simplifying restriction does not reduce generality, for if a Wang tile protoset admits a tiling in which all rigid motions are allowed, then another larger Wang protoset containing reflected and/or rotated copies of the tiles in the original protoset can be found that admits the same tilings as the original protoset, but using only translated copies. Since Berger's original discovery, much lower-order aperiodic Wang tile protosets have been discovered, culminating in 2015 with the discovery by E. Jeandel and M. Rao \cite{JR1} of an order-11 aperiodic Wang tile protoset. In this same work the authors proved that 11 is the minimal order for an aperiodic Wang tile protoset. The aperiodic Jeandel-Rao protoset is seen in Figures \ref{fig:T0_tiles} and \ref{fig:T0_shapes}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{paper/figures/T0_tiles.pdf}\caption{The Jeandel-Rao aperiodic Wang protoset $\mathcal{T}_0$}\label{fig:T0_tiles}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{paper/figures/T0_shapes.pdf}\caption{$\mathcal{T}_0$ with geometric edge-matching rules}\label{fig:T0_shapes}
\end{center}
\end{figure}
IN 1971, R. Robinson developed a way to encode the idea of Berger's original aperiodic protoset into an order-6 aperiodic protoset consisting of shapes with notched edges (Figure \ref{fig:robinson}). In 1974, R. Penrose discovered an order-6 aperiodic protoset that he was later able to reduce to an aperiodic protoset of order 2. One form of order-2 aperiodic protoset discovered by Penrose is depicted in Figure \ref{fig:penrose_rhombs}; we refer to this protoset as the Penrose rhombs, and this protoset will be the main subject of this article. The Penrose rhombs have been studied extensively, and Penrose tilings have many special properties (e.g., 5-fold rotational symmetry, substitutive/hierachical structure) and applications (e.g., crystallography, cut-and-project schemes) that make them a central example in the theory of tilings.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{.4\textwidth}
\centering
\includegraphics[width=.8\textwidth]{paper/figures/robinson}
\caption{Robinson's order-6 aperiodic protoset}
\label{fig:robinson}
\end{subfigure}
\begin{subfigure}[b]{.4\textwidth}
\centering
\includegraphics[width=.8\textwidth]{paper/figures/penrose}
\caption{The Penrose rhombs}
\label{fig:penrose_rhombs}
\end{subfigure}
\label{fig:aperiodic}\caption{Two low-order aperiodic protosets}\end{figure}
\subsection{Labb{\'{e}}'s Markov Partition for the Jeandel-Rao Protoset}
Let $\mathcal{T}_0$ be the aperiodic Jeandel-Rao Wang tile protoset (Figures \ref{fig:T0_tiles}). The set of all tilings admitted by $\mathcal{T}_0$ is called the Jeandel-Rao Wang shift and is denoted $\Omega_0$. In \cite{Labb2021}, S. Labb\'{e} presented a Markov partition $\mathcal{P}_0$ (Figure \ref{fig:P0}) for the unique minimal subshift $\mathcal{X}_{\mathcal{P}_0,R_0}$ of $\Omega_0$. $\mathcal{P}_0$ is a partition of the torus $\boldsymbol{T}=\R^2 / \Gamma_0$ where $\Gamma_0$ is the lattice spanned by the vectors $(0, \varphi)$ and $(1,\varphi+3)$, where $\varphi = (1 + \sqrt{5})/2$ is the golden mean. The labels of the atoms of $\mathcal{P}_0$ correspond to the labels of the tiles in $\mathcal{T}_0$. The significance of this Markov partition is remarkable in that it encodes almost all tilings in $\Omega_0$\footnote{Labb\'{e} conjectures that the set of tilings in $\Omega_0$ that are not encoded by the partition is a set of measure 0} in the following way: tilings in $\Omega_0$ correspond to orbits of points in $\boldsymbol{T}$ under the $\ZZ^2$ shift action $R_0$ defined on $\boldsymbol{T}$ by $R_{0}^{\boldsymbol{n}}(\boldsymbol{x}) = \boldsymbol{x} + \boldsymbol{n}$; the Wang tile from $\mathcal{T}_0$ placed at position $\boldsymbol{n}$ is the tile with the same label as the atom of $\mathcal{P}_0$ in which $R_{0}^{\boldsymbol{n}}(\boldsymbol{x})$ lies. In this way, the Markov partition $\mathcal{P}_0$ serves as a sort of DNA for $\Omega_0$, encoding the instructions for how to grow tilings from a seed.
The major focus of this article is to apply Labb\'{e}'s symbolic dynamical insight to Wang tilings associated with the Penrose rhombs. More precisely, we shall show that the Penrose rhombs can be encoded as a set $\mathcal{T}$ of Wang tiles, and then we will demonstrate the existence of a Markov partition for the set of all Wang tilings admitted by $\mathcal{T}$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{paper/figures/JR-partition_to_tiling.pdf}\caption{Labb\'{e}'s partition $\mathcal{P}_0$ for the minimal subshift $\mathcal{X}_{\mathcal{P}_0,R_o}$ of the Jeandel-Rao Wang shift $\Omega_0$ is outlined by the heavy black line. The orbit of the red point $\boldsymbol{p}$ under the $\ZZ^2$-action $R_0$ definted by $R_{0}^{\boldsymbol{n}}(\boldsymbol{p}) = \boldsymbol{p} + \boldsymbol{n}$ (the set of white points) corresponds to a Wang tiling in $\Omega_0$, depicted at right.} \label{fig:P0}
\end{center}
\end{figure}
\subsection{Results}
\begin{flushleft}
First we define a ``Wangified" version of the Penrose rhombs (Section \ref{sec:Penrose_shift}). This Wang tile protoset, $\mathcal{T}$, we call the Penrose Wang protoset, and it is shown in Figure \ref{fig:protoset}. Let $\Omega_{\mathcal{T}}$ be the set of all Wang tilings admitted by $\mathcal{T}$. In Section \ref{sec:Wang_shifts}, we discuss how $\Omega_{\mathcal{T}}$ be viewed through the lens of dynamical systems; $(\Omega_{\mathcal{T}},\ZZ^2,\sigma)$ is called the Wang shift of $\mathcal{T}$, where $\sigma$ is the $\ZZ^2$-shift action such that $\sigma^{\boldsymbol{n}}(\tau)$ is the tiling $\tau\in \Omega_{\mathcal{T}}$ translated by $\boldsymbol{n} \in \ZZ^2$. We aim to establish to following:
\begin{enumerate}
\item We define a partition $\mathcal{P}$ of the torus $\boldsymbol{T} = \RR^2 / \ZZ^2$ that serves as a Markov partition for a symbolic dynamical system on $\boldsymbol{T}$ (Theorem \ref{thm:Markov}).
\item $\mathcal{P}$ and a $\ZZ^2$-shift action $R$ give rise to a minimal subshift $\mathcal{X}_{\mathcal{P},R} \subseteq \Omega_{\mathcal{T}}$.
\item As an application of this Markov partition $\mathcal{P}$, we identify that the Penrose Wang shift has 5 nonexpansive directions. See Corollary \ref{cor:nonexpansive_dir}.
\end{enumerate}
\end{flushleft}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\textwidth]{paper/figures/partitionwticks2.pdf}\caption{A Markov partition $\mathcal{P}$ of the unit square for the Penrose Wang shift $\Omega_{\mathcal{T}}$. With the $\ZZ^2$-action $R$ is defined on the torus $\boldsymbol{T} = \RR^2/\ZZ^2$ by $R^{\boldsymbol{n}}(\boldsymbol{x}) = \boldsymbol{x} + n_1 (\varphi - 1,\varphi-1) + n_2 (2-\varphi,0)$ where $\boldsymbol{n} = (n_1,n_2)$, $\mathcal{P}$ gives a symbolic representation of $(\boldsymbol{T},\ZZ^2,R)$.} \label{fig:square}
\end{center}
\end{figure}
\section{Background}
In this section we provide some definitions from the theory of topological dynamical systems that we will use to describe Wang tilings and properties thereof.
\subsection{Topological Dynamical Systems}\label{section:dyn_sys}
We begin by briefly describing a useful way to understanding the spaces of Wang tilings (Wang shifts) as dynamical systems and refer the reader to \cite{Walt1982} for more details. A \emph{dynamical system} is a triple $(X, G, S)$, where $X$ is a topological space, $G$ is a topological group, and $S$ is a continuous function $G \times X \rightarrow X$ defining a left action of $G$ on $X$: if $x \in X$, $e$ is the identity element of $G$, and $g, h \in G$, then using additive notation for the operation in $G$ we have $S(e, x) = x$ and $S(g + h, x) = S(g, T(h, x))$. That is, if we denote the transformation $x \mapsto S(g, x)$ by $S^g$, then $S^{g+h} = S^gS^h$. The \emph{orbit} of a point $x \in X$ under the left action of $G$ by $S$ is the set $\mathcal{O}_S(x,G) = \{S^g(x):g \in G\}$, and when the group $G$ is understood by context, we will shorten this notation to just $\mathcal{O}_{S}(x)$.
If $Y \subset X$, $\overline{Y}$ denotes the topological closure of $Y$ and let $S(Y ) := \cup_{g \in G}S^g(Y)$ denote the $S$-closure of $Y$. A subset $Y \subset X$ is $S$\emph{-invariant} if $S(Y ) = Y$. A dynamical system $(X, G, S)$ is called \emph{minimal} if $X$ does not contain any nonempty, proper, closed $S$-invariant subset. The left action of $G$ on $X$ is \emph{free} if $g = e$ whenever there exists $x \in X$ such that $S^g(x) = x$.
Let $(X, G, R)$ and $(Y, G, S)$ be two dynamical systems with the same topological group $G$. A \emph{homomorphism} $\theta \!\!:\!\! (X, G, R) \rightarrow (Y, G, S)$ is a continuous function $\theta \!\!:\!\! X \rightarrow Y$ satisfying the commuting property that $R^g \circ \theta = \theta \circ S^g$ or every $g \in G$. A homomorphism $\theta \!\!:\!\! (X, G, R) \rightarrow (Y, G, S)$ is called an \emph{embedding} if it is one-to-one, a \emph{factor map} if it is onto, and a \emph{topological conjugacy} if it is both one-to-one and onto and its inverse map is continuous. If $\theta \!:\! (X, G, R) \rightarrow (Y, G, S)$ is a factor map, then $(Y, G, S)$ is called a \emph{factor} of $(X, G, R)$ and $(X, G, R)$ is called an \emph{extension} of $(Y, G, S)$. Two dynamical systems are \emph{topologically conjugate} if there is a topological conjugacy between them.
\subsection{Subshifts and Subshifts of Finite Type}\label{subsec:shifts}
Here we follow the notation of \cite{Schmidt2001}. Let $\mathcal{A}$ be a finite set, $d \geq 1$, and let $\mathcal{A}^{\ZZ^d}$ be the set of all maps $x : \ZZ^d \rightarrow \mathcal{A}$, equipped with the compact product topology. An element $x \in \mathcal{A}^{\ZZ^d}$ is called a \emph{configuration} and we write it as $x = (x_{\boldsymbol{m}}) = (x_{\boldsymbol{m}} :\boldsymbol{m} \in \ZZ^d)$, where $x_{\boldsymbol{m}} \in \mathcal{A}$ denotes the value of $x$ at $\boldsymbol{m}$. The topology on $\mathcal{A}^{\ZZ^d}$ is compatible with the metric defined for all configurations $x, x' \in \mathcal{A}^{\ZZ^d}$ by $\text{dist}(x, x') = 2^{-\min\{\|\boldsymbol{n}\|:x_{\boldsymbol{n}} \neq x'_{\boldsymbol{n}}\}}$ where $\|\boldsymbol{n}\| = |n_1| + \cdots + |n_d|$. The \emph{shift action} $\sigma : \boldsymbol{n} \mapsto \sigma^{\boldsymbol{n}}$ of $\ZZ^d$ on $\mathcal{A}^{\ZZ^d}$ is defined by \begin{equation} (\sigma^{\boldsymbol{n}}(x))_{\boldsymbol{m}} = x_{\boldsymbol{m}+\boldsymbol{n}} \label{eq:shift_action}\end{equation} for every $x = (x_{\boldsymbol{m}}) \in \mathcal{A}^{\ZZ^d}$ and $\boldsymbol{n} \in \ZZ^d$. A subset $X \in \mathcal{A}^{\ZZ^d}$ is \emph{shift-invariant} if $\sigma(X) = X$ and a closed, shift-invariant subset $X \subset \mathcal{A}^{\ZZ^d}$ is a \emph{subshift}. If $X \subset \mathcal{A}^{\ZZ^d}$ is a subshift, we write $\sigma = \sigma^X$ for the restriction $\sigma$ to $X$. When $X$ is a subshift, the triple $(X, \ZZ^d, \sigma)$ is a dynamical system.
A configuration $x \in X$ is \emph{periodic} if there is a nonzero vector $\boldsymbol{n} \in \ZZ^d \setminus \{0\}$ such that $x = \sigma^{\textbf{n}}(x)$ and otherwise it is said \emph{nonperiodic}. We say that a nonempty subshift $X$ is \emph{aperiodic} if the shift action $\sigma$ on $X$ is free.
For any subset $S \subset \ZZ^d$, let $\pi_S : \mathcal{A}^{\ZZ^d} \rightarrow \mathcal{A}^{S}$ denote the projection map which restricts every $x \in \mathcal{A}^{\ZZ^d}$ to $S$. A pattern is a function $p \in \mathcal{A}^{S}$ for some finite subset $S \subset \ZZ^d$. To every pattern $p \in \mathcal{A}^{\ZZ^d}$ corresponds a subset $\pi_{S}^{-1}(p) \subset \mathcal{A}^{\ZZ^d}$ called a \emph{cylinder}. A subshift $X \subset \mathcal{A}^{\ZZ^d}$ is a \emph{shift of finite type} (SFT) if there exists a finite set $\mathcal{F}$ of \emph{forbidden patterns} such that \begin{equation} X = \{x \in \mathcal{A}^{\ZZ^d} \,|\, \pi_S \circ \sigma^{\boldsymbol{n}}(x) \notin \mathcal{F} \text{ for all } \boldsymbol{n} \in \ZZ^d \text{ and } S \subset \ZZ^d\} \label{eqn:SFT}\end{equation} In this case, we write $X = SFT(\mathcal{F})$. In this article, we consider shifts of finite type on $\ZZ \times \ZZ$; that is, the case $d = 2$. In particular, we will consider 2-dimensional Wang shifts (defined in Section \ref{sec:Wang_shifts}), which are shifts of finite type on $\ZZ^2$.
\subsection{Symbolic dynamical systems}
This section is based on ideas from \cite[\S6.5]{LindDouglas1995} on Markov partitions. A \emph{topological partition} of a metric space $M$ is a finite collection $\{P_0, P_1,\ldots, P_{r-1}\}$ of disjoint open sets such that $M = \overline{P_0} \cup \overline{P_1} \cup \cdots \cup \overline{P_{r-1}}.$ Suppose that $M$ is a compact metric space, $(M, \ZZ^2,R)$ is a dynamical system, and that $\mathcal{P} = \{P_0, P_1,\ldots, P_{r-1}\}$ is a topological partition of $M$. Let $\mathcal{A} = \{0,1,\ldots,r-1\}$ and $S \subset \ZZ^2$ be a finite set. We say that a \emph{pattern} $w \in \mathcal{A}^S$ is \emph{allowed} for $\mathcal{P},R$ if
\[
\bigcap_{\boldsymbol{k}\in S}R^{-\boldsymbol{k}}(P_{w_{\boldsymbol{k}}}) \neq \emptyset.
\]
Let $\mathcal{L}_{\mathcal{P},R}$ be the collection of all allowed patterns for $\mathcal{P},R$.
The set $\mathcal{L}_{\mathcal{P},R}$ is the language of a subshift
$\mathcal{X}_{\mathcal{P},R}\subseteq\mathcal{A}^{\Z^2}$ defined as follows,
see \cite[Prop.~9.2.4]{MR3525488},
\begin{definition} \label{dfn:symb_dyn_sys} Let \[
\mathcal{X}_{\mathcal{P},R} =
\{x\in\mathcal{A}^{\Z^2} \mid \pi_S\circ\sigma^{\boldsymbol{n}}(x)\in\mathcal{L}_{\mathcal{P},R}
\text{ for every } {\boldsymbol{n}}\in\Z^2 \text{ and finite subset } S\subset\Z^2\}.\] We call $\mathcal{X}_{\mathcal{P},R}$ the \textbf{symbolic dynamical system} corresponding to $\mathcal{P},R$.
\end{definition}
For each $w \in \mathcal{X}_{\mathcal{P},R} \subset \mathcal{A}^{\ZZ^2}$ and $n \geq 0$ there is a corresponding nonempty open set
\[D_n(w) = \bigcap_{\|\boldsymbol{k}\| \leq n} R^{-\boldsymbol{k}}(P_{w_{\boldsymbol{k}}}) \subseteq M.\]
The closures $\overline{D}_n(w)$ of these sets are compact and decrease with $n$ in the sense that that $\overline{D_0}(w) \supseteq \overline{D_1}(w) \supseteq \overline{D_2}(w) \supseteq \cdots$. It follows that $\cap_{n = 0}^{\infty} \overline{D_n}(w) \neq \emptyset$. In order for configurations in $\mathcal{X}_{\mathcal{P},R}$ to correspond to points in $M$, this intersection should contain only one point. This leads to the following definition.
\begin{definition} A topological partition $\mathcal{P}$ of $M$ gives a \textbf{\emph{symbolic representation}} of $(M, \ZZ^2,R)$ if for every $w \in \mathcal{X}_{\mathcal{P},R}$, the intersection $\cap_{n = 0}^{\infty}\overline{D_n}(w)$ consists of exactly one point $m \in M$. We call $w$ a \textbf{\emph{symbolic representation}} of $m$. \label{dfn:symb_rep}\end{definition}
Labb{\'{e}} provides the following alternative criterion for determining when a partition gives a symbolic representation. This lemma utilizes algebraic constructs in place of topological ones, and will therefore become useful in our later proofs.
\begin{lemma}[\cite{Labb2021}]\label{lem:LabbeMinimality}Let $(M,\ZZ^2,R)$ be a minimal dynamical system and $\mathcal{P} = \{P_0,P_1,\ldots,P_{r-1}\}$ be a topological partition of $M$. If there exists an atom $P_i$ which is invariant only under the trivial translation in $M$, then $\mathcal{P}$ gives a symbolic representation of $(M,\ZZ^2,R)$.\end{lemma}
\begin{proposition}\label{prop:factor-map}{\rm\cite[Prop.~5.1]{Labb2021}}
Let $\mathcal{P}$ give a symbolic representation of the dynamical system
$(\boldsymbol{T},\Z^2,R)$ such that
$R$ is a $\ZZ^2$-rotation on $\boldsymbol{T}$.
Let $f:\mathcal{X}_{\mathcal{P},R}\to \boldsymbol{T}$ be defined
such that $f(w)$ is the unique point
in the intersection $\cap_{n=0}^{\infty}\overline{D}_n(w)$.
The map $f$ is a factor map from
$(\mathcal{X}_{\mathcal{P},R},\Z^2,\sigma)$ to $(\boldsymbol{T},\Z^2,R)$
such that $R^{\boldsymbol{k}}\circ f = f\circ\sigma^{\boldsymbol{k}}$
for every ${\boldsymbol{k}}\in\Z^2$.
The map $f$ is one-to-one on
$f^{-1}(\boldsymbol{T}\setminus\Delta_{\mathcal{P},R})$.
\end{proposition}
\subsection{Markov Partitions}
The following definition is taken from \cite{Labb2021}.
\begin{definition}\label{def:Markov} A topological partition $\mathcal{P}$ of $M$ is a \textbf{Markov partition} for $(M,\ZZ^2,R)$ if
\begin{itemize}
\item $\mathcal{P}$ gives a symbolic representation of $(M,\ZZ^2,R)$ and
\item $\mathcal{X}_{\mathcal{P},R}$ is a shift of finite type (SFT).
\end{itemize}\end{definition}
The symbolic representation condition guarantees that every configuration in $\mathcal{X}_{\mathcal{P},R}$ corresponds to a single point in $M$. The shift of finite type condition guarantees that the set of patterns can be expressed using a using a finite set of forbidden patterns, mirroring the structure of a Wang shift.
\section{Wang Shifts}\label{sec:Wang_shifts}
A \emph{\textbf{Wang tile}} is a square with colored edges where the colors serve as edge matching rules; two Wang tiles may meet along an edge if the colors match. More formally, a Wang tile can be represented as a tuple of four colors $(r, t, \ell, b) \in V \times H \times V \times H$ where $V$ and $H$ are two finite sets (the vertical and horizontal colors, respectively). For any Wang tile $\tau = (r, t, \ell, b)$, let $\text{Right}(\tau) = r$, $\text{Top}(\tau) = t$, $\text{Left}(\tau) = \ell$, and $\text{Bottom}(\tau) = b$ be the colors of the right, top, left, and right sides of $\tau$, respectively.
Let $\mathcal{T}$ be a collection of Wang tiles (called a \emph{\textbf{protoset}}). A configuration $x:\ZZ^2 \rightarrow \mathcal{T}$ in $\mathcal{T}^{\ZZ^2}$ is \emph{\textbf{valid}} if for each $\boldsymbol{n} \in \ZZ^2$, $\text{Right}(x_{\boldsymbol{n}}) = \text{Left}(x_{\boldsymbol{n}+(1,0)})$ and $\text{Top}(x_{\boldsymbol{n}}) = \text{Bottom}(x_{\boldsymbol{n}+(0,1)})$. The \emph{\textbf{Wang shift}} of $\mathcal{T}$ is the set $\Omega_{\mathcal{T}}$ of all valid configurations in $\mathcal{T}^{\ZZ^2}$. Thus, $\Omega_{\mathcal{T}}$ captures in a symbolic way the set of all valid Wang tilings admitted by the Wang tile protoset $\mathcal{T}$.
In a natural way, $(\Omega_{\mathcal{T}},\ZZ^2,\sigma)$ (where $\sigma$ is defined as in Section \ref{subsec:shifts}) is a dynamical system, and also notice that a Wang shift $\Omega_{\mathcal{T}}$ is an SFT since there is a natural set of forbidden patterns defined as the set of all invalid horizontal and vertical dominoes formed from Wang tiles in $\mathcal{T}$ whose shared edge are marked by nonmatching colors.
We say that a Wang protoset $\mathcal{T}$ is \emph{\textbf{aperiodic}} if the Wang shift $\Omega_{\mathcal{T}}$ is nonempty and aperiodic. Notice that this definition for aperiodic protoset agrees with with more general definition given in Section \ref{sec:tiling} if we restrict the allowable symmetries to translations only.
\section{The Penrose Wang Shift}\label{sec:Penrose_shift}
In \cite[Section 11.1, Exercise 2]{GS1987}, the authors outline a way to create an order-24 aperiodic Wang tile protoset by cutting up a tiling by the Penrose rhombs into 24 distinct patches, the unions of which tile the plane as topological squares. In Figure \ref{fig:patch-proto} we see these 24 patches. Since this set of 24 patches tile the plane using only translates, they may be viewed as a protoset of Wang tiles. The equivalence between these patches and Wang tiles is shown in Figure \ref{fig:patch-proto}. We refer to this collection of Wang tiles as the \emph{Penrose Wang tile protoset} (Figure \ref{fig:protoset}) and, from this point forward, $\mathcal{T}$ shall specifically denote the Penrose Wang tile protoset, and the Wang shift $\Omega_{\mathcal{T}}$ will be refered to as the \emph{Penrose Wang Shift}.
In H. Jang's recent Ph.D. thesis \cite[Lemma 6.2.2, p. 75]{Jang2021}, it was shown that any tiling by Penrose rhombs is equivalent to a Wang tiling in the Penrose Wang shift $\Omega_{\mathcal{T}}$. This is a nice result in that it allows us to gain a fuller understanding of the dynamics of Penrose tilings by applying Labb\'{e}'s program of study to the Penrose Wang shift $\Omega_{\mathcal{T}}$. In Section \ref{sec:symbrep}, we outline the methodology and pertinent results from \cite{Labb2021}, and in Section \ref{sec:markov} we use Labb'{e}'s results to verify a number of facts concerning $\Omega_{\mathcal{T}}$; most centrally, we give a description of a Markov partition for $\Omega_{\mathcal{T}}$, which provides a very nice object with which to study the dynamics of Penrose tilings.
\begin{figure}[H]
\centering
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p0.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p1.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p2.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p3.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p4.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p5.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p6.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p7.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p8.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p9.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p10.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p11.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p12.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p13.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p14.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p15.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p16.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p17.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p18.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p19.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p20.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p21.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p22.pdf}
\end{subfigure}
\begin{subfigure}[H]{.32\textwidth}
\centering
\includegraphics[width=.87\textwidth]{paper/figures/p23.pdf}
\end{subfigure}
\caption{Patches of Penrose rhomb tilings and their corresponding Wang tiles. Red arcs with the same label are translates of one another, and so these labels correspond to colors the sides of the Wang tiles.}\label{fig:patch-proto}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.9\textwidth]{paper/figures/Penrose_Wang_Protoset.pdf}\caption{The Penrose Wang Protoset} \label{fig:protoset}
\end{center}
\end{figure}
\section{From Symbolic Representations to Wang Shifts}\label{sec:symbrep}
Here we borrow just enough of the general theory in Section 4 and 8 of \cite{Labb2021} to establish that the partition $\mathcal{P}$ of the torus $\boldsymbol{T}$ in Figure \ref{fig:square} gives rise to a Wang protoset $\mathcal{T}$ so that the symbolic dynamical system $\mathcal{X}_{\mathcal{P},R}$ is a subset of the Wang shift $\Omega_{\mathcal{T}}$.
Suppose that a partition $\mathcal{P} = \{P_a\}_{a \in \mathcal{A}}$ of the 2-dimensional torus $\boldsymbol{T}$, and suppose that $\mathcal{P}$ gives a symbolic representation of $(\boldsymbol{T},\ZZ^2,R)$. The \textbf{boundary} of $\mathcal{P}$ is the set \[\Delta = \bigcup_{a \in \mathcal{A}}\partial P_a,\] and \[ \Delta_{\mathcal{P},R} = \bigcup_{\boldsymbol{n}\in \ZZ^2} R^{\boldsymbol{n}}(\Delta) \subset \boldsymbol{T}. \] is the set of points in $\boldsymbol{T}$ whose orbits intersect $\Delta$. It can be seen that $\boldsymbol{T} \setminus \Delta_{\mathcal{P},R}$ is dense in $\boldsymbol{T}$. We will assume that $\Delta$ consists of straight line segments.
Following Section 4 of \cite{Labb2021}, for each $\boldsymbol{x} \in \boldsymbol{T} \setminus \Delta_{\mathcal{P},R}$, we define a map $\textsc{Config}_{\boldsymbol{x}}^{\mathcal{P},R} \,:\, \ZZ^2 \rightarrow \mathcal{A}$ by \[\textsc{Config}_{\boldsymbol{x}}^{\mathcal{P},R}(\boldsymbol{n}) = a \text{\hspace{.2in}if and only if \hspace{.2in} } R^{\boldsymbol{n}}(\boldsymbol{x}) \in P_a.\] $\textsc{Config}_{\boldsymbol{x}}^{\mathcal{P},R}$ gives rise to a map $\textsc{SymbRep}\,:\, \boldsymbol{T} \setminus \Delta_{\mathcal{P},R} \rightarrow \mathcal{A}^{\ZZ^2}$ defined by \[\textsc{SymbRep}(\boldsymbol{x}) = \textsc{Config}_{\boldsymbol{x}}^{\mathcal{P},R}.\] Next, we wish to extend $\textsc{SymbRep}$ to a map on all $\boldsymbol{T}$, including $\Delta_{\mathcal{P},R}$. This is accomplished by choosing a direction $\boldsymbol{v}$ that is not parallel to any of the lines comprising $\Delta$, and then defining $\textsc{SymbRep}^{\boldsymbol{v}}\,:\, \boldsymbol{T} \rightarrow \mathcal{A}^{\ZZ^2}$ by \[\textsc{SymbRep}^{\boldsymbol{v}}(\boldsymbol{x}) = \lim_{\varepsilon \rightarrow 0^{+}} \textsc{SymbRep}(\boldsymbol{x} + \varepsilon \boldsymbol{v}).\] The idea here is that if $R^{\boldsymbol{n}}(\boldsymbol{x}) \in \Delta$, there is ambiguity as to which atom's label is assigned to $\boldsymbol{n}$ by $\textsc{Config}$; the direction $\boldsymbol{v}$ settles that ambiguity by assigning to $\boldsymbol{n}$ the label of the atom on the $\boldsymbol{v}$ side of $\Delta$ where $R^{\boldsymbol{n}}(\boldsymbol{x})$ lies.
The following lemma from \cite{Labb2021} characterizes $\mathcal{X}_{\mathcal{P},R}$ in terms of map $\textsc{SymbRep}$.
\begin{lemma}
For each dirction $\boldsymbol{v}$ not parallel to a line segment in $\Delta$, we have \[\overline{\textsc{SymbRep}^{\boldsymbol{v}}(\boldsymbol{T})} = \overline{\textsc{SymbRep}(\boldsymbol{T} \setminus \Delta_{\mathcal{P},R})} = \mathcal{X}_{\mathcal{P},R}\] where the overline indicates topological closure in the compact product topology on $\mathcal{A}^{\ZZ^2}$.
\end{lemma}
We summarize further pertinent results from \cite{Labb2021} here: \begin{enumerate}
\item $\textsc{SymbRep}^{\boldsymbol{v}} \,:\, \boldsymbol{T} \rightarrow \mathcal{X}_{\mathcal{P},R}$ is 1-1, and so
\item the inverse map can be extended to the factor map $f:\mathcal{X}_{\mathcal{P},R} \rightarrow \boldsymbol{T}$ such that for each configuration $\omega \in \mathcal{X}_{\mathcal{P},R}$, $f(\omega) = \cap_{n=0}^{\infty}\overline{D_\omega}$ (so $f$ is as defined in Proposition \ref{prop:factor-map}).
\item $f$ is 1-1 on $f^{-1}(\boldsymbol{T} \setminus \Delta_{\mathcal{P},R})$.
\item The factor map commutes the shift actions $R$ on $\boldsymbol{T}$ and $\sigma$ on $\mathcal{X}_{\mathcal{P},R}$; that is, $R^{\boldsymbol{n}}f = f \sigma^{\boldsymbol{n}}$.
\end{enumerate}
Using the factor map $f$ and these properties of $f$, Labb\'{e} proved the following important fact \cite{Labb2021}:
\begin{lemma}
Suppose $\mathcal{P}$ gives a symbolic representation of $(\boldsymbol{T},\ZZ^2,R)$. Then \begin{enumerate}
\item if $(\boldsymbol{T},\ZZ^2,R)$ is minimal, then $(\mathcal{X}_{\mathcal{P},R},\ZZ^2,\sigma)$ is minimal, and
\item if $R$ is a free $\ZZ^2$-action on $\boldsymbol{T}$, then $\mathcal{X}_{\mathcal{P},R}$ is aperiodic.
\end{enumerate}
\label{lem:aperiodic}\end{lemma}
The last results we quote from \cite{Labb2021} pertain to understanding how a topological partition $\mathcal{P}$ gives rise to Wang tile protoset $\mathcal{T}$ such that $\mathcal{X}_{\mathcal{P},R} \subset \Omega_{\mathcal{T}}$. To that end, suppose that $(\boldsymbol{T},\ZZ^2,R)$ is a dynamical system, let $\mathcal{P}_r = \{P_i\}_{i \in I}$ (the \emph{right side partition}) and $\mathcal{P}_b = \{P_{j}\}_{j \in J}$ (the \emph{bottom side partition}) be two finite topological partitions of $\boldsymbol{T}$, and then let $\mathcal{P}_l = \{R^{(1,0)}(P_a)\,:\,P_a \in \mathcal{P}_r\}$ (the \emph{left side partition}) and $\mathcal{P}_b = \{R^{(0,1)}(P_a)\,:\,P_a \in \mathcal{P}_t\}$ (the \emph{top side partition}); the labels of the atoms of $\mathcal{P}_l$ and $\mathcal{P}_r$ (i.e. the set $I$) are the colors of the right and left sides of tiles in a soon-to-be-defined Wang tile protoset, and the labels of the atoms of $\mathcal{P}_b$ and $\mathcal{P}_t$ (i.e. the set $J$) are the colors of the top and bottom sides in that same Wang tile protoset. For each $(i,j,k,\ell) \in I \times J \times I \times J$, let \[P_{(i,j,k,\ell)} = P_i \cap P_j \cap P_k \cap P_{\ell}.\] Next, define \[\mathcal{T} = \{\tau \in I \times J \times I \times J \,:\, P_{\tau} \neq \emptyset\}.\] We interpret each $\tau \in \mathcal{T}$ as a Wang tile as described in Section \ref{sec:Wang_shifts}. $\mathcal{T}$ is naturally associated with the \emph{tile partition} $\mathcal{P} = \{P_{\tau} \,:\, \tau \in \mathcal{T}\}$ which is the refinement of $\mathcal{P}_l$, $\mathcal{P}_r$, $\mathcal{P}_b$ and $\mathcal{P}_t$, and each point $\boldsymbol{x} \in \boldsymbol{T} \setminus \Delta$ corresponds to a unique tile in $\mathcal{T}$. With $\mathcal{P}$ defined as a partition of the torus $\mathcal{T}$ in this way, we find the following lemmas in \cite{Labb2021}.
\begin{lemma}
With $\mathcal{P}$ defined as above as a refinement of $\mathcal{P}_l$, $\mathcal{P}_r$, $\mathcal{P}_b$, and $\mathcal{P}_t$, we have $\mathcal{X}_{\mathcal{P},R} \subseteq \Omega_{\mathcal{T}}$. \label{lem:Chi_in_Omega}
\end{lemma}
\begin{lemma}
For every direction $\boldsymbol{v}$ not parallel to a line segment in $\Delta$, $\textsc{SymbRep}^{\boldsymbol{v}}\,:\, \boldsymbol{T} \rightarrow \Omega_{\mathcal{T}}$ is a one-to-one map.
\end{lemma}
\section{A Markov Partition for the Penrose Wang Shift}\label{sec:markov}
First, we see how the previously described Penrose Wang protoset $\mathcal{T}$ from Figure \ref{fig:protoset} correspond to a tile partition, as described in Section \ref{sec:symbrep}: This explanation is provided by Figure \ref{fig:partition_refinement} where we give the side partitions $\mathcal{P}_l$, $\mathcal{P}_r$, $\mathcal{P}_b$, and $\mathcal{P}_t$, whose refinement is exactly the tile partition $\mathcal{P}$ from Figure \ref{fig:square}.
\pagebreak
\begin{figure}[H]
\centering
\begin{subfigure}[h]{.49\textwidth}
\centering
\includegraphics[width=.95\textwidth]{paper/figures/Left_Partition.pdf}
\caption{$\mathcal{P}_l$ - the left side partition}
\label{fig:P_l}
\end{subfigure}
\begin{subfigure}[h]{.49\textwidth}
\centering
\includegraphics[width=.95\textwidth]{paper/figures/Right-Partition.pdf}
\caption{$\mathcal{P}_r$ - the right side partition}
\label{fig:P_r}
\end{subfigure}
\begin{subfigure}[h]{.49\textwidth}
\centering
\includegraphics[width=.95\textwidth]{paper/figures/Bottom_Partition.pdf}
\caption{$\mathcal{P}_b$ - the bottom side partition}
\label{fig:P_b}
\end{subfigure}
\begin{subfigure}[h]{.49\textwidth}
\centering
\includegraphics[width=.95\textwidth]{paper/figures/Top_Partition.pdf}
\caption{$\mathcal{P}_t$ - the top side partition}
\label{fig:P_t}
\end{subfigure}
\begin{subfigure}[h]{.49\textwidth}
\centering
\includegraphics[width=.95\textwidth]{paper/figures/Penrose_Partition_red_dot.pdf}
\caption{$\mathcal{P}$ - the tile partition}
\label{fig:P}
\end{subfigure}\caption{Refining the side partitions to obtain the tile partition $\mathcal{P}$. The red dot corresponds to the tile $(r,t,l,b) = (16,5,9,13)$, which is tile 5 from Figure \ref{fig:protoset}, so the corresponding region in $\mathcal{P}$ is labeled 5.}\label{fig:partition_refinement}
\end{figure}
\pagebreak
In Figure \ref{fig:small_tiling} we see how $\textsc{SymbRep}^{\boldsymbol{v}}$ takes a point in $\boldsymbol{T}$ to a tiling in $\Omega_{\mathcal{T}}$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{paper/figures/Penrose_Partition_to_Wang_Tiling_2.pdf}\caption{The $\ZZ^2$ action $R$ is defined on the Markov partition $\mathcal{P}$ (outlined in heavy black at left) is defined by $R^{(n_1,n_2)}(\boldsymbol{p}) = \boldsymbol{p} + n_1 \gamma_1 + n_2 \gamma_2 \pmod{\boldsymbol{T}}$ where $\boldsymbol{\gamma}_1=(\varphi-1,\varphi-1)$ (the green arrow) and $\boldsymbol{\gamma}_2=(2-\varphi,0)$ (the blue arrow). The white points in the partition are a portion of the orbit of the red point under $R$. To see the correspondence between the orbit points and the tiles, know that $\boldsymbol{\gamma}_1$ (green) corresponds to $(1,0)$ in the Wang tiling at right, and $\boldsymbol{\gamma}_2$ (blue) corresponds to $(0,1)$ in the Wang tiling at right.} \label{fig:small_tiling}
\end{center}
\end{figure}
Next will will show that $\mathcal{P}$ is a Markov partition for $(\boldsymbol{T},\ZZ^2,R)$. Let $\mathcal{P}$ be the partition of the torus $\boldsymbol{T} = \RR^2 / \ZZ^2$ shown in Figure \ref{fig:square}. Let $\boldsymbol{\gamma}_1=(\varphi-1,\varphi-1)$ and $\boldsymbol{\gamma}_2=(2-\varphi,0)$ and define the $\ZZ^2$ action $R$ on $\boldsymbol{T}$ by $R^{\boldsymbol{n}}(\boldsymbol{x}) = \boldsymbol{x} + n_1\boldsymbol{\gamma}_1 + n_2\boldsymbol{\gamma}_2 \pmod{\boldsymbol{T}}$. In order to demonstrate that $\mathcal{P}$ gives a symbolic representation of $(T,\Z^2,R)$, it is first necessary to show that the dynamical system $(\boldsymbol{T},\Z^2,R)$ is minimal, and we take advantage of the fact that $(\boldsymbol{T}, \ZZ^2, R)$ is minimal if for all $\boldsymbol{x} \in \boldsymbol{T}$, $\mathcal{O}_R(\boldsymbol{x})$ is dense in $\boldsymbol{T}$.
Lemmas \ref{lem:minimality1} and \ref{lem:minimality2} allow us to establish the minimality of $(\boldsymbol{T},\ZZ^2,R)$ (Proposition \ref{prop:min}).
\begin{lemma}\label{lem:minimality1}
Let $\boldsymbol{v} = (v_x,v_y) \in \boldsymbol{T}$ and let $x_0,x_1 \in [0,1)$ with $x_0<x_1$. Then, there exists some $\boldsymbol{v}' = (v_x',v_y) \in \mathcal{O}_R(\boldsymbol{v})$ with $x_0<v'_x < x_1$. \end{lemma}
\begin{proof}
Consider the set $\{R^{(0,n)}(\boldsymbol{v}) \,:\, n \in \ZZ\} = \{(v_x - n\varphi + 2n \pmod{1},v_y) \,:\,n \in \ZZ\} = \{(v_x - n\varphi \pmod{1},v_y) \,:\,n \in \ZZ\} \subset \mathcal{O}_{R}(\boldsymbol{v})$, which we see is an irrational rotation of the point $\boldsymbol{v}$ horizontally around $\boldsymbol{T}$. Consequently, $\{R^{(0,n)}(\boldsymbol{v}) \,:\, n \in \ZZ\}$ is dense on the horizontal line through $\boldsymbol{v}$, and it follows that there exists some $\boldsymbol{v}' = (v_x',v_y) \in \mathcal{O}_R(\boldsymbol{v})$ with $x_0<v'_x < x_1$.
\end{proof}
\begin{lemma}\label{lem:minimality2}
Let $\boldsymbol{v} = (v_x,v_y) \in \boldsymbol{T}$ and let $y_0,y_1 \in [0,1)$ with $y_0<y_1$. Then there exists some $\boldsymbol{v}' = (v_x',v_y') \in \mathcal{O}_R(\boldsymbol{v})$ with $y_0 < v_y' < y_1$. \end{lemma}
\begin{proof}
Consider the set $R^{(n,0)}(\boldsymbol{v}) = \{(v_x + n(\varphi - 1) \pmod{1},v_y + n(\varphi-1)\pmod{1}) \,:\, n \in \ZZ \} = \{(v_x + n\varphi \pmod{1},v_y + n\varphi\pmod{1}) \,:\, n \in \ZZ \}\subset \mathcal{O}_R(\boldsymbol{v})$. Note that we can interpret the second component $v_y+n\varphi \pmod{1}$ as an irrational rotation of the of the unit interval, and it follows that there exists some $n \in \ZZ$ such that $\boldsymbol{v}' = R^{(n,0)}(\boldsymbol{v}) = (v_x',v_y')$ and $v_y' = v_y + n\varphi \pmod{1}$ satisfies $y_0 < v_y' < y_1$.
\end{proof}
These lemmas have demonstrated that the orbit of any point in $T$ has a non-empty intersection with any horizontal or vertical strip on the torus, which allows us to prove the following theorem.
\begin{proposition}
$(\boldsymbol{T}, \ZZ^2, R)$ is minimal. \label{prop:min}
\end{proposition}
\begin{proof}
We show that for arbitrary $\boldsymbol{p} \in \boldsymbol{T}$, $\mathcal{O}_R(\boldsymbol{p})$ is dense in $\boldsymbol{T}$. Let $\boldsymbol{a} \in \boldsymbol{T} / \partial \boldsymbol{T}$ and let $U$ be an open square of side length $2\varepsilon$ centered at $\boldsymbol{a} = (a_x,a_y)$ in $\boldsymbol{T}$, and without loss of generality, choose $\varepsilon$ sufficiently small so that $0 < a_x - \varepsilon < a_x + \varepsilon < 1$ and $0 < a_y - \varepsilon < a_y + \varepsilon < 1$. By Lemma \ref{lem:minimality2}, there exists $m \in \Z$ such that $\boldsymbol{v'} = (v_x',v_y')= R^{(m,0)}(\boldsymbol{p})$ lies in the horizontal strip bounded by lines $y=a_y - \varepsilon$ and $y = a_y + \varepsilon$. Next, by Lemma \ref{lem:minimality1}, there exists some $n \in \Z$ such that $\boldsymbol{v''}=R{(0,n)}(\boldsymbol{v'}) = (v_x'',v_y')$ with $a_x - \varepsilon < v_x'' < a_x + \varepsilon$, and so we see that $\boldsymbol{v''} \in U$, and so $\mathcal{O}_R(\boldsymbol{p}) \cap U \neq \emptyset$.
If $a \in \partial \boldsymbol{T}$, obvious modifications of the argument above give the same result, that $\mathcal{O}_R(\boldsymbol{p}) \cap U \neq \emptyset$, and so $\mathcal{O}_R(\boldsymbol{p})$ is dense in $\boldsymbol{T}$.\end{proof}
To prove that $\mathcal{P}$ gives a symbolic representation of $(\boldsymbol{T},\ZZ^2,R)$, We apply Lemma \ref{lem:LabbeMinimality}, showing that some atom of $\mathcal{P}$ is invariant only under the trivial translation.
\begin{proposition} \label{prop:symbrep}
$\mathcal{P}$ gives a symbolic representation of $(T,\Z^2,R)$.
\end{proposition}
\begin{proof}
Suppose that $p_{17} \in \mathcal{P}$ (see Figure \ref{fig:square}) is invariant under a translation $R^{\boldsymbol{n}}$ for some $\boldsymbol{n} = (n_1,n_2) \in \ZZ^2$. $R^{\boldsymbol{n}}$ must then fix the point $(0,0) \in \boldsymbol{T}$, from which we obtain $n_1(\varphi - 1) + n_2 (2 - \varphi) = 0 \pmod{1}$ and $n_1(\varphi-1) = 0 \pmod{1}$. Due to the irrationality of $\varphi$, we conclude that $n_1 = n_2 = 0$, so $R^{\boldsymbol{n}}$ is the trivial translation.
\end{proof}
Having now established that $\mathcal{P}$ gives a symbolic representation of $(\boldsymbol{T},\ZZ^2,R)$, we want to establish that $\mathcal{X}_{\mathcal{P},R}$ is an SFT. Once we have established this, we have that $\mathcal{P}$ is a Markov partition for $(\boldsymbol{T},\ZZ^2,R)$. according to Definition \ref{def:Markov}.
\begin{proposition} $\mathcal{X}_{\mathcal{P},R}$ is a shift of finite type. \label{prop:SFT}\end{proposition}
\begin{proof}
There is a natural set of forbidden patterns defined as the set of all invalid horizontal and vertical ``dominoes.'' In the context of configurations $\omega \in \mathcal{X}_{\mathcal{P},R}$, horizontal and vertical dominoes correspond to subwords of the form $\omega_{\boldsymbol{n}} \omega_{\boldsymbol{n}+(1,0)}$ and $\omega_{\boldsymbol{n}} \omega_{\boldsymbol{n}+(0,1)}$. Indeed, the factor map $f:\mathcal{X}_{\mathcal{P},R}\to \boldsymbol{T}$ takes $\omega$ to a point $\boldsymbol{p} \in \boldsymbol{T}$ such that the labels of the atoms in $\mathcal{P}$ containing $R^{\boldsymbol{n}}(\boldsymbol{p})$ and $R^{\boldsymbol{n}+ (1,0)}(\boldsymbol{p})$ are exactly $\omega_{\boldsymbol{n}}$ and $\omega_{\boldsymbol{n}+(1,0)}$, and the atoms containing $R^{\boldsymbol{n}}(\boldsymbol{p})$ and $R^{\boldsymbol{n}+ (0,1)}(\boldsymbol{p})$ are exactly $\omega_{\boldsymbol{n}}$ and $\omega_{\boldsymbol{n}+(1,0)}$. Thus, the forbidden patterns in $\mathcal{X}_{\mathcal{P},R}$ are the subwords $\omega_{\boldsymbol{n}} \omega_{\boldsymbol{n}+(1,0)}$ and $\omega_{\boldsymbol{n}} \omega_{\boldsymbol{n}+(0,1)}$ that do not correspond to any points $\boldsymbol{p}$ such that $R^{\boldsymbol{n}}(\boldsymbol{p})$ is an atom labeled $\omega_{\boldsymbol{n}}$ and $R^{\boldsymbol{n}+ (1,0)}(\boldsymbol{p})$ is in an atom labeled $\omega_{\boldsymbol{n}+(1,0)}$. Given that there are only finitely many labels for atoms in $\mathcal{P}$, it is clear that there are finitely many forbidden patterns in $\mathcal{X}_{\mathcal{P},R}$.\end{proof}
The next theorem follows directly from Propositions \ref{prop:min}, \ref{prop:symbrep}, and \ref{prop:SFT}.
\begin{theorem} $\mathcal{P}$ is a Markov partition for $(\boldsymbol{T},\ZZ^2,R)$. \label{thm:Markov}\end{theorem}
Finally, by Lemma \ref{lem:Chi_in_Omega}, we see that $\mathcal{X}_{\mathcal{P},R} \subseteq \Omega_{\mathcal{T}}$. Because the action $R$ is clearly free on $\boldsymbol{T}$, we see by Lemma \ref{lem:aperiodic} that $\mathcal{X}_{\mathcal{P},R}$ is an aperiodic minimal subshift of $\Omega_{\mathcal{T}}$.
\begin{theorem} $\mathcal{X}_{\mathcal{P},R}$ is an aperiodic minimal subshift of $\Omega_{\mathcal{T}}$. \end{theorem}
Next we consider the question of whether or not $\mathcal{X}_{\mathcal{P},R}$ is equal to all of $\Omega_{\mathcal{T}}$. We believe that $\Omega_{\mathcal{T}}$ is itself a minimal shift; since $\mathcal{X}_{\mathcal{P},R}$ is minimal, this would imply that $\mathcal{X}_{\mathcal{P},R} = \Omega_{\mathcal{T}}$, which means that the partition $\mathcal{P}$ encodes all Penrose Wang tilings through the $\ZZ^2$-action of $R$ (and hence, all Penrose rhomb tilings). We plan to address this in a later revision of this article, so for now we leave it as a conjecture:
\begin{conjecture} $(\Omega_{T},\ZZ^2,\sigma)$ is minimal. \end{conjecture}
\section{Nonexpansive Directions in Wang Shifts}
In this section apply the methods described in \cite{Mann2022} for determining the nonexpansive directions of the Penrose shift. These methods are based on definitions from \cite[\S 2]{colle2019} and \cite{MR1355295}. Let $F$ be a subspace of $\RR^d$. For each $g\in\ZZ^d$, let $\text{dist}(g,F)=\inf\{\Vert g-u\Vert\colon u\in F\}$, where $\Vert\cdot\Vert$ is the Euclidean norm in $\RR^d$. Given $t>0$, the \emph{$t$-neighbourhood} of $F$ is defined by $F^t:=\{g\in\ZZ^d : \text{dist}(g,F)\leq t\}$. Let $X\subset\mathcal{A}^{\mathbb{Z}^d}$ be a subshift. We say that a subspace $F\subset\RR^d$ is \emph{expansive} on $X$ if there exists $t>0$ such that for any $x,y \in X$, $x|_{F^t}=y|_{F^t}$ implies that $x=y$. Thus, a subspace $F$ is \emph{nonexpansive} if for all $t > 0$, there exist $x,y \in X$ such that $x|_{F^t} = y|_{F^t}$ but $x \neq y$. Additionally, we see that if $F$ is expansive, then every translate of $F$ is expansive since $F^t \subset (F + \boldsymbol{v})^{t + \Vert\boldsymbol{v}\Vert}$ for any $\boldsymbol{v} \in \RR^d$. Thus, in the 2-dimensional case, which will be the focus of this article, we may refer to \emph{nonexpansive directions}. Equivalently, the notion of expansiveness can be defined on half-spaces in
$\RR^d$\cite[\S 2]{MR1869066}. Let $\mathsf{S}_{d-1}=\{{\boldsymbol{v}}\in\RR^d\colon\Vert{\boldsymbol{v}}\Vert=1\}$ be the unit $(d-1)$-sphere.
For ${\boldsymbol{v}}\in \mathsf{S}_{d-1}$ define $H_{\boldsymbol{v}}=\{{\boldsymbol{x}}\in\RR^d\colon {\boldsymbol{x}}\cdot{\boldsymbol{v}}\leq0\}$ to be the
half-space with outward unit normal ${\boldsymbol{v}}$. Let $\mathsf{H}_d$ be the set of half-spaces in $\RR^d$,
which are identified with $\mathsf{S}_{d-1}$ via the parametrization ${\boldsymbol{v}}\leftrightarrow H_{\boldsymbol{v}}$.
For $H\in\mathsf{H}_d$, we denote its outward unit normal vector by ${\boldsymbol{v}}_H$.
Let $\sigma$ be a $\Z^d$-action on the subshift $X$.
We say that a half-space $H\in\mathsf{H}_d$ is \emph{nonexpansive} for $\sigma$ if there exist
$x,y \in X$ such that $x|_{\Z^d\cap H} = y|_{\Z^d\cap H}$ but $x \neq y$.
\begin{lemma}{\rm\cite[Lemma 2.9]{MR1869066}}
\label{lem:half-space-is-good}
Let $\sigma$ be a $\Z^d$-action and $V$ be a codimension 1 subspace of $\R^d$.
Then $V$ is nonexpansive for $\sigma$ if and only if
there is a half-space $H\in\mathsf{H}_d$
which is nonexpansive for $\sigma$ with $\partial H=V$.
\end{lemma}
Thus if $F$ is a nonexpansive codimension 1 subspace for a subshift $X$, then,
there exist $x,y\in X$ such that $x|_{\Z^d\cap H} = y|_{\Z^d\cap H}$ but $x\neq
y$ where $H$ is the half-space on one side of the space $F$.
The next lemma shows that the set of nonexpansive directions of a subshift is a topological
invariant.
\begin{lemma}\cite{Mann2022}
Let
$(X,\Z^d,f)$
and $(Y,\Z^d,g)$ be two topologically conjugate subshifts
and $F\subset\R^d$ be a codimension 1 subspace.
If $F$ is a nonexpansive in $X$, then $F$ is nonexpansive in $Y$.
\label{lem:top_invar}
\end{lemma}
The following lemma shows that nonexpansiveness in $\mathcal{X}_{\mathcal{P},R}$ comes from points in $\boldsymbol{T}$ whose orbits intersect the boundary $\Delta$ of $\mathcal{P}$. By Lemma \ref{prop:factor-map}, when $x \in \boldsymbol{T} \setminus \Delta_{\mathcal{P},R}$, $f^{-1}(x)$ is a singleton and so $x$ corresponds to a unique configuration in $\mathcal{X}_{\mathcal{P},R}$. Points in $\Delta_{\mathcal{P},R}$, on the other hand, correspond to multiple configurations in $\mathcal{X}_{\mathcal{P},R}$ that are equal on half planes.
\begin{lemma}\cite{Mann2022}
Let $H$ be a nonexpansive half-space
for the subshift $\mathcal{X}_{\mathcal{P},R}$.
Then
there exist $x,y\in\mathcal{X}_{\mathcal{P},R}$
such that
$x|_{H\cap \Z^2}=y|_{H\cap \Z^2}$,
$x\neq y$,
and
$f(x)=f(y)\in\Delta_{\mathcal{P},R}$.\label{lem:on_boundary}
\end{lemma}
\subsection{Nonexpansiveness in the Penrose Wang Shift}
The section follows the methodology described in \cite{Mann2022}. Let $\mathcal{P}$ be the Markov partition for the Penrose Wang shift and let $\boldsymbol{T}$ be the torus $\RR^2 / \ZZ^2$. Observe that the line segments in $\Delta$ are of five distinct slopes: $0,\infty,1, \varphi+1$ and $-\varphi$. For convenience, we choose the origin $\boldsymbol{0} \in \boldsymbol{T}$ to be the point $(0,2-\varphi)$ in Figure \ref{fig:square} because lines of all 5 slopes intersect at this point. For $i \in \{0,\infty,1,\varphi+1,-\varphi\}$, let \begin{itemize}
\item $\Delta_i$ denote the union of the slope-$i$ line segments in $\Delta$ (the \emph{\textbf{slope-$i$ part}} of $\Delta$), and
\item $\overline{\Delta}_i$ be the segment of slope $i$ with endpoint at $\boldsymbol{0}$, (the \textbf{\emph{slope-$i$ base segment}}).
\end{itemize}
\begin{lemma} For each $\boldsymbol{x} \in \Delta_{i}$ where $i \in \{0,\infty,1,\varphi+1,-\varphi\}$, there exists $\boldsymbol{n} \in \ZZ^2$ with $\|\boldsymbol{n}\| < 5$ such that $R^{\boldsymbol{n}}(\boldsymbol{x}) \in \overline{\Delta}_i$ \label{lem:move_to_origin}\end{lemma}
\begin{proof}
The proof is by inspection of each of the numbered segments labeled in Figure \ref{fig:delta_lines}, and the corresponding shifts $\boldsymbol{n}$ are giving in Figure \ref{tab:move_to_0}.
\end{proof}
\begin{figure}[h]
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[scale=0.8]{paper/figures/lines.pdf}
\caption{$\Delta$-lines in $\mathcal{P}$}\label{fig:delta_lines}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.45\linewidth}
\centering
\footnotesize
\begin{tabular}{|c|c|c|c|}\hline
Slope & $\Delta$-line $\ell$ & Domain Restriction & $(a,b)$ \\ \hline\hline
\multirow{5}{*}{$\varphi+1$} & \Circled{1} & $0 \leq x \leq 2\varphi-3$ & $(0,0)$ \\
& \Circled{2} & $2\varphi-3 \leq x \leq 2-\varphi $ & $(-1,1)$\\
& \Circled{2} & $2-\varphi \leq x \leq \varphi-1 $ & $(0,-1)$\\
& \Circled{3} & $\varphi-1 \leq x \leq 3\varphi-4 $ & $(0,1)$\\
& \Circled{4} & $3\varphi-4 \leq x \leq 1 $ & $(-1,2)$\\\hline
\multirow{2}{*}{$0$} & \Circled{5} & $0 \leq x \leq \varphi - 1$ & $(-1,-1)$\\
& \Circled{6} & $0 \leq x \leq \varphi-1$ & $(0,0)$\\\hline
\multirow{6}{*}{$\infty$} & \Circled{7} & $0 \leq y \leq 2-\varphi$ & $(-1,-1)$\\
& \Circled{7} & $2-\varphi \leq y \leq 1$ & $(0,0)$\\
& \Circled{8} & $0 \leq y \leq 2-\varphi$ & $(-1,1)$\\
& \Circled{8} & $2-\varphi \leq y \leq 1$ & $(0,2)$\\
& \Circled{9} & $0 \leq y \leq 2-\varphi$ & $(-1,0)$\\
& \Circled{9} & $2-\varphi \leq y \leq 1$ & $(0,1)$\\\hline
\multirow{6}{*}{$1$} & \Circled{10} & $0 \leq x \leq 2\varphi-3$ & $(1,1)$\\
& \Circled{11} & $0 \leq x \leq 2\varphi-3$ & $(0,0)$\\
& \Circled{11} & $2\varphi-3 \leq x \leq \varphi-1$ & $(-2,0)$\\
& \Circled{12} & $2\varphi-3 \leq x \leq \varphi-1$ & $(0,-2)$\\
& \Circled{12} & $\varphi-1 \leq x \leq 1$ & $(0,1)$\\
& \Circled{13} & $\varphi-1 \leq x \leq 1$ & $(-1,0)$\\\hline
\multirow{5}{*}{$-\varphi$} & \Circled{14} & $0 \leq x \leq 2\varphi-3$ & $(0,0)$\\
& \Circled{15} & $2\varphi-3 \leq x \leq 2-\varphi$ & $(0,1)$\\
& \Circled{15} & $2-\varphi \leq x \leq \varphi-1$ & $(0,-1)$\\
& \Circled{16} & $\varphi-1 \leq x \leq 4-2\varphi$ & $(0,0)$\\
& \Circled{16} & $4-2\varphi \leq x \leq 1$ & $(0,0)$\\\hline
\end{tabular}
\caption{Shifts $\boldsymbol{n} = (a,b)$ such that $R^{\boldsymbol{n}}(\ell)$ has endpoint at $(0,2-\varphi)$. }
\label{tab:move_to_0}
\end{minipage}
\end{figure}
\normalsize
The next lemma says that only the orbit of the origin can touch line segments in $\Delta$ of unequal slopes, and thus the orbit of a point in $\Delta$ that is not in the orbit of the origin will touch lines in $\Delta$ of only one kind of slope.
\begin{lemma} Let $\boldsymbol{p} \in \Delta \subset \boldsymbol{T}$ and let $\boldsymbol{0} = (0,0) \in \boldsymbol{T}$. If $\mathcal{O}_{R}(\boldsymbol{p})$ contains points from $\Delta_i$ and $\Delta_j$ with $i \neq j$, then $\mathcal{O}_{R}(\boldsymbol{p}) = \mathcal{O}_{R}(\boldsymbol{0})$.\label{lem:origin_orbit}\end{lemma}
\begin{proof} Because $\{\mathcal{O}_{R}(\boldsymbol{x}) : \boldsymbol{x} \in \boldsymbol{T} \}$ partitions $\boldsymbol{T}$, we need only show that $\mathcal{O}_{R}(\boldsymbol{p}) \cap \mathcal{O}_{R}(\boldsymbol{0}) \neq \emptyset$. To that end, let $\boldsymbol{x},\boldsymbol{y} \in \mathcal{O}_{R}(\boldsymbol{p})$ be such that $\boldsymbol{x} \in \Delta_i$ and $\boldsymbol{y} \in \Delta_j$ where $i \neq j$, and without loss of generality, suppose that $i \neq \infty$. By Lemma \ref{lem:move_to_origin}, we may assume without loss of generality that $\boldsymbol{x} \in \overline{\Delta}_i$ and $\boldsymbol{y} \in \overline{\Delta}_j$. Then $\boldsymbol{x} = \alpha(1,i)$ for some $\alpha \in \RR$ and $\boldsymbol{y} = \beta(d_1,d_2)$ for some $\beta \in \RR$, where $d_1 = 0$ and $d_2 = 1$ if $j = \infty$ and $d_1 = 1$ and $d_2 = j$ if $j \neq \infty$. We know that $\boldsymbol{y} \in \mathcal{O}_{R}(\boldsymbol{p})
= \mathcal{O}_{R}(\boldsymbol{x})$, so there exists some and $\boldsymbol{n} = (n_1,n_2) \in \ZZ^2$
such that $\boldsymbol{y} = R^{\boldsymbol{n}}(\boldsymbol{x}) = \boldsymbol{x} + n_1(\varphi-1,\varphi -1) + n_2(2-\varphi,0) \pmod{\boldsymbol{T}}$. Thus there exists some and $\boldsymbol{g} = (g_1,g_2) \in \ZZ^2$ such that \[\beta(d_1,d_2) = \alpha(1,i) + n_1(\varphi-1,\varphi-1)+n_2(2-\varphi,0) + (g_1,g_2), \] from which we obtain the equation \[
\left<\alpha(1,i)+n_1(\varphi-1,\varphi-1)+n_2(2-\varphi,0) + (g_1,g_2),(d_2,-d_1)\right> = 0.\] Solving this equation for $\alpha$ gives \begin{equation*}\label{eqn:alpha_vals}
\alpha = \frac{d_1(n_1(\varphi-1) + g_2) - d_2(n_1(\varphi-1)+n_2(2- \varphi) + g_1)}{d_2-d_1 i}.
\end{equation*} Next, to have $\boldsymbol{x} = \alpha(1,i) \in \mathcal{O}_R(\boldsymbol{0})$, there must exist $(a,b),(z_1,z_2) \in \ZZ^2$ such that \begin{equation}\label{eq:origin_orbit}\alpha(1,i) = a(\varphi-1,\varphi-1) + b(2-\varphi,0) + (z_1,z_2).\end{equation} In Table \ref{tab:orbit_of_orig}, we give the various possible values of $i$, $j$, $d_1$, and $d_2$ and the resulting integers $a,b,z_1,z_2$ making Equation \ref{eq:origin_orbit} valid, and thus we have $\mathcal{O}_R(\boldsymbol{p})= \mathcal{O}_R(\boldsymbol{x}) = \mathcal{O}_R(\boldsymbol{y})) = \mathcal{O}_R(\boldsymbol{0})$.
\begin{table}[H]\centering \footnotesize
\begin{tabular}{c|c|c|c||c|c|c|c}
$i$ & $j$ & $d_1$ & $d_2$ & $a$ & $b$ & $z_1$ & $z_2$ \\\hline
$0$ & $\infty$ & $0$ & $1$ & $0$ & $n_1 - n_2$ & $-n_1-g_1$ & $0$ \\
$1$ & $\infty$ & $0$ & $1$ & $-n_1 + n_2$ & $0$ & $-n_2-g_1$ & $-n_2-g_1$ \\
$\varphi+1$ & $\infty$ & $0$ & $1$ & $-n_1-g_1$ & $-n_2-g_1$ & $0$ & $-n_1-n_2-2g_1$ \\
$-\varphi$ & $\infty$ & $0$ & $1$ & $n_2+g_1$ & $n_1 + g_1$ & $-n_1-n_2-2g_1$ & $n_1+g_1$ \\\hline
$1$ & $0$ & $1$ & $0$ & $-n_1$ & $0$ & $-g_2$ & $-g_2$ \\
$\varphi+1$ & $0$ & $1$ & $0$ & $-n_1$ & $n_2-g_2$ & $2n_1-2n_2$ & $-g_2$ \\
$-\varphi$ & $0$ & $1$ & $0$ & $-n_1$ & $-g_2$ & $n_1+g_2$ & $-g_2$ \\\hline
$\varphi+1$ & $1$ & $1$ & $1$ & $n_2+g_1-g_2$ & $-n_2$ & $0$ & $g_1-g_2$ \\
$-\varphi$ & $1$ & $1$ & $1$ & $2n_2+g_1-g_2$ & $-n_2$ & $-n_2-g_2+g_2$ & $-n_2$ \\\hline
$-\varphi$ & $\varphi+1$ & $1$ & $\varphi+1$ & $-n_1-n_2+g_2$ & $n_1+n_2+g_1-g_2$ & $-g_1$ & $n_1+n_2+g_1-g_2$ \end{tabular}\caption{} \label{tab:orbit_of_orig}\end{table}
\end{proof}
\normalsize
Thus, our strategy for identifying all of the nonexpansive directions in $\mathcal{X}_{\mathcal{P},R}$ is as follows. First, we consider a point $\boldsymbol{p} \in \Delta$. By Lemma \ref{lem:origin_orbit}, there are two cases: \begin{enumerate}
\item $\mathcal{O}_R(\boldsymbol{p}) \cap \mathcal{O}_R(\boldsymbol{0}) = \emptyset$, in which case we have $\mathcal{O}_{R}(\boldsymbol{p}) \cap \Delta \subset \Delta_{i}$ for exactly one $i \in \{0,\infty,1,\varphi+1,-\varphi\}$, or
\item $\mathcal{O}_{R}(\boldsymbol{p}) = \mathcal{O}_{R}(\boldsymbol{0})$, in which case $\mathcal{O}_{R}(\boldsymbol{p}) \cap \Delta_{i} \neq \emptyset$ for each $i \in \{0,\infty,1,\varphi+1,-\varphi\}$ (since $\boldsymbol{0}$ is at the intersection of segments of all 5 slopes in $\Delta$).
\end{enumerate}
For the first case, we shall argue that the set $B_{\boldsymbol{p}} = \{\boldsymbol{n} \in \ZZ^2: R^{\boldsymbol{n}}(\boldsymbol{p}) \in \Delta\}$ lies in a finite-width strip $S_i \subset \RR^2$ whose direction is determined by $i$. By Lemma \ref{lem:origin_orbit}, we know that $\{R^{\boldsymbol{n}}(\boldsymbol{p}) : \boldsymbol{n} \in B_{\boldsymbol{p}}\} \subset \Delta_i$. If we consider the factor map $f:\mathcal{X}_{\mathcal{P},R}\!\to\! \boldsymbol{T}$, $f^{-1}(\boldsymbol{p})$ contains exactly 2 configurations $x,y \in \mathcal{X}_{\mathcal{P},R}$ with $x \neq y$, and we observe that $B_{\boldsymbol{p}} = \{\boldsymbol{n} \in \ZZ^2 : x_{\boldsymbol{n}} \neq y_{\boldsymbol{n}} \}$. Consequently, the direction of the strip $S_i$ containing $B_{\boldsymbol{p}}$ will be a nonexpansive direction for the Penrose Wang shift.
In the second case (where $\mathcal{O}_{R}(\boldsymbol{p}) = \mathcal{O}_{R}(\boldsymbol{0})$), we see that $B_{\boldsymbol{p}}$ lies in the union of 5 finite width strips $S_i$ with $i \in \{0,\infty,1,\varphi+1,-\varphi\}$, exactly those strips $S_i$ in the first case, owing to the fact that $\boldsymbol{0} \in \boldsymbol{T}$ is at the intersection of lines of all 5 slopes in $\Delta$. Thus, $f^{-1}(\boldsymbol{p})$ contains $5$ pairs of configurations $\boldsymbol{x},\boldsymbol{y} \in \mathcal{X}_{\mathcal{P},R}$, one pair for each $i \in \{0,\infty,1,\varphi+1,-\varphi\}$, and for each such pair $\boldsymbol{x},\boldsymbol{y}$, just as in the first case, we have $\{\boldsymbol{n} \in \ZZ^2 : x_{\boldsymbol{n}} \neq y_{\boldsymbol{n}} \}$ is contained in one of the strips $S_i$, and consequently the direction of $S_i$ is a nonexpansive direction. Thus, the second case does not contribute any new nonexpansive directions that were not already found in the first case.
To illustrate examples of the two cases, in Figure \ref{fig:0_orbit}, we depict the set $B_{\boldsymbol{p}}$ for $\boldsymbol{p} = (0,0)$ and in Figure \ref{fig:vert_pt_orbit} we see $B_{\boldsymbol{p}}$ where $\boldsymbol{p} = (0,1/2)$; because $(0,1/2) \notin \mathcal{O}_{R}(\boldsymbol{0})$, we see that only one nonexpansive direction emerges.
\begin{figure}[h]
\centering
\begin{subfigure}[h]{.45\textwidth}
\centering
\includegraphics[scale = .5]{paper/figures/0_orbit.pdf}
\caption{$B_{(0,0)}$}
\label{fig:0_orbit}
\end{subfigure}
\begin{subfigure}[h]{.45\textwidth}
\centering
\includegraphics[scale = .5]{paper/figures/vert_pt_orbit.pdf}
\caption{$B_{(0,1/2)}$}\label{fig:vert_pt_orbit}
\end{subfigure}\caption{Examples of sets $B_{\boldsymbol{p}}$ when $\boldsymbol{p} \in \Delta$}\label{fig:delta_pts}
\end{figure}
\begin{definition}
Let $\boldsymbol{x} \in \Delta_i \setminus \mathcal{O}_R(\boldsymbol{0})$ for some $i \in \{0,\infty,1,\varphi+1,-\varphi\}$. The \textbf{$\Delta_i$ base orbit} of $\boldsymbol{x}$ is the set $\overline{\mathcal{O}}_{R,i}(\boldsymbol{x}) = \mathcal{O}_R(\boldsymbol{x}) \cap \overline{\Delta_i}$ \label{dfn:base_orb}\end{definition}
The next lemma says that points in the orbit of a boundary point $\boldsymbol{x}$ are uniformly near $\overline{\mathcal{O}}_{R,i}(\boldsymbol{x})$.
\begin{lemma} Let $\boldsymbol{x} \in \Delta_i \setminus \mathcal{O}_{R}(\boldsymbol{0})$ where $i \in \{0,\infty,1,\varphi+1,-\varphi\}$. For every $\boldsymbol{v} \in \mathcal{O}_R(\boldsymbol{x}) \cap \Delta$, there exists $\boldsymbol{n} \in \ZZ^2$ with $\|n\| < 5$ such that $R^{\boldsymbol{n}}(v) \in \overline{\mathcal{O}}_{R,i}(\boldsymbol{x})$ \label{lem:cloud}\end{lemma}
\begin{proof} This follows immediately from Lemmas \ref{lem:move_to_origin} and \ref{lem:origin_orbit}. \end{proof}
The next lemma says that if $\boldsymbol{x} \in \Delta_i \setminus \mathcal{O}_R(\boldsymbol{0})$, then $\{ \boldsymbol{n} \in \ZZ^2 \,|\, R^{\boldsymbol{n}}(\boldsymbol{x}) \in \overline{\mathcal{O}}_{R,i}(\boldsymbol{x})\}$ lies in a finite width strip $S_i \subset \R^2$.
\begin{theorem}\label{thm:base_orbits}
For $i \in \{0,\infty,1,\varphi+1,-\varphi\}$, let $\boldsymbol{p} \in \Delta_i \setminus \mathcal{O}_{R}(\boldsymbol{0})$ and define $N_i(\boldsymbol{p}) = \{ \boldsymbol{n} \in \ZZ^2 \,|\, R^{\boldsymbol{n}}(\boldsymbol{p}) \in \overline{\mathcal{O}}_{R,i}(\boldsymbol{x})\} \subset B_{\boldsymbol{x}}.$
\begin{enumerate}
\item $N_0(\boldsymbol{p})$ lies in a finite width strip $\overline{S_0} \subset \R^2$ that is parallel to the vector $\boldsymbol{r}_{0} = (0,1)$ of slope $m_0=\infty$.
\item $N_{\infty}(\boldsymbol{p})$ lies in a finite width strip $\overline{S_{\infty}} \subset \R^2$ that is parallel to the vector $\boldsymbol{r}_{\infty} = (1,1)$ of slope $m_{\infty}=1$.
\item $N_1(\boldsymbol{p})$ lies in a finite width strip $\overline{S_1} \subset \R^2$ that is parallel to the vector $\boldsymbol{r}_{1} = (2,0)$ of slope $m_{1}=0$.
\item $N_{\varphi + 1}(\boldsymbol{p})$ lies in a finite width strip $\overline{S_{\varphi+1}} \subset \R^2$ that is parallel to the vector $\boldsymbol{r}_{\varphi + 1} = (-1,\varphi)$ of slope $m_{\varphi+1}=-\varphi$.
\item $N_{-\varphi}(\boldsymbol{p})$ lies in a finite width strip $\overline{S_{-\varphi}} \subset \R^2$ that is parallel parallel to the vector $\boldsymbol{r}_{-\varphi} = (4-3\varphi,7-4\varphi)$ of slope $m_{-\varphi}=-1/\varphi$.
\end{enumerate}
\end{theorem}
\begin{proof} Let $\boldsymbol{p} \in \overline{\Delta_i} \setminus \mathcal{O}_R(\boldsymbol{0},\ZZ^2)$, where $i \in \{0,\infty,1,\varphi+1,-\varphi\}$, let $\boldsymbol{\omega} = (\omega_1,\omega_2) \in \RR^2$ be the vector with the same direction and length as $\overline{\Delta_i}$, let $\boldsymbol{n} \in N_i(\boldsymbol{p})$, and let $\boldsymbol{v} = R^{\boldsymbol{n}}(\boldsymbol{p}) = \boldsymbol{p} + n_1 \boldsymbol{\gamma}_1 + n_2 \boldsymbol{\gamma}_2$ where $\gamma_1 = (\varphi -1,\varphi-1)$ and $\gamma_2 = (2-\varphi,0)$. Referring to Figure \ref{fig:p_orbit_pt}, $\boldsymbol{p} = \alpha \boldsymbol{\omega}$ for some $\alpha \in (0,1)$, and since $\boldsymbol{v} \in \overline{\Delta_i}$ we have $\boldsymbol{v} = \boldsymbol{g} + \beta \boldsymbol{\omega} \pmod{T} $ for some $\boldsymbol{g} = (g_1,g_2) \in \ZZ^2$ and $\beta \in (0,1)$. Thus, we have \begin{equation}
\boldsymbol v = \boldsymbol{p} + n_1\boldsymbol{\gamma}_1 + n_2\boldsymbol{\gamma}_2 = \boldsymbol{g} + \beta \boldsymbol{\omega}.
\end{equation} Additionally, for every $\boldsymbol{n} \in N_i$ the following hold. \begin{align}
\label{eqn:perp}\langle \boldsymbol{v}-\boldsymbol{g}, \boldsymbol{\omega}^{\bot} \rangle & = 0,\\
\label{eqn:par}\langle \boldsymbol{v}-\boldsymbol{g}, \boldsymbol{\omega}^{} \rangle & = \beta |\boldsymbol{\omega}|^2.
\end{align} Development of these equations yields the following. \begin{align}
\label{eqn:perp_dev}(n_1 (\varphi - 1) + n_2(2 - \varphi) - g_1)\omega_2 &= (n_1(\varphi -1) - g_2)\omega_1\\
\label{eqn:par_dev}(n_1 (\varphi - 1) +n_2(2- \varphi) - g_1)\omega_1 + (n_1(\varphi-1) - g_2)\omega_2 &= (\beta - \alpha)|\boldsymbol{\omega}|^2
\end{align}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\textwidth]{paper/figures/p_orbit_pt_original.pdf}\caption{Because $\boldsymbol{v} \in N_i$, $v = R^{\boldsymbol{n}}(\boldsymbol{p})$ and $\boldsymbol{v} = \boldsymbol{g} + \beta \boldsymbol{\omega} \pmod{T}$ for some $\boldsymbol{n},\boldsymbol{g} \in \ZZ^2$ and $\beta \in (0,1)$.}\label{fig:p_orbit_pt}
\end{center}
\end{figure}
We have $\boldsymbol{p} \in \overline{\Delta_i}$, so the five parts of Theorem \ref{thm:base_orbits} correspond to the five values of $i \in \{0,\infty,1,\varphi+1,-\varphi\}$, which we inspect as separate cases:
\mbox{}\\
\noindent \textbf{Case:} $\boldsymbol{p} \in \overline{\Delta_{0}}$.\\
In this case, $\boldsymbol{\omega} = (\omega_1,\omega_2) = (1,0)$ and substitution into Equation \ref{eqn:perp_dev} gives $n_1 (\varphi - 1) - g_2 = 0$, and because $\varphi - 1 \notin \QQ$, we get $n_1=g_2 = 0$. In particular, $n_1 = 0$, and consequently we see that $N_0(\boldsymbol{p}) = \{(0,n_2) \in \ZZ^2 \,|\, R^{(0,n_2)} \in \overline{\mathcal{O}}_{R,0}(\boldsymbol{p})\}$ lies on a vertical line. Thus, it is certainly true that $N_0(\boldsymbol{p})$ lies in a finite width strip $\overline{S_0}$ in the direction of the vector $\boldsymbol{r}_0 = (0,1)$ with slope $m_0=\infty$.
\mbox{}\\
\noindent \textbf{Case:} $\boldsymbol{p} \in \overline{\Delta_{\infty}}$.\\
With $\boldsymbol{\omega} = (\omega_1,\omega_2) = (0,1)$, Equation \ref{eqn:perp_dev} provides $n_1(\varphi-1) + n_2(2-\varphi)-g_1 = 0$ from which we see that $n_1 = n_2 = g_1$. Equation \ref{eqn:par_dev} then gives $g_1(\varphi-1) - g_2 = \beta - \alpha$, or equivalently, $\left<\boldsymbol{g},(\varphi-1,-1) \right> = \beta - \alpha$. Because $0 \leq \beta - \alpha \leq 1$, we see that the points $\boldsymbol{g}$ satisfying the previous equation lie in a width 1 strip in the direction of $\boldsymbol{u} = (\varphi - 1, -1)_{\perp} = (1,\varphi - 1)$.
Let \[M = \left(\begin{tabular}{cc}
$\varphi - 1$ & $2-\varphi$\\
$\varphi - 1$ & $0$ \end{tabular}\right),\] and notice that notice that since $R^{\boldsymbol{n}}(\boldsymbol{p}) = \boldsymbol{p} + M\boldsymbol{n} = \boldsymbol{g} + \beta\boldsymbol{\omega}$, so $\boldsymbol{n} = M^{-1}\boldsymbol{g} + M^{-1}(\beta \boldsymbol{\omega} - \boldsymbol{p})$ where $M^{-1}(\beta \boldsymbol{\omega} - \boldsymbol{p})$ is a fixed vector. Because the vectors $\boldsymbol{g}$ lie in a width 1 strip in the direction of $\boldsymbol{u} = (1,\varphi - 1)$, then the vectors $\boldsymbol{n} \in N_{\infty}(\boldsymbol{p})$ lie in a finite width strip $\overline{S_{\infty}}$ in the direction of $\boldsymbol{r}_{\infty} = M^{-1}\boldsymbol{u} = (1,1)$ having slope $m_{\infty} = 1$.
\mbox{}\\
\noindent \textbf{Case:} $\boldsymbol{p} \in \overline{\Delta_1}$.\\
In this case, $\boldsymbol{\omega} = (\omega_1,\omega_2) = (4-2\varphi,4-2\varphi)$. Substitution into Equation \ref{eqn:perp_dev} gives $n_2 (2-\varphi) = g_1 - g_2$, but since $2-\varphi \notin \QQ$, we obtain $n_2 = 0$ and $g_1 = g_2$. In particular, $n_2 = 0$ tells us that $N_1(\boldsymbol{p})$ lies on a horizontal line, and it follows that $N_1(\boldsymbol{p})$ is contained in a finite width strip $\overline{S_1}$ in the direction of $\boldsymbol{r}_1 = (1,0)$ having slope $m_1 = 0$.
\mbox{}\\
\noindent \textbf{Case:} $\boldsymbol{p} \in \overline{\Delta_{\varphi +1}}$.\\
Here we have $\boldsymbol{\omega} = (\omega_1,\omega_2) = (2-\varphi,1)$. Substitution into Equation \ref{eqn:perp_dev} and simplifying gives $\varphi g_1 = -n_1 - n_2 + g_1 - g_2$, and since $\varphi \notin \QQ$ we see that $g_1 = 0$ and $g_2 = -n_1 - n_2$. By substituting these values into Equation \ref{eqn:par_dev} and simplifying we determine that $n_1\varphi + n_2 = (1/2)\varphi^2(\beta-\alpha)|\boldsymbol{\omega}|^2$, which can be expressed as the inner product $\left<\boldsymbol{n},(\varphi,1)\right> = (1/2)\varphi^2(\beta-\alpha)|\boldsymbol{\omega}|^2$. Because $0 \leq \beta - \alpha \leq 1$, we see that $N_{\varphi+1}(\boldsymbol{p})$ lies in a finite width strip $\overline{S_{\varphi+1}}$ in the direction of $\boldsymbol{r}_{\varphi+1} = (\varphi,1)_{\perp} = (-1,\varphi)$ which has slope $m_{\varphi+1} = -\varphi$.
\mbox{}\\
\noindent \textbf{Case:} $\boldsymbol{p} \in \overline{\Delta_{-\varphi}}$.\\
With $\boldsymbol{\omega} = (\omega_1,\omega_2) = (\varphi-2,\varphi-1)$, substitution into Equation \ref{eqn:perp_dev} and simplifying yields $\varphi(n_2-g_1+n_1) = n_2+g_2$, and because $\varphi \notin \QQ$, we obtain $g_1 = n_1 + n_2$ and $g_2 = -n_2$. Substituting these values into Equation \ref{eqn:par_dev} and simplifying reveals that $n_1(3 - \varphi) + n_2(2\varphi -1) = (\beta-\alpha)|\boldsymbol{\omega}|^2$, or equivalently, $\left<\boldsymbol{n},(3-\varphi,2\varphi-1)\right> = (\beta-\alpha)|\boldsymbol{\omega}|^2$. Because $0 \leq \beta - \alpha \leq 1$, we see $N_{-\varphi}(\boldsymbol{p})$ is contained in a finite width strip $\overline{S_{-\varphi}}$ in the direction of $\boldsymbol{r}_{-\varphi} = (3-\varphi,2\varphi-1)_{\perp} = (1-2\varphi,3-\varphi)$ having slope $m_{-\varphi} = -1/\varphi$.
\end{proof}
\begin{corollary}There are five nonexpansive directions for the Penrose Wang shift, the slopes of which are $-\varphi$, $-1/\varphi$, $0$, $1$, and $\infty$. \label{cor:nonexpansive_dir}\end{corollary}
\begin{proof} Let $\boldsymbol{p} \in \Delta \setminus \mathcal{O}_R(\boldsymbol{0})$. By Lemma \ref{lem:origin_orbit}, $\mathcal{O}_R(\boldsymbol{p})$ must intersect one and only one of the sets $\Delta_i$ where $i \in \{0,\infty,1,\varphi+1,-\varphi\}$. By Theorem \ref{thm:base_orbits}, $N_i(\boldsymbol{p})$ is contained in a finite width strip $\overline{S_i}$ in the direction of $\boldsymbol{r}_i$, and by Lemma \ref{lem:cloud}, we see that the set of points $B_{\boldsymbol{p}}=\{ n\in \ZZ^2 \,|\, R^{\boldsymbol{n}}(p) \in \Delta_i \}$ lie in the finite width strip $S_i = \overline{S_i}+[-5,5]^2$ parallel to $\boldsymbol{r}_i$ with slope $m_i \in \{-\varphi, -1/\varphi, 0, 1,\infty\}$. Now if we consider the two configurations $x,y \in f^{-1}(\boldsymbol{p})$ where $f$ is the factor map $f:\mathcal{X}_{\mathcal{P},R} \to \boldsymbol{T}$, $x$ and $y$ agree outside of the set $N_i(\boldsymbol{p})$, and consequently $x$ and $y$ agree on a half-plane, and thus by Lemma \ref{lem:half-space-is-good}, $\boldsymbol{r}_i$ is a nonexpansive direction for $\mathcal{X}_{\mathcal{P},R}$ of slope $m_i$.
If $\boldsymbol{p} \in \mathcal{O}_R(\boldsymbol{0})$, then $B_{\boldsymbol{p}}$ is contained in the union of the five strips $S_i$, due to the fact that $\boldsymbol{0}$ is at the intersection of lines of all five slopes in $\Delta$. Thus, in this case, $\boldsymbol{p}$ gives rise to all of the 5 nonexpansive directions already found and no more.
\end{proof}
\section*{Acknowledgements}
The authors would like to thank S\'{e}bastien Labb\'{e} for his helpful suggestions.
\printbibliography
\end{document}
|
1211.4929
|
\section{Introduction}
Online reviews of products and services are an important source of knowledge for people to make their purchasing decisions. They contain a wealth of information on various product/service aspects from diverse perspectives of consumers. However, it is a challenge for stakeholders to retrieve useful information from the enormous pool of reviews. Many automatic systems were built to address this challenge including generating aspect-based sentiment summarization of reviews \cite{Blair:google,Hu:2004,Popescu:2005} and comparing and ranking products with regard to their aspects \cite{Liu:nlphandbook}. In this study we focus on the problem of review summarization, which takes as input a set of user reviews for a specific product or service entity and produces a set of representative text excerpts from the reviews.
Most work on summarization so far used sentence as the unit of summary. However, we do not need a complete
sentence to understand its main communicative point. Consider the following sentence from review of a coffee maker:
`My mum bought me this one, and I have to say it makes really awful tasing coffee'. To a buyer looking for an opinion
about the coffee maker, only the part `makes really awfultasing coffee' is helpful. Being able to extract such short
and meaningful segments from lengthy sentences can bring significant utilities to users. It reduces their reading load as well as presents more readable summaries on devices with limited screen size such as smart phones.
This motivates our main research question of how to extract concise and informative text from reviews of products and services that can be used for summarization. Previous work has ignored the differences in product and service reviews, which is questionable. To the best of our knowledge, this is the first work that studies and compares summarization for the two domains in details. We propose to to extract text segments that match against pre-defined syntactic patterns that occur frequently in reviews of both products and services. However, the extracted segments should be subjected to some selection or filtering procedure as not all matching candidates are likely to contain rich information. Our proposed selection mechanism is based on the observation that segments containing users' opinions and evaluations about product and service aspects carry valuable information. This motivates the use of output of joint sentiment topic models to discriminate between desirable and non-desirable text segments. Since joint sentiment topic models capture sentiments that are highly associative with aspects, they are well suited for selecting informative segments from the pool of extracted candidates.
The major contributions of our work are as follows.
\begin{enumerate}
\item A new joint sentiment-topic model that automatically
learns polarities of sentiment lexicons from reviews.
\item Identification of five frequently occuring syntactic patterns for extracting concise segments from reviews of both products and services.
\item Demonstration of the effective application of topic models to select informative variable-length
segments for review summarization.
\item Production of summaries that recall important information from review entities' characteristics.
\end{enumerate}
The rest of the paper is structured as follows. We begin with the related literature in review summarization and joint
sentiment topic models in Sect. 2. Next we describe our extension to a topic model and its improvements over previous models in Sect. 3. We then introduce our proposed extraction patterns and procedures for segment selection in Sect. 4. We present our experiments and evaluation in Sect. 5 and 6 and conclude in Sect. 7.
\section{Related work}
We first look at how text excerpts are extracted from reviews in the existing literature. Previous studies mainly generated aspect-based summary for products and services by aggregating subjective text excerpts related to each aspect. Different forms of the excerpts include sentence \cite{Hu:2004}, concise phrase composing of a modifier and a header term \cite{Lu:shortcomments}, adjective-noun pair extracted based on POS tagging and the term-frequency of the pair \cite{Yatani:spotlight}, and phrase generated by rules \cite{Liu:paraphrase}. Some limitations of these previous work are i) they only worked with the simplistic adjective-noun pairs or specific form of reviews such as short comments, and ii) experiments were carried out with reviews of services only. Our approach to extract text segments by matching variable-length linguistic patterns overcome these shortcomings and can generalize well for free-text reviews of both products and services.
Various methods for selecting informative text fragments were applied in previous research, such as matching against pre-defined or frequently occurring aspects \cite{Blair:google,Hu:2004}, ranking frequency \cite{Yatani:spotlight}, and topic models \cite{Mei:tsm,Titov:mas,Xu:2011}. We are interested in the application of joint sentiment topic models as they can infer sentiment words that are closely associative with an aspect. This is an important property of polarity of sentiment words as pointed out in \cite{Fahrni:oldwine,Lin:jst,Liu:nlphandbook,Pang:thumbsup}, and recently several joint topic models have been proposed to unify the treatment of sentiment and topic (aspect) \cite{Jo:asum,Lin:jst,Mei:tsm,Titov:mga}. Applications of these models have been limited to sentiment classification for reviews, but we hypothesize that they can also be helpful in summarization. We focus our next discussion on previous joint models in comparison to our proposed model.
One of the earliest work is the Topic-Sentiment Model (TSM) \cite{Mei:tsm}, which generates a word either from a topic or one
of the two additional subtopics -- sentiments, but it fails to account for the intimate interplay between a topic/aspect and a sentiment. TSM is based on pLSI whereas more recent work (\cite{Jo:asum,Lin:jst,Titov:mas}) uses or extends Latent Dirichlet Allocation (LDA) \cite{Blei:lda}. In the Multi-Aspect Sentiment (MAS) model \cite{Titov:mas}, customer ratings are incorporated as signals to guide the formation of pre-defined aspects, which can then be used to extract sentences from reviews that are related to each aspect.
In the Joint Sentiment/Topic (JST) model \cite{Lin:jst}, and the Aspect and Sentiment Unification Model (ASUM) \cite{Jo:asum}, each word is assumed to be generated from a distribution jointly defined by a topic and a sentiment (either positive or negative). As a result, JST and ASUM learn words that are commonly associated with an aspect although the models are incapable of distinguishing between sentiment and non-sentiment lexicons.
We propose a new model that leverages syntactic information to identify sentiment lexicons and automatically learn their polarities from the co-occurrences of words in a sentence. This allows the model to bootstrap using a minimum set of sentiment seed words, thereby alleviating the need for information that is expensive to obtain
such as ratings of users for reviews \cite{Titov:mas} or large lists of sentiment lexicons \cite{Lin:jst}.
\section{A Topic Model for Learning Polarity of Sentiment Lexicons}
Our key modelling assumption for reviews is that a sentence expresses an opinion toward an aspect via its sentiment component. For example, in the sentence `The service was excellent', only the word `excellent' carries the positive sentiment. This is not a new assumption as adjectives and adverbs are commonly considered the main source of sentiment in a sentence in existing literature. Our model leverages on this type of knowledge to locate sentiment words in a sentence with relatively high confidence.
\begin{figure}
\centering
\includegraphics[scale=0.2]{model.pdf}
\caption{Graphical representation of the model.}
\label{fig:model}
\end{figure}
\subsection{Generative Process}
The formal generative process of our model for the graphical representation in Fig. ~\ref{fig:model} is as follows (see Table ~\ref{tab:notations} for the list of notations).
\begin{itemize}
\renewcommand{\labelitemii}{$\diamond$}
\renewcommand{\labelitemiii}{\bf --}
\item{For every aspect $k$, draw a distribution of non-sentiment words, $\bm{\phi_{k}} \sim \mathrm{Dir}(\beta),$ } and two distributions of sentiment words, $\bm{\phi'_{jk}} \sim \mathrm{Dir}(\mathbf{\beta'_{jk}})$,
where $j = 0$ denotes positive polarity and $j=1$ denotes negative polarity.
\item{For each review $d$,
\begin{itemize}
\item{Draw a sentiment distribution $\bm{\pi}_{d} \sim \mathrm{Dir}(\gamma)$}
\item{Draw a topic distribution $\bm{\theta}_{d} \sim \mathrm{Dir}(\alpha)$}
\item{For each sentence $c$ in document $d$,
\begin{itemize}
\item{Choose a topic $z = k \sim \mathrm{Mult}(\bm{\theta}_{d})$
and a sentiment $s = j \sim \mathrm{Mult}(\bm{\pi}_{d})$}
\item{Choose words $w \sim \mathrm{Mult}(\bm{\phi}_{k})$ to discuss aspect $k$ and sentiment words $w' \sim \mathrm{Mult}(\bm{\phi'}_{jk})$
to convey the sentiment $j$ toward $k$.}
\end{itemize}
}
\end{itemize}
}
\end{itemize}
Notice in the graphical model that the part of a sentence which emanates the sentiment is observed. In our implementation, we treat all adjectives and adverbs as $w'$ and remaining words as $w$ in the generative procedure, but this is not a restriction imposed on the model. It is easy to incorporate prior knowledge about words that convey sentiment into the model. For example, we can instruct the model that words such as \textit{love, hate, enjoy, worth, disappoint} are sentiment words, even though they are not adjective nor adverb.
Our main extension deals with the word smoother $\bm{\beta'}$ for sentiment words. Each sentiment word $i$ is associated with a topic dependent smoothing coefficient
$y_{ki}$ for topic $k$ and a sentiment dependent smoothing coefficient $y_{ji}$ for sentiment $j$.
We then impose that
\begin{alignat}{3}
\beta'_{jki} &= \exp(y_{ki} + y_{ji}),
&\quad y_{ki} &\sim N(0, \sigma^{2}_{1}),
&\quad y_{ji} &\sim N(0, \sigma^{2}_{2}).
\end{alignat}
This modeling allows us to incorporate polarity of sentiment words as side information.
The polarity of sentiment lexicon $i$ in a corpus is represented by the values of $y_{ji}$; this is to assume that
the polarity of $i$ is its intrinsic property as the corpus is about a specific domain \cite{Choi:2009}.
The topic dependent smoother $y_{ki}$ is introduced to accommodate the different frequency of association between
the sentiment word $i$ and different aspects.
\begin{table}
\centering
\caption{List of notations used in the paper (senti = sentiment, dist. = distribution)}
\label{tab:notations}
\begin{tabular}{|llp{10cm}|} \hline
$d, c, w, w', k, j$&:& review, sentence, non-senti word, senti word, topic/aspect, sentiment \\
$T, S, V, V'$&:& number of topics, sentiments, non-senti words, senti words\\
$\bm{\pi}_d, \bm{\theta}_d$&:&sentiment distribution, topic distribution of the review $d$ \\
$\bm{\phi}_k, \bm{\phi'}_{jk}$&:&word dist. of topic $k$, senti word dist. of topic $k$ and senti $j$ \\
$y_{ji}$&:&polarity of senti word $i$ with sentiment $j$ \\
$y_{ki}$&:&smoother for dependency between topic $k$ and senti word $i$\\
$\beta'_{jki}$&:&word smoother for senti word $i$ with topic $k$ and senti $j$ \\
$\alpha,\beta,\gamma,\mu,\sigma$&:& hyperparameters \\
$n^{TW}_{ki}, n^{STW}_{jki}$&:& counts of word $i$ being assigned topic $k$, senti word $i$ being assigned topic $k$ and senti $j$\\
$n^{DT}_{dk}, n^{DS}_{dj}$&:& counts of sentences in $d$ being assigned topic $k$, sentences in $d$ being assigned senti $j$\\
\hline
\end{tabular}
\end{table}
\subsection{Inference}
In order to perform inference we alternate between two procedures: sampling and maximum a posteriori.
The sampler assigns values for the latent variables: the topics and sentiments of sentences.
Using a collapsed Gibbs sampler \cite{Griffiths:gibbs}, new values for the topic and sentiment of a sentence $c$ in document $d$
are drawn from the conditional probability
\begin{eqnarray} \nonumber
p(z_{dc} = k, s_{dc} = j|\mathrm{rest}) \propto
\frac{\prod_{i \in A(dc)} (n^{TW}_{\backslash ki} + \beta)}{\prod_{x = 0}^{\lvert A(dc)\rvert - 1}\sum_{i = 1}^{V}n_{\backslash ki}^{TW} + V\beta + x} \\
\frac{\prod_{i \in S(dc)} (n^{STW}_{\backslash jki} + \beta'_{jki})}{\prod_{x = 0}^{|S(dc)| - 1}\sum_{i = 1}^{V'}(n_{\backslash jki}^{STW} + \beta'_{jki}) + x}
(n^{DT}_{\backslash dk} + \alpha) (n^{DS}_{\backslash dj} + \gamma)
\end{eqnarray}
where $S(dc)$ is the set of sentiment words in $c$ and $A(dc)$ is the set of remaining words. The '\textbackslash' notation means not counting the sentence being sampled.
We estimate the value for $\bm{\beta'}$ and $\bm{y}$ from a maximum a posteriori procedure,
optimizing $\bm{\beta'}$ over $y$ and the assigned values of the latent variables. The negative log prior is
\begin{equation}
-\log p(\beta') = S\sum_{k,i}{y_{ki}} + T\sum_{j,i}{y_{ji}} + \sum_{j,k,i}{\frac{(y_{ki} + y_{ji})^2}{2\sigma^2}}
\end{equation}
where $\sigma^2 = \sigma_{1}^2 + \sigma_{2}^2$.
The collapsed negative log likelihood (dependent on sentiment words only) is
\begin{equation}
L_{\beta'} = \sum_{j,k}{\left[ \log\Gamma({\bar n_{jk} + \bar \beta'_{jk}}) - \log\Gamma(\bar \beta'_{jki})\right]}
+ \sum_{j,k,i}{\left[\log\Gamma(\beta'_{jki}) - \log\Gamma(n^{STW}_{jki} + \beta'_{jki}) \right]}
\end{equation}
where $\bar n_{jk} = \sum_{i = 1}^{V'}{n^{STW}_{jki}}$, $\bar \beta'_{jk} = \sum_{i = 1}^{V'}{\beta'_{jki}}$, and $\Gamma$ is the Gamma function. We use the L-BFGS optimizer \cite{Liu:lbfgs} to minimize the objective function $L_{\beta'} - \log p(\beta')$ by taking its partial derivatives with respect to $y_{ki}$ and $y_{ji}$.
A sample from the Markov chain in the sampler can be used to estimate the distributions of interest. The approximate probabilities of sentiment $j$ in document $d$ ($\hat\pi_{dj}$), topic $k$ in document $d$ ($\hat\theta_{dk}$), non-sentiment word $i$ in topic $k$ ($\hat\phi_{ki}$), and sentiment word $i$ in topic $k$ and sentiment $j$ ($\hat\phi'_{jki}$) are
\begin{alignat}{2}\nonumber
\hat\pi_{dj} &= \frac{n_{dj}^{DS} + \gamma}{\sum_{j' = 1}^{S}n_{dj'}^{DS} + S\gamma} \enspace,
&\quad
\hat\theta_{dk} &= \frac{n_{dk}^{DT} + \alpha}{\sum_{k' = 1}^{T}n_{dk'}^{DT} + T\alpha} \enspace, \\
\hat\phi_{ki} &= \frac{n_{ki}^{TW} + \beta}{\sum_{i' = 1}^{V}n_{ki'}^{TW} + V\beta} \enspace,
&\quad
\hat\phi'_{jki} &= \frac{n_{jki}^{STW} + \beta'_{jki}}{\sum_{i = 1}^{V'}(n_{jki'}^{STW} + \beta'_{jki'})} \enspace.
\end{alignat}
\subsection{Aspect and Sentiment Classification Using Output of the Model}
\label{sec:classifiers}
As stated in the introduction, we attempt to use the outputs of this model to improve the selection of informative segments for
summarization. We define the topic classifier of an arbitrary segment of $n$ words $G = (w_{1}, w_{2},$ $\ldots, w_{n})$ as
\begin{equation}
\label{eq:segtopic}
\arg\max_{k} p(k|G) = \arg\max_{k} \sum_{i = w_{1}}^{w_{n}} (\log\hat\phi_{ki} + \sum_{j}\log\hat\phi'_{jki}) \enspace.
\end{equation}
To classify the sentiment of a segment G, we use the sentiment value $y_{ji}$ learned from the model. We define the polarity of G as
\begin{equation}
polarity(G) := \sum_{\text{sentiment word i} \in G} polarity(i) = \sum_{\text{sentiment word i} \in G}y_{0i} - y_{1i}\enspace.
\end{equation}
G is classified as positive if $polarity(G) >= 0$ and as negative if $polarity(G) < 0$.
\section{Summarization Using Syntactic Patterns and Topic Models}
In this section we present our framework for variable-length segment-based summarization of reviews.
We first describe the five frequently occuring syntatic patterns in reviews that are used to extract candidate text segments.
We then discuss the use of topic models in selecting meaningful segments from the set of extracted candidates.
We also present an independent framework for evaluation of the summaries comprising segments
regardless of the approaches.
\subsection{Extraction Patterns}
Central to our summarization system is how to extract meaningful, informative text segments out of a sentence.
We use sentence syntax to guide the extraction process by defining patterns of lexical classes for matching against
text segments. The purpose is to extract semantically meaningful unit of text in a sentence that can be understood without extra context. In the particular task of summarizing reviews for products and services, we want to capture units that contain sentiments toward aspects. This type of segments is important because it expresses and formulates opinions about the entity being reviewed.
Based on the above observation, we identify five most common extraction patterns to capture a variety of text
segments in both product and service reviews as follows. First we use POS tagger to tag all pros and cons items available in our data
sets of restaurant and coffee maker reviews (see Sect. 5.1). The pros and cons are relatively short and meaningful, and can therefore be suitable representatives of the text segments that we want to generate. The resulting sequences of tags are then ranked based on their frequency. After carefully studying the top ranked patterns we select the five most productive ones listed in Table ~\ref{tab:patterns}.
\begin{table}
\centering
\caption{Extraction patterns and their occurrences in data sets}
\begin{tabular}{c|c|p{6.5cm}|c|c}
\textbf{no.} & \textbf{the pattern} & \textbf{example} & \textbf{restaur} & \textbf{coff makers} \\ \hline
1&nn? vb dt? rb* jj nn & instruction booklet includes clear instruction &56468 &23210 \\ \hline
2&nn? vb rb* jj to vb & filter basket is simple to remove &4226&3770 \\ \hline
3&nn? vb rb* jj & design is striking, tasted fresh &130853&30449\\ \hline
4&rb* jj to vb nn? & easy to clean, wide enough to insert a K-Cup &5937&5288\\ \hline
5&rb* jj nn & very good food, most expensive pod brewer &197123&69273\\ \hline
\end{tabular}
\label{tab:patterns}
\end{table}
We use the same notations as in regular expression, where the constituent parts correspond to lexical categories as specified by the PennTree bank. For simplicity, a single tag is used to represent different forms of a category; i.e., \textbf{jj} represents adjective and matches all of JJ, JJR and JJS.
Also, \textbf{nn} matches a noun phrase rather than just a single word. We further restrict that each segment must match the longest pattern.
This means, for example, a segment matching pattern 1 in a sentence is consumed and no longer available for matching pattern 5.
Each pattern also has its negation form easily constructed from its positive form, hence we do not show in the table.
\subsection{Selecting Informative Segments using Topic Models}
\label{sec:selection}
Candidate segments can be meaningless even if they match
the defined extraction patterns. For example, `final thought'
and `several hour' are instances of pattern 5, but they
reveal no interesting information.
Furthermore, the sheer number of text segments
matching the patterns (Table ~\ref{tab:patterns})
requires us to be selective in finding segments to include in summaries.
We observe that informative segments often contain words that convey opinions about aspects of entities. Since the aspect-sentiment intimate interplay is modeled and learned by our joint sentiment-topic model, we propose the following filters to prune less informative segments using the output the model.
\begin{description}
\item[Baseline] No filtering, i.e., keep all matching segments.
\item[AW\quad\quad] Eliminate a segment if it does not contain one of the top $X$ most probable words of the segment's inferred aspect.
\item[SW\quad\quad] Eliminate a segment if it does not contain one of the top $Y$ most probable sentiment words of the segment's inferred sentiment and aspect.
\item[RANK\quad] Rank all segments having the same inferred sentiment and aspect in order of their probabilities and eliminate the bottom half segments.
\end{description}
It is possible to use previous joint sentiment topic models, such as ASUM \cite{Jo:asum} and JST \cite{Lin:jst}, for the filtering purpose.
Note that ASUM and JST output word distributions for each pair of sentiment and aspect; hence, \textbf{ASUM} and \textbf{JST} are in effect both sentiment classifier and filter:
\begin{description}
\item[ASUM] Eliminate a segment if it does not contain one of the top $Z$ most probable words of the segment's inferred senti-aspect using the ASUM model.
\item[JST] Same as \textbf{ASUM} except that JST is used in placed of ASUM.
\end{description}
A complete procedure for summarization would need a sentiment classifier component for segments as sentiment-based summaries are preferred by users [10]. In addition to our model-based sentiment classifier, we introduce another
sentiment classifier based on SentiWordNet (SWN) [4], a popular lexical resource for opinion mining, using the same approach as in \cite{Fahrni:oldwine}. For convenience, we call our model-based classifier \textbf{SEN} and the SWN-based classifier \textbf{SWN}.
Various procedures for retaining quality segments can then be constructed by combining different sentiment classifiers and filters.
For example, we may first use \textbf{SEN} to classify sentiment of a segment and then use both \textbf{AW} and \textbf{SW} to
discard non-qualified segments. We name such procedure \textbf{SEN+AW+SW}, with the convention that the output of a
preceding classifier/filter is the input to the next classifier/filter whenever applicable.
\subsection{A Framework for Segment-based Summary Evaluation}
\label{sec:framework}
We now introduce a framework for automatically evaluating the extraction patterns at the levels of segment and
entity (a specific product or service). This framework is independent of the way segments are generated and therefore
can be applied to any method that uses segment as the unit of summary.
Each entity E has a candidate summary $E^{C} = \{Y | Y$ matches one of the patterns $\}$ and a reference summary $E^{R} = \{X | X$ is in the gold standard summary of E $\}$.
For $Y \in E^{C}$ and $X \in E^{R}$, we measure the similarity of their content using precision and recall
\begin{alignat*}{2}
P(X, Y) &:= \frac{skip2(X, Y)}{\binom{|Y|}{2}}\enspace,
&\quad\quad
R(X, Y) &:= \frac{skip2(X, Y)}{\binom{|X|}{2}}
\end{alignat*}
where $skip2(X, Y)$ is the number of skip-bigram matches between X and Y (termed ROUGE-SU in \cite{Lin:rouge}).
For a candidate segment $Y \in E_{C}$, define $R(Y) = R(X_{max},Y) \mbox{ and } P(Y) = P(X_{max}, Y)$
where $X_{max} = \arg \max_{X \in E^{R}} R(X,Y)$.
For an entity E, the average precision $P_{skip}(E)= \sum_{Y \in E^{C}} P(Y) / |E^{C}|$ and recall $R_{skip}(E) = \sum_{Y \in E^{C}} R(Y) / |E^{C}|$ tells us how similar the content of extracted segments is to a reference set of segments on average.
We also want to assess how many portion of the reference summary is recovered and what percentage of the candidate summary is useful.
For this reason, we introduce P(E) and R(E) to measure the precision and recall for the candidate summary set $E^{C}$ of $E$:
\begin{alignat}{2}
P(E) &:= \frac{\sum_{Y \in E^{C}} \mathbf{1_{A}\{Y\}}}{|E^{C}|},
&\quad
R(E) &:= \frac{\sum_{X \in E^{R}} \mathbf{1_{B} \{X\} }} {|E^{R}|}
\label{eq:pr}
\end{alignat}
where $\mathbf{1_{A}\{Y\}}$ and $\mathbf{1_{B} \{X\} }$ are indicator functions;
$\mathbf{A} = \{ Y | $ $R(Y) \ge \alpha\}$ and $\mathbf{B} = \{X | \text{ } \exists Y \in \mathbf{A}$ s.t $R(X, Y) = R(Y) \}$ where $\alpha$ is a recall threshold for a candidate summary to be considered useful.
A good measure for a reference summary of an entity $E$ must be a combination of the segment-level recall (precision),
$R_{skip}(E)$ and the entity-level recall (precision), $R(E)$. A simple combination is the average of the two, i.e, $R_{cb}(E) = (R_{skip}(E) + R(E))/2$ and $P_{cb}(E) = (P_{skip}(E) + P(E))/2$.
Since we typically work with data that contains a large set of review entities, it is convenient to report the results using the following summarization statistics:
\begin{alignat*}{3}
P_{s} &= \frac{\sum_{i = 1}^{N}\sum_{Y \in E^{C}_{i}}P(Y)}{\sum_{i = 1}^{N}|E_{i}^{C}|}\enspace,
&\quad
P_{e} &= \sum_{i = 1}^{N} P(E_{i}) / N \enspace,
&\quad
R_{e} &= \sum_{i = 1}^{N} R(E_{i}) / N \enspace,\\
R_{s} &= \frac{\sum_{i = 1}^{N}\sum_{Y \in E^{C}_{i}}R(Y)}{\sum_{i = 1}^{N}|E_{i}^{C}|}\enspace,
&\quad
P &= \sum_{i = 1}^{N} P_{cb}(E_{i}) / N \enspace,
&\quad
R &= \sum_{i = 1}^{N} R_{cb}(E_{i}) / N\enspace.
\end{alignat*}
\section{Experiments}
We experimented using reviews of coffee makers as representative for the product domain and reviews of restaurants
as representative for the service domain. We describe our data sets and experimental set-ups in ~\ref{sec:data}.
In ~\ref{sec:topicsandpolarities} we give example of the topics and sentiment words learned by the model. We
analyze the effectiveness of extraction patterns in ~\ref{sec:patternsevaluation} and compare the performance of different sentiment classifiers and segments filters in ~\ref{sec:selectionevaluation}.
\subsection{Data Sets and Experimental Set-ups}
\label{sec:data}
Our data sets consist of restaurant reviews and coffee maker reviews. For each review, we collected its free-format text content and its pros and cons lists if available.
\begin{itemize}
\item{\textbf{RESTAURANTS} 50,000 reviews of 5,532 restaurants collected from Citysearch New York. This data is provided by Ganu, et al. \cite{Ganu:ursa}}.
\item{\textbf{COFFEEMAKERS} 23,411 reviews of 534 coffee makers collected from epinions.com.}
\end{itemize}
Our first step is to fit the joint sentiment topic model to each data set.
Data is pre-processed as in other standard topic models, in which sentences are tokenized by the punctuations: `.', `!', and `?'. The hyperparameters are set as $\alpha$ = 0.1, $\beta$ = 0.01,
$\gamma$ = 0.1 for both positive and negative sentiment; the number of aspects is 7 for both corpora.
We incorporated prior sentiment information into the model using sentiment seed words in analogy to [9].
After running the sampler for a burnin period of 500 iterations, we interleaved it with the optimizer,
optimizing over $y_{ki}$ and $y_{ji}$ every $100^{th}$ step of sampling. We
trained the model in 2000 iterations for both data sets and
used the last sample in the chain in all of our experiments.
In the segments selection step, the maximum number of words in
a sequence is set to 7 and the number of top words for \textbf{AW}, \textbf{SW}, \textbf{ASUM}, and \textbf{JST} is set to 200, 100, 300, and 300, respectively. We used a value of 0.25 for the recall threshold
$\alpha$ in Eq.~\ref{eq:pr}. All parameters were set empirically after
many experiments.
In order to evaluate the quality of segments and summaries using the framework in ~\ref{sec:framework},
a reference summary must be obtained for each review entity. We aggregate the pros written by all reviewers for an entity as its pros gold standard and similarly for its cons standard (duplicated entries are removed). To construct an entity's candidate summaries, the procedures in ~\ref{sec:selection} are applied to the segments extracted from all of its reviews. The sentiment classifier in a procedure partitions the
entity's segments into a positive candidate summary and a negative candidate summary. The candidates are evaluated against their counterpart reference summaries independently.
\begin{table}
\centering
\caption{Example inferred topics (restaurants: row 1-3, coffee makers: row 4-6)}
\begin{tabular}{p{4.4cm}|p{4.45cm}|p{4.4cm}}
\textbf{top aspect words} & \textbf{top positive words} & \textbf{top negative words} \\ \hline
sauc, chicken, chees, salad, shrimp, soup, fri, potato, rice & good, delici, best, great, fresh, love, perfect, excel, amaz, tasti & dri, disappoint, tasteless, cold, soggi, bad, fri, rare, medium\\ \hline
music, place, bar, decor, room, table, wall, seat, atmosphere & great, nice, good, love, beauti, enjoy, romant, perfect, friend, & loud, noisi, bad, littl, small, crowd, dark, expens, back\\ \hline
wait, table, waiter, seat, minut, reserv, order, ask, told, manag & friend, nice, worth, great, attent, prompt, long, enjoy, quick & rude, bad, wrong, final, empti, horribl, terribl, poor, worst \\ \hline
coffe, bean, cup, grind, ground, brew, grinder, espresso, tast & good, like, fresh, great, best, hot, strong, fine, french, perfect & weak, bad, disappoint, wast, grind, wrong, unfortun, bitter \\ \hline
filter, clean, basket, water, paper, rins, dishwash, gold, use & easi, clean, perman, like, remov, easili, good, recommend, safe & difficult, clean, wet, bad, imposs, perman, wast, not easi \\ \hline
servic, game, custom, warranti, repair, ship, product, send, call & good, new, back, great, free, thank, well, happi, local, origin, & back, disappoint, poor, bad, wrong, negative, defect, sorri
\end{tabular}
\label{tab:topics}
\end{table}
\subsection{Topics and Polarities of Sentiment Words Learned by the Model}
\label{sec:topicsandpolarities}
Example of topics inferred by the model is given in Table
~\ref{tab:topics}. Each topic has three
distributions where one distribution (first column) consists
descriptive words about the aspect and two distributions
(remaining columns) consist evaluative words directing the aspect.
Except the common sentiment words such as \textit{good}, \textit{great}, \textit{bad}, \textit{wrong}
that are associated with most aspects due to their frequent usage, positive and negative sentiment lexicons look highly related to their corresponding aspects. For example, the model discovers that people
are more likely to praise the food with \textit{delicious}, \textit{best}, \textit{fresh}, and \textit{tasty}
and disapprove food that is \textit{dry}, \textit{tasteless}, \textit{cold} or \textit{soggy}. Such results can be very helpful
for the exploratory purpose of understanding what aspects reviewers care and comment about.
Table ~\ref{tab:lexicons} demonstrates the effectiveness of our model in learning the polarities of domain-specific sentiment lexicons
(the seed words used for bootstrapping are excluded). To verify this claim we compare with the \textbf{SWN} classifier described in ~\ref{sec:selection} in a classification task for noun phrases. SWN leverages synsets in WordNet and so, in some sense, it captures the context-dependent sentiment of a word.
We used a set of 929 positive and 236 negative noun phrases obtained from an external set of restaurant reviews
in [5]. All phrases are unique and manually annotated with their true sentiments. Our classifier outperforms \textbf{SWN} in classification accuracy for both the positive (90.1\% vs. 83.4\%) and and negative (74.6\% vs. 66.1\%) categories. This shows that our model is quite accurate in assigning sentiment score to domain-specific lexicons compared to the more general propagation approach in SWN.
\begin{table}
\centering
\caption{Selected lexicons with their sentiment polarities}
\begin{tabular}{p{12cm}} \hline
\textbf{positive lexicons in restaurant reviews} \\ \hline
knowledgeable, helpful, unique, courteous, prompt, cozy, terrific, wonderful, affordable, superb, warm, impeccable, outstanding, elegant, consistent, fabulous, charming \\ \hline
\textbf{negative lexicons in restaurant reviews} \\ \hline
tasteless, mediocre, bland, inedible, dry, ridiculous, lousy, overpriced, flavorless, average, unacceptable, obnoxious, soggy, bare, bore, tough, unfriendly, horrendous, stale \\ \hline
\textbf{positive lexicons in coffee maker reviews} \\ \hline
simple, ready, fresh, correct, removable, automatic, impressive, stainless, large, free, light, strong, rich, reasonable, amazing, fast, clear, wonderful, delicious, quick, sturdy\\ \hline
\textbf{negative lexicons in coffee maker reviews} \\ \hline
difficult, impossible, inferior, loud, lousy, dull, defective, stupid, sticky, dirty, faulty, uneven, weak, noisy, stiff, frustrating, dissatisfied, smelly, unclear, erratic, leak, slow\\ \hline
\end{tabular}
\label{tab:lexicons}
\end{table}
\subsection{Evaluation of Extraction Patterns}
\label{sec:patternsevaluation}
We now analyze how different extraction patterns behave when applied to the service domain and the product domain
(Table ~\ref{tab:individual1} and ~\ref{tab:individual2}). We use \textbf{AW+SEN+SW} procedure because it produced the best result among all methods.
For reviews of restaurants, pattern 3 and 5 are the most
productive with superior average precision and recall at both
segment-level and entity-level compared to the rest. They account for more than
half of an entity's pros and cons reference. This is probably due to the prevalence of sentences such as `the service was good' in restaurant reviews. The result is consistent with the current literature where
adjectives and nouns are commonly used to detect sentiments and aspects in reviews for services.
It is worth noticing that extracting any thing other than adjective-noun pairs may degrade the quality of summarization as
the scores for pattern 2 and 4 are overwhelmingly low.
\begin{table}
\centering
\caption{Comparison of extraction patterns for services.}
\begin{tabular}{c|l|l|l|l|l|l|l|l|} \cline{2-9}
\label{tab:individual1}
&\multicolumn{4}{|c|}{pros} & \multicolumn{4}{|c|}{cons}\\ \hline
patt& $\mathrm{P_{s}}$ & $\mathrm{R_{s}}$ & $\mathrm{P_{e}}$ & $\mathrm{R_{e}}$ & $\mathrm{P_{s}}$ & $\mathrm{R_{s}}$ & $\mathrm{P_{e}}$ & $\mathrm{R_{e}}$\\ \hline
1&20.2 &51.1 &74.4 &30.4 &14.6 &\textbf{39.6} &55.7 &14.8\\ \hline
2&23.7 &26.4 &54.6 &1.9 &7.9 &22.4 &63.6 &0.5 \\ \hline
3&31.8 &\textbf{65.7} &83.3 &\textbf{40.6} &26.4 &\textbf{57.6} &70.1 &\textbf{21.5} \\ \hline
4&21.3 &28.8 &56.1 &2.6 &6.3 &15.5 &35.7 &0.37 \\ \hline
5&25.9 &\textbf{53.5} &72.7 &\textbf{49.7} &18.1 &37.8 &52.7 &\textbf{31.2} \\ \hline
\end{tabular}
\end{table}
The behaviors of extraction patterns are trickier for the product domain as can be seen in Table ~\ref{tab:individual2}.
There is no dominating pattern in terms of high precision and recall
at both segment and entity level. In particular, pattern 3
and 5 still recover a large portion of an entity's reference
summary; however, the average quality of their matching
segments ($R_{s}$) is the lowest among all patterns. Pattern 2
and 4 perform badly when used with the service domain but
are more useful in the product domain, producing
the highest quality segments ($R_{s} = 66.3 \text{ and } 69.4$ for positive; $R_{s} = 48.6 \text{ and } 43.8$ for negative).
Although they do not appear as frequently in reviews as other patterns, they
tend to carry more meaning in their words that it is hard to
ignore them. Hence, all five patterns can contribute to the
extraction of informative segments for summarization. This
shows that doing summarization for products is harder than
for services; and, care should be exercised when generalizing
results from one domain to the other.
\begin{table}
\centering
\caption{Comparison of extraction patterns for products.}
\begin{tabular}{c|l|l|l|l|l|l|l|l|} \cline{2-9}
\label{tab:individual2}
&\multicolumn{4}{|c|}{pros} & \multicolumn{4}{|c|}{cons}\\ \hline
patt& $\mathrm{P_{s}}$ & $\mathrm{R_{s}}$ & $\mathrm{P_{e}}$ & $\mathrm{R_{e}}$ & $\mathrm{P_{s}}$ & $\mathrm{R_{s}}$ & $\mathrm{P_{e}}$ & $\mathrm{R_{e}}$\\ \hline
1&22.1 &59.6 &70.8 &\textbf{30.1} &19.5 &\textbf{44.2} &59.7 &\textbf{17.5} \\ \hline
2&45.5 &\textbf{66.3} &78.7 &6.6 &30.2 &\textbf{48.6} &64.3 &2.9\\ \hline
3&33.2 &54.4 &65.9 &26.7 &27.0 &37.7 &51.1 &15.0\\ \hline
4&52.7 &\textbf{69.4} &78.5 &8.6 &30.9 &43.8 &61.3 &3.5\\ \hline
5&26.0 &50.8 &59.6 &\textbf{38.9} &26.2 &40.8 &56.3 &\textbf{25.0}\\ \hline
\end{tabular}
\end{table}
\subsection{Evaluation of Sentiment Classifiers and Segment Filters}
\label{sec:selectionevaluation}
Results in the previous section suggest to use different syntactic patterns
for summarization of the service and product
domains. We used patterns 1, 3, and 5 for services and all
patterns for products in all of our experiments in this section.
We applied seven different procedures for selecting candidate segments to compare the effects of 2 sentiment classifiers
(\textbf{SEN} and \textbf{SWN}) and 5 filters (\textbf{AW}, \textbf{SW}, \textbf{RANK}, \textbf{ASUM}, and \textbf{JST}).
The results are depicted in Table ~\ref{tab:services} and ~\ref{tab:products}. The good overall performance of the \textbf{Baseline+SWN} procedure in both domains indicates that the proposed patterns extract good segments for summarization.
\begin{table}
\centering
\caption{Comparison of classifiers and filters for services.}
\begin{tabular}{c|l|l|l|l|l|l|l|l|l|l|l|l|} \cline{2-13}
\label{tab:services}
&\multicolumn{6}{|c|}{pros} & \multicolumn{6}{|c|}{cons}\\ \hline
procedure & $\mathrm{P_{s}}$ & $\mathrm{R_{s}}$ & $\mathrm{P_{e}}$ & $\mathrm{R_{e}}$ & P & R & $\mathrm{P_{s}}$ & $\mathrm{R_{s}}$ & $\mathrm{P_{e}}$ & $\mathrm{R_{e}}$ & P & R\\ \hline
Baseline+SWN &24.4 &48.6 &64.8 &65.4 &44.5 &56.7 & 17.6 &29.0 &42.1 &45.0 &29.7 &36.8\\ \hline
AW+SWN &24.8 &55.5 &71.4 &60.4 &47.9 &57.6 &19.0 &33.8 &47.6 &37.8 &33.1 &35.5\\ \hline
AW+SEN &23.8 &52.3 &67.5 &62.8 &45.5 &57.2 &21.3 &47.6 &61.5 &36.9 &41.3 &42.1\\ \hline
AW+SEN+RANK &\textbf{29.3} &52.6 &66.7 &44.5 &47.8 &48.2 &\textbf{26.0} &46.8 &60.4 &23.1 &43.0 &34.8 \\ \hline
AW+SEN+SW & 25.8 & \textbf{56.4} & \textbf{73.3} & 59.9 & \textbf{49.3} & \textbf{57.8} & 25.4& \textbf{58.2}& \textbf{73.5}& 30.2& \textbf{49.4}& \textbf{44.0} \\ \hline
ASUM& 27.4& 49.7& 66.7& \textbf{65.8}& 46.8& 57.4& 22.0& 33.5& 47.0& \textbf{47.4}& 34.3& 40.2 \\ \hline
JST &24.6 &44.7 &59.8 &60.4 &42.0 &51.9 &21.3 &37.3 &49.4 &52.3 &35.2 &43.9 \\ \hline
\end{tabular}
\centering
\caption{Comparison of classifiers and filters for products.}
\begin{tabular}{c|l|l|l|l|l|l|l|l|l|l|l|l|} \cline{2-13}
\label{tab:products}
&\multicolumn{6}{|c|}{pros} & \multicolumn{6}{|c|}{cons}\\ \hline
procedure & $\mathrm{P_{s}}$ & $\mathrm{R_{s}}$ & $\mathrm{P_{e}}$ & $\mathrm{R_{e}}$ & P & R & $\mathrm{P_{s}}$ & $\mathrm{R_{s}}$ & $\mathrm{P_{e}}$ & $\mathrm{R_{e}}$ & P & R\\ \hline
Baseline+SWN&23.1 &49.2 &60.9 &44.8 &41.1 &43.9 &23.9 &34.5 &47.3 &\textbf{32.6} &34.6 &30.7 \\ \hline
AW+SWN&21.8 &52.6 &65.4 &42.7 &42.8 &44.7 &23.4 &39.0 &52.7 &30.8 &37.6 &31.6 \\ \hline
AW+SEN &23.9 &56.3 &68.5 &47.0 &45.3 &48.2 &26.1 &47.5 &62.9 &24.7 &43.5 &\textbf{32.1} \\ \hline
AW+SEN+RANK&\textbf{29.7} &51.2 &61.0 &32.4 &44.0 &38.4 &\textbf{32.6} &37.3 &48.9 &21.2 &39.4 &25.8 \\ \hline
AW+SEN+SW &25.5 &\textbf{59.8} &\textbf{71.4} &43.8 &\textbf{47.4} &48.2 &24.3 &\textbf{51.4} &\textbf{65.4} &16.7 &\textbf{44.2} &29.4 \\ \hline
ASUM &25.3 &54.8 &68.2 &\textbf{49.3} &45.7 &\textbf{48.8} & 26.9 &37.8& 49.0 &28.7& 36.7 &29.8 \\ \hline
JST &27.1 & 52.0& 64.3 &44.9 &44.7 &45.2 &25.9 &35.7 &46.4& 28.8 &34.5 &28.5 \\ \hline
\end{tabular}
\end{table}
Comparing \textbf{AW+SWN} and \textbf{AW+SEN}, we see that \textbf{SEN} is better than \textbf{SWN} at sentiment classification.
This result agrees with previous section, again confirming the effectiveness of our model in learning sentiments
of domain-specific lexicons. \textbf{AW+SWN} performs better than \textbf{Baseline+SWN},
suggesting that top aspect words can be used to identify more informative segments.
The best procedure is \textbf{AW+SEN+SW}, which tops all other procedures especially in the cons case.
It favors segments that contain common aspect-related words and its associated sentiment lexicons,
which are likely to be predominant in the pros and cons lists.
ASUM has similar modelling assumption as ours and so it also produces relatively good results.
However, the ability of our model to optimize sentiment polarities creates the improvement in performance.
\textbf{JST} is even inferior to \textbf{Baseline+SWN} for half of the cases. This is not surprising given
that the JST model is not intended for sentiment lexicons discovery; in contrast, it requires a large list of sentiment
seed words to function well. Finally, \textbf{AW+SEN+RANK} always has highest precision for segments but many segments
are eliminated and that hurts its performance.
\section{Qualitative Evaluation}
\label{sec:qualitative}
In this section we complement our results in previous section by
qualitatively evaluating the quality of extracted segments
with a user study and present example of summaries generated by our approach.
\subsection{Quality of Extracted Segments}
We carried out an user study with 130 workers from the
Amazon Mechanical Turk service. We randomly selected 123
short passages each has 4 to 6 sentences from reviews of
coffee makers. The user's task is to read a passage and rate each text item as `very useful',
`useful', `somewhat useful', or `useless' with reference to the passage. We included two types of items for each passage: segments
extracted by our approach using the \textbf{AW} filter and adjective-noun phrases extracted using tagging and term-frequency as in \cite{Yatani:spotlight}.
Each user performs 6 tasks in which half of them are repetitions
of others, thereby allowing us to detect users that give inconsistent ratings. We discarded users who completed their tasks in less than 90 seconds or rated half of the items inconsistently. Of the remaining 90 qualified users, 13 have not used coffee makers before whereas 60 have used for more than two months.
In total there were 358 unique segments, each rated 5.3 times and 470 unique word pairs, each rated 5.7 times. We converted the ratings into a numeric scale from 4 to 1 with 4 being 'very useful' and 1 being 'useless'. On average, users rated the segments extracted by our method as 3.01 compared to 2.53 for the adjective-noun phrases. The higher rating is not merely due to segments having more words, as we observed that users typically give an adjective-noun word pair a same or higher rating than
a segment if the two carry the same message. For example, `carafe stays hot' and `hot carafe' are same but the former
has a rating of 2.7 whereas the latter has a rating of 3.1. Therefore the higher average rating for segments is
a strong evidence that they convey more valuable information than adjective-noun word pairs.
Table ~\ref{tab:highlyrated} elaborates further on this evidence by showing
example of the segments and phrases rated as very useful by users.
As can be seen, the segments are quite complete semantically
whereas the phrases can be rather short in their meaning, which may require interpretation from users.
\begin{table}
\centering
\caption{Example of highly rated segments and phrases}
\begin{tabular}{p{12cm}} \hline
\textbf{segment} \\ \hline \hline
very easy instruction, almost completely unscrewed to pour, buttons are easy to press, cup is always fresh, coffee pot is very hard to take, closing is easy, makes really awful tasting coffee, feature works fine, machine brews a great cup every time, machine is very simple to use, machine is programmable, carafe is dripless \\ \hline
\textbf{adjective-noun phrase} \\ \hline \hline
affordable maker, better tasting, correct time, filtered water, finished quality, difficult place, fresh tasting, good customer, good tasting, new recipe, removable basket, optimal temperature, cheap use, easy closing, darker flavor, hot cup, great pot \\ \hline
\end{tabular}
\label{tab:highlyrated}
\end{table}
\subsection{Example Summaries}
Below we show examples of a restaurant review and a coffee maker review together
with the segments extracted as their summaries.
\textbf{Review of restaurant}: \textit{The space is small but cozy, and the staff is friendly and knowledgeable. There was some great music playing, which kind of made me feel like I was on vacation some place far away from Astoria. There are a lot of really great vegetarian options, as well as several authentic Turkish dishes. If you're still wasting time reading this review, stop now and head straight for Mundo. Your stomach could already be filled with tons of deliciousness.}
\textbf{Summary}: staff is friendly, space is small, some great music playing , several authentic Turkish dishes, really great vegetarian options.
\textbf{Review of coffee maker}: \textit{I bought this machine about a week ago. I did not know which machine in the store to get, but the sales clerk helped me make the decision to buy this one. It is incredibly simple to use and the espresso is great. The crema is perfect too. My latte's rival those in coffee houses and I am saving a ton of money. The "capsules" must be ordered from the Nespresso website, but they are usually at your door in 48 hours via UPS...}
\textbf{Summary}: incredibly simple to use, espresso is great, crema is perfect.
In both cases the summaries express the gist of each review
relatively well. Looking at the sentence where a segment is
extracted from, it can be seen that the segment conveys the
main talking point of the sentence. Additionally,
each segment does express an opinion about some aspect of the coffee
maker or the restaurant. Recall that our key assumption in
modeling reviews is that each sentence has a sentiment and
an aspect. Therefore extracting segments the way we propose
is likely to capture the main content of a sentence.
\section{Conclusions}
In this paper we have describe a framework for extracting and selecting informative
segments for review summarization of products and services. We extract candidate segments by
matching against variable-length syntactic patterns and select the segments that contain top sentiment
and aspect words learned by topic models. We proposed a new joint sentiment topic model that learns the
polarity of aspect dependent sentiment lexicons. Qualitative and quantitative experiments verify that our model outperforms previous approaches in improving the quality of the extracted segments as well as the generated summaries.
\bibliographystyle{splncs03}
|
2103.16901
|
\section{Introduction}
\vspace*{-0.35cm}
Divergences are non-negative measures of dissimilarity between pairs
of probability measures which are defined on the same measurable space. They
play a key role in the development of information theory, probability theory,
statistics, learning, signal processing, and other related fields. One important
class of divergence measures is defined by means of convex functions $f$, and it
is called the class of $f$-divergences. It unifies fundamental and
independently-introduced concepts in several branches of mathematics such as the
chi-squared test for the goodness of fit in statistics, the total variation distance
in functional analysis, the relative entropy in information theory and statistics,
and it is closely related to the R\'{e}nyi divergence which generalizes the
relative entropy. The class of $f$-divergences satisfies pleasing features such as
the data-processing inequality, convexity, continuity and duality properties,
finding interesting applications in information theory and statistics.
Majorization theory is a simple and productive concept in the theory of inequalities,
which also unifies a variety of familiar bounds (see the book by \cite{MarshallOA}).
The concept of majorization finds various applications in diverse fields of pure
and applied mathematics, including information theory and communication.
This work, presented in the papers by \cite{IS18, IS19}, is focused on
new data-processing and majorization inequalities for $f$-divergences
and the R\'{e}nyi entropy.
The reason for discussing both types of inequalities in this work
is the interplay which exists between majorization and data processing
where a probability mass function $P$, defined over a finite set, is
majorized by another probability mass function $Q$ which is defined
over the same set if and only if there exists a doubly-stochastic
transformation $W_{Y|X}$ such that an input distribution that is equal
to $Q$ yields an output distribution that is equal to $P$ (denoted by,
$Q \rightarrow W_{Y|X} \rightarrow P$).
We consider applications of the inequalities which are derived in this work to
information theory, statistics, and coding problems.
One application refers to the performance analysis of list decoding with either fixed
or variable list sizes; some earlier bounds on the list decoding error probability are
reproduced in a unified way, and new bounds are obtained and exemplified numerically.
A second application, covered in \cite{IS19}, is related to a study of the quality of
approximating a probability mass function, induced by the leaves of a Tunstall tree,
by an equiprobable distribution. The compression rates of finite-length Tunstall codes
are further analyzed for asserting their closeness to the Shannon entropy of a memoryless
and stationary discrete source. A third application of our bounds relies on our tight
bounds for the R\'{e}nyi entropy (see \cite{IS18}) and the source coding theorem by
\cite{Campbell65} to obtain tight non-asymptotic bounds for lossless compression of
discrete memoryless sources.
\section{Coding Problems and main Results}
\subsection{Bounds on the List Decoding Error Probability
with $f$-divergences}
\label{subsection: Fano - list decoder}
The minimum probability of error of a random variable $X$ given
$Y$, denoted by $\varepsilon_{X|Y}$,
can be achieved by a deterministic function
(\textit{maximum-a-posteriori} decision rule)
$\mathcal{L}^\ast \colon \mathcal{Y} \to \mathcal{X}$ (see \cite{ISSV18}):
\begin{align}
\varepsilon_{X|Y}
&= \min_{\mathcal{L} \colon \mathcal{Y} \to \mathcal{X}}
\mathbb{P} [ X \neq \mathcal{L} (Y) ] \label{20170904} \\
&= \mathbb{P} [ X \neq \mathcal{L}^\ast (Y) ] \label{eq:MAP}\\
&= 1- \mathbb{E} \left[ \max_{x \in \mathcal{X}}
P_{X|Y}(x|Y) \right].
\label{eq1: cond. epsilon}
\end{align}
Fano's inequality gives an upper bound on the
conditional entropy $H(X|Y)$ as a function of $\varepsilon_{X|Y}$
(or, otherwise, providing a lower bound on $\varepsilon_{X|Y}$ as
a function of $H(X|Y))$ when $X$ takes a finite number of possible values.
The list decoding setting, in which the hypothesis tester is allowed
to output a subset of given cardinality, and an error occurs if the
true hypothesis is not in the list, has great interest in
information theory. A generalization of Fano's inequality to list decoding,
in conjunction with the blowing-up lemma \cite[Lemma~1.5.4]{Csiszar_Korner},
leads to strong converse results in multi-user information theory.
The main idea of the successful combination
of these two tools is that, given a code, it is possible to blow-up the
decoding sets in a way that the probability of decoding error can be as small
as desired for sufficiently large blocklengths; since the blown-up decoding
sets are no longer disjoint, the resulting setup is a list decoder with
sub-exponential list size (as a function of the block length).
In this section, we further study the setup of list decoding, and derive
bounds on the average list decoding error probability. We first
consider the special case where the list size is fixed, and then
consider the more general case of a list size which depends on the channel
observation. All of the following bounds on the list decoding error probability
are derived in the paper by \cite{IS19}.
\subsubsection{Fixed-Size List Decoding}
\label{subsubsection: fixed-size list decoding}
The next result provides a generalized Fano's inequality for fixed-size
list decoding, expressed in terms of an arbitrary $f$-divergence. Some
earlier results in the literature are reproduced from the next result.
\begin{thm}
\label{theorem: generalized Fano Df}
Let $P_{XY}$ be a probability measure defined on
$\mathcal{X} \times \mathcal{Y}$ with $|\mathcal{X}|=M$. Consider
a decision rule $\mathcal{L} \colon \mathcal{Y} \to \binom{\mathcal{X}}{L}$,
where $\binom{\mathcal{X}}{L}$ stands for the set of subsets
of $\mathcal{X}$ with cardinality $L$, and $L < M$ is fixed.
Denote the list decoding error probability by
$P_{\mathcal{L}} := \ensuremath{\mathbb{P}} \bigl[ X \notin \mathcal{L}(Y) \bigr]$.
Let $U_M$ denote an equiprobable probability mass
function on $\mathcal{X}$. Then, for every convex function
$f \colon (0, \infty) \to \ensuremath{\mathbb{R}}$ with $f(1)=0$,
\begin{align}
\label{generalized Fano Df}
& \ensuremath{\mathbb{E}}\Bigl[D_f \bigl(P_{X|Y}(\cdot|Y) \, \| \, U_M \bigr) \Bigr] \nonumber \\
& \geq \frac{L}{M} \; f\biggl(\frac{M \, (1-P_{\mathcal{L}})}{L} \biggr)
+ \biggl(1-\frac{L}{M}\biggr) \; f\biggl(\frac{M P_{\mathcal{L}}}{M-L} \biggr).
\end{align}
\end{thm}
The special case where $L=1$ (i.e., a decoder with a single
output) gives \cite[(5)]{Guntuboyina11}.
As consequences of Theorem~\ref{theorem: generalized Fano Df}, we first
reproduce some earlier results as special cases.
\begin{thm} \cite[(139)]{ISSV18}
\label{corollary: Fano - list}
Under the assumptions in Theorem~\ref{theorem: generalized Fano Df},
\begin{align}
\label{ISSV18 - Fano}
H(X|Y) \leq \log M - d\biggl(P_{\mathcal{L}} \, \| \, 1-\frac{L}{M} \biggr)
\end{align}
where $d(\cdot \| \cdot) \colon [0,1] \times [0,1] \to [0, +\infty]$ denotes
the binary relative entropy, defined as the continuous extension of
$D([p, 1-p] \| [q, 1-q]) := p \log \frac{p}{q} + (1-p) \log \frac{1-p}{1-q}$
for $p,q \in (0,1)$.
\end{thm}
\vspace*{0.2cm}
The following refinement of the generalized Fano's inequality in
Theorem~\ref{theorem: generalized Fano Df} relies on the version
of the strong data-processing inequality for $f$-divergences in
\cite[Theorem~1]{IS19}.
\begin{thm}
\label{theorem: refined Fano's inequality}
Under the assumptions in Theorem~\ref{theorem: generalized Fano Df},
let $f \colon (0, \infty) \to \ensuremath{\mathbb{R}}$ be twice
differentiable, and assume that there exists a constant $m_f>0$ such that
\begin{align}
\label{m_f}
f''(t) \geq m_f, \quad \forall \,
t \in \mathcal{I}(\xi_1^\ast, \xi_2^\ast),
\end{align}
where
\begin{align}
\label{28062019a1}
& \xi_1^\ast := M \inf_{(x,y) \in \mathcal{X} \times \mathcal{Y}} P_{X|Y}(x|y), \\
\label{28062019a2}
& \xi_2^\ast := M \sup_{(x,y) \in \mathcal{X} \times \mathcal{Y}} P_{X|Y}(x|y),
\end{align}
and the interval $\mathcal{I}(\cdot, \cdot)$ is the interval
\begin{align}
\label{I_interval}
\mathcal{I} := \mathcal{I}(\xi_1, \xi_2) = [\xi_1, \xi_2] \cap (0, \infty).
\end{align}
Let $u^+ := \max\{u, 0\}$ for $u \in \ensuremath{\mathbb{R}}$. Then,
\begin{enumerate}[a)]
\item \label{Part a - refined Fano's inequality}
\begin{align}
\label{list dec.-26062019a}
& \ensuremath{\mathbb{E}}\Bigl[D_f \bigl(P_{X|Y}(\cdot|Y) \, \| \, U_M \bigr) \Bigr] \\
& \geq \frac{L}{M} \; f\biggl(\frac{M \, (1-P_{\mathcal{L}})}{L} \biggr)
+ \left(1-\frac{L}{M}\right) \; f\biggl(\frac{M P_{\mathcal{L}}}{M-L} \biggr) \nonumber \\
& \hspace*{0.4cm} + \tfrac12 m_f \, M \left( \ensuremath{\mathbb{E}}\bigl[P_{X|Y}(X|Y)\bigr]
-\frac{1-P_{\mathcal{L}}}{L} - \frac{P_{\mathcal{L}}}{M-L} \right)^+. \nonumber
\end{align}
\item \label{Part b - refined Fano's inequality}
If the list decoder selects the $L$ most probable elements from $\mathcal{X}$,
given the value of $Y \in \mathcal{Y}$, then \eqref{list dec.-26062019a} is
strengthened to
\begin{align}
& \ensuremath{\mathbb{E}}\Bigl[D_f \bigl(P_{X|Y}(\cdot|Y) \, \| \, U_M \bigr) \Bigr] \nonumber \\
& \geq \frac{L}{M} \; f\biggl(\frac{M \, (1-P_{\mathcal{L}})}{L} \biggr)
+ \biggl(1-\frac{L}{M}\biggr) \; f\biggl(\frac{M P_{\mathcal{L}}}{M-L} \biggr) \nonumber \\
\label{list dec.-26062019b}
& \hspace*{0.4cm} + \tfrac12 m_f \, M \left( \ensuremath{\mathbb{E}}\bigl[P_{X|Y}(X|Y)\bigr]
-\frac{1-P_{\mathcal{L}}}{L} \right),
\end{align}
where the last term in the right side of \eqref{list dec.-26062019b} is
necessarily non-negative.
\end{enumerate}
\end{thm}
Discussions and numerical experimentation of these proposed bounds are provided
in the paper by \cite{IS19}, showing the obtained improvement over Fano's inequality.
\subsubsection{Variable-Size List Decoding}
\label{subsubsection: variable-size list decoding}
In the more general setting of list decoding where the size of the list
may depend on the channel observation, Fano's inequality has been
generalized as follows.
\begin{thm} (\cite{AhlswedeK75} and \cite[Appendix~3.E]{RS_FnT19})
\label{prop: Fano-Ahlswede-Korner}
Let $P_{XY}$ be a probability measure defined on $\mathcal{X} \times \mathcal{Y}$
with $|\mathcal{X}|=M$. Consider a decision rule $\mathcal{L} \colon \mathcal{Y} \to
2^{\mathcal{X}}$, and let the (average) list decoding error probability be
given by $P_{\mathcal{L}} := \ensuremath{\mathbb{P}} \bigl[ X \notin \mathcal{L}(Y) \bigr]$ with
$|\mathcal{L}(y)| \geq 1$ for all $y \in \mathcal{Y}$. Then,
\begin{align}
\label{Fano-Ahlswede-Korner 1}
H(X|Y) \leq h(P_\mathcal{L}) + \ensuremath{\mathbb{E}}[\log |\mathcal{L}(Y)|] + P_{\mathcal{L}} \log M,
\end{align}
where $h \colon [0,1] \to [0, \log 2]$ denotes the binary entropy function.
If $|\mathcal{L}(Y)| \leq N$ almost surely, then also
\begin{align}
\label{Fano-Ahlswede-Korner 2}
H(X|Y) \leq h(P_\mathcal{L}) + (1-P_{\mathcal{L}}) \log N + P_{\mathcal{L}} \log M.
\end{align}
\end{thm}
By relying on the data-processing inequality for $f$-divergences, we derive in
the following an alternative explicit lower bound on the average list decoding
error probability $P_{\mathcal{L}}$. The derivation relies on the $E_\gamma$ divergence
(see, e.g., \cite{LCV17}), which forms a subclass of the $f$-divergences.
\begin{thm}
\label{theorem: LB - variable list size}
Under the assumptions in \eqref{Fano-Ahlswede-Korner 1}, for all $\gamma \geq 1$,
\begin{align}
\label{LB - variable list size}
P_{\mathcal{L}} \geq \frac{1+\gamma}{2} - \frac{\gamma \ensuremath{\mathbb{E}}[|\mathcal{L}(Y)|]}{M}
- \frac12 \, \ensuremath{\mathbb{E}} \left[ \, \sum_{x \in \mathcal{X}} \, \biggl| P_{X|Y}(x|Y)
- \frac{\gamma}{M} \biggr| \right].
\end{align}
Let $\gamma \geq 1$, and let $|\mathcal{L}(y)| \leq \frac{M}{\gamma}$ for all
$y \in \mathcal{Y}$. Then, \eqref{LB - variable list size} holds with equality if,
for every $y \in \mathcal{Y}$, the list decoder selects the $|\mathcal{L}(y)|$ most
probable elements in $\mathcal{X}$ given $Y=y$; if $x_\ell(y)$ denotes
the $\ell$-th most probable element in $\mathcal{X}$ given $Y=y$, where ties
in probabilities are resolved arbitrarily, then \eqref{LB - variable list size}
holds with equality if
\begin{align}
& P_{X|Y}(x_\ell(y) \, | y) \nonumber \\
\label{02072019a19}
&=
\begin{dcases}
\alpha(y), \quad & \forall \, \ell \in \bigl\{1, \ldots, |\mathcal{L}(y)| \bigr\}, \\
\frac{1-\alpha(y) \, |\mathcal{L}(y)|}{M-|\mathcal{L}(y)|},
\quad & \forall \, \ell \in \bigl\{|\mathcal{L}(y)|+1, \ldots, M\},
\end{dcases}
\end{align}
with $\alpha \colon \mathcal{Y} \to [0,1]$ being an arbitrary function which satisfies
\begin{align}
\label{02072019a20}
\frac{\gamma}{M} \leq \alpha(y) \leq \frac1{|\mathcal{L}(y)|},
\quad \forall \, y \in \mathcal{Y}.
\end{align}
\end{thm}
As an example,
let $X$ and $Y$ be random variables taking their values in
$\mathcal{X} = \{0, 1, 2, 3, 4\}$ and $\mathcal{Y} = \{0, 1\}$, respectively,
and let $P_{XY}$ be their joint probability mass function, which is given by
\begin{align}
\label{03072019a1}
\begin{dcases}
& P_{XY}(0,0) = P_{XY}(1,0) = P_{XY}(2,0) = \tfrac18, \\[0.1cm]
& P_{XY}(3,0) = P_{XY}(4,0) = \tfrac1{16}, \\[0.1cm]
& P_{XY}(0,1) = P_{XY}(1,1) = P_{XY}(2,1) = \tfrac1{24}, \\[0.1cm]
& P_{XY}(3,1) = P_{XY}(4,1) = \tfrac3{16}.
\end{dcases}
\end{align}
Let $\mathcal{L}(0) := \{0,1,2\}$ and $\mathcal{L}(1) := \{3,4\}$ be the lists in $\mathcal{X}$,
given the value of $Y \in \mathcal{Y}$. We get $P_Y(0) = P_Y(1) = \tfrac12$, so the
conditional probability mass function of $X$ given $Y$ satisfies
$P_{X|Y}(x|y) = 2 P_{XY}(x,y)$ for all $(x,y) \in \mathcal{X} \times \mathcal{Y}$.
It can be verified that, if $\gamma = \tfrac54$, then
$\max\{|\mathcal{L}(0)|, |\mathcal{L}(1)|\} = 3 \leq \frac{M}{\gamma}$, and also
\eqref{02072019a19} and \eqref{02072019a20} are satisfied
(here, $M:=|\mathcal{X}|=5$, $\alpha(0) = \tfrac14 = \frac{\gamma}{M}$
and $\alpha(1) = \tfrac38 \in \bigl[\tfrac14, \tfrac12\bigr]$). By
Theorem~\ref{theorem: LB - variable list size}, it follows that
\eqref{LB - variable list size} holds in this case with equality,
and the list decoding error probability is equal to
$P_{\mathcal{L}}=1-\ensuremath{\mathbb{E}}\bigl[ \alpha(Y) \, |\mathcal{L}(Y)| \bigr]=\tfrac14$
(i.e., it coincides with the lower bound in the right side of
\eqref{LB - variable list size} with $\gamma = \tfrac54$).
On the other hand, the generalized Fano's inequality in
\eqref{Fano-Ahlswede-Korner 1} gives that $P_\mathcal{L} \geq 0.1206$
(the left side of \eqref{Fano-Ahlswede-Korner 1} is
$H(X|Y) = \tfrac52 \, \log 2 - \tfrac14 \, \log 3 = 2.1038$~bits);
moreover, by letting $N := \underset{y \in \mathcal{Y}}{\max} \, |\mathcal{L}(y)| = 3$,
\eqref{Fano-Ahlswede-Korner 2} gives the looser bound
$P_\mathcal{L} \geq 0.0939$. This exemplifies a case where the lower bound in
Theorem~\ref{theorem: LB - variable list size} is tight, whereas the
generalized Fano's inequalities in \eqref{Fano-Ahlswede-Korner 1} and
\eqref{Fano-Ahlswede-Korner 2} are looser.
\subsection{Lossless Source Coding}
\label{subsubsection: lossless source coding}
For uniquely-decodable (UD) source codes, \cite{Campbell65}
proposed the cumulant generating function of the codeword lengths as a
generalization to the frequently used design criterion of average code
length. The motivation in the paper by \cite{Campbell65} was to control
the contribution of the longer codewords via a free parameter in the
cumulant generating function: if the value of this parameter tends to zero,
then the resulting design criterion becomes the average code length per
source symbol; on the other hand, by increasing the value of the free
parameter, the penalty for longer codewords is more severe, and the resulting
code optimization yields a reduction in the fluctuations of the codeword lengths.
We introduce the coding theorem by \cite{Campbell65} for lossless compression
of a discrete memoryless source (DMS) with UD codes, which serves for our
analysis (see \cite{IS18}).
\begin{thm
\label{theorem: Campbell}
Consider a DMS which emits symbols with a probability
mass function $P_X$ defined on a (finite or countably infinite) set $\mathcal{X}$.
Consider a UD fixed-to-variable source code operating on
source sequences of $k$ symbols with an alphabet of the codewords of size $D$.
Let $\ell(x^k)$ be the length of the codeword which corresponds to the source
sequence $x^k := (x_1, \ldots, x_k) \in \mathcal{X}^k$. Consider the scaled
{\em cumulant generating function} of the codeword lengths:
\begin{align}
\label{eq: cumulant generating function}
\Lambda_k(\rho) := \frac1{k} \, \log_D \left( \, \sum_{x^k \in \mathcal{X}^k}
P_{X^k}(x^k) \, D^{\rho \, \ell(x^k)} \right), \quad \rho > 0
\end{align}
where
\begin{align}
\label{eq: pmf}
P_{X^k}(x^k) = \prod_{i=1}^k P_X(x_i), \quad \forall \, x^k \in \mathcal{X}^k.
\end{align}
Then, for every $\rho > 0$, the following hold:
\begin{enumerate}[a)]
\item Converse result:
\begin{align}
\label{eq: Campbell's converse result}
\frac{\Lambda_k(\rho)}{\rho} \geq \frac{1}{\log D} \; H_{\frac1{1+\rho}}(X).
\end{align}
\item Achievability result:
there exists a UD source code, for which
\begin{align}
\label{eq: Campbell's achievability result}
\frac{\Lambda_k(\rho)}{\rho} \leq \frac{1}{\log D} \; H_{\frac1{1+\rho}}(X) + \frac{1}{k}.
\end{align}
\end{enumerate}
\end{thm}
The bounds in Theorem~\ref{theorem: Campbell}, expressed in terms of the R\'{e}nyi entropy,
imply that for sufficiently long source sequences, it is possible to make the scaled
cumulant generating function of the codeword lengths approach the R\'{e}nyi entropy as closely
as desired by a proper fixed-to-variable UD source code; moreover, the converse result
shows that there is no UD source code for which the scaled cumulant generating function
of its codeword lengths lies below the R\'{e}nyi entropy.
By invoking L'H\^{o}pital's rule, one gets from \eqref{eq: cumulant generating function}
\begin{align}
\label{eq: limit rho tends to zero}
\lim_{\rho \downarrow 0} \frac{\Lambda_k(\rho)}{\rho}
= \frac1k \sum_{x^k \in \mathcal{X}^k} P_{X^k}(x^k) \, \ell(x^k) = \frac1k \, \ensuremath{\mathbb{E}}[\ell(X^k)].
\end{align}
Hence, by letting $\rho$ tend to zero in \eqref{eq: Campbell's converse result} and
\eqref{eq: Campbell's achievability result}, it follows
that Campbell's result in Theorem~\ref{theorem: Campbell} generalizes the well-known bounds
on the optimal average length of UD fixed-to-variable source codes:
\begin{align}
\label{eq: Shannon}
\frac{1}{\log D} \; H(X) \leq \frac1k \; \ensuremath{\mathbb{E}}[\ell(X^k)] \leq \frac{1}{\log D} \; H(X) + \frac1k,
\end{align}
and \eqref{eq: Shannon} is satisfied by Huffman coding.
Campbell's result therefore generalizes Shannon's fundamental result for the average
codeword lengths of lossless compression codes, expressed in terms of the Shannon entropy.
Following the work by \cite{Campbell65}, non-asymptotic bounds were derived by
\cite{CV2014a} for the scaled cumulant
generating function of the codeword lengths for $P_X$-optimal variable-length
lossless codes. These bounds were used by \cite{CV2014a}
to obtain simple proofs of the asymptotic normality of the distribution of
codeword lengths, and the reliability function of memoryless sources allowing
countably infinite alphabets.
The analysis which leads to the following result for lossless source compression
with uniquely-decodable (UD) codes is provided in the paper by \cite{IS18}.
Let $X_1, \ldots, X_k$ be i.i.d. symbols which are emitted from a DMS according to a probability
mass function $P_X$ whose support is a finite set $\mathcal{X}$ with $|\mathcal{X}|=n$.
In order to cluster the data, suppose that
each symbol $X_i$ is mapped to $Y_i = f(X_i)$ where $f \in \mathcal{F}_{n,m}$ is an arbitrary
deterministic function (independent of the index $i$) with $m<n$. Consequently, the i.i.d.
symbols $Y_1, \ldots, Y_k$ take values on a set $\mathcal{Y}$ with $|\mathcal{Y}|=m<|\mathcal{X}|$.
Consider two UD fixed-to-variable source codes: one operating on the sequences $x^k \in \mathcal{X}^k$,
and the other one operates on the sequences $y^k \in \mathcal{Y}^k$; let $D$ be the size of
the alphabets of both source codes.
Let $\ell(x^k)$ and $\overline{\ell}(y^k)$ denote the length of the codewords for the
source sequences $x^k$ and $y^k$, respectively, and let $\Lambda_k(\cdot)$ and
$\overline{\Lambda}_k(\cdot)$ denote their corresponding scaled cumulant generating functions
(see \eqref{eq: cumulant generating function}).
Relying on our tight bounds on the R\'{e}nyi entropy (of any positive order) in
\cite[Theorems~1, 2]{IS18} and Theorem~\ref{theorem: Campbell}, we obtain upper
and lower bounds on $\frac{\Lambda_k(\rho) - \overline{\Lambda}_k(\rho)}{\rho}$
for all $\rho > 0$ (see \cite[Theorem~5]{IS18}). To that end, for
$m \in \{2, \ldots, n-1\}$, if $P_X(1) < \frac1m$, let $\widetilde{X}_m$ be the equiprobable
random variable on $\{1, \ldots, m\}$; otherwise, if $P_X(1) \geq \frac1m$,
let $\widetilde{X}_m \in \{1, \ldots, m\}$ be a random variable with the probability mass function
\begin{align*}
P_{\widetilde{X}_m}(i) =
\begin{dcases}
P_X(i), & i \in \{1, \ldots, n^\ast\}, \\
\frac1{m-n^\ast} \sum_{j = n^\ast+1}^n P_X(j), & i \in \{n^\ast+1, \ldots, m\},
\end{dcases}
\end{align*}
where $n^\ast$ is the maximal integer $i \in \{1, \ldots, m-1\}$ such that
\begin{align}
\label{eq: n ast}
P_X(i) \geq \frac1{m-i} \sum_{j=i+1}^n P_X(j).
\end{align}
The result in \cite[Theorem~5]{IS18} is of interest since it provides upper and lower
bounds on the reduction in the cumulant generating function of close-to-optimal UD source
codes as a result of clustering data, and \cite[Remark~11]{IS18} suggests an algorithm to
construct such UD codes which are also prefix codes. For long enough sequences (as $k \to \infty$),
the upper and lower bounds on the difference between the scaled cumulant generating functions
of the suggested source codes for the original and clustered data almost match, being roughly equal to
$\rho \left( H_{\frac1{1+\rho}}(X)- H_{\frac1{1+\rho}}(\widetilde{X}_m) \right)$ (with logarithms
on base $D$, which is the alphabet size of the source codes), and
as $k \to \infty$,
the gap between these upper and lower bounds is less than $0.08607 \log_D 2$.
Furthermore, in view of \eqref{eq: limit rho tends to zero},
\begin{align}
\lim_{\rho \downarrow 0} \frac{\Lambda_k(\rho) - \overline{\Lambda}_k(\rho)}{\rho}
= \frac1k \left( \ensuremath{\mathbb{E}}[\ell(X^k)] - \ensuremath{\mathbb{E}}[\overline{\ell}(Y^k)] \right),
\end{align}
so, it follows from \cite[Theorem~5]{IS18} that the difference between the average code
lengths (normalized by~$k$) of the original and clustered data satisfies
\begin{align}
- \frac1k & \leq \frac{\ensuremath{\mathbb{E}}[\ell(X^k)] - \ensuremath{\mathbb{E}}[\overline{\ell}(Y^k)]}{k}
- \frac{H(X) - H(\widetilde{X}_m)}{\log D} \nonumber \\
\label{eq: 20181030e}
& \leq 0.08607 \log_D 2,
\end{align}
and the gap between the upper and lower bounds is small.
|
2002.10183
|
\section{Motivation and significance}
\label{motivation}
Positron Emission Tomography is one of the most popular methods for tomographic imaging used in nuclear medicine.
In contrast to other techniques such as Computed Tomography that can detect anatomical changes,
PET provides information about metabolic processes in the patient's body even at the cell level~\cite{dbailey}.
This allows detection of pathological symptoms that usually precede the anatomical changes.
PET tomography has a wide range of research and clinical applications e.g.\ it is commonly used for diagnosis of cancer, neurological disorders, heart diseases and many others~\cite{Slomka}.
Although the PET technique is well established for clinical usage, there are ongoing efforts in the scientific community that would overcome the limits of the commercial scanners
and improve the quality of the image~\cite{Slomka,Borghi_2016,8049484,Karpetas,Cates_2016} or even enrich the available information by introducing new diagnostic methods.
Whole-body or total-body PET scanner projects~\cite{Badawi2019,Cherry2017,Cherry2018} propose tomographs that improve the sensitivity of the measurement in order to shorten the time of a scan or alternatively require a smaller radiation exposure for the patients~\cite{NATURE_EXPLORER}.
The transformation of the data acquired by a PET scanner from the \textit{raw} binary level till the final patient image analysed by physicians is a complex, multi-stage process involving low- and high-level reconstruction algorithms.
The associated data handling and reconstruction is an especially hard task in case of the whole-body scanners due to large data volume~\cite{Zhang2017}.
The J-PET collaboration aims at providing a low-cost, modular, whole-body PET scanner based on detection of photon interactions in plastic scintillators~\cite{NIM2014,Szymon-Acta, PMB2018} with a view to its application in both medical diagnostics~\cite{PMB2018, moskal:pmb2019} and in proton therapy monitoring~\cite{rucinski2020}.
The \jpet{} prototype is a research device which not only demonstrates the new operating principle for its use in standard PET tomography but also explores new imaging modalities such as spatially-resolved determination of properties of positronium atoms produced in a patient's body~\cite{imaging_patent, Jasinska-Moskal2017, daria_epjc, NATURE}.
The exploratory nature of the \jpet{} device results in its operation
with much more flexible data registration conditions than used in commercial PET solutions.
In order to allow for classical PET imaging without discrimination of signals, which may be used in the novel diagnostic methods, \jpet{} operates
in a trigger-less data acquisition mode~\cite{Korcyl-IEEE}, resulting in a volume of recorded data unprecedented in medical imaging technologies.
From the software point of view, development and testing of novel PET modalities and tomography methods become challenging as the standard approaches must be either extended or entirely replaced by new algorithms.
Moreover, at the prototyping stage, multiple elements of the detector, its geometrical setup and the data acquisition chain are subject to change and various reconstruction procedures may be tested in parallel.
The software framework used to analyze data from evolving prototypes and to implement and test new reconstruction algorithms must follow these changes dynamically.
At the same time, however, the need to efficiently process the data stream from trigger-less acquisition requires that the performance may not be compromised when asserting flexibility.
The \jpet{} Framework package has been developed as an answer to the aforementioned challenges, providing a dynamically adjustable environment for development and efficient implementation of new algorithms.
The basic idea is to provide a set of generic building blocks, allowing a quick implementation of data processing chains to be used by analysts with even little programming experience through a convenient and simple API.
The \jpet{} Framework is used for analysis of data recorded by the tomograph prototype from the level of raw data saved by its data acquisition system, through assembly of higher-level data structures representing the logic needed for reconstruction of the physical properties of electron-positron annihilation into photons, up to the level of medical image reconstruction and statistical analysis of the data.
Notably, the data acquisition system of \jpet{} is based on the TRB3 hardware platform~\cite{Traxler_2011, Neiser_2013} which is widely used by experimental setups both in the fields of medical imaging and particle physics experiments \cite{TRB3users}. Consequently, the \jpet{} Framework can be easily adopted for data analysis in other TRB3-based experiments.
From the point of view of the full data reconstruction flow, the usage scope of the \jpet{} Framework is different compared with the existing tomography image software packages such as STIR~\cite{stir}, CASTOR~\cite{castor} or QETIR~\cite{qetir}, since
it also allows to implement low-level reconstruction and calibration algorithms which operate before the formation of Line-of-Responses (LORs), while typical input data for image algorithms consists of higher-level structures such as sinograms or list of LORs. At the same time, \jpet{} environment provides tools for the implementation of typical image reconstruction algorithms and effectively such procedures e.g. Time-of-Flight Filtered-Backprojection, have been implemented within the Framework.
However, the aim of the \jpet{} Framework package is not
to replace the existing image tomography toolkits, which offer well tested and proven solutions, but rather to provide a possibility for passing the transformed data to the external packages.
While an early version of the \jpet{} Framework is described in Ref.~\cite{Krzemien2015Framework},
this article is intended to present its architecture and functionality available in its current mature form,
which allows to extend the scope of its usage beyond the \jpet project. Therefore, we focus on the properties of the core \jpet{} Framework library~\cite{framework-repo} rather than on the particular \jpet-specific reconstruction algorithms developed using the framework which are available in a separate repository~\cite{examples-repo}.
\section{Software description}
\label{soft-desc}
The design of the \jpet{} Framework originated from the necessity of performing reconstruction and analysis of PET data from a prototype tomography scanner. It has become a more generalized environment for execution of tasks which could be adapted to multi-step analysis of various kinds of data. All the features naturally followed the implementation: managing input/output, incorporating palette of configurations, adapting data and parameter structures, user interface and task handling.
The core of the J-PET Framework is
constituted by
a dynamic library that can be linked to user applications. The library provides tools for loading, analysing and saving transformed data as well as for implementation of transformation algorithms that can be further connected in chains and finally executed.
The \jpet{} environment can be used to e.g. develop a reconstruction chain for the real data collected by a PET scanner or to implement a calibration procedure, an image reconstruction method or any kind of a multi-step analysis. Other typical applications consist of comparative studies of prototype PET scanners performance based on the Monte Carlo (MC) simulations.
\subsection{User Application Programming Interface}
\label{api}
The core library of the J-PET Framework provides the users with an Application Programming Interface (API) presented schematically in Figure~\ref{fig:api_scheme}.
The API is concentrated on giving the user access to data (structured as a stream of subsequent \textit{"events"}) read from various sources as well to parameters of the experimental setup (read from external configuration files) in order to combine
the event data and setup details in user-defined algorithms assembling higher-level
data structures and filtering them based on custom conditions (Listing~\ref{lst:event_finder} presents an simple example of such procedure).
The API further allows for grouping of such user-provided logic into analysis modules which can then be chained to constitute a complete analysis workflow (as demonstrated in Listing~\ref{lst:main}).
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{api_scheme}
\caption{Scheme of the Application Programming Interface exposed to the user for creation of data reconstruction and analysis workflows. Elements of the workflow exposed to and modified by the user are contained in the middle gray region whereas the elements outside it are handled by the framework transparently to the user. Main functionalities of the API exposed to the user are marked with blue arrows and blue text.}
\label{fig:api_scheme}
\end{figure}
\subsection{Software Architecture}
\label{arch}
The library is written in C++ using object-oriented paradigm. The core components are implemented as classes with well-defined responsibilities e.g. computing task execution, input/output operations, logging, option parsing, option validation. Moreover, the package contains a set of classes representing physical entities e.g. part of the scanner or PET-specialized data structures such Line-of-Response, which form a language that can be used to express the domain-specific concepts (see more details in section~\ref{sec:param-data}).
The basic concept of the \jpet{} Framework is the decomposition of a data processing chain into a series of standardized modular blocks. Each module corresponds to a particular computing task, e.g.\ a reconstruction algorithm or a calibration procedure, with well-defined input and output. The processing chain is built by registration of chosen modules in the \texttt{JPetManager}, responsible for synchronization of the data flow between the modules (see Figure~\ref{fig:manager}). This approach
permits to quickly interchange modules and to create processing chains
for different experimental setups.
\begin{figure}[h]
\includegraphics[width=340pt,height=220pt]{scheme_manger.png}
\centering
\label{fig:manager}
\caption{Scheme of Frameworks \texttt{JPetManager} structure, showing order of initialization and execution of tasks.}
\end{figure}
\subsection{Software Functionalities}\label{se:func}
\jpet{} Framework provides a set of functionalities that helps in rapid data reconstruction and analysis prototyping.
In this section, we list the most useful features and present selected usage examples. More applications can be found in the repository~\cite{examples-repo}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{scheme.pdf}
\caption{Scheme of the data processing paths realized with the J-PET Framework for different cases of analysis of data from the \jpet{} prototypes and the corresponding Monte Carlo simulations. Each gray rectangle represents a single module whereas arrows denote the flow of data represented as abstract objects.}
\label{fig:large_scheme}
\end{figure}
\subsubsection{Handling of multiple data sources}\label{sec:inputs}
The Framework provides a generic input-output mechanism that through simple extensions allow for processing of data from various sources e.g. low-level data in a binary format from a tomographic data acquisition system, textual representations of complete photomultiplier (PMT) signal waveforms collected using a serial data analyzer at detector testing stages, as well as high-level tomography-specific structures e.g. sinograms or lines-of-response. Other extensions of the input interface feature using results of Monte Carlo simulations in place of data as described in section~\ref{sec:mc}.
Figure~\ref{fig:large_scheme} presents how inputs from various data sources are unified at higher analysis levels so that more abstract steps of reconstruction can act on data independently of their origin. PMT signals recorded with a serial data analyzer, for example, correspond to PMT signal representations already assembled from single Time-to-Digital Converter (TDC) signals in case of the TRB3-based data acquisition system and are thus injected to the analysis chain at the corresponding level, i.e.\ before a module pairing PMT signals from the same detection module to identify photon interactions.
\subsubsection{Input/Output mechanisms}\label{sec:io}
Besides the source-specific data formats handled by dedicated wrapper modules, the \jpet{} Framework relies on binary internally-compressed data format provided by the ROOT package, widely adopted in both particle physics and nuclear medicine research. The framework provides automatic handling of input and output files for standard data analysis modules, abstracting the actual storage away from the analysis or reconstruction logic. User code is only responsible for deciding whether an entry processed by a module should be preserved or discarded. Depending on the option chosen by the user, output from every analysis module is either saved to a separate file in the ROOT format or directly fed as input to the subsequent module in the chain. While the former is useful at the stages of testing the analysis, the latter approach allows to create a pipeline of analysis modules minimizing I/O load as the only output saved to disk is the one produced by the last module in the chain which typically corresponds to the most filtered data stream where the data volume is reduced by 1-2 orders of magnitude with respect to the raw input. This is particularly important when multiple analysis processes are operating on the same disk space, which is a common use case in data-driven parallel computing specific to particle physics and low-level PET tomography event filtering and reconstruction.
\subsubsection{Options}
The library provides multiple manners of loading optional information for any custom processing task. These collections of various parameters would be required for a successful reconstruction of PET data, giving i.e. descriptions of a experimental setup, measurement conditions, necessary calibrations or desired form of the output. The Framework provides the following interfaces for dynamic configuration:
\begin{itemize}
\item command line options (e.g. input file, configuration files, progress display)
\item \texttt{JSON} file with the description of experimental setup - parametrization of objects, that serve as data schema,
\item \texttt{JSON} file with user-provided options -
any custom settings to be used during execution of tasks, passed as named parameters of elementary C++ types
\end{itemize}
All the provided options are parsed and validated before execution of chain of tasks and
are accessible
during its processing.
\subsubsection{Data and parameter structures}\label{sec:param-data}
The library includes classes representing abstract entities
common for analysis of data from J-PET measurements.
Parameter Objects represent hardware parts of the detector,
along with their working parameters, in-setup placement and connections with other parts,
e.g.\
a single object per each plastic scintillator strip with its location in the detector or
a photomultiplier coupled to a given scintillator.
Data Objects are structures representing subsequent stages of reconstructed data --
from elementary ones containing only TDC time and data acquisition channel number, to a detailed reconstruction of a physical event or a line-of-response.
Data Objects refer to particular elements of the setup encapsulated in Parameter Objects where the physical signals have originated. Moreover, mapping of connections between such components imposes relations between Parameter Objects themselves. These relations are implemented using persistent object references (\textit{TRef}) provided by the ROOT libraries~\cite{root} which ensure $\mathcal{O}(1)$ lookup of corresponding elements as well as persistence of the relations across file storage.
On the user side, encapsulation of data and setup properties into abstract objects allows
for definition of reconstruction and analysis logic even by users without programming proficiency which is one of the objectives of the \jpet{} Framework. Listing~\ref{lst:event_finder} demonstrates the interplay between Data (\texttt{JPetHit} and \texttt{JPetEvent}) and Parameter Objects (\texttt{Scintillator} and \texttt{BarrelSlot}) in a simple task.
\subsubsection{Setup description}\label{sec:setup}
Since experimental setup and its conditions can change from one measurement to another, the set of parameters describing it must be generated dynamically.
The library provides dedicated tools to handle a setup representation in a form of a configuration \texttt{JSON} file.
Based on its content, the Framework generates the collection of Parameter Objects (see section~\ref{sec:param-data}) together with relations between them expressed in a standardized format.
Once this file is parsed, a Parameter Bank encapsulating the latter is embedded in all output data files to allow for their further stand-alone analysis.
\subsubsection{Processing control and logging}
Each execution instance of any application based on the Framework environment, generates a log file with a unique name. By default all input parameters and options are stored in the log file, moreover each task can produce custom messages with one of appropriate tags: \texttt{INFO}, \texttt{DEBUG}, \texttt{ERROR}.
User Tasks classes can use tools for creating control histograms. All such objects are then automatically stored in the output file. An usage example can be found in code snippet \ref{lst:stats}.
\subsubsection{Compressed input files}
It is also possible to provide a raw data file in a compressed format; in that case a task is added by default, that simply decompresses the input file before any other procedures begin. Supported formats are: \texttt{xz}, \texttt{gz}, \texttt{bz2}, \texttt{zip}.
\subsubsection{Handling of binary data format}
The library can read raw data input files provided from the scanner data acquisition or from digital oscilloscope measurements. Binary data is transformed with dedicated tasks in the \texttt{ROOT} format, making it available for further processing by the consecutive tasks in the stream.
\subsubsection{Iterative tasks}
The structure of task chain allows the implementation of iterative tasks schemes, in which a module can be executed in a loop till a given condition is fulfilled. The stopping condition can be based on desired number of consecutive iterations or on the return value of the function defined by the user. This functionality is especially useful for optimization goals, i.e. refining detector calibration constants or estimation of event classification parameters.
\subsubsection{Interfaces to Monte Carlo simulation packages}
\label{sec:mc}
Testing and debugging of data analysis modules is often supported by using Monte Carlo-simulated events in place of actual data. To this end, the \jpet{} Framework offers interfaces to two Monte Carlo simulation packages: the custom \jpet{} MC simulation software~\cite{mc-repo} based on the Geant4 toolkit~\cite{geant4} as well as the GATE package for simulation of PET and SPECT tomography~\cite{Gate_2004}.
MC-simulated events are wrapped into the same data structures as data so that analysis modules intended to process experimental data can be applied transparently to the simulation results. At the same time, all MC-specific event information is preserved and accessible on demand.
\subsubsection{Event Display}
\label{sec:event-display}
J-PET Event Display~\cite{event-display-repo} is a visualization tool based on the \jpet{} library. It can load files with the Framework data structures to visualize the reconstructed PET data in an event-by-event manner at different phases of the processing. Information on input geometry of the detector is provided by the same configuration files in the \texttt{JSON} format used for reconstruction of data described in section~\ref{sec:setup}. A usage example of the Event Display is shown in Fig.~\ref{fig:eventDisplay}.
\subsection{Development philosophy, testing and continuous integration}
The J-PET Framework developer community is trying to consistently adopt good coding rules and practices in the development routine to assure the quality of the software. In particular, any new code before being merged into official repository must be reviewed and accepted by at least one person not being the author. Moreover, it must pass a set of unit and integration tests defined for the platform.
The contributors are strongly encouraged to add unit tests together with new classes and to format the code consistently using the clang-format tool.
The Continuous Integration process is integrated with the project Github repository. Any new pull request launches automatic set of tests based on the Travis~\cite{travis} and Jenkins~\cite{jenkins} services.
The unit tests are operated by the Travis system, while larger integration tests, which typically require some input data, are run by a dedicated Jenkins server. Both services deliver a detailed report about possible failures.
The testing system is fully automatized on the servers and can be launched manually for local testing.
Issue and bug tracking is performed with the Redmine service which also serves also as a user support forum. The reference guide is automatically generated from the code using the Doxygen tool
and is available online~\cite{framework_doxygen}.
Additionally, an analysis user guide is provided in the repository and it is being updated with every new version of the Framework.
\subsection{Sample code snippets analysis}
\label{snippets}
Listing~\ref{lst:main} presents the instance of \texttt{JPetManager} registering user tasks to form a chain of procedures. With the following \texttt{useTask} method, the user is specifying the input and output data format of each tasks. In the example, the output of the first task serves as input for the second one. The processing of all algorithms with the provided arguments is triggered by the \texttt{run} method. This simple construction allows to create an analysis from custom building blocks even for a user with little programming experience.
Listing~\ref{lst:event_finder} presents a snippet of an analysis module identifying 2-photon coincidence events in a stream of single recorded photon interactions (referred to as \textit{hits}).
Listing~\ref{lst:stats} shows a basic usage of the statistics facilities for creation of histograms to be filled during data analysis.
\begin{lstlisting}[language=C++, label={lst:main}, caption=Exemplary main class of a program based on J-PET Framework library.]
#include <JPetManager/JPetManager.h>
#include "Task1.h"
#include "Task2.h"
using namespace std;
int main (int argc, const char * argv []) {
JPetManager& manager = JPetManager::getManager();
manager.registerTask<Task1>("Task1");
manager.registerTask<Task2>("Task2");
manager.useTask("Task1", "data.input", "data.type1");
manager.useTask("Task2", "data.type1" , "data.type2");
manager.run(argc, argv);
}
\end{lstlisting}
\begin{lstlisting}[
language=C++,
label={lst:event_finder},
caption=Exemplary naive procedure of finding 2-photon coincidence events demonstrating the ease of operations on the data structures provided by the Framework.
]
for(JPetHit& hit_1: gamma_hits){
for(JPetHit& hit_2: gamma_hits){
// find double coincidences within 5000 ps
if(hit_2.getTime() - hit_1.getTime() < 5000.){
// check if the two gamma interactions were recorded
// in distinct scintillators of the setup
if(hit_1.getScintillator() != hit_2.getScintillator()){
// check if locations of the two detection modules
// differ by more than 160 degrees in azimuthal angle
if(fabs(hit_1.getBarrelSlot().getTheta() -
hit_2.getBarrelSlot().getTheta()) > 160.){
// reconstruct e+e- -> 2gamma annihilation point
TVector3 point =
EventCategorizerTools::calculateAnnihilationPoint(hit_1,hit_2);
// assemble an event containing the two hits
JPetEvent event;
event.addHit(hit_1);
event.addHit(hit_2);
event.setEventType(JPetEventType::k2Gamma);
// automatically store the event in the output file
// or pass on to the next analysis module
fOutputEvents->add<JPetEvent>(event);
}
}
}
}
}
\end{lstlisting}
\begin{lstlisting}[language=C++, label={lst:stats}, caption={Example of using tools for creating filling histograms, that are stored in output files.}]
// Creating histogram with JPetStatistics class
getStatistics().createHistogram(
new TH1F("hit_z_pos", "Z-axis position of photon interaction in plastic scintillator", 200, -25.0, 25.0));
getStatistics().getHisto1D("hit_z_pos")->GetXaxis()->SetTitle("Z-axis position [cm]");
getStatistics().getHisto1D("hit_z_pos")->GetYaxis()->SetTitle("Number of Hits");
// Invoking a histogram by title from statistics interface for filling
getStatistics().getHisto1D("hit_z_pos")->Fill(hit.getPosZ())
\end{lstlisting}
\section{Illustrative Examples}
\label{examples}
In this section we present two examples developed with the Framework library.
The first application can be used to perform tests of a prototype PET scanner based on the Monte Carlo simulations of various phantoms.
The simulated scanner was built from a cylindrical layer (radius of 42.5, length of 50 cm) of 384 plastic strips (more details about the MC simulations of the \jpet scanners can be found in~\cite{PMB2018}).
The program loads the data sample generated by the GATE Monte Carlo simulation package and transforms it by smearing the measured observables such as time, energy and position
based on the parametrization of experimental uncertainties determined for a given prototype scanner. This procedures mimics the real measurement effects. Next, the data is reconstructed and finally transformed to a sinogram, which
serves as an input to the image reconstruction task implementing the Time-of-Flight Filtered-Backprojection algorithm (see Figure~\ref{fig:toffbp}) or can be send to an external image reconstruction package such as STIR~\cite{stir}, CASTOR~\cite{castor} or QETIR~\cite{qetir}. All operations are implemented as consecutive tasks executed by the framework.
\begin{figure}[ht]
\includegraphics[width=0.32\textwidth]{TOFFBP-RamLak.png}
\includegraphics[width=0.32\textwidth]{TOFFBP-SheppLogan.png}
\includegraphics[width=0.32\textwidth]{TOFFBP-Hamming.png}
\centering
\caption{Example image reconstructed with the Time-of-Flight Filtered-Backprojection algorithm with various filters: Ramlak (left), Shepp-Logan (center), Hamming (right). The input sample is based on Monte Carlo simulations of NEMA IEC phantom performed with the GATE package~\cite{Gate_2004},and further processed by the Framework-based parser which applies the experimental parametrizations to fully imitate a measurement of the scanner.}
\label{fig:toffbp}
\end{figure}
The second program implements a full reconstruction chain for the real data collected by the 3-layer J-PET scanner. The example reconstruction and analysis is based on the test measurement with the radioactive source placed in the center
of the scanner. The reconstructed results are visualized with the J-PET Event Display tool (see Figure~\ref{fig:eventDisplay}).
\begin{figure}[ht]
\includegraphics[width=0.33\textwidth]{display-unrolled.png}
\includegraphics[width=0.33\textwidth]{display-front.png}
\includegraphics[width=0.32\textwidth]{display-3d.png}
\centering
\caption{Screenshots of J-PET Event Display \cite{event-display-repo}, (see \ref{sec:event-display}). This example visualizes result of data reconstruction acquired from a test measurement with radioactive source in the center of the scanner. Figures show the model of the J-PET detector, consisting of plastic strips, arranged as 3 concentric cylinders. The right image shows a 3D view, center is a frontal view and left shows these detector elements with positions of reconstructed interactions marked with red stars. In the center and right figures red lines connect pairs of reconstructed positions, that satisfy selection criteria aiming in preparing the sample of events of electron-positron annihilation.}
\label{fig:eventDisplay}
\end{figure}
\section{Impact}
\label{sec:impact}
Flexibility and robustness of the J-PET Framework library allowed it to be adopted as the main software platform of the J-PET project. The software and applications constructed based on this package have been used for many kinds of scientific studies involving data analyses from the J-PET tomography scanner and will be utilized for future analyses in the fundamental resarch and in the development of various PET scanners prototypes.
\begin{itemize}
\item performance assessment of novel PET scanners ~\cite{Monika-Acta},
\item time calibration techniques for PET scanners~\cite{MS2-time-calib, kd_calib:2020},
\item parametrization of deposited energy in plastic scintillators by Time-over-Threshold measurements~\cite{sushil-tot},
\item implementation of PET image reconstruction techniques such as Kernel Density Estimation, Maximum Likelihood Expectation Maximization ~\cite{MLEM2015} and Time-of-Flight Filtered-Backprojection,
\item development of plastic-based prototype a Positron Emission Mammography scanner~\cite{Shivani:2019mab},
\item studies in positronium annihilation reconstruction and imaging~\cite{alek:pra2016, Jasinska-Moskal2017, NATURE},
\item fundamental research on photon polarization and quantum entanglement~\cite{entanglement2,bHiesmayr},
\item tests of discrete symmetries~\cite{juhi-tsymmetry, alek-tsymmetry},
\item mirror matter searches~\cite{mirror:2019}.
\end{itemize}
The Framework software platform is currently used by scientists from the Jagiellonian University in Kraków, National Centre for Nuclear Research in Warsaw and INFN Laboratori Nazionali di Frascati and has been successfully deployed on different scales starting from laptops and personal PC-s, through mid-size computing clusters to HPC Swierk cluster.
\section{Conclusions}
\label{sec:conclusions}
In this article we presented the features and range of possible applications of the \jpet{} Framework, a C++ based library for data processing and analysis for PET tomography and for fundamental searches.
The Framework provides tools for the implementation of a wide range
of data reconstruction and calibration procedures
as well as user-level data analyses
and preparation of input for higher-level medical imaging software.
The platform is focused on flexibility in adjusting to dynamically changing prototyping environments and asserting ease of implementation of the required logic by users without programming proficiency while maintaining high processing performance.
Currently, use cases of the \jpet{} Framework span among various data analyses and imaging application of the first \jpet{} device.
In the near future, a new generation light-weight modular \jpet{} scanner with fully digital readout and high mobility will be commissioned along with sibling devices such as a similar-technology-based mammography scanner. The software platform is currently being extended with modules specific to the new devices, which will allow for reusing the higher-level analysis steps with data from new hardware setups.
Despite
having originated solely for the purpose of analysis of data from a single
setup,
the recent expansion of the scope of its usage in the context of \jpet{} demonstrates that
the flexibility of its architecture allows for use in a wider range of experiments related to nuclear medicine and fundamental studies.
.
\section{Conflict of Interest}
No conflict of interest exists:
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
\section*{Acknowledgements}
This work was supported in part by the Foundation for Polish Science through the Grant No. TEAM POIR.04.04.00-00-4204/17.
\bibliographystyle{elsarticle-num}
|
2002.10101
|
\section{Introduction}
Transformer~\cite{vaswani2017attention} has outperformed other methods on several neural language generation (NLG) tasks, like machine translation~\cite{deng2018alibaba}, text summarization~\cite{chang2018hybrid}, \textit{etc}.
Generally, Transformer is based on the \textit{encoder-decoder framework} which consists of two modules: an encoder network and a decoder network. The encoder encodes the input sentence into a sequence of hidden states, each of which corresponds to a specific word in the sentence. The decoder generates the output sentence word by word. At each decoding time-step, the decoder performs attentive read~\cite{Luong2015Effective,vaswani2017attention} to fetch the input hidden states and decides which word to generate.
As mentioned above, the decoding process of Transformer only relies on the representations contained in these hidden states. However, there is evidence showing that hidden states from the encoder in Transformer only contain local representations which focus on word level information.
For example, previous work~\cite{vaswani2017attention,devlin2018bert,song2020alignment} showed that these hidden states pay much attention to the word-to-word mapping; and the weights of attention mechanism, determining which target word will be generated, is similar to word alignment.
As \newcite{frazier1987sentence} pointed, the global information, which is about the whole sentence in contrast to individual words, should be involved in the process of generating a sentence. Representation of such global information plays an import role in neural text generation tasks.
In the recurrent neural network (RNN) based models~\cite{Bahdanau2015Neural}, \newcite{Chen2018} showed on text summarization task that introducing representations about global information could improve quality and reduce repetition. \newcite{lin2018de} showed on machine translation that the structure of the translated sentence will be more correct when introducing global information. These previous work shows global information is useful in current neural network based model. However, different from RNN~\cite{sutskever2014sequence,Cho2014Learning,Bahdanau2015Neural} or CNN~\cite{gehring2016convolutional,gehring2017convolutional}, although self-attention mechanism can achieve long distance dependence, there is no explicit mechanism in the Transformer to model the global representation of the whole sentence. Therefore, it is an appealing challenge to provide Transformer with such a kind of global representation.
In this paper, we divide this challenge into two issues that need to be addressed:
1). \textit{how to model the global contextual information?} and 2). \textit{how to use global information in the generation process?}, and propose a novel global representation enhanced Transformer (GRET) to solve them.
For the first issue, we propose to generate the global representation based on local word level representations by two complementary methods in the encoding stage.
On one hand, we adopt a modified \textit{capsule network}~\cite{sabour2017dynamic} to generate the global representation based the features extracted from local word level representations.
The local representations are generally related to the word-to-word mapping, which may be redundant or noisy. Using them to generate the global representation directly without any filtering is inadvisable.
Capsule network, which has a strong ability of feature extraction~\cite{zhao2018investigating}, can help to extract more suitable features from local states. Comparing with other networks, like CNN~\cite{krizhevsky2012imagenet}, it can see all local states at one time, and extract feature vectors after several times of deliberation.
On the other hand, we propose a \textit{layer-wise recurrent structure} to further strengthen the global representation. Previous work shows the representations from each layer have different aspects of meaning~\cite{peters2018deep,dou2018exploiting}, e.g. lower layer contains more syntactic information, while higher layer contains more semantic information. A complete global context should have different aspects of information.
However, the global representation generated by the capsule network only obtain intra-layer information.
The proposed layer-wise recurrent structure is a helpful supplement to combine inter-layer information by aggregating representations from all layers.
These two methods can model global representation by fully utilizing different grained information from local representations.
For the second issue, we propose to use \textit{a context gating mechanism} to dynamically control how much information from the global representation should be fused into the decoder at each step.
In the generation process, every decoder states should obtain global contextual information before outputting words. And the demand from them for global information varies from word to word in the output sentence. The proposed gating mechanism could utilize the global representation effectively to improve generation quality by providing a customized representation for each state.
Experimental results on four WMT translation tasks, and LCSTS text summarization task show that our \method model brings significant improvements over a strong baseline and several previous researches.
\section{Approach}
Our \method model includes two steps: modeling the global representation in the encoding stage and incorporating it into the decoding process. We will describe our approach in this section based on Transformer~\cite{vaswani2017attention}.
\subsection{Modeling Global Representation}
In the encoding stage, we propose two methods for modeling the global representation at different granularity.
We firstly use capsule network to extract features from local word level representations, and generate global representation based on these features.
Then, a layer-wise recurrent structure is adopted subsequently to strengthen the global representation by aggregating the representations from all layers of the encoder. The first method focuses on utilizing word level information to generate a sentence level representation, while the second method focuses on combining different aspects of sentence level information to obtain a more complete global representation.
\paragraph{Intra-layer Representation Generation} We propose to use \textit{capsules with dynamic routing} to extract the specific and suitable features from the local representations for stronger global representation modeling, which is an effective and strong feature extraction method \cite{sabour2017dynamic,zhang2018sentence}\footnote{Other details of the Capsule Network are shown in \newcite{sabour2017dynamic}.}.
Features from hidden states of the encoder are summarized into several capsules, and the weights (routes) between hidden states and capsules are updated by dynamic routing algorithm iteratively.
\begin{algorithm}[tb]
\caption{Dynamic Routing Algorithm}
\label{alg:dynamic routing}
\begin{algorithmic}[1]
\State \textbf{procedure}: \textsc{Routing}($\textbf{H}$, $r$)
\For{$i$ in input layer and $k$ in output layer }
\State $b_{ki}\leftarrow0$;
\EndFor
\For{$r$ iterations}
\For{$k$ in output layer}
\State $\textbf{c}_{k}\leftarrow\text{softmax}(\textbf{b}_{k})$;
\EndFor
\For{$k$ in output layer}
\State $\textbf{u}_{k}\leftarrow q(\sum_{i}^{I}c_{ki}\textbf{h}_{i})$;
\\ \Comment{$\textbf{H}=\{\textbf{h}_{1},\cdots,\textbf{h}_{i},\cdots\}$}
\EndFor
\For{$i$ in input layer and $k$ in output layer}
\State $b_{ki}\leftarrow b_{ki}+\textbf{h}_{i}\cdot\textbf{u}_{k}$;
\EndFor
\EndFor
\State \textbf{return} $\textbf{U}$;
\Comment{$\textbf{U}=\{\textbf{u}_{1},\cdots,\textbf{u}_{k},\cdots\}$}
\end{algorithmic}
\end{algorithm}
Formally, given an encoder of the Transformer which has $M$ layers and an input sentence $\textbf{x}=\{x_{1}, \cdots, x_{i}, \cdots, x_{I}\}$ which has $I$ words.
The sequence of hidden states $\textbf{H}^{m}=\{\textbf{h}^{m}
_{1},\cdots,\textbf{h}^{m}_{i},\cdots,\textbf{h}^{m}_{I}\}$ from the $m^\text{th}$ layer of the encoder is computed by
\begin{align}
\textbf{H}^{m}=\text{LN}(\text{SAN}(\textbf{Q}^{m}_{e},\textbf{K}_{e}^{m-1},\textbf{V}_{e}^{m-1})),
\end{align}
where the $\textbf{Q}^{m}_{e}$, $\textbf{K}^{m-1}_{e}$ and $\textbf{V}^{m-1}_{e}$ are query, key and value vectors which are same as $\textbf{H}^{m-1}$, the hidden states from the $m-1^\text{th}$ layer. The $\text{LN}(\cdot)$ and $\text{SAN}(\cdot)$ are layer normalization function~\cite{ba2016layer} and self-attention network~\cite{vaswani2017attention}, respectively. We omit the residual network here.
Then, the capsules $\textbf{U}^{m}$ with size of $K$ are generated by $\textbf{H}^{m}$. Specifically, the $k^\text{th}$ capsule $\textbf{u}^m_{k}$ is computed by
\begin{align}
\textbf{u}^m_{k}&=q(\sum_{i}^{I}c_{ki}\hat{\textbf{h}}^{m}_{i}),~c_{ki} \in \textbf{c}_{k}, \\
\hat{\textbf{h}}^{m}_{i}& = \textbf{W}_{k}\textbf{h}^{m}_{i},
\end{align}
where $q(\cdot)$ is non-linear squash function~\cite{sabour2017dynamic}:
\begin{align}
\text{squash}(\textbf{t})=\frac{||\textbf{t}||^{2}}{1+||\textbf{t}||^{2}}\frac{\textbf{t}}{||\textbf{t}||},
\end{align}
and $\textbf{c}_{k} $ is computed by
\begin{align}
\textbf{c}_{k} = \text{softmax}(\textbf{b}_{k}),~\textbf{b}_{k} \in \textbf{B},
\end{align}
where the matrix $\textbf{B}$ is initialized by zero and whose row and column are $K$ and $I$, respectively. This matrix will be updated when all capsules are produced.
\begin{align}
\textbf{B} = \textbf{B} + \textbf{U}^{m\top} \cdot \textbf{H}^m.
\end{align}
The algorithm is shown in Algorithm \ref{alg:dynamic routing}.
The sequence of capsules $\textbf{U}^{m}$ could be used to generate the global representation.
\begin{figure}[t]
\centering
\includegraphics[scale = 0.40]{capsule.pdf}
\caption{\label{fig: sentence representation} The overview of generating the global representation with capsule network.}
\end{figure}
Different from the original capsules network which use a concatenation method to generate the final representation, we use an \textit{attentive pooling method} to generate the global representation\footnote{Typically, the concatenation and other pooling methods, e.g. mean pooling, could be used here easily, but they will decrease 0.1$\sim$0.2 BLEU in machine translation experiment.}. Formally, in the $m^\text{th}$ layer, the global representation is computed by
\begin{align}
\textbf{s}^{m}&=\text{FFN}(\sum_{k=1}^{K}a_{k}\textbf{u}^{m}_{k}),
\label{eq: ffn} \\
a_{k}&=\frac{\text{exp}(\hat{\textbf{s}}^{m} \cdot \textbf{u}^{m}_{k})}{\sum_{t=1}^{K}\text{exp}(\hat{\textbf{s}}^{m} \cdot \textbf{u}^{m}_{t})},
\end{align}
where $\text{FFN}(\cdot)$ is a feed-forward network and the $\hat{\textbf{s}}^{m}$ is computed by
\begin{align}
\textbf{s}^{m}=\text{FFN}(\frac{1}{K}\sum^{K}_{k=1}{\textbf{u}^{m}_{k}}). \label{eq: average}
\end{align}
This \textit{attentive} method can consider the different roles of the capsules and better model the global representation. The overview of the process of generating the global representation are shown in Figure~\ref{fig: sentence representation}.
\paragraph{Inter-layer Representation Aggregation} Traditionally, the Transformer model only fed the last layer's hidden states $\textbf{H}^{M}$ as representations of input sentence to the decoder to generate the output sentence. Following this, we can feed the last layer's global representation $\textbf{s}^{M}$ into the decoder directly. However, current global representation only contain the intra-layer information, the other layers' representations are ignored, which were shown to have different aspects of meaning in previous work~\cite{wang2018multi,dou2018exploiting}.
Based on this intuition, we propose a \textit{layer-wise recurrent structure} to aggregate the representations generated by employing the capsule network on all layers of the encoder to model a complete global representation.
The layer-wise recurrent structure aggregates each layer's intra global state by a gated recurrent unit~\cite[GRU]{Cho2014Learning} which could achieve different aspects of information from the previous layer's global representation.
Formally, we adjust the computing method of $\textbf{s}^{m}$ by
\begin{align}
\textbf{s}^{m}=\text{GRU}(\text{ATP}(\textbf{U}^{m}),\textbf{s}^{m-1}),
\end{align}
where the $\text{ATP}(\cdot)$ is the attentive pooling function computed by Eq \ref{eq: ffn}-\ref{eq: average}.
The GRU unit can control the information flow by forgetting useless information and capturing suitable information, which can aggregate previous layer's representations usefully.
The layer-wise recurrent structure could achieve a more exquisite and complete representation. Moreover, the proposed structure only need one more step in the encoding stage which is not time-consuming.
The overview of the aggregation structure is shown in Figure \ref{fig: deep}.
\begin{figure}
\centering
\includegraphics[scale = 0.40]{figure_deep.pdf}
\caption{\label{fig: deep} The overview of the layer-wise recurrent structure.}
\end{figure}
\subsection{Incorporating into the Decoding Process}
Before generating the output word, each decoder state should consider the global contextual information. We combine the global representation in decoding process with an additive operation to the last layer of the decoder guiding the states output true words. However, the demand for the global information of each target word is different. Thus, we propose \textit{a context gating mechanism} which can provide specific information according to each decoder hidden state.
\begin{figure}[t]
\centering
\includegraphics[scale = 0.40]{figure_gate.pdf}
\caption{\label{fig: dg} The context gating mechanism of fusing the global representation into decoding stage.}
\end{figure}
Specifically, given an decoder which has $N$ layers and the target sentence $\textbf{y}$ which has $J$ words in the training stage, the hidden states $\textbf{R}^{N}=\{\textbf{r}^{N}_{1},\cdots,\textbf{r}^{N}_{j},\cdots,\textbf{r}^{N}_{J}\}$ from the $N^\text{th}$ layer of the decoder is computed by
\begin{align}
\textbf{R}^{N}=\text{LN}(&\text{SAN}(\textbf{Q}^{N}_{d},\textbf{K}_{d}^{N-1},\textbf{V}_{d}^{N-1}) \nonumber
\\&+\text{SAN}(\textbf{Q}^{N}_{d},\textbf{K}_{e}^{M},\textbf{V}_{e}^{M})),
\end{align}
where $\textbf{Q}^{N}_{d}$, $\textbf{K}^{N-1}_{d}$ and $\textbf{V}^{N-1}_{d}$ are hidden states $\textbf{R}^{N-1}$ from $N-1^{\text{th}}$ layer. The $\textbf{K}_{e}^{M}$ and $\textbf{V}_{e}^{M}$ are same as $\textbf{H}^{M}$. We omit the residual network here.
For each hidden state $\textbf{r}^{N}_{j}$ from $\textbf{R}^{N}$, the context gate is calculated by:
\begin{align}
\textbf{g}_{j}=\text{sigmoid}(\textbf{r}^{N}_{j},\textbf{s}^{M}).
\end{align}
The new state, which contains the needed global information, is computed by:
\begin{align}
\overline{\textbf{r}}^{N}_{j} = \textbf{r}^{N}_{j} + \textbf{s}^{M}_{j}*\textbf{g}.
\end{align}
Then, the output probability is calculated by the output layer's hidden state:
\begin{align}
P(y_{j}|y_{<j},\textbf{x})&=\text{softmax}(\text{FFN}(\overline{\textbf{r}}^{N}_{j})). \label{prob}
\end{align}
This method enables each state to achieve it's customized global information. The overview is shown in Figure \ref{fig: dg}.
\subsection{Training}
The training process of our \method model is same as the standard Transformer. The networks is optimized by maximizing the likelihood of the output sentence $\textbf{y}$ given input sentence $\textbf{x}$, denoted by $\mathcal{L}_\text{trans}$.
\begin{align}
\mathcal{L}_{\text{trans}}=\frac{1}{J}\sum_{j=1}^{J}\log P(y_{j}|y_{<j},\textbf{x}),
\end{align}
where $P(y_{j}|y_{<j},\textbf{x})$ is defined in Equation~\ref{prob}.
\section{Experiment}
\subsection{Implementation Detail}
\paragraph{Data-sets} We conduct experiments on machine translation and text summarization tasks. In machine translation, we employ our approach on four language pairs: Chinese to English (ZH$\rightarrow$EN), English to German (EN$\rightarrow$DE), German to English (DE$\rightarrow$EN), and Romanian to English (RO$\rightarrow$EN)~\footnote{http://www.statmt.org/wmt17/translation-task.html}. In text summarization, we use LCSTS~\cite{hu2015lcsts}~\footnote{http://icrc.hitsz.edu.cn/Article/show/139.html} to evaluate the proposed method. These data-sets are public and widely used in previous work, which will make other researchers replicate our work easily.
In machine translation, on the ZH$\rightarrow$EN task, we use WMT17 as training set which consists of about 7.5M sentence pairs. We use {\texttt{newsdev2017}} as validation set and \texttt{newstest2017} as test set which have 2002 and 2001 sentence pairs, respectively.
On the EN$\rightarrow$DE and DE$\rightarrow$EN tasks, we use WMT14 as training set which consists of about 4.5M sentence pairs. We use {\texttt{newstest2013}} as validation set and \texttt{newstest2014} as test set which have 2169 and 3000 sentence pairs, respectively.
On the RO$\rightarrow$EN task, we use WMT16 as training set which consists of about 0.6M sentence pairs. We use {\texttt{newstest2015}} as validation set and \texttt{newstest2016} as test set which has 3000 and 3002 sentence pairs, respectively.
In text summarization, following in \newcite{hu2015lcsts}, we use PART I as training set which consists of 2M sentence pairs. We use the subsets of PART II and PART III scored from 3 to 5 as validation and test sets which consists of 8685 and 725 sentence pairs, respectively.
\begin{table*}[ht]\footnotesize
\centering
\begin{tabular}{l||c|c|c|c}
\toprule
Model & ZH$\rightarrow$EN&EN$\rightarrow$DE&DE$\rightarrow$EN&RO$\rightarrow$EN \\
\hline
$\text{Transformer}^{*}$~\cite{vaswani2017attention}&$-$&27.3&$-$&$-$\\
$\text{Transformer}^{*}$~\cite{hassan2018achieving}& 24.13&$-$&$-$&$-$\\
$\text{Transformer}^{*}$~\cite{gu2017non}&$-$&27.02&$-$&31.76 \\
\hline
$\text{DeepRepre}^{*}$~\cite{dou2018exploiting} & 24.76&28.78&$-$&$-$ \\
$\text{Localness}^{*}$~\cite{yang2018modeling} &24.96&28.54&$-$&$-$ \\
$\text{RelPos}^{*}$~\cite{shaw2018self} & 24.53&27.94&$-$&$-$ \\
$\text{Context-aware}^{*}$~\cite{yang2019context}& 24.67&28.26&$-$&$-$ \\
$\text{GDR}^{*}$~\cite{zheng2019dynamic}&$-$&28.10&$-$&$-$\\
\hline
Transformer &24.31&27.20&32.34&32.17 \\
\method & 25.53$^{\ddagger}$&28.46$^{\dagger}$&33.79$^{\ddagger}$&33.06$^{\ddagger}$ \\
\bottomrule
\end{tabular}
\caption{The comparison of our \method, Transformer baseline and related work on the WMT17 Chinese to English (ZH$\rightarrow$EN), WMT14 English to German (EN$\rightarrow$DE) and German to English (DE$\rightarrow$EN), and WMT16 Romania to English (RO$\rightarrow$EN) tasks (* indicates the results came from their paper, $\dagger/\ddagger$ indicate significantly better than the baseline ($p<0.05/0.01$)).}
\label{tab:mt_r}
\end{table*}
\begin{table*}[t]\footnotesize
\centering
\begin{tabular}{l||c|c|c}
\toprule
Model & ROUGE-1&ROUGE-2&ROUGE-L \\
\hline
$\text{RNNSearch}^{*}$~\cite{hu2015lcsts}&30.79&$-$&$-$ \\
$\text{CopyNet}^{*}$~\cite{gu2016incorporating}&34.4&21.6&31.3 \\
$\text{MRT}^{*}$~\cite{ayana2016neural} & 37.87&25.43&35.33 \\
$\text{AC-ABS}^{*}$~\cite{li2018actor} & 37.51&24.68&35.02 \\
$\text{CGU}^{*}$~\cite{Lin2018GlobalEF}&39.4&26.9&36.5\\
$\text{Transformer}^{*}$~\cite{chang2018hybrid}& 42.35 & 29.38&39.23\\
\hline
Transformer &43.14&29.26&39.72\\
\method & 44.77 & 30.96&41.21 \\
\bottomrule
\end{tabular}
\caption{The comparison of our \method, Transformer baseline and related work on the LCSTS text summarization task (* indicates the results came from their paper).}
\label{tab: lcsts}
\end{table*}
\paragraph{Settings} In machine translation, we apply byte pair encoding (BPE)~\cite{sennrich2015neural} to all language pairs and limit the vocabulary size to 32K. In text summarization, we limit the vocabulary size to 3500 based on the character level. Out-of-vocabulary words and chars are replaced by the special token \emph{UNK}.
For the Transformer, we set the dimension of the input and output of all layers as 512, and that of the feed-forward layer to 2048.
We employ 8 parallel attention heads. The number of layers for the encoder and decoder are 6.
Sentence pairs are batched together by approximate sentence length. Each batch has 50 sentence and the maximum length of a sentence is limited to 100.
We set the value of dropout rate to 0.1. We use the Adam~\cite{kingma2014adam} to update the parameters, and the learning rate was varied under a warm-up strategy with 4000 steps~\cite{vaswani2017attention}. Other details are shown in \newcite{vaswani2017attention}. The number of capsules is set 32 and the default time of iteration is set 3. The training time of the Transformer is about 6 days on the DE$\rightarrow$EN task. And the training time of the \method model is about 12 hours when using the parameters of baseline as initialization.
After the training stage, we use beam search for heuristic decoding, and the beam size is set to 4.
We measure translation quality with the NIST-BLEU~\cite{Papineni2002bleu} and summarization quality with the ROUGE~\cite{lin2004rouge}.
\subsection{Main Results}
\paragraph{Machine Translation} We employ the proposed \method model on four machine translation tasks. All results are summarized in Table \ref{tab:mt_r}.
For fair comparison, we reported several Transformer baselines with same settings reported by previous work~\cite{vaswani2017attention,hassan2018achieving,gu2017non} and researches about enhancing local word level representations~\cite{dou2018exploiting,yang2018modeling,shaw2018self,yang2019context}.
The results on the WMT17 ZH$\rightarrow$EN task are shown in the second column of Table \ref{tab:mt_r}.
The improvement of our \method model could be up to 1.22 based on a strong baseline system, which outperforms all previous work we reported.
To our best knowledge, our approach attains the state-of-the-art in relevant researches.
Then, the results on the WMT14 EN$\rightarrow$DE and DE$\rightarrow$EN tasks, which is the most widely used data-set recently, are shown in the third and fourth columns.
The \method model could attain 28.46 BLEU (+1.26) on the EN$\rightarrow$DE and 33.79 BLEU (+1.45) on the DE$\rightarrow$EN, which are competitive results compared with previous studies.
To verify the generality of our approach, we also experiment it on low resource language pair of the WMT16 RO$\rightarrow$EN task. Results are shown in the last column.
The improvement of the \method is 0.89 BLEU, which is a material improvement in low resource language pair. And it shows that proposed methods could improve translation quality in low resource scenario.
Experimental results on four machine translation tasks show that modeling global representation in the current Transformer network is a general approach, which is not limited by the language or size of training data, for improving translation quality.
\begin{table*}[t]\footnotesize
\centering
\begin{tabular}{l|ccc||cccc}
\toprule
Model&Capsule&Aggregate&Gate&\#Param&Inference&BLEU&$\Delta$\\
\hline
Transformer & $-$& $-$& $-$& 61.9M& 1.00x &27.20 & $-$\\
\hline
\multirow{8}{*}{\textit{Our Approach}}& & & & 61.9M& 0.99x&27.39&+0.19\\
&\checkmark & & & 63.6M& 0.87x&28.02&+0.82\\
&\checkmark &\checkmark & & 68.1M& 0.82x&28.32&+1.02\\
&\checkmark & &\checkmark & 63.6M& 0.86x&28.23&+1.03\\
&&\checkmark & & 66.6M& 0.95x&27.81&+0.61\\
&&\checkmark &\checkmark & 66.8M& 0.93x&27.76&+0.56\\
&&&\checkmark & 62.1M& 0.98x&27.53&+0.33\\
&\checkmark &\checkmark &\checkmark & 68.3M& 0.81x&28.46&+1.26\\
\bottomrule
\end{tabular}
\caption{Ablation study on the WMT14 English to German (EN$\rightarrow$DE) machine translation task.}
\label{tab: ablation}
\end{table*}
\paragraph{Text Summarization} Besides machine translation, we also employ proposed methods in text summarization, a monolingual generation task, which is an important and typical task in natural language generation.
The results are shown in Table \ref{tab: lcsts}, we also reports several popular methods in this data-set as a comparison. Our approach achieves considerable improvements in ROUGE-1/2/L (+1.63/+1.70/+1.49) and outperforms other work with same settings. The improvement on text summarization is even more than machine translation. Compared with machine translation, text summarization focuses more on extracting suitable information from the input sentence, which is an advantage of the \method model.
Experiments on the two tasks also show that our approach could work on different types of language generation task and may improve the performance of other text generation tasks.
\begin{table}[t]\footnotesize
\centering
\begin{tabular}{l||c|c|c}
\toprule
Model &\#Param&Inference&BLEU \\
\hline
Transformer-Base &61.9M&1.00x&27.20 \\
\hline
GTR-Base &68.3M&0.81x&28.46\\
\hline
\hline
Transformer-Big &249M&0.59x&28.47 \\
\hline
\textsc{GReT}-Big &273M&0.56x&29.33\\
\bottomrule
\end{tabular}
\caption{The comparison of \method and Transformer with \textit{big} setting~\cite{vaswani2017attention} on the EN$\rightarrow$DE task.}
\label{tab: bigsize}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[scale = 0.25]{figure_it.png}
\caption{\label{fig: it} The comparison of the GTR with different number of capsules at different iteration times on the EN$\rightarrow$DE task.}
\end{figure}
\subsection{Ablation Study}
To further show the effectiveness and consumption of each module in our \method model, we make ablation study in this section.
Specifically, we investigate how the \textit{capsule network}, \textit{aggregate structure} and \textit{gating mechanism} affect the performance of the global representation.
The results are shown in Table \ref{tab: ablation}. Specifically, without the capsule network, the performance decreases 0.7 BLEU , which means extracting features from local representations iteratively could reduce redundant information and noisy. This step determines the quality of global representation directly. Then, aggregating multi-layers' representations attains 0.61 BLEU improvement. The different aspects of information from each layer is an excellent complement for generating the global representation.
Without the gating mechanism, the performance decreases 0.24 BLEU score which shows the context gating mechanism is important to control the proportion of using the global representation in each decoding step. While the \method model will take more time, we think it is worthwhile to improve generation quality by reducing a bit of efficiency in most scenario.
\begin{table}[t]\footnotesize
\centering
\begin{tabular}{l||ccc}
\toprule
\multirow{2}{*}{Model}&\multicolumn{3}{c}{Precision}\\
\cline{2-4}
&Top-200&Top-500&Top-1000 \\
\hline
\textit{Last}&43\% &52\%&64\% \\
\hline
\textit{Average}&49\%& 57\%&69\%\\
\hline
\method&63\%&74\%&81\%\\
\bottomrule
\end{tabular}
\caption{The precision from the bag-of-words predictor based on \method, last encoder state (\textit{Last}) and averaging all local states (\textit{Average}) on the EN$\rightarrow$DE task.}
\label{tab: pooling}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[scale = 0.25]{figure_sen_leng.png}
\caption{\label{fig: sl} The comparison of the GTR with different number of capsules at different iteration times on the EN$\rightarrow$DE task.}
\end{figure}
\subsection{Effectiveness on Different Model Settings}
We also experiment the \method model with \textit{big} setting on the EN$\rightarrow$DE task.
The \textit{big} model is far larger than above \textit{base} model and get the state-of-the-art performance in previous work~\cite{vaswani2017attention}.
The results are shown in Table \ref{tab: bigsize}, Transformer-Big outperforms Transformer-Base, while the GRET-Big improves 0.86 BLEU score comparing with the Transformer-Big.
It is worth to mention that our model with base setting could achieve a similar performance to the Transformer-Big, which reduces parameters by almost 75\% (68.3M VS. 249M) and inference time by almost 27\% (0.81x VS. 0.56x).
\begin{figure*}[t]
\centering
\includegraphics[scale = 0.9]{case.pdf}
\caption{\label{translation samples}Translation cases from Transformer and our \method model on the ZH$\rightarrow$EN task.}
\end{figure*}
\subsection{Analysis of the Capsule}
The number of capsules and the iteration time from dynamic routing algorithm may affect the performance of the proposed model. We evaluate the \method model with different number of capsules at different iteration times on the EN$\rightarrow$DE task. The results are shown in Figure \ref{fig: it}.
We can get two empirical conclusions in this experiment. First, the first three iterations can significantly improve the performance, while the results of more iterations (4 and 5) tend to stabilize. Second, the increase of capsule number (48 and 64) doesn't get a further gain. We think the reason is that most sentences are shorter than 50, just the suitable amount of capsules can extract enough features.
\subsection{Probing Experiment}
What does the global representation learn is an interesting question. Following \newcite{weng2017neural}, we do a probing experiment here. We train a \textit{bag-of-words predictor} by maximizing $ P(\textbf{y}_{bow}|\textbf{s}^{M})$, where $\textbf{y}_{bow}$ is an unordered set containing all words in the output sentence. The structure of the predictor is a simple feed-forward network which maps the global state to the target word embedding matrix.
Then, we compare the precision of target words in the top-\textit{K} words which are chosen through the predicted probability distribution\footnote{Experiment details are shown in \newcite{weng2017neural}.}. The results are shown in Table \ref{tab: pooling}, the global state from \method can get higher precision in all conditions, which shows that the proposed method can obtain more information about the output sentence and partial answers why the \method model could improve the generation quality.
\subsection{Analysis of Sentence Length}
To see the effectiveness of the global representation, we group the EN$\rightarrow$DE test set by the length of the input sentences to re-evaluate the models. The set is divided into 4 sets. Figure \ref{fig: sl} shows the results.
We find that our model outperforms the baseline in all categories, especially in the longer sentences, which shows that fusing the global representation may help the generation of longer sentences by providing more complete information.
\subsection{Case Study}
We show two real-cases on the ZH$\rightarrow$EN task to see the difference between the baseline and our model. These cases are shown in Figure \ref{translation samples}. The “Source” indicates the source sentence and the “Reference” indicates the human translation. The \textbf{bold font} indicates improvements of our model; and the \textit{italic font} indicates translation errors.
Each output from \method is decided by previous state and the global representation. So, it can avoid some common translation errors like over/under translation, caused by the strong language model of the decoder which ignores some translation information.
For example, the over translation of ``\textit{the cities of Hefei}'' in case 1 is corrected by the \method model. Furthermore, providing global information can avoid current state only focuses on the word-to-word mapping. In case 2, the vanilla Transformer translates the ``Moscow Travel Police'' according to the source input ``mosike lvyou jingcha''', but omits the words ``de renyuan zhaolu'', which leads it fails to translate the target word ``\textit{recruiting}''.
\section{Related Work}
Several work also try to generate global representation. In machine translation, \newcite{lin2018de} propose a deconvolutional method to obtain global information to guide the translation process in RNN-based model. However, the limitation of CNN can not model the global information well and there methods can not employ on the Transformer.
In text summarization, \newcite{Chen2018} also propose to incorporate global information in RNN-based model to reduce repetition. They use an additional RNN to model the global representation, which is time-consuming and can not get the long-dependence relationship, which hinders the effectiveness of the global representation.
\newcite{zhang2018sentence} propose a sentence-state LSTM for text representation. Our method shows an alternative way of obtaining the representation, on the implementation of the Transformer.
Many previous researches notice the importance of the representations generated by the encoder and focus on making full use of them. \newcite{wang2018towards} propose to use Capsule network to generate hidden states directly, which inspire us to use capsules with dynamic routing algorithm to extract specific and suitable features from these hidden states.
\newcite{wang2018multi,dou2018exploiting} propose to utilize the hidden states from multiple layers which contain different aspects of information to model more complete representations, which inspires us to use the states in multiple layers to enhance the global representation.
\section{Conclusion}
In this paper, we address the problem that Transformer doesn't model global contextual information which will decrease generation quality. Then, we propose a novel \method model to generate an external state by the encoder containing global information and fuse it into the decoder dynamically. Our approach solves the both issues of how to model and how to use the global contextual information.
We compare the proposed \method with the state-of-the-art Transformer model. Experimental results on four translation tasks and one text summarization task demonstrate the effectiveness of the approach. In the future, we will do more analysis and combine it with the methods about enhancing local representations to further improve generation performance.
\section*{Acknowledgements}
We would like to thank the reviewers for their insightful comments. Shujian Huang is the corresponding author. This work is supported by the National Key R\&D Program of China (No. 2019QY1806), the National Science Foundation of China (No. 61672277), the Jiangsu Provincial Research Foundation for Basic Research (No. BK20170074).
|
1403.2801
|
\section{Introduction}
Spectral-imaging data-cubes (SIDCs), from the new radio telescopes that are currently in various stages of construction or
commissioning -- Australian Square Kilometre Array Pathfinder (ASKAP) \citep{ASKAP..09},
Murchison Widefield Array (MWA) \citep{TINGAY}, LOFAR \citep{2013A&A...556A...2V},
MeerKAT~\citep{2009arXiv0910.2935B}, eVLA ~\citep{2011ApJ...739L...1P}
-- are expected to be in the range of tens of GBs to several TBs. The Square Kilometre
Array (SKA) Design Reference Mission, SKA Phase 1 \citep{SPDO2011}, defines at least
one survey, namely the ``Galaxy Evolution in the Nearby Universe: HI Observations", for
which the SKA pipeline will produce hundreds of SIDCs, of tens of terabytes each. In its first
year the SKA Phase 1 is expected to collect over 8 EB of data. The data volumes for the full
SKA are expected to be by at least an order of magnitude larger.
Even taking into account projected advances in HDD/SSD and network technologies, such
large SIDCs cannot be processed or stored on local user computers. Most of the imaging
data will be never seen by a human, but rather processed automatically \citep{2012MNRAS.421.3242W,
2012PASA...29..318P, 2012PASA...29..352J, 2012PASA...29..371W}. However,
there will still be a number of cases where visualisation is going to be
required, e.g. data quality control/assessment or detailed studies of individual objects.
Visual exploration of such large data volumes requires a new paradigm for the generation
and servicing of the higher level data products to the end-user. In this paper
we present a straw man of the functionality required to enable working with extremely
large radio astronomy imagery. We consider the JPEG2000 industry
standard as a suitable example that addresses many similar requirements, even though it was originally
developed for medical and remote sensing imagery.
Currently, most radio astronomy imaging data is stored and distributed in one of three
formats: FITS (Flexible Image Transport System) \citep{2010A&A...524A..42P};
CASA Image Tables~\footnote{\href{http://ascl.net/1107.013}{http://ascl.net/1107.013}} and newly developed by
LOFAR HDF5-based format \citep{2011ASPC..442...53A}.
FITS and HDF5 are, in general, single self-describing files containing the image data, as
well as metadata. CASA, on the other hand, uses a different approach representing any
data as a hierarchical structure of directories and files. CASA data is usually distributed
as an archived file created by using common archiving software, such as
\texttt{tar}\footnote{\href{http://en.wikipedia.org/wiki/Tar(computing)/}{http://en.wikipedia.org/wiki/Tar(computing)/}}. These formats provide
both, portability and access to image data. Currently,
image files or CASA tar-balls are normally retrieved from an archive and stored on a local
computer for exploration, analysis or processing purposes. Alternatively, a specified part
of an image-cube (cutout) is produced in one of the image formats, and presented to the
user as a download. If coterminous regions are required, several cutout files would be
produced and downloaded. The example of such a framework is Simple Image Access Protocol (SIAP)\footnote{\href{http://www. ivoa.net/Documents/SIA/}{http://www. ivoa.net/Documents/SIA/}}
of the International Virtual Observatory Alliance (IVOA)\footnote{\href{http://www.ivoa.net}{http://www.ivoa.net}} that provides a
uniform interface for retrieving image
data from a variety of astronomical image repositories.
By using SIAP the user can query compliant archives in a standardised manner and
retrieve image files in one or more formats, depending on the archive capabilities (e.g. FITS,
PNG or JPEG). The resulting files can then be stored on a local computer or a virtual network
storage device that is provided through VOSpace, which is another IVOA standard.
In the paper we discuss the use case of extremely large SIDCs in the context
of the limitations of the current standard astronomy file formats. We present the analysis
of the applicability of the approach taken in developing JPEG2000 standards to addressing
the new requirements of extremely large astronomical imagery. We also present some interesting
benchmarks from using JPEG2000 on large radio astronomy images.
The rest of paper is structured as follows. In Section~\ref{cha:use-case}, we discuss the specific requirements
of extremely large imaging. Section~\ref{cha:JPEG2000} discusses JPEG2000 standards, and
how they have addressed the requirements of extremely large imaging. We specifically discuss the image interaction
protocol in detail as the alternative to the used in astronomy cutout framework. Section~\ref{cha:Bench1}
presents benchmarks for JPEG2000 compression for radio astronomy images. In Section~\ref{cha:adopting}
we discuss the strategic approaches for improving the existing astronomy standards or
the adoption of new industry standards. Finally, we conclude in Section~\ref{cha:conc}.
\section{Use case for extremely large images} \label{cha:use-case}
\textit{ASKAP Science Data Archive: Requirements and Use Cases}\footnote{\href{http://www.atnf.csiro.au/management/atuc/2013dec/docs/ASKAP\_SW\_0017\_v0.8\_fordistribution.pdf}{CSIRO: ASKAP Science Data Archive: Requirements and Use Cases}}
indicates the individual data product sizes up to ~2.24TB for some science cases at the maximum interferometer
baseline 6 km. By 2020 ASKAP will be incorporated into the SKA1-Survey increasing the number of antennas from 36 to 96
and the maximum baseline to 50 km. The individual data products can be expected as large as 32TB.
Figure~\ref{fig:HDD-capacity} shows the capacity of HDD over the years. The projection assumes
the currently observed average growth of disc capacity at 32\% rate. The disc capacity increases
by factor of 20 every 10 years. If the same rate is sustained, by the time SKA1 is constructed
(2020-23), the individual disc capacity can be expected to be about 32TB.
Figure~\ref{fig:Max-sustained-bandwidth} shows the maximum sustained bandwidth of HHDs over the years \citep{Freitas2011}.
One can see that the improvement rate is rather moderate, about 4--5 times per decade. Figure~\ref{fig:hdd-read-time} shows the increasing read time of the entire HDD over the years.
Of course, such a read only indicates the time need to read the data sequentially. In many cases,
during the scientific data analysis, the data is accessed randomly. Been a mechanical device, HDD requires a time
to relocate the head to the required position, and wait until the disc turns into the right position before the needed data
can be accessed. This delay translates into a latency when the data need to be accessed randomly. Figure~\ref{fig:average-seek-time} shows the average seek time trend in HDD over the years. The improvement is very moderate, factor about 1.7 per decade.
Solid State Drives (SSD) is a promising technology that may help to overcome the I/O
bandwidth and latency problems in the future, though, at the time of writing this paper,
the market only offers 1TB SSD for the desktop/laptop computers, while the
largest HDD is 6TB.
These all means that working with the increasing in size datasets is likely going to be
increasingly difficult on personal computers if feasible at all. The software technologies allowing
an interactive work with the data stored on a server that can provide a fast parallel access
to the data are going to be important for the projects like SKA.
\begin{figure}
\centering
\includegraphics[width=90mm]{images/HDD-capacity.pdf}
\caption{Historical (blue diamonds) and projected (red squares) HDD capacity. Based on \href{http://www.storagenewsletter.com/rubriques/hard-disk-drives/milestones-in-hdd-capacity/}{http://www.storagenewsletter.com/rubriques/hard-disk-drives/milestones-in-hdd-capacity/} (accessed on 6/05/2014).}
\label{fig:HDD-capacity}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=90mm]{images/Max-sustained-bandwidth.jpg}
\caption{Maximum sustained bandwidth of HHDs over the years. Adopted from \citealt{Freitas2011}.}
\label{fig:Max-sustained-bandwidth}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=90mm]{images/hdd-read-time.pdf}
\caption{Read time of the entire HDD over the years with the projected time for 2020.}
\label{fig:hdd-read-time}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=90mm]{images/average-seek-time.jpg}
\caption{Average seek time trend. Adopted from \citealt{Freitas2011}.}
\label{fig:average-seek-time}
\end{figure}
In many cases, images are not required at their full resolution or fidelity. It should be possible
to access images at any of a multitude of reduced resolutions and/or reduced fidelities, according
to need, all from a single master image. Such a multi-resolution representation should not lead to
increased storage requirements, which are already high. For example, pyramid representations \citep{adelson1984pyramid}
possess the desired multi-resolution accessibility attributes, but inevitably expands the data.
We also note that not all of the image might be required at the same level of quality. Particular
regions of interest (ROI), e.g. containing an object to be studied, such as a galaxy or a nebula, may
need to be of much higher quality or resolution than others. Producing a cutout or many cutouts
is a limited solution, as cutouts completely remove the surrounding area. This is problematic
because the surrounding area provides context for reconstructing the relationship between
multiple objects in the field of view; moreover, the imagery within the surrounding area may
be of interest in its own right. A much better approach, in this case, would be to have an adaptively
encoded image, in which the regions of interest are encoded with higher fidelity/resolution than
the surrounding areas.
Even combining such advanced techniques as multiple resolution/fidelity and adaptive
encoding/transferring of ROI, the images can still be very large and require time to be
transferred to the client. In the case of visual exploration of data, it would make sense to
immediately transfer only the data that is required for display. Other parts of an image could
be requested and transferred on demand. The protocol should be intelligent enough for such a use case.
It would also be very useful to support the progressive transfer of an image from a server to the client. That is, the user should be
able to see the whole image of the selected region queried as soon as a first portion of the
data is transferred, while each successive portion of the data that is transferred should serve
to improve the quality of the displayed imagery. By contrast, many ``pyramid" techniques
possess only multi-resolution access, without progressive transfer, so that higher quality
representations must completely replace the lower quality ones, leading to substantial inefficiencies
and much higher transfer bandwidths. The client-server framework should be intelligent enough
not to transfer more data than is necessary for displaying or processing the content that is of interest.
Further, we will demonstrate that radio astronomy imaging data can be effectively compressed,
and the error due to the compression can be controlled. Compression significantly reduces the cost of storage,
operations and network bandwidth. However, it should be possible to access image regions,
resolutions and qualities directly from the compressed representation. If the imagery must
first be decompressed, and then re-compressed to address a users needs, this will place
unreasonable computational and memory demands upon the server, leading to a large
latency in service time and limited ability to serve a variety of users. Ideally, decompression
should occur only at the point where an image is to be displayed or used. Some usage cases
can expect large ratios in compression; examples include visual data exploration, draft mosaicing,
etc. Other use cases may be less tolerant to the loss of fidelity in the data, e.g. source finding. It
follows that multiple levels of compression should be available:
\begin{itemize}
\item high fidelity, potentially even numerically lossless compression, in which the decompressed
image is either an exact reproduction of the original uncompressed image, or differs by
considerably less than the intrinsic uncertainties in the imaging process; and
\item lossy compression, where the decompressed image exhibits higher levels of distortion that
are considered acceptable in exchange for corresponding reductions in communication bandwidth
or storage requirements.
\end{itemize}
As is suggested by the last point above, distortion metrics need to be defined and made available
to a user, so that the impact of lossy compression can be controlled. Such metrics involve:
\begin{itemize}
\item statistical characterisation of how the decompressed image can be expected to differ
from the original image; and
\item measures of the impact that different levels of distortion can be expected to have on some
specific purposes of data exploration, e.g. source finding.
\end{itemize}
The second point is especially important, given that much of the new Radio Astronomy science is done at a very low
signal-to-noise ratio (SNR).
\section{Case study: JPEG2000} \label{cha:JPEG2000}
In developing a contemporary protocol for working with extremely large astronomical images it is
useful to study how other communities have approached this problem. Indeed, large images are
not unique to astronomy, though new telescopes such as the SKA will be at the very extreme end of
the spectrum. Medical imaging, remote sensing, geographic information systems, virtual microscopy,
high definition video and other applications have long histories of development in the imaging domain.
The large size of images is not the only similarity. Multi-frequency, multi-component, volumetric data
sets, and metadata are common attributes in a range of existing imaging fields. A number of advanced
image/metadata formats and access layer protocols have been developed over the years\footnote{\href{http://en.wikipedia.org/wiki/Comparison\_of\_graphics\_file\_formats}{http://en.wikipedia.org/wiki/Comparison\_of\_graphics\_file\_formats}}. Some of the formats use wavelet encoding
that enables not only efficient compression but also advanced options for interaction with the image
data (e.g. MrSID\footnote{\href{http://en.wikipedia.org/wiki/MrSID}{http://en.wikipedia.org/wiki/MrSID}}, JPEG2000\footnote{\href{http://www.jpeg.org/jpeg2000/}{http://www.jpeg.org/jpeg2000/}}, or ECW\footnote{\href{http://en.wikipedia.org/wiki/ECW\_(file\_format)}{http://en.wikipedia.org/wiki/ECW\_(file\_format)}}).
One of those, namely JPEG2000, has been developed into a comprehensive royalty free industry standard -- ISO/IEC 15444.
Due to the specific focus of the standard on the large imagery, instead of the consumer photography,
the standard has become widely adopted by the industries, such as medical imaging \citep{1028146},
meteorology and remote sensing \citep{1294863}, Sun \citep{JHelioviewer} and planetary imaging \citep{Powell2010},
microscopy~\citep{2014Microscopy}, etc. We believe that the astronomy community may benefit from this development,
and learn from those industries that had faced similar challenges before astronomy.
JPEG2000 is an image compression standard and coding system created by the Joint Photographic
Experts Group committee and published as the international standard JPEG2000 \citep{Taubman02} in 2000.
The standard was developed to address weaknesses in existing image compression standards and
provide new features, specifically addressing the issue of working with large images. Considerable
effort has been made to ensure that the JPEG2000 codec can be implemented free of royalties.
Today, there is a growing level of support for the JPEG2000 standard, through both proprietary and
open source software libraries such as: OpenJPEG\footnote{\href{http://www.openjpeg.org}{http://www.openjpeg.org}}, JasPer\citep{JasPer},
Aware\footnote{\href{http://www.aware.com/imaging/jpeg2000sdk.html}{http://www.aware.com/imaging/jpeg2000sdk.html}}. JPEG2000 has been successfully used
in a number of astronomy applications already, including the HiRISE (high resolution Mars imaging)
project \citep{Powell2010} and JHelioviewer (high resolution Sun images) \citep{JHelioviewer}.
The following key objectives were considered during the development of the standard. It was expected
to allow efficient lossy and lossless compression within a single unified coding framework as well as to
provide superior image quality, both objectively and subjectively, at
high and low bit rates. It was expected to support additional features such as: ROI coding, a more
flexible file format, and, at the same time, to avoid excessive computational and memory complexity,
and excessive need for bandwidth to view an image remotely.
The main advantage offered by the approach used in JPEG2000 is the significant flexibility of its
codestream. The codestream obtained after compression of an image with JPEG2000 is scalable,
meaning that it can be decoded in a number of different ways. For instance, by truncating the
codestream at any point, a lower resolution or signal-to-noise ratio representation of the image
can be attained; moreover, the truncated representation remains efficient, in terms of the tradeoff
that it represents between fidelity and compressed size. By ordering the codestream in various
ways, applications can exploit this so-called ``scalability" attribute to achieve significant performance
benefits \citep{Taubman02}.
The following main features of JPEG2000 make it an attractive approach for astronomy:
\begin{itemize}[leftmargin=*]
\item High compression performance, substantially superior to JPEG.
\item Availability of multi-component transforms, including arbitrary inter-component wavelet transforms
and arbitrary linear transforms (e.g., KLT, block-wise KLT, etc.), with both reversible and irreversible versions.
\item Multiple resolution representation.
\item Progressive transmission (or recovery) by fidelity or resolution, or both.
\item Lossless and lossy compression in a single compression architecture. Lossless compression
is provided by the use of a reversible integer wavelet transform and progressive transmission of a lossless
representation provides lossy to lossless refinement.
\item Random codestream access and processing, also identified as \emph{ROI: JPEG2000} codestreams, offer
several mechanisms to support spatial random access to regions of interest, at varying degrees of granularity.
These allow different parts of the same picture to be stored and/or retrieved at different quality levels.
\item Error resilience -- JPEG2000 is robust to bit errors introduced by communication channels, due to
the coding of data in relatively small independent blocks within the transform domain.
\item Flexible file format -- The JPX file format, in particular, allows for rich and flexible description
and composition of components. It allows images to be composed from any number of independently
compressed codestreams.
\item Extensive metadata support and handling.
\item Support for volumetric image cubes, either through the specific set of extensions in Part 10 (a.k.a. ``JP3D")
or by using the extensive set of multi-component transforms provided with Part 2 of the standard.
\item Interactivity in networked applications, as developed in the JPEG2000 Part 9 JPIP protocol.
\end{itemize}
\subsection{Encoding/decoding}
Unlike the binary compression available through \texttt{cfitsio} or HDF5, JPEG2000 is a true image
compression that takes advantage of the multidimensionality of data. Figure~\ref{fig:jpeg2000-encoding} depicts
the stages of encoding in JPEG2000.
In the first stage, pre-processing is performed. Pre-processing actually contains three substages:
Tiling, Level Offset, Reversible/Irreversible Color Transform. This stage prepares the data to correctly perform
the Wavelet Transform. During the Wavelet Transform, image components are passed recursively through the
low pass and high pass Wavelet filters. This enables an intra-component decorrelation that
concentrates the image information in a small and very localised area. It enables the multi-resolution
image representation. The result is that 4 sub-bands with the upper left one $LL$ on Figure~\ref{fig:jpeg2000-encoding} containing
all low frequencies (low resolution image), $HL$ containing vertical high frequencies, LH containing horizontal high frequencies, and $HH$
containing diagonal high frequencies. Successive decompositions are applied on the low
frequencies $LL$ recursively as many times as desired.
By itself the Wavelet Transform does not compress the image data; it restructures the image information so
that it is easier to compress. Once the Discrete Wavelet Transform (DWT) has been
applied, the output is quantified in Quantisation unit.
Before coding is performed, the sub-bands of each tile are further partitioned into small code-blocks (e.g. 64x64
or 32$x$32 samples) such that code blocks from a sub-band have the same size. Code-blocks
are used to permit a flexible bit stream organisation.
The quantised data is then encoded in the Entropy Coding unit.
The Entropy Coding unit is composed of a Coefficient Bit Modeller and the Arithmetic Coder itself.
The Arithmetic Coder removes the redundancy in the encoding of the data. It assigns short code-words to
the more probable events and longer code-words to the less probable ones.
The Bit Modeller estimates the probability of each possible event at each point in the coding stream.
At the same time as embedded block coding is being performed, the resulting bit streams
for each code-block are organised into quality layers. A quality layer is a collection of some
consecutive bit-plane coding passes from all code-blocks in all sub-bands and all components,
or simply stated, from each tile. Each code-block can contribute an arbitrary number
of bit-plane coding passes to a layer, but not all coding passes must be assigned to a quality
layer. Every additional layer successively increases the image quality.
Once the image has been compressed, the compressed blocks are passed over to the Rate Control unit
that determines the extent to which each block's embedded bit stream should be truncated in order to achieve
the target bit rate. The ideal truncation strategy is one that minimises distortion while still reaching the target bit-rate.
In Data Ordering unit, the compressed data from the bit-plane coding passes are first separated into packets. One packet is generated for each precinct in a tile. A precinct is
essentially a grouping of code blocks within a resolution level. Then, the packets are multiplexed
together in an ordered manner to form one code-stream. There are five built-in ways to order the packets, called
progressions, where position refers to the precinct number:
\begin{itemize}[leftmargin=*]
\item Quality: layer, resolution, component, position
\item Resolution 1: resolution, layer, component, position
\item Resolution 2: resolution, position, component, layer
\item Position: position, component, resolution, layer
\item Component: component, position, resolution, layer
\end{itemize}
The decoder basically performs the opposite operations of the encoder.
The details and mathematics of JPEG2000 encoding can be found in \citealt{Gray}, \citealt{Adams2001}, or \citealt{Li2003}.
\begin{figure}
\centering
\includegraphics[width=90mm]{images/jpeg2000-encoding.pdf}
\caption{JPEG2000 encoding is based on discrete wavelet transformation, scalar quantisation,
context modelling, arithmetic coding and post-compression rate allocation.}
\label{fig:jpeg2000-encoding}
\end{figure}
\subsection{File format and metadata}
The \textit{JP2} file format is organised as a sequence of "boxes", as depicted
in Figure~\ref{fig:jp2-file}. Boxes play a role in the file format similar to that of marker segments
in the code-stream syntax, and they appear consecutively in the file.
There are four required required boxes: \textit{JPEG200 Signature}, \textit{File Type}, \textit{JP2 Header},
and \textit{Contiguous Code-Stream} boxes.
\textit{IPR}, \textit{XML}, \textit{UUID}, and \textit{UUID Info} boxes are all optional and may appear in
any order, anywhere after the \textit{File Type} box. There may be multiple instances of these three boxes.
The \textit{JPEG2000 Signature} box identifies the file as belonging to the JPEG2000 family of file formats.
The \textit{File Type} box identifies the file specifically as a JP2 file. The \textit{JP2 Header} box contains information
such as image size, bit-depth, resolution, and colour space. The \textit{Contiguous Code-Stream} box contains
a single valid JPEG2000 code-stream. \textit{IPR contains} Intellectual Property Rights information. \textit{XML} boxes
provide for the inclusion of additional structured information, while \textit{UUID} and \textit{UUID} Info boxes provide
a mechanism for defining vendor specific extensions.
Each of the boxes has an internal structure and sub-boxes containing the information about the image. The
details can be found in e.g. \citealt{Taubman02}.
The \textit{XML} box may contain any information whatsoever, provided that it complies to the XML
(extensible Markup Language). For example, the discussed later \texttt{SkuareView} software uses \textit{XML}
box to contain FITS header "as is" wrapped in a simple XML envelop. Alternatively, one of the IVOA's data models
or some proprietary custom information could be placed in a single or multiple XML boxes.
The \textit{JPX} file format provides even more advanced metadata handling \citep{15444-2}.
\begin{figure}
\centering
\includegraphics[width=90mm]{images/jp2-file.pdf}
\caption{JP2 file format structure. Rounded conners indicate optional boxes.}
\label{fig:jp2-file}
\end{figure}
\subsection{JPIP}
JPIP protocol deserves a special consideration as it offers significantly richer functionality compared to IVOA SIAP.
JPIP (JPEG2000 Interactive Protocol)) is a client/server communication protocol that
enables a server to transmit only those portions of a JPEG2000 image that are applicable to
the client's immediate needs. However, this is achieved in a different way compared to a traditional
cutout service, such as IVOA SIAP\footnote{\href{http://www.ivoa.net/documents/latest/SIA.html}{http://www.ivoa.net/documents/latest/SIA.html}}.
Using an HTTP-based query syntax, together with
TCP or UDP based transport protocols, JPIP enables the client to selectively access content of
interest from the image file, including metadata of interest. This capability results in a vast improvement
in bandwidth efficiency and speed when performing some very important and valuable image viewing
tasks in a client/server environment, while reducing the storage and processing requirements of the
client. The larger the images -- and the more constrained the bandwidth between client and server --
the greater are the benefits brought by JPIP.
JPIP clients access imagery on the basis of a so-called ``Window of Interest" (WOI). The WOI
consists of a spatial region, at a given resolution, within one or more image components in one
or more underlying compressed codestreams, optionally limited to a desired quality level or amount
of communicated data. In advanced applications, the WOI may also be expressed relative to one or
more higher level composited representations whose definition depends on metadata. JPEG2000
enables the efficient identification and extraction of elements from the compressed codestream(s)
that intersect with the WOI. This means that from a single compressed image, a user can remotely
extract a particular region of the image, a larger or smaller version of the image, a higher or lower
quality version of the image, or any combination of these. JPIP can be used to progressively forward
images of increasing quality, giving the client a view of the image as quickly as possible, which
improves as rapidly as possible, along the direction of interest.
Such features are most desirable for extremely large radio astronomy images, which can hardly
be used without examining the metadata and previewing the image at low resolution first, transferring
only the selected parts of the image to a user's computer. This would normally require generating low
resolution images, thumbnails and metadata and linking them all together in a database. In a system
equipped with JPEG2000 and JPIP, however, it is only necessary to store a single file per image;
lower resolutions and thumbnails can be extracted directly out of this high-resolution JPEG2000
``master" image and streamed or downloaded to the client. This removes the need to store, manage, and link images of
different resolutions in the database, which can be cumbersome.
In a typical application, when the user chooses to view a particular image, only the resolution layer required
to view the entire image on the screen need be transferred at first. Quality layers are downloaded progressively
to give the user an image as quickly as possible. When the user zooms into a particular region of interest in the
image, only that portion of the image is transferred by the server, and only at the resolution that is of interest.
Again, the image can be transferred progressively by quality layers. The user can continue to zoom into the image
until the maximum quality/resolution is reached, and pan across the image; each time, transferred content is limited
to the area of the image being viewed. An interactive user might then scan across different images of a series,
maintaining the same region and resolution of interest. Again, only the relevant content is actually transferred.
The result is a dramatic increase in speed of viewing, and significant increase in the quality and efficiency of the
viewing experience.
\subsection{JPIP Stream Type}
The JPIP allows three different types of image data to be transmitted between the server and client: 1) full,
self-contained compressed images (typically, but not necessarily, in the JPEG2000 format); 2) tile data; and 3)
precinct data \citep{Taubman}.
\textit{Full JPEG2000 Images}. For this data type the server sends the client
complete JPEG2000 images, at the requested resolution. The resolution level
is selected to fit in the display window. Because the JPEG2000 images are
self-contained, they do not require any additional metadata or headers during
transmission; the images are simply sent to the client and the client decodes them.
\textit{Tiles}. Tiles are rectangular spatial regions within an image that are independently encoded. It can be useful
to encode a large image as multiple independent tiles, but even huge images can be encoded as a single tile. A tile-based
JPIP service is useful where numerous small tiles have been used during the encoding process; this allows the server to
send only the relevant tiles to the client, for decoding. Because tile data is not a self-contained image, additional
JPIP messaging headers are attached to convey to the client the contents of the messages. Tiling has been used
in a number of image formats (e.g. TIFF). It has been introduced in FITS along with the compression applied
to the tiles rather than to the entire image \citep{2010A&A...524A..42P}.
However, the use of small tiles reduces compression efficiency and can have a large adverse effect upon the
service of reduced resolution imagery, since the effective size of the tiles within reduced resolutions can become
very small. In JPEG2000 tiles are not considered as a preferred method of structuring an image, as
\textit{precincts} offer more advanced solution.
\textit{Precincts}. Precincts are fundamental spatial groupings within a JPEG2000 codestream. Unlike tiles,
which represent independently coded regions in space, precincts are defined in the wavelet transform domain.
The detail subbands at each resolution level are partitioned into independently coded blocks, which are assembled
into precincts. Each precinct represents a limited spatial region within the detail bands that are used to augment
the displayed imagery from a given resolution to the next. Since precincts are defined in the transform domain,
their contributions to the reconstructed imagery are overlapping. This means that a server which sends the precincts
that are relevant to a particular WOI is also sending some content that belongs to surrounding regions, whose
extent is resolution dependent. Precincts are the providers of ROI functionality in JPEG2000. The content of a
precinct can be sent progressively, so as to incrementally refine the quality of the associated imagery. Additional
JPIP messaging headers are attached to the precinct data to convey to the client their contents. This image type
is often the most efficient, as it requires the smallest amount of data to be transmitted; moreover, it is equally
efficient at all spatial resolutions, unlike tiles, whose size can be optimized only for at a pre-determined resolution.
An interesting potential mechanism for exploiting precincts within ASKAP and SKA applications, would be to use
source finding algorithms to automatically generate a catalogue of the most relevant precincts, as part of the
telescope pipeline. This would enable the selective storage of precinct data based on relevance (from lossy up
to potentially numerically lossless), as well as the selective delivery of those precincts to a JPIP client; ``empty"
parts of an image can be sent at much lower quality or resolution, saving the bandwidth, storage/archive space,
and increasing the speed of fetching and viewing the data.
\subsection{JPIP Operation and Features}
The client application generates and sends to the server a properly formatted JPIP
WOI request, containing information about the specific region of the image that the
user wishes to view, along with the desired resolution, image components of interest
and optionally explicit quality constraints -- alternatively, the client may request everything
and expect to receive a response with progressively increasing quality. The JPIP server
parses the request, calls the JPEG2000 library to extract the relevant image data, and
sends back to the client a formatted JPIP response. When the response data is received,
the JPIP client extracts the codestream elements and inserts them into a sparse local
cache, from which the imagery of interest is decompressed, rendered and/or further
processed on demand. Importantly, JPEG2000 codestreams have such a high degree
of scalability that any image region of interest can be successfully decoded from almost
any subset of the original content on the server, albeit at a potentially reduced quality.
This means that decompression and rendering/processing from a local JPIP cache is
an asynchronous activity that depends only loosely on the arrival of suitable data from
the server. To the extent that such data becomes available, the quality of the
rendered/processed result improves.
Tile and precinct ``databins" are the basic elements of a JPEG2000 image used by JPIP.
JPEG2000 files can be disassembled into individual finer elements, called \textit{databins},
and then reassembled. Each databin is uniquely identified and has a unique place within
a JPEG2000 file. Full or partial databins are transmitted from the server to the client in
response to a JPIP request. The JPIP client can decode these databins and generate
a partial image for display at any point while still receiving data from the server.
JPIP provides a structure and syntax for caching of databins at the client, and for communication
of the contents of this cache between the client and the server. A client may wish to transmit
a summary of the contents of its cache to the server with every request, or allow the server
to maintain its own model of the client cache by maintaining a stateful session. In either case,
a well behaved server should reduce the amount of data it is transmitting in response to a JPIP
request by eliminating the databins that the client has already received in previous transmissions.
In this way, JPIP provides a very efficient means for browsing large images in a standards-compliant
fashion.
Both precinct and tile databins have the property that they may be incrementally communicated,
so that the quality of the associated imagery improves progressively. JPIP also provides for the
partitioning of metadata into databins, which can also be communicated incrementally. This allows
large metadata repositories to be organised and delivered on demand, rather than as monolithic
data sets. Moreover, metadata can be used to interpret imagery requests and the image WOI can
also be used to implicitly identify the metadata that is of interest in response to a JPIP request.
While databins are being transferred between the server and the client, they usually get split
up into smaller chunks, called \textit{messages}. The JPIP server decides the JPIP message
size. This flexibility to transmit partial databins enables one to vary the progressive nature of
the data being sent to the client. If entire databins are sent, first for the lower resolution levels
in the codestream and then for the higher resolution levels, the imagery pertaining to the
requested WOI will be received in a resolution-progressive fashion; if messages from different
databins at the same resolution level are interlaced, the data will be received by the client in
a quality-progressive order. This flexibility allows applications to control the user experience,
depending on the application requirements \citep{Taubman}.
There are numerous implementations of JPIP servers and client SDK available:
OpenJPEG JPIP\footnote{\href{https://code.google.com/p/openjpeg/wiki/JPIP}{https://code.google.com/p/openjpeg/wiki/JPIP}},
LEADTOOLS\footnote{\href{http://www.leadtools.com/sdk/jpip/}{http://www.leadtools.com/sdk/jpip/}},
KDU SDK from Kakadu Software\footnote{\href{http://www.kakadusoftware.com/index.php?option=com\_content\&task=view\&id=25}{http://www.kakadusoftware.com}},
2KAN\footnote{\href{http://www.2kan.org/demonstrator.html}{http://www.2kan.org/demonstrator.html}},
JPIPKAK as part of Geospatial Data Abstraction Library\footnote{\href{http://www.gdal.org/frmt\_jpipkak.html}{http://www.gdal.org/frmt\_jpipkak.html}},
and other.
\section{Benchmarking of JPEG2000 compression on radio astronomy images} \label{cha:Bench1}
As yet, JPEG2000 has not been used in astronomy very widely. Most of the accessible radio astronomy images are stored
in FITS or CASA Image Tables. At the time when this investigation started there was
no software available to convert FITS or CASA Image Tables to JPEG2000 images
with a sufficient range of encoding parameters. To begin with, we limited ourselves to
encoding FITS images only, as the most common image format currently used in astronomy.
\subsection{Software}
\texttt{f2j} software was developed to convert FITS files to JPEG2000 images.
The software has been written in C using the open source OpenJPEG\footnote{\href{http://www.openjpeg.org}{http://www.openjpeg.org}}
codec version 1.0\footnote{v2.0 was already available at the time when the paper was written.} for JPEG2000 compression and NASA's
\texttt{cfitsio}(\href{http://ascl.net/1010.001}{ascl:1010.001})
library for reading FITS files\footnote{\texttt{f2j} does not transfer FITS headers to
JPEG2000 files, however, the software described in \citealt{2014arXiv1401.7433P} does
transfer FITS headers into metadata boxes of JPX.}. \texttt{f2j} is an open source software, and can be
downloaded from the Github\footnote{\href{https://github.com/ICRAR/f2j.git}{https://github.com/ICRAR/f2j.git}}.
\texttt{f2j} encodes FITS files as JPEG2000 images with a single component consisting
of greyscale pixel intensities stored as 16 bit unsigned integers. Each plane
of a data-cube is written to a separate JPEG2000 image. \texttt{f2j} reads a full plane from a FITS
file into an array and then processes each raw value in this array into a greyscale pixel intensity.
This results in, what is essentially, a bitmap image being passed to the JPEG2000 encoder.
There are multiple options as to how raw FITS data may be transformed into pixel intensities.
The particular transformation applied depends on the data type used to store the raw FITS values.
In the case of 8 or 16 bit integer data, raw values may be used directly as pixel intensities.
At the time of the first trials OpenJPEG v1.0 codec did not support floating point data directly, so floating point values had to
be converted to integers in order to create a JPEG2000 image from such data. Later releases of OpenJPEG,
however, already support a full range of data types that includes double and single precision floating point.
Arbitrary transformations may be defined in \texttt{f2j} for this purpose and it is relatively easy to add
new transformations to the program. The floating point transformations currently implemented
work by assigning the smallest and largest raw data values in the FITS file to the lowest and
highest possible pixel intensities respectively and then scaling the intermediate data in various ways.
The logarithmic, square root and power scales are available.
The JPEG2000 standard specifies that image components may be represented with arbitrary precisions
up to 38 bits \citep{Schelkens..2009}, however OpenJPEG stores pixel intensities using 32 bit integers
(in the internal structure it uses to represent an image prior to passing it to the JPEG2000 encoder),
limiting the precision attainable. Through experimentation it was also determined that OpenJPEG
could not correctly encode and decode images using 32 bit precision due to an error in the library.
However, the used imaged had the dynamic range that could be mostly sufficiently accurately represented
by 16 bit precision.
In the case of floating point data, it was observed that files would often use only a tiny
portion of the full range of values supported by this data type. As the data is scaled to the minimum
and maximum of the allowable 16 bit integer range, the small range of values being scaled would lessen
the loss of precision as a result of this quantisation.
There are many compression options that may be specified affecting how an image is encoded
using JPEG2000, such as the number of resolutions in the file, tile sizes, compression ratios and
the use of lossy or lossless compression. \texttt{f2j} supports almost
all of the compression options supported by OpenJPEG codec v1.0.
As we were interested in testing the quality of JPEG2000 compression on radio astronomy images,
as well as converting FITS files to JPEG2000, \texttt{f2j} had been equipped with some benchmarking and
experimentation features. The software is capable of adding varying amounts of Gaussian noise to an image
to investigate the effects of noise on the compression process. It can perform quality benchmarks
to examine how lossy compression degrades an image, by decompressing an encoded JPEG2000
image from a file and calculating quality metrics comparing it to the uncompressed image.
While we have acknowledged the usefulness of JPIP protocol, and it's ability to significantly extend the \emph{cutout}
method of interrogation of image data, which is currently the mainstream method in astronomy, in all
further presented tests in this paper we benchmarked only image files stored on a local drive leaving
benchmarking of JPIP for a future investigation.
\subsubsection {Metrics}
We have built-in \texttt{f2j} the options to calculate the mean squared error (MSE), root mean
squared error (RMSE), peak signal to noise ratio (PSNR), mean absolute error (MAE), fidelity
and maximum absolute distortion (MAD) metrics (as well as intermediate data for these metrics). These
metrics are recommended for compression benchmarks \citep{Delcourt:2011:EFB:2436496.2436501}.
In practice, we've found the fidelity metric to be unhelpful, as in none of the tests conducted
did it drop below 0.98, even for badly distorted images. MSE, RMSE and PSNR are all re-expressions
of the same information and thus interchangeable. PSNR was found to be the most intuitive to work
with and was therefore used in most of our tests.
MAE is not directly related to RMSE, but one would intuitively expect these metrics to be closely correlated.
This was verified in practice. Figure~\ref{fig:MAEvsRMSE} shows the values of MAE collected in
quality versus compression ratio benchmark tests (vertical axis) plotted against RMSE values
(horizontal axis) -- a clear linear relationship is visible. The correlation coefficient between the
two variables is 0.992. The close correlation between RMSE and MAE supports the conclusion that
MAE offers little information beyond RMSE (and thus PSNR). Thus our discussions of results will
focus on PSNR and MAD mostly.
\begin{figure}
\centering
\includegraphics[width=90mm]{images/image11.png}
\caption{Mean absolute error versus root mean squared error (both axes use units of 16 bit pixel intensities).}
\label{fig:MAEvsRMSE}
\end{figure}
\subsubsection{Test images}
A large number of publicly available radio astronomy FITS files were examined to come up with a representative set of
test images representing features and attributes that radio astronomers would expect to encounter.
These include sparsely and densely populated images, dominant and diffuse features, high or low
noise and regular or random noise. A final test set of 11 images was selected, including 9 planar
images and 2 data cubes. All images contained floating point data. These images were used in the
benchmarking described in the following sections.
\subsection{Lossless compression benchmarking}
These benchmarks involved encoding the test images losslessly and observing
the compression ratios attainable.
There are many parameters that might be specified when encoding to an JPEG2000 image.
These allow the image compression to be fine-tuned for a particular purpose, \emph{i.e.} for
distribution as part of a JPIP system and have the potential to affect compression ratios and
image quality (for lossily encoded images). For the initial benchmarking, other than altering the
compression ratios and target quality for the lossy compression benchmarks (below), the
default OpenJPEG settings were used as typical parameters. Therefore, while our project
provides a guide to the compression performance possible using JPEG2000, best results for
any practical application will result from optimising the compression process for a particular use.
\begin{table*}[!t]
\centering
\resizebox{18cm}{!} {
\begin{tabular} { l || r | r | c | c}
\hline
File & \parbox[t]{2.7cm}{Size of JPEG2000 file (bytes)} & \parbox[t]{2.7cm}{Size of FITS file (bytes)} & \parbox[t]{2cm}{Compression Ratio} & Disk Space Saved (\%)\\
\hline
1.45I1.50\_AM0381\_1992DEC14\_1\_125.U50.7S.imfits & 110,689 & 406,080 & 1:3.67 & 73\\
1.45I4.68\_AK456\_1998AUG28\_1\_76.1U2.95M.imfits & 120,021 & 406,080 & 1:3.38 & 70\\
1.45I4.70\_AK456\_1998SEP04\_1\_131.U1.68M.imfits & 86,841 & 406,080 & 1:4.68 & 79 \\
1.45I6.65\_TESTS\_1994JUL24\_1\_120.U2.61M.imfits & 74,199 &406,080 & 1:5.47 & 82\\
1.45I9.04\_AB778\_1996JAN29\_1\_42.6U4.91M.imfits & 103,796 &406,080 & 1:3.91 & 74\\
1.45I10.1\_AK456\_1998NOV15\_1\_23.3U4.63M.imfits & 118,756 & 406,080 & 1:3.41 & 71\\
00015+00390Z.fits & 2,459,591 & 7,145,280 & 1:2.90 & 66\\
22.4I0.94\_AF350\_1998DEC24\_1\_3.41M55.7S.imfits & 209,432 & 898,560 & 1:4.29 & 77\\
CYG.ICLN.FITS & 1,696,799 & 16977600 & 1:10.01 & 90\\
M31\_5Mpc\_dirty\_6km.fits & 635,258,960 & 2,073,605,760 & 1:3.26 & 69\\
M31\_model\_5Mpc.fits & 12,854,386 & 652,916,160 & 1:50.79 & 98\\
\hline
\end{tabular}}
\caption{Lossless compression benchmarking results. The compression ratios are true ratios
for all the images including those that had been truncated from 32 bit to 16 bit integers.}
\label{tab:LL_br}
\end{table*}
\begin{table*}[!t]
\centering
\resizebox{15cm}{!} {
\begin{tabular} { l || l | c | c}
\hline
\parbox[t]{2cm}{Compression Ratio} & File & \parbox[t]{2.7cm}{PSNR (bB)} & \parbox[t]{2.7cm}{MAD}\\
\hline
1:15 & 1.45I1.50\_AM0381\_1992DEC14\_1\_125.U50.7S.imfits & 46.3 & 2241\\
& 1.45I4.68\_AK456\_1998AUG28\_1\_76.1U2.95M.imfits & 41.4 & 3295\\
& 1.45I10.1\_AK456\_1998NOV15\_1\_23.3U4.63M.imfits & 42.0 & 3590\\
& 00015+00390Z.fits & 45.5 & 2031\\
& M31\_5Mpc\_dirty\_6km.fits (110) & 49.2 & 3942\\
\hline
1:20 & 1.45I9.04\_AB778\_1996JAN29\_1\_42.6U4.91M.imfits & 46.3 & 1859\\
& M31\_5Mpc\_dirty\_6km.fits (40) & 47.4 & 2621\\
& M31\_5Mpc\_dirty\_6km.fits (75) & 46.4 & 3941\\
\hline
1:25 & 1.45I4.70\_AK456\_1998SEP04\_1\_131.U1.68M.imfits & 51.4 & 1760\\
& M31\_5Mpc\_dirty\_6km.fits (5) & 43.4 & 3792\\
& M31\_5Mpc\_dirty\_6km.fits (145) & 43.8 & 3932\\
\hline
1:30 & 1.45I6.65\_TESTS\_1994JUL24\_1\_120.U2.61M.imfits & 54.5 & 1987\\
& 22.4I0.94\_AF350\_1998DEC24\_1\_3.41M55.7S.imfits & 48.8 & 3160\\
\hline
\end{tabular}}
\caption{Quality benchmarks for lossy compression at the compression ratio of first visual degradation. The compression ratios in the
table are those supplied to the JPEG2000 encoder. The numbers in brackets next to the file names
indicate a particular plane (frequency channel) of a data cube. }
\label{tab:QB}
\end{table*}
Table~\ref{tab:LL_br} shows the lossless compression ratio attained for each of the 11 test
images, the space saved as a result of compression, and the sizes (in bytes) of the JPEG2000 images
and original FITS files.
In terms of lossless compression ratios, there are two obvious outliers in this table:
M31\_model\_5Mpc.fits and CYG.ICLN.FITS. The first is a very clean data cube
containing a simulated ASKAP image of the M31
galaxy\footnote{\href{http://www.atnf.csiro.au/people/Matthew.Whiting/ASKAPsimulations.php}{http://www.atnf.csiro.au/people/Matthew.Whiting/ASKAPsimulations.php}},
which achieved a compression ratio of 1:50.79. The second image is of Cygnus A
observed on the EVLA\footnote{Credit to Richard Dodson of ICRAR for the original FITS file},
which achieved a more modest compression ratio of 1:10.01
(see Figure~\ref{fig:CygA}). This image contained a reasonable amount of instrumental
noise, but nevertheless this noise could be represented efficiently using the JPEG2000
lossless algorithm.
\begin{figure}
\centering
\includegraphics[width=90mm]{images/image2.png}
\caption{Cygnus A as observed on the EVLA converted to JPEG2000.}
\label{fig:CygA}
\end{figure}
Not taking in account the outliers, the mean compression ratio was 1:3.89 with a standard deviation of 0.80.
Of the remaining images, the worst compression ratio, of 1:2.90, occurred with the file 00015+00390Z.fits
(see Figure~\ref{fig:VLA00015}). This was a very noisy (mostly instrumental) image as observed on the VLA array.
The worst compression ratio this image was achieved despite the fact that the instrumental noise has a regular
but very finely gridded structure.
\begin{figure}
\centering
\includegraphics[width=90mm]{images/image3.png}
\caption{00015+00 as observed on the VLA array converted to JPEG2000.}
\label{fig:VLA00015}
\end{figure}
Of the remaining files, the best compression ratio of 1:5.47 was achieved on the file
1.45I6.65\_TESTS\_1994JUL24\_1\_120.U2.61M.imfits, which contained a relatively
clean image of RC2357 as observed from the VLA array (see Figure~\ref{fig:RC2357}).
The image also has constant values in all four corners that contributes to the high
compression ratio.
\begin{figure}
\centering
\includegraphics[width=90mm]{images/image4.png}
\caption{RC2357 as observed on the VLA array.}
\label{fig:RC2357}
\end{figure}
In all of our test cases lossless compression gave a significant disc space saving (see Table~\ref{tab:LL_br}).
\subsection {Lossy compression benchmarking}
\subsubsection {Quality versus compression ratio benchmarks}
These benchmarks involved compressing the test images lossily by specifying a particular
compression ratio to the JPEG2000 encoder. Compression and quality metrics were recorded
for each of the compressed images. The compression ratios at which compression artefacts first
became visually noticeable (relative to the losslessly compressed version) were recorded.
The residual images resulting from the lossy compression process were written to files
and were visually examined for features of interest.
Compression ratios of 1:$X$ were used, where $X$ took the values \emph{25, 20, 15, 10, 5, 2, 1.5}.
Higher compression ratios were examined if there were no visible compression artefacts at the
1:25 compression ratio.
Table~\ref{tab:QB} shows the nominal compression ratio at which each file first showed
\emph{visual} degradation and quality metrics at this point. Note that the compression ratios in the
table are those supplied to the JPEG2000 encoder. The numbers in brackets next to the file names
indicate a particular plane (frequency channel) of a data cube.
The first point to note is that every file could be compressed lossily to a nominal 1:10 ratio without showing
visual degradation, which is 2.6 times greater than the average 1:3.89 compression ratio attainable
using lossless compression.
Figure~\ref{fig:PSNRdegrad_2} shows the PSNR values recorded at the compression ratios that visual
degradation first occurred.
\begin{figure}
\centering
\includegraphics[width=90mm]{images/image5.png}
\caption{PSNR at the nominal compression ratio at which visual degradation first occurred.}
\label{fig:PSNRdegrad_2}
\end{figure}
From the graph, it is obvious that while visual degradation first occurred over a
relatively narrow PSNR range, it occurred over a relatively wide nominal
compression ratio range. This observation motivated the next set of benchmarks.
\subsubsection{Compression ratio versus quality benchmarks}
These benchmarks investigated the opposite side of the equation to the previous set of benchmarks.
The tests proceeded as in the previous section, except that the test images were compressed lossily
by specifying a particular target quality, expressed as a peak signal to noise ratio (PSNR), to the
JPEG2000 encoder, rather than specifying a compression ratio. For these benchmarks, FITS files
were encoded lossily with a particular targeted quality (PSNR).
The quality metrics here are thus largely influenced by this compression parameter -- therefore
it is the compression metrics that are of interest in these tests.
\begin{table*}[!t]
\centering
\resizebox{12cm}{!} {
\begin{tabular} { l | l | l }
\hline
\parbox[t]{2.7cm}{PSNR (bB)} & File & \parbox[t]{2cm}{Compression Ratio}\\
\hline
50 & 1.45I1.50\_AM0381\_1992DEC14\_1\_125.U50.7S.imfits & 1:24\\
& 1.45I4.70\_AK456\_1998SEP04\_1\_131.U1.68M.imfits & 1:62\\
& 1.45I6.65\_TESTS\_1994JUL24\_1\_120.U2.61M.imfits & 1:80\\
& 1.45I9.04\_AB778\_1996JAN29\_1\_42.6U4.91M.imfits & 1:32\\
& 22.4I0.94\_AF350\_1998DEC24\_1\_3.41M55.7S.imfits & 1:58\\
40 & 1.45I4.68\_AK456\_1998AUG28\_1\_76.1U2.95M.imfits & 1:41\\
& 1.45I10.1\_AK456\_1998NOV15\_1\_23.3U4.63M.imfits & 1:36\\
& 00015+00390Z.fits & 1:111\\
& M31\_5Mpc\_dirty\_6km.fits (5)\\
& M31\_5Mpc\_dirty\_6km.fits (40)\\
& M31\_5Mpc\_dirty\_6km.fits (75)\\
& M31\_5Mpc\_dirty\_6km.fits (110)\\
& M31\_5Mpc\_dirty\_6km.fits (145) & 1:102\\
\hline
\end{tabular}}
\caption{Compression benchmarks at the quality (PSNR) of first visual degradation.}
\label{tab:QB_PSNR}
\end{table*}
Table~\ref{tab:QB_PSNR} shows the compression ratios achieved at the PSNR (quality)
that files first showed visual degradation.
\begin{figure}
\centering
\includegraphics[width=90mm]{images/image7.png}
\caption{Compression ratio at the PSNR at which visual degradation first occurred.}
\label{fig:degrad_vis_M51less}
\end{figure}
Figure~\ref{fig:degrad_vis_M51less} shows the compression ratios attained
at the PSNR that visual degradation first occurred. Again, it is obvious that visual degradation first occurs
over a relatively wide range of compression ratios but a relatively narrow PSNR range. Thus the
target PSNR appears to predict visual image quality far better than compression ratio. When working
with lossily compressed images, it would thus be advisable to encode images for a particular quality
using PSNR as a metric rather than a particular compression ratio.
\section{Improving existing formats, adopting other technologies or starting from scratch?} \label{cha:adopting}
\subsection{Improving of existing formats}
The FITS standard, in its present form, is clearly unable to support such cases as ASKAP or SKA.
Can it be improved? \citealt{white2012tiled} have made perhaps the most significant attempt
to improve FITS for the large images use case. The convention suggests that compressed tiles of image
are stored in a binary table extension which is hidden from the end-user as the image is
accessed through the same image interface that is used to access normal raw images. However,
this convention does not offer any new framework to work with the imagery data. As before,
a cutout needs to be produced as a separate file and downloaded to to the client. Comparing to the
described JPIP client-server framework the cutout framework is clearly limiting for many use cases, especially
that involve visualisation.
Another problem for improving existing standards such as FITS, is its expected and important property
of backwards compatibility. Such legacy often conflicts with the modern
performance and flexibility requirements \citep{2011ASPC..442...53A}. On the other hand, FITS rather loosely specifies
the formats for various uses, and conservatively defines the metadata. As the result, the actual use of FITS
often deviates from the original specifications in order to accommodate the specific needs of
projects or to extend the functionality in general. This creates an illusion of a standard, while in reality
there are many proprietary cases of FITS that software can not universally interpret.
These are factors, that, in our view, significantly limiting the opportunity to improve FITS in particularly
to address large data issue. This looming predicament is especially relevant to such projects as SKA,
wherein certain operational modes will be generating datasets comprising tens of terabytes of individual
data products.
\subsection{Adopting technologies vs starting from scratch}
A standard like JPEG2000 requires many years of development by the top experts in the field.
The amount of investment in both, time and money required to implement the standard are much greyer.
Many widely adopted standards are often supported by both commercial and open source developments.
The downsides are that not all standards are royalty free to use in development, and that
it might be more difficult to influence the development
of a standard to accommodate the needs of rather small astronomy community.
The industry interest to collaboration in developing standards can be piqued by the high public profile
astronomy projects that can be used as a vehicle for promotion of a standard
or technology. Projects like SKA may represent the unique opportunities for effective collaboration
between the astronomy community and the industry R\&D.
Clearly, no single industry standard/technology can address all the needs of astronomy. JPEG2000
is not universal, and only limitedly suitable for handling other types of data e.g. visibilities
for radio interferometers. The functionality required for other types of data is significantly different to the
functionality required for visual exploration of image data. However, we considered JPEG2000
in detail to demonstrate how powerfully an industry standard can address the requirements for
large astronomical imagery. Use of a suitable standard opens access to many tools that are
readily available providing a shortcut to the solutions that would take many years to achieve
otherwise.
Moreover, we would like to argue that the ability to exchange and correctly interpret the
data is more important than: the format of data that needs to be optimised; a particular
use case; or an optimisation to a hardware platform. As long as there is a clear description of data
in an universal way, the data can be extracted from or ported to any particular format as necessary.
We would like to urge the astronomical community to begin the work of defining the new set of standards
to provide a guidance for new developing instruments to efficiently store and exchange the staggering
amount of data that are going to be generated in the next decade. The work that has been done by IVOA
over the last decade can be a great asset.
\section{Conclusion} \label{cha:conc}
New telescopes, such as the SKA, will produce images of extreme sizes. Providing adequate
performance and level of convenience when serving such images to the end-user is going
to be beyond the capabilities of current astronomy image formats. Improvements
of the existing image and data formats can not solve the deficiencies inherently there, due to the fundamental
limitations at the time of development.
New advanced technologies
are necessary. Technologies such as JPEG2000 have the potential to powerfully pave the way to
a contemporary solution that will adequately address the challenges of extremely large imagery.
Substantial reductions in storage/archive requirements can be achieved by losslessly encoding data into
JPEG2000 images. Even greater saving may be achieved through lossy compression.
JPIP provides a standard powerful way for interaction with the imagery data reducing the bandwidth,
storage, and memory requirements, and increasing the mobility of future astronomy application.
The results of our benchmarks demonstrate the viability of JPEG2000 compression for storing
and distributing radio astronomy images. JPEG2000 is not just about compression -- it has
the potential to enable an entirely new paradigm for working with radio astronomy imagery data.
|
2008.04921
|
\section{Introduction}
Time-domain astrophysics has entered a new era of large photometric datasets thanks to on-going and upcoming wide-field surveys, including Pan-STARRS (PS; \citealt{kaiser2010}), the Asteroid Terrestrial-impact Last Alert System (ATLAS; \citealt{jedicke2012atlas}), the All-Sky Automated Survey for SuperNovae (ASASSN; \citealt{shappee2014all}), the Zwicky Transient Facility (ZTF; \citealt{kulkarni2018zwicky}), the Legacy Survey of Space and Time (LSST; \citealt{ivezic2011large}), and the Roman Space Telescope \citep{spergel2015wide}. LSST, to be conducted by the Vera C. Rubin Observatory between 2023 and 2033, is expected to discover roughly one million SNe per year, a more than two orders of magnitude increase compared to the current rate.
Historically, SNe and other optical transients have been classified primarily based on their optical spectra. Class labels are largely phenomenological, dependent on the presence of various elements in the photospheric-phase spectra (see e.g., \citealt{filippenko1997optical} for a review). SNe, for example, have historically been classified as Type I (equivalent to today's Type Ia) or Type II based on the absence or presence of strong hydrogen Balmer lines, respectively. As the number of events increased, further classes were created to account for the increased diversity (e.g., \citealt{uomoto1985peculiar}). Type Ib and Type Ic designations were created to indicate the presence and absence of helium, respectively. Today, semi-automated software such as {\tt GELATO} \citep{2008Harutyunyan}, {\tt SNID} \citep{blondin2007determining} and {\tt Superfit} \citep{howell2005gemini} are used to match SN spectra to a library of previously classified events to determine the spectroscopic class. More recently, \cite{2019Muthukrishna} utilized a convolutional neural network to classify SN spectra.
However, spectroscopic follow up remains an expensive endeavor, taking up to an hour on 8-meter class telescopes to classify a single object given the depth wide-field surveys can now achieve. As a result, only $\sim 10\%$ of the $\sim 10^4$ transients currently discovered each year are spectroscopically classified\footnote{Based on data from the public Open Supernova Catalog \citep{guillochon2017open} and the Transient Name Server.}. Spectroscopic follow up is not expected to significantly increase when the LSST commences, meaning that only $\sim 0.1\%$ of events will be spectroscopically classified.
Given the growing rate of discovery and limited spectroscopic resources, classification of transients based on their photometric light curves is becoming essential. Luckily, the phenomenological labels often correspond to unique underlying processes that are also encoded in the light curve behavior. For example, while Type Ia SNe are spectroscopically classified by strong Si II absorption and lack of hydrogen, these features distinctly originate from the thermonuclear detonations or deflagrations of carbon-oxygen white dwarfs, which also lead to specific light curve evolution \citep{hillebrandt2000type}. Generally, unique progenitor system and explosion mechanisms likely lead to other observable features, some of which are captured in broadband optical light curves. Said features allow transients to be classified into their traditional subclasses (based on spectroscopy and photometry) using \textit{only} their broadband, optical light curves.
There is a growing literature on light curve classifiers that rely on data-driven and machine learning algorithms. Most studies use \textit{supervised} learning, in which the training set consists of SNe with known classes (e.g., \citealt{lochner2016, 2017ApJ...837L..28C,boone2019,villar2019,moller2020supernnova}). However, SN classification can benefit from \textit{semi-supervised} methods, in which the training set contains both labelled and unlabelled SNe. The unlabelled set is used to better understand low-dimensional structure in the SN dataset to improve classification. \citet{richards2012semi}, for example, created a diffusion map (a nonlinear dimensionality reduction technique) based on light curve similarities in shape and color using unlabelled data from the Supernova Photometric Classification Challenge (SPCC; \citealt{kessler2010results}). They use the diffusion map to extract 120 nonlinear SN features from each labelled SN, which are then used to train a random forest classifier. More recently, \citet{pasquet2018deep} introduced the {\tt PELICAN} classifier, also trained on synthetic SPCC data. {\tt PELICAN} uses a convolutional autoencoder to encode nonlinear SN features and a set of fully connected neural network layers to classify the full set of simulated SPCC light curves as Ia or non-Ia SNe.
\defcitealias{hossen20}{H20}
Here we introduce a new semi-supervised classification method for SNe, which utilizes a recurrent autoencoder neural network (RAENN). This method is uniquely trained on real (rather than simulated) data from the Pan-STARRS1 Medium Deep Survey (PS1-MDS) and is optimized for general SN classification (as opposed to Ia versus non-Ia classification). Our method has been trained on a combination of 557\ spectroscopically-classified SNe and 2,328 additional SN-like events. We then use RAENN and hand-selected features with a random forest to classify the PS1-MDS sample of \newsne\ previously unclassified SN-like transients with host galaxy spectroscopic redshifts. We publish the full set of light curves and associated labels for community use. We present an open source code listed on the Python Package Index as {\tt SuperRAENN} \citep{villar_superraenn}. A companion paper, \citet[][hereafter \citetalias{hossen20}]{hossen20}, presents and compares photometric classifications of the same dataset using an independent classification method (following the supervised methods of our previous work in \citealt{villar2019}).
The paper is organized as follows. In \S\ref{sec:ps1}, we review the PS1-MDS and associated sample of SN-like transients. In \S\ref{sec:semisup} we introduce the RAENN architecture and training procedure. We present the classification results and discuss implications in \S\ref{sec:red} and \S\ref{sec:dis}, respectively. We conclude in~\S\ref{sec:con}. Throughout this paper, we assume a flat $\Lambda$CDM cosmology with $\Omega_M=0.307$, and $H_0 = 67.8$ km s$^{-1}$ Mpc$^{-1}$ \citep{ade2014planck}.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{peaks.pdf}
\caption{Peak apparent $r$-band magnitude of the full SN-like dataset (grey), objects used in our unsupervised method (orange) and the spectroscopic sample (blue). The spectroscopic dataset is roughly one magnitude brighter than the full dataset. \label{fig:peaks}}
\end{figure}
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{red_hist.pdf}
\caption{Histogram of the redshifts of the full SN-like dataset (grey line; 4,055 objects), the subset of host redshift measurements for objects used in our unsupervised learning algorithm (black line;2,885\ objects), and the subset with spectroscopic classification (colored lines; 557\ objects). The shaded grey region represents the summed, spectroscopically classified objects. The full sample and spectroscopic distribution peak at $z\approx 0.25$, although the spectroscopic sample has an additional peak near $z \approx 0.1$. At $z\gtrsim 0.75$, the spectroscopic sample is limited to SLSNe. \label{fig:f1}}
\end{figure}
\section{The PS1-MDS Supernova Sample}\label{sec:ps1}
PS1 is a wide-field survey telescope located near the summit of Haleakala, Maui with a 1.8 m diameter primary mirror and a 1.4 gigapixel camera (GPC1) \citep{kaiser2010}. PS1-MDS, one of several PS1 surveys \citep{2016chambers}, was conducted between July 2009 and July 2014. It consisted of 10 single-pointing fields, each of approximately 7.1 deg$^2$, with a pixel-scale of $0.''25$. The survey was conducted in five broadband filters \citep{stubbs2010precise,tonry2012pan} with a nominal cadence of $3$ days per filter in four filters ($g_\mathrm{P1}r_\mathrm{P1}i_\mathrm{P1}z_\mathrm{P1}$), and a $5\sigma$ limiting magnitude of $\approx 23.3$ per visit. In practice, \cite{scolnic2018complete} finds a cadence of roughly $6-7$ days per filter. In general, PS1-MDS observed a field in $g_\mathrm{P1}$ and $r_\mathrm{P1}$ on the same night, followed by $i_\mathrm{P1}$ and then $z_\mathrm{P1}$ on subsequent nights. PS1-MDS also included observations in the $y_\mathrm{P1}$-band, primarily near full moon and with a shallower $5\sigma$ limiting magnitude of $\approx 21.7$. Due to the poor cadence and shallow depth, we do not use the $y_{\mathrm{P1}}$ data here.
During its 5-year survey, PS1-MDS discovered 5,243\ SN-like objects, defined as events with at least three observersations with a signal-to-noise ratio (SNR) $>4$ in any filter and no previous detection within the survey \citep{jones2018measuring, jones2019foundation}. We obtain data for these events via the PS Data Processing System \citep{2016magnier,magnier2016panstarrs, waters2016panstarrs}. The photometric pipeline is based on {\tt photpipe} \citep{rest2005testing,rest2014cosmological} with improvements made in \cite{scolnic2018complete}. Images and templates, used for image subtraction, are re-sampled and aligned to match a ``skycell" in the PS1 sky tessellation. Image zeropoints are determined by comparing point spread function (PSF) photometry of stars to PS1 stellar catalogs \citep{2016chambers}. PS1 templates are convolved to match nightly images and then subtracted using {\tt HOTPANTS} \citep{becker2015hotpants}. For each event, a flux-weighted centroid is calculated and forced PSF photometry is performed at the centroid. Finally, a nightly zeropoint is applied.
Of the 5,243\ SN-like objects, 4,090 host galaxies were targeted through a concerted observational effort. To identify the most likely host galaxy for each SN, we used the galaxy size and orientation-weighted R-parameter from \citet{sullivan2006rates}, as outlined in \citet{jones2017measuring}. The majority (3,321 objects) were observed using the Hectospec multifiber instrument on MMT \citep{fabricant2005hectospec,mink2007automating}. Additional host redshifts were obtained with the Anglo-Australian Telescope (AAT; 290 objects), the WIYN telescope (217 objects), and the Apache Point Observatory 3.5m telescope (APO; 5 objects). Host galaxies selected for follow-up were largely unbiased in terms of transient properties (e.g., we did not prioritize SNe based on luminosity, color or amount of additional followup). Additional host redshifts were obtained from archival survey data: 2dFGRS \citep{colless2003}, 6dFGS \citep{jones20096df}, DEEP2 \citep{newman2013deep2}, SDSS \citep{smee2013}, VIPERS \citep{scodeggio2018}, VIMOS \citep{le2005vimos}, WiggleZ \citep{blake2008wigglez} and zCOSMOS \citep{lilly2009zcosmos}.
We use the {\tt RVSAO} package \citep{kurtz1998rvsao} to determine the spectroscopic redshifts through cross-correlation with galaxy templates. We use the standard {\tt RVSAO} galaxy templates (including spiral and elliptical galaxies and quasars), as well as galaxy templates provided by SDSS \citep{sdss5}\footnote{\url{http://classic.sdss.org/dr5/algorithms/spectemplates/}}. We quantify the quality of the template matches using the \citet{tonry1979survey} cross-correlation parameter, $R_\mathrm{CC}$. Following \citet{jones2017measuring}, we remove host galaxies with $R_\mathrm{CC}<4$, ensuring that the vast majority ($\approx98$\%) of the remaining galaxies have accurate redshift measurements. This cut removes 1,084 SNe with redshift measurements.
To ensure that our final set of redshift measurements is robust, we identify a subset of spectra to be manually validated. Of the remaining redshifts which we initially estimate using {\tt RVSAO}, we accept the redshift of the best-matching template without visual inspection if the median redshift estimate across templates is equal to both the most-likely redshift and the mode of the template matches \textit{and} more than two templates match this redshift estimate. We (VAV and GH) visually inspected $\sim600$ redshift spectra to ensure that our final redshift estimates are as accurate as possible. In total, 2,487 redshifts (of 3,056 redshift estimates with $R_\mathrm{CC}\ge4$) match the most-likely redshift provided by {\tt RVSAO}. Of the remaining hosts, we remove 393 redshift estimates which we could not validated manually. A total of 145 redshifts ($\sim4$\%) which were measured manually do not match the median or mode of the {\tt RVSAO} redshift estimates. The galaxy spectra and further details are presented in \citetalias{hossen20}.
We additionally remove events with $z<0.005$, which are unlikely to be SNe given the peak absolute magnitudes (e.g., \citealt{2010ATel.2680....1C}). We visually inspect the light curves which have quasar-like hosts (based on template matching) or which overlap with the host galaxy's center. We remove events which are clearly variable over multiple seasons and lack a transient spectrum. Our final sample includes 2,885\ transients with redshifts measurements (from the hosts or transients themselves), including spectroscopically-identified SNe.
\subsection{Spectroscopic versus Photometric SN Sample}
Approximately 10\% of the PS1-MDS transients were spectroscopically observed in real time throughout the survey, without a specific selection function (although brighter objects were more likely to be targeted). For this work, we limit our spectroscopic sample (557\ objects) to five potential classes:
\begin{enumerate}
\item Type I SLSNe (17\ objects), which are thought to arise from the birth of highly magnetized neutron stars \citep{quimby2007sn,chomiuk2011pan,nicholl2017magnetar}
\item Type II SNe (94\ objects; including Type IIP and Type IIL SNe\footnote{Type IIP and Type IL are thought to arise from the same progenitor population. See e.g., \cite{sanders2015toward}}), which arise from red supergiant progenitors
\item Type IIn SNe (24\ objects), powered by the interaction the SN ejecta with pre-existing circumstellar material (e.g. \citealt{smith2014sn})
\item Type Ia SNe (404\ objects), which are the thermonuclear explosions of white dwarfs
\item Type Ibc SNe (19\ objects), which arise from the core-collapse of massive stars that have lost their hydrogen (Ib) and helium (Ic) envelopes. Due to the small sample size we consider Type Ib and Type Ic SNe as a single class.
\end{enumerate}
The SLSN and Type Ia SN light curve samples have been previously published in \citet{lunnan2018hydrogen} and \citet{jones2017measuring}, respectively. Model fits to the Type II light curves were presented in \citet{sanders2015toward}. For four objects, the transient spectra yield a reliable redshift but an ambiguous classification. A fifth object, PSc130816, has previously been identified as both a Type IIP/L SN \citep{sanders2015toward} and a Type IIn SN \citep{drout2016peculiar}. We do not include these five objects in our spectroscopic sample. An additional 15 objects are spectroscopically identified but do not fall in on of our five classes, including two tidal disruption events (TDEs), a lensed Type Ia, a Type Ibn, a Type Iax and ten fast evolving luminous transients (FELTs). All except the TDEs are included in our photometric sample for training purposes, but not included in our spectroscopic sample. These objects are discussed in more detailed in \S\ref{sec:dis1}
Our photometric sample contains \newsne\ objects with host galaxy spectroscopic redshifts, that are independent of the 557\ SNe which are spectroscopically classified. We refer to the union of the photometric and spectroscopic samples (the full set of 2,885\ events), as the ``complete" photometric dataset. We summarize the PS1-MDS SN-like objects, their associated hosts and redshift information in Table~\ref{tab:one}. We also specify which SNe are used in the supervised/unsupervised portions of our classification algorithm.
Our spectroscopic dataset is brighter than our complete photometric dataset. As shown in Figure \ref{fig:peaks}, the spectroscopic sample has a median peak $r$-band magnitude of $\sim-21$ mag, about 1 magnitude brighter than the photometric sample. We directly compare the redshift distributions in Figure \ref{fig:f1}. The spectroscopic sample peaks at a slightly lower redshift compared to the photometric dataset($z\approx0.27$ versus $z\approx0.35$), with a tail extending to $z\approx1.0$. The lack of confident high-redshift measurements is likely due to the key spectroscopic lines shifting out of the optical and due to the peak absolute magnitudes of most SNe falling below our limiting magnitude. The mismatch between the spectroscopic and photometric samples may translate to biases in our classification pipeline, which we explore in more detail in \S~\ref{sec:dis}. The complete $griz$ light curves of our sample are available on Zenodo \citep{villar_lc_2020}.
We explore the overall data quality of our sample in Figure~\ref{fig:sig}, finding that the majority of events have $\sim20$ data points across all filters with signal-to-noise ratios of $\gtrsim 3$. Given a typical SN duration of a month and our typical cadence of a few days, we expect the majority (but not all) SNe to have fairly complete light curves.
\begin{figure}
\includegraphics[width=0.5\textwidth]{num_sig.pdf}
\caption{ Histogram of the number of SN light curves with $N$ data points with SNR of $\ge 3$ (blue), $\ge 5$ (orange), and $\ge 10$ (green) from the complete sample of SN-like objects (5,243\ events). Most events have $\approx10-20$ $3\sigma$ data points, with only a handful having $>100$ points. \label{fig:sig}}
\end{figure}
\section{A Semi-supervised Classification Pipeline}\label{sec:semisup}
About 10\% of our SN sample is spectroscopically classified. Traditional supervised classification methods are strictly limited to this subset of our data, as they require labelled SN examples. However, information about SN subtypes exists as substructure in the unlabelled dataset as well. For example, SN classes may be clustered in duration and luminosity (e.g., \citealt{kasliwal2012systematically,villar2017theoretical}). Because we would like to leverage the information in both the labelled and unlabelled subsets of the training set, we use a recurrent autoencoder neural network (RAENN) paired with a random forest classifier for a semi-supervised classification approach. In this section, we describe the complete algorithm and training process.
Our pipeline is composed of three steps: (1) a pre-processing and interpolation step using Gaussian processes (GP); (2) an unsupervised step in which we train a RAENN on the complete photometric set (labelled and unlabelled); and (3) a supervised step in which we train a random forest on the spectroscopically labelled set of SNe. The complete pipeline, dubbed {\tt SuperRAENN} \citep{villar_superraenn}, is available via GitHub\footnote{\url{https://github.com/villrv/SuperRAENN}}.
\subsection{Pre-processing with Gaussian Processes}
\begin{figure*}
\includegraphics[width=\textwidth]{gp.pdf}
\caption{Examples of three spectroscopically classified SNe and their associated GP-interpolated light curves in the four PS1 filters ($g$: green; $r$: red; $i$: orange; $z$: purple). Solid lines represent the mean GP prediction, while the shaded regions represent the $1\sigma$ estimated uncertainties. \label{fig:gp}}
\end{figure*}
We generate and pre-process absolute magnitude light curves before extracting features. We correct each light curve for Milky Way reddening using the extinction map of \citet{schlafly2011measuring}. We estimate and normalize the absolute magnitude using the measured host redshift:
\begin{equation}
\begin{split}
M_\mathrm{norm} = m - 5\log_{10}(d_\mathrm{L}/10\mathrm{pc})\\ + 2.5\log_{10}(1+z) - m_\mathrm{lim} - A_\lambda
\end{split}
\end{equation}
where $m_\mathrm{lim}$ is a chosen limiting magnitude, which we take to be $m_\mathrm{lim}=25$. This value is dimmer than the $5\sigma$-limiting magnitude of PS1-MDS. We choose a dimmer magnitude to ensure that even marginal detections will be included in the light curve. We perform the re-normalization so that the GP mean will be zero (i.e., the light curve will be zero when no data is available). Finally, we correct all light curves for time-dilation based on the measured redshifts. We do not attempt to make a wavelength-dependent $k$-correction to the rest-frame data given the complicated, diverse, and time-evolving spectral energy distributions (SEDs) of the various SN types.
We do not correct the SN light curves for host galaxy reddening. The intrinsic reddening of SNe adds an additional scatter in our feature space. Correcting for host galaxy reddening would require estimating both the color excess and dust law, which is not possible given our current dataset.
The PS1-MDS light curves are irregularly sampled across the four filters (see \S\ref{sec:ps1} for the PS1 observing strategy). The architecture of the RAENN does \textit{not} require uniformly sampled light curves. However, it does require that each observation is made in all four filters. For example, if an observation is made in $g$-band, we need to provide interpolated values for $riz$-bands for that time.
To interpolate the $griz$ light curves, we fit a GP using the open-source Python package {\tt George} \citep{foreman2015george}. GPs are a non-parametric model that has been previously used to interpolate and classify SN light curves (see e.g., \citealt{lochner2016,revsbech2018,boone2019}). GPs define a prior over a family of functions, which is then conditioned on the light curve observations. A key assumption is that the posterior distribution describing the light curve is Gaussian, described by a mean, $\mu(\vec{t})$, and a covariance matrix, $\Sigma(\vec{t})$, given by $\Sigma_{i,j}=\kappa(\vec{x_i},\vec{x_j})$ with kernel $\kappa$. We use a 2D squared exponential kernel to simultaneously fit all four filtered light curves:
\begin{equation}
\begin{split}
\kappa(\vec{t_i}\vec{t_j}\vec{f_i}\vec{f_j}; \sigma, l_{t}l_{f})= \sigma^2 \exp\Big[-\frac{(t_i-t_j)^2}{2l_t^2}\Big]\\
\times\exp\Big[-\frac{(f_i-f_j)^2}{2l_f^2}\Big]
\end{split}
\end{equation}
where $f$ is an integer between 0 and 3 that represents the $griz$ filters, and the parameters $l_t$ and $l_f$ are characteristic correlation length scales in time and filter integer, respectively. This fitting process accounts for the measured data uncertainties, making it robust to low-confidence outliers.
We independently optimize the kernel parameters for each SN using the {\tt minimize} function implemented in {\tt scipy}, with initial values of $l_t=100$ days and $l_f=1$. We find that our choice of initialization values has little effect on the resulting best fit. We find that $l_t$ is typically about one week, and $l_f$ is typically $2-3$, indicating that the filters are highly correlated. Examples of the GP interpolation for Type Ia, Type Ic and Type II SNe are shown in Figure~\ref{fig:gp}. The GP is able to produce reasonable interpolated light curves even in cases with sparse and noisy data and provide reasonable error estimates.
A similar GP method was implemented by \citet{boone2019} to classify a variety of SN types in the Photometric LSST Astronomical Time-series Classification (PLAsTiCC; \citealt{allam2018photometric,kessler2019models}) dataset. Instead of an integer, \citealt{boone2019} used the rest frame central wavelength of each filter for each object. We avoid this added layer of complexity because the $k$-corrections and time-evolving SN spectral energy distribution (SED) change the weighted central filter wavelength. However, the simple 2D kernel still allows the four bands to share mutual information.
Our light curves contain several years of data, most of which are non-detections. To limit our input data, we keep datapoint (of any significance) within 100 days of peak flux (in whichever filter is brightest). For ease of optimization, the light curves need to contain the same number of data points. The data must be a consistent size during the back-propagation step of optimization for the RAENN for each iteration (see next section). Our longest light curve contains 169 data points, so we pad all light curves to match this length. We do so by appending a value dimmer than the estimated absolute limiting flux (we use $m_\mathrm{lim}=25$) to 100 days after the last detection in the light curve.
We note that using luminosity-based light curves (rather than magnitudes) is an alternative pre-processing choice. Luminosity-based light curves would remove the need to re-normalize the light curves to a chosen limiting magnitude. We find that using luminosity-based light curves results in worse performance of the RAENN, likely due to the orders-of-magnitude differences in scale between events.
\subsection{Unsupervised Learning: A Recurrent Autoencoder Neural Network (RAENN)}
\begin{figure*}
\includegraphics[width=\textwidth]{NNdiagram.pdf}
\caption{Diagram of the RAENN architecture. The pre-processed GP-interpolated light curves are fed into the encoder, which encodes the light curve into an encoding vector. This vector is then repeated, and new time values are appended to each copy. The final light curve is then predicted at each new time value. The RAENN is trained by comparing the input light curve with the predicted light curve. The values from the encoded layer are inputted into the random forest as features and used to classify the SN light curves. \label{fig:nnd}}
\end{figure*}
To extract unique features from the complete (unlabelled and labelled) PS1-MDS photometric sample, we construct a RAENN, inspired by the work of \citet{naul2018recurrent}, who uses a similar method to classify variable stars.
Neural networks are a class of machine learning algorithms that use many latent layers to model complex functions. These and other machine learning algorithms are becoming increasingly common in astronomy (see \citealt{ntampaka2019role} for an overview). Autoencoders (AEs, \citealt{kramer1991nonlinear}) are a class of neural network architectures that learn a compressed representation of input data. By training an AE to return the original data given a limited set of variables, it learns an ``encoded" version of the data.
In astrophysics, AEs have been used for feature-learning in galaxy spectral energy distributions (SEDs, \citealt{frontera2017unsupervised}), image de-noising \citep{ma2018radio,lucas2018recovering}, and event classification \citep{naul2018recurrent,pasquet2019pelican}. AEs are also increasingly being used in the astrophysics literature for dimensionality reduction (see e.g., \citealt{ralph2019radio} and \citealt{portillo2020dimensionality} for recent examples).
Here, our model is designed to address several concerns of SN light curves: (1) the temporal irregularity of data; (2) data across multiple filters; and (3) streaming data that update on a given cadence. The last point is not a concern for our PS1-MDS archival dataset, but it will become important as LSST comes online and discovers thousands of SNe nightly.
The RAENN uses the GP light curves as input, by codifying the light curves as matrices of size $9\times T_0$, where $T_0=169$, as described in the previous section. The $9$ values are: one time value, relative to maximum (in whichever filter is brightest); four magitude values ($griz$) at that time; and four magnitude uncertainties. Recall that the magnitude values are either measured or estimated from the GP. For the uncertainties, we use the $1\sigma$ errors for the measured points. For the GP points, we use a large error of 1 mag. We note that the GP produces estimates errors, but we find that, in practice, using this larger error bar leads to better performance. We leave exploration of utilizing error bars to future work. We emphasize that, while $T_0=169$ for training, the RAENN architecture allows a user to input a light curve of any size without needing to pad the light curve.
The RAENN architecture is divided into an encoder and a decoder. Our encoder is a series of fully-connected layers that decrease in size until the final encoded layer with size $N_E$ (i.e., the number of neurons used to fully encode the SN light curve). We note that $N_E$ is a free parameter of our model that needs to be optimized. Similarly, the fully-connected layer has $N_N$ neurons, where $N_N>N_E$ and is also a tunable parameter. Following the encoded layer, the decoder half of the architecture mirrors the encoder with increasing layer sizes.
A novel feature of our architecture is the inclusion of a repeat layer immediately after our final encoding layer (the layer of size $N_E$). In this layer, we repeat the encoded version of the light curve $T_N$ times. To each copy, we append the time of each data point, relative to peak brightness in one filter (whichever filter happens to be brightest). One way to view the purpose of this layer is to imagine the autoencoder as two functions. The first function (the encoder) takes in the original data points, including observation times, and outputs a set of $N_E$ values. This is similar to the idea of taking a light curve and fitting it to a model with $
N_E$ free parameters. A second function (the decoder) takes in a set of $N_E$ values and $T_N$ times to generate a light curve at the $T_N$ times. This architecture allows us to generate a light curve at different $T_N$ times; e.g., interpolated or extrapolated light curves, which is further explored in \S\ref{sec:dis}. In this work, we choose $T_N=T_0$; namely, we repeat the encoded values to match the original light curve length.
Our autoencoder utilizes gated recurrent neurons (GRUs; \citealt{cho2014learning,rumelhart1988learning}). In addition to the typical hidden weights that are optimized during training, recurrent neurons have additional weights that act as ``memory'' of previous input. GRUs in particular utilize an \textit{update} value (called a gate) and a \textit{reset} gate. The values of these neurons determine how the current and previous input affect the value of the output. With each light curve data point, the gates become updated with new information that informs the next prediction. This class of neurons is useful for our light curves with various numbers of observed data points. Our GRU neurons use the $\tanh$ activation functions with a hard sigmoid for the gate activation function.
Our RAENN is implemented in {\tt Keras} \citep{chollet2015} with a Tensorflow backend \citep{abadi2016tensorflow}. A diagram of the architecture is shown in Figure~\ref{fig:nnd}, and is outlined as follows:
\begin{enumerate}
\item \textbf{Input Layer}: Input light curve of size $T_0\times 9$ with each $griz$ data point labelled with a time (1 value) in days relative to light curve peak (4 values) and an uncertainty (4 values).
\item \textbf{Encoding Layer}: Encoding layer with $N_N$ neurons, where $N_N$ is a hyperparameter.
\item \textbf{Encoded Layer}: Encoded light curve with $N_E$ neurons, where $N_E$ is a hyperparameter.
\item \textbf{Repeat Layer}: Layer to repeat encoded light curve to match with new time-array, with size $T_0\times N_E$.
\item \textbf{Concatenate Layer}: Layer to concatenate new times to encoded light curve, with size $T_0\times (N_E+1)$.
\item \textbf{Decoding Layer}: Decoding layer with $N_N$ neurons.
\item \textbf{Decoded Layer}: $T_0\times 4$ decoded $griz$ light curve.
\end{enumerate}
To optimize the free parameters (the weights) of the RAENN model, we must define a \textit{loss function}. Our loss function is a simple mean square error function:
\begin{equation}
\mathcal{L}=
\sum_{i=0}^N\frac{\Big[F_{i,\mathrm{True}}(t,f) - F_{i,\mathrm{Predicted}}(t,f)\Big]^2}{N},
\end{equation}
where $F$ is the SN flux as a function of time $t$ and filter $f$. Although we feed unvertainties into the network, we find that excluding flux errors in our loss function substantially improved the ability of the RAENN to match the input light curves. We minimize our loss function using the gradient descent-based optimizer, {\tt Adam} \citep{kingma2014adam}, finding an optimal learning rate of $10^{-4}$, which is a typical value.
\begin{figure*}
\includegraphics[width=0.98\textwidth]{conf.pdf}
\caption{Confusion matrices for the full set of 557\ spectroscopically classified SNe. In the bottom panel, we include only objects where the maximum probability is $\ge0.7$ (438 events). \textit{Left panels:} Completeness-based confusion matrices, in which each row is normalized to equal one. Completeness quantifies how much of a spectroscopic class the classifier has correctly classified. \textit{Right panels:} Purity-based confusion matrices, in which each column is normalize to equal one. Purity quantifies how much a photometric class is comprised of the true spectroscopic class. By restricting our classes to the high-confidence objects (bottom panels), both our completeness and purity increase. \label{fig:conf}}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.7\textwidth]{cumlaplot.pdf}
\caption{\textit{Top:} Cumulative fraction of the spectroscopic SN sample as a function of classification confidence (left) and number of $>5\sigma$ data points (right), grouped by spectroscopic class. Misclassifications are marked with an ``x". \textit{Middle:} Cumulative fraction of the spectroscopic SN sample, grouped by photometrically-identified class. As expected, most misclassifications occur at low-confidence. At our chosen high-confidence cutoff ($p>0.7$), we find that the samples are largely pure. \textit{Bottom:} Cumulative fraction of the photometric SN sample, grouped by photometrically-identified class. The distributions based on classification confidence follow a similar trend to those seen in the spectroscopic sample, with Type Ia SNe and SLSNe having the highest fraction of high-confidence events. However, the photometric set has significantly more points on average when compared to the spectroscopic dataset. \label{fig:cumula}}
\end{figure*}
We randomly split our unlabeled dataset into training (2/3) and test (1/3) sets. We optimize the number of neurons in both the encoding and decoding layers (fixed to be the same number, $N_N$) and the number of encoding neurons ($N_E$) through a grid search, allowing $N_N$ to vary from 20 to 160 in intervals of 20, and $N_E$ to vary between 2 and 24 in intervals of 2. We find that, when optimizing over final classification F1-score (defined below), purity and completeness, our results are relatively insensitive to $N_E$ and $N_N$ for values of $N_E\sim10$ and $N_N\sim100$. For our final model, we use $N_E=8$ and $N_N=120$, which is \textit{not} our optimal model but a \textit{representative} model. Utilizing our optimal model without creating a valid test set (in addition to a training and validation set) would likely overestimate performance. Given our limited dataset, we are unable to properly optimize our hyperparameters and thus present representative results. We note that $N_N$ is slightly below the maximum number of data points in our set of light curves (where the longest light curve has 169 observed data points). The number of encoding neurons $N_E$ is similar to the number of free parameters for the analytical model used in \citet{villar2019} to capture the shape of a single-filter SN light curve.
We contrast our architecture with methods from \citet{naul2018recurrent} and \citet{pasquet2019pelican}, who present similar methodologies. \citet{naul2018recurrent} uses a similar GRU-based RAENN to classify variable stars with unevenly sampled light curves in one filter from the All Sky Automated Survey Catalog of Variable stars \citep{pojmanski2002all}. The flux and time since last observation ($\Delta t$) is sequentially read into the recurrent layers. The same time array is fed into the decoder for output. In our case, we feed in a time series across four filters and give a time array relative to peak rather than relative to the previous data point. This is more natural in our problem, in which the SNe have a clear beginning and end, versus the periodic signals of variable stars. Additionally, our architecture allows us to give the decoder a different time series to allow for interpolation or extrapolation of the data.
\citet{pasquet2019pelican} uses a semi-supervised method to classify simulated SN light curves from the SPCC \citep{kessler2010results}. They use an AE with convolutional layers by transforming the light curves into ``light curve images'' (see \citealt{pasquet2018deep}). Rather than interpolate the light curves, \citet{pasquet2019pelican} applies a mask to filters that are missing data at a certain time. In contrast, we interpolate our light curves but assign interpolated values a large uncertainty of 1 mag, as explained above. We found that the method of transforming light curves into images and masking across four filters led to unstable training and poorer performance. This is likely due to the large data gaps in the {\it real} PS1-MDS light curves, compared to the high-cadence (2-days for each filter) {\it simulated} light curves of SPCC. Since the LSST data are expected to more closely resemble the PS1-MDS light curves than the SPCC simulated events, we expect our method to be more robust in a real-life application.
\begin{figure}
\includegraphics[width=0.45\textwidth]{conf_other_1.pdf}
\caption{ Completeness and purity confusion matrices, generated from classifying the spectroscopic dataset using only RAENN features and leave-one-out cross-validation. Even without additional features, the classifier performs similarly to other simulation-based classifiers such as those presented in \citet{muthukrishna2019rapid} and \citet{boone2019}. \label{fig:conf21}}
\end{figure}
\subsection{Supervised Learning: Random Forest Classifier}
As a final step, we use the encoded light curves as features for a supervised classification method. Following \citet{villar2019}, we train a random forest (RF) classifier on the PS1-MDS spectroscopically classified SNe, including the RAENN encodings as features.
In addition to the encoding (8 features), we use the following 36 features based on the GP-interpolated light curves:
\begin{itemize}
\item The $griz$ rise times in the rest frame, calculated 1, 2, and 3 mag below peak (12 features).
\item The $griz$ decline times in the rest frame, calculated 1-, 2- and 3-magnitudes below peak (12 features).
\item The $griz$ peak absolute magnitudes (4 features)
\item The median $griz$ slope between 10 and 30 days post-peak in observer frame. This area was chosen by eye to specifically help the model differentiate between Type II and Type Ibc SNe (4 features).
\item The integral of $griz$ light curves (4 features).
\end{itemize}
We measure these values from the GP-interpolated light curves rather than the decoded light curves. The decoded light curves are, at best, approximations of the GP-interpolated light curves. Therefore, using them would only result in noisier features. The decoded light curves are necessary, however, as a means to train the RAENN to extract the $N_E$ encoding neuron values. We note that for some features, e.g., the rise and decline times, the feature values are heavily dependent on the GP extrapolation in cases where there is no measured data. Including GP errors in the supervised portion of our analysis could help capture this intrinsic uncertainty in the underlying light curve, but we leave that exploration to future work.
These features were chosen through trial-and-error while optimizing classification accuracy. We find that inclusion of all features leads to our optimal classification accuracy, although we do explore how well our classifier performs with the RAENN features alone in the following section.
We pass these features through a RF classifier, utilizing 350 trees in the random forest and the Gini-information criterion. The number of trees was determined based on trial-and-error optimization. To counteract the imbalance across the five spectroscopic classes, we tested several algorithms to generate synthetic data to augment our training set. Following \citealt{villar2019}, we use a Synthetic Minority Over-sampling Technique (SMOTE; \citealt{chawla2002smote}) and a multivariate Gaussian (MVG) fit. We additionally test using a Kernel Density Estimate (KDE) of the training set, using a Gaussian kernel with bandwidth equal to 0.2 (or 20\% of the whitened feature standard deviation). We find that the MVG with a halved covariance matrix performs best. We test our classifier using leave-one-out cross validation, in which we remove one SN from the sample, oversample the remaining objects by generating new objects using the MVG, and then apply the trained RF to the single, removed event and recording the result. For each object, our RF reports probabilities associated to each class, which are calculated using the fraction of trees which vote for each class. We take the class with the highest probability as the predicted SN type.
\section{Classification Results}\label{sec:red}
\begin{figure}
\includegraphics[width=0.45\textwidth]{conf_other_2.pdf}
\caption{\textit{Bottom:} Confusion matrices for a simpler Type Ia SN versus non-Ia (CCSN) classification, generated by collapsing the complete confusion matrices. \label{fig:conf22}}
\end{figure}
There are several metrics to measure the success of a classifier. We focus on three metrics: the purity, completeness and accuracy. We define the three, calculated for a single class, below:
\begin{equation}
\begin{split}
\mathrm{Purity} &= \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}}\\
\mathrm{Completeness} &= \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}\\
\mathrm{Accuracy} &= \frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{TS}}
\end{split}
\end{equation}
where TP (FP) is the number of true (false) positives, TN (FN) is the number true (false) negatives, and TS is the total sample size. We optimize the hyperparameters of our classifier using the F1-score, defined here as the class-averaged harmonic mean of the purity and completeness.
\subsection{Spectroscopic Sample}
We visualize the completeness and purity of the spectroscopic sample using confusion matrices in Figure~\ref{fig:conf}. A confusion matrix compares our RAENN label (horizontal axis, in the case of the completeness matrix) with the spectroscopic label (vertical). Results are shown for leave-one-out cross validation, in which one event is removed from the sample for training and the trained model is applied to the left out event. As with \citet{villar2019}, we find that our classifier performs best for Type Ia SNe (92\% completenss), SLSNe (76\%), and Type II SNe (82\%), and worst for Type Ibc SNe (37\%). Our class-averaged classification completeness is 69\% across the 5 SNe types. This is worse than the performance of \citet{villar2019}, who find a class-averaged completeness of 80\%. Our class-averaged purity is 66\%, again slightly worse than the average purity of 72\% found in \citet{villar2019}. When limiting the sample to only objects in which the classification probability is $\ge 0.7$ (a total of 438 objects), we find that our performance increases, with a class-averaged completeness of 75\% and a class-averaged purity of 74\% with a loss of 20\% of the sample size.
Next, we explore the classification confidence reported by our algorithm. The confidence estimates are directly outputted by the RF. With larger datasets, one can calibrate the outputted uncertainties using e.g., an additional logistic function. Given our small dataset, we do not perform any additional calibration. In Figure~\ref{fig:cumula} we plot the cumulative fractions of SNe in our training set, grouped by their spectroscopic and photometric classifications. The majority of high-confidence objects are Type Ia SNe, with nearly half of the spectroscopic Type Ia SNe having a confidence (p$>0.98$). Similarly, half of the SLSNe have high confidence identifications ($p>0.8$). Type Ibc SNe and Type IIn SNe have the lowest confidence on average, with the majority of events having $p<0.5$. This is likely reflective of the fact that Type Ibc and Type IIn SNe span a wide range of observed properties, including overlap with Type Ia SNe.
Figure~\ref{fig:cumula} also indicates the misclassified objects. Ideally, we want our misclassifications to largely occur in low-confidence objects. This is the case for SLSNe, Type Ia SNe and Type II SNe. For Type IIn and Type Ibc SNe, the misclassifications occur even for high-confidence events. This indicates that for Type Ia, Type II and SLSNe, misclassifications are likely tied to data quality. In contrast, misclassifications of Type IIn and Type Ibc SNe seem to be due to intrinsic overlap of the classes in feature-space with other SNe (mainly Type Ia SNe).
We additionally attempt to sort events based on the number of data points, rather than classification confidence (see the right column of Figure~\ref{fig:cumula}). Our photometric dataset has, on average, fewer $>5\sigma$ datapoints compared to our spectroscopic dataset ($\sim15$ versus $\sim30$ data points on average). Because of this mismatch and the lack of a strong correlation between number of points and classification confidence, we do not further explore how cutting sparse light curves affects our final classification accuracy.
We next turn our attention to the performance of our classifier when constrained to only data-driven (RAENN) features. Using the same set of RAENN features \textit{without} any additional information, we produce the confusion matrices shown in Figure~\ref{fig:conf21}. We find a class-averaged completeness of 53\%, approximately 20\% worse than including the additional features. The overall breakdown is similar to our final confusion matrix, with the worst-performing classes being Type Ibc and Type IIn. We find that our RAENN-only classification is more inclined to label events as Type Ia SNe, likely a bias from the fact that our SN dataset used to train the RAENN is highly dominated by Type Ia SNe. If we run our classification algorithm \textit{without} the RAENN features, we find that {\tt SuperRAENN} performs similarly (slightly worse), implying that the RAENN has not picked up on uniquely helpful features independent from our hand-selected feature set. To be clear: the intent of RAENN is not necessarily to outperform hand-selected features but to create model-independent features in real time. In this work, we determine final classifications with the RAENN and hand-selected features to provide the highest confidence photometric classifications. Improvements to classifications based solely on RAENN features is left to future work.
While not optimized for Type Ia versus non-Type Ia SN classification, we explore how well our classifier (using the full set of features) performs when we collapse the confusion matrix into just two classes. In Figure~\ref{fig:conf22}, we show the completeness and purity confusion matrices for Type Ia versus non-Ia (CCSN) classifications, finding $\approx90$\% completeness and $>80$\% purity in both classes.
The random forest classifier allows us to measure the relative ``importance" of the 44 features used to classify the SNe. We define importance as the decrease in the Gini impurity, which accounts for how often a feature is used to split a node and how often a node is reached in the forest \citep{breiman1984classification}. We show the importance of each RF feature in Figure~\ref{fig:featimport}, along with the measured importance for a Normal random variable. The peak magnitudes and decline rates are the most important features for classification. However, the RAENN features also have significant influence on the final classifications, with two RAENN features appearing in the top ten important features.
The feature importance unfortunately loses some quantitative meaning if the features are correlated, which is the case with our features. When two features are highly correlated, one may be arbitrarily measured as more important, so the general trends are more meaningful than precise order. We show the magnitudes of the feature correlations in Figure~\ref{fig:dendro} to better understand the underlying correlations. There are clear correlations between features derived in multiple bands (e.g., the peak magnitude in $g$-band is highly correlated to that in $r$-band). However, we also see correlations between the RAENN features and the more traditional light curve features. About half of the RAENN features seem strongly correlated with the peak magnitudes, while two others seem well-correlated with rise and decline times. A more detailed exploration of the physical interpretation of the RAENN feature-space may be worthwhile but is beyond the scope of this work.
\begin{figure*}
\includegraphics[width=\textwidth]{featimport.pdf}
\caption{Feature importance (grey). The blue horizontal line shows the importance measure for a normally-distributed random variable; features at or below this line can be considered largely unimportant to the final classification. In our case, all featured are considered important by the RF. \label{fig:featimport}}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{samplelc.pdf}
\caption{A sample of SNe from our photometric sample, sorted by low (left column) versus high (right column) confidence and photometrically identified SN class (rows). Here we show only $>3\sigma$ detections and otherwise show magnitudes as upper limits (triangles). Low-confidence in classification appears to be both due either poor data quality or confusion between multiple classes. \label{fig:samplelc}}
\end{figure*}
\begin{figure}
\includegraphics[width=0.5\textwidth]{dendro.pdf}
\caption{Absolute values of the covariance matrix of the various features used in our classification method, where darker blue represents a stronger absolute correlation. Unsurprisingly, the same features derived from different bands (e.g., the peak $g$-band flux versus the peak $r$-band flux) are highly correlated. The RAENN features are also correlated to the physically-motivated parameters, with some being strongly correlated to peak magnitudes, some to rise and decline times, and some to the post-peak slope. \label{fig:dendro}}
\end{figure}
\subsection{Classifying the Complete Photometric Dataset}
\begin{figure*}
\includegraphics[width=\textwidth]{bar.pdf}
\caption{Breakdown of SN subclasses in the spectroscopic and photometric samples. There is a significantly smaller fraction of Type Ia SNe in our photometric sample (orange) versus our spectroscopic sample (blue), implying we have misclassified Type Ia SNe as CCSNe. If we limit our photometric sample to the high-confidence ($p>0.7$) events (green), the class breakdowns are better aligned. Using our confusion matrix, we can correct the photometric class breakdown for known biases (see text for details; red), which also better aligns our class breakdowns. Finally, we compare our results to the ZTF Bright Transient Survey, finding good agreement between the spectroscopic class breakdown and corrected photometric class breakdown \citep{fremling2019zwicky}.\label{fig:pie}}
\end{figure*}
We apply our trained classification algorithm to the PS1-MDS dataset of SN-like transients that pass our quality cuts described in \S~\ref{sec:ps1}. We report the probabilities of each class type for each light curve in Table~\ref{tab:2}. Error bars for each class probability are calculated by running the trained RF classifier 25 times with unique random seeds. We show the class breakdown of the complete photometric set (2,885\ SNe) in Figure~\ref{fig:pie}. Excluding the spectroscopic sample, we present \newsne\ new SNe with 1435\ (61.9\%) Type Ia SNe, 459\ (19.9\%) Type IIP SNe, 272\ (11.7\%) Type Ibc SNe, 112\ (4.8\%) Type IIn SNe, and 37\ (1.6\%) SLSNe. Of these, 1,311 are high-confidence ($p>0.7$) photometric classifications. A cumulative plot of the confidences grouped by each photometric class is shown in Figure~\ref{fig:cumula}; the distribution of these probabilities largely match the spectroscopic sample.
A sample of SNe from each photometric class is shown in Figure~\ref{fig:samplelc}, including high- and low-probability examples. For the low-probability examples, it seems that even well-sampled light curves can have low confidence scores, likely because the features of their light curves reside on a region of feature space in which various SN classes reside.
The redshift distributions of the new, photometrically-classified events is roughly consistent with that seen in the spectroscopic sample (Figure~\ref{fig:f1}), with Type Ibc peaking at $z\sim0.19$, Type II peaking at $z\sim0.21$, Type Ia and Type IIn peaking at $z\sim0.42$ and SLSNe peaking at $z\sim0.58$.
We compare the overall photometric breakdown of SN types to that of the ZTF Bright Transient Survey \citet{fremling2019zwicky}, which spectroscopically classified 761 SNe with peak $g$- or $r$-band magnitude of $<18.5$. \citet{fremling2019zwicky} find that their magnitude-limited survey breaks down into 72\% Type Ia SNe, 16\% ``normal" Type II SNe (Type IIP/L), 3\% Type IIn SNe (including their Type IIn and SLSN-II category), 5\% Type Ibc SNe, and 1.6\% Type I SLSNe. This is a similar breakdown found in our spectroscopic sample. Comparing to our photometric set, we find a slightly higher fraction of Type II and Type IIn SNe and a lower fraction of Type Ia SNe (all within $\sim20$\% of the ZTF BTS values), as shown in Figure \ref{fig:pie}. For our high-confidence ($p>0.7$) sample (also shown in Figure~\ref{fig:pie}), our class breakdowns are closer to those of our spectroscopic and the ZTF BTS sample, with a slight over-abundance of Type Ia SNe ($\approx78$\%). Based on our understanding of how our classifier performs on the training set, we can understand the biases present (e.g., that some spectroscopic Type Ibc SNe are classified photometrically as Type Ia SNe). We can use these known biases, encoded within the confusion matrices, to correct our class breakdown. Mathematically, this is calculated as the dot product of the purity matrix and our original class breakdown. Applying this correction to the photometric dataset, the class breakdown is well aligned with the breakdown of our spectroscopic sample, as shown in Figure~\ref{fig:pie}. This study should \textit{not} be used to rigorously study the observational breakdown of SN classes; however, the fact that our $p>0.7$ sample is in relatively good agreement with the ZTF BTS provides some evidence that our photometric sample is correctly labelled.
\section{Discussion}\label{sec:dis}
\subsection{Classification of Other Transients}\label{sec:dis1}
Our algorithm assumes that every SN belongs in one of five classes: SLSNe, Type II SNe, Type IIn SNe, Type Ia SNe and Type Ibc SNe. Yet what does our algorithm do for transients which do not fall in these five classes? Here we address this question for a number of spectroscopically classified extragalactic transients. We summarize the photometric classification for these rare transients in Table~\ref{tab:3}.
\citet{drout2014rapidly} presented a sample of ten extragalactic transients discovered with PS1-MDS with redshift measurements which rise too rapidly to be powered solely with $56$Ni\footnote{\citet{drout2014rapidly} presents an additional four objects which lack a confident redshift estimate (the ``bronze" sample), which we exclude from our analysis.}. Following \citet{rest2018fast}, we refer to these as FELTs. FELTs have a broad range of peak magnitude (-16 $\gtrsim M\gtrsim$ -20), which is reflected in the distribution of photometric classifications. Of these ten objects, six objects have ``high confidence" ($p>0.7$) classifications in one of our five categories: four of which are Type Ia SNe and two of which are Type II SNe. The other four objects are classified as low confidence Type Ia (one object), Type II (two objects) and Type Ibc (one object). As expected, the higher-luminosity objects are those classified as Type Ia, while the lower-luminosity objects are classified at Type II. The majority of objects have Type Ibc as their second-highest classification. Based on this analysis, FELTs are likely a (small) contaminant of both Type II and Type Ia SNe in our sample, and our algorithm would need to be retrained to specifically classify FELTs.
Two known TDEs were discovered in PS1-MDS: PS1-10jh (PSc040777, \citealt{gezari2012ultraviolet} and PS1-11af (PSc120170, \citealt{chornock2014tde}. Both objects are classified as Type IIn SNe with $p\sim0.8$ and $p\sim0.6$, respectively. This makes intuitive sense, as the light curves tend to be long-lived and bright like some Type IIn SNe. Both objects have Type Ia and Type II as their next most likely classifications. Based on these, it may be possible to search for TDEs in our sample within the photometric Type IIn sample.
We highlight four other SNe which do not fit in our five categories. PS1-10afx (PSc080333) is a lensed Type Ia SN \citep{chornock2013lensed,quimby2014lensed}, which peaks at -22 mag. We classify PS1-10afx as a high probability ($p\sim0.9$) SLSN. PS1-12sk (PSc370290) is a Type Ibn SN \citep{Sanders2013ibn} which peaks at $M\sim-19$. We classify PS1-12sk as a low probability Type Ia ($p\sim0.6$) or Type IIn ($p\sim0.4$). We classify PS1-12sz (PSc370330) as a likely IIb SN using SNID; PS1-12sz peaks at $M\sim-18.5$. We photometrically classify this object as a low probability Type Ibc ($p\sim0.6$). Finally, SN 2009ku (PS0910012) is a spectroscopically identified Type Iax \citep{narayan2011displaying} which peaks at $M\sim-18.5$. We classify this object as a low probability Type Ia ($p\sim0.5$) or Type Ibc ($p\sim0.3$).
\subsection{Potential Biases}
As discussed in \S~\ref{sec:ps1}, our spectroscopic sample is somewhat brighter and at a lower redshift than our test set. This difference may introduce biases in our final classifications, although this effect should be minimal considering the small ($\sim 1$ mag) difference between the two sets. De-redshifting the SNe removes some of this bias, by removing knowledge of the underlying redshift as a feature.
The relative fractions of SN subtypes may evolve with redshift as host properties change (see e.g., \citealt{graur2017loss} for an exploration of the correlations between host properties and SN type). Our spectroscopic and photometric sets differs most greatly at $z\gtrsim 0.5$ (see Fig.~\ref{fig:f1}). In this redshift range, average host metallicity is not expected to drastically shift \citep{lilly2003metallicities}, implying a small potential bias. A separate bias may arise from the fact that our photometric sample relies on a measured spectroscopic redshift. At higher redshift, our galaxy redshift measurements become increasingly uncertain as dominant emission lines shift out of the optical band and intrinsically dim hosts fall below our observational limits. In contrast, rest-frame UV features of SNe (especially SLSNe) remain in the optical band, making it easier to confidently measure a distance from SNe spectra. In the future, this problem can be mitigated with photometrically derived host galaxy redshifts.
As expected, the relative observed fraction of SN subtypes evolves with redshift due to the magnitude limit of the survey. We trace this evolution in Figure~\ref{fig:zfrac}. We show the cumulative fraction (integrating from $z=0$) of each subclass as a function of redshift. Each subclass peaks in order of luminosity function. The dimmest subclass, Type II SNe, dominates the sample for $z<0.3$, peaking near $z\sim 0$.
Using the high redshift ($z>0.75$) sample, we can test if redshift information is playing an unwanted role in our training. The spectroscopic sample at $z\gtrsim0.75$ is solely made up of SLSNe; however, we do not expect \textit{all} high-$z$ objects to be SLSNe. Given a typical limiting magnitude of $m_\mathrm{r,lim}\sim23.3$, the corresponding absolute magnitude is $\sim-20$ at $z=0.75$. At this sensitivity, we expect to find SLSNe, Type IIn SNe and potentially bright Type Ia SNe (if the limiting magnitude is slightly deeper). For $z>0.75$, we find that our photometric sample (a total of 28 SNe) is 68\% SLSNe, 18\% Type IIn SNe and 14\% Type Ia SNe (with all Ia SNe occurring at $z<0.85$), implying our classifier has not learned to simply classify all high-$z$ events as SLSNe. The high-$z$ Type Ia SNe, in particular, have noisy light curves which peak at $M\sim-20$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{z_frac.pdf}
\caption{Observed SN subclass cumulative fraction as a function of redshift (colored) and the overall cumulative distribution (grey). \label{fig:zfrac}}
\end{figure}
\subsection{Comparison to Other Works}
We first compare our results to \citetalias{hossen20}, which extends the work of \citet{villar2019} to classify the PS1-MDS photometric sample using features extracted from analytical fits to the light curves. Overall, \citetalias{hossen20} (and \citealt{villar2019}) achieve better performance at the cost of a more computationally-expensive feature extraction method. We agree with 74\% of the photometric classifications of \citetalias{hossen20}. If we compare the top two labels, the algorithms agree on 95\% of classifications. Indeed, often the top two classification choices are flipped for either algorithm, occurring most often with Type II/Ibc SNe and Type IIn/Ia SNe. We find stronger agreement if we exclude objects with low classification confidence; namely, using only $p>0.7$ in both algorithms, our classifications agree 84\% of the time (with 1,597 objects remaining after the cut, i.e., a loss of $\sim50$\% of the sample). The agreement increases further for even higher probability cuts of $p>0.8$ ($>0.9$), with 88\% (92\%) agreement with 1249 (888) objects remaining. Most classification disagreements lie in the Type Ibc/IIn categories, which have low confidence classifications. We find that our algorithm is more likely to classify SNe as Type Ia, likely a bias built into the unsupervised step of training on the complete dataset (which is dominated by Type Ia SNe). A more detailed comparison of these two results is provided in \citetalias{hossen20}.
\citet{villar2019} discusses the difficulty in comparing our results to the broader literature. In short, previous works have largely focused on Type Ia versus CCSN classification (e.g., \citealt{ishida2013kernel,jones2017measuring,brunel2019cnn}) or have been trained and tested on simulated data (e.g., \citealt{kessler2010results,muthukrishna2019rapid,moller2020supernnova}). In the case of Type Ia versus CCSN classification, we achieve an accuracy of $\approx92$\%, similar to (but somewhat worse than) specialized classifiers \citep{jones2017measuring,brunel2019cnn}.
When comparing to works based on simulated data, we caution that not all simulated datasets are suitable for multi-class SN classification. In particular, the Supernova Photometric Classification Challenge (SNPCC) training set \citep{kessler2010results} lacks the SN diversity necessary to accurately train classifiers and will lead to artificially promising results. PLAsTiCC \citep{allam2018photometric,kessler2019models} is better suited for this task, and we encourage future work to be built on this dataset or the PS1-MDS dataset presented here.
We next compare our results to \citealt{jones2017measuring}, who presented a PS1-MDS sample of 1,169 likely Type Ia SNe, focusing on Type Ia versus non-Ia classification. \citealt{jones2017measuring} used four classification algorithms: the template-matching algorithm PSNID \citep{sako2011photometric}, a nearest neighbor approach using the PSNID templates; an algorithm based on fitting light curves to SALT2 templates \citep{guy2007salt2}; and a method, GALSNID \citep{foley2013classifying}, which only utilizes host galaxy properties. \citet{jones2017measuring} similarly removed objects with unreliable host redshifts and potential AGN hosts, but unlike our analysis they removed objects at $z>0.75$. Of their 1,169 identified Type Ia SNe, only 1,046 SNe pass our quality cuts to be classified in this work. For these, we find 95\% agreement. Of the remaining 48 SNe, we identified Type Ia as the second highest choice in 24 cases. Of the remaining 24 cases, 15 have low Type Ia probabilities ($p<0.8$ from \citealt{jones2017measuring}) or classification probabilities based entirely on host galaxy. It is worth noting that our classifier, similar to \citet{jones2017measuring}, achieves 96\% purity in Type Ia SNe, making it likely usable for cosmological studies \citep{jones2018measuring}.
We compare our results to those trained on PLAsTiCC -- in particular, \citealt{boone2019,muthukrishna2019rapid} and \citealt{Gabruseva2020}. These classifiers present average completenesses of $\approx0.88$ for SLSNe (higher than our score), $\approx0.5$ for Type II/IIn SNe (lower than our averaged Type II/IIn score), $\approx0.92$ for Type Ia SNe (similar to our score), and $\approx0.46$ for Type Ibc SNe (similar to our score given low-number statistics). These results are based on simulated data which lack the complexity of real data, so it is encouraging that our algorithm performs similarly or outperforms these works. It would be interesting and useful to the community to know how these algorithms perform on the PS1-MDS dataset, but we leave this for future work.
\begin{figure*}
\hspace*{-2cm}\includegraphics[width=1.25\textwidth]{extrap.pdf}
\caption{Examples of a Type Ia SN (\textit{top row}), Type II SN (\textit{middle row}) and Type Ibc SN (\textit{bottom row}). Filled points represent observations used to generate the RAENN model (colored lines), while empty points are the full data set to guide the eye. In the right-most column, we show the root-mean-squared (RMS) error as a function of SN phase, as more data are being included in the RAENN model. Interestingly, the RMS reaches $\sim1$ near peak for all SNe shown. We emphasize that the RAENN model has been optimized to classify complete SN light curves rather than partial light curves.\label{fig:extrap}}
\end{figure*}
\subsection{RAENN Architecture: Limitations and Benefits}
We now turn to the architecture of the RAENN itself and its use in future surveys. The recurrent neurons allow our neural network to generate light curve features that can be updated in real time, in addition to extrapolating and interpolating light curves. We highlight the accuracy of the RAENN light curve model as a function of light curve completeness in Figure~\ref{fig:extrap}. We track how well the RAENN is able to both model the complete light curve and accurately classify the SNe with limited data by providing a partial light curve into the RAENN. For each step, we hold the other features (e.g., peak luminosity and duration) constant. This is not a completely robust method, as some features (e.g., decline time) cannot be measured before peak. We leave the optimization of {\tt SuperRAENN} for real time data streams to future work. We find that {\tt SuperRAENN} performance drastically improves post-peak, but that it can provide accurate classifications and light curves somewhat before peak. To explore why {\tt SuperRAENN} improves near-peak, we track how the RAENN features change as the light curves evolve. In Figure~\ref{fig:en1}, we plot the values of representative encoding values of a Type Ia SN. The encodings vary smoothly until settling on the correct final values $\sim10$ days post-peak.
The ability of the RAENN to extrapolate light curves without built-in physical assumptions allows it to search for anomalous events in real time for the purpose of spectroscopic and multi-wavelength follow-up. Given the millions of events expected from LSST, it is essential to search for unexpected or previously unknown physical effects that. One concern is that our algorithm is potentially not robust to noisy live-streaming data; in other words, our algorithm must be able to distinguish between anomalous data and noisy data. We check the stability of our encoded values as a function of scaled white noise by adding white noise to a light curve. We then use our RAENN to encode the noisy light curve and record the scatter of the encoded values. We report the results of this test in Figure~\ref{fig:en2}, in which we show the scaled scatter of the encoded values as a function of the magnitude of the injected noise. The scatter grows linearly with noise; however, even with one magnitude of scatter added to the light curve, the overall scatter of the encoded values only increases to 30\% of the overall spread of class's features. This implies that the RAENN is largely robust to noise
Several steps need to be taken to allow our architecture to work on streaming data. First, we use phases relative to maximum light, which will be unavailable during the rise of the SN. A shift to a time measurable early in the light curve, like time of first detection, will allow the RAENN to otherwise perform as designed. Similarly, the features utilized during the supervised portion of our classifier rely on the full light curve being available. All features can be estimated from extrapolated RAENN light curves or a new set of features may be used on streaming data. Finally, although not necessary, our RAENN could output uncertainties on the SN light curves by converting the network into a \textit{variational} AE, which is designed to simultaneously find an encoding space and uncertainties on the encoded data. This more complex architecture would likely require a larger training set to be reliable. Finally, we note that an algorithm like RAENN could be used in conjunction with an active oracle (a software which recommends new observations to improve classification) such as REFITT \citep{2020Sravan}, in order to actively optimize classification accuracy.
\section{Conclusions}\label{sec:con}
Deep learning-based classifiers are becoming increasingly important for classification of archival SN light curves. In this paper we present a novel, semi-supervised approach to light curve classification, which utilizes spectroscopically labelled and unlabelled SN data from the PS1-MDS. Our key conclusions are as follows:
\begin{enumerate}
\item We present the light curves of 5,243\ SN-like events discovered with PS1-MDS.
\item We present the spectroscopic classifications of 557\ SNe, including 17\ Type I SLSNe, 94\ Type II, 24\ Type IIn, 404\ Type Ia and 19\ Type Ibc SNe.
\item We measure and report the spectroscopic redshifts for 2,885\ SN-like events used in our unsupervised training set.
\item We present a new, open source photometric classification algorithm, {\tt SuperRAENN}. {\tt SuperRAENN} uses a semi-supervised approached and novel neural network architecture to classify irregularly-sampled SN light curves.
\item Using {\tt SuperRAENN}, we extract learned, nonlinear features from the sparse light curves. We use these features and others to classify the complete set of 2,885\ SN-like objects in the PS1-MDS dataset with host galaxy redshifts.
\item We achieve high (87\%) accuracy for our spectroscopically labelled sample. We find best performance for SLSNe, Type Ia, and Type II SNe due to their distinctive regions of feature space. We find worst performance for Type Ibc SNe, likely due to the small sample size (just 19 events) and their significant overlap with Type Ia SNe and the subset of rapidly-declining Type II SNe (formerly, IIL).
\item Compared to previous studies, we find that our general classifier performs as well or can outperform classifiers trained on synthetic data sets.
\item We perform simple tests for classification bias and method robustness to noise, finding our method robust to both.
\end{enumerate}
In addition to these key results, we highlight several lessons learned from this study. We find that both Type IIn and Type Ibc classes suffer from poor accuracy likely due to substantial overlap with Type Ia SNe in feature space. This finding has also been shown in \citet{villar2019} and \citetalias{hossen20}, implying this is a general problem for classifiers. Additionally, rare transients, e.g. FELTs, abnormal Type Ia classes, etc., can be hidden as high-confidence events in another class \textit{or} low-confidence events across several classes. Adapting pre-existing classifiers to new classes should be taken on a case-by-case basis. Finally, we find that a mixture of hand-selected and data-driven (in our case, RAENN) features can improve classification accuracy, but hand-selected features seem to generally out perform data-driven features.
Finally, we note that several modifications to our presented classifier will allow it to work with live, rather than archival, data streams such as ZTF and LSST. We perform simple tests and find that our classifier performs optimally around peak, although we have not optimized for this purpose. Finally, the RAENN architecture may also be utilized to search for anomalous events in real time. We plan to explore this in future work.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{feature.pdf}
\caption{\textit{Top:} Normalized, GP-interpolated $r$-band light curve of a spectroscopically-classified Type Ia SN. \textit{Bottom:} Representative set of three (orange, purple, blue) normalized AE features as a function of SN phase. To generate these features, we run the light curve data through the RAENN up to a certain phase. As shown, the values vary smoothly and then settle on the final values about one week post-peak. \label{fig:en1}}
\end{figure}
\begin{figure}[b]
\includegraphics[width=0.5\textwidth]{encoding_noise.pdf}
\caption{Average spread of the RAENN features for a spectroscopic Type Ia SN as a function of light curve noise. For every noise scale, we run 100 simulations, adding random noise to the light curve. We then track the average spread of each parameter. We scale this spread by the total spread in the Type Ia class. Even with an injected error of 0.5 mag, the spread in the RAENN feature space only reaches 30\% of the total spread throughout the Type Ia class in feature space, implying the method is robust to noise. \label{fig:en2}}
\end{figure}
\input{table1.tex}
\input{table2.tex}
\input{table3.tex}
\facilities{ADS, NED, PS1, TNS}
\defcitealias{astropy_collaboration_astropy_2018}{Astropy Collaboration 2018}
\software{Astropy \citepalias{astropy_collaboration_astropy_2018}, extinction \citep{barbary_extinction_2016}, keras \citep{chollet2015keras}, Matplotlib \citep{hunter_matplotlib:_2007}, NumPy \citep{oliphant_guide_2006}, RVSAO \citep{kurtz_rvsao_1998}, scikit-learn \citep{pedregosa_scikit-learn_2011}, SciPy \citep{virtanen_scipy_2020}}
\acknowledgements
We thank Jessica Mink and Brian Hsu for providing assistance with host galaxy redshifts estimates. V.A.V.~acknowledges support by the Ford Foundation through a Dissertation Fellowship and the Simons Foundation through a Simons Junior Fellowship (\#718240). G.H. thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant \#1829740, the Brinson Foundation, and the Moore Foundation; his participation in the program has benefited this work. The Berger Time-Domain Group is supported in part by NSF grant AST-1714498 and the Harvard Data Science Initiative. D.O.J. is supported by a Gordon and Betty Moore Foundation postdoctoral fellowship at the University of California, Santa Cruz. R.L. is supported by a Marie Sk\l{}odowska-Curie Individual Fellowship within the Horizon 2020 European Union (EU) Framework Programme for Research and Innovation (H2020-MSCA-IF-2017-794467). D.M. acknowledges NSF support from from grants PHY-1914448 and AST-2037297. The UCSC team is supported in part by NASA grants 14-WPS14-0048, NNG16PJ34C, NNG17PX03C, NSF grants AST-1518052 and AST-1815935, NASA through grant number AR-14296 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555, the Gordon \& Betty Moore Foundation, the Heising-Simons Foundation, and by fellowships from the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation to R.J.F. Some of the computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University. The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.
|
1511.07491
|
\section{Introduction}
\noindent
In the spring of 1996 I was visiting the City College of New York for a month,
in order to pursue a research project with Stuart Samuel, who was a professor
at City University of New York at the time, and to run in the 100th Boston marathon.
Several evenings and part of weekends I'd spend with our mutual friend Pascal Gharemani,
a tennis coach and instructor at Trinity School (a private high school on West
91st Street in Manhattan). Typically we would go dining, visit places or fly kites.
Pascal had an Iranian background but grew up in Versailles near Paris before
moving to the US. My wife and I had come to know him during my postdoc years
at City College (1987--90), when we would meet weekly at various restaurants in the
Columbia University neighborhood for an evening of French conversation.
He was important for our socialization in Manhattan and had grown into a good friend.
Pascal was a very curious individual, with a great sense of humor and always ready
to engage in discussions about savoir vivre, philosophy, and the natural sciences.
Regarding the latter, he regularly pondered phenomena and questions which involved physics.
Lacking a formal science training, he would go to great lengths and try his physicist
friends for explanations.
So one evening in 1996 he shared his musings about the gravitational force of a long
and homogeneous rod, as it is felt by a (say, minuscule) creature crawling on its surface.
Clearly, the mass points in its neighborhood are mainly responsible for creating the force.
On one hand, at the end of the rod, the nearby mass is fewer than elsewhere, but it is all
pulling roughly in the same direction. On the other hand, around the middle part of the rod,
twice as much mass points are located near the creature, yet their gravitational forces point
to almost opposing directions and hence tend to cancel each other out. So which location
gives more weight to the mini-bug? Where along the rod is its surface gravity largest?
This was a typical `Pascal question', and my immediate response was: ``That's an easy one.
Let me just compute it.'' Well, easier said then done. For the mid-rod position
the resulting integrals were too tough to perform on the back of an envelope.
To simplify my life, I persuaded Pascal to modify the problem.
Let us vary not the position of the bug but the geometry of its planet:
keep the bug sitting on the top of a cylinder, and compare a long rod with
a slim disk of the same volume and mass. Then it was not too hard to calculate the surface
gravity as a function of the ratio of the cylinder's diameter to its length.
To our surprise, in a narrow window of this parameter the weight of the bug exceeds the
value for a spherical ball made from the same material. This finding inspired us to
generalize the question to another level: Given a bunch of homogeneous material
(fixed volume and density, hence total mass), for which shape is the gravitational
force somewhere on its surface maximized? Thus, the idea of ``asteroid engineering'' was born.
After solving the problem and comparing the result with a few other geometries,
I put the calculations aside and forgot about them. Four years later, when teaching
Mathematical Methods for physics freshmen, I was looking for a good student exercise
in variational calculus. Coming across my notes from 1996, I realized they can be turned
into an unorthodox, charming and slightly challenging homework problem.
And so I did, posing the challenge in the summer of 2000~\cite{exercise1} and again
in 2009~\cite{exercise2}, admittedly with mixed success.\footnote{
In 2002, the problem also occurred in a physics quizz page~\cite{kantor} and later
in the textbook~\cite{morin}.}
But let the reader decide!
\newpage{}
\begin{figure}[!ht]
\centerline
\includegraphics[width=8cm]{pascalfig1.eps}
\caption{Geometry of massive cylinder}
\label{fig:1}
\end{figure}
\vspace{1cm}
\section{Surface gravity of a homogeneous massive cylinder}
\noindent
It is textbook material how to compute the Newtonian gravitational field~$\vec G(\!\vec{\,r})$
generated by a given three-dimensional static mass distribution~$\rho(\!\vec{\,r}')$.
In the absence of symmetry arguments, it involves a three-dimensional integral collecting
the contributions
\begin{equation}
{\mathrm{d}}\vec G(\!\vec{\,r},\!\vec{\,r}') \=
\gamma\,\rho(\!\vec{\,r}')\,\frac{\!\vec{\,r}'\!-\!\vec{\,r}}{|\!\vec{\,r}'\!-\!\vec{\,r}|^3}\,{\mathrm{d}}^3\!\vec{\,r}'
\end{equation}
produced by the masses at positions $\!\vec{\,r}'$, with $\gamma$ denoting the gravitational constant.
For the case of a solid homogeneous body~$B$ of volume~$V$ and total mass~$M$, clearly
$\rho(\!\vec{\,r}')=M/V$ is constant, and one gets
\begin{equation} \label{surfacegravity}
\vec G(\vec r) \= \gamma\,\sfrac{M}{V} \int_B\!{\mathrm{d}}^3\!\vec{\,r}' \frac{\vec{e}_{\!\vec{\,r}'\!-\!\vec{\,r}}}{(\!\vec{\,r}'\!-\!\vec{\,r})^2}\ ,
\end{equation}
where $\vec{e}_{\!\vec{\,r}'\!-\!\vec{\,r}}$ is the unit vector pointing from the observer (at $\!\vec{\,r}$) to the mass point
at~$\!\vec{\,r}'$. The surface gravity (specific weight of a probe) located somewhere on the surface
$\partial B$ of my solid is obtained by simply restricting $\!\vec{\,r}$ to $\partial B$.
One might think of simplifying the task by computing the gravitational potential rather than the field,
since the corresponding integral is scalar and appears to be easier. However, evaluating the surface
gravity then requires taking a gradient in the end and thus keeping at least an infinitesimal dependence
on a coordinate normal to the surface. Retaining this additional parameter until finally computing the
derivative of the potential with respect to it before setting it to zero yields no calculational gain
over a direct computation of~$\vec G$.
The original question of Pascal concerned a cylindrical rod, whose length and radius I denote
by $\ell$ and $a$, respectively, so that $V = \pi a^2 \ell$. The integral above has dimension of length,
and I shall scale out a factor of~$\ell$ to pass to dimensionless quantities. For the remaining
dimensionless parameter I choose the ratio of diameter to length of the cylinder, $t:=2a/\ell$, see Fig.~1.
I shall frequently have to express some of the four quantities $a$, $\ell$, $t$ and $V$ in terms of
a pair of the others, so let me display the complete table of the relations,
\begin{equation}
\begin{aligned}
a &\= \ell\,t/2 \= \sqrt{V/(\pi\ell)} \= \root3\of{V\,t/(2\pi)} \\[4pt]
\ell &\= 2a/t \= V/(\pi a^2) \ \,\= \root3\of{4 V/(\pi t^2)} \\[4pt]
t &\= 2a/\ell \= 2\pi a^3/V \ \;\= \sqrt{4\,V/(\pi\ell^3)} \\[6pt]
V &\= \pi a^2\ell \= 2\pi a^3/t \ \ \ \= \pi\ell^3 t^2/4\ .
\end{aligned}
\end{equation}
Pascal's problem was to compare for this cylinder the surface gravity at the symmetry axis point
to the one at a point on the mid-circumference or equator. Let me treat both cases in turn.
\subsection{Surface gravity at the axis}
\noindent
Naturally I employ cylindrical coordinates $(z,\rho,\phi)$ for $\!\vec{\,r}'$ and put the symmetry axis point
in the origin. With $\!\vec{\,r}=0$ the expression~(\ref{surfacegravity}) then becomes
\begin{equation}
\begin{aligned}
\vec G(0) &\= \gamma\,\sfrac{M}{V} \int_0^\ell\!{\mathrm{d}}{z}\int_0^a\!{\mathrm{d}}{\rho}\,\rho\int_0^{2\pi}\!\!\!{\mathrm{d}}\phi\
(z^2+\rho^2)^{-3/2}\ \Bigl( \begin{smallmatrix} \rho\cos\phi \\ \rho\sin\phi \\ -z \end{smallmatrix} \Bigr)
\\[4pt]
&\= -2\pi\,\gamma\,\sfrac{M}{V} \int_0^\ell\!{\mathrm{d}}{z}\int_0^a\!{\mathrm{d}}{\rho}\
\frac{\rho\,z}{(z^2+\rho^2)^{3/2}}\ \vec{e}_z
\ =:\ -G_a\,\vec{e}_z\ .
\end{aligned}
\end{equation}
The $\rho$ and $z$ integrals are elementary,
\begin{equation}
\begin{aligned}
G_a
&\= 2\pi\gamma\,\sfrac{M}{V} \int_0^\ell\!{\mathrm{d}}{z}\int_0^a\!{\mathrm{d}}{\rho}\ \frac{\rho\,z}{(z^2+\rho^2)^{3/2}}
\= 2\pi\gamma\,\sfrac{M}{V} \int_0^\ell\!{\mathrm{d}}{z}\ \Bigl[ \frac{z}{\sqrt{z^2+\rho^2}} \Bigr]_0^a \\[4pt]
&\= 2\pi\gamma\,\sfrac{M}{V} \int_0^\ell\!{\mathrm{d}}{z}\ \Bigl\{ 1 - \frac{z}{\sqrt{z^2+a^2}} \Bigr\}
\= 2\pi\gamma\,\sfrac{M}{V} \,\Bigl[ z - \sqrt{z^2+a^2} \Bigr]_0^\ell \\[4pt]
&\= 2\pi\gamma\,\sfrac{M}{V} \,\Bigl\{ \ell + a - \sqrt{\ell^2+a^2} \Bigr\}
\= 2\pi\gamma\,\sfrac{M}{V}\,\ell\,\Bigl\{ 1 + \sfrac{t}{2} - \sqrt{1+\sfrac{t^2}{4}} \Bigr\}\ .
\end{aligned}
\end{equation}
It is a bit curious that the result is symmetric under the exchange of $\ell$ and~$a$,
and so in the thin rod ($a\to0$) and thin disk ($\ell\to0$) limits one finds that
\begin{equation}
G_a \= 2\pi\gamma\,\sfrac{M}{V}\,a\,\bigl\{ 1 - \sfrac{a}{2\ell} + \ldots \bigr\} {\qquad{\rm and}\qquad}
G_a \= 2\pi\gamma\,\sfrac{M}{V}\,\ell\,\bigl\{ 1 - \sfrac{\ell}{2a} + \ldots \bigr\}\ ,
\end{equation}
respectively, with $a^2\ell=V/\pi$ fixed of course.
Apart from the linear dependence on the gravitational constant~$\gamma$
and the mass density~$\sfrac{M}{V}$, the surface gravity must carry a dimensional length factor,
which choose to be the cylinder length~$\ell$.
However, $\ell$, $t$ and $V$ are obviously related, and for comparing different shapes
of the same mass and volume it is preferable to eliminate $\ell$ in favor of $V$ and~$t$.
The resulting expression for the surface gravity has the universal form
\begin{equation}
G \= \textrm{(numerical factor)}\ \times\ \gamma\,M\,V^{-2/3}\ \textrm{(shape function)}\ ,
\end{equation}
where the shape function depends on dimensionless parameters like~$t$ only.
For the case at hand, I obtain
\begin{equation}
G_a \= 2^{5/3} \pi^{2/3}\,\gamma\,M\,V^{-2/3}\,t^{-2/3}\,
\Bigl\{ 1 + \sfrac{t}{2} - \sqrt{1+\sfrac{t^2}{4}} \Bigr\}\ .
\end{equation}
The asymptotic behavior for a thin rod ($t\to0$) and for a thin disk ($t\to\infty$) takes the form
\begin{equation}
G_a \= 2^{5/3} \pi^{2/3}\,\gamma\,M\,V^{-2/3}\,\times\, \begin{cases}
\sfrac{1}{2}t ^{1/3}- \sfrac{1}{8}t^{4/3} + \sfrac{1}{128}t^{10/3} + O(t^{16/3})
& \textrm{for} \quad t\to 0 \\[8pt]
t^{-2/3} - t^{-5/3} + t^{-11/3} + O(t^{-17/3})
& \textrm{for} \quad t\to\infty \end{cases}\ .
\end{equation}
\subsection{Surface gravity at the equator}
\noindent
This is the harder case, as it lacks the cylindrical symmetry.
Naturally putting the origin of the cylindrical coordinate system at the
cylinder's center of mass, hence $\!\vec{\,r}=(a,0,0)^\top$, the surface gravity integral~(\ref{surfacegravity}) reads
\begin{equation}
\begin{aligned}
\vec G(a)
&\= \gamma\,\sfrac{M}{V} \int_{-\ell/2}^{\ell/2}\!\!\!{\mathrm{d}}{z}\int_0^a\!{\mathrm{d}}{\rho}\,\rho\int_0^{2\pi}\!\!\!{\mathrm{d}}\phi\
\bigl([\rho\cos\phi-a]^2+[\rho\sin\phi]^2+z^2\bigr)^{-3/2}\
\Bigl( \begin{smallmatrix} \rho\cos\phi-a \\ \rho\sin\phi \\ z \end{smallmatrix} \Bigr) \\[4pt]
&\= \gamma\,\sfrac{M}{V} \int_{-\ell/2}^{\ell/2}\!\!\!{\mathrm{d}}{z}\int_0^a\!{\mathrm{d}}{\rho}\,\rho\int_0^{2\pi}\!\!\!{\mathrm{d}}\phi\
\frac{\rho\cos\phi-a}{(z^2+a^2+\rho^2-2\,a\rho\cos\phi)^{3/2}}\ \vec{e}_x \\[4pt]
&\= 2\,\gamma\,\sfrac{M}{V}\,\ell\, \int_0^{1/2}\!\!{\mathrm{d}}{u}\int_0^1\!{\mathrm{d}}{v}\int_0^{2\pi}\!\!\!{\mathrm{d}}\phi\
\frac{v\,(v\cos\phi-1)}{u^2\ell^2/a^2+1+v^2-2\,v\cos\phi)^{3/2}}\ \vec{e}_x
\ =:\ -G_m\,\vec{e}_x\ ,
\end{aligned}
\end{equation}
where I employed the $z\leftrightarrow-z$ symmetry and substituted $z=u\,\ell$ and $\rho=v\,a$
for a dimensionless integral.
The $u$ integration is elementary,
\begin{equation} \label{hardintegral}
\begin{aligned}
G_m
&\= \gamma\,\sfrac{M}{V}\,\ell\,\int_0^1\!{\mathrm{d}}{v}\int_0^{2\pi}\!\!\!{\mathrm{d}}\phi\
\frac{v\,(1-v\cos\phi)/(1+v^2-2\,v\cos\phi)}{\sqrt{\ell^2/(4a^2)+1+v^2-2\,v\cos\phi}} \\[4pt]
&\= 2\,\gamma\,\sfrac{M}{V}\,\ell\,\int_0^1\!{\mathrm{d}}{v}\int_{-1}^{1}\!\!{\mathrm{d}}{w}\
\frac{v\,(1-v\,w)/(1+v^2-2\,v\,w)}{\sqrt{(1-w^2)(t^{-2}+1+v^2-2\,v\,w)}}\ ,
\end{aligned}
\end{equation}
after substituting $\cos\phi=w$ and using the definition $2a/\ell=t$.
The remaining double integrals leads to lengthy expressions
in terms of complete elliptic integrals, which I do not display here.
For $t\to\infty$ it diverges logarithmically.
It is possible, however, to extract the limiting behavior for $t\to0$ as
\begin{equation}
G_m \= 2\pi\,\gamma\,\sfrac{M}{V}\,a\,\bigl\{ 1 - O(\sfrac{a}{\ell}) \bigr\}\ ,
\end{equation}
which in leading order surprisingly agrees with that of $G_a$.
\newpage{}
\begin{figure}[!h]
\includegraphics[width=16cm]{pascalfig2.eps
\caption{Cylinder surface gravity on symmetry axis and mid-circumference}
\label{fig:2}
\end{figure}
\vspace{0.5cm}
\subsection{Comparison with a spherical ball}
\noindent
To get a feeling for these results, it is natural to compare them with the surface gravity of a homogeneous ball
of the same mass~$M$ and density, thus of radius
\begin{equation}
r_b \= \bigl(\sfrac{4\pi}{3}\bigr)^{-1/3} V^{1/3}\ .
\end{equation}
The surface gravity $\vec G(r_b)=-G_b\,\vec{e}_r$ of the latter is well known,
\begin{equation}
G_b \= \gamma\,M/r_b^2 \= \gamma\,\sfrac{M}{V}\,\sfrac{4\pi}{3}\,r_b \=
\bigl(\sfrac{4\pi}{3}\bigr)^{2/3} \gamma\,M\,V^{-2/3}\ .
\end{equation}
Hence, the relation of the cylindrical to the spherical surface gravity is
\begin{equation}
\frac{G_a}{G_b} \= 2\pi\,\bigl(\sfrac{\pi}{4}\bigr)^{-1/3} t^{-2/3}\,
\Bigl\{ 1 + \sfrac{t}{2} - \sqrt{1+\sfrac{t^2}{4}} \Bigr\} \Big/ \bigl(\sfrac{4\pi}{3}\bigr)^{2/3}
\= \root 3 \of {18}\;t^{-2/3}\,\Bigl\{ 1 + \sfrac{t}{2} - \sqrt{1+\sfrac{t^2}{4}} \Bigr\}\ ,
\end{equation}
for the axis position, see~Fig.~2.
Surprisingly, in the interval
\begin{equation}
t\ \in\ \bigl[ \sfrac49(2\sqrt{13}{-}5)\ ,\ \sfrac32 \bigr] \ \approx\
\bigl[ 0.98271\ ,\ 1.50000 \bigr]
\end{equation}
the weight on the cyclinder's axis exceeds that on the reference ball!
Indeed, its maximal value is attained at
\begin{equation}
t_a \= \sfrac14(9-\sqrt{17}) \ \approx\ 1.21922 \qquad\Rightarrow\qquad
\sfrac{G_a}{G_b}\big|_{\textrm{max}} \= \sfrac{G_a}{G_b}(t_a) \ \approx\ 1.00682\ .
\end{equation}
The asymptotic behavior is easily deduced to be
\begin{equation}
\frac{G_a}{G_b} \= \root3\of{\sfrac{9\pi}{2}}\,\frac{a}{V^{1/3}}\,\Bigl(1-O\bigl(\sfrac{a^3}{V}\bigr)\Bigr) {\qquad{\rm and}\qquad}
\frac{G_a}{G_b} \= \root3\of{\sfrac{9\pi}{2}}\,\frac{\ell}{V^{1/3}}\,\Bigl(1-O\bigl(\sqrt{\sfrac{\ell^3}{V}}\bigr)\Bigr)\ ,
\end{equation}
for $a\to0$ and $\ell\to0$, respectively.
For the equatorial position's surface gravity I do not have an analytic expression,
only its limiting forms
\begin{equation}
\frac{G_m}{G_b} \ \sim\ \root3\of{\sfrac{9\pi}{2}}\,\frac{a}{V^{1/3}} \ \approx\
2.41799\,\frac{a}{V^{1/3}} {\qquad{\rm and}\qquad}
\frac{G_m}{G_b} \ \sim\ 0.36813\,\frac{\ell}{V^{1/3}}\Bigl|\log\frac{\ell}{V^{1/3}}\Bigr|
\end{equation}
for $a\to0$ and $\ell\to0$, respectively.
Numerical analysis shows that $G_m/G_b$ (see Fig.~2) attains a maximum at
\begin{equation}
t_m \ \approx \ 1.02928 \qquad\Rightarrow\qquad
\sfrac{G_m}{G_b}\big|_{\textrm{max}} \ \approx \ 1.00619\ .
\end{equation}
Furthermore, for any given shape in an asymptotic regime, the equatorial position is superior to the axis one.
Only in the interval $1.10948 \lesssim t \lesssim 2.82154$ is our mini-bug heavier on the axis.
\section{Which shape maximizes the surface gravity?}
\noindent
This finding suggests the question: Can one do better than the cylinder with a clever choice of shape?
It turns the problem into a variational one. Suppose I have by some means discovered the homogeneous
body~$\bar B$ which, for fixed mass and volume, yields the maximally possible gravitational pull in some
location on its surface.
Without loss of generality I can put this point to the origin of my coordinate system and orient the solid
in such a way that its outward normal in this point aims in the positive $z$~direction,
so gravity pulls downwards as is customary.
Expressing the surface gravity at this position for an arbitrary body~$B$ as a functional of its shape,
then~$\bar B$ must maximize this functional, under the constraint of fixed mass and volume.
The following three features of the optimal shape are evident:
\begin{itemize}
\addtolength{\itemsep}{-8pt}
\item It does not have any holes, so has just a single boundary component
\item It is convex
\item It is rotationally symmetric about the normal at the origin
\end{itemize}
\vspace{-1cm}
\begin{figure}[!ht]
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,260)(60,65)
\put(0,0){\includegraphics[width=12cm,trim=0 0 0 70,clip]{pascalfig3a.eps}}
\end{picture}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,180)(-50,0)
\put(0,0){\includegraphics[width=6cm]{pascalfig3b.eps}}
\end{picture}
\end{minipage}
\caption{ Parametrization of surface of revolution $\partial B$}
\label{fig:3}
\end{figure}
\vspace{1cm}
These facts imply that the surface~$\partial B$ may be parametrized as in Fig.~3,
\begin{equation}
\partial B \= \bigl\{ R(\th) (\sin\th\cos\phi,\sin\th\sin\phi,-\cos\th)^\top \ \big| \
0\le\th\le\sfrac{\pi}{2}\;,\ 0\le\phi<2\pi \bigr\} \ ,
\end{equation}
with $R(\th)\ge0$ and $R(\sfrac{\pi}{2})=0$.
The function~$R(\th)$ (which may be extended via $R(-\th)=R(\th)$) completely describes
the shape of the solid of revolution~$B$.
It may be viewed as the boundary curve of the intersection of~$B$ with the $xz$~plane.
Its convexity implies the condition
\begin{equation}
\bigl(\sfrac1{R(\th)}\bigr)''+\sfrac1{R(\th)}\ \ge\ 0\ .
\end{equation}
Employing the symmetry under reflection on the rotational axis,
\begin{equation}
S\,:\ (\th,\phi)\ \mapsto\ (\th,\phi{+}\pi)\ \sim\ (-\th,\phi)\ ,
\end{equation}
the surface gravity functional (\ref{surfacegravity}) then reads
\begin{equation}
\vec{G}[R] \= \gamma\,\sfrac{M}{V} \int_B\!\frac{{\mathrm{d}}^3\!\vec{\,r}}{r^2}\ \sfrac12(\vec{e}_{\!\vec{\,r}}+S\vec{e}_{\!\vec{\,r}})
\ =:\ -G[R]\,\vec{e}_z\ ,
\end{equation}
\begin{equation} \label{functional}
G[R]\= 2\pi\,\gamma\,\sfrac{M}{V} \int_0^1\!{\mathrm{d}}\cos\th \int_0^{R(\th)}\!\!{\mathrm{d}}{r}\;\cos\th
\= 2\pi\,\gamma\,\sfrac{M}{V} \int_0^1\!{\mathrm{d}}\cos\th\ R(\th) \cos\th\ .
\end{equation}
It is to be maximized with the mass (and thus the volume) kept fixed,
\begin{equation} \label{constraint}
M[R] \= \sfrac{M}{V} \int_B\!{\mathrm{d}}^3\!\vec{\,r}
\= 2\pi\,\sfrac{M}{V} \int_0^1\!{\mathrm{d}}\cos\th \int_0^{R(\th)}\!\!r^2{\mathrm{d}}{r}
\= \sfrac{2\pi}{3}\,\sfrac{M}{V} \int_0^1\!{\mathrm{d}}\cos\th\ R(\th)^3
\ \buildrel ! \over = \ M\ .
\end{equation}
Such constrained variations are best treated by the method of Lagrange multipliers,
which here instructs me to combine the two functionals to
\begin{equation}
2\pi\,\sfrac{M}{V}\,U[R,\lambda] \= G[R]\ -\ \lambda\bigl(M[R]-M\bigr)\ ,
\end{equation}
introducing a Lagrange multiplier~$\lambda$ (a real parameter to be fixed subsequently).
More explicitly,
\begin{equation}
U[R,\lambda] \= \int_0^1\!{\mathrm{d}}\cos\th\ \bigl[ \gamma\,R(\th)\cos\th\ -\ \sfrac13\lambda\,R(\th)^3 \bigr]
\ -\ \lambda\,\sfrac{V}{2\pi}\ ,
\end{equation}
so $\partial_\lambda U=0$ clearly fixes the volume of $B$ to be equal to $V$.
Demanding that, for $\lambda$ fixed but arbitrary, $U$ is stationary under any variation of the boundary curve,
$R\mapsto R+\delta R$, determines $R=R_\lambda$:
\begin{equation}
0 \= \delta U[R_\lambda,\lambda]
\= \int_0^1\!{\mathrm{d}}\cos\th\ \delta R(\th)\ \bigl[ \gamma\,\cos\th\ -\ \lambda\,R_\lambda(\th)^2 \bigr]\ ,
\end{equation}
so I immediately read off
\begin{equation}
R_\lambda(\th) \= \sqrt{\sfrac{\gamma}{\lambda}\,\cos\th}\ .
\end{equation}
It remains to compute the value $\bar\lambda$ of the Lagrange multiplier by inserting the solution $R_\lambda$ into
the constraint~(\ref{constraint}),
\begin{equation}
M\ \buildrel ! \over = \ M[R_{\bar\lambda}]
\= \sfrac{2\pi}{3}\,\sfrac{M}{V} \int_0^1\!{\mathrm{d}}\cos\th\ \bigl( \sfrac{\gamma}{\bar\lambda}\,\cos\th \bigr)^{3/2}
\= \sfrac{4\pi}{15}\,\sfrac{M}{V}\,\bigl(\sfrac{\gamma}{\bar\lambda}\bigr)^{3/2}\ ,
\end{equation}
yielding \ $\bar\lambda=\bigl(\sfrac{4\pi}{15\,V}\bigr)^{2/3}\gamma$ \ and hence the complete solution
as displayed in Fig.~4,
\begin{equation} \label{solution}
\bar{R}(\th)\ :=\ R_{\bar\lambda}(\th) \= 2\,R_0\,\sqrt{\cos\th}
\qquad\textrm{with}\qquad (2\,R_0)^3 \= \sfrac{15}{4\pi}\,V \ .
\end{equation}
What does this curve look like?
Let me pass to Cartesian coordinates in the $xz$~plane,
\begin{equation}
\bar{R}^2 \= (2\,R_0)^2\,\cos\th \= x^2+z^2 {\qquad{\rm and}\qquad}
\cos\th \= \sfrac{z}{\sqrt{x^2+z^2}} \ ,
\end{equation}
which yields the sextic curve (cubic in squares)
\begin{equation}
(x^2+z^2)^3 \= (2\,R_0)^4\,z^2 \qquad\textrm{with}\qquad
R_0^3 \= \sfrac{15}{32\pi}\,V \ .
\end{equation}
The parameter $R_0$ only takes care of the physical dimensions
and determines the overall size of the solid.
In dimensionless coordinates it may be put to unity,
which fixes the vertical diameter to be equal to~2 and allows
for a comparison of my optimal curve with the unit circle,
\begin{equation}
r(z) \= 2\,|z|^{1/3}
\qquad\textrm{versus}\qquad
r(z)\= 2\,|z|^{1/2}
\qquad\textrm{for}\quad r(z)^2\=x^2+z^2\ ,
\end{equation}
with $-z\in[0,2]$ and $r(z)\in[0,2]$.
Since $|z|^{1/3}\ge|z|^{1/2} $ in the interval of question,
my curve lies entirely outside the reference circle, touching it only
twice on the $z$~axis.
(Note that $R_0\neq r_b$ so the corresponding volumes differ.)
Other than the sphere, my curve has a critical point:
Due to $z\sim x^3$ near the origin, the curvature vanishes there.
Clearly, the vertical extension of~$\bar{B}$ is $\Delta z=2R_0$
while its width is easily computed to be
\begin{equation}
\Delta x \= 2\,\root 4 \of {\sfrac{4}{27}}\;2R_0 \ \approx\ 2.48161\,R_0
\qquad\textrm{at}\qquad
z_0 \= -\root 4 \of {\sfrac{1}{27}}\;2R_0 \ \approx\ -0.87738\,R_0 \ .
\end{equation}
The shape of my optimal body~$\bar{B}$ vaguely resembles an apple,
with the flatter side up.
My final goal is to calculate the maximal possible weight $G_{\textrm{max}}$, or
\begin{equation}
G[\bar R] \= 2\pi\,\gamma\,\sfrac{M}{V}\,2R_0 \int_0^1\!{\mathrm{d}}\cos\th\ \bigl( \cos\th \bigr)^{3/2}
\= 2\pi\,\gamma\,\sfrac{M}{V}\,\root 3 \of {\sfrac{15\,V}{4\pi}}\;\sfrac25
\= \bigl( \sfrac{4\pi\sqrt{3}}{5} \bigr)^{2/3} \gamma\,M\,V^{-2/3}\ .
\end{equation}
Comparing with the spherical shape,
\begin{equation} \label{maximum}
\frac{G[\bar R]}{G_b} \= 3\cdot 5^{-2/3} \= \sfrac35\root 3 \of 5 \ \approx\ 1.02599 \ .
\end{equation}
I conclude that by homogeneous reshaping it is possible to increase the surface gravity of a spherical ball
by at most $2.6\%$ !
\newpage{}
\begin{figure}[!ht]
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,180)(60,165)
\put(0,0){\includegraphics[width=12cm,trim=0 0 0 60,clip]{pascalfig4a.eps}}
\end{picture}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,180)(-15,20)
\put(0,0){\includegraphics[width=8cm]{pascalfig4b.eps}}
\end{picture}
\end{minipage}
\vspace{0.3cm}
\caption{ Optimal asteroid surface $\partial \bar{B}$}
\label{fig:4}
\end{figure}
\section{Other shapes}
\noindent
Since the cylinder shape is already superior to the spherical one for maximizing surface gravity,
it is interesting to explore a few other more or less regular bodies,
to see how close they can get to the optimal value of $\sfrac35\root 3\of 5\approx1.02599$.
Let me discuss three cases which are fairly easy to parametrize in the cylindrical coordinates chosen.
\begin{figure}[!ht]
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,140)(0,0)
\put(0,0){\includegraphics[width=7cm,trim=0 0 0 0,clip]{pascalfig5a.eps}}
\end{picture}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,140)(0,0)
\put(0,0){\includegraphics[width=8cm]{pascalfig5b.eps}}
\end{picture}
\end{minipage}
\caption{ Conical segment of a spherical ball}
\label{fig:5}
\end{figure}
\vspace{0.5cm}
First, I consider a conical segment of a spherical ball centered in the origin,
with opening angle $2\alpha<\pi$ and radius $r_c$, see~Fig.~5.
Here, one simply has
\begin{equation}
0 \le \th \le \alpha {\qquad{\rm and}\qquad} R_c(\th) \= r_c\ ,
\end{equation}
thus the surface gravity~(\ref{functional}) reduces to
\begin{equation}
G_c\= 2\pi\,\gamma\,\sfrac{M}{V} \int_{\cos\alpha}^1\!{\mathrm{d}}\cos\th \int_0^{r_c}\!{\mathrm{d}}{r}\;\cos\th
\= 2\pi\,\gamma\,\sfrac{M}{V}\,r_c\,\sfrac12 (1-\cos^2\alpha) \ .
\end{equation}
\newpage{}
\begin{figure}[!ht]
\centerline
\includegraphics[width=8cm]{pascalfig6.eps}
\caption{ Surface gravity on the apex of a conical segment of a spherical ball}
\label{fig:6}
\end{figure}
\vspace{0.5cm}
Since at the same time,
\begin{equation}
M\= 2\pi\,\sfrac{M}{V} \int_{\cos\alpha}^1\!{\mathrm{d}}\cos\th \int_0^{r_c}\! r^2{\mathrm{d}}{r}
\= 2\pi\,\sfrac{M}{V}\,\sfrac13 r_c^3\, (1-\cos\alpha) \ ,
\end{equation}
one gets
\begin{equation}
G_c \= \bigl( \sfrac{\sqrt{3}\pi}{\sqrt{2}} \bigr)^{2/3} \gamma\,M\,V^{-2/3}\,
(1-\cos^2\alpha)\,(1-\cos\alpha)^{-1/3}\ ,
\end{equation}
leading to the curve in Fig.~6,
\begin{equation}
\frac{G_c}{G_b} \= \sfrac34 \root 3\of 2\,(1-\cos^2\alpha)\,(1-\cos\alpha)^{-1/3}\ .
\end{equation}
The best opening angle occurs at an angle of about $78.5^\circ$,
\begin{equation}
\cos\alpha \= \sfrac15 \ \approx\ 1.36944 \qquad\Rightarrow\qquad
\frac{G_c}{G_b}\bigg|_{\textrm{max}} \= 2^{2/3}\cdot 9\cdot 5^{-5/3}
\ \approx\ 0.97719\ .
\end{equation}
Clearly, the spherical ball beats any cone.
The value $\alpha=\sfrac{\pi}{2}$ describes a semi-ball, which yields
\begin{equation}
\frac{G_c}{G_b}\bigg|_{\alpha=\pi/2} \= 2^{-8/3}\cdot 3\ \approx\ 0.94494\ .
\end{equation}
\begin{figure}[!ht]
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,240)(-20,-10)
\put(0,0){\includegraphics[width=7cm,trim=0 0 0 0,clip]{pascalfig7a.eps}}
\end{picture}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,240)(-20,0)
\put(0,0){\includegraphics[width=7cm]{pascalfig7b.eps}}
\end{picture}
\end{minipage}
\caption{ Shape for radius function $R(\th)\sim\cos^2\th$}
\label{fig:7}
\end{figure}
\vspace{0.5cm}
Second, let me try out the radius function $R(\th)$
being an arbitrary power $n$ of~$\cos\th$,
\begin{equation}
R_n(\th) \= 2\,r_n\,(\cos\th)^n \qquad\textrm{with}\quad n>0 \ ,
\end{equation}
displayed in Fig.~7 for $n{=}2$.
This produces
\begin{equation}
G_n \= 2\pi\,\gamma\,\sfrac{M}{V}\,2r_n\int_0^1\!{\mathrm{d}}\cos\th\ (\cos\th)^{n+1}
\= 2\pi\,\gamma\,\sfrac{M}{V}\,2r_n\,\sfrac1{n+2}\ .
\end{equation}
The special value of $n{=}1$ yields a spherical ball, which separates
squashed forms ($n{<}1$) from elongates ones ($n{>}1$). With
\begin{equation}
M\= \sfrac{2\pi}{3}\,\sfrac{M}{V} \int_0^1\!{\mathrm{d}}\cos\th\ \bigl(2r_n\,(\cos\th)^{n}\bigr)^3
\= \sfrac{2\pi}{3}\,\sfrac{M}{V}\,(2r_n)^3\,\sfrac{1}{3n+1}
\end{equation}
I can eliminate $r_n$ and find
\begin{equation}
\frac{G_n}{G_b} \= 3\,\bigl( \sfrac14(3n+1)\bigr)^{1/3} \big/ (n+2)\ ,
\end{equation}
which is shown in Fig.~8.
This is indeed maximized for
\begin{equation}
n=\sfrac12 \qquad\Rightarrow\qquad
\sfrac{G_{1/2}}{G_b} \= 3\cdot 5^{-2/3}\ ,
\end{equation}
as was already found in (\ref{solution}) and (\ref{maximum}).
It exceeds unity in the interval $0.17424\lesssim n < 1$.
\vspace{0.5cm}
\begin{figure}[!ht]
\centerline
\includegraphics[width=8cm]{pascalfig8.eps}
\caption{ Surface gravity on a body with radius function $R(\th)\sim\cos^n\th$}
\label{fig:8}
\end{figure}
\newpage{}
\begin{figure}[!ht]
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,150)(-20,10)
\put(0,0){\includegraphics[width=7cm,trim=0 0 0 0,clip]{pascalfig9a.eps}}
\end{picture}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}
\begin{picture}(140,150)(-20,0)
\put(0,0){\includegraphics[width=7cm]{pascalfig9b.eps}}
\end{picture}
\end{minipage}
\caption{ Oblate ellipsoid with eccentricity $\epsilon=0.8$}
\label{fig:9}
\end{figure}
\vspace{0.5cm}
Third, I look at an oblate ellipsoid of revolution
with minor semi-axis length $r_e$ and eccentricity $\epsilon$,
see Fig.~9. In this case,
\begin{equation}
R_e(\th) \= \frac{2\,r_e\,\cos\th}{1-\epsilon^2\sin^2\th}
\= \frac{2\,r_e\,\cos\th}{1{-}\epsilon^2+\epsilon^2\cos^2\th}
\qquad\textrm{with}\quad \epsilon\in[0,1)\ ,
\end{equation}
which includes the sphere for $\epsilon{=}0$.
(The prolate case corresponds to imaginary~$\epsilon$.)
The surface gravity and mass integrals then become
\begin{equation}
G_e \= 2\pi\,\gamma\,\sfrac{M}{V}\,2r_e\int_0^1\!{\mathrm{d}}{y}\
\frac{y^2}{1{-}\epsilon^2+\epsilon^2 y^2}
\= 2\pi\,\gamma\,\sfrac{M}{V}\,2r_e\,\frac1{\epsilon^2}
\Bigl( 1 - \sqrt{\sfrac{1-\epsilon^2}{\epsilon^2}}\arctan\sqrt{\sfrac{\epsilon^2}{1-\epsilon^2}} \Bigr)\ ,
\end{equation}
\begin{equation}
M \= \sfrac{2\pi}{3}\,\sfrac{M}{V}\,(2r_e)^3\int_0^1\!{\mathrm{d}}{y}\
\frac{y^3}{(1{-}\epsilon^2+\epsilon^2 y^2)^3}
\= \sfrac{2\pi}{3}\,\sfrac{M}{V}\,(2r_e)^3\,\frac{1}{4(1{-}\epsilon^2)}\ ,
\qquad\qquad\qquad\quad{}
\end{equation}
respectively. From this I conclude that
\begin{equation}
\frac{G_e}{G_b} \= 3\,\bigl(1-\epsilon^2\bigr)^{1/3}\,\frac1{\epsilon^2}
\Bigl( 1 - \sqrt{\sfrac{1-\epsilon^2}{\epsilon^2}}\arctan\sqrt{\sfrac{\epsilon^2}{1-\epsilon^2}} \Bigr)\ ,
\end{equation}
shown in Fig.~10.
This is larger than one for $\epsilon\lesssim0.85780$ and is maximized numerically at
\begin{equation}
\epsilon\ \approx\ 0.69446 \qquad\Rightarrow\qquad
\frac{G_e}{G_b}\bigg|_{\textrm{max}} \ \approx\ 1.02204 \ .
\end{equation}
Hence, I can come to within less than $0.4\%$ of the optimal surface gravity by
engineering an appropriate ellipsoid.
\newpage{}
\begin{figure}[!ht]
\centerline
\includegraphics[width=8cm]{pascalfig0.eps}
\caption{ Surface gravity on an ellipsoid with eccentricity~$\epsilon$}
\label{fig:10}
\end{figure}
\section{Conclusions}
\noindent
The main result of this short paper is a universal sixth-order planar curve,
\begin{equation}
{\cal C}_{\textrm{Gh}}: \quad
(x^2+z^2)^3-(4\,z)^2\=0 \qquad\Leftrightarrow\qquad r(\th)\=2\sqrt{\cos\th}\ ,
\end{equation}
which characterizes the shape of the homogeneous body admitting the maximal possible
surface gravity in a given point, for unit mass density and volume.
It is amusing to speculate about its use for asteroid engineering in an advanced civilization
or our own future. This curve seems not yet to have occurred in the literature,
and so I choose to name it ``Gharemani curve'' after my deceased friend who
initiated the whole enterprise.
The maximally achievable weight on bodies of various shapes is listed in the following table.
It occurs at the intersection of the rotational symmetry axis with the body's surface
and is normalized to the value on the spherical ball.
\begin{center}
\begin{tabular}{|c|ccccc|}
\hline
shape & cone & ball & cylinder & ellipse & Gharemani \\
\hline
maximum of $G/G_b$ & 0.97719 & 1.00000 & 1.00682 & 1.02204 & 1.02599 \\
\hline
\end{tabular}
\end{center}
It can pay off to get inspired by the curiosity of your non-scientist friends.
The result is a lot of fun and may even lead to new science!
\begin{figure}[!ht]
\centerline
\includegraphics[width=6cm]{pascalcurve.eps}
\end{figure}
\newpage
\section*{Acknowledgments}
\noindent
I thank Michael Flohr for help with Mathematica and the integral~(\ref{hardintegral}).
\bigskip
\small{
|
1511.07856
|
\section{Introduction}
Primordial inflation provides solutions for cosmological puzzles such as the flatness and horizon problems and also explains the emergence of the primordial density fluctuations essential for the formation of the large scale structure observed today \cite{Guth:1980zm,Linde:1983gd}. Inflation is typically studied considering a self interacting scalar field and has been widely studied in the literature (see \cite{Bassett:2005xm,Martin:2013tda} for reviews).
The possibility of the energy source of the inflationary expansion
to be of a non-scalar nature has, however, never been excluded.
It is, therefore, important to understand the nature of higher spin fields and how robust they are
in order to fully test their applications in cosmology.
Inflation considering higher spinor fields has been investigated in the past and these
models are also important due to their connection to string theory
scenarios \cite{Frey:2002qc,Gubser:2000vg,Groh:2012tf}. Vector inflation has been studied in Ref.~\cite{Ford:1989me}, however, for inflation to proceed, the vector needs a nonminimal coupling and the model appears to feature some instabilities. Inflation with a 2-form field
resembles much the vector inflation with the same problems \cite{Germani:2009iq,Koivisto:2009sd}.
A 3-form has been shown to present viable solutions, not only for inflation \cite{Koivisto:2009ew,Koivisto:2009fb,Mulryne:2012ax,DeFelice:2012jt}, but also for describing dark energy \cite{Koivisto:2012xm}.
Inflation driven by two 3-form fields has also been studied and does presents interesting results \cite{Kumar:2014oka}.
The natural question that arises now is how these properties translate to an extra-dimensional cosmological scenario. For example, in the Randall-Sundrum II model, proposed in 1999 \cite{Randall:1999vf}, our universe is confined to a four dimensional 3-brane, where the standard model particles reside, embedded in a five dimensional slice of an anti-de Sitter (AdS) space-time, the bulk. The presence of the bulk modifies the evolution equations \cite{Brax:2003fv}, more specifically, the Friedmann equation leads to a non-standard expansion law of the universe at high energies, while reproducing the standard four dimensional cosmology at low energies. One particular feature of the RSII model is that the tensor modes are enhanced due to the presence of the five dimensional bulk \cite{Langlois:2000ns,Langlois:2002bb}. Chaotic inflation on the brane has been investigated in Ref. \cite{Maartens:1999hf} and it was shown that the inflationary predictions are modified from those in the four dimensional standard cosmology. Quintessential inflation from brane worlds has also been explored in \cite{Nunes:2002wz} and also inflation in the context of a Gauss-Bonnet brane cosmology \cite{Lidsey:2003sj}.
More recently, simple inflationary models in the context of braneworld cosmology were analysed against the 2015 Planck data \cite{Okada:2014eva,Okada:2015bra}.
It is important to compare the dynamics of inflation with scalar fields with the dynamics where higher order fields are considered. The purpose of this work is, therefore, to study braneworld inflationary models driven by a single 3-form, confined to the brane, in the light of the Planck 2015 results \cite{Ade:2015lrj,Ade:2015xua}. In Sec. \ref{RSII} we introduce the 3-form model in the Randall Sundrum II braneworld. We follow to rewrite the equations of motion in terms of a first order dynamical system for which we identify the critical points and analyse their stability for a specific form of the potential. We explore the main differences of the dynamics compared with the four dimensional case. In Sec. \ref{perturbations} we write the power spectra for the scalar and tensor perturbations, calculate the cosmological parameters tensor to scalar ratio and spectral index and evaluate how sensitive they are to small changes in the brane tension. We find a lower bound on this parameter for a particular potential given the recent Planck data \cite{Ade:2015lrj,Ade:2015xua}. Finally in Sec. \ref{conclusions} we summarize and discuss our results.
\section{3-form in Randall-Sundrum II}\label{RSII}
In the RSII scenario, our universe is confined to a single positive tension four dimensional 3-brane embedded in a five dimensional Anti de Sitter spacetime with a negative (bulk) cosmological constant.
A single 3-form field $A_{\mu\nu\rho}$ minimal coupled to Einstein gravity is confined to the brane,
\begin{eqnarray}
\label{action}
S &=& -\int d^5 x \sqrt{-g^{(5)}} \left( \frac{R}
{2\kappa_5^2} + \Lambda_5 \right) \nonumber \\
&-& \int d^4 x \sqrt{-g^{(4)}} \left(\lambda -\frac{1}{48}F^2 -V(A^2)\right).
\end{eqnarray}
Here, $R$ is the Ricci scalar, $\Lambda_5$ is the bulk´s cosmological constant, $\lambda$ is the brane tension, $g^{(4)}$ and $g^{(5)}$ are the determinants of the four and five dimensional metrics, respectively. $\kappa^2=8\pi G$ and $F_{\alpha\beta\gamma\delta}$ is the Maxwell tensor given by,
\begin{equation}
F_{\alpha\beta\gamma\delta} = 4 \nabla_{[\alpha} A_{\beta\gamma\delta]},
\end{equation}
where square brackets denote antisymmetrization.
In order to avoid an excessive use of indices, we use the notation in which squaring means contracting all the indices, $A^2=A_{\mu\nu\rho} A^{\mu\nu\rho}$, and dotting means contracting the first index, $(\nabla \cdot A )_{\alpha\beta} = \nabla^{\mu} A_{\mu\alpha\beta}$.
We consider a Friedmann-Robertson-Walker Universe and take the scalar function $\chi (t)$ to parametrize the background contribution of the 3-form $A_{\mu\nu\rho}$. Thus, the non-vanishing components are given by,
\begin{equation}
A_{ijk}=a^3 (t) \epsilon_{ijk} \chi(t),
\end{equation}
and therefore, $A^2=6\chi^2 (t)$, where $\epsilon_{ijk}$ is the standard Levi-Civita symbol and $i$,$j$ and $k$ denote spatial indices.
The action (\ref{action}) leads to the equations of motion for the 3-form,
\begin{equation}
\label{mot}
\nabla\cdot F= 12V'(A^2)A,
\end{equation}
and, due to antisymmetry, implies the additional set of constraints,
\begin{equation}
\nabla\cdot V'(A^2)A = 0.
\end{equation}
The equations of motion in terms of the comoving field, $\chi$, are unmodified with respect to the previously studied four dimensional case because the matter fields are confined to the brane,
\begin{equation}
\label{motion}
\ddot{\chi} + 3H\dot{\chi} + 3\dot{H}\chi + V_{,\chi} =0,
\end{equation}
where the third term is a new feature from the 3-form model, not present in the standard scalar field theory.
The generalization of the equations of motion to multiple 3-forms was done in Ref. \cite{Kumar:2014oka}.
The presence of the bulk, however, modifies Einstein's equations \cite{Brax:2003fv}. The five-dimensional Einstein's equations lead to the Friedmann equation,
\begin{equation}
H^2=\frac{\kappa ^2}{3} \rho\left[ 1 + \frac{\rho}{2 \lambda}\right] + \frac{\Lambda_4}{3} + \frac{\mu}{a^4},
\end{equation}
where $\Lambda_4$ is the brane four-dimensional cosmological constant and the last term represents the influence of the bulk gravitons on the brane. In what follows we will use units where $\kappa^2=1$ and we will assume that $\Lambda_4=\mu=0$, leaving us with,
\begin{equation}
\label{fridmannRSII}
H^2=\frac{1}{3} \rho\left[ 1 + \frac{\rho}{2 \lambda}\right].
\end{equation}
When we inspect Eq.~(\ref{fridmannRSII}), we note that the expansion rate is larger at high energies ($\rho \gg 2\lambda$), which means that the friction term in Eq.~(\ref{motion}) is larger in that regime. This means that the field $\chi(t)$ rolls slower and, for the same initial conditions, inflation can last longer in this five-dimensions set up than in the four-dimensional case. The Friedmann equation in the standard cosmology is reproduced in the limit of low energies, $\rho \ll 2\lambda$.
We can define the energy density and pressure for the field in the form,
\begin{eqnarray}
\rho &=& \frac{1}{2} (\dot{\chi} + 3H\chi)^2 + V,\label{energy} \\
p &=& -\frac{1}{2} (\dot{\chi} + 3H\chi)^2 -V + V_{,\chi}\chi.
\end{eqnarray}
\subsection{Dynamics of the 3-form on the brane}\label{dynamics}
In order to study the dynamics of the 3-form on the brane we introduce the dimensionless variables,
\begin{eqnarray}
x &\equiv& \kappa \chi, \label{x}\\
y^2 &\equiv& \frac{V}{\rho} \label{y}, \\
w &\equiv& \frac{\dot{\chi} + 3 H \chi}{\sqrt{2\rho}}, \label{w}\\
\Theta &\equiv& \left( 1+ \frac{\rho}{2 \lambda} \right)^{-1/2}, \label{theta}
\end{eqnarray}
where $x$ represents the comoving field $\chi$, $y$ and $w$ are, respectively, the normalized potential and kinetic energies and $\Theta$ represents the correction term in Eq.~(\ref{fridmannRSII}). These variables are subject to the constraint, that follows from Eq.~(\ref{energy}),
\begin{equation}
\label{constrangimento}
w^2 + y^2 =1.
\end{equation}
Using Eqs.~(\ref{energy}), (\ref{x}), (\ref{w}) and (\ref{theta}), the modified Friedmann and Raychaudhuri equations can be written as,
\begin{eqnarray}
H^2 &=& \frac{1}{3} \frac{V}{(1-w^2)} \Theta^{-2}, \label{friedMod}\\
\dot{H} &=& -V_{,x} x \left(\Theta^{-2} - \frac{1}{2}\right). \label{rayMod}
\end{eqnarray}
Substituting for $\rho$ in Eq.~(\ref{constrangimento}) using Eqs.~(\ref{y}) and (\ref{theta}), we obtain the useful relation for $\Theta$ in terms of the $x$ and $w$ variables,
\begin{equation}
\Theta^2= \frac{1-w^2}{1-w^2 + \frac{V}{2\lambda}}.
\end{equation}
Next we follow to rewrite the equation of motion Eq.~(\ref{motion}) in terms of a system of first order differential equations for the new variables such that,
\begin{eqnarray}
x' &=& 3 \left( \sqrt{\frac{2}{3}} \Theta w - x \right), \label{xeq}
\\
w' &=& \frac{3}{2} \frac{V_{,x}}{V} (1-w^2) \left[ xw - \Theta \sqrt{\frac{2}{3}} \right], \label{weq}
\end{eqnarray}
where a prime means differentiating in respect to the number of e-folds $N=\ln a(t)$, so that $x'=dx/dN$.
This system of equations closes as $\Theta$ depends only on $x$ and $w$.
We immediately note that at low energies ($\rho\ll 2\lambda$ and therefore, $\Theta\approx1$) we end up recovering the
four-dimensional equations studied in Ref.~\cite{Kumar:2014oka} even though the variables were normalized to $H^2$ instead of $\rho$ as we do here. We would like to see now, how the presence of this correction term, $\Theta$, affects the dynamics of the system in comparison with the evolution in the four-dimensional case.
\subsection{Critical points}\label{critpoints}
Let us assume for now that $\Theta$ evolves sufficiently slow such that we can take it to be a constant within a few $e$-folds. We will see later that this assumption is actually supported by the numerical solutions. We can then identify the {\it instantaneous} critical points of the dynamical system established by Eqs.~(\ref{xeq}) and (\ref{weq}). These are shown in Table \ref{tabela}.
\begin{table}[ht]
\centering
\begin{tabular}{ c|c c c c }
& $x$ & $w$ & $V_{,x}/V$ & Description \\ \hline
A & $\pm \sqrt{\frac{2}{3}} \Theta$ & $\pm 1$ & any & kinetic domination\\
B & $x_{\rm ext}$ & $\sqrt{\frac{3}{2}}\frac{1}{\Theta} x_{\rm ext}$ & 0 & potential extrema
\end{tabular}
\caption{\label{tabela} Instantaneous critical points of the dynamical system.}
\end{table}
The critical points A do not exist for the standard scalar field models \cite{Copeland:1997et} and result from the extra $3 H \chi$ term in the equation of motion (\ref{motion}). One of the eigenvalues vanishes, hence, we cannot infer anything regarding its stability from the linear analysis without specifying the form of the potential. The critical point B corresponds to the value of the field at the extrema of the potential, therefore, its stability is strongly dependent on whether it corresponds to a minimum or a maximum of the potential.
From the analysis of the critical points we can see that, in the five dimensional set up, the critical points have a dependence on the correction term $\Theta$. This means that as the energy decreases, the instantaneous critical points move along the phase space and approach the four dimensional case at low energies, $\Theta =1$.
In Figs.~\ref{phase1} and \ref{phase2} is shown the phase space portrait for a potential of the form $V=e^{\chi^2} -1$.
Comparing these figures we, again, note that the critical points A (upper and lower dots) are shifted along the $x$ axis as the system evolves and will eventually end at $x=\pm \sqrt{2/3}$ (4 dim case). As we will see in Sec.~\ref{inflation}, at the critical points A (top and bottom dots), the universe inflates and critical point B (central dots) corresponds to the attractor and potential minimum for this potential where reheating happens as usual \cite{DeFelice:2012wy}.
\begin{figure}[ht]
\includegraphics[width=8.5cm]{phase1.png}
\caption{\label{phase1}Phase space $(\tanh (x),w)$ for $V=e^{\chi^2} -1$ at $\Theta=0.3$. }
\end{figure}
\begin{figure}[ht]
\includegraphics[width=8.5cm]{phase2.png}
\caption{\label{phase2}Phase space $(\tanh (x),w)$ for $V=e^{\chi^2} -1$ at $\Theta=0.9$. }
\end{figure}
An alternative way to study the stability of the critical points is by defining the effective potential,
\begin{equation}
V_{{\rm eff},\chi}= 3 \dot{H}\chi + V_{,\chi}.
\end{equation}
We illustrate the potential and the corresponding effective potential for $V=e^{\chi^2} -1$ in Fig.~\ref{effective1}.
\begin{figure}[ht]
\includegraphics[width=8.5cm]{effective.png}
\caption{\label{effective1} Potential $V(\chi)$ (solid line) and effective potential $V_{eff}$ (dashed lines) for the potential $V=e^{\chi^2} -1$ for different values of $\Theta$.}
\end{figure}
We can observe the shift in the value of the instant critical points as the energies decrease, i.e., as $\Theta$ approaches unity, where the critical points are $x = \pm \sqrt{\frac{2}{3}}$ as we can also verify in Table \ref{tabela}.
One interesting feature regarding the dynamics of a 3-form in RSII is that the $\Theta$ dependence of the dynamics can change the stability of the critical points as the energy decreases.
For example, in Fig.~\ref{effmexican}, we traced the Landau-Ginzburg potential
\begin{equation}
V(\chi)=(\chi^2-c^2)^2,
\end{equation}
with $c=0.5$ (solid), and its effective potential (dashed) at different values of $\Theta$ and we observe that at early times the potential minima at $x = \pm 0.5$ are initially unstable and, as the energy decreases, they become stable.
%
\begin{figure}[ht]
\includegraphics[width=8.5cm]{effmexican.png}
\caption{\label{effmexican} Potential $V(\chi)$ (solid line) and effective potential $V_{eff}$ (dashed lines) for the potential $V=(\chi^2-0.5^2)^2$ for different values of $\Theta$.}
\end{figure}
\subsection{Initial conditions and slow roll regime}
In order to study inflation we need to understand how the slow-roll parameters are modified in this set up. Analogously to the scalar field as well as 3-forms \cite{Koivisto:2009ew,DeFelice:2012jt} the parameters are defined by $\epsilon \equiv -\dot{H} / H^2 = -d\ln H/ dN$ and $\eta=\epsilon ' / \epsilon - 2\epsilon$. One manner to establish a sufficient condition for inflation is, $\epsilon \ll 1$ and
$|\eta|\ll 1$,
which must last for at least $\approx 50$ $e$-folds. For our RSII model we have,
\begin{eqnarray}
\epsilon &=& \frac{3}{2} x \frac{V_{,x}}{V} (1-w^2) (2 - \Theta^2), \\
\eta &=& \frac{x'(V_{,x} + V_{,xx}x)}{V_{,x}x} + 6x \frac{V_{,x}}{V} (1-w^2) \frac{\Theta^2 -1}{2-\Theta^2},
\end{eqnarray}
where the terms in $\Theta$ signal the new contributions to the slow-roll parameters.
\subsection{3-form inflation on the brane}\label{inflation}
In this subsection we present inflationary solutions for the system (\ref{xeq})--(\ref{weq}). We also compare the evolutions between the four and five dimensional cases.
Inspecting Fig.~\ref{exp} and Fig.~\ref{epsilon} we note that inflation happens when the field is on the plateau of the evolution that for the four dimensional case is flat and corresponds to the critical point $\chi=\pm \sqrt{2/3}$ \cite{Kumar:2014oka}. For the RSII case, however, the plateau has a gentle slope due to the dependence of the instantaneous critical points on $\Theta$ (we saw that $\chi = \pm \sqrt{2/3} \Theta$) up to the point in which $\chi=\pm \sqrt{2/3}$. We can also note that, for the same initial conditions, inflation lasts about 30 $e$-folds longer in the five dimensional set up due to the fact that there is additional friction to the field's evolution. When inflation ends, the field goes to the attractor $\chi=0$ which is the potential minimum (critical point B in Table \ref{tabela}).
\begin{figure}[ht]
\includegraphics[width=8.5cm]{exp.png}
\caption{\label{exp} Solutions for the system (\ref{xeq})--(\ref{weq}) for the four dimensional case (dashed line) i.e. for $\Theta=1$ already studied in \cite{Kumar:2014oka} and for the RSII model (solid line) when $\Theta$ is given by Eq.~(\ref{theta}) for $V=V_0 (e^{\chi^2} -1)$, $V_0=10^{-14}$, $\lambda=10^{-12}$ and for the initial conditions $(x_0,w_0)=(2,0.9055)$. The smaller panel shows the change in $\Theta$, for the RSII model, as the system evolves. }
\end{figure}
\begin{figure}[ht]
\includegraphics[width=7.5cm]{epsilon.png}
\caption{\label{epsilon} Change in the slow roll parameter $\epsilon$ for the solutions for the system (\ref{xeq})--(\ref{weq}) for the RSII model for $V=V_0 (e^{\chi^2} -1)$, $V_0=10^{-14}$, $\lambda=10^{-12}$ and for the initial conditions $(x_0,w_0)=(2,0.9055)$. The dashed line marks $\epsilon=1$ just for reference.}
\end{figure}
\section{Cosmological perturbations}
\label{perturbations}
Since the 3-form is confined to the brane and neglecting any backreaction effects of the metric fluctuationsb in the fifth dimension \cite{Maartens:1999hf}, the power spectrum of the curvature perturbations reads,
\begin{equation}
\label{power}
\mathcal{P}_{\zeta} = \left.\frac{2 H^4}{m_{\rm pl}^2 \pi V_{,\chi} \chi c_s} \right|_*,
\end{equation}
where,
$*$ indicates horizon crossing $ c_s k=aH$, and the sound speed is given by \cite{Koivisto:2009fb,Mulryne:2012ax},
\begin{equation}
c_s^2 = \frac{V_{,\chi\chi} \chi}{V_{,\chi}}.
\end{equation}
From the Planck 2015 results \cite{Ade:2015xua}, we fix the power spectrum of scalar perturbations as $\mathcal{P}_{\zeta}(k_0) = 2.196 \times 10^{-9}$ for the pivot scale chosen at $k_0 = 0.002$ Mpc$^{-1}$.
The spectral index is given b
\begin{equation}
\label{spectral}
n_s -1 = -5\epsilon - \frac{\dot{c}_s}{c_s H} - \epsilon c_s^2 + \frac{V_{,\chi}}{3\chi H^2} (1+c_s^2),
\end{equation}
which, as the power spectrum, also has a dependence on the speed of sound.
In the Randall-Sundrum model, however, the amplitude of the tensor modes are modified and the respective power spectrum reads \cite{Langlois:2000ns},
\begin{equation}
\label{at}
\mathcal{P}_T = \frac{64\pi}{m_{\rm pl}^2} \left( \frac{H}{2\pi} \right)^2 F^2(x_0) |_*,
\end{equation}
where $F$ is a correction function,
\begin{equation}
\label{f}
F(x)= \left[ \sqrt{1+x^2} - x^2 \ln \left( \frac{1}{x} + \sqrt{1+ \frac{1}{x^2}} \right) \right]^{-1/2},
\end{equation}
and
\begin{equation}
x_0 = \left(\frac{3}{4\pi\lambda} \right)^{1/2} H M_{\rm Pl}.
\end{equation}
For $x_0 \ll 1$, $F(x_0) \simeq 1$ and Eq.~(\ref{at}) reduces to the standard cosmology formula, and for $x_0 \gg 1$, $F(x_0) \simeq \sqrt{3x_0 /2}$. Finally, the tensor to scalar ratio is then,
\begin{equation}
\label{tsratio}
r\equiv \frac{\mathcal{P}_T}{\mathcal{P}_{\zeta}} = \frac{8}{H^2} V_{,\chi} \chi c_s F^2 (x_0).
\end{equation}
We are now ready to compare the cosmological parameters, scalar to tensor ratio and spectral index, of our inflationary setting with the 2015 Planck data \cite{Ade:2015lrj}.
First we consider a form of the scalar potential which has been proven in Ref. \cite{Kumar:2014oka} to lead to a viable cosmology in the four dimensional set up (although for a two 3-form system) and to produce a good fit to the Planck 2013 results,
\begin{equation}
\label{pot}
V=V_0 (\chi^2 + b\chi^4),
\end{equation}
where $V_0$ and $b$ are free parameters.
In Fig.~\ref{results} the bottom bar represents the prediction for the five dimensional case with
$\lambda=10^{-5}$. With this value of the brane tension, the evolution quickly reaches $\Theta \approx 1$ which means that this case is practically indistinguishable from the four dimensional solution.
When we lower the brane tension and consequently increase the five dimensional effects, we observe that the predictions worsen due to the presence of the correction $F^2(x_0)$ in Eq.~(\ref{at}), which enhances the tensor to scalar ratio. For $\lambda =10^{-10}$, corresponding to $\lambda \simeq (3.9 \times 10^{16}\,\,{\rm GeV})^4$ (corresponding to the upper bar) the predictions are beyond the Planck TT+lowP contour limits. We find a lower bound, for 60 $e$-folds, of $\lambda \simeq 1.5 \times 10^{-9}$, corresponding to $\lambda \geq (7.6\times 10^{16}\,\,{\rm GeV})^4$, for the inflationary predictions to be within the Planck TT,TE,EE+lowP contour limits.
\begin{figure}[ht]
\includegraphics[width=8.5cm]{results.png}
\caption{\label{results} Comparison of the spectral index and the tensor to scalar ratio against the recent Planck 2015 data \cite{Ade:2015lrj} for 50 (small dot) and 60 (large dot) $e$-folds for different values of the brane tension $\lambda$. We considered the potential in Eq.~(\ref{pot}) with $b=-0.245$. The bars represent, from bottom to top, the solutions with $\lambda=10^{-5}$, $\lambda = 3 \times 10^{-9}$ and $\lambda =10^{-10}$ in units
$\kappa^2=1$).}
\end{figure}
In Fig.~\ref{r1} we analyse how the brane tension and the tensor to scalar ratio are related as $\lambda$ is lowered for 60 $e$-folds. For $\lambda < 10^{-7}$, $r$ quickly increases due to the presence of $F^2$ in Eq.~(\ref{tsratio}), making the predictions worse as we also saw in Fig.~\ref{results}. In Fig.~\ref{ns} we present the relation between the spectral index and the logarithm of the brane tension $\lambda$. As expected, $n_s$ is almost insensitive to $\lambda$ for large values of this quantity. This is the case because
at large $\lambda$ the standard scenario is recovered and as in the scalar picture of the three-form the scalar potential is quadratic, the spectral index must be close to $n_s \sim 0.967$ \cite{Mulryne:2012ax}.
When we lower the brane tension, in order to keep the power spectrum of scalar perturbations fixed as $\mathcal{P}_{\zeta}(k_0) = 2.196 \times 10^{-9}$, for the pivot scale chosen at $k_0 = 0.002$ Mpc$^{-1}$, we also have to change $V_0$ in order to
ensure this normalization. This relation is shown in Fig.~\ref{vzero}.
\begin{figure}[ht]
\includegraphics[width=7.5cm]{r1.png}
\caption{\label{r1} $\log \lambda$ vs $r$, for the potential (\ref{pot}), with $b=-0.245$, for 60 $e$-folds, for different values of the brane tension $\lambda$. }
\end{figure}
\begin{figure}[ht]
\includegraphics[width=7.5cm]{ns.png}
\caption{\label{ns} $\log \lambda$ vs $n_s$, for the potential (\ref{pot}), with $b=-0.245$, for 60 $e$-folds, for different values of the brane tension $\lambda$. }
\end{figure}
\begin{figure}[ht]
\includegraphics[width=7.5cm]{vzero.png}
\caption{\label{vzero} $\log \lambda$ vs $V_0^* = V_0 \times 10^{12}$, for the potential (\ref{pot}), with $b=-0.245$, for 60 $e$-folds, for different values of the brane tension $\lambda$. }
\end{figure}
\section{Summary and discussion}
\label{conclusions}
In this work we explored the main differences between the dynamics of a single 3-form in the Randall-Sundrum II braneworld and the standard four dimensional case \cite{Koivisto:2009ew}. We followed to write the equations of motion for the 3-form model in terms of a system of first order differential equations (\ref{xeq})--(\ref{weq}). By defining a set of useful variables $(x,y,w,\Theta)$ we identified what we called the instantaneous critical points which now have a dependence on the correction term, $\Theta$, arising from the modified Friedmann equation.
We illustrated the effects that take place at high energies by showing the phase space of the system at different stages of the universe, or in other words, for different values of $\Theta$, and by interpreting them as a modification to the effective potential.
It was observed that in five dimensions the stability of some instantaneous critical points can change with the energy.
We presented an inflationary solution for the potential in Eq.~(\ref{pot}) and computed the respective
tensor to scalar ratio (\ref{tsratio}) and spectral index (\ref{spectral}).
We were able to fit the cosmological predictions with the recent Planck 2015 data \cite{Ade:2015lrj} for a choice of parameters and saw that the effects of the braneworld bring the observables away from the central region of the data contours. By performing this study, we found a lower bound for the brane tension for the potential (\ref{pot}) such that the observables' values remain inside the contours of the Planck TT,TE,EE+lowP.
\begin{acknowledgments}
The authors thank Carsten van de Bruck and Tomi Koivisto for comments on the manuscript.
N.J.N was supported by the Funda\c{c}\~{a}o para a Ci\^{e}ncia e Tecnologia
(FCT) through the grants EXPL/FIS-AST/1608/2013 and UID/FIS/04434/2013.
\end{acknowledgments}
|
2110.00142
|
\section{Introduction}
The difference between the structure of dark matter (DM) halos as predicted by the Lambda-Cold Dark Matter cosmological model ($\Lambda$CDM) and that which is inferred by observations of gas rotational profiles in galaxies is a long-standing problem in modern cosmology \citep[][]{Moore1994,FloresPrimack1994} with a wide range of postulated solutions. The structure of DM halos as predicted by DM-only simulations \citep[e.g.][]{white1978,Springel2008} is characterized by steeply rising density profiles (`cusps') in the inner regions of halos, parameterized by the NFW profile \citep[][]{navarro1996CDMhalos} which gives a power-law slope $\alpha$ of this inner profile of $-1$. Early measurements of rotation curves in dwarf galaxies have shown regions of constant density known as `cores' with power-law slopes of $\alpha \sim 0$ \citep[e.g.][]{Burkert1995,deblok2001,deBlok2008,KuzioDeNaray2008}. While there is substantial evidence for the existence of cores in dwarf galaxies \citep[e.g.][]{Gilmore2007,Kormendy2009,Oh2011,Oh2015,Lelli2016}, there is debate over the reliability of certain techniques for the inference of the true dark matter potential \citep{Genina2018,Oman2019}.
Another complication to this dilemma is that observed rotation curves in dwarf galaxies exhibit a wide variety of behavior, including rotation curves that rise more rapidly than the NFW profile, consistent with a contraction of the halo, and those that rise significantly more slowly, consistent with expansion. Despite their success in reproducing many observed properties of galaxies, both local and statistical, \citep[][]{Vogelsberger2020}, hydrodynamical simulations of galaxy formation have consistently predicted a uniform shape for rotation curves, posing a problem in replicating the observed diversity \citep{Oman2015,Oman2019,read2016rc,Santos-Santos2018,Santos-Santos2020}.
A theoretically appealing solution to these discrepancies is that the nature of DM is more complex than proposed in $\Lambda$CDM. Proposed models include warm dark matter \citep{Dodelson1994,Bode2001} and self-interacting dark matter \citep[SIDM,][]{Yoshida2000,Spergel2000,Vogelsberger2012,Vogelsberger2019,Rocha2013,TulinYu2018}. SIDM has been fairly successful in reproducing diverse rotation curves \citep[e.g.][]{Creasey2017,Ren2019,Kaplinghat2020} and explaining the diversity of MW satellites \citep[][]{Zavala2019}. It is worth noting, however, that results for SIDM can depend strongly on the adopted cross-section. Another interesting proposal includes a new hypothetical ultra-light scalar particle with a de Broglie wavelength on astrophysical scales, forming a Bose-Einstein condensate the size of the DM halo, known as fuzzy dark matter \citep[][]{Hu2000,Mocz2017,Lancaster2020,Burkert2020}. While these models prove viable alternatives to $\Lambda$CDM~with testable predictions \citep[][]{Robles2017,Bozek2019}, they may remain difficult to distinguish from CDM on small scales, especially when the effects of galaxy formation are taken into account \citep{Elbert2018,Fitts2019}.
It has also been proposed that the feedback-driven motion of baryons within the halo can gravitationally perturb the dark matter potential, leading to expansion \citep{navarro1996}. The repeated outflow of gas following bursts of star formation (SF) has been demonstrated to be a more realistic mechanism for core formation than single, highly violent outbursts \citep[][]{ReadGilmore2005,Governato2010}. This framework was theoretically quantified by \citet{pontzengovernato2012} who introduced an analytical model for core formation in which dark matter particles acquire energy and migrate to more distant orbits via repeated oscillations in the central gravitational potential, driven by supernova (SN) feedback.
Since the physics of star formation and feedback have not been fully constrained, different effective models of interstellar medium (ISM) physics implemented across the literature have produced different outcomes. For example, the Illustris simulations have been successful in reproducing many properties of galaxies \citep[][]{Genel2014,v14illustris,v14nature}, but have not been able to produce DM cores \citep[][]{Chua2019}. The EAGLE simulations \citep[][]{Schaye2015} have also been shown to not produce DM cores under their fiducial model \citep[][]{Schaller2015,benitezllambay2019}. Zoom-in simulations using the same prescriptions as EAGLE and Illustris have been performed and similarly demonstrate an inability to induce expansion in the DM halo \citep[e.g.][]{bose2019}, indicating that resolution is not responsible for this effect in these models. Meanwhile, other simulations, including \citet{Zolotov2012}, the FIRE project \citep[][]{hopkins2014fire1,hopkins2018fire2,chan2015,wetzel2016LATTE,fitts2017}, and NIHAO \citep[][]{wang2015nihao,tollet2016,Dutton2016}, have been able to produce cores in dwarf galaxies that more closely match observations, indicating that the prediction of DM cores is model-dependent to some degree.
Differences in the modeling of baryonic physics have long been quantified by the SF density threshold, which is the minimum gas density required to form a star particle. \citet{pontzengovernato2012} showed that cosmological zoom-in simulations run with the \textsc{Gasoline} code were unable to induce core formation when using a value of $\rho_\text{th}$\,$=0.1$~cm$^{-3}$, but cores did indeed form when increased to $\rho_\text{th}$\,$=100$~cm$^{-3}$, a value consistent with the observed densities of molecular clouds \citep[][]{Ferriere2001}. Recent work has therefore focused heavily on this parameter, arriving at similar conclusions within the EAGLE simulations \citep[][]{benitezllambay2019}, and NIHAO \citep[][]{Dutton2019}. It has long been reported that `bursty' SF drives repeated outflows, thereby expanding the DM halo by driving mass to the outer regions \citep[][]{Brooks2014}. \citet{benitezllambay2019} conclude from their numerical tests on the density threshold that rapid fluctuations in gas content resulting from bursty SF are insufficient to alter the inner DM halo, but that gas must accrete to high levels of density, dominating the inner gravitational potential before being blown away in order to induce core formation. They also make note that there is no simple relation between SF history and core formation. \citet{Dutton2019} also find that a higher value of $\rho_\text{th}$~induces cores in the NIHAO simulations, but their analysis suggests that fluctuations in SF feedback (and therefore gas content) must occur on \textit{sub-dynamical} timescales in order to induce core formation. Both authors agree that SF burstiness is insufficient to fully explain halo expansion, and that the density threshold is strongly indicative of a resulting flattened inner DM distribution.
The energetics of core formation discussed in \citet{pontzengovernato2012} require
rapid motion of sufficiently dense gas clouds in the inner regions of galaxies in order to perturb the gravitational potential and transfer DM to larger orbits.
High resolution simulations that lack detailed physical modelling are unable to capture the small-scale effects of energetic coupling between SF and the ISM due to the use of low star formation threshold, often with $\rho_\text{th}$~$= 0.1 \text{cm}^{-3}$, as well as effective equations of state rather than explicitly implemented cooling physics. Meanwhile, detailed ISM models that self-consistently treat a multiphase, structured ISM are relatively new and have not been directly applied to the problem of core formation. In short, the majority of models that have been used to study this problem are empirically calibrated to reproduce scaling relations of populations of galaxies and implemented in large-volume simulations. These models have been adapted to high resolution zoom-in simulations, with mixed results \citep[][]{benitezllambay2019,bose2019}. Fewer studies have focused on studying core formation as a thoroughly small-scale problem, requiring both high resolution zoom-in simulations and models that capture the local details of physical processes relevant to the state of the ISM. More details of the varying approaches to galaxy modeling are given in a recent review of cosmological simulations \citep[][]{Vogelsberger2020}.
While there is broad agreement in the literature that high thresholds induce cores \citep[e.g.][among the previously listed]{Governato2010,Maccio2012,Teyssier2013,dicintio2014SMHM} and low thresholds do not do not \citep[][]{Oman2015,Schaller2015,bose2019}, there have been limited systematic investigations of the physical outcomes of modeling choices, including comparative analyses of parameters within the same overall modeling scheme. The consistency of models with similar $\rho_\text{th}$~does not rule out the possibility that other modeling choices contribute to halo expansion, including ones that cannot be neatly quantified by a single parameter.
Beyond the physical effects of baryons, difficulties in observing and modeling gas rotation curves in galaxies have led to speculation that large uncertainties might be partially responsible for the observed diversity of galactic rotation curves. While extensive work has been done to improve observational techniques for estimating velocity profiles \citep[][]{KuzioDeNaray2006,KuzioDeNaray2008,Adams2014}, techniques based on alignment of metallicity populations \citep[e.g.][]{Walker2011} and tilted-ring modeling \citep[e.g.][]{Rogstad1974,Iorio2017} have been recently been demonstrated via application to the APOSTLE simulations to predict DM cores when none actually exist \citep[][]{Genina2018,Oman2019}. This, combined with the large degenerecies in modeling rotational velocities in the presence of non-circular motions \citep[][]{Marasco2018,Santos-Santos2020} suggest that the observed diversity of rotation curves might not be solely a result of physical processes within galaxies, be they baryonic or dark.
In this study, we compare the novel Stars and MUltiphase Gas in GaLaxiEs (\texttt{SMUGGLE}) feedback model \citep[][]{Marinacci2019} to the classic \citeauthor{SpringelHernquist2003} (2003; SH03 hereafter) model, as they represent two paradigms of galaxy formation modeling (i.e. top-down -- SH03, and bottom-up -- \texttt{SMUGGLE}) while implementing the same method of solving gravity+hydrodynamics \citep[\textsc{Arepo,}][]{arepo}. We aim to investigate the differences in and relationship between galaxy formation and DM distribution within these two modeling paradigms in a controlled environment through the use of idealized simulations of a single dwarf galaxy. We also implement variations in model parameters (density threshold and local SH efficiency) within \texttt{SMUGGLE} to shed light on their relevance to core formation within this model, and how their differential effects within this model compare to previous numerical experiments.
The paper is organized as follows: in Section \ref{sec:methods}, we discuss the set up of our isolated dwarf galaxy simulations; in Section \ref{sec:smuggle_and_SH03}, we compare the phenomenological differences between an isolated dwarf galaxy ($M_\star$~$\sim 10^8$ M$_\odot$, $M_{200}$~$\sim 10^{10}$ M$_\odot$) run with each model, and then introduce variations in the \texttt{SMUGGLE} model to investigate the physical nature of core formation in Section \ref{sec:ISM}. We conclude in Section \ref{sec:structure} by examining the morphology of each run, including an investigation of the variation of rotational velocity curves of gas. We summarize our findings in Section \ref{sec:summary}.
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth]{figures/rho_400.pdf}
\caption{Face-on and edge-on surface density projections of the isolated dwarf galaxy on the fiducial \texttt{SMUGGLE} model generated with the Cosmic Ly$\alpha$ Transfer code \citep[COLT,][]{Smith2015,Smith2017}. Stars formed during the duration of the simulation are shown in white, with gas color-weighted according to surface mass density. The width of each frame is 20 kpc. The richly structured ISM is a result of the detailed ISM physics included in the \texttt{SMUGGLE} model.}
\label{fig:my_label}
\end{figure*}
\section{Methods}
\label{sec:methods}
We analyze a set of high-resolution, idealized simulations of isolated dwarf galaxies of $M_\star$~$\approx 10^8$ M$_\odot$~in halos of mass $M_{200}$~$\approx 10^{10}$ M$_\odot$~run with the moving mesh code \textsc{Arepo} \citep{arepo,Weinberger2020}. This scale of stellar mass to halo mass has been demonstrated to form feedback-driven cores in other simulation codes \citep[e.g.][]{dicintio2014SMHM,tollet2016}.
Initial conditions were generated via the method described in \citet{Springel2005}, while star formation and feedback were subsequently enabled via the \texttt{SMUGGLE} model \citep{Marinacci2019}. \texttt{SMUGGLE} implements a wide variety of sub-resolution processes, including gas heating and cooling from which a detailed, multiphase inter-stellar medium (ISM) emerges, a stochastic formation process for stars, and feedback via supernovae (SNe), radiation, and stellar winds.
Previous work with \texttt{SMUGGLE} includes \citet{Li2020}, who study the formation of giant molecular clouds (GMCs) in Milky-Way mass galaxies, in particular the response of GMCs to various choices of the local star formation efficiency - a parameter we study here as well. They find that \texttt{SMUGGLE} is able to regulate star formation through feedback, with a 3-fold increase in star formation rate (SFR) in runs with no feedback processes enabled. This result is encouraging as it enables self-consistent prediction of kpc-scale galaxy properties. Further, they demonstrate that SN feedback disrupts the spatial correlation of GMCs on scales $> 0.2$ kpc, which is relevant to our discussion on core formation later on. In addition, the \texttt{SMUGGLE} has been further refined with the development of a state-of-the-art model for the treatment of radiation fields, dust physics, molecular chemistry, and metal cooling by \citet{Kannan2020}. This model is able to produce a more complex picture of the ISM of galaxies while maintaining consistent global properties, such as SFR.
\subsection{The \texttt{SMUGGLE} ISM Model}
\label{sec:SMUGGLEmodel}
In this work, we implement the standard \texttt{SMUGGLE} model as described in \citet{Marinacci2019}.
Here we summarize the main physical modeling choices. The primary processes include gravitational hydrodynamics, which is solved by \textsc{Arepo} \citep[][]{arepo}, gas heating and cooling which produce an emergent multiphase ISM, the stochastic formation of star particles, and feedback from stars and SNe.
\subsubsection{Heating and cooling}
One of the biggest differences in \texttt{SMUGGLE} compared to previously implemented ISM models in {\sc arepo} \citep[e.g. SH03,][]{v14illustris,Pillepich2018} is its ability to explicitly model a cold gas phase with temperatures falling below T$_\text{gas} \sim 10^4$ K. First, a primordial mix of Hydrogen and Helium is modeled by a network of two-body processes including collisions, recombination, Compton cooling via CMB photons \citep[][]{Ikeuchi1986}, and UV-background photoionization \citep[][]{faucher2009}.
Cooling has two main regimes, metal-line cooling driving the gas temperature down to the warm phase (T$_\text{gas} \sim 10^4$ K) -- which was included in previous ISM models -- while fine structure and molecular cooling implemented in \texttt{SMUGGLE} allows the gas to further cool to $T \sim 10$ K. Cooling rates are calculated in a UV background with the \citet{hopkins2018fire2} fit as a function of temperature, metallicity, gas density, and redshift, with self-shielding taken into account at $z \leq 6$ as in \citet{rahmati2013}. The calculated rates are then scaled to the metallicity of the gas cell. By default, metallicities are updated self-consistently as the simulation evolves in {\sc arepo}. However, for idealized set-ups metallicity can be fixed to offset the lack of replenishment of pristine gas from cosmological infall. For simplicity, in this paper we fix the metallicity of our idealized runs to the solar value.
\subsubsection{Star formation}
\label{sec:starformation}
Star particles representing single stellar populations with a \citet{chabrier2001} initial mass function are formed probabilistically in cold, dense gas. Gas is determined to be eligible for star formation based on several criteria. The first is the gas density threshold, below which no gas can be converted into a star particle. \texttt{SMUGGLE} adopts a value of 100 cm$^{-3}$, in line with observations of giant molecular clouds \citep[][]{Ferriere2001}. Star formation is also restricted to gravitationally self-bound regions \citep[see][]{hopkins2018fire2}.
Additionally, star formation rates may be computed according to the H$_2$ fraction, though it is usually $\sim 1$ in sufficiently dense gas.
The probability of an eligible gas cell to be turned into stars is determined via $\dot{M}_\star = $ $\varepsilon_\text{sf}$~$ M_\text{gas}/t_\text{dyn}$, where $t_\text{dyn}$ is the gravitational dynamical time of the gas and $\dot{M}_\star$ its star formation rate. In its default mode, the local SF efficiency parameter $\varepsilon_\text{sf}$~is assigned a value of 0.01 to match the low efficiencies measured in observations \citep{smith2018}, although \citet{hopkins2018fire2} showed that the exact level of feedback-regulated star formation is independent of $\varepsilon_\text{sf}$. We explore in Section \ref{sec:ISM} the effect of $\varepsilon_\text{sf}$~on our \texttt{SMUGGLE} simulations.
In addition to the default mode described above, \texttt{SMUGGLE} can also be run using the variable efficiency model (\texttt{vareff}), which implements a variable star formation timescale ($t_\text{sfr}$). This quantity, defined as $t_\text{sfr}$ = $M_\text{gas}$ / $\dot{M}_\star$, is varied for each grid cell according to its virial parameter ($\alpha$), which quantifies the cell's ability to resist gravitational collapse via thermal support and gas pressure. The exact parameterization is given by Eqn. \ref{eqn:vareff} below \citep[][]{Padoan2012,Semenov2016}.
\begin{equation}
\label{eqn:vareff}
t_\text{sfr} = \frac{M_\text{gas}}{\dot{M}_\star}\text{min}\Big(\exp{\big(1.6\sqrt{\alpha/1.35}\big)}, 10^{30}\Big)
\end{equation}
\noindent This model prioritizes star formation efficiency in highly dense regions. In Section \ref{sec:smuggle_vars}, we investigate both a variable efficiency model, and one that maximizes the local star formation efficiency. Note that since $t_\text{sfr} = t_\text{dyn}/\varepsilon_\text{sf} = M_\text{gas}/\dot{M}_\star$, a parameterization on $t_\text{sfr}$ is equivalent to a parameterization of the efficiency $\varepsilon_\text{sf}$, all other quantities being the same for a given cell.
\subsubsection{Feedback}
Stellar feedback is modeled locally according to several sources including stellar winds, radiation from young stars and supernovae (SNe). Stellar winds due to massive OB and AGB stars contribute to the mass return to the ISM and are taken into account during the pre-processing of the gas.
Cumulative mass loss from OB stars, as well as energy and momentum returned from both OB and AGB stars are determined via the parameterizations presented in \citet{hopkins2018fire2}, while AGB wind mass transfer is given by \citet{v13feedback}. All the properties determined from the different feedback channels are then injected with corresponding metallicity to the surrounding gas in the rest frame of the star. Stellar winds are a continuous process, and are thus treated continuously across each time step for each star particle.
Radiative feedback from young stars change the ionization, thermal, and dynamical state of the ISM, pre-processing the media where later SNe will go off. \texttt{SMUGGLE} includes a treatment of photoionization aimed at capturing the formation of HII regions by young, massive stars. Ionizing photon rates from young stellar particles are calculated by choosing a mass-to-light ratio and average ionizing photon ($> 13.6$\,eV) energy to correspond with a $T = 4\times10^4$\,K blackbody spectrum, consistent with OB type stars. The number of available photons in a given timestep is used to stochastically photoionize neighboring gas cells after accounting for the expected number of recombinations. Photoionized cells are then updated to be fully ionized and placed at a temperature $T=1.7 \times 10^4$\,K. In addition to photoionization, young stars exert radiation pressure on neighboring gas cells, which is calculated according to their optical depth and position within the kernel. Multiple IR scattering is included, by assuming an average opacity $\tau = 10\,Z/Z_\odot$\,cm$^2$\,g$^{-1}$ \citep{hopkins2018fire2}. In the regime of small mass galaxies explored here, photoionization is expected to dominate among the radiation effects on the ISM, lowering the density of gas in the neighborhood of massive stars \citep[][]{Sales2014,hopkins2018fire2}.
Lastly, we stochastically model the injection of energy and momentum by discrete SN events onto neighbouring gas cells. It is important to note that \texttt{SMUGGLE} resolves individual SN explosions, and as such, the injected rates of energy and momentum are not continuous. The temporal distribution of individual Type Ia events is found by integrating the delay time distribution, which accounts for the approximate lifespan of an 8\,M$_\odot$~main sequence star, with rates and energetics consistent with observations \citep[][]{greggio2005} as well as previous implementations in \textsc{Arepo} \citep{v13feedback}, with each event releasing the same mass of ejecta \citep{thielemann2003}. The total number of Type II SNe is found by integrating the Chabrier IMF of each stellar particle. If necessary, we account for PdV work in the (unresolved) Sedov-Taylor phase by applying a momentum boost to match the terminal momentum per SN, which depends primarily on local density and metallicity \citep[e.g. ][]{Cioffi1988}. Energy and momentum are distributed to surrounding gas cells following a kernel weighting and a maximum coupling radius, as described in detail in \citet{Marinacci2019}.
\begin{table*}
\centering
\begin{tabular*}{\textwidth}{l @{\extracolsep{\fill}} c c c c c}
\hline
Name & $r_\text{core}$~[pc] & $\alpha$ & $M_\star$~[M$_\odot$] & Model description \\
\hline \\
\texttt{SMUGGLE} / \texttt{fiducial} & 431.3 & $-0.13$ & 7.76\e{7} M$_\odot$ & default \texttt{SMUGGLE} model \\
SH03 & 160.2 & $-0.52$ & 4.29\e{7} M$_\odot$ & \citet{SpringelHernquist2003} model\\
\texttt{rho0.1} & 324.2 & $-0.05$ & 9.69\e{7} M$_\odot$ & \texttt{SMUGGLE} with reduced gas density threshold, $\rho_\text{th}$~$=0.1$ cm$^{-3}$\\
\texttt{eSF100} & 490.7 & $-0.03$ & 8.39\e{7} M$_\odot$ & \texttt{SMUGGLE} with maximized local SF efficiency, $\varepsilon_\text{sf}$~$= 1$ \\
\texttt{vareff} & 528.3 & $-0.03$ & 9.12\e{7} M$_\odot$ & \texttt{SMUGGLE} with the variable efficiency model, see Sec. \ref{sec:starformation} \\
\\
\hline
\end{tabular*}
\caption{List of simulations used in this study. All initial conditions were generated according to \citet{Springel2005} and run for 2 Gyr $h^{-1}$, where we take $h = 0.7$. Our standard resolution initializes a $2.17\times10^{10}$ M$_\odot$~ halo with 3\e{7} dark matter particles, and $10^6$ gas particles, corresponding to a baryonic mass per cell of $\sim$850 M$_\odot$~and DM mass per cell of $\sim$7200 M$_\odot$. We adopt a gravitational force softening of $\epsilon = 16$ pc for all particle types. Also listed are the core radius (measured as described in Section \ref{sec:coresize}), inner DM power law slope $\alpha$, and stellar mass formed (i.e. not including the disk and bulge from initial conditions), all taken at final time.}
\label{tab:simtable}
\end{table*}
\subsubsection{Variations on the fiducial SMUGGLE model}
\label{sec:methods_variations}
We will explore in Section \ref{sec:ISM} the effect of changing some of the default choices in \texttt{SMUGGLE} and how this affects the formation of dark matter cores and the properties of our simulated dwarfs. The changes will be inspired by results presented previously in the literature, including \citet{read2016rc,benitezllambay2019,bose2019}. More specifically, we choose to vary the star formation gas density threshold $\rho_\text{th}$~and the local star formation efficiency $\varepsilon_\text{sf}$.
Table \ref{tab:simtable} summarizes our runs, which include the fiducial \texttt{SMUGGLE} run, SH03, and three variations on \texttt{SMUGGLE} as discussed in Section \ref{sec:smuggle_vars}: (i) \texttt{rho0.1}, using a reduced star formation density threshold of $\rho_\text{th}$~$= 0.1$ cm$^{-3}$; (ii) \texttt{eSF100}, which maximizes the local star formation efficiency to 100 per cent, $\varepsilon_\text{sf}$ $=1$; and (iii) \texttt{vareff}, a variable efficiency model which chooses a value between $\varepsilon_\text{sf}$~$=0.01$ and $\varepsilon_\text{sf}$~$=1$ depending on the density of the surrounding ISM. The fiducial \texttt{SMUGGLE} model implements these parameters with values of $\rho_\text{th}$~$= 100$ cm$^{-3}$ and $\varepsilon_\text{sf}$~$=0.01$.
\subsection{The Springel and Hernquist Model}
In addition to the fiducial \texttt{SMUGGLE} model, we run a simulation with the SH03 model \citep{SpringelHernquist2003}, which forms the basis for the ISM treatments in Illustris \citep{v14illustris,v14nature}, Auriga \citep{Grand2017} simulations, EAGLE \citep{Schaye2015}, APOSTLE \citep[][]{sawala2016}, HorizonAGN \citep[][]{Dubois2014}, SIMBA \citep[][]{Dave2019}, and others. The SH03 model, also run with the {\sc Arepo} gravity and hydrodynamics solver, uses an equation of state treatment of cold gas modelled with a two-fluid approach (cold clouds embedded in a lower density hot gas bath) to describe the interstellar medium. This approach, which has been demonstrated to be successful at modeling the kpc-scale properties of galaxies, has been found to not form dark matter cores \citep[][]{v14dwarfs,bose2019}.
We explicitly include stellar winds in the SH03 run with the wind velocity calculated directly from the energy and momentum summation of all SN in a given timescale and independent of halo properties. This is different from, for instance, the Illustris or Auriga projects, where the wind velocity is scaled to the dark matter velocity dispersion of the subhalo. Although such scheme is {\it de-facto} closer to the scalings expected for momentum-driven winds \citep{Norman2005} and shown to more accurately reproduce some galaxy and CGM observables \citep[e.g. ][]{Dave2011}, we choose a simpler wind model where no pre-assumptions are made with respect to the properties of the host halo, in an attempt to establish a fairer comparison with the \texttt{SMUGGLE} runs where no input information is required about the galaxy host. Ultimately, the impact of the exact modeling of the winds in our SH03 run is subdominant to the differences imprinted by the modeling of the ISM itself. As is the case in Illustris, Auriga, and other projects mentioned above, the wind particles in the SH03 model are artificially decoupled from the hydrodynamics for a short period of time, while such a treatment is not necessary in our new \texttt{SMUGGLE} prescription where outflows naturally arise from the kinematics and thermodynamics of stellar winds and SN explosions.
\subsection{Isolated Galaxy Setup}
Throughout this paper, we analyze simulations run with different ISM models applied to the same initial conditions (ICs). We initialize an isolated, idealized dwarf galaxy with $M_{200}$~$= 2.17 \times 10^{10}$\,M$_\odot$\ using the method outlined in \citet{Springel2005}. The distribution of dark matter is initialized according to a Hernquist profile \citep[][]{Hernquist1990},
\begin{equation}
\label{eqn:hernquist}
\rho_\text{dm}(r) = \frac{M_\text{dm}}{2\pi}\frac{a}{r(r+a)^3} \, ,
\end{equation}
\noindent where $a$ is a concentration-dependent scale length. This model is identical to the widely used NFW profile \citep[][]{navarro1996CDMhalos} at small radii ($\rho \propto r^{-1}$), while the power law exponent differs at large radii: $\rho_\text{NFW} \propto r^{-3}$ versus $\rho_\text{Hernquist} \propto r^{-4}$. Both models have been shown to accurately describe the distribution of DM for halos in a cosmological context.
The galaxy itself is initialized with an exponential disk of scale length $h$ for both stars and gas, in addition to a spherical stellar bulge modeled by the Hernquist profile. See Section 2 of \citet{Springel2005} for more details on the model galaxy setup. We choose parameters for our model galaxy consistent with the `Dwarf/SMC' setup described in \citet{Hopkins2011}, which gives a DM dominated dwarf galaxy similar to the pre-infall Small Magellanic Cloud with total baryonic mass $M_\text{bary} = 8.9\times10^8$ M$_\odot$, gaseous disk with $M_\text{gas} = 7.5\times10^8$ M$_\odot$, and DM halo with $M_{200} = 2\times10^{10}$ M$_\odot$~and concentration parameter $c = 15$.
The partitioning of cells in the initial conditions is done by setting the number of gas particles, $N_\text{gas}$, with $N_\text{DM} = 30 N_\text{gas}$, $N_\text{disk} = 0.2 N_\text{gas}$, and $N_\text{bulge} = 0.02 N_\text{gas}$. For the runs analyzed herein, we choose $N_\text{gas} = 10^6$, resulting in a particle mass of $m_\text{bary}\approx 850$ M$_\odot$. We choose the same value of gravitational softening for all particle types, with $\epsilon = 16$ pc. We have also run a set of simulations with an order of magnitude lower resolution ($N_\text{gas} = 10^5, \epsilon = 32$ pc) for convergence testing. We find excellent agreement between the two resolution levels tested, as shown in Figure \ref{fig:res_converge}.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/four_rho_ratio_slope_ffill1e6.pdf}
\caption{Dark matter density profiles of the isolated dwarf galaxy run with the fiducial \texttt{SMUGGLE} feedback model (black) and the SH03 feedback model (green) at selected times. The top row shows the DM density at each labeled time. Light grey lines represent the DM density profiles at $t=0$, and the blue dashed line is the NFW profile fit to $r$ > 3 kpc to account for variations in the inner region. The core radius $r_\text{core}$~is defined as the radius where $\rho_\text{NFW}$ / $\rho_\text{DM}$ = 2, and which can be seen in the bottom panels, including the horizontal line at 2. The vertical dashed-dot lines in each panel represented our measured $r_\text{core}$, which consistently capture the changes in DM density. In addition, power law slopes ($\rho \propto r^\alpha$) are shown in yellow, and are fit for $r_\text{DM}^\text{conv} < r < r_\text{core}$. Values for the convergence radius $r_\text{DM}^\text{conv}$ are typically around 50 pc.}
\label{fig:rho_panel}
\end{figure*}
\section{Forming Dark Matter cores in SMUGGLE}
\label{sec:smuggle_and_SH03}
We explore the evolution of the dark matter density profile in our simulated dwarf galaxy in Figure \ref{fig:rho_panel}, where each panel corresponds to different times, as labeled. The results of the default \texttt{SMUGGLE} model are shown in the solid black line, which demonstrates a clear flattening in the inner regions corresponding to the formation of a dark matter core in our initially cuspy halo. For reference, we include the initial dark matter distribution in each panel with a solid gray line.
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{figures/rcore_etc_ff_ill_1e6.pdf}
\caption{Time-evolving properties of the simulated isolated dwarf galaxy run with the fiducial \texttt{SMUGGLE} model in black and the SH03 feedback model in green. (\textit{a}) Measured core radius $r_\text{core}$~versus time. Black squares indicate timestamps of density profiles shown in Figure \ref{fig:rho_panel}. See text for definition of $r_\text{core}$. (\textit{b}) Power law slopes $\alpha$ fitted to $r_\text{DM}^\text{conv} < r < r_\text{core}$, binned with $\Delta t$ = 25 Myr. Dashed lines indicate the average slope for $t > $ 0.5 Gyr to account for initial relaxation effects. The \texttt{SMUGGLE} model results in a very flat inner profile ($\alpha\sim-0.1$) which extends over a larger portion of the galaxy with $r_\text{core}$ $\sim$ 400 pc, in contrast to the steeper ($\alpha\sim-0.6$), more concentrated ($r_\text{core}$ $\sim$ 150 pc) profile formed by SH03. (\textit{c}) Star formation rate (SFR) versus time. The SFR is smoothed over $\Delta t = $ 25 Myr bins. We find that \texttt{fiducial} produces a substantially burstier star formation history (SFH) than SH03, and that the average magnitude of SFR for SH03 agrees with that of \texttt{fiducial} in early times, but declines to much smaller levels after $t\approx$ 1.5 Gyr. (\textit{d}) Stellar mass ($M_\star$, dashed), gas mass within $r < 5$ kpc (solid, thick), and gas mass within $r < 1$ kpc (solid, thin). \texttt{SMUGGLE} results in frequent and significant changes in gas mass in the inner regions, while the gas mass < 1 kpc in SH03 smoothly decreases.}
\label{fig:rcore_sfr}
\end{figure}
\subsection{A consistent method for core size measurements}
\label{sec:coresize}
\subsubsection{Caveats \& numerical effects}
Figure \ref{fig:rho_panel} shows density profiles for various runs implementing the same ICs. We find the best fit NFW profile to the outer ($r > r_{\rm fit} = 3$\,kpc) dark matter distribution. The bottom panels in Figure \ref{fig:rho_panel} show the ratio between the analytic NFW fit and the measured DM density in the fiducial \texttt{SMUGGLE} simulations (solid black lines). Although in the outskirts the simulated profiles are very well described by the NFW fits ($\rho_{\rm NFW}/\rho_{\rm dm} \sim 1$), in the inner regions the analytic profile clearly overestimates the dark matter density in all cases. This is partially due to adiabatic contraction, demonstrated by the magenta line. In the case of SH03, feedback is not capable of producing further changes in the DM distribution, resulting in a profile almost identical to the adiabatic run, while the \texttt{SMUGGLE} model is able to produce an extended region of constant density by later times. Additionally, the shape of the galaxy can affect the resultant DM distribution. In the case of disks, this can lead to shallower central profiles \citep[][]{Burger2021}.
We note that numerical effects can spuriously transfer kinetic energy between particles of different masses, such as our gas and DM particles \citep[][]{Ludlow2019a}. A thorough investigation of the effects of gravitational softening and `numerical feedback' have been presented in \citet{Ludlow2019b,Ludlow2020}. While we adopt softening on the order suggested by \citet{vandenBosch2018} -- approximately three times lower than the convergence radius $r_{\rm dm}^{\rm conv}$ -- it is possible that spurious energy transfer between DM and baryonic particles via 2-body interactions contributes to the observed halo expansion. However, our tests are designed to isolate the effects of feedback. Numerical effects will be present in all our simulations, including the adiabatic and SH03 runs, but the methods of feedback coupling to the ISM vary. As such, our claims are about the differential effects between feedback implementations, not predictions of the absolute core sizes expected within dwarf galaxies in a cosmological context.
\subsubsection{Core size measurement}
Following \citet{benitezllambay2019}, we define $r_\text{core}$~as the location where the simulated dark matter density is a factor of 2 lower than the extrapolated best-fit NFW profile, $\rho_{\rm NFW}/\rho_{\rm dm} = 2$. However, we note that the authors compare against a low-threshold run rather than an NFW. Hydrodynamic relaxation may lead to a difference in predicted core radius. The measured $r_\text{core}$~is indicated with a vertical dashed line and listed in the lower panels.
This definition is robust to variations on $r_{\rm fit}$ in the range $1 - 10$ kpc (see Figure \ref{fig:rcore_slope_compare_nfwdcut} in the Appendix). Figure \ref{fig:rho_panel} shows that the density profile within $r_\text{core}$~for the fiducial \texttt{SMUGGLE} run is nearly flat at later times. We quantify this by finding the slope $\alpha$ of a power-law fit to the dark matter density between the convergence radius $r_{\rm dm}^{\rm conv}$ and $r_\text{core}$, where $r_{\rm dm}^{\rm conv}$ is defined as the radius containing 200 DM particles \citep[as in ][]{Klypin2001,hopkins2018fire2}, and is typically around 50 pc in size. For reference, the measured slopes $\alpha$ are quoted in each panel.
While the initial DM distributions of our simulations follow a Hernquist profile, we find no difference in measured core radius when using Hernquist or NFW parameterizations, consistent with the intended similarity between the fits for $r \ll r_{200}$. While some choices of our methodology are arbitrary, we find that it consistently produces an accurate characterization of the physical extent and slope of the constant density inner regions. We show in the Appendix that core formation is well converged and robust to numerical choices, such as resolution and $r_{\rm fit}$ (see Figs.~\ref{fig:res_converge} and \ref{fig:rcore_slope_compare_nfwdcut}).
\subsection{Halo response to SMUGGLE versus SH03 models}
Interestingly, and in contrast to previous results of model implementations within {\sc Arepo} \citep[e.g.][]{Marinacci2014,Chua2019,bose2019}, we find that the new \texttt{SMUGGLE} model develops a well-defined constant-density core with radius $200 - 600$\,pc in our idealized $M_{200} \sim 10^{10}$ M$_\odot$~dwarf halo. In comparison, the same initial setup run with the SH03 model does not robustly form a core.
In practice, our method suggests $r_\text{core}$~$\approx 175$\,pc (see bottom panels) for the SH03 run, although this is more consistent with a relaxation effect than a true dark matter core achieved by repeated perturbation of the potential. This is further supported by the inner slope $\alpha$, which is far from being a flat constant density distribution ($\alpha \sim 0$) as found for our fiducial \texttt{SMUGGLE} run and instead remains steep ($\alpha \sim -0.55$), consistent with that of the initial condition over a similar distance range. In addition, we have run an adiabatic (i.e. no star formation or feedback) version of the same initial setup for $t \sim 0.7$\,Gyr. By our methods, we calculate time-averaged values of $r_\text{core}$~$= 150$\,pc and $\alpha = -0.57$ for the adiabatic run, indicating that the behavior seen in SH03 is consistent with relaxation and is not representative of a feedback-induced core. Note the similarity between the green SH03 and magenta adiabatic curves in Figure \ref{fig:rho_panel}.
We therefore find that the SH03 ISM treatment does not create a core, in agreement with previous studies implementing similar models \citep[e.g.][]{Marinacci2014,bose2019} while the new ISM treatment \texttt{SMUGGLE} results in clear halo expansion. The measured core extends over several hundred pc, which is well beyond the gravitational softening for the dark matter $\epsilon=16$\,pc or the convergence radius $r_{\rm DM}^{\rm conv} \approx 50$\,pc.
A more detailed description of the time evolution for the core is shown in the panels (a) and (b) of Figure \ref{fig:rcore_sfr}, showing the core radius $r_\text{core}$~and the power law slope $\alpha$ of the inner region $r^{\rm DM}_{\rm conv} < r < r_\text{core}$ of the dark matter density profile. In \texttt{SMUGGLE}, the core radius grows during the first Gyr, after which it settles on an average $r_\text{core}$~$\sim 400$\,pc with fluctuations. The slope flattens from $\alpha = -0.55$ to $-0.09$ in the first half Gyr, where it remains for the rest of the simulation. In contrast, SH03 relaxes into a stable density distribution with $r_\text{core}$ $\sim 160$\,pc and no significant change in slope, resulting in a cusp rather than a core.
Panel (c) of Figure \ref{fig:rcore_sfr} compares the star formation histories in the \texttt{SMUGGLE} and SH03 runs. The rapid fluctuations in the \texttt{SMUGGLE} run are sustained throughout the $\sim 3$\,Gyr of run time, though with decreased burstiness after $t \sim 1$\,Gyr. This contrasts the smoother SFR from the SH03 ISM model. In fact, SH03 shows a declining SFR, likely due to the lack of cold inflows and depletion of all eligible star forming gas. The cooling implementation of SH03 results in an effective temperature floor of $\sim 10^4$\,K, such that, with the lack of cold inflows, no new gas is able to condense to sufficiently high densities to fuel star formation. As a result, the final stellar mass formed in \texttt{SMUGGLE} is $\sim 50\%$ larger compared to SH03.
Note that this burstiness in the star formation of \texttt{SMUGGLE} is associated to fluctuations on the gas mass in the inner 1 kpc (Figure \ref{fig:rcore_sfr}, panel d), while SH03 simply depletes the gas content in this region. As discussed in \citet{pontzengovernato2012}, such mass fluctuations in short timescales can cause the local gravitational potential to non-adiabatically change resulting in the expansion of dark matter orbits and, consequently, on the formation of a lower density core.
In the case of \texttt{SMUGGLE}, although the gas content is changing very abruptly in the very inner regions (thin) and less so outwards, the mass fluctuation can be discerned quite far out into the main body of the galaxy, $r \sim 5$\,kpc.
What is driving these differences between the ISM models? Discussions in the literature have cited rapid fluctuations of the potential in the inner regions \citep{navarro1996,pontzengovernato2012}, burstiness of star formation rates \citep{madau2014,chan2015,tollet2016,Dutton2019}, and high gas densities such that it dominates the central potential \citep{benitezllambay2019}. These features are all present in the \texttt{SMUGGLE} treatment but not in the SH03-like models, explaining why core formation is achieved in \texttt{SMUGGLE} but not in previous ISM treatments in {\sc Arepo}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/rho_hist_timeavg_median_ff_ill_1e6.pdf}
\caption{Median mass-weighted probability density function of gas density for $t > 0.75$ Gyr for the inner 1 kpc, with shaded regions representing the 68 per cent confidence interval in each $\rho$ bin. The fiducial \texttt{SMUGGLE} run is able to achieve gas densities of $>10^3$ cm$^{-3}$, while SH03 is unable to obtain densities $> 1$ cm$^{-3}$. The higher densities achieved by \texttt{SMUGGLE} allow its gas to gravitationally influence the DM to a stronger degree than in SH03. }
\label{fig:rhogas1kpc_fidill}
\end{figure}
Figure \ref{fig:rhogas1kpc_fidill} shows the time-averaged gas density distribution within 1 kpc for each run. This distribution is calculated with equal logarithmically spaced bins between $\rho_\text{gas}=10^{-6}\,\text{cm}^{-3}$ and $\rho_\text{gas}=10^{6}\,\text{cm}^{-3}$ at each snapshot. The median gas density is then calculated for each bin to construct the final gas density distribution, with standard deviation about the median shown as shaded regions.
As a result of the molecular cooling and other physics modeled in \texttt{SMUGGLE}, the typical gas densities achieved in \texttt{SMUGGLE} can be several orders of magnitude higher than in SH03. This run results in very few gas particles denser than $\rho = 1$\,cm$^{-3}$ (green curve) while about half of the gas in the \texttt{SMUGGLE} run is above that threshold and up to $\sim 10^4$\,cm$^{-3}$. The high gas densities achieved by \texttt{SMUGGLE} are instrumental in gravitationally perturbing the dark matter to create cores, while the wide range of densities reached in the inner 1\,kpc indicates repeated disruption of dense gas from feedback in central star forming regions, maintaining a multi-phase nature that compares well with observations of real galaxies. While models based on an equation of state ISM treatment might be able to reproduce and predict statistical properties of galaxy populations as well as large-scale structure with remarkable success \citep[e.g. ][]{v14illustris, Marinacci2014, Schaye2015, sawala2016, Grand2017, Pillepich2018}, they cannot capture the interplay between DM and baryons on small scales, where the contribution of baryons to the gravitational potential is significant.
\section{The Effect of the ISM Model Parameters}
\label{sec:ISM}
\subsection{Variations on \texttt{SMUGGLE}}
\label{sec:smuggle_vars}
In addition to the fiducial \texttt{SMUGGLE} model and SH03, we have run three simulations using the same initial conditions with variations on key parameters in the \texttt{SMUGGLE} ISM model: (i) \texttt{rho0.1} reduces the star formation gas density threshold\footnote{The H$_2$ star formation requirement discussed in Section \ref{sec:SMUGGLEmodel} was lifted to allow the density threshold to take full effect.} from the fiducial value of $\rho_\text{th}$~= 100 cm$^{-3}$ to $\rho_\text{th}$~= 0.1 cm$^{-3}$ to mimic the value used in simulations such as SH03 and EAGLE \citep[][respectively]{v13feedback,crain2015}; (ii) \texttt{eSF100} increases the star formation efficiency from the fiducial value of $\varepsilon_\text{sf}$~= 0.01 to $\varepsilon_\text{sf}$~= 1 to compare with FIRE \citep{hopkins2018fire2}; and (iii) \texttt{vareff}, which parameterizes $\varepsilon_\text{sf}$~(see Section \ref{sec:methods_variations}, Eqn \ref{eqn:vareff}) to maximize star formation in dense, self-gravitating gas clouds. Table \ref{tab:simtable} summarizes these runs and their key features.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/rcore_etc_main_1e6.pdf}
\caption{Selected properties for \texttt{rho0.1} (purple), \texttt{eSF100} (orange), and \texttt{vareff} (blue), as in Figure \ref{fig:rcore_sfr}, including faint lines for \texttt{fiducial} and SH03. All variations on the \texttt{SMUGGLE} model are able to form flattened DM cores between approximately 250--400\,pc in extent and with $\alpha\sim -0.1$--$0$. \texttt{rho0.1} shows the least bursty SFR of the \texttt{SMUGGLE} runs, while both \texttt{eSF100} and \texttt{vareff} have SFRs that are significantly burstier than the fiducial \texttt{SMUGGLE} model. Remarkably, all \texttt{SMUGGLE} runs converge in $M_\star$~within $\sim$20\%, despite differences in SFR and gas content. The effect of different SFRs can be seen in the bottom panel as sharp jumps in $M_\star$~and decreases in $M_\text{gas}$ (outflows), or the lack thereof. We see that the high efficiency runs undergo repeated outflows, slowly depleting their gas reserves, while \texttt{fiducial}, \texttt{rho0.1}, and SH03 retain a majority of their original gas content.}
\label{fig:rcore_varISM}
\end{figure}
Figure \ref{fig:rcore_varISM} shows time-dependent properties of the variations on \texttt{SMUGGLE}, with the original two runs shown in faded, thin lines. The core radius and slope are shown in Panels (a) and (b). We find that all \texttt{SMUGGLE} runs form clearly defined cores, with shallow slopes and core sizes larger than demonstrated in SH03. We find that time-averaged ($t>0.75$ Gyr) values of $r_\text{core}$~vary from $275 - 400$ pc in extent, with slopes of $\alpha=-0.07\pm0.06$. This is within the range of core sizes observed for dwarf galaxies in the literature, with typical values of $\alpha = -0.2\pm0.2$ \citep{deblok2001,Oh2011,Oh2015}.
We find variation between the different \texttt{SMUGGLE} runs. The low threshold \texttt{rho0.1} forms the smallest $r_\text{core}$, as expected, though much more of a robust core than the mild expansion seen in SH03. Interestingly, the high efficiency run \texttt{eSF100} appears to form its core slower than \texttt{fiducial}, but ends up with a larger core by the final time. The variable efficiency run \texttt{vareff} forms its core early on -- similar to \texttt{fiducial} -- but continues to grow at later times. These variations, however, are relatively minor. The primary distinction between the fiducial \texttt{SMUGGLE} model and its two increased efficiency variations appears to the continued growth of the core over time as a result of the sustained burstiness of their star formation. This is likely due to the increased energy injection into the ISM via the efficient star formation and SN feedback. That is, a much higher fraction of gas that is eligible to turn into star particles is converted. For contrast, the fiducial \texttt{SMUGGLE} model only turns $\sim$ 1 per cent of the eligible gas into stars (on an average, not per-particle basis), in accordance with observations of GMCs \citep[][]{smith2018}. These strong blow-outs represent a somewhat different, more violent mode of core formation than exhibited in the fiducial run, which experiences smaller, more frequent outbursts. Convergence among runs to universally shallow slopes is notable. However, we do still observe that the higher efficiency runs \texttt{eSF100} and \texttt{vareff} form slightly shallower cores with $\alpha\sim -0.03$, while \texttt{rho0.1} and \texttt{fiducial} form cores with $\alpha\sim -0.1$.
Panels (c) and (d) of Figure \ref{fig:rcore_varISM} show the SFR, stellar mass, and gas mass versus time for all runs. The SFRs we observe in the new \texttt{SMUGGLE} models are within expectation. The \texttt{rho0.1} run maintains a higher average SFR due to a lower $\rho_\text{th}$, which effectively increases the amount of gas that is eligible for SF at any given timestep. Meanwhile, the higher efficiency runs see extremely bursty star formation histories due to a cycle of intense star formation, feedback that blows gas out of the inner regions, and re-accretion of gas to eligible SF densities. Despite these differences in star formation, we find excellent convergence in $M_\star$~for all \texttt{SMUGGLE} runs, with all runs reaching a final value within $\sim$20\% of one another.
However, we do find differences in gas content and nature of outflows between these runs. We see that \texttt{rho0.1} retains more of its gas within 5 kpc than \texttt{fiducial} while also undergoing fewer and shallower outflows (seen as dips in the gas mass). In stark contrast, the highly efficient runs lose a majority of their initial gas content by the end of the simulation, undergoing frequent and larger outflows than either \texttt{rho0.1} or \texttt{fiducial}, retaining only $\sim$20\% of their original gas mass by $t=2.0$ Gyr$h^{-1}$.
Figure \ref{fig:sigma_dm} shows the DM velocity dispersion for all runs, averaged over the final 0.5 Gyr of the simulations. We find results roughly as expected: the velocity dispersion of SH03 is consistent with a cuspy NFW profile, while the \texttt{SMUGGLE} runs form ever-flatter inner profiles, approaching the constant-$\sigma$ signature of an isothermal profile with the higher efficiency runs, as expected from self-interacting dark matter models \citep[][]{Vogelsberger2012,Rocha2013,TulinYu2018,Burger2019}.
While it is interesting to see isothermal velocity dispersion profiles generated as a result of baryonic feedback, these results are not identical with expectation from SIDM. For example, profiles in SIDM are isothermal to much larger radii, then immediately decline, whereas the contribution from baryons results in a sizable bump at intermediate radii with a smoother tail. This may a possible avenue to distinguish SIDM from baryonic feedback \citep[][]{Fitts2019}. Additionally, the isothermal profiles seen in the \texttt{SMUGGLE} runs demonstrate that they are not in dynamical equilibrium, an effect we discuss in Section \ref{sec:diversity}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/sigma_DM_median_main_1e6.pdf}
\caption{Time-averaged dark matter velocity dispersion profiles for each run. We find that the high efficiency variations on \texttt{SMUGGLE} approximately reproduce an isothermal (constant $\sigma_v$) core in the inner regions, while the SH03 run produces a decreasing profile similar to an NFW.}
\label{fig:sigma_dm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/rho_hist_timeavg_median_main_1e6.pdf}
\caption{Median gas density distribution for each run over the run time of the simulation after $t=0.75$ Gyr, with shaded regions representing the 68 per cent confidence interval in each $\rho$ bin. Both \texttt{fiducial} and \texttt{rho0.1} are able to produce an ISM with a substantial fraction of the gas above their star formation thresholds, while the median gas densities achieved by \texttt{eSF100} and \texttt{vareff} demonstrate a more rapidly decreasing high density tail. This is a result of different star formation efficiencies: in the high efficiency runs, gas that reaches $\rho_\text{th}$~is quickly turned into stars, while low efficiency preserves a component of highly dense gas.}
\label{fig:rhohist1kpc_varISM}
\end{figure}
As discussed previously, examining average gas densities can be a useful exercise to understand the behavior of both the DM and the baryons. Figure \ref{fig:rhohist1kpc_varISM} shows the same time-averaged gas density calculation as Figure \ref{fig:rhogas1kpc_fidill}, but for all runs, including shaded regions for standard deviations.
Interestingly, we find that \texttt{rho0.1} is able to produce gas densities well above its star formation threshold of $\rho_\text{th}$$=0.1$ cm$^{-3}$, with an almost identical distribution to \texttt{fiducial}, though slightly favoring lower densities. In contrast, the runs with higher efficiencies (\texttt{eSF100} and \texttt{vareff}) are limited to gas densities at or near the standard value of $\rho_\text{th}$~= 100 cm$^{-3}$, with slightly lower values in the fully efficient \texttt{eSF100} than in the selectively efficient \texttt{vareff}.
The changes in the high-density tail between fiducial \texttt{SMUGGLE} model and \texttt{eSF100} are consistent with results from \citet{Li2020}, who investigated the effects of this parameter on GMCs in MW-mass galaxies.
\subsection{The role of modeling parameters}
As discussed in Section \ref{sec:smuggle_and_SH03}, we find that the same isolated galaxy setup run with the SH03 feedback model does not form cores due to its relatively low density gas and its lack of bursty star formation. It is generally claimed that these features are governed by the choice of $\rho_\text{th}$~in the model \citep{pontzengovernato2012,bose2019,benitezllambay2019}, however, the clear differences between SH03 and \texttt{rho0.1}, both of which implement a low density threshold of $\rho_\text{th}$~= 0.1 cm$^{-3}$, demonstrate that the physics of core formation is dependent on factors beyond this parameter.
The physical differences between these runs is clear: \texttt{rho0.1} has somewhat bursty star formation, dense gas, and SN-driven outflows of gas from the central regions, while SH03 has monotonically decreasing SFR, sparse gas, and no discernible feedback-driven outflows. If both runs implement $\rho_\text{th}$~= 0.1 cm$^{-3}$ yet achieve such different outcomes, other differences in subgrid physics must be to blame. The unstructured ISM of the SH03 feedback model is a result of its conception as a model for large-scale structure simulations, and is not particularly well suited for studying small-scale structures of galaxies and their halos, such as DM cores. The detailed ISM model implemented in \texttt{SMUGGLE} is able to achieve much higher gas densities, resolving multiple physical gas phases at smaller scales, as well as achieving the bursty star formation necessary to form cores.
The difference in density achieved by these two runs (Figure \ref{fig:rhohist1kpc_varISM}) therefore points to two facts: (1) the physical gas density achieved by a simulation is not solely governed by $\rho_\text{th}$, especially when using local star formation efficiencies lower than 100 per cent and (2) gas density and star formation burstiness (which drive outflows and subsequently core formation) are a product of the subgrid physics model as a holistic enterprise, including processes such as cooling physics and self-shielding, as well as resolution to the extent that such processes are resolution-dependent, rather than any individual parameter. However, changes in relevant parameters, as demonstrated here and in many other works, \citep[e.g.][, Burger et al. \textit{in prep.}]{pontzengovernato2012,benitezllambay2019} do indeed produce observable differences within the same overall modeling scheme.
In their seminal work, \citet{pontzengovernato2012} compare cosmological zoom simulations run with the SPH \textsc{gasoline} code \citep{Wadsley2004,Stinson2006} run with two different value of $\rho_\text{th}$, corresponding to our fiducial value of $\rho_\text{th}$ = 100 cm$^{-3}$ and a low threshold run with $\rho_\text{th}$ = 0.1 cm$^{-3}$, as in our \texttt{rho0.1} run. They find that the low threshold run does not form a core, yet the high threshold run does, comparing the same overall ISM model in both cases. They point out that fluctuations in potential result in the expansion of the orbits of DM particles in the inner halo. We emphasize in this discussion that it is the ability of a model to create these physical density fluctuations that matters in producing DM cores.
As noted by \citet{benitezllambay2019}, it is indeed surprising that few systematic tests of the star formation density threshold have been conducted by this time. The authors investigate the effect of a variety of values for $\rho_\text{th}$~spanning 0.1 cm$^{-3}$ up to 640 cm$^{-3}$ for cosmological halos in the EAGLE simulations \citep{crain2015}. They find that core formation is maximized for values between 1 cm$^{-3}$ and 160 cm$^{-3}$, but find smaller cores for smaller values of $\rho_\text{th}$~due to the lack of gravitationally dominant gas, and also for larger values due to the inefficiency of EAGLE's feedback model in this regime. They identify that core formation in dwarf galaxies is not sufficiently explained by either burstiness of star formation or strong outflows of gas within the EAGLE model. Instead, they point to features in the SFH of different halos that produce differences in outcomes of core sizes.
A similar investigation, though over a smaller range of threshold values, was conducted by \citet{Dutton2019,Dutton2020} for the NIHAO simulation project \citep{wang2015nihao}. They find that, of their halos run with $\rho_\text{th}$ = 0.1 cm$^{-3}$, 1 cm$^{-3}$, and 10 cm$^{-3}$, only those with $\rho_\text{th}$ = 10 cm$^{-3}$ and stellar mass to halo mass ratio of 0.1--1\% underwent strong expansion, in agreement with the trend pointed out in \citet{dicintio2014SMHM}. Further, they identify that variability in star formation feedback must occur at sub-dynamical time-scales to produce expansion of the DM halo.
In the case of \textsc{gasoline}, a change in density threshold was able to predictably alter the outcome of core formation. The picture is somewhat more complex for EAGLE and NIHAO, which find that core formation, while increasing with $\rho_\text{th}$, is further dependent on SFH, timescale of burstiness, and halo mass, among other things. All these studies examined cosmological simulations. Our idealized numerical experiments seek to eliminate the complexities of cosmological runs, which produce substantial halo-to-halo variations in $M_\star$/$M_{200}$, SFH, merger histories, gas fractions, etc. These are all important factors in understanding the diversity of observed galaxies, but can serve to obscure the impact of modeling choices.
Our idealized \texttt{SMUGGLE} runs produced cores for both the fiducial threshold of $\rho_\text{th}$~= 100 cm$^{-3}$ and the lowered threshold of $\rho_\text{th}$~= 0.1 cm$^{-3}$, though \texttt{rho0.1} did produce a somewhat smaller core radius ($\sim 300$ pc, versus $\sim 400$ pc for the fiducial run). When compared to the cuspy profiles of SH03, the core size within these two variations of \texttt{SMUGGLE} can be considered quite similar. This similarity in core size and shape between the two \texttt{SMUGGLE} variations makes sense in light of their achieved physical gas density distributions (Figure \ref{fig:rhohist1kpc_varISM}) versus the highly truncated distribution of SH03, which is incapable of producing $\rho_\text{gas} \gtrsim 1 $cm$^{-3}$. With an initial mean DM density of $\sim$ 4 $m_{\rm p}$cm$^{-3}$ within 1 kpc, it is clear that, even if SH03 produced fluctuations in gas mass within this region, it would be insufficient to perturb the DM potential.
Another factor that impacts the physics of core formation is the ability of the gravity solver to resolve the free-fall timescale of gas in the centermost star-forming regions of the galaxy. When larger softening lengths are used, the collapse of gas into dense clouds is delayed, and the resulting star formation process will be smoothed out. This leads to fewer discrete star formation events, and a reduction in both the burstiness of star formation and maximum gas density achieved in star forming regions, limiting the growth of cores.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/10panelproj_main_1e6.pdf}
\caption{Face-on (top row) and edge-on (bottom row) projections of stars (orange) and gas (blue) for all runs. We only include star particles that were formed after the simulation began, not the old disk and bulge components from the initial condition. Each panel edge is 15 kpc in length, with the half stellar mass radius $r_{h\star}$ shown in black (again, only new stars), and the core radius $r_\text{core}$~shown in red. Both the fiducial run and \texttt{rho0.1} maintain fairly well behaved disks, though with somewhat more disturbance and fragmentation in \texttt{fiducial} as well as a more compact distribution of stars, while \texttt{rho0.1} has a more extended stellar distribution with a smaller core radius. Both \texttt{eSF100} and \texttt{vareff} show a highly disturbed ISM, with gas extending much further in the $z$-direction (perpendicular to the disk). Both galaxies have more compact stellar distributions than the fiducial run.}
\label{fig:10panelprojection}
\end{figure*}
We emphasize that it is the ability of a model to produce both sufficiently dense gas and sufficient variation in baryonic mass in the inner regions of a halo that will allow it to form cores. The ability of $\rho_\text{th}$~to affect these physical phenomena depends (i) on how the chosen modeling prescriptions modulate the effect of that parameter on star formation, (ii) on how energy injection and dissipation distribute energy throughout the ISM, and (iii) on the interplay between resolution and all of the above. In short, the precise role of $\rho_\text{th}$~in core formation is model-dependent. For example, \texttt{SMUGGLE} produces similar inner gas density distributions regardless of the adopted value of $\rho_\text{th}$, and forms a feedback-induced core in all our explored variations. While the density threshold parameter is a commonly used parameter in ISM models, making it an appealing avenue for study, more attention must be given to the differences between modeling prescriptions with respect to their resulting physical properties (such as the gas density distribution and fluctuations in baryonic mass) before the effects of individual parameters can be understood in proper context.
For example, most treatments of star formation use relatively low values when implementing fixed local star formation efficiencies: $\varepsilon_\text{sf}$ $ = 0.01 - 0.1$ \citep{Stinson2006,wang2015nihao}. As in our \texttt{rho0.1}, the density threshold is therefore not necessarily an accurate tracer of the actual density achieved by the gas. The actual distribution of gas density will depend more complexly on modeling prescriptions (i.e. realistic versus effective cooling treatments) when using $\varepsilon_\text{sf}$~$\ll 1$. For this reason, comparing simulations run with distinct modeling treatments but similar $\rho_\text{th}$~does not make sense when considering the dependence of core formation on $\rho_\text{th}$, as the resulting distribution of gas density and its sensitivity to feedback can vary substantially between models.
\section{Galaxy Structure}
\label{sec:structure}
Figure \ref{fig:10panelprojection} shows face-on and edge-on projections of the four alterations of the \texttt{SMUGGLE} model we consider, with the stellar half-mass radius ($r_{h\star}$) shown in green and $r_\text{core}$~shown in magenta. The SH03 model shows a uniform disk with an unstructured ISM, along with large $r_{h\star}$~and small $r_\text{core}$, while \texttt{fiducial} and \texttt{rho0.1} show a much more structured ISM, with clear fragmentation containing regions of both dense and rarefied gas. In addition, small pockets representing SN shock fronts can be seen in the face-on image. The disk remains well-behaved, with clear rotation and a roughly even distribution of gas throughout the disk. The ISM of \texttt{fiducial} is somewhat less evenly distributed than \texttt{rho0.1}, resulting in larger pockets of dense and rarefied gas, with an overall more centrally concentrated ISM (as seen in the edge-on projections), though it does maintain a disk morphology with clear cohesive rotation. Conversely, both \texttt{eSF100} and \texttt{vareff} have highly disturbed gaseous components with no clear rotation and strong radial outflows from more energetic SN feedback. Even the edge-on projections show little traditional disk structure, with the galaxies appearing irregular in structure. In addition, they are much more compact, with $r_{h\star}$~roughly half the size of those of SH03 or \texttt{rho0.1}. The core radii of the three \texttt{SMUGGLE} models are larger than that of the SH03 model (as shown previously).
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/rhalf_rcore_dots_main_1e6.pdf}
\caption{Core radius versus stellar half-mass radius for each run, with each point indicating a different snapshot after $t = 0.75$ Gyr. As in Figure \ref{fig:10panelprojection}, $r_{h\star}$~only includes star particles that were formed after the simulation began. Naturally, SH03 forms the tightest grouping, while the \texttt{SMUGGLE} runs are stratified according to galaxy size. It is a clear consequence of \texttt{vareff}'s prioritization of star formation in dense gas that it forms the most compact galaxy, while the global high efficiency of \texttt{eSF100} produces large fluctuations in galaxy size (and core radius). The low threshold of \texttt{rho0.1} allows for less dense gas in the outer regions to form stars, resulting in a more extended galaxy.}
\label{fig:rcore_r50}
\end{figure}
\subsection{Morphology and cores}
Figure \ref{fig:rcore_r50} shows the core radius versus the half stellar mass radius for each run at $t > 0.75$ Gyr. We find a fair degree of stratification of the runs with $r_{h\star}$, indicating the effects of different modeling choices on galaxy structure. The variable efficiency run demonstrates the most compact galaxy size overall, mostly hovering aroung $r_{h\star}$~= 1 kpc. This concentrated morphology is a result of the maximized star formation efficiency in dense regions (which tend to be near the center of the galaxy) used in this model. The globally maximized star formation efficiency in \texttt{eSF100} produces a more concentrated galaxy than the fiducial \texttt{SMUGGLE} model, though it also has more variation. This run experienced a large burst of star formation at early times, expanding the initial galaxy, only contracting at later times. This expansion and contraction is seen in the orange dots that extend to the right of $r_{h\star}$~= 1.5 kpc, overlapping somewhat with our largest galaxy, \texttt{rho0.1}.
Interestingly, the large core sizes and compact galaxies seen in \texttt{eSF100} and \texttt{vareff} are contrary to the observed trend in which large cores are expected in low surface-brightness galaxies \citep[][]{Santos-Santos2020}. This may indicate that cores can form in high surface brightness galaxies, but have not yet been detected (either due to incompleteness or the disruption of gas kinematics in systems that may mimic these runs), or it may indicate that high star formation efficiency is not an empirically consistent modeling choice. The latter may be more likely, since most ISM treatments that calibrate this parameter to observed data choose values in the range 0.01 -- 0.1 \citep{Stinson2006,wang2015nihao}, while models that implement such high efficiencies have other strict criteria on star formation \citep[][]{hopkins2014fire1}. Again, the effect of this parameter is indeed model-dependent. At least within \texttt{SMUGGLE}, an increased local SF efficiency parameter produces a trend counter to what is currently expected from observational data.
The large extent of \texttt{rho0.1} is a result of the reduced density threshold, which allows more rarefied gas in the outskirts of the galaxy to form stars, rather than concentrating star formation to the dense gas which collects near the center. The fiducial \texttt{SMUGGLE} model balances each of these effects, producing an intermediate-size galaxy, with $r_{h\star}$~$\approx 1.5$ kpc throughout its evolution. Each \texttt{SMUGGLE} model produces variation in both the core size and stellar half-mass radius. The SH03 model on the other hand maintains the same core radius and galaxy size throughout its evolution, forming a tight cluster of points. We note again that SH03 did not form a robust feedback-induced core. We include the data here only for contrast with our \texttt{SMUGGLE} runs which did form robust cores.
The variation in both core size and half stellar mass radius is worth noting. Observed galaxies can effectively only be measured at one point in their evolution. While a large sample of observed galaxies helps to sample the variation, it is still impossible to take into account the variation in these properties over a given galaxy's lifetime. It is certainly possible that extreme values of inner DM density from highly overdense cusps to underdense cores represent local maxima or minima in their fluctuations. We emphasize that a given observation is not necessarily representative of the property's expectation value. Numerically constraining the predicted fluctuation in such properties like DM core sizes may be a worthwhile addition to the discussion on diversity of rotation curves.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/sigma_all_avg_main_1e6.pdf}
\caption{Time-averaged ($t > 0.75$ Gyr) stellar velocity dispersion $\sigma_\star$ in cylindrical coordinates. Standard errors are shown but appear smaller than the width of the lines. We find that SH03 preserves disk coherence better than the \texttt{SMUGGLE} models, which all produce stronger feedback that disrupts the rotational structure of the galaxy. The high efficiency variations distribute stellar motion more evenly between all three cylindrical components, indicating a dispersion-dominated galactic structure.}
\label{fig:sigma}
\end{figure}
To quantify differences in the kinematic structure between our runs, Figure \ref{fig:sigma} shows time-averaged (for $t > 0.75$ Gyr) stellar velocity dispersion profiles in cylindrical coordinates, with $\sigma_R$ (radial direction) on the top panel, $\sigma_\phi$ (direction of disk rotation) in the center, and $\sigma_z$ (direction perpendicular to the disk plane) on the bottom panel. We see that all four \texttt{SMUGGLE} runs have higher $\sigma_R$ than SH03. The grouping of models echos that of their density distributions in Figure \ref{fig:rhohist1kpc_varISM}: \texttt{fiducial} and \texttt{rho0.1} have similar $\sigma_R$ profiles, and smaller than both \texttt{eSF100} and \texttt{vareff}, which are also similar to each other. This is a natural result of their higher star formation efficiencies, which results in stronger feedback, disturbing the ISM and causing increased radial motion into the gas due to increase SN activity. The center panel shows $\sigma_\phi$, representing the rotation of the disk. Disks with coherent rotation exhibit a typical ``S''-shaped curve, such as that of SH03, indicating a smooth increase in rotational velocity towards the outskirts of the galaxy. We see that \texttt{fiducial} and \texttt{rho0.1} exhibit this characteristic shape, but to a lesser degree as a result of their increased feedback. Naturally, the high efficiency models with their disrupted morphology show a near-constant $\sigma_\phi$ profile, indicating little to no rotational support. We observe a similar stratification of behavior in the bottom panel, where SH03 shows little gas motion in the $z$-direction, while \texttt{fiducial} and \texttt{rho0.1} show an increased amount, and \texttt{eSF100} and \texttt{vareff} show a stronger increase in gas disruption in this direction as a result of the strong feedback that injects a large amount of momentum in the local radial direction, resulting in increased gas velocity dispersions in all directions. Due to the broad similarity in core formation between the four \texttt{SMUGGLE} runs, this implies that the choice of star formation efficiency has little impact on the dark matter content while drastically affecting the gas content and morphology of dwarf galaxies.
\subsection{Diversity of rotation curves}
\label{sec:diversity}
Figure \ref{fig:vphi} shows the rotational velocity $v_\phi$ of the gas as well as $v_\text{circ}$~= $\sqrt{\text{GM}(<r)/r}$ for each run, averaged over the final 0.5 Gyr of each run. We find that the ISM of SH03 traces the potential of the galaxy remarkably well. In contrast, the high efficiency \texttt{SMUGGLE} runs \texttt{eSF100} and \texttt{vareff} are so kinematically disrupted that there is little to no measurable rotation. \citet{elbadry2018b} found similarly dispersion-supported gas in dwarf galaxies within the FIRE simulations \citep{hopkins2018fire2}, and that rotational support increases with increasing mass. Further, they find that the majority of FIRE galaxies across 6.3 < $\log_{10}(M_\star / \text{M}_\odot)$ < 11.1 have little rotational support, and while the higher mass galaxies have morphological gas discs, only a fraction of the dwarf galaxies ($M_\star$~$\lesssim 10^9$ M$_\odot$) host this feature.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figures/vphi_median_main_1e6.pdf}
\caption{Median rotational velocity ($v_\phi$) profile of gas (dashed line) and total circular velocity $v_\text{circ}$~= $\sqrt{\text{GM}(<r)/r}$ (solid line). Shaded regions represent the $1\sigma$ deviation from median (inner 68 per cent confidence interval) within each $r$ bin across time.
We see that the (relatively) well-behaved ISM of \texttt{fiducial} and \texttt{rho0.1} trace the gravitational potential of the galaxy much better than the disturbed ISMs of \texttt{eSF100} and \texttt{vareff}. The large shaded errors indicate substantial variations in rotational velocity profiles over the course of the simulation.}
\label{fig:vphi}
\end{figure}
It is notable that even within the `well-behaved' variations on \texttt{SMUGGLE}, we find that the rotational velocity of the gas does not accurately trace the $v_\text{circ}$~implied by the gravitational potential. A naive reading of the gas $v_\phi$ distribution in Figure \ref{fig:vphi} could suggest a core radius of $\gtrsim$2 kpc for the fiducial \texttt{SMUGGLE} model and \texttt{rho0.1}, while our method of core size measurement relying only on DM density profiles (see Section \ref{sec:smuggle_and_SH03}) results in values of a few hundred parsecs. Interestingly, this 2 kpc figure is consistent with the fiducial radius used to compare well-resolved rotation curves between simulations and observations \citep[as in][]{Santos-Santos2018,Oman2019}. This result supports the notion that non-circular motion of gas in the inner regions of galaxies limits the ability of observational analyses to accurately recreate the DM profile, potentially contributing to the diversity of rotation curves in observed galaxies \citep{Oman2015,Oman2019,Santos-Santos2020}.
The variability in the $v_\phi$ distribution indicates another problem of time sampling bias. The measured gas rotational velocity is subject to frequent and substantial variation as a result of energy injection via feedback, as depicted by the shaded regions on Figure \ref{fig:vphi} representing the RMS error due to time-averaging. Measurements of the $v_\phi$ distribution taken at the extrema of the error range could either produce rapidly rising rotation curves implying a mass distribution consistent with $\Lambda$CDM, or a slowly rising rotation curve implying an inner mass deficit and substantial core.
Figure \ref{fig:vphi_all} plots the individual $v_\phi$ profiles for each snapshot of the fiducial \texttt{SMUGGLE} run over the final 0.5 Gyr. Here we see that, while the majority of rotation curves are below the actual DM $v_\text{circ}$, there are a handful of profiles that demonstrate rotation speeds faster than the DM in the inner regions, i.e. profiles that would be interpreted as cuspy. Based on the number of snapshots with rotation curves that rise faster than an NFW, we place an upper limit on the presence of highly cuspy rotation curves at approximately 10 per cent. While this is an unlikely result, it indicates that cuspy profiles as a result of gas kinematics are indeed possible.
The discrepancy between the rotational velocity $v_\phi$ profile and the circular velocity profile $v_\text{circ}$ indicates that the rotation of gas is rarely an accurate tracer of the DM potential in dwarf galaxies due to its sensitivity to energy injection via feedback. Our simulations predict that substantial diversity of rotation curves should be expected within the same dwarf galaxy across time. The variability of gas content and velocity in the inner regions of the galaxy on timescales $\lesssim 100$ Myr poses a challenge to the assumption of virial equilibrium (i.e. `steady-state') that underlies the inference of DM distributions from gas velocity profiles. As suggested by \citet{read2016rc}, expanding bubbles of HI can be used to identify post-starburst galaxies which are likely out of equilibrium. Collisionless stars may be a better tracer of the inner gravitational potential than gas.
Overall, \texttt{SMUGGLE} produces rotational profiles that systematically underestimate the DM content of the inner regions, consistent with previous attempts to reconcile the observed diversity of rotation curves with baryonic physics \citep[][]{Oman2015,Oman2019,read2016rc,tollet2016,Santos-Santos2018,Santos-Santos2020}. This indicates either that our understanding of small-scale ISM physics within galaxies is incomplete, or that another mechanism is responsible for creating rapidly rising rotation curves. It is possible that higher mass systems with stronger potentials are less susceptible to this effect, but we emphasize that this must be demonstrated explicitly rather than taken as an assumption.
The above considerations are only a result of ISM kinematics within an idealized, non-cosmological simulation and do not take into account additional bias introduced by observational measurement techniques, such as tilted-ring modeling and Jeans modeling, nor do they take into account evolutionary histories consistent with real galaxies or effects of cosmological environments such as mergers and infall of cold gas from filaments. Rather, these idealized tests isolate the effects of ISM modeling from other complex phenomena, allowing us to directly test the effects of baryonic feedback on the dark matter distribution of dwarf galaxies.
\section{Summary \& Conclusions}
\label{sec:summary}
We study the behavior of the \texttt{SMUGGLE} \citep{Marinacci2019} feedback and ISM model for the \textsc{Arepo} \citep{arepo} moving mesh simulation code. In particular, we investigate the formation of dark matter `cores' in idealized (non-cosmological) dwarf galaxies with $M_\star$~$\approx$ 8\e{7} M$_\odot$~and $M_{200}$~$\approx$ 2\e{10} M$_\odot$~by comparing runs with identical initial conditions under both \texttt{SMUGGLE} and the SH03 model \citep{SpringelHernquist2003}, a precursor to the model used in Illustris and Auriga \citep[][]{v14illustris,Grand2017} simulations, among others. We develop a self-consistent method of measuring the core radius to track its evolution over time. We define the core radius to be the location where the actual DM density falls below the predicted DM density of an NFW profile fit to the outer regions of the halo ($r > 3$ kpc) by a factor or 2 (Figure \ref{fig:rho_panel}). We then measure the slope of a power law fit to the resolved region within the measured core radius. Tracing these metrics over time, we find that SH03 does not produce a constant-density DM core, while the fiducial \texttt{SMUGGLE} model creates a flattened core of radius $\sim$ 350 pc within the first 0.75 Gyr. We show that the origin of these cores is linked to the successful self-regulation of the star formation history in \texttt{SMUGGLE} which establishes a bursty star formation mode. These bursty cycles then create significant variations in the enclosed gas mass within 1 kpc, resulting in non-adiabatic expansion of the inner DM distribution. Contrary to the self-regulation seen in \texttt{SMUGGLE}, SH03 produces a steadily declining SFH, with a constant mass of gas reached after most of the originally eligible gas for star formation has been transformed into stars. This equilibrium state then preserves the steep inner density profiles that have been reported previously in the literature (Figure \ref{fig:rcore_sfr}).
In addition, we run three simulations of identical initial set up including alterations to key feedback parameters: (i) \texttt{rho0.1} changes the star formation density threshold from the fiducial value of $\rho_\text{th}$~$=100$ cm$^{-3}$ to a reduced value of $\rho_\text{th}$~$=0.1$ cm$^{-3}$; (ii) \texttt{eSF100} changes the local star formation efficiency (the mass fraction of eligible star forming gas that is converted into stars) from the fiducial value of $\varepsilon_\text{sf}$~$=0.01$ to an increased value of $\varepsilon_\text{sf}$~$=1$; and (iii) \texttt{vareff}, which implements a parameterization of the star formation efficiency based on the virial parameter (a measure of local self-gravitation; see Section \ref{sec:methods_variations}). We find that the formation of a core is robust to these changes in \texttt{SMUGGLE} (though \texttt{rho0.1} does form a $\sim 25$ per cent smaller core, and the high efficiency models exhibit stronger growth over time).
It is significant that \texttt{rho0.1} forms a feedback-induced core while SH03 does not. Since both implementations use the same star formation density threshold $\rho_\text{th}$~$=0.1$ cm$^{-3}$, this is an indication that the density threshold alone is not a good predictor of core formation for detailed ISM models such as \texttt{SMUGGLE}. It is important to note that while SH03 does not generate a core through feedback, it does experience a halo expansion due to relaxation effects and the influence of the baryonic component \citep[][]{Burger2021}. Its expansion was smaller than in all \texttt{SMUGGLE} runs and was shown to be consistent with an adiabatic run, indicating that feedback was not a relevant factor. In contrast, \texttt{rho0.1} demonstrates large fluctuations of baryonic matter in the inner regions of the halo, linking feedback to core formation.
We find that the ability to resolve dense gas ($\rho_\text{gas} \gtrsim 10^2$ cm$^{-3}$; see Figure \ref{fig:rhohist1kpc_varISM}) is more predictive of core formation, in agreement with findings from \citep{benitezllambay2019}. For example, \texttt{rho0.1} resolves gas up to $\rho\sim 10^4$ cm$^{-3}$ while SH03 only resolves gas up to $\rho\sim 1$ cm$^{-3}$. This indicates that the SF density threshold is not a good proxy for actual gas density when using low local star formation efficiencies $\varepsilon_\text{sf}$~$\ll 1$. This then allows the dense gas to linger around and affect locally the gravitational potential even if the density threshold for star formation is nominally low. Note that this is different from predictions in other ISM implementations, such as NIHAO \citep{Dutton2019,Dutton2020}.
Our high efficiency runs \texttt{eSF100} and \texttt{vareff} have more bursty star formation than \texttt{fiducial} or \texttt{rho0.1}, yet they do not form substantially larger cores (Figure \ref{fig:rcore_varISM}). This indicates that core size and burstiness are not tightly correlated, but that sufficiently bursty star formation, like sufficiently high gas density, are necessary conditions for core formation, as predicted previously \citep{pontzengovernato2012,benitezllambay2019,Dutton2019}. All \texttt{SMUGGLE} variations also demonstrate mild time-dependence over the course of our runs, indicating that core expansion should continue over cosmological timescales. We hypothesize that the source of this continued expansion is the continued bursty star formation in these runs. The core evolution in the fiducial \texttt{SMUGGLE} run is inconclusive in its time-dependence due to the short runtime of these simulations. Density profiles of the \texttt{SMUGGLE} variations can be found in Figure \ref{fig:fourrho_varISM}.
While there is broad agreement in core formation between the \texttt{SMUGGLE} variations, there are still differences between the models: \texttt{rho0.1} forms the smallest core of the \texttt{SMUGGLE} models, with $r_\text{core}$ $\sim$300 pc by final time, while \texttt{vareff} and \texttt{eSF100} reach final core radii of $\sim$500 pc. Despite this difference, we maintain that \texttt{rho0.1} does indeed form a comparable core due to its highly flattened inner slope consistent with the fiducial \texttt{SMUGGLE} model. Interestingly, the fluctuations in gas mass within 1 kpc for all \texttt{SMUGGLE} runs are comparable (though with \texttt{rho0.1} having less frequent outflows). This is likely the source of the similarity in core sizes and shapes between the runs.
This similarity between variations of \texttt{SMUGGLE} indicates that the physical consequences of changing parameters such as the SF density threshold $\rho_\text{th}$~are highly model dependent. As mentioned, $\rho_\text{th}$~is not an accurate tracer of physical gas densities achieved by simulations when using empirically calibrated models that limit the local SF efficiency $\varepsilon_\text{sf}$~to values $\ll 1$. Local gas densities will be highly dependent on implementations of subgrid physics. In particular, molecular and fine-structure cooling allows gas to naturally reach temperatures far lower than $10^4$ K and achieve densities comparable to or higher than the average density of DM in the inner regions. The implemented modes of feedback-driven energy injection into the ISM allow this dense gas to be disrupted and flow to outer regions of the halo, repeatedly perturbing the DM potential as suggested by \citet{pontzengovernato2012}. That is to say, changes in model parameters must result in the required physical changes within the simulation to accurately capture the details of baryon-induced core formation. Simulations that do not produce sufficiently dense gas (due either to modeling choices or resolution) are simply unable to follow the physics expected to lead to core formation.
We also investigate the implications various modeling choices have on morphology. The fiducial \texttt{SMUGGLE} model and \texttt{rho0.1} both form rotationally supported disks with structured ISMs, while SH03 naturally produces a stable galaxy with featureless ISM (see Figures \ref{fig:10panelprojection} and \ref{fig:sigma}). On the other hand, the high efficiency models produce dispersion-dominated spheroid galaxies with lower gas fractions. This is a natural result of the increased burstiness and feedback of these models, and is in agreement with the FIRE simulations \citep{hopkins2018fire2}, which implement $\varepsilon_\text{sf}$ $=1$ and also observe dwarf galaxies with spheroid morphology and dispersion supported ISM \citep{elbadry2018b}. Interestingly, we find that the most compact galaxies (\texttt{eSF100} and \texttt{vareff}) form the largest cores, while the most diffuse galaxies (\texttt{rho0.1}) form the smallest cores (Figure \ref{fig:rcore_r50}), in agreement with \citet{Burger2021}.
Our examination of the rotational velocity ($v_\phi$) profiles of the gas content (Figure \ref{fig:vphi}) indicates that the ISM does not trace the potential of the DM in the inner regions ($r < 2$ kpc). This is true for all \texttt{SMUGGLE} variations, though the fiducial model and \texttt{rho0.1} are better able to trace the DM $v_\text{circ}$~in the outskirts, while \texttt{eSF100} and \texttt{vareff} demonstrate almost no rotational velocity component of the gas at any radius. Further, we find significant variations in the $v_\phi$ profiles across time, suggesting that a diverse morphology of rotation curves can be observed at different times within the same galaxy. We find that individual $v_\phi$ profiles can vary between exceeding the expected DM circular velocity and drastically underestimating it (Figure \ref{fig:vphi_all}). However, we find that the ISM in \texttt{SMUGGLE} systematically falls below the $v_\text{circ}$~of the halo within the inner regions, consistent with previous work \citep[][]{Santos-Santos2020,Oman2019}, further indicating that the baryon-induced core formation scheme struggles to reproduce the steep end of the diversity of rotation curves problem.
Our analysis indicates that feedback-induced core formation is fundamentally a small-scale problem. Its effects may be observed on the scale of a few kpc, but the physics which generates these observables occur on the scales of star formation and feedback, i.e. $10 - 100$ pc, as well as sub-pc processes that are yet unresolved and only implemented through sub-grid modeling. Lack of cores in models which are not able (and do not attempt) to produce this microphysics is not evidence against the validity of baryon-induced core formation, but evidence against the suitability of such models to study this process.
Finally, our results suggest that even if perfect observations of gas rotation curves are obtained, these do not necessarily trace the DM potential in non-equilibrium systems such as dispersion-dominated dwarf galaxies. Caution is needed when attempting to infer DM distributions from gas rotation. It is important to investigate the assumption of equilibrium for dwarf galaxies, whose small sizes make them susceptible to large fluctuations in gas content and velocity.
\section*{Acknowledgements}
EDJ and LVS acknowledge support from the NASA ATP 80NSSC20K0566, NSF AST 1817233 and NSF CAREER 1945310 grants.
FM acknowledges support through the program "Rita Levi Montalcini" of the Italian MIUR.
MV acknowledges support through NASA ATP grants 16-ATP16-0167, 19-ATP19-0019, 19-ATP19-0020, 19-ATP19-0167, and NSF grants AST-1814053, AST-1814259, AST-1909831, AST-2007355 and AST-2107724.
PT and JQ acknowledge support from NSF grants AST-200849. PT acknowledges support from NASA ATP grant 80NSSC20K0502.
AS and HL acknowledge support for Program numbers HST-HF2-51421.001-A and HST-HF2-51438.001-A provided by NASA through a grant from the Space Telescope Science Institute,
which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.
JB acknowledges support by a Grant of Excellence from the Icelandic Center for Research (Rann\'is; grant number 173929)
JZ acknowledges support by a Grant of Excellence from the Icelandic Research fund (grant number 206930).
\section*{Data Availability}
The data that support the findings of this study are available from the corresponding author, upon reasonable request.
\bibliographystyle{mnras}
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\aistatstitle{Sequential Informed Federated Unlearning: Efficient and Provable Client Unlearning in Federated Optimization}
\aistatsauthor{
$\text{Yann Fraboni}^{1,2}$
\And $\text{Richard Vidal}^{2}$
\And $\text{Laetitia Kameni}^{2}$
\And $\text{Marco Lorenzi}^{1}$ }
\aistatsaddress{ ${}^1$ Universit\'e C\^{o}te d’Azur, Inria Sophia Antipolis,
Epione Research Group, France\\
\and
${}^2$ Accenture Labs, Sophia Antipolis, France\\ } ]
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\section*{Acknowledgments}
This work has been supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002, and by the ANR JCJC project Fed-BioMed 19-CE45-0006-01. The project was also supported by Accenture.
The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.
\subsubsection*{\bibname}}
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\usepackage{amsthm}
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\usepackage{amsmath}
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\input{./tex/abbreviations}
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\begin{document}
\twocolumn[
\aistatstitle{Sequential Informed Federated Unlearning: Efficient and Provable Client Unlearning in Federated Optimization}
\aistatsauthor{
$\text{Yann Fraboni}^{1,2}$
\And $\text{Richard Vidal}^{2}$
\And $\text{Laetitia Kameni}^{2}$
\And $\text{Marco Lorenzi}^{1}$ }
\aistatsaddress{ ${}^1$ Universit\'e C\^{o}te d’Azur, Inria Sophia Antipolis,
Epione Research Group, France\\
\and
${}^2$ Accenture Labs, Sophia Antipolis, France\\ } ]
\input{./tex/abstract}
\input{./tex/introduction}
\input{./tex/theory}
\input{./tex/experiments}
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\section*{Acknowledgments}
This work has been supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002, and by the ANR JCJC project Fed-BioMed 19-CE45-0006-01. The project was also supported by Accenture.
The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.
\subsection{Influence of the initial model on the bias}
Let us consider $X$ the feature matrix and $Y$ the ground truth with loss function
\begin{equation}
f({\bm{\theta}}) = \frac{1}{N} \frac{1}{2} \left[{\bm{Y}} - {\bm{X}} {\bm{\theta}} \right]^T \left[{\bm{Y}} - {\bm{X}} {\bm{\theta}} \right]
\end{equation}
We assume that there are more features than data samples. Therefore, $X^TX$ is singular and all the model ${\bm{\theta}}^*$ satisfying
\begin{equation}
{\bm{X}}^T {\bm{X}} {\bm{\theta}}^* = {\bm{X}}^T {\bm{Y}}
\end{equation}
are optimal model. $f$ is non convex.
The SGD process can be rewritten such that
\begin{equation}
{\bm{\theta}}^N
= \underbrace{\left[I - \eta {\bm{X}}^T {\bm{X}}\right]^N}_{A({\bm{X}}, N)} {\bm{\theta}}^0
+ \underbrace{\eta \sum_{n =0}^{N-1} \left[I - \eta {\bm{X}}^T {\bm{X}}\right]^n {\bm{X}}^T {\bm{Y}}}_{B({\bm{X}}, {\bm{Y}}, N)}
\end{equation}
We consider a learning rate small enough such that the highest eigen value of ${\bm{X}}^T {\bm{X}}$ is strictly inferior to 1. We also consider that its smallest eigen value is non-negative. Therefore, $I - \eta {\bm{X}}^T {\bm{X}} $ is positive definite. We get
\begin{equation}
{\bm{\theta}}^N
= \underbrace{\left[I - \eta {\bm{X}}^T {\bm{X}}\right]^N}_{A({\bm{X}}, N)} {\bm{\theta}}^0
+ \underbrace{\eta \sum_{n =0}^{N-1} \left[I - \eta {\bm{X}}^T {\bm{X}}\right]^n {\bm{X}}^T {\bm{Y}}}_{B({\bm{X}}, {\bm{Y}}, N)}
\end{equation}
The second term is independent from the initial model and only depends of the amount of SGD $N$. When ${\bm{X}}^T{\bm{X}}$ i non singular, the first term converges to 0.
We differentiate now 2 flows. One with all the samples and another one with one removed sample.
\begin{equation}
{\bm{\theta}}_2^N
= A({\bm{X}}_2, N) {\bm{\theta}}^0
+ B({\bm{X}}_2, {\bm{Y}}_2, N)
\end{equation}
If unlearning with $\tilde{N}$ SGD, we have
\begin{align}
\tilde{{\bm{\theta}}}_2^{\tilde{N}}
&= A({\bm{X}}_2, \tilde{N}) {\bm{\theta}}_1^N
+ B({\bm{X}}_2, {\bm{Y}}_2, \tilde{N})
\\
&= A({\bm{X}}_2, \tilde{N}) \left[A({\bm{X}}_1, N) {\bm{\theta}}^0
+ B({\bm{X}}_1, {\bm{Y}}_1, N)\right]
+ B({\bm{X}}_2, {\bm{Y}}_2, \tilde{N})
\\
&= A({\bm{X}}_2, \tilde{N}) A({\bm{X}}_1, N) {\bm{\theta}}^0
+ A({\bm{X}}_2, \tilde{N}) B({\bm{X}}_1, {\bm{Y}}_1, N)
+ B({\bm{X}}_2, {\bm{Y}}_2, \tilde{N})
\end{align}
We always have
\begin{equation}
\lim_{\tilde{N} \to \infty}
B({\bm{X}}_2, {\bm{Y}}_2, \tilde{N})
=
\lim_{N \to \infty}
B({\bm{X}}_2, {\bm{Y}}_2, N)
\end{equation}
and if ${\bm{X}}_2^T {\bm{X}}_2$ is non singular, we retrieve
\begin{equation}
\lim_{\tilde{N} \to \infty}
A({\bm{X}}_2, \tilde{N})
=0
\end{equation}
which gives
\begin{align}
\lim_{\tilde{N} \to \infty}
\tilde{{\bm{\theta}}}_2^{\tilde{N}}
=
\lim_{N \to \infty}
{\bm{\theta}}_2^N
\end{align}
In practice, in Machine Learning, the initial model ${\bm{\theta}}_0$ is obtained with a normal distribution. Setting the initial model at $0$ enables to cancel the first step, still, the second term cannot be mitigated by setting the initial model to a certain value and depends of the optimum of the previous optimum model.
\subsection{Theoretical Work}
$\EE{S}{X}$ is the expected value of $X$ over $S$ the random set of sampled clients.
If we consider the Lipschitz smoothness of $f$, we get
\begin{equation}
\EE{S}{f(x^{t+1}) - f(x^{t})}
\le - \gamma \EE{S}{\inner{\nabla f(x^t)}{g(x^t)}}
+ \frac{L}{2}\gamma \EE{S}{\norm{g_t}^2}
\label{app:eq:A0}
\end{equation}
\subsubsection*{1st term}
We define $\alpha(n) \coloneqq a_n p_n - a_{n+1} p_{n+1}$ which gives
\begin{equation}
\EE{S}{g(x^t)}
= \nabla f ( x^t)
+ \frac{1}{n-1} \frac{1}{m} \alpha(n) \nabla l_n(x^t)
,
\end{equation}
and
\begin{equation}
\inner{\nabla f(x^t)}{\EE{S}{g(x^t)}}
= \norm{\nabla f ( x^t)}^2
+ \frac{1}{n-1}\frac{1}{m} \alpha(n)
\inner{\nabla f(x^t)}{\nabla l_n(x^t)}
\label{app:eq:A1}
,
\end{equation}
\subsubsection*{2nd term}
Considering
\begin{align}
\mathbb{E}{\omega_i^2}
&
= \frac{1}{(n-1)^2} \left[\frac{m-1}{m} + \frac{1}{mp_i}\right]
\\
\mathbb{E}{\omega_i\omega_j}
&
= \frac{1}{(n-1)^2} \frac{m- 1}{m}
\\
\mathbb{E}{\left(\omega_n + \omega_{n+1}\right)^2 }
&= \mathbb{E}{\omega_n^2}
+ \mathbb{E}{ \omega_{n+1}^2}
+ 2 \mathbb{E}{ \omega_n \omega_{n+1} }
\\
&= \frac{1}{(n-1)^2}
\left[\frac{m-1}{m}\alpha^2(n)
+ \frac{1}{m}a_n^2 p_n
+ \frac{1}{m}a_{n+1}^2 p_{n+1}\right]
\eqqcolon \beta(n)
,
\end{align}
we get
\begin{align}
\EE{S}{\norm{g_t}^2}
& = \sum_{i =1}^{n-1} \sum_{ j = 1 }^{ n - 1 } \mathbb{E}{\omega_i \omega_j} \nabla l_i(x^t)^T \nabla l_j(x^t)
+ \mathbb{E}{(\omega_n + \omega_{n+1})^2} \norm{\nabla l_n(x^t)}^2
\nonumber\\
& + 2 \sum_{i =1}^{n-1} \mathbb{E}{\omega_i (\omega_n + \omega_{n+1})} \nabla l_i(x^t)^T \nabla l_n(x^t)
\\
& = \frac{1}{(n-1)^2}\frac{1}{m}\sum_{i =1}^{n-1} \frac{1}{p_i} \norm{\nabla l_i(x^t)}^2
+ \frac{m-1}{m} \norm{\nabla f(x^t)}^2
+ \beta(n) \norm{\nabla l_n(x^t)}^2
\nonumber\\
& + 2 \frac{1}{(n-1)^2}\frac{m-1}{m}\alpha(n)\nabla f(x^t)^T \nabla l_n(x^t)
\label{app:eq:A2}
,
\end{align}
\subsubsection*{Rewrite inner product}
We can rewrite the gradient of the previous loss function as
\begin{equation}
\nabla \Lcal_1(x^t)
= \frac{n-1}{n}\nabla f(x^t)
+ \frac{1}{n}\nabla l_n(x^t)
,
\end{equation}
which gives
\begin{align}
\inner{\nabla f(x^t)}{\nabla l_n(x^t)}
& = \frac{n^2}{n-1} \inner{\frac{n-1}{n}\nabla f(x^t)}{\frac{1}{n}\nabla l_n(x^t)}
\\
& = \frac{n^2}{n-1} \frac{1}{2} \left[ \norm{\nabla \Lcal_1(x^t)}^2
-\frac{(n-1)^2}{n^2}\norm{\nabla f(x^t)}^2
-\frac{1}{n^2}\norm{\nabla l_n(x^t)}^2
\right]
\end{align}
\subsubsection*{Merge}
Substituting equation (\ref{app:eq:A1}) and (\ref{app:eq:A2}) in (\ref{app:eq:A0}) gives
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[1 - \frac{L}{2}\gamma \frac{m-1}{m}\right] \norm{\nabla f(x^t)}^2
\nonumber\\
&- \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{1}{n-1}\frac{1}{m}\alpha(n) \nabla f(x^t)^T \nabla l_n(x^t)
\nonumber\\
&+ \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \sum_{i =1}^{n-1} \frac{1}{p_i}\norm{\nabla l_i(x^t)}^2
+ \frac{L}{2}\gamma \beta(n) \norm{\nabla l_n(x^t)}^2
.
\label{app:eq:B1}
\end{align}
We note at this point that the only influence of $p_i$ lies in $\sum_{i =1}^{n-1}\frac{1}{p_i}\norm{\nabla l_i(x^t)}^2$ which for all $p_n$ and $p_{n+1}$ is minimized when
\begin{equation}
p_i = (1 - p_n - p_{n+1})\frac{\norm{\nabla l_i(\theta)}}{\sum_{i=1}^{n-1}\norm{\nabla l_i(\theta)}}
.
\end{equation}
Therefore equation (\ref{app:eq:B1}) cn be simplified as
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[1 - \frac{L}{2}\gamma \frac{m-1}{m}\right] \norm{\nabla f(x^t)}^2
\nonumber\\
&- \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{1}{n-1}\frac{1}{m}\alpha(n) \nabla f(x^t)^T \nabla l_n(x^t)
\nonumber\\
&+ \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \frac{1}{1 - p_n - p_{n+1}} \left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x^t)}\right]^2
+ \frac{L}{2}\gamma \beta(n) \norm{\nabla l_n(x^t)}^2
.
\label{app:eq:B2}
\end{align}
In the rest of this work, we assume that $1 - L\gamma \frac{m-1}{m}>0$.
For the biased gradient estimator to provide a faster convergence process, the deleted sample needs to be considered for descent or ascent such that $\alpha(n) \nabla f(x^t)^T \nabla l_n(x^t) \ge 0 $ is satisfied which can be rewritten as
\begin{align}
&\alpha(n) \inner{ \nabla f(x^t)^T }{\nabla l_n(x^t) }
\nonumber\\
&\alpha(n) \frac{n^2}{n-1}\inner{\frac{n-1}{n}\nabla f(x^t)^T }{\frac{1}{n}\nabla l_n(x^t) }
\\
&=
\alpha(n) \frac{n^2}{n-1}
\left[ \norm{\nabla \Lcal_1(x^t)}^2 - \left[\frac{n-1}{n}\right]^2\norm{\nabla f(x^t)}^2 - \frac{1}{n^2}\norm{\nabla l_n(x^t)}^2 \right]
\ge 0
\end{align}
We are interested in improving the unlearning process. Especially to leave faster the initial model $\theta_1^*$. for which, we have $\nabla \Lcal_1(\theta_1^*)= 0$. Therefore, regardless of whether a deleted sample is considered to obtain the biased gradient estimator, we have $p_n =0$. Therefore $\alpha(n) = - a_{n+1} p_{n+1}$ and $\beta(n) = \frac{1}{(n-1)^2}\left[\frac{m-1}{m}a_{n+1}^2 p_{n+1}^2 + \frac{1}{m}a_{n+1}^2 p_{n+1} \right]$. We obtain
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[\left[1 - \frac{L}{2}\gamma \frac{m-1}{m}\right] + \left[1 - L\gamma \frac{m-1}{m}\right] \frac{1}{m} a_{n+1}p_{n+1} \right] \norm{\nabla f(x^t)}^2
\nonumber\\
&- \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{n^2}{(n-1)^2}\frac{1}{m} a_{n+1}p_{n+1} \norm{\nabla \Lcal_1(x^t) }^2
\nonumber\\
&+ \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \frac{1}{1 - p_{n+1}} \left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x^t)}\right]^2
\nonumber\\
&+ \left[\frac{L}{2}\gamma \beta(n) + \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{1}{(n-1)^2}\frac{1}{m} a_{n+1}p_{n+1}\right]\norm{\nabla l_n(x^t)}^2
.
\label{app:eq:B3}
\end{align}
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[1 + \frac{1}{m} a_{n+1}p_{n+1} \right] \norm{\nabla f(x^t)}^2
- \frac{n^2}{(n-1)^2}\frac{1}{m} a_{n+1}p_{n+1} \norm{\nabla \Lcal_1(x^t) }^2
\nonumber\\
&+ \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \frac{1}{1 - p_{n+1}} \left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x^t)}\right]^2
\nonumber\\
&+ \left[\frac{L}{2}\gamma \beta(n) +
\frac{1}{(n-1)^2}\frac{1}{m} a_{n+1}p_{n+1}\right]\norm{\nabla l_n(x^t)}^2
.
\label{app:eq:B4}
\end{align}
We also define
\begin{equation}
A(t)
= \frac{1}{\gamma}\mathbb{E}{f(x^{t}) - f(x^{t+1})}
,\
B(t)
= \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{1}{n-1}\frac{1}{m} \nabla f(x^t)^T \nabla l_n(x^t)
\end{equation}
\begin{equation}
C(t)
= \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x^t)}\right]^2
,\
D(t)
= \frac{L}{2}\gamma \frac{1}{(n-1)^2} \frac{m-1}{m}\norm{\nabla l_n(x^t)}^2
\end{equation}
\begin{equation}
E(t)
= \frac{L}{2}\gamma \frac{1}{(n-1)^2} \frac{1}{m}\norm{\nabla l_n(x^t)}^2
\end{equation}
which gives
\begin{align}
\left[1 - \frac{L}{2}\gamma \frac{m-1}{m}\right] \norm{\nabla f(x^t)}^2
&\le A(t) - B(t) a_{n+1} p_{n+1}
\nonumber\\
&+ \frac{1}{1 - p_{n+1}} C(t)
+ D(t) a_{n+1}^2 p_{n+1}^2
+ E(t) a_{n+1}^2 p_{n+1}
.
\label{app:eq:B3}
\end{align}
We consider the loss function where we replace $\frac{1}{1-p_{n+1}} = 1 + p_{n+1} + p_{n+1}^2$.
Let us consider
\begin{equation}
f(x^t)
= \left[ - B(t) a_{n+1}+ C(t) + E(t) a_{n+1}^2\right]p_{n+1}
+ \left[ C(t) + D(t) a_{n+1}^2\right] p_{n+1}^2
\end{equation}
We therefore have
\begin{equation}
content...
\end{equation}
\subsubsection*{Not Kept}
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \rho \left[1 - \frac{1}{2}\frac{1}{m}\alpha(n)\right]\norm{\nabla f(x^t)}^2
- \rho \alpha(n) \frac{1}{2} \frac{n^2}{(n-1)^2} \frac{1}{m}\norm{\nabla \Lcal_1(x^t)}^2
\nonumber\\
&+ \frac{L}{2}\gamma\frac{1}{m}\frac{1}{(n-1)^2} \sum_{i =1}^{n-1} \frac{1}{p_i}\norm{\nabla l_i(x^t)}^2
\nonumber\\
&+ \frac{1}{(n-1)^2}\left[\frac{1}{2}\frac{1}{m}\alpha(n)\rho + \frac{L}{2}\gamma \beta'(n) \right] \norm{\nabla l_n(x^t)}^2
\end{align}
We note that the importance $p_i$ given to each sample $i \le n$ can already be allocated such that
\begin{equation}
p_i = (1 - p_n - p_{n+1})\frac{\norm{\nabla l_i(\theta)}}{\sum_{i=1}^{n-1}\norm{\nabla l_i(\theta)}}
\end{equation}
Lastly, the bound is proportional to $p_n$ and $p_{n+1}$. Therefore, only $p_n$ or $p_{n+1}$ is positive.
If we consider that we are at the optimum, one gets
\begin{equation}
\nabla \Lcal_1(x_1^*)
= \frac{n-1}{n}\nabla f(x_1^*)
+ \frac{1}{n}\nabla l_n(x_1^*)
=0
,
\end{equation}
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[ \rho \left[ 1
- \frac{1}{m}\alpha(n) \right]
- \frac{L}{2}\gamma \beta'(n)\right]
\norm{\nabla f(x_1^*)}^2
\nonumber\\
&+ \frac{L}{2}\gamma\frac{1}{m}\frac{1}{(n-1)^2} \frac{1}{1- p_n - p_{n+1}}\left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x_1^*)}\right]^2
\end{align}
where the weights when $i\le n-1$ are assigned such that $\mathbb{E}{\omega_i} = \frac{1}{n-1}$ and $\mathbb{E}{\omega_n + \omega_{n+1}} = 0$. As such, we have $a_n p_n= a_{n+1} p_{n+1}$ which gives
\begin{align}
\VAR{\omega_i}
&
= \frac{1}{(n-1)^2}\frac{1}{mp_i}(1-p_i)
\\
\COV{\omega_i}{\omega_j}
&
= - \frac{1}{(n-1)^2} \frac{m- 1}{m}
\\
\mathbb{E}{\left(\omega_n + \omega_{n+1}\right)^2 }
&= \mathbb{E}{\omega_n^2}
+ \mathbb{E}{ \omega_{n+1}^2}
+ 2 \mathbb{E}{ \omega_n \omega_{n+1} }
\\
&= \frac{1}{m}a_n^2 \left[p_n^2 (m-1) + p_n\right]
+ \frac{1}{m}a_{n+1}^2 \left[p_{n+1}^2 (m-1) + p_{n+1}\right]
- 2\frac{m-1}{m}a_n a_{n+1}p_n p_{n+1}
\\
&= \frac{m-1}{m}(a_n p_n - a_{n+1} p_{n+1})^2
+ \frac{1}{m}a_n^2 p_n
+ \frac{1}{m}a_{n+1}^2 p_{n+1}
\\
&=
\frac{1}{m}a_n^2 p_n
+ \frac{1}{m}a_{n+1}^2 p_{n+1}
\\
\end{align}
In that case, we have
\begin{align}
\sigma(\theta)
&= \frac{1}{m(n-1)^2}\left[\sum_{i=1}^{n-1} \norm{\nabla l_i(\theta)}\right]^2
+ \frac{m-2}{m}\sum_{i=1}^{n-1} \norm{ \frac{1}{n-1}\nabla l_i(\theta) - \nabla \Lcal_2(\theta)}^2
- \frac{1}{m}\norm{\nabla \Lcal_2(\theta)}^2
,
\end{align}
More generally,
we have
\begin{equation}
p_i = ( 1 - p_n - p_{n+1})\frac{\norm{\nabla l_i(\theta)}}{\sum_{i=1}^{n-1}\norm{\nabla l_i(\theta)}}
\end{equation}
which gives
\begin{align}
\sigma(\theta)
&= \frac{1}{1 - p_n - p_{n+1}} \frac{1}{m(n-1)^2}\left[\sum_{i=1}^{n-1} \norm{\nabla l_i(\theta)}\right]^2
+ \frac{m-2}{m}\sum_{i=1}^{n-1} \norm{ \frac{1}{n-1}\nabla l_i(\theta) - \nabla \Lcal_2(\theta)}^2
\nonumber\\
&- \frac{1}{m}\norm{\nabla \Lcal_2(\theta)}^2
+ \beta(n) \norm{\nabla \Lcal_2(\theta) - \nabla \Lcal_1(\theta)}^2
,
\end{align}
Therefore we have
\begin{align}
&\sigma_2(\theta) - \sigma_1(\theta)
\nonumber\\
& = \left[\frac{1}{1 - p_n - p_{n+1}} - 1\right]\frac{1}{m(n-1)^2}\left[\sum_{i=1}^{n-1} \norm{\nabla l_i(\theta)}\right]^2
+ \beta(n) \norm{\nabla l_n(\theta)}^2
\end{align}
\section{Applications}\label{sec:applications}
\subsection{Centralized Learning}\label{subsec:centralized}
\input{./tex/application_centralized}
\subsection{DL and FL}\label{subsec:DL}
\input{./tex/application_FL}
\section{New Initial Model}
Let us consider $N_t$ the SGD budget for training and $N_m$ the SGD budget for adapting the model.
DP privacy relies on noising the last model such that for any combination of samples, a model cannot know whether a sample was used to train it or not.
We have an history of models $N_t$ models. How to obtain a new model forgetting a given data samples?
What kind of procedure can be proposed to train the learning model such that a sample is forgotten?
The method currently used is to retrain everything. Why ? Because the initial model has no information on the sample to forget. We propose in this work instead to have a better starting point than the initial model to obtain the new model when trying to forget some samples.
We propose a generalization of DP-Privacy. With DP, only one training is needed and a noise is added to prevent specific information recovery from one client. What should be the noise to ensure DP? Is DP equivalent to ensuring forgetting of a data sample?
What is the impact of the amount of samples to forget? e.g. forgetting a class.
We propose the following two scenarios which we compare theoretically and experimentally to the following two baselines
\begin{enumerate}
\item noising the trained model before performing the budget of SGD
\item noising the model obtained after $N_t - N_m$ SGD before performing the budget of SGD
\item noising the trained model with no retraining
\item retraining from scratch with the available SGD budget
\end{enumerate}
\section*{Acknowledgements and Disclosure of Funding}
\label{sec:ack}
This work has been supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002, and by the ANR JCJC project Fed-BioMed 19-CE45-0006-01. The project was also supported by Accenture.
The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.
\subsection{Intermediate results}
\begin{property}\label{app:prop:SIFU_increasing_t}
If there exists $\nu$, $s$, $u$ such that $s < u$, $(\nu, t_s) \in O(s)$ and $(\nu, t_u) \in O(u)$, then $t_s \ge t_u$.
\end{property}
\begin{proof}
We first assume that $s$ and $u$ satisfy $u = s + 1$.
Considering that $(\nu, t_s) \in O(s)$ and $(\nu, t_u) \in O(u)$, we have, by definition of $\zeta_u$ in equation (\ref{eq:zeta_r}), $\nu \le \zeta_u$.
\begin{itemize}
\item $\zeta_u>\nu$. Considering that $u = s + 1$, we have $t_s = t_u$, equation (\ref{eq:oracle_recurrent}).
\item $\zeta_u = \nu$. Considering that $(\nu, t_s) \in O(s)$ and $(\nu, t_u) \in O(u)$, then we have $\nu \le s-1$. Therefore, by definition of $\zeta_u$, we have $\Psi_{\zeta_u}(t_s, W_u) > \Psi^*$. By construction of $T_u$, equation (\ref{eq:T_SIFU}), we have $t_u = T_u < t_s$.
\end{itemize}
When considering the more general case where there exists an integer $k$ such that $u = s +k$ while $(\nu, t_s) \in O(s)$ and $(\nu, t_u) \in O(u)$, then it is sufficient to consider iteratively an integer $j$ ranging from 1 to $k$. Considering $(\nu, t_u) \in O(u)$, there exists $t_{s+j}$ such that $(\nu, t_{s+j}) \in O(s+j)$. In that case, using the same reasoning as for $k=1$, we have $t_s \le t_{s+1} \le \ldots \le t_{s+k-1} \le t_u$.
\end{proof}
\subsection{Proof of Theorem \ref{theo:zeta}}
\begin{proof}
Proving that ${\bm{\theta}}_r^{N_r}$ $(\epsilon, \delta)$-unlearns every client in $F_r$, equation (\ref{eq:F_r}), reduces to proving that ${\bm{\theta}}_r^0$ $(\epsilon, \delta)$-unlearns every client in $F_r$, equation (\ref{eq:F_r}). Indeed, the data of clients in $F_r$ are not used on the noised perturbed model ${\bm{\theta}}_r^0 = {\bm{\theta}}_{\zeta_r}^{T_r} + \Ncal(0, \sigma^2 {\bm{I}}_{\bm{\theta}})$.
We prove by induction that ${\bm{\theta}}_r^0$ $(\epsilon, \delta)$-unlearns every client in $F_r$, equation (\ref{eq:F_r}).
The initialization ($r=1$) directly follows from IFU, Algorithm \ref{alg:unlearning_ours}, with Theorem \ref{theo:noise_DP}. We now assume that for every $s$ such that $s\le r-1$, ${\bm{\theta}}_s^0$ $(\epsilon, \delta)$-unlearns every client in $F_s$ and prove that ${\bm{\theta}}_r^0$ $(\epsilon, \delta)$-unlearns every client in $F_r$.
\begin{itemize}
\item \underline{$s \le \zeta_r$}.
Using the induction property, ${\bm{\theta}}_{\zeta_r}^0$ $(\epsilon, \delta)$-unlearns every clients in $W_s$. Clients in $W_s$ are not used for training on ${\bm{\theta}}_{\zeta_r}^0$.
Hence, ${\bm{\theta}}_{\zeta_r}^{T_r}$ and ${\bm{\theta}}_r^0$ also $(\epsilon, \delta)$-unlearns every client in $W_s$.
\item \underline{$s = r$.}
By definition of $\zeta_r$, equation (\ref{eq:zeta_r}), the noise perturbations for every model in $O(r)$ is such that ${\bm{\theta}}_{\zeta_r}^0$ $(\epsilon, \delta)$-unlearns every client in $W_r$.
Hence, by definition of $T_r$ on the bounded sensitivity of clients in $W_r$ at unlearning request $\zeta_r$, equation (\ref{eq:T_SIFU}), the noised perturbed model ${\bm{\theta}}_r^0$ $(\epsilon, \delta)$-unlearns every client in $W_r$, Theorem \ref{theo:noise_DP}.
\item \underline{$\zeta_r < s \le r-1$.}
The successive update of the oracle, equation (\ref{eq:oracle_recurrent}), from $O(\zeta_r)$ to $O(s)$ gives, by construction, that there exists $t_s$ such that the coordinates $(\zeta_r, t_s)$ are in $O(s)$. Hence, by definition of $\zeta_s$, equation (\ref{eq:zeta_r}), we have $\zeta_s \ge \zeta_r$ and the successive noise perturbations to obtain ${\bm{\theta}}_{\zeta_r}^0$ $(\epsilon, \delta)$-unlearns every client in $W_s$.
Also, while we have the coordinates $(\zeta_r, t_s)$ in $O(s)$, we also have the coordinates $(\zeta_r, T_r)$ in $O(r)$, equation (\ref{eq:oracle_recurrent}). Therefore, using property \ref{app:prop:SIFU_increasing_t}, we have $t_s \ge T_r$. Hence, we have $\Psi_{\zeta_r}(T_r, W_s) \le \Psi^*$. Therefore, with the noise perturbation of SIFU, clients in $W_s$ are $(\epsilon, \delta)$-unlearned in ${\bm{\theta}}_r^0$.
\end{itemize}
\end{proof}
\section{Litterature Review}
\textbf{\cite{MUnlearningviaAlgorithmicStability}.}
This paper proposes a new MU algorithm for the centralized setting. The model is retrained on a fixed amount of SGD on the new dataset.
In more details, the authors propose a forgetting definition called $\rho$-TV-stability. This definition is very interesting and this work is built around it. However, their algorithm outperforms recomputation from scratch only if $\rho < 1$ and the authors show that their method has $\rho = \frac{mT}{n}$ where $m$ is the batch size, $n$ is the number of samples, and $T$ is the SGD budget. I find this point under discussed in this work and in my opinion voluntarily hidden by setting $m = \rho n/T$ which gives $m=0$ in most cases.
\textbf{\cite{DescentToDelete}.}
This work is theoretical and propose forgetting clients by adding a noise at the end of the retraining. This work uses a privacy criteria based on DP and interestingly provides an upper bound for the probability of the resulting model to be in the neighborhood of the new optimum.
Authors also propose to train different models on slices of the split dataset before aggregating the results. However, it requires slices to have iid data distributions to prevent convergence to a sub-optimum point \cite{FedNova}. Also this method requires a lot of computation to find the appropriate dataset split.
\textbf{\cite{FederatedUnlearning}.}
This works considers MU for the federated setting but only provides a new MU algorithm and experimental results.
The method is based on reusing the clients update of FedAvg. Yet, no theoretical justification nor any intuition behind why to use their method is provided.
They verify the correctness of their method on the training loss and testing accuracy of the resulting model. However, how well a client is forgotten is not considered ?
\textbf{\cite{MachineUnlearning}.}
This work considers splitting the dataset into subset and training a model per subset. The resulting model is the aggregation of all the trained models on each subset. This can be seen as FL where $N=1$ and $K\gg 1$. Therefore for this method to give the true optimum, we need all the shards to have the same data distribution which is not possible in practice even for the iid case. This work is only experimental and does not consider this point. The experimental setting only considers the testing accuracy of their method without considering if a client has been forgotten or not.
An interesting point is to consider that every sample has not the same probability to be forgotten. As such, putting them in the same shard could be useful. While it does not solve the problems considered before, associating a forgetting probability per sampling is interesting.
\textbf{\cite{TowardsProbabilisticVerification}.}
Very interesting work proposing a framework to verify that the samples of a given owner were removed from a given model. The framework is based on an attack scheme. The attacker modifies part of its samples and add to it a marker. Through the predictions returned for the true et modified inputs, the server is able to infer if the samples of an owner were forgotten or not.
\textbf{\cite{MU_LinearFiltration}.}
Experimental work that introduce a linear filtration to remove a class from a predictive model. The authors consider models that can be decomposed as a logistic regression and a feature extraction function. The focus of this work is for a model to forget one of its class.
\textbf{\cite{ApproximateDataDeletion}.}
This work investigates MU for linear regression. The method proposed is based on the introduction of synthetic points for the deleted points. The method proposed by the authors is linear with the dimension of the model contrarily to current quadratic complexity.
Authors also verify their work on \cite{TowardsProbabilisticVerification} and not just on testing and training accuracy.
This work is not based on SGD approaches. These methods gives directly the close form of the new model.
\textbf{\cite{VariationalBayesianUnlearning}.}
This work considers the problem of unlearning a model trained with a variational Bayesian approach. They propose a new lower bound to obtain the resulting posterior when forgetting some samples. Experiments compare the posterior distance of the new posterior and the one obtained with retraining which enables to measure the forgetting guarantees of their method.
\textbf{\cite{CertifiedDataRemoval}.}
DP is used to ensure unlearning like in \cite{DescentToDelete}. The authors theoretical work assume bounded gradients and Lipschitz second derivative for samples loss function.
Authors assume that the gradient of the trained model contains too much information about the dataset used. They thus consider adding a linear perturbation to change the optimum and thus reduce the data information in the final model.
The authors measure the effectiveness of their method on testing accuracy and norm of the gradient.
\textbf{\cite{Cao2015TowardsMS}.}
This paper introduces 4 MU algorithms (See Table 1).
However no theoretical guarantees are given for any of them. Correctness of unlearning algorithms are measures by comparison of the unlearned model true and false positive with respect to the ones when retraining from scratch.
\textbf{\cite{Golatkar_2020_CVPR}.}
Definition of forgetting data based on KL divergence.
This paper has similar elements to \cite{VariationalBayesianUnlearning}.
Authors consider the problem of forgetting while keeping at heart a low training loss by proposing the forgetting Lagrangian. A loss function composed of the new loss function with the remaining samples regularized with the KL divergence between the current model posterior and the theoretical one.
The formalization of the problem is very interesting.
\textbf{\cite{MakingAIForgetYou}.}
This work proposed two MU algorithms for the problem of k-means. No theoretical guarantees are given for any of these algorithms. Experiments are conducted in term of standard metric and does not able to know if a client has been forgotten or not. Major speed-up has to be noticed for the proposed algorithms.
\textbf{\cite{WhenMachineUnlearning}.} This work considers the amount of information that can be recovered by an attacker by using the trained and modified models.
\cite{Golatkar_2021_CVPR}
\cite{golatkar2020forgetting}
\cite{shibata2021learning}
\cite{gong2021bayesian}
\cite{wang2021federated}
\cite{DeltaGrad}
\cite{WhenMachineUnlearning}
\cite{AdaptiveMachineUnlearning}
\cite{CertifiableMchineUnlearning}
\cite{FedEraser}
\cite{zeroshotMU}
\section{Noise data point to forget}
\section{Formalizing the Unlearning problem}
\subsection{Optimization}
We consider a dataset $\Dcal$ composed of $|\Dcal| = N$ data samples $({\bm{x}}_i, {\bm{y}}_i)$ where ${\bm{x}}_i$ is the feature vector of a data sample and ${\bm{y}}_i$ its prediction or label.
For unsupervised learning, it suffices to consider that data samples in $\Dcal$ are only made with ${\bm{x}}_i$. We denote by $\Scal$ the set of $S = |\Scal|$ indices we want to forget from $\Dcal$ and $\Dcal_{-\Scal} = \Dcal\backslash\{({\bm{x}}_i, {\bm{y}}_i) | i \in \Scal \}$ the dataset without the forgotten data samples.
We consider a function $f$ depending on one data sample and model parameters ${\bm{\theta}}$.
For example, $f$ can be the loss of one data sample or its activation vector. We define the associated normalized functions
\begin{equation}
F({\bm{\theta}}) \coloneqq \frac{1}{|\Dcal|} \sum_{i \in \Dcal} f({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
\text{ and }
F_{-\Scal}({\bm{\theta}}) \coloneqq \frac{1}{|\Dcal_{-\Scal}|} \sum_{i \in \Dcal_{-\Scal}} f({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
.
\end{equation}
We consider $F$ is a $K$ dimensional vector where $K=1$ if $f$ is the loss function of one data samples and $K$ is the amount of classes in a classification problem.
For ease of writing, we consider $F_{-i}$ to be the resulting function when forgetting data sample $i$.
\subsection{$(\epsilon, \delta)$-indistinguishable}
We first define $(\epsilon, \delta)$-indistinguishable in Definition \ref{def:DP_forget}.
\begin{definition}[$(\epsilon, \delta)$-indistinguishable] Let $\epsilon$ and $\delta$ be two positive real numbers. Two distributions $X$ and $Y$ are $(\epsilon, \delta)$-indistinguishable if for all subset $A \subseteq \Acal$, we have
\label{def:DP_forget}
\begin{equation}
\mathbb{P}(X \in A)
\le e^\epsilon \mathbb{P}(Y \in A) + \delta
\text{ and }
\mathbb{P}(Y \in A)
\le e^\epsilon \mathbb{P}(X \in A) + \delta
.
\end{equation}
\end{definition}
${\bm{\theta}}$ is the $d$ dimensional vector of the trained model parameters. We consider a stochastic perturbation ${\bm{\phi}} \sim N( {\bm{\mu}}, {\bm{\Sigma}})$, where ${\bm{\Sigma}}$ is positive definite and can thus be expressed as ${\bm{\Sigma}} = {\bm{S}}^T {\bm{S}}$, such that $F({\bm{\theta}} + {\bm{\phi}})$ and $F_{-\Scal}({\bm{\theta}} + {\bm{\phi}})$ are $(\epsilon, \delta)$-indistinguishable.
Definition \ref{def:DP_forget} is linked to differential privacy(DP). With DP, we are looking at the noise ${\bm{\phi}}$ such that for any sample $i$, $F({\bm{\theta}} + {\bm{\phi}})$ and $F_{-i}({\bm{\theta}} + {\bm{\phi}})$
are $(\epsilon, \delta)$-indistinguishable. In that case, no information specific to any data sample can be retrieve from $\hat{{\bm{\theta}}} \coloneqq {\bm{\theta}} + {\bm{\phi}}$. With Machine Unlearning, we are instead looking at ${\bm{\phi}}$ such that $\hat{{\bm{\theta}}}$ does not contain specific information from any forgotten data sample. As such, DP implies Unlearning but Unlearning is weaker than DP.
\begin{definition}[$(\epsilon, \delta, \Scal)$-unlearning]
If $F({\bm{\theta}} + {\bm{\phi}})$ and $F_{-\Scal}({\bm{\theta}} + {\bm{\phi}})$ are $(\epsilon, \delta)$-indistinguishable, $\hat{{\bm{\theta}}} \coloneqq {\bm{\theta}}+ {\bm{\phi}}$ contains no information from $\Scal$.
\end{definition}
\section{Complete Unlearning}
We assume that $f$ satisfies Assumption \ref{ass:linear_theta}.
\begin{assumption}
\label{ass:linear_theta}
For any data point $({\bm{x}}, {\bm{y}})$ and model with parameters ${\bm{\theta}}$,
we consider that $f$ can be linearly approximated in ${\bm{\theta}}$ for a perturbation ${\bm{\epsilon}}$, i.e.
\begin{equation}
f({\bm{x}}, {\bm{y}}, {\bm{\theta}} + \epsilon)
= f({\bm{x}}, {\bm{y}}, {\bm{\theta}}) + {\bm{J}}_{{\bm{\theta}}}({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
where ${\bm{J}}_{{\bm{\theta}}}$ is the Jacobian matrix of $f$ with respect to ${\bm{\theta}}$.
\end{assumption}
With Assumption \ref{ass:linear_theta} and the stochastic perturbation ${\bm{\phi}}$, we have
\begin{align}
F({\bm{\theta}} + {\bm{\phi}})
= F({\bm{\theta}})
+ \frac{1}{N}\sum_{ i = 1 }^{N} {\bm{J}}_{\bm{\theta}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}) {\bm{\phi}}
= F({\bm{\theta}})
+ {\bm{J}}_{\bm{\theta}} ( \Dcal, {\bm{\theta}}) {\bm{\phi}}
,
\end{align}
where we define $J_{\bm{\theta}} ( \Dcal, {\bm{\theta}}) = \frac{1}{N}\sum_{ i = 1 }^{N} {\bm{J}}_{\bm{\theta}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})$
with a similar derivation for $F_{-\Scal}$. We remind that ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{\Sigma}} = {\bm{S}}^T {\bm{S}})$. Hence, we have
\begin{equation}
F({\bm{\theta}} + {\bm{\phi}} )
\sim \mathcal{N}( F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}){\bm{\mu}}, {\bm{J}}_{\bm{\theta}} ( \Dcal, {\bm{\theta}})^T{\bm{\Sigma}} {\bm{J}}_{\bm{\theta}}(\Dcal, {\bm{\theta}}))
,
\end{equation}
with a similar derivation for $F_{-\Scal}$. With a non zero expected perturbation ${\bm{\mu}}$, a perfect unlearning can be obtained with $F({\bm{\theta}} + {\bm{\phi}}) = F_{-\Scal}({\bm{\theta}} + {\bm{\phi}})$ by choosing ${\bm{\mu}}$ and ${\bm{\Sigma}}$ such that the probability density functions of these two queries are identical. Considering the dimensions of the Jacobian, numerous ${\bm{\mu}}$ can satisfy this condition, we are thus interested in the ones with the smallest norm where Assumption \ref{ass:linear_theta} is the most valid.
Hence, we have
\begin{alignat}{2}
&\min_{{\bm{\mu}}} &\qquad& \norm{{\bm{\mu}}}^2
\\
&\text{subject to} & & \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]{\bm{\mu}} = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}),
\nonumber\\
& & & \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]^T{\bm{\Sigma}} \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right] = 0 \label{eq:constraint_sigma}.
\nonumber
\end{alignat}
We define ${\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal}) = \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]$.
When this optimization problem is feasible, then its optimum ${\bm{\mu}}^*$ satisfies
\begin{equation}
{\bm{\mu}}^* = {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal}) \left[ {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal})^T {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}})\right].
\end{equation}
Any $\Sigma$ such that equation(\ref{eq:constraint_sigma}) is satisfied works including ${\bm{\Sigma}} =0$. Yet, considering a positive ${\bm{\Sigma}}$ to mitigate the linear approximation would be helpful. We note that when $K=1$, we have ${\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) = \nabla_{\bm{\theta}} F({\bm{\theta}})$ and can simplify the close form of ${\bm{\mu}}^*$ with
\begin{equation}
{\bm{\mu}}^* = \frac{\nabla_{\bm{\theta}} F({\bm{\theta}}) - \nabla_{\bm{\theta}} F_{-\Scal}({\bm{\theta}})}{\norm{\nabla_{\bm{\theta}} F({\bm{\theta}}) - \nabla_{\bm{\theta}} F_{-\Scal}({\bm{\theta}})}^2} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}})\right].
\end{equation}
\subsubsection{Interpretability using Local Perturbation}
Perturbating the model ${\bm{\theta}}$ with ${\bm{\phi}}$ impacts the performance of all the data samples. As such, even with Definition \ref{def:DP_forget} only based on the data points to forget, all the samples are impacted. For interpretability concerns, we consider a local perturbation ${\bm{\psi}}$ to the parameters to forget and look at the resulting pdf.
\begin{assumption}
For any data point $({\bm{x}}, {\bm{y}})$ and model parameters ${\bm{\theta}}$,
we consider that $f$ can be linearly approximated in ${\bm{x}}$ such that for a feature perturbation $\epsilon$ we have
\begin{equation}
f({\bm{x}}+ {\bm{\epsilon}}, {\bm{y}}, {\bm{\theta}} )
= f({\bm{x}}, {\bm{y}}, {\bm{\theta}})
+ J_{\bm{x}} ({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
where ${\bm{J}}_{{\bm{x}}}$ is the Jacobian matrix of $f$ with respect to ${\bm{x}}$.
\end{assumption}
We consider clients are given the same Gaussian Noise ${\bm{\psi}} \sim N({\bm{\nu}}, {\bm{\Phi}})$ which gives the following optimization problem and results ${\bm{\nu}}^*$.
\begin{equation}
F(\Dcal, {\bm{\theta}})
= F({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} {\bm{J}}_{{\bm{x}}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}){\bm{\psi}}
= F({\bm{\theta}}) + \frac{|\Scal|}{|\Dcal|}{\bm{J}}_{{\bm{x}}} (\Dcal_\Scal, {\bm{\theta}}) {\bm{\psi}}
\end{equation}
and
\begin{alignat}{2}
&\min_{{\bm{\nu}}} &\qquad& \norm{{\bm{\nu}}}^2
\\
&\text{subject to} & & {\bm{J}}_{\bm{x}} (\Dcal_\Scal, {\bm{\theta}}){\bm{\nu}} = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^*,
\nonumbe
\end{alignat}
\begin{equation}
{\bm{\nu}}^*
= \frac{|\Dcal|}{|\Scal|}{\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})
\left[ {\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})^T {\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) - {\bm{J}}_{\bm{\theta}}(\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^* \right].
\end{equation}
We can also give every client their own perturbation ${\bm{\psi}}_i \sim N({\bm{\nu}}_i, {\bm{\Phi}}_i)$ which gives the following optimization problem and results ${\bm{\nu}}_i^*$.
\begin{equation}
F(\Dcal, {\bm{\theta}})
= F({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} {\bm{J}}_{{\bm{x}}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}){\bm{\psi}}_i
\end{equation}
which gives the following optimization problem
\begin{alignat}{2}
&\min_{{\bm{\nu}}_i} &\qquad& \frac{1}{|\Scal|}\sum_{ i \in S}\norm{{\bm{\nu}}_i}^2
\\
&\text{subject to} & & \frac{1}{|\Dcal|}\sum_{ i \in \Scal} {\bm{J}}_{\bm{x}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}) {\bm{\nu}}_i = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^*,
\nonumbe
\end{alignat}
\begin{equation}
{\bm{\nu}}_i^*
= |\Dcal| {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
\left[ \sum_{ i \in S} {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})^T {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) - {\bm{J}}_{\bm{\theta}}(\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^* \right].
\end{equation}
\section{Differential Privacy}
Let us consider
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
We provide sufficient conditions for ${\bm{\mu}}$ and ${\bm{S}}$ such that ${\bm{\theta}} + {\bm{\phi}}$ satisfies Definition \ref{def:DP_forget}.
${\bm{\phi}}$ is a multinomial Gaussian distribution ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{S}}^T {\bm{S}})$. Therefore,
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
We are investigating the probability, given that the database is $\mathcal{D}$, of observing an output that occurs with a very different probability under $\mathcal{D}$ than under and adjacent database $\Dcal_{-\Scal}$, where the probability space is $\mathbb{R}^d$.
We define $\Delta \Lcal ({\bm{\theta}}) = \Lcal({\bm{\theta}}) - \Lcal_{-\Scal}({\bm{\theta}})$ and the privacy loss $R$ as
\begin{equation}
R(\xi)
\coloneqq \ln \left[ \frac{\mathbb{P}\left(\Lcal({\bm{\theta}} + {\bm{\phi}}) = \xi \right)}{\mathbb{P}\left(\Lcal_{-N}({\bm{\theta}} + {\bm{\phi}}) = \xi \right)} \right]
= \ln \left[ \frac{ \mathbb{P} \left(\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}} = \xi - \Lcal({\bm{\theta}}) \right)}
{\mathbb{P} \left(\inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}}) }{{\bm{\phi}}} = \xi - \Lcal_{-N}({\bm{\theta}})\right)} \right]
.
\end{equation}
The numerator in the ratio describes the probability of seeing $\Lcal({\bm{\theta}}) + \xi$ when the database is $\mathcal{D}$, the denominator corresponds to the probability of seeing this same value when the database is $\Dcal_{-\Scal}$.
${\bm{\phi}}$ is a multinomial Gaussian distribution ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{S}}^T {\bm{S}})$. Therefore,
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
Therefore, we have
\begin{equation}
\mathbb{P}\left(\Lcal({\bm{\theta}} + {\bm{\phi}}) = \xi \right)
=
\left[2 \pi \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2 \right]^{-1/2} e^{ - \frac{1}{2} \left(\xi - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}} \right)^2 \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^{-2}}
.
\end{equation}
With similar results for $\Lcal_{-\Scal}$. We can thus rewrite the privacy loss $R$ as
\begin{align}
R
& = \ln \left[ \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}} \right]
+ \frac{1}{2} \left[ \left(\frac{\xi - \Lcal_{-N}({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal_{-N}({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}\right)^2
- \left(\frac{\xi - \Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}\right)^2 \right]
\end{align}
We define $Q \coloneqq \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}} $.
\subsubsection*{$Q = 1 $}
We define $\alpha(\Lcal, {\bm{\theta}}) \coloneqq \frac{\Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}$.
Therefore, we can simplify $R(\xi)$ as
\begin{align}
R (\xi)
& =
\left[\alpha(\Lcal, {\bm{\theta}}) - \alpha(\Lcal_{-\Scal}, {\bm{\theta}})\right]\xi
+
\frac{1}{2} \left[\alpha^2(\Lcal_{-\Scal}, {\bm{\theta}})
- \alpha^2(\Lcal, {\bm{\theta}}) \right]
\end{align}
Therefore, we have $R(\xi) \le \epsilon$ when
\begin{equation}
R(\xi) \le \epsilon
\Leftrightarrow
\xi \le \frac{\epsilon - \frac{1}{2} \left[\alpha^2(\Lcal_{-\Scal}, {\bm{\theta}})
- \alpha^2(\Lcal, {\bm{\theta}}) \right]}
{\alpha(\Lcal, {\bm{\theta}}) - \alpha(\Lcal_{-\Scal}, {\bm{\theta}})}
\eqqcolon \xi_1
\end{equation}
Let us assume that $\xi_1 > 0$.
To satisfy DP privacy, it suffices
\begin{equation}
\mathbb{P}(\Lcal({\bm{\theta}} + {\bm{\phi}}) > \xi_1)
\le \mathbb{P}(\inner{\nabla_{\bm{\theta}} \Lcal({\bm{\theta}})}{{\bm{\phi}}} > \xi_1 - \Lcal({\bm{\theta}}))
\end{equation}
We define $Q \coloneqq \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}} $, $U \coloneqq \Delta \Lcal ({\bm{\theta}})
+ \left[ \nabla_{\bm{\theta}} \Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal_{-N}({\bm{\theta}}) \right]^T {\bm{\mu}} $,
$
S_1 =
\{
\xi \ | \ R(\xi) \le \epsilon
\}$
, and
$
S_2 =
\{
\xi \ | \ R(\xi) \ge - \epsilon
\}$
.
\subsubsection*{1st case}
We change variable for ease of writing. We set $y = (\xi - \Lcal({\bm{\theta}}) - \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}) / \norm{{\bm{S}} \nabla \Lcal ({\bm{\theta}})}$ which gives $\xi = \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y + \Lcal({\bm{\theta}}) + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}} $. We also consider $U_{-N} = \frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})} } U$.
\begin{align}
R
& = - \ln Q
+ \frac{1}{2} \left[ \left(Q y + U_{-N} \right)^2
- y^2 \right]
\end{align}
which can be developed as
\begin{align}
R
& = \frac{1}{2}\left[Q^2 - 1\right] y^2
+ U_{-N} Q y
+ \frac{1}{2}U_{-N}^2
- \ln Q
\end{align}
To find $S_1$, we consider $R_\epsilon$ defined as
\begin{align}
R_{\epsilon}
& = \frac{1}{2}\left[Q^2 - 1\right] y^2
+ \Delta({\bm{\theta}}) Q y
+ \frac{1}{2}\Delta^2({\bm{\theta}})
- \ln Q
- \epsilon
.
\end{align}
We define $\Delta_{\epsilon}$ the associated characteristic polynomial of $R_\epsilon$, i.e.
\begin{equation}
\Delta_{ \epsilon}
= U_{-N}^2 Q^2 - 4 \frac{1}{2}\left[Q^2 - 1\right]\left[\frac{1}{2}U_{-N}^2 - \ln Q - \epsilon\right]
= U_{-N}^2 + 2\left[Q^2 - 1\right]\left[\ln Q + \epsilon\right]
\end{equation}
Let us call $y_1$ and $y_2$ the two roots of $\Delta_{\epsilon}$ such that $y_{1} < y_{2}$.
Therefore, we have
\begin{equation}
y_{1}
= \frac{- \Delta({\bm{\theta}}) Q - \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
\text{ and }
y_{2}
= \frac{- \Delta({\bm{\theta}}) Q + \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
.
\end{equation}
Therefore, we have
\begin{equation}
S_1
= \left[
\xi_1 \coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_1 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}},
\xi_2
\coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_2 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}}
\right]
\end{equation}
We consider that $\Delta$ is such that $y_1 < 0$ and $y_2>0$.
For DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\eta_1 \notin S_1
\right)
\le \delta
\Leftrightarrow
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \delta
\end{equation}
Let us consider $X \sim N(\mu, \sigma)$, then we have
\begin{equation}
p(X < x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x <0
\end{equation}
\begin{equation}
p(X > x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x >0
\end{equation}
We can bound the cumulative distribution function of a normal distribution using the complementary error function which gives
\begin{equation}
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )}
+ \sqrt{2}\sigma\frac{e^{-(x_2 - \mu)^2/2 \sigma^2}}{2 (x_2 - \mu_1 )}
\end{equation}
Finally, for DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\Lcal({\bm{\theta}} + {\bm{\phi}} ) - \Lcal({\bm{\theta}})\in S_1
\right)
\ge 1- \delta
\end{equation}
\subsubsection*{2nd case}
We change variable for ease of writing. We set
$z = (\xi - \Lcal_{-N}({\bm{\theta}}) - \inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}{{\bm{\mu}}}) / \norm{{\bm{S}} \nabla \Lcal_{-N} ({\bm{\theta}})}$
which gives
$\xi = \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N} ({\bm{\theta}})} z + \Lcal_{-N}({\bm{\theta}}) + \inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})} {{\bm{\mu}}} $.
We also consider $\frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})} } U
= \frac{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})} } \frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}U
= \frac{1}{Q}U_{-N}$.
\begin{align}
R
& = - \ln Q
+ \frac{1}{2} \left[ z^2
- \left( \frac{1}{Q} z - \frac{1}{Q} U_{-N} \right)^2 \right]
\end{align}
which we can develop as
\begin{align}
R
& = \frac{1}{2}\left[1 - \frac{1}{Q^2}\right] z^2
+ \frac{1}{Q^2} U_{-N} z
- \ln Q
- \frac{1}{2} \frac{1}{Q^2}U_{-N}^2
\end{align}
We define $\Delta_{\epsilon}$ the associated characteristic polynomial of $R_\epsilon$, i.e.
\begin{equation}
\Delta_{ \epsilon}
= \frac{1}{Q^4} U_{-N}^2
- 4 \frac{1}{2}\left[ 1 - \frac{1}{Q^2}\right]\left[- \frac{1}{2}\frac{1}{Q^2}U_{-N}^2 - \ln Q + \epsilon\right]
= \frac{1}{Q^2} U_{-N}^2 + 2\left[1 - \frac{1}{Q^2}\right]\left[\ln Q - \epsilon\right]
\end{equation}
Let us call $y_1$ and $y_2$ the two roots of $\Delta_{\epsilon}$ such that $y_{1} < y_{2}$.
Therefore, we have
\begin{equation}
y_{1}
= \frac{- \Delta({\bm{\theta}}) Q - \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
\text{ and }
y_{2}
= \frac{- \Delta({\bm{\theta}}) Q + \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
.
\end{equation}
Therefore, we have
\begin{equation}
S_1
= \left[
\xi_1 \coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_1 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}},
\xi_2
\coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_2 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}}
\right]
\end{equation}
We consider that $\Delta$ is such that $y_1 < 0$ and $y_2>0$.
For DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\eta_1 \notin S_1
\right)
\le \delta
\Leftrightarrow
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \delta
\end{equation}
Let us consider $X \sim N(\mu, \sigma)$, then we have
\begin{equation}
p(X < x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x <0
\end{equation}
\begin{equation}
p(X > x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x >0
\end{equation}
We cn bound the cumulative distribution function of a normal distribution using the complementary error function which gives
\begin{equation}
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )}
+ \sqrt{2}\sigma\frac{e^{-(x_2 - \mu)^2/2 \sigma^2}}{2 (x_2 - \mu_1 )}
\end{equation}
Finally, for DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\Lcal({\bm{\theta}} + {\bm{\phi}} ) - \Lcal({\bm{\theta}})\in S_1
\right)
\ge 1- \delta
\end{equation}
\subsubsection{Interpretability with local perturbation}
Perturbating the model ${\bm{\theta}}$ with ${\bm{\phi}}$ impacts the performance of all the data samples. As such, even with Definition \ref{def:DP_forget} only based on the data points to forget, all the samples are impacted. For interpretability concerns, we consider a local perturbation ${\bm{\psi}}$ to the parameters to forget and look at the resulting pdf.
We also consider the following linear approximation
\begin{assumption}
For any data point $({\bm{x}}, {\bm{y}})$ and model with parameters ${\bm{\theta}}$,
we consider that $l$ can be linearly approximated in ${\bm{x}}$, i.e.
\begin{equation}
l({\bm{x}}+ {\bm{\epsilon}}, {\bm{y}}, {\bm{\theta}} ) = l({\bm{x}}, {\bm{y}}, {\bm{\theta}}) + \nabla_{\bm{x}} l({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
with $\norm{{\bm{\epsilon}}} \ll \norm{{\bm{x}}}$.
\end{assumption}
Therefore, we consider
\begin{equation}
\Lcal(\Dcal, {\bm{\theta}})
= \Lcal({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} \inner{\nabla_{\bm{x}} l({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})}{{\bm{\psi}}_i}
\end{equation}
\subsection{Practical Implementation}
\section{Formalizing the Unlearning problem}
\subsection{Optimization}
We consider a dataset $\Dcal$ composed of $|\Dcal| = N$ data samples $({\bm{x}}_i, {\bm{y}}_i)$ where ${\bm{x}}_i$ is the feature vector of a data sample and ${\bm{y}}_i$ its prediction or label.
For unsupervised learning, it suffices to consider that data samples in $\Dcal$ are only made with ${\bm{x}}_i$. We denote by $\Scal$ the set of $S = |\Scal|$ indices we want to forget from $\Dcal$ and $\Dcal_{-\Scal} = \Dcal\backslash\{({\bm{x}}_i, {\bm{y}}_i) | i \in \Scal \}$ the dataset without the forgotten data samples.
We consider a function $f$ depending on one data sample and model parameters ${\bm{\theta}}$.
For example, $f$ can be the loss of one data sample or its activation vector. We define the associated normalized functions
\begin{equation}
F({\bm{\theta}}) \coloneqq \frac{1}{|\Dcal|} \sum_{i \in \Dcal} f({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
\text{ and }
F_{-\Scal}({\bm{\theta}}) \coloneqq \frac{1}{|\Dcal_{-\Scal}|} \sum_{i \in \Dcal_{-\Scal}} f({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
.
\end{equation}
We consider $F$ is a $K$ dimensional vector where $K=1$ if $f$ is the loss function of one data samples and $K$ is the amount of classes in a classification problem.
For ease of writing, we consider $F_{-i}$ to be the resulting function when forgetting data sample $i$.
\section{Scrubbing Methods}
We consider a quadratic approximation of every sample loss function.
Therefore, the scrubbing method $h$ satisfies
\begin{equation}
h(w) = w - H_w^{-1} \nabla f_{\Scal}(w)
\end{equation}
We can also analyze this transformation in function of the samples features on the scrubbed model
\begin{equation}
t(x_i) = x_i - H_x^{-1}(x_i, h(w)) \nabla_{x} f(x_i, h(w))
\end{equation}
The learnt model is not necessarily the best model to start the new learning process.
We can bound the distance between a given model $w^t$ and the optimum model $w^*$ as :
We consider $H$ to be symmetric positive definite matrix.
\begin{align}
\norm{w^t - w^*}_2^2
&= \norm{w^t - h(w^t)}_2^2
= \norm{H^{-1}\nabla f}_2^2
\le \norm{H^{-1}}_F^2 \norm{\nabla f}_2^2
\\
&= \Tr{\left(H^{-2}\right)} \norm{\nabla f}_2^2
\le \Tr{\left(H\right)}^{-2} \norm{\nabla f}_2^2
= \left(\sum_{ i = 1 }^M \Tr H_i \right)^{-2} \norm{\nabla f}_2^2
\end{align}
\section{Complete Unlearning}
We assume that $f$ satisfies Assumption \ref{ass:linear_theta}.
\begin{assumption}
\label{ass:linear_theta}
For any data point $({\bm{x}}, {\bm{y}})$ and model with parameters ${\bm{\theta}}$,
we consider that $f$ can be linearly approximated in ${\bm{\theta}}$ for a perturbation ${\bm{\epsilon}}$, i.e.
\begin{equation}
f({\bm{x}}, {\bm{y}}, {\bm{\theta}} + \epsilon)
= f({\bm{x}}, {\bm{y}}, {\bm{\theta}}) + {\bm{J}}_{{\bm{\theta}}}({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
where ${\bm{J}}_{{\bm{\theta}}}$ is the Jacobian matrix of $f$ with respect to ${\bm{\theta}}$.
\end{assumption}
With Assumption \ref{ass:linear_theta} and the stochastic perturbation ${\bm{\phi}}$, we have
\begin{align}
F({\bm{\theta}} + {\bm{\phi}})
= F({\bm{\theta}})
+ \frac{1}{N}\sum_{ i = 1 }^{N} {\bm{J}}_{\bm{\theta}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}) {\bm{\phi}}
= F({\bm{\theta}})
+ {\bm{J}}_{\bm{\theta}} ( \Dcal, {\bm{\theta}}) {\bm{\phi}}
,
\end{align}
where we define $J_{\bm{\theta}} ( \Dcal, {\bm{\theta}}) = \frac{1}{N}\sum_{ i = 1 }^{N} {\bm{J}}_{\bm{\theta}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})$
with a similar derivation for $F_{-\Scal}$. We remind that ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{\Sigma}} = {\bm{S}}^T {\bm{S}})$. Hence, we have
\begin{equation}
F({\bm{\theta}} + {\bm{\phi}} )
\sim \mathcal{N}( F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}){\bm{\mu}}, {\bm{J}}_{\bm{\theta}} ( \Dcal, {\bm{\theta}})^T{\bm{\Sigma}} {\bm{J}}_{\bm{\theta}}(\Dcal, {\bm{\theta}}))
,
\end{equation}
with a similar derivation for $F_{-\Scal}$. With a non zero expected perturbation ${\bm{\mu}}$, a perfect unlearning can be obtained with $F({\bm{\theta}} + {\bm{\phi}}) = F_{-\Scal}({\bm{\theta}} + {\bm{\phi}})$ by choosing ${\bm{\mu}}$ and ${\bm{\Sigma}}$ such that the probability density functions of these two queries are identical. Considering the dimensions of the Jacobian, numerous ${\bm{\mu}}$ can satisfy this condition, we are thus interested in the ones with the smallest norm where Assumption \ref{ass:linear_theta} is the most valid.
Hence, we have
\begin{alignat}{2}
&\min_{{\bm{\mu}}} &\qquad& \norm{{\bm{\mu}}}^2
\\
&\text{subject to} & & \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]{\bm{\mu}} = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}),
\nonumber\\
& & & \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]^T{\bm{\Sigma}} \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right] = 0 \label{eq:constraint_sigma}.
\nonumber
\end{alignat}
We define ${\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal}) = \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]$.
When this optimization problem is feasible, then its optimum ${\bm{\mu}}^*$ satisfies
\begin{equation}
{\bm{\mu}}^* = {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal}) \left[ {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal})^T {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}})\right].
\end{equation}
Any $\Sigma$ such that equation(\ref{eq:constraint_sigma}) is satisfied works including ${\bm{\Sigma}} =0$. Yet, considering a positive ${\bm{\Sigma}}$ to mitigate the linear approximation would be helpful. We note that when $K=1$, we have ${\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) = \nabla_{\bm{\theta}} F({\bm{\theta}})$ and can simplify the close form of ${\bm{\mu}}^*$ with
\begin{equation}
{\bm{\mu}}^* = \frac{\nabla_{\bm{\theta}} F({\bm{\theta}}) - \nabla_{\bm{\theta}} F_{-\Scal}({\bm{\theta}})}{\norm{\nabla_{\bm{\theta}} F({\bm{\theta}}) - \nabla_{\bm{\theta}} F_{-\Scal}({\bm{\theta}})}^2} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}})\right].
\end{equation}
\subsubsection{Interpretability using Local Perturbation}
Perturbating the model ${\bm{\theta}}$ with ${\bm{\phi}}$ impacts the performance of all the data samples. As such, even with Definition \ref{def:DP_forget} only based on the data points to forget, all the samples are impacted. For interpretability concerns, we consider a local perturbation ${\bm{\psi}}$ to the parameters to forget and look at the resulting pdf.
\begin{assumption}
For any data point $({\bm{x}}, {\bm{y}})$ and model parameters ${\bm{\theta}}$,
we consider that $f$ can be linearly approximated in ${\bm{x}}$ such that for a feature perturbation $\epsilon$ we have
\begin{equation}
f({\bm{x}}+ {\bm{\epsilon}}, {\bm{y}}, {\bm{\theta}} )
= f({\bm{x}}, {\bm{y}}, {\bm{\theta}})
+ J_{\bm{x}} ({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
where ${\bm{J}}_{{\bm{x}}}$ is the Jacobian matrix of $f$ with respect to ${\bm{x}}$.
\end{assumption}
We consider clients are given the same Gaussian Noise ${\bm{\psi}} \sim N({\bm{\nu}}, {\bm{\Phi}})$ which gives the following optimization problem and results ${\bm{\nu}}^*$.
\begin{equation}
F(\Dcal, {\bm{\theta}})
= F({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} {\bm{J}}_{{\bm{x}}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}){\bm{\psi}}
= F({\bm{\theta}}) + \frac{|\Scal|}{|\Dcal|}{\bm{J}}_{{\bm{x}}} (\Dcal_\Scal, {\bm{\theta}}) {\bm{\psi}}
\end{equation}
and
\begin{alignat}{2}
&\min_{{\bm{\nu}}} &\qquad& \norm{{\bm{\nu}}}^2
\\
&\text{subject to} & & {\bm{J}}_{\bm{x}} (\Dcal_\Scal, {\bm{\theta}}){\bm{\nu}} = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^*,
\nonumbe
\end{alignat}
\begin{equation}
{\bm{\nu}}^*
= \frac{|\Dcal|}{|\Scal|}{\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})
\left[ {\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})^T {\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) - {\bm{J}}_{\bm{\theta}}(\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^* \right].
\end{equation}
We can also give every client their own perturbation ${\bm{\psi}}_i \sim N({\bm{\nu}}_i, {\bm{\Phi}}_i)$ which gives the following optimization problem and results ${\bm{\nu}}_i^*$.
\begin{equation}
F(\Dcal, {\bm{\theta}})
= F({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} {\bm{J}}_{{\bm{x}}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}){\bm{\psi}}_i
\end{equation}
which gives the following optimization problem
\begin{alignat}{2}
&\min_{{\bm{\nu}}_i} &\qquad& \frac{1}{|\Scal|}\sum_{ i \in S}\norm{{\bm{\nu}}_i}^2
\\
&\text{subject to} & & \frac{1}{|\Dcal|}\sum_{ i \in \Scal} {\bm{J}}_{\bm{x}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}) {\bm{\nu}}_i = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^*,
\nonumbe
\end{alignat}
\begin{equation}
{\bm{\nu}}_i^*
= |\Dcal| {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
\left[ \sum_{ i \in S} {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})^T {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) - {\bm{J}}_{\bm{\theta}}(\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^* \right].
\end{equation}
\section{Differential Privacy}
Let us consider
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
We provide sufficient conditions for ${\bm{\mu}}$ and ${\bm{S}}$ such that ${\bm{\theta}} + {\bm{\phi}}$ satisfies Definition \ref{def:DP_forget}.
${\bm{\phi}}$ is a multinomial Gaussian distribution ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{S}}^T {\bm{S}})$. Therefore,
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
We are investigating the probability, given that the database is $\mathcal{D}$, of observing an output that occurs with a very different probability under $\mathcal{D}$ than under and adjacent database $\Dcal_{-\Scal}$, where the probability space is $\mathbb{R}^d$.
We define $\Delta \Lcal ({\bm{\theta}}) = \Lcal({\bm{\theta}}) - \Lcal_{-\Scal}({\bm{\theta}})$ and the privacy loss $R$ as
\begin{equation}
R(\xi)
\coloneqq \ln \left[ \frac{\mathbb{P}\left(\Lcal({\bm{\theta}} + {\bm{\phi}}) = \xi \right)}{\mathbb{P}\left(\Lcal_{-N}({\bm{\theta}} + {\bm{\phi}}) = \xi \right)} \right]
= \ln \left[ \frac{ \mathbb{P} \left(\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}} = \xi - \Lcal({\bm{\theta}}) \right)}
{\mathbb{P} \left(\inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}}) }{{\bm{\phi}}} = \xi - \Lcal_{-N}({\bm{\theta}})\right)} \right]
.
\end{equation}
The numerator in the ratio describes the probability of seeing $\Lcal({\bm{\theta}}) + \xi$ when the database is $\mathcal{D}$, the denominator corresponds to the probability of seeing this same value when the database is $\Dcal_{-\Scal}$.
${\bm{\phi}}$ is a multinomial Gaussian distribution ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{S}}^T {\bm{S}})$. Therefore,
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
Therefore, we have
\begin{equation}
\mathbb{P}\left(\Lcal({\bm{\theta}} + {\bm{\phi}}) = \xi \right)
=
\left[2 \pi \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2 \right]^{-1/2} e^{ - \frac{1}{2} \left(\xi - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}} \right)^2 \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^{-2}}
.
\end{equation}
With similar results for $\Lcal_{-\Scal}$. We can thus rewrite the privacy loss $R$ as
\begin{align}
R
& = \ln \left[ \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}} \right]
+ \frac{1}{2} \left[ \left(\frac{\xi - \Lcal_{-N}({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal_{-N}({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}\right)^2
- \left(\frac{\xi - \Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}\right)^2 \right]
\end{align}
We define $Q \coloneqq \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}} $.
\subsubsection*{$Q = 1 $}
We define $\alpha(\Lcal, {\bm{\theta}}) \coloneqq \frac{\Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}$.
Therefore, we can simplify $R(\xi)$ as
\begin{align}
R (\xi)
& =
\left[\alpha(\Lcal, {\bm{\theta}}) - \alpha(\Lcal_{-\Scal}, {\bm{\theta}})\right]\xi
+
\frac{1}{2} \left[\alpha^2(\Lcal_{-\Scal}, {\bm{\theta}})
- \alpha^2(\Lcal, {\bm{\theta}}) \right]
\end{align}
Therefore, we have $R(\xi) \le \epsilon$ when
\begin{equation}
R(\xi) \le \epsilon
\Leftrightarrow
\xi \le \frac{\epsilon - \frac{1}{2} \left[\alpha^2(\Lcal_{-\Scal}, {\bm{\theta}})
- \alpha^2(\Lcal, {\bm{\theta}}) \right]}
{\alpha(\Lcal, {\bm{\theta}}) - \alpha(\Lcal_{-\Scal}, {\bm{\theta}})}
\eqqcolon \xi_1
\end{equation}
Let us assume that $\xi_1 > 0$.
To satisfy DP privacy, it suffices
\begin{equation}
\mathbb{P}(\Lcal({\bm{\theta}} + {\bm{\phi}}) > \xi_1)
\le \mathbb{P}(\inner{\nabla_{\bm{\theta}} \Lcal({\bm{\theta}})}{{\bm{\phi}}} > \xi_1 - \Lcal({\bm{\theta}}))
\end{equation}
We define $Q \coloneqq \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}} $, $U \coloneqq \Delta \Lcal ({\bm{\theta}})
+ \left[ \nabla_{\bm{\theta}} \Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal_{-N}({\bm{\theta}}) \right]^T {\bm{\mu}} $,
$
S_1 =
\{
\xi \ | \ R(\xi) \le \epsilon
\}$
, and
$
S_2 =
\{
\xi \ | \ R(\xi) \ge - \epsilon
\}$
.
\subsubsection*{1st case}
We change variable for ease of writing. We set $y = (\xi - \Lcal({\bm{\theta}}) - \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}) / \norm{{\bm{S}} \nabla \Lcal ({\bm{\theta}})}$ which gives $\xi = \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y + \Lcal({\bm{\theta}}) + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}} $. We also consider $U_{-N} = \frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})} } U$.
\begin{align}
R
& = - \ln Q
+ \frac{1}{2} \left[ \left(Q y + U_{-N} \right)^2
- y^2 \right]
\end{align}
which can be developed as
\begin{align}
R
& = \frac{1}{2}\left[Q^2 - 1\right] y^2
+ U_{-N} Q y
+ \frac{1}{2}U_{-N}^2
- \ln Q
\end{align}
To find $S_1$, we consider $R_\epsilon$ defined as
\begin{align}
R_{\epsilon}
& = \frac{1}{2}\left[Q^2 - 1\right] y^2
+ \Delta({\bm{\theta}}) Q y
+ \frac{1}{2}\Delta^2({\bm{\theta}})
- \ln Q
- \epsilon
.
\end{align}
We define $\Delta_{\epsilon}$ the associated characteristic polynomial of $R_\epsilon$, i.e.
\begin{equation}
\Delta_{ \epsilon}
= U_{-N}^2 Q^2 - 4 \frac{1}{2}\left[Q^2 - 1\right]\left[\frac{1}{2}U_{-N}^2 - \ln Q - \epsilon\right]
= U_{-N}^2 + 2\left[Q^2 - 1\right]\left[\ln Q + \epsilon\right]
\end{equation}
Let us call $y_1$ and $y_2$ the two roots of $\Delta_{\epsilon}$ such that $y_{1} < y_{2}$.
Therefore, we have
\begin{equation}
y_{1}
= \frac{- \Delta({\bm{\theta}}) Q - \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
\text{ and }
y_{2}
= \frac{- \Delta({\bm{\theta}}) Q + \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
.
\end{equation}
Therefore, we have
\begin{equation}
S_1
= \left[
\xi_1 \coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_1 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}},
\xi_2
\coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_2 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}}
\right]
\end{equation}
We consider that $\Delta$ is such that $y_1 < 0$ and $y_2>0$.
For DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\eta_1 \notin S_1
\right)
\le \delta
\Leftrightarrow
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \delta
\end{equation}
Let us consider $X \sim N(\mu, \sigma)$, then we have
\begin{equation}
p(X < x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x <0
\end{equation}
\begin{equation}
p(X > x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x >0
\end{equation}
We can bound the cumulative distribution function of a normal distribution using the complementary error function which gives
\begin{equation}
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )}
+ \sqrt{2}\sigma\frac{e^{-(x_2 - \mu)^2/2 \sigma^2}}{2 (x_2 - \mu_1 )}
\end{equation}
Finally, for DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\Lcal({\bm{\theta}} + {\bm{\phi}} ) - \Lcal({\bm{\theta}})\in S_1
\right)
\ge 1- \delta
\end{equation}
\subsubsection*{2nd case}
We change variable for ease of writing. We set
$z = (\xi - \Lcal_{-N}({\bm{\theta}}) - \inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}{{\bm{\mu}}}) / \norm{{\bm{S}} \nabla \Lcal_{-N} ({\bm{\theta}})}$
which gives
$\xi = \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N} ({\bm{\theta}})} z + \Lcal_{-N}({\bm{\theta}}) + \inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})} {{\bm{\mu}}} $.
We also consider $\frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})} } U
= \frac{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})} } \frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}U
= \frac{1}{Q}U_{-N}$.
\begin{align}
R
& = - \ln Q
+ \frac{1}{2} \left[ z^2
- \left( \frac{1}{Q} z - \frac{1}{Q} U_{-N} \right)^2 \right]
\end{align}
which we can develop as
\begin{align}
R
& = \frac{1}{2}\left[1 - \frac{1}{Q^2}\right] z^2
+ \frac{1}{Q^2} U_{-N} z
- \ln Q
- \frac{1}{2} \frac{1}{Q^2}U_{-N}^2
\end{align}
We define $\Delta_{\epsilon}$ the associated characteristic polynomial of $R_\epsilon$, i.e.
\begin{equation}
\Delta_{ \epsilon}
= \frac{1}{Q^4} U_{-N}^2
- 4 \frac{1}{2}\left[ 1 - \frac{1}{Q^2}\right]\left[- \frac{1}{2}\frac{1}{Q^2}U_{-N}^2 - \ln Q + \epsilon\right]
= \frac{1}{Q^2} U_{-N}^2 + 2\left[1 - \frac{1}{Q^2}\right]\left[\ln Q - \epsilon\right]
\end{equation}
Let us call $y_1$ and $y_2$ the two roots of $\Delta_{\epsilon}$ such that $y_{1} < y_{2}$.
Therefore, we have
\begin{equation}
y_{1}
= \frac{- \Delta({\bm{\theta}}) Q - \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
\text{ and }
y_{2}
= \frac{- \Delta({\bm{\theta}}) Q + \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
.
\end{equation}
Therefore, we have
\begin{equation}
S_1
= \left[
\xi_1 \coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_1 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}},
\xi_2
\coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_2 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}}
\right]
\end{equation}
We consider that $\Delta$ is such that $y_1 < 0$ and $y_2>0$.
For DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\eta_1 \notin S_1
\right)
\le \delta
\Leftrightarrow
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \delta
\end{equation}
Let us consider $X \sim N(\mu, \sigma)$, then we have
\begin{equation}
p(X < x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x <0
\end{equation}
\begin{equation}
p(X > x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x >0
\end{equation}
We cn bound the cumulative distribution function of a normal distribution using the complementary error function which gives
\begin{equation}
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )}
+ \sqrt{2}\sigma\frac{e^{-(x_2 - \mu)^2/2 \sigma^2}}{2 (x_2 - \mu_1 )}
\end{equation}
Finally, for DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\Lcal({\bm{\theta}} + {\bm{\phi}} ) - \Lcal({\bm{\theta}})\in S_1
\right)
\ge 1- \delta
\end{equation}
\subsubsection{Interpretability with local perturbation}
Perturbating the model ${\bm{\theta}}$ with ${\bm{\phi}}$ impacts the performance of all the data samples. As such, even with Definition \ref{def:DP_forget} only based on the data points to forget, all the samples are impacted. For interpretability concerns, we consider a local perturbation ${\bm{\psi}}$ to the parameters to forget and look at the resulting pdf.
We also consider the following linear approximation
\begin{assumption}
For any data point $({\bm{x}}, {\bm{y}})$ and model with parameters ${\bm{\theta}}$,
we consider that $l$ can be linearly approximated in ${\bm{x}}$, i.e.
\begin{equation}
l({\bm{x}}+ {\bm{\epsilon}}, {\bm{y}}, {\bm{\theta}} ) = l({\bm{x}}, {\bm{y}}, {\bm{\theta}}) + \nabla_{\bm{x}} l({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
with $\norm{{\bm{\epsilon}}} \ll \norm{{\bm{x}}}$.
\end{assumption}
Therefore, we consider
\begin{equation}
\Lcal(\Dcal, {\bm{\theta}})
= \Lcal({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} \inner{\nabla_{\bm{x}} l({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})}{{\bm{\psi}}_i}
\end{equation}
\subsection{Practical Implementation}
\subsection{Proof of Theorem \ref{theo:diff_bound} for $K=1$}\label{app:subsec:proof_DL}
\begin{proof}
We define by ${\bm{\theta}}^N = \textsc{FedAvg}(I, N)$ and ${\bm{\phi}}^N = \textsc{FedAvg}(I_{-c}, N)$ the models trained with $\textsc{FedAvg}$ on ${\bm{\theta}}_0$ with respectively all the clients, i.e. $I$, and all the clients but client $c$, i.e. $I_{-c}$, performing $K=1$ GD step.
When clients perform $K=1$ GD step, two consecutive global models can be related, when training with clients in $I$ as a simple GD step, i.e.
\begin{equation}
{\bm{\theta}}^{n+1}
= {\bm{\theta}}^n
- \eta \nabla f_I({\bm{\theta}}^n)
.
\end{equation}
By considering the same process for $I_{-c}$ and with Assumption \ref{ass:linear_approx}, we get
\begin{align}
{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}
& = {\bm{\phi}}^n - {\bm{\theta}}^n
- \eta \left[ \nabla f_{I_{-c}}({\bm{\phi}}^n) - \nabla f_I({\bm{\theta}}^n) \right]
\\
& = \left[I - \eta H_{I_{-c}}({\bm{\theta}}^n) \right] \left[{\bm{\phi}}^n - {\bm{\theta}}^n \right]
- \eta \left[ \nabla f_{I_{-c}}({\bm{\theta}}^n) - \nabla f_I({\bm{\theta}}^n) \right]
\label{eq:diff}
.
\end{align}
$H_{I_{-c}}({\bm{\theta}}^n)$ is semi-positive, Assumption \ref{ass:linear_approx}. Let us define $\sigma_\text{max} (H_{I_{-c}}({\bm{\theta}}^n))$ the highest eigenvalue of $H_{I_{-c}}({\bm{\theta}}^n)$. When consider that $\eta \le 1/\sigma_\text{max} (H_{I_{-c}}({\bm{\theta}}^n))$, and due to the subadditivity of the norm, we get the following recurrent inequality
\begin{align}
\norm{{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}}_2
&\le
\eta \norm{\nabla f_I({\bm{\theta}}^n) - \nabla f_{I_{-c}}({\bm{\theta}}^n)}_2
+ \norm{{\bm{\phi}}^n - {\bm{\theta}}^n}_2
,
\end{align}
which when developed completes the proof when clients perform $K=1$ GD.
\end{proof}
\subsection{Proof of Theorem \ref{theo:diff_bound} for $K\ge 1$}\label{app:subsec:proof_FL}
\begin{proof}
We maintain the definitions of ${\bm{\theta}}^n$ and ${\bm{\phi}}^n$ introduced in Section \ref{app:subsec:proof_DL}. To account for the amount of local work $K$, we introduce ${\bm{\theta}}_i^{n , k}$ the model of client $i$ after $k$ GD steps performed on global model ${\bm{\theta}}^n$. We apply a similar reasoning for ${\bm{\phi}}_i^{n , k}$.
With Assumption \ref{ass:linear_approx}, we have
\begin{equation}
\nabla f_i ({\bm{\phi}}_i^{n, k})
= \nabla f_i ({\bm{\theta}}_i^{n, k})
+ H_i ({\bm{\theta}}_i^{n, k}) \left({\bm{\phi}}_i^{n, k} - {\bm{\theta}}_i^{n, k}\right)
\label{eq:rewrite_linear}
,
\end{equation}
which gives
\begin{align}
{\bm{\phi}}_i^{n, k+1} - {\bm{\theta}}_i^{n, k+1}
& = \left( {\bm{\phi}}_i^{n, k+1} - {\bm{\phi}}_i^{n, k}\right)
- \left( {\bm{\theta}}_i^{n, k+1 } - {\bm{\theta}}_i^{n, k} \right)
+ \left({\bm{\phi}}_i^{n, k} - {\bm{\theta}}_i^{n, k}\right)
\\
& = - \eta \left[\nabla f_i \left({\bm{\phi}}_i^{n, k}\right)
- \nabla f_i \left({\bm{\theta}}_i^{n, k} \right) \right]
+ \left({\bm{\phi}}_i^{n, k} - {\bm{\theta}}_i^{n, k}\right)
\\
& = \left[I - \eta H_i ({\bm{\theta}}_i^{n, k}) \right]\left({\bm{\phi}}_i^{n, k} - {\bm{\theta}}_i^{n, k}\right)
\\
& = \left[\prod_{r=0}^{k}\left[I - \eta H_i ({\bm{\theta}}_i^{n, r}) \right]\right]\left({\bm{\phi}}^n - {\bm{\theta}}^n \right)
\label{eq:diff_two_locals}
,
\end{align}
where the third equality follows from equation (\ref{eq:rewrite_linear}), and the fourth from expanding the recurrent equation. For the rest of this work, we define $Q_i^n = \prod_{k=0}^{K-1}\left[I - \eta H_i ({\bm{\theta}}_i^{n, k}) \right]$.
Using equation (\ref{eq:diff_two_locals}), we relate the difference between two global models with every client in $I$ and in $I_c$. When removing client $c$ the clients' importance changes. We consider importance $p_i$ when training with $I$. Instead, when training with clients in $I_c$, we consider the regularized importance $q_i = p_i / (1 - p_c)$ for the remaining clients and $q_c = 0$. We have
\begin{align}
{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}
& = \sum_{i=1}^M q_i \left({\bm{\phi}}_i^{n+1} - {\bm{\phi}}^n \right)
- \sum_{i=1}^M p_i \left({\bm{\theta}}_i^{n+1} - {\bm{\theta}}^n\right)
\\
& = \sum_{i=1}^M q_i \left[ \left({\bm{\phi}}_i^{n+1} - {\bm{\theta}}_i^{n+1}\right) + \left({\bm{\theta}}_i^{n+1} - {\bm{\theta}}^n \right) \right]
- \sum_{i=1}^M p_i \left({\bm{\theta}}_i^{n+1} - {\bm{\theta}}^n\right)
\\
& = \left(\sum_{i=1}^M q_i Q_i^n \right) \left({\bm{\phi}}^n - {\bm{\theta}}^n\right)
+ \Delta(I_{-c}, {\bm{\theta}}^n)
- \Delta(I, {\bm{\theta}}^n)
.
\label{app:eq:diff_one}
\end{align}
We consider a learning rate $\eta$ such that $\eta \le 1/\sigma_\text{max} (H_i({\bm{\theta}}^{n , k}))$. Hence, $\norm{Q_i^n}_2 \le 1$. With equation (\ref{app:eq:diff_one}), we get the following inequality
\begin{align}
\norm{{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}}_2
& \le \norm{{\bm{\phi}}^n - {\bm{\theta}}^n}_2
+ \norm{ \Delta(I, {\bm{\theta}}^n)
- \Delta(I_{-c}, {\bm{\theta}}^n)}_2
,
\end{align}
which expansion completes the proof.
\end{proof}
\subsection{Local Loss Functions' Regularization and Strong Convexity, Proof of Corollary \ref{cor:tighter}}
\label{app:subsec:regularization}
\begin{proof}
Under L2 regularization, every client's regularized loss function $F_i$ is expressed as
\begin{equation}
F_i({\bm{\theta}})
= f_i({\bm{\theta}})
+ \frac{\lambda}{2}\norm{{\bm{\theta}}}^2
\text{ and }
\nabla F_i({\bm{\theta}})
= \nabla f_i({\bm{\theta}})
+ \lambda{\bm{\theta}}
.
\end{equation}
When clients perform $K=1$ GD step, equation (\ref{app:eq:diff_one}) reduces to
\begin{align}
{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}
& = \eta \left[\nabla f_I({\bm{\theta}}^n) - \nabla f_{I_{-c}}({\bm{\theta}}^n) \right]
+ \left[(1 - \eta \lambda) I -\eta H_{I_{-c}}(\theta^n) \right]({\bm{\phi}}^n - \theta^n)
,
\end{align}
which, if $\eta \le 1 / (\lambda + \sigma_{\max}(H_i({\bm{\theta}}^n))$, gives
\begin{align}
\norm{{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}}_2
&\le
\eta \norm{\nabla f_I({\bm{\theta}}^n) - \nabla f_{I_{-c}}({\bm{\theta}}^n)}_2
+ (1- \eta \lambda -\eta \mu)\norm{{\bm{\phi}}^n - {\bm{\theta}}^n}_2
.
\end{align}
When clients perform $K\ge 1$ GD steps, we have $ {\bm{\phi}}_i^{n + 1} - {\bm{\theta}}_i^{n + 1} = Q_i^n \left[{\bm{\phi}}^n - {\bm{\theta}}^n\right]$ with
\begin{equation}
Q_i^n
= \prod_{r=0}^{K-1}\left[(1 - \eta \lambda)I - \eta H_i ({\bm{\theta}}_i^{n, r}) \right]
.
\end{equation}
Hence, we retrieve equation (\ref{app:eq:diff_one}). We consider the local learning rate satisfy $\eta \le 1 /( \lambda + \sigma_{\max} (H_i ({\bm{\theta}}^n)))$. Hence, considering that $Q_i^n$ can be bounded with the $\mu$-strong convexity of the Hessian, we get
\begin{align}
\norm{{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}}_2
&\le
\eta \norm{\Delta(I, {\bm{\theta}}^n) - \Delta(I_{-c}, {\bm{\theta}}^n)}_2
+ (1- \eta \lambda -\eta \mu)^K \norm{{\bm{\phi}}^n - {\bm{\theta}}^n}_2
.
\end{align}
Developing this recurrent equation completes the proof.
\end{proof}
\subsection{Generalization}
The proof of Theorem \ref{theo:diff_bound} can be also extended to account for FL regularization methods \citep{FedProx, FedDane, FedDyn}, other SGD solvers \citep{Adam, AdaGrad,pmlr-v89-li19c, OnTheLienarSpeedUp, Yu_Yang_Zhu_2019, haddapour2019trading}, client sampling \citep{FedProx, OnTheConvergence, TheoryClientSampling} and/or gradient compression/quantization \citep{FedPaq, QSparse, Atomo}.
\subsection{Calculus simplification with uniform importance}
For computation purposes, we propose the following expression to estimate a client bounded sensitivity, equation (\ref{eq:def_Psi}.
When removing client $c$, each client has new importance $q_i = p_i /(1 - p_c)$ for the remaining clients and $q_c = 0$. Hence, we have
\begin{align}
\norm{\Delta(I, {\bm{\theta}}^n) - \Delta({\bm{\theta}}^n, \Dcal_{-c})}_2
& = \norm{\left[{\bm{\theta}}^{n+1} - {\bm{\theta}}^n\right] - \left[\sum_{i=1}^{M}q_i {\bm{\theta}}_i^{n+1} - {\bm{\theta}}^n\right]}_2
\\
& = \norm{{\bm{\theta}}^{n+1} - \frac{1}{1 - p_c}\left[{\bm{\theta}}^{n+1} - p_c {\bm{\theta}}_i^{n+1}\right]}_2
\\
& = \frac{p_c}{1 - p_c} \norm{ {\bm{\theta}}_i^{n+1} - {\bm{\theta}}^{n+1} }_2
\end{align}
In the special case where clients have identical importance, we have $p_c/(1 - p_c) = 1/ (M -1)$.
\section{Introduction}
With the emergence of new data regulations, such as the EU General Data Protection Regulation (GDPR) \citep{GDPR} and the California Consumer Privacy Act (CCPA) \citep{CCPA}, the storage and processing of sensitive personal data is often subject of strict constraints and restrictions. In particular, the “right to be forgotten” states that personal data must be erased upon request from the concerned individuals, with subsequent potential implications on machine learning models trained by using this data.
Machine Unlearning (MU) is an emerging discipline that studies methods to ideally remove the contribution of a given data instance used to train a machine learning model.
Current MU approaches are essentially based on routines that modify the model weights in order to guarantee the “forgetting" of a given data point, i.e. to obtain a model equivalent to an hypothetical one trained without this data point \citep{Cao2015TowardsMS, SISA}.
Motivated by data governance and confidentiality concerns,
Federated learning (FL) has gained popularity in the last years to allow data owners to collaboratively learn a model without sharing their respective data.
Among the different FL approaches, federated averaging (\textsc{FedAvg}) has emerged as the most popular optimization scheme \citep{FedAvg}. An optimization round of \textsc{FedAvg} requires data owners, also called clients, to receive from the server the current global model which they update
before sending it back to the server. The new global model is then created as the weighted average of the client updates, according to their data ratio.
FL communication design guarantees to clients that their data is solely used to compute their model update, while theoretical work guarantees FL convergence to a stationary point of the clients' joint optimization problem \citep{FedNova, OnTheConvergence}.
With the current deployments of FL in the real-world, it is of crucial importance to extend MU to guarantee the unlearning of clients wishing to opt-out from a collaborative training routine.
This is not straightforward, since current MU schemes have been proposed essentially in the centralized learning setting, and cannot be seamlessly applied to the federated one.
For example, several MU methods require the estimation of the Hessian of the loss function \citep{CertifiedDataRemoval, ApproximateDataDeletion, Golatkar_2020_CVPR, golatkar2020forgetting, Golatkar_2021_CVPR}, an operation which is notoriously computationally heavy and intractable for high dimensional models. Moreover, sharing the Hessian would require clients to share with the server additional information about their data, thus exposing the federated setting to information leakage and attacks, for example under the form of model inversion \citep{Fredrikson-MI-2015}.
Alternative MU methods draw from the concept of differential privacy \cite{DP_book} and are based on a Gaussian noise perturbation of the trained model \citep{DescentToDelete, CertifiedDataRemoval, AdaptiveMachineUnlearning}. The magnitude of the noise perturbation should be estimated directly from the clients data, which is by construction inaccessible to the server in the FL regime. We also note that while recent federated unlearning (FU) methods have been proposed to unlearn a client from the global FL model \citep{FedEraser, wang2021federated, halimi2022federated, wu2022federated}, these approaches do not come with theoretical guarantees on the effectiveness of the unlearning.
The main contribution of this work consists in Informed Federated Unlearning (IFU), a novel efficient FU approach to unlearn a client's contribution with quantifiable unlearning guarantees. IFU requires minimal additional computations to the server in a standard \textsc{FedAvg} procedure. Specifically, the server quantifies at every optimization round each client's contribution to the global model.
Upon receiving an unlearning request from a client, the server identifies in the FL training history the optimal FL iteration and associated intermediate global model from which re-initializing the unlearning procedure. Unlearning guarantees are provided by introducing a novel randomized mechanism to perturb the selected intermediate model with client-specific noise.
We also extend IFU to Sequential Informed Federated Unlearning (SIFU), to account for realistic unlearning scenarios where the server receives sequential unlearning requests from one or more clients at different times \citep{DescentToDelete, AdaptiveMachineUnlearning}.
This manuscript is structured as follows.
In Section \ref{sec:background}, we provide formal definitions for MU, FL, and FU, and introduce the randomized mechanism with associated unlearning guarantees.
In Section \ref{sec:theory}, we introduce sufficient conditions for IFU to unlearn a client from the FL routine (Theorem \ref{theo:noise_DP}). In Section \ref{sec:SIFU}, we extend IFU to the sequential unlearning setting with Sequential IFU (SIFU).
Finally, in Section \ref{sec:experiments}, we experimentally demonstrate on different tasks and datasets that SIFU leads to more efficient unlearning procedures as compared to basic re-training and state-of-the-art FU approaches.
\section{Conclusions}
In this work, we introduce informed federated unlearning (IFU), a novel federated unlearning scheme to unlearn a client's contribution from a model trained with federated learning. Upon receiving an unlearning request from a given client, IFU identifies the optimal FL iteration from which FL has to be reinitialised, with statistical unlearning guarantees defined by Definition \ref{def:DP_adpated}. We extend the theory of IFU to account for the practical scenario of sequential unlearning (SIFU), where the server receives a series of forgetting request of one or more clients. We prove that SIFU can unlearn a series of forgetting requests while satisfying our unlearning guarantees, and demonstrate the effectiveness of our methods on a variety of tasks and dataset.
An additional contribution of this work consists in a new theory for bounding the clients contribution in FL. The server can compute this bound for every client without asking for any additional computation and communication. The theoretical justification of this approach relies on the linear approximation of the clients' loss function, and its relevance is here demonstrated across several benchmarks. Future extensions of the work will focus on generalizing our unlearning framework to more general settings.
\section{Alternative SGD}
We first formalize the learning and unlearning problem. We consider $\Lcal_1$ the averaged loss over a dataset of $n$ samples. Without loss of generality we consider removing the sample with index $n$ from the dataset which gives the new loss function $\Lcal_2$. Hence, we have
\begin{equation}
\Lcal_1(\theta)
= \frac{1}{n}\sum_{i=1}^{n} l(\theta, X_i, y_i)
\text{ and }
\Lcal_2(\theta)
= \frac{1}{n-1}\sum_{i=1}^{n-1} l(\theta, X_i, y_i)
.
\end{equation}
We denote by $\theta_1^*$ and $\theta_2^*$ the respective minimum of $\Lcal_1$ and $\Lcal_2$. For simplicity, we denote $l_i(\theta) \coloneqq l(\theta, X_i, y_i)$. Retraining is done on amount $K$ of gradient descents. In practice for computation concerns, optimizations steps have to use GD instead of SGD. Let us consider $m$ to be the batch size. The variance between the gradient estimator and the true gradient is shown to slow down the learning process. We thus propose another gradient method enabling faster optimization and thus faster unlearning.
We can express the gradient of $\Lcal_2$ in function of the one of $\Lcal_1$ as
\begin{equation}
\nabla\Lcal_2(\theta)
= \frac{1}{n-1}\sum_{i=1}^{n-1} \nabla l_i(\theta)
= \frac{n}{n-1} \nabla\Lcal_1 (\theta) - \frac{1}{n-1}\nabla l_n(\theta)
.
\end{equation}
We note that if the training has been properly done then the learnt model is $\theta_1^*$ and therefore, the unlearning process starts with $\nabla\Lcal_2(\theta_1^*) = - \frac{1}{n-1}\nabla l_n(\theta_1^*)$ showing that the deleted data is the only sample with importance when forgetting this sample from the learnt model.
\subsection*{Expansion}
Let us consider $\theta$ the model obtained by training with algorithm $\Acal$, and the perturbation $\delta$ brought to $S$, the resulting trained model is
\begin{equation}
\Acal(S + \delta)
= \Acal(S)
+ \delta \nabla \Acal(S)
\end{equation}
$\delta_i$ is the modification of sampled $i$ and $S_i=(X_i, y_i)$ is sample $i$. We define $u$ the set of samples to forget. We consider there are $K$ classes. We define
\begin{equation}
\delta_i =
\begin{cases}
\EE{X \sim X_i | y_i =y_k, i \in S'}{X} - X_k
& \text{if } k\in u\\
0
& \text{otherwise}
\end{cases}
\end{equation}
\section{Experiments}
\label{sec:experiments}
\begin{figure*}[ht]
\begin{centering}
\includegraphics[width=\linewidth]{./plots/fig_SIFU_every_dataset.pdf}
\end{centering}
\caption{
Total amount of aggregation rounds (1\textsuperscript{st} row) and model accuracy of unlearned clients (2\textsuperscript{nd} row) for MNIST, FashionMNIST, CIFAR10, CIFAR100, and CelebA (the lower the better).
The server runs a federated routine with $M=100$ clients, and unlearns 10 of them at each unlearning request ($R=3$).
}
\label{fig:SIFU}
\end{figure*}
In this section, we experimentally demonstrate the effectiveness of SIFU on a series of benchmarks introduced in Section \ref{subsec:experimental_setup}. In Section \ref{subsec:experimental_results}, we illustrate and discuss our experimental results.
Results and related code are publicly available at URL.
\subsection{Experimental Setup}
\label{subsec:experimental_setup}
\textbf{Datasets.} We report experiments on reduced versions of MNIST \citep{MNIST}, FashionMNIST \citep{FashionMNIST}, CIFAR-10 \citep{CIFAR-10}, CIFAR-100 \citep{CIFAR-10}, and CelebA \citep{CelebA}. For each dataset, we consider $M=100$ clients, with 100 data points each. For MNIST and FashionMNIST, each client has data samples from only one class, so that each class is represented in 10 clients only. For CIFAR10 and CIFAR100, each client has data samples with ratio sampled from a Dirichlet distribution with parameter 0.1 \citep{FL_and_CIFAR_dir}.
Finally, in CelebA, clients own data samples representing the same celebrity. With these five datasets, we consider different level of heterogeneity based on label and feature distribution.
\textbf{Models.}
For MNIST, we train a logistic regression model to consider a convex classification problem, while, for the other datasets, we train a neural network with convolutional layers followed by fully connected ones. More details on the networks are available in Appendix \ref{app:sec:experiments}.
\textbf{Unlearning schemes.}
In addition to SIFU, we consider the following unlearning schemes from the state-of-the-art: \textsc{Scratch}, where retraining of a new initial model is performed on the remaining clients; \textsc{Fine-Tuning}, where retraining is performed on the current global model with the remaining clients; \textsc{Last} \citep{DescentToDelete}, where retraining is performed on the remaining clients via perturbation of the final FL global model; \textsc{DP} \citep{DP_book}, where training with every client is performed with differential privacy, and \textsc{FedAccum} \citep{FedEraser}, where retraining is performed on the current global model from which the server removes the updates of the clients to unlearn, by re-aggregating the parameter updates
of clients that were stored by the server
across FL iterations. We provide in Appendix \ref{app:sec:FedAccum} the pseudo-code of \textsc{FedAccum} with the notation of our paper. We remind that \textsc{FedAccum} does not provide quantitative guarantees of the unlearning procedure, and requires the server to store the full sequence of models during the FL procedure.
\textbf{Experimental scenario.}
We consider a sequential unlearning scenario in which the server performs the FL training procedure and then receives $R=3$ sequential unlearning requests to unlearn 10 random clients per request. In the special case of MNIST and FashionMNIST, the server must unlearn 10 clients owning the same class.
The server orchestrates each unlearning scheme through retraining until the global model accuracy on the remaining clients exceeds a fixed value specific to each dataset. We set the minimum number of 50 aggregation rounds, and a maximum budget of 10000 rounds when the stopping accuracy criterion is not met.
Each unlearning method is applied with the same hyperparameters, i.e. stopping accuracy, local learning rate $\eta$, and amount of local work $K$ (Appendix \ref{app:sec:experiments}). We define the set of clients requesting unlearning as:
\begin{equation}
F_r
= \cup_{s=1}^r W_s
.
\label{eq:F_r}
\end{equation}
In our experimental scenario, we have $|F_0|=0$ during training and $|F_1|=10$, $|F_2|=20$, and $|F_3|=30$ after each unlearning request.
\textbf{Unlearning quantification.}
We verify the success of an unlearning scheme with two metrics: (a) the amount of server aggregation rounds needed for retraining, and (b) the resulting model accuracy on the unlearned clients. we note that, by construction, \textsc{Scratch} perfectly unlearns the clients from a request $W_r$. Therefore, we consider an unlearning scheme successful if it reaches similar accuracy of \textsc{Scratch} with less aggregation rounds, when tested on the data samples of $F_r$.
\subsection{Experimental Results}
\label{subsec:experimental_results}
\begin{figure}
\begin{centering}
\includegraphics[width=\linewidth]{./plots/fig_SIFU_small_backdoored.pdf}
\end{centering}
\caption{Total amount of aggregation rounds (1\textsuperscript{st} row) and model accuracy of unlearned clients (2\textsuperscript{nd} row) for the unlearning of watermarked data from CIFAR100 and CelebA.
}
\label{fig:SIFU_small_backdoored}
\end{figure}
Figure \ref{fig:SIFU} shows that for every dataset and unlearning index, \textsc{Fine-Tuning}, \textsc{FedAccum}, and DP provide similar model accuracy for the unlearned clients in $F_r$ (Figure \ref{fig:SIFU}-2\textsuperscript{nd} row), albeit significantly higher than for \textsc{Scratch}, the unlearning standard. Noteworthy, unlearning with \textsc{Fine-Tuning}, \textsc{FedAccum}, and DP results in significantly less aggregation rounds than \textsc{Scratch} (Figure \ref{fig:SIFU}-1\textsuperscript{st} row).
We note that SIFU and \textsc{Scratch} lead to similar unlearning results, quantified by low accuracy on the unlearned clients $F_r$ (Figure \ref{fig:SIFU}-2\textsuperscript{nd} row), while SIFU unlearns these clients in roughly half the amount of aggregation rounds needed for \textsc{Scratch} (Figure \ref{fig:SIFU}-1\textsuperscript{st} row).
However, the model accuracy of SIFU is slightly higher than the one of \textsc{Scratch}, with perfect overlap only for FashionMNIST. This behavior is natural and can be explained by our privacy budget $(\epsilon, \delta)$, which trades unlearning capabilities for effectiveness of the retraining procedure. With highest unlearning budget, i.e. $\epsilon =0$ and $\delta=0$, SIFU would require to retrain from the initial model ${\bm{\theta}}_0^0$, thus reducing to \textsc{Scratch}.
Finally, we observed that when unlearning with \textsc{Last}, the retrained model always converged to a local optimum with accuracy inferior to our target after $10000$ aggregation rounds. This behavior is likely due to the difficulty of calibrating the noise perturbation due to the numerous heterogeneous contributions of the clients. For this reason, we decided to exclude \textsc{Last} from the plots of Figure \ref{fig:SIFU}.
\subsection{Verifying Unlearning through Watermarking}
\label{subsec:exp_watermark}
The work of \cite{TowardsProbabilisticVerification} proposes an adversarial approach to verify the efficiency of an unlearning scheme based on watermarking.
We apply here this method to our federated setting, in which watermarking is operated by each client by randomly assigning on all its data samples the maximum possible value to 10 given pixels.
To ensure that clients' data heterogeneity is only due to the modification of the pixels, we define heterogeneous data partitioning across clients by randomly assigning the data according to a Dirichlet distribution with parameter 1. Figure \ref{fig:SIFU_small_backdoored} shows our results for this experimental scenario on CIFAR100 and CelebA, while Appendix \ref{app:sec:experiments} provides similar results for MNIST, FashionMNIST and CIFAR10. We retrieve the same conclusions drawn from Figure \ref{fig:SIFU}. SIFU and \textsc{Scratch} have similar accuracies on the unlearned clients in $F_r$, to demonstrate the effectiveness of the unlearning. Moreover, SIFU unlearns these clients in significantly less aggregation rounds than \textsc{Scratch}.
\subsection{Impact of the noise perturbation on SIFU}
\label{subsec:exp_nosie_std}
Appendix \ref{app:sec:experiments} illustrates the impact of the perturbation amplitude $\sigma$ on convergence speed when unlearning with SIFU. We note that when unlearning with a small $\sigma$, SIFU has identical behavior to \textsc{Scratch} as the unlearning is applied to the initial random model ${\bm{\theta}}_0^0$. With large values of $\sigma$, SIFU performs instead identically to \textsc{Last} and applies the unlearning to the finale global model ${\bm{\theta}}_r^{N_r}$.
\section{Sequential FU with SIFU}
\label{sec:SIFU}
\input{./tex/SIFU_algo}
\input{./tex/example_SIFU}
In this section, we extend IFU to the sequential unlearning setting with Sequential IFU (SIFU). With Algorithm \ref{alg:SIFU}, SIFU is designed to satisfy a series of $R$ unlearning requests $\{W_r\}_{r=1}^R$, where $W_r$ is the set of clients to unlearn at request index $r$.
SIFU generalizes IFU for which $R=1$ and $W_1 = \{c\}$. We provide an illustration of SIFU with an example in Figure \ref{fig:example_with_R3}.
The notations introduced thus far need to be generalized to account for our series of unlearning requests $W_1, W_2, \ldots, W_R$. Global models are now referenced by their coordinates $(r, n)$, i.e. ${\bm{\theta}}_r^n$, which represent the unlearning request index $r$ and the amount of server aggregations $n$ performed during the retraining. Hence, ${\bm{\theta}}_r^0$ is the initialization of the model when unlearning the clients in $W_r$.
Also, we consider that the retraining at request index $r$ requires $N_r$ server aggregations on the remaining clients.
Therefore, by construction, ${\bm{\theta}}_r^{N_r}$ is the model obtained after using SIFU to $(\epsilon, \delta)$-unlearn the sequence of unlearning requests $\{W_s\}_{s=1}^r$. Finally, we define $I_r$ as the set of remaining clients after unlearning request $r$, i.e. $I_r \coloneqq I\setminus \cup_{s=1}^r W_s = I_{r-1} \setminus W_r$ with $I_0 = I$.
We extend the bounded sensitivity (\ref{eq:def_Psi}) with $\Psi_r(n, i)$ to compute the metric of client $i$ at unlearning index $r$ with
\begin{equation}
\Psi_r(n, i)
\coloneqq \sum_{s=0}^{n-1}\norm{\Delta(I_r, {\bm{\theta}}_r^s) - \Delta(I_r\setminus\{i\}, {\bm{\theta}}_r^s)}_2
\label{eq:def_Psi_extended}
.
\end{equation}
When unlearning client $c$ at $r=1$, the metric at $r=0$ is equivalent to the previous definition of $\Psi$. Also, when computing the metric on a client already unlearned, i.e. $i \notin I_r$, we retrieve $\Psi_r(n, i) = 0$.
Finally, for a set of clients $S$, we generalize the bounded sensitivity (\ref{eq:def_Psi_extended}) to
\begin{equation}
\Psi_r(n, S)
= \max_{i \in S}\Psi_r(n, i)
\label{eq:def_Psi_extended_set}
.
\end{equation}
With SIFU, the selection of the unlearning index $T$ for a request $r$ depends of the past history of unlearning requests.
To keep track of the unlearning history, we introduce the oracle $O(r)$ which returns at each request $r$ the coordinates of the history of global models where unlearning has been applied. These coordinates represent the nodes of the training history across unlearning requests (Figure \ref{fig:example_with_R3}). With reference to Figure \ref{fig:example_with_R3}, we start with the original sequence of global models obtained at each FL round, i.e. (${\bm{\theta}}_0^0, \ldots, {\bm{\theta}}_0^{N_0})$. Similarly to IFU, the first unlearning request requires to identify the unlearning index $T_1$ for which the corresponding global model ${\bm{\theta}}_0^{T_1}$ must be perturbed to obtain ${\bm{\theta}}_1^0$ and retrained until convergence, i.e. up to ${\bm{\theta}}_1^{N_1}$. The oracle is updated with the coordinates of the branching $O(1) = \{(0, T_1)\}$, and the current training history is now $({\bm{\theta}}_0^0, \ldots, {\bm{\theta}}_0^{T_1}, {\bm{\theta}}_1^0, \ldots, {\bm{\theta}}_1^{N_1})$. At the next unlearning request, the server needs to identify the coordinates $(\zeta_r, T_r)$ in the new training history for which unlearning must be applied on the model ${\bm{\theta}}_{\zeta_r}^{T_r}$ to obtain ${\bm{\theta}}_r^0 = {\bm{\theta}}_{\zeta_r}^{T_r} + \Ncal(0, \sigma^2{\bm{I}}_{\bm{\theta}})$. The oracle is subsequently updated with the new set of nodes describing the new branching in the training history.
By construction, we have $\zeta_r \le r-1$ and $T_r \le N_{\zeta_r}$.
More precisely, we define the index $\zeta_r$ associated to the first coordinate in $O(r-1)$ for which the bounded sensitivity (\ref{eq:def_Psi_extended}) of clients in $W_r$ exceeds $\Psi^*$.
Formally, we have
\begin{align}
\zeta_r
\coloneqq \min_s \{ &s : \Psi_s(n, W_r) > \Psi^* \text{ and } (s, n) \in O(r-1), \nonumber\\
&r-1 \}
.
\label{eq:zeta_r}
\end{align}
The definition of $T_r$ follows directly from the one of $\zeta_r$. Similarly as for IFU, the unlearning index $T_r$ quantifies the maximum amount of server aggregations starting from the unlearning request index $\zeta_r$ such that the bounded sensitivity $\Psi_{\zeta_r}(n, W_r)$ on this global model is inferior to $\Psi^*$, i.e.
\begin{equation}
T_r
\coloneqq \argmax_n \{\Psi_{\zeta_r}(n, W_r) \le \Psi^* \}
.\label{eq:T_SIFU}
\end{equation}
Finally, we update the oracle $O(r-1)$ to $O(r)$ with the following recurrent equation
\begin{equation}
O(r) = \{(s, n) \in O(r-1) \text{ s.t. } s < \zeta_r, (\zeta_r, T_r) \}
\label{eq:oracle_recurrent}
.
\end{equation}
Theorem \ref{theo:zeta} shows that for a model trained with SIFU after a given training request $r$, $(\epsilon, \delta)$-unlearning is guaranteed for every client belonging to the sets $W_s$, $s\leq r$.
\begin{theorem}
\label{theo:zeta}
The model ${\bm{\theta}}_r^{N_r}$ obtained with SIFU satisfies $(\epsilon, \delta)$-unlearning for every client in current and previous unlearning requests, i.e. clients in $\cup_{s=1}^r W_s$.
\end{theorem}
\begin{proof}
See Appendix \ref{app:sec:SIFU_convergence}.
\end{proof}
\subsection{A more general case}
Let us consider that clients loss functions are $\mu$ strongly convex. We denote by ${\bm{\theta}}^*$ and ${\bm{\theta}}_{-k}^*$ respectively the optimum of the loss function with all the clients and all but client $k$. We can obtain the following convergence guarantees
\begin{equation}
\norm{{\bm{\theta}}^N - {\bm{\theta}}^*}^2
\le ( 1 - \eta \mu)^N \norm{{\bm{\theta}}^0 - {\bm{\theta}}^*}^2
\end{equation}
As such, we have
\begin{equation}
\norm{{\bm{\theta}}_{-k}^N - {\bm{\theta}}^N}
\le
\norm{{\bm{\theta}}_{-k}^N - {\bm{\theta}}_{-k}^*}
+ \norm{{\bm{\theta}}^N - {\bm{\theta}}^*}
+ \norm{{\bm{\theta}}_{-k}^* - {\bm{\theta}}^*}
\end{equation}
which gives
\begin{align}
\norm{{\bm{\theta}}_{-k}^N - {\bm{\theta}}^N}
&\le
\sqrt{2} ( 1 - \eta \mu)^{N/2}
\max(\norm{{\bm{\theta}}^0 - {\bm{\theta}}^*}^2, \norm{{\bm{\theta}}^0 - {\bm{\theta}}_{-k}^*}^2)
\nonumber\\
&+ \norm{{\bm{\theta}}_{-k}^* - {\bm{\theta}}^*}
\end{align}
While this result holds for more functions the main issue lies in the provided bound. Indeed, it depends of the new optimum which we do not know.
\section{Background and Related Work}
\label{sec:background}
In Section \ref{subsec:unlearning_baselines}, we introduce the state-of-the art behind Machine Unlearning, while in Section \ref{subsec:FedAvg}, we introduce FL and \textsc{FedAvg}.
Finally, we introduce Federated Unlearning (FU) in Section \ref{subsec:federated_unlearning}.
\subsection{Machine Unlearning}
\label{subsec:unlearning_baselines}
\input{./tex/unlearning_baselines}
\subsection{Federated Optimization and \textsc{FedAvg}}
\label{subsec:FedAvg}
\input{./tex/formalization}
\subsection{Federated Unlearning}
\label{subsec:federated_unlearning}
\input{./tex/background_federated_unlearning}
\section{Unlearning a FL client with IFU}
\label{sec:theory}
In this section, we develop our theory for the scenario where a model is trained with \textsc{FedAvg} on the set of clients $I$, after which a client $c$ requests unlearning of its own data.
In Section \ref{subsec:bounding_drift}, we define the sensitivity of the global model with respect to a client's contribution, and provide a bound relating this sensitivity to the FL procedure.
In Section \ref{subsec:tighter_sensitivity}, we provide a tighter model sensitivity for some specific FL applications.
Using Theorem \ref{theo:diff_bound}, we introduce in Section \ref{subsec:unlearning_guarantees} the perturbation procedure to unlearn a client $c$ from the model trained with \textsc{FedAvg} (Theorem \ref{theo:noise_DP}).
Finally, using Theorem \ref{theo:noise_DP}, we introduce Informed Federated Unlearning (IFU) (Algorithm \ref{alg:unlearning_ours}).
\subsection{Theorem \ref{theo:diff_bound}, Bounding the Model Sensitivity}
\label{subsec:bounding_drift}
\input{./tex/diff_grad}
\subsection{Improving the Tightness of the Sensitivity Bound}
\label{subsec:tighter_sensitivity}
\input{./tex/corollary}
\subsection{Satisfying Definition \ref{def:DP_adpated}}
\label{subsec:unlearning_guarantees}
\input{./tex/unlearning_client}
\subsection{Informed Federated Unlearning (IFU)}
\label{subsec:unlearning_ours}
\input{./tex/unlearning_procedure}
\input{./tex/generalization}
\subsection{Influence of the initial model on the bias}
Let us consider $X$ the feature matrix and $Y$ the ground truth with loss function
\begin{equation}
f({\bm{\theta}}) = \frac{1}{N} \frac{1}{2} \left[{\bm{Y}} - {\bm{X}} {\bm{\theta}} \right]^T \left[{\bm{Y}} - {\bm{X}} {\bm{\theta}} \right]
\end{equation}
We assume that there are more features than data samples. Therefore, $X^TX$ is singular and all the model ${\bm{\theta}}^*$ satisfying
\begin{equation}
{\bm{X}}^T {\bm{X}} {\bm{\theta}}^* = {\bm{X}}^T {\bm{Y}}
\end{equation}
are optimal model. $f$ is non convex.
The SGD process can be rewritten such that
\begin{equation}
{\bm{\theta}}^N
= \underbrace{\left[I - \eta {\bm{X}}^T {\bm{X}}\right]^N}_{A({\bm{X}}, N)} {\bm{\theta}}^0
+ \underbrace{\eta \sum_{n =0}^{N-1} \left[I - \eta {\bm{X}}^T {\bm{X}}\right]^n {\bm{X}}^T {\bm{Y}}}_{B({\bm{X}}, {\bm{Y}}, N)}
\end{equation}
We consider a learning rate small enough such that the highest eigen value of ${\bm{X}}^T {\bm{X}}$ is strictly inferior to 1. We also consider that its smallest eigen value is non-negative. Therefore, $I - \eta {\bm{X}}^T {\bm{X}} $ is positive definite. We get
\begin{equation}
{\bm{\theta}}^N
= \underbrace{\left[I - \eta {\bm{X}}^T {\bm{X}}\right]^N}_{A({\bm{X}}, N)} {\bm{\theta}}^0
+ \underbrace{\eta \sum_{n =0}^{N-1} \left[I - \eta {\bm{X}}^T {\bm{X}}\right]^n {\bm{X}}^T {\bm{Y}}}_{B({\bm{X}}, {\bm{Y}}, N)}
\end{equation}
The second term is independent from the initial model and only depends of the amount of SGD $N$. When ${\bm{X}}^T{\bm{X}}$ i non singular, the first term converges to 0.
We differentiate now 2 flows. One with all the samples and another one with one removed sample.
\begin{equation}
{\bm{\theta}}_2^N
= A({\bm{X}}_2, N) {\bm{\theta}}^0
+ B({\bm{X}}_2, {\bm{Y}}_2, N)
\end{equation}
If unlearning with $\tilde{N}$ SGD, we have
\begin{align}
\tilde{{\bm{\theta}}}_2^{\tilde{N}}
&= A({\bm{X}}_2, \tilde{N}) {\bm{\theta}}_1^N
+ B({\bm{X}}_2, {\bm{Y}}_2, \tilde{N})
\\
&= A({\bm{X}}_2, \tilde{N}) \left[A({\bm{X}}_1, N) {\bm{\theta}}^0
+ B({\bm{X}}_1, {\bm{Y}}_1, N)\right]
+ B({\bm{X}}_2, {\bm{Y}}_2, \tilde{N})
\\
&= A({\bm{X}}_2, \tilde{N}) A({\bm{X}}_1, N) {\bm{\theta}}^0
+ A({\bm{X}}_2, \tilde{N}) B({\bm{X}}_1, {\bm{Y}}_1, N)
+ B({\bm{X}}_2, {\bm{Y}}_2, \tilde{N})
\end{align}
We always have
\begin{equation}
\lim_{\tilde{N} \to \infty}
B({\bm{X}}_2, {\bm{Y}}_2, \tilde{N})
=
\lim_{N \to \infty}
B({\bm{X}}_2, {\bm{Y}}_2, N)
\end{equation}
and if ${\bm{X}}_2^T {\bm{X}}_2$ is non singular, we retrieve
\begin{equation}
\lim_{\tilde{N} \to \infty}
A({\bm{X}}_2, \tilde{N})
=0
\end{equation}
which gives
\begin{align}
\lim_{\tilde{N} \to \infty}
\tilde{{\bm{\theta}}}_2^{\tilde{N}}
=
\lim_{N \to \infty}
{\bm{\theta}}_2^N
\end{align}
In practice, in Machine Learning, the initial model ${\bm{\theta}}_0$ is obtained with a normal distribution. Setting the initial model at $0$ enables to cancel the first step, still, the second term cannot be mitigated by setting the initial model to a certain value and depends of the optimum of the previous optimum model.
\subsection{Theoretical Work}
$\EE{S}{X}$ is the expected value of $X$ over $S$ the random set of sampled clients.
If we consider the Lipschitz smoothness of $f$, we get
\begin{equation}
\EE{S}{f(x^{t+1}) - f(x^{t})}
\le - \gamma \EE{S}{\inner{\nabla f(x^t)}{g(x^t)}}
+ \frac{L}{2}\gamma \EE{S}{\norm{g_t}^2}
\label{app:eq:A0}
\end{equation}
\subsubsection*{1st term}
We define $\alpha(n) \coloneqq a_n p_n - a_{n+1} p_{n+1}$ which gives
\begin{equation}
\EE{S}{g(x^t)}
= \nabla f ( x^t)
+ \frac{1}{n-1} \frac{1}{m} \alpha(n) \nabla l_n(x^t)
,
\end{equation}
and
\begin{equation}
\inner{\nabla f(x^t)}{\EE{S}{g(x^t)}}
= \norm{\nabla f ( x^t)}^2
+ \frac{1}{n-1}\frac{1}{m} \alpha(n)
\inner{\nabla f(x^t)}{\nabla l_n(x^t)}
\label{app:eq:A1}
,
\end{equation}
\subsubsection*{2nd term}
Considering
\begin{align}
\mathbb{E}{\omega_i^2}
&
= \frac{1}{(n-1)^2} \left[\frac{m-1}{m} + \frac{1}{mp_i}\right]
\\
\mathbb{E}{\omega_i\omega_j}
&
= \frac{1}{(n-1)^2} \frac{m- 1}{m}
\\
\mathbb{E}{\left(\omega_n + \omega_{n+1}\right)^2 }
&= \mathbb{E}{\omega_n^2}
+ \mathbb{E}{ \omega_{n+1}^2}
+ 2 \mathbb{E}{ \omega_n \omega_{n+1} }
\\
&= \frac{1}{(n-1)^2}
\left[\frac{m-1}{m}\alpha^2(n)
+ \frac{1}{m}a_n^2 p_n
+ \frac{1}{m}a_{n+1}^2 p_{n+1}\right]
\eqqcolon \beta(n)
,
\end{align}
we get
\begin{align}
\EE{S}{\norm{g_t}^2}
& = \sum_{i =1}^{n-1} \sum_{ j = 1 }^{ n - 1 } \mathbb{E}{\omega_i \omega_j} \nabla l_i(x^t)^T \nabla l_j(x^t)
+ \mathbb{E}{(\omega_n + \omega_{n+1})^2} \norm{\nabla l_n(x^t)}^2
\nonumber\\
& + 2 \sum_{i =1}^{n-1} \mathbb{E}{\omega_i (\omega_n + \omega_{n+1})} \nabla l_i(x^t)^T \nabla l_n(x^t)
\\
& = \frac{1}{(n-1)^2}\frac{1}{m}\sum_{i =1}^{n-1} \frac{1}{p_i} \norm{\nabla l_i(x^t)}^2
+ \frac{m-1}{m} \norm{\nabla f(x^t)}^2
+ \beta(n) \norm{\nabla l_n(x^t)}^2
\nonumber\\
& + 2 \frac{1}{(n-1)^2}\frac{m-1}{m}\alpha(n)\nabla f(x^t)^T \nabla l_n(x^t)
\label{app:eq:A2}
,
\end{align}
\subsubsection*{Rewrite inner product}
We can rewrite the gradient of the previous loss function as
\begin{equation}
\nabla \Lcal_1(x^t)
= \frac{n-1}{n}\nabla f(x^t)
+ \frac{1}{n}\nabla l_n(x^t)
,
\end{equation}
which gives
\begin{align}
\inner{\nabla f(x^t)}{\nabla l_n(x^t)}
& = \frac{n^2}{n-1} \inner{\frac{n-1}{n}\nabla f(x^t)}{\frac{1}{n}\nabla l_n(x^t)}
\\
& = \frac{n^2}{n-1} \frac{1}{2} \left[ \norm{\nabla \Lcal_1(x^t)}^2
-\frac{(n-1)^2}{n^2}\norm{\nabla f(x^t)}^2
-\frac{1}{n^2}\norm{\nabla l_n(x^t)}^2
\right]
\end{align}
\subsubsection*{Merge}
Substituting equation (\ref{app:eq:A1}) and (\ref{app:eq:A2}) in (\ref{app:eq:A0}) gives
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[1 - \frac{L}{2}\gamma \frac{m-1}{m}\right] \norm{\nabla f(x^t)}^2
\nonumber\\
&- \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{1}{n-1}\frac{1}{m}\alpha(n) \nabla f(x^t)^T \nabla l_n(x^t)
\nonumber\\
&+ \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \sum_{i =1}^{n-1} \frac{1}{p_i}\norm{\nabla l_i(x^t)}^2
+ \frac{L}{2}\gamma \beta(n) \norm{\nabla l_n(x^t)}^2
.
\label{app:eq:B1}
\end{align}
We note at this point that the only influence of $p_i$ lies in $\sum_{i =1}^{n-1}\frac{1}{p_i}\norm{\nabla l_i(x^t)}^2$ which for all $p_n$ and $p_{n+1}$ is minimized when
\begin{equation}
p_i = (1 - p_n - p_{n+1})\frac{\norm{\nabla l_i(\theta)}}{\sum_{i=1}^{n-1}\norm{\nabla l_i(\theta)}}
.
\end{equation}
Therefore equation (\ref{app:eq:B1}) cn be simplified as
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[1 - \frac{L}{2}\gamma \frac{m-1}{m}\right] \norm{\nabla f(x^t)}^2
\nonumber\\
&- \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{1}{n-1}\frac{1}{m}\alpha(n) \nabla f(x^t)^T \nabla l_n(x^t)
\nonumber\\
&+ \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \frac{1}{1 - p_n - p_{n+1}} \left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x^t)}\right]^2
+ \frac{L}{2}\gamma \beta(n) \norm{\nabla l_n(x^t)}^2
.
\label{app:eq:B2}
\end{align}
In the rest of this work, we assume that $1 - L\gamma \frac{m-1}{m}>0$.
For the biased gradient estimator to provide a faster convergence process, the deleted sample needs to be considered for descent or ascent such that $\alpha(n) \nabla f(x^t)^T \nabla l_n(x^t) \ge 0 $ is satisfied which can be rewritten as
\begin{align}
&\alpha(n) \inner{ \nabla f(x^t)^T }{\nabla l_n(x^t) }
\nonumber\\
&\alpha(n) \frac{n^2}{n-1}\inner{\frac{n-1}{n}\nabla f(x^t)^T }{\frac{1}{n}\nabla l_n(x^t) }
\\
&=
\alpha(n) \frac{n^2}{n-1}
\left[ \norm{\nabla \Lcal_1(x^t)}^2 - \left[\frac{n-1}{n}\right]^2\norm{\nabla f(x^t)}^2 - \frac{1}{n^2}\norm{\nabla l_n(x^t)}^2 \right]
\ge 0
\end{align}
We are interested in improving the unlearning process. Especially to leave faster the initial model $\theta_1^*$. for which, we have $\nabla \Lcal_1(\theta_1^*)= 0$. Therefore, regardless of whether a deleted sample is considered to obtain the biased gradient estimator, we have $p_n =0$. Therefore $\alpha(n) = - a_{n+1} p_{n+1}$ and $\beta(n) = \frac{1}{(n-1)^2}\left[\frac{m-1}{m}a_{n+1}^2 p_{n+1}^2 + \frac{1}{m}a_{n+1}^2 p_{n+1} \right]$. We obtain
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[\left[1 - \frac{L}{2}\gamma \frac{m-1}{m}\right] + \left[1 - L\gamma \frac{m-1}{m}\right] \frac{1}{m} a_{n+1}p_{n+1} \right] \norm{\nabla f(x^t)}^2
\nonumber\\
&- \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{n^2}{(n-1)^2}\frac{1}{m} a_{n+1}p_{n+1} \norm{\nabla \Lcal_1(x^t) }^2
\nonumber\\
&+ \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \frac{1}{1 - p_{n+1}} \left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x^t)}\right]^2
\nonumber\\
&+ \left[\frac{L}{2}\gamma \beta(n) + \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{1}{(n-1)^2}\frac{1}{m} a_{n+1}p_{n+1}\right]\norm{\nabla l_n(x^t)}^2
.
\label{app:eq:B3}
\end{align}
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[1 + \frac{1}{m} a_{n+1}p_{n+1} \right] \norm{\nabla f(x^t)}^2
- \frac{n^2}{(n-1)^2}\frac{1}{m} a_{n+1}p_{n+1} \norm{\nabla \Lcal_1(x^t) }^2
\nonumber\\
&+ \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \frac{1}{1 - p_{n+1}} \left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x^t)}\right]^2
\nonumber\\
&+ \left[\frac{L}{2}\gamma \beta(n) +
\frac{1}{(n-1)^2}\frac{1}{m} a_{n+1}p_{n+1}\right]\norm{\nabla l_n(x^t)}^2
.
\label{app:eq:B4}
\end{align}
We also define
\begin{equation}
A(t)
= \frac{1}{\gamma}\mathbb{E}{f(x^{t}) - f(x^{t+1})}
,\
B(t)
= \left[1 - L\gamma \frac{m-1}{m}\right]
\frac{1}{n-1}\frac{1}{m} \nabla f(x^t)^T \nabla l_n(x^t)
\end{equation}
\begin{equation}
C(t)
= \frac{L}{2} \gamma \frac{1}{(n-1)^2} \frac{1}{m} \left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x^t)}\right]^2
,\
D(t)
= \frac{L}{2}\gamma \frac{1}{(n-1)^2} \frac{m-1}{m}\norm{\nabla l_n(x^t)}^2
\end{equation}
\begin{equation}
E(t)
= \frac{L}{2}\gamma \frac{1}{(n-1)^2} \frac{1}{m}\norm{\nabla l_n(x^t)}^2
\end{equation}
which gives
\begin{align}
\left[1 - \frac{L}{2}\gamma \frac{m-1}{m}\right] \norm{\nabla f(x^t)}^2
&\le A(t) - B(t) a_{n+1} p_{n+1}
\nonumber\\
&+ \frac{1}{1 - p_{n+1}} C(t)
+ D(t) a_{n+1}^2 p_{n+1}^2
+ E(t) a_{n+1}^2 p_{n+1}
.
\label{app:eq:B3}
\end{align}
We consider the loss function where we replace $\frac{1}{1-p_{n+1}} = 1 + p_{n+1} + p_{n+1}^2$.
Let us consider
\begin{equation}
f(x^t)
= \left[ - B(t) a_{n+1}+ C(t) + E(t) a_{n+1}^2\right]p_{n+1}
+ \left[ C(t) + D(t) a_{n+1}^2\right] p_{n+1}^2
\end{equation}
We therefore have
\begin{equation}
content...
\end{equation}
\subsubsection*{Not Kept}
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \rho \left[1 - \frac{1}{2}\frac{1}{m}\alpha(n)\right]\norm{\nabla f(x^t)}^2
- \rho \alpha(n) \frac{1}{2} \frac{n^2}{(n-1)^2} \frac{1}{m}\norm{\nabla \Lcal_1(x^t)}^2
\nonumber\\
&+ \frac{L}{2}\gamma\frac{1}{m}\frac{1}{(n-1)^2} \sum_{i =1}^{n-1} \frac{1}{p_i}\norm{\nabla l_i(x^t)}^2
\nonumber\\
&+ \frac{1}{(n-1)^2}\left[\frac{1}{2}\frac{1}{m}\alpha(n)\rho + \frac{L}{2}\gamma \beta'(n) \right] \norm{\nabla l_n(x^t)}^2
\end{align}
We note that the importance $p_i$ given to each sample $i \le n$ can already be allocated such that
\begin{equation}
p_i = (1 - p_n - p_{n+1})\frac{\norm{\nabla l_i(\theta)}}{\sum_{i=1}^{n-1}\norm{\nabla l_i(\theta)}}
\end{equation}
Lastly, the bound is proportional to $p_n$ and $p_{n+1}$. Therefore, only $p_n$ or $p_{n+1}$ is positive.
If we consider that we are at the optimum, one gets
\begin{equation}
\nabla \Lcal_1(x_1^*)
= \frac{n-1}{n}\nabla f(x_1^*)
+ \frac{1}{n}\nabla l_n(x_1^*)
=0
,
\end{equation}
\begin{align}
\frac{1}{\gamma}\mathbb{E}{f(x^{t+1}) - f(x^{t})}
&\le - \left[ \rho \left[ 1
- \frac{1}{m}\alpha(n) \right]
- \frac{L}{2}\gamma \beta'(n)\right]
\norm{\nabla f(x_1^*)}^2
\nonumber\\
&+ \frac{L}{2}\gamma\frac{1}{m}\frac{1}{(n-1)^2} \frac{1}{1- p_n - p_{n+1}}\left[\sum_{i =1}^{n-1} \norm{\nabla l_i(x_1^*)}\right]^2
\end{align}
where the weights when $i\le n-1$ are assigned such that $\mathbb{E}{\omega_i} = \frac{1}{n-1}$ and $\mathbb{E}{\omega_n + \omega_{n+1}} = 0$. As such, we have $a_n p_n= a_{n+1} p_{n+1}$ which gives
\begin{align}
\VAR{\omega_i}
&
= \frac{1}{(n-1)^2}\frac{1}{mp_i}(1-p_i)
\\
\COV{\omega_i}{\omega_j}
&
= - \frac{1}{(n-1)^2} \frac{m- 1}{m}
\\
\mathbb{E}{\left(\omega_n + \omega_{n+1}\right)^2 }
&= \mathbb{E}{\omega_n^2}
+ \mathbb{E}{ \omega_{n+1}^2}
+ 2 \mathbb{E}{ \omega_n \omega_{n+1} }
\\
&= \frac{1}{m}a_n^2 \left[p_n^2 (m-1) + p_n\right]
+ \frac{1}{m}a_{n+1}^2 \left[p_{n+1}^2 (m-1) + p_{n+1}\right]
- 2\frac{m-1}{m}a_n a_{n+1}p_n p_{n+1}
\\
&= \frac{m-1}{m}(a_n p_n - a_{n+1} p_{n+1})^2
+ \frac{1}{m}a_n^2 p_n
+ \frac{1}{m}a_{n+1}^2 p_{n+1}
\\
&=
\frac{1}{m}a_n^2 p_n
+ \frac{1}{m}a_{n+1}^2 p_{n+1}
\\
\end{align}
In that case, we have
\begin{align}
\sigma(\theta)
&= \frac{1}{m(n-1)^2}\left[\sum_{i=1}^{n-1} \norm{\nabla l_i(\theta)}\right]^2
+ \frac{m-2}{m}\sum_{i=1}^{n-1} \norm{ \frac{1}{n-1}\nabla l_i(\theta) - \nabla \Lcal_2(\theta)}^2
- \frac{1}{m}\norm{\nabla \Lcal_2(\theta)}^2
,
\end{align}
More generally,
we have
\begin{equation}
p_i = ( 1 - p_n - p_{n+1})\frac{\norm{\nabla l_i(\theta)}}{\sum_{i=1}^{n-1}\norm{\nabla l_i(\theta)}}
\end{equation}
which gives
\begin{align}
\sigma(\theta)
&= \frac{1}{1 - p_n - p_{n+1}} \frac{1}{m(n-1)^2}\left[\sum_{i=1}^{n-1} \norm{\nabla l_i(\theta)}\right]^2
+ \frac{m-2}{m}\sum_{i=1}^{n-1} \norm{ \frac{1}{n-1}\nabla l_i(\theta) - \nabla \Lcal_2(\theta)}^2
\nonumber\\
&- \frac{1}{m}\norm{\nabla \Lcal_2(\theta)}^2
+ \beta(n) \norm{\nabla \Lcal_2(\theta) - \nabla \Lcal_1(\theta)}^2
,
\end{align}
Therefore we have
\begin{align}
&\sigma_2(\theta) - \sigma_1(\theta)
\nonumber\\
& = \left[\frac{1}{1 - p_n - p_{n+1}} - 1\right]\frac{1}{m(n-1)^2}\left[\sum_{i=1}^{n-1} \norm{\nabla l_i(\theta)}\right]^2
+ \beta(n) \norm{\nabla l_n(\theta)}^2
\end{align}
\section{Applications}\label{sec:applications}
\subsection{Centralized Learning}\label{subsec:centralized}
\input{./tex/application_centralized}
\subsection{DL and FL}\label{subsec:DL}
\input{./tex/application_FL}
\section{New Initial Model}
Let us consider $N_t$ the SGD budget for training and $N_m$ the SGD budget for adapting the model.
DP privacy relies on noising the last model such that for any combination of samples, a model cannot know whether a sample was used to train it or not.
We have an history of models $N_t$ models. How to obtain a new model forgetting a given data samples?
What kind of procedure can be proposed to train the learning model such that a sample is forgotten?
The method currently used is to retrain everything. Why ? Because the initial model has no information on the sample to forget. We propose in this work instead to have a better starting point than the initial model to obtain the new model when trying to forget some samples.
We propose a generalization of DP-Privacy. With DP, only one training is needed and a noise is added to prevent specific information recovery from one client. What should be the noise to ensure DP? Is DP equivalent to ensuring forgetting of a data sample?
What is the impact of the amount of samples to forget? e.g. forgetting a class.
We propose the following two scenarios which we compare theoretically and experimentally to the following two baselines
\begin{enumerate}
\item noising the trained model before performing the budget of SGD
\item noising the model obtained after $N_t - N_m$ SGD before performing the budget of SGD
\item noising the trained model with no retraining
\item retraining from scratch with the available SGD budget
\end{enumerate}
\section*{Acknowledgements and Disclosure of Funding}
\label{sec:ack}
This work has been supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002, and by the ANR JCJC project Fed-BioMed 19-CE45-0006-01. The project was also supported by Accenture.
The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.
\subsection{Intermediate results}
\begin{property}\label{app:prop:SIFU_increasing_t}
If there exists $\nu$, $s$, $u$ such that $s < u$, $(\nu, t_s) \in O(s)$ and $(\nu, t_u) \in O(u)$, then $t_s \ge t_u$.
\end{property}
\begin{proof}
We first assume that $s$ and $u$ satisfy $u = s + 1$.
Considering that $(\nu, t_s) \in O(s)$ and $(\nu, t_u) \in O(u)$, we have, by definition of $\zeta_u$ in equation (\ref{eq:zeta_r}), $\nu \le \zeta_u$.
\begin{itemize}
\item $\zeta_u>\nu$. Considering that $u = s + 1$, we have $t_s = t_u$, equation (\ref{eq:oracle_recurrent}).
\item $\zeta_u = \nu$. Considering that $(\nu, t_s) \in O(s)$ and $(\nu, t_u) \in O(u)$, then we have $\nu \le s-1$. Therefore, by definition of $\zeta_u$, we have $\Psi_{\zeta_u}(t_s, W_u) > \Psi^*$. By construction of $T_u$, equation (\ref{eq:T_SIFU}), we have $t_u = T_u < t_s$.
\end{itemize}
When considering the more general case where there exists an integer $k$ such that $u = s +k$ while $(\nu, t_s) \in O(s)$ and $(\nu, t_u) \in O(u)$, then it is sufficient to consider iteratively an integer $j$ ranging from 1 to $k$. Considering $(\nu, t_u) \in O(u)$, there exists $t_{s+j}$ such that $(\nu, t_{s+j}) \in O(s+j)$. In that case, using the same reasoning as for $k=1$, we have $t_s \le t_{s+1} \le \ldots \le t_{s+k-1} \le t_u$.
\end{proof}
\subsection{Proof of Theorem \ref{theo:zeta}}
\begin{proof}
Proving that ${\bm{\theta}}_r^{N_r}$ $(\epsilon, \delta)$-unlearns every client in $F_r$, equation (\ref{eq:F_r}), reduces to proving that ${\bm{\theta}}_r^0$ $(\epsilon, \delta)$-unlearns every client in $F_r$, equation (\ref{eq:F_r}). Indeed, the data of clients in $F_r$ are not used on the noised perturbed model ${\bm{\theta}}_r^0 = {\bm{\theta}}_{\zeta_r}^{T_r} + \Ncal(0, \sigma^2 {\bm{I}}_{\bm{\theta}})$.
We prove by induction that ${\bm{\theta}}_r^0$ $(\epsilon, \delta)$-unlearns every client in $F_r$, equation (\ref{eq:F_r}).
The initialization ($r=1$) directly follows from IFU, Algorithm \ref{alg:unlearning_ours}, with Theorem \ref{theo:noise_DP}. We now assume that for every $s$ such that $s\le r-1$, ${\bm{\theta}}_s^0$ $(\epsilon, \delta)$-unlearns every client in $F_s$ and prove that ${\bm{\theta}}_r^0$ $(\epsilon, \delta)$-unlearns every client in $F_r$.
\begin{itemize}
\item \underline{$s \le \zeta_r$}.
Using the induction property, ${\bm{\theta}}_{\zeta_r}^0$ $(\epsilon, \delta)$-unlearns every clients in $W_s$. Clients in $W_s$ are not used for training on ${\bm{\theta}}_{\zeta_r}^0$.
Hence, ${\bm{\theta}}_{\zeta_r}^{T_r}$ and ${\bm{\theta}}_r^0$ also $(\epsilon, \delta)$-unlearns every client in $W_s$.
\item \underline{$s = r$.}
By definition of $\zeta_r$, equation (\ref{eq:zeta_r}), the noise perturbations for every model in $O(r)$ is such that ${\bm{\theta}}_{\zeta_r}^0$ $(\epsilon, \delta)$-unlearns every client in $W_r$.
Hence, by definition of $T_r$ on the bounded sensitivity of clients in $W_r$ at unlearning request $\zeta_r$, equation (\ref{eq:T_SIFU}), the noised perturbed model ${\bm{\theta}}_r^0$ $(\epsilon, \delta)$-unlearns every client in $W_r$, Theorem \ref{theo:noise_DP}.
\item \underline{$\zeta_r < s \le r-1$.}
The successive update of the oracle, equation (\ref{eq:oracle_recurrent}), from $O(\zeta_r)$ to $O(s)$ gives, by construction, that there exists $t_s$ such that the coordinates $(\zeta_r, t_s)$ are in $O(s)$. Hence, by definition of $\zeta_s$, equation (\ref{eq:zeta_r}), we have $\zeta_s \ge \zeta_r$ and the successive noise perturbations to obtain ${\bm{\theta}}_{\zeta_r}^0$ $(\epsilon, \delta)$-unlearns every client in $W_s$.
Also, while we have the coordinates $(\zeta_r, t_s)$ in $O(s)$, we also have the coordinates $(\zeta_r, T_r)$ in $O(r)$, equation (\ref{eq:oracle_recurrent}). Therefore, using property \ref{app:prop:SIFU_increasing_t}, we have $t_s \ge T_r$. Hence, we have $\Psi_{\zeta_r}(T_r, W_s) \le \Psi^*$. Therefore, with the noise perturbation of SIFU, clients in $W_s$ are $(\epsilon, \delta)$-unlearned in ${\bm{\theta}}_r^0$.
\end{itemize}
\end{proof}
\section{Litterature Review}
\textbf{\cite{MUnlearningviaAlgorithmicStability}.}
This paper proposes a new MU algorithm for the centralized setting. The model is retrained on a fixed amount of SGD on the new dataset.
In more details, the authors propose a forgetting definition called $\rho$-TV-stability. This definition is very interesting and this work is built around it. However, their algorithm outperforms recomputation from scratch only if $\rho < 1$ and the authors show that their method has $\rho = \frac{mT}{n}$ where $m$ is the batch size, $n$ is the number of samples, and $T$ is the SGD budget. I find this point under discussed in this work and in my opinion voluntarily hidden by setting $m = \rho n/T$ which gives $m=0$ in most cases.
\textbf{\cite{DescentToDelete}.}
This work is theoretical and propose forgetting clients by adding a noise at the end of the retraining. This work uses a privacy criteria based on DP and interestingly provides an upper bound for the probability of the resulting model to be in the neighborhood of the new optimum.
Authors also propose to train different models on slices of the split dataset before aggregating the results. However, it requires slices to have iid data distributions to prevent convergence to a sub-optimum point \cite{FedNova}. Also this method requires a lot of computation to find the appropriate dataset split.
\textbf{\cite{FederatedUnlearning}.}
This works considers MU for the federated setting but only provides a new MU algorithm and experimental results.
The method is based on reusing the clients update of FedAvg. Yet, no theoretical justification nor any intuition behind why to use their method is provided.
They verify the correctness of their method on the training loss and testing accuracy of the resulting model. However, how well a client is forgotten is not considered ?
\textbf{\cite{MachineUnlearning}.}
This work considers splitting the dataset into subset and training a model per subset. The resulting model is the aggregation of all the trained models on each subset. This can be seen as FL where $N=1$ and $K\gg 1$. Therefore for this method to give the true optimum, we need all the shards to have the same data distribution which is not possible in practice even for the iid case. This work is only experimental and does not consider this point. The experimental setting only considers the testing accuracy of their method without considering if a client has been forgotten or not.
An interesting point is to consider that every sample has not the same probability to be forgotten. As such, putting them in the same shard could be useful. While it does not solve the problems considered before, associating a forgetting probability per sampling is interesting.
\textbf{\cite{TowardsProbabilisticVerification}.}
Very interesting work proposing a framework to verify that the samples of a given owner were removed from a given model. The framework is based on an attack scheme. The attacker modifies part of its samples and add to it a marker. Through the predictions returned for the true et modified inputs, the server is able to infer if the samples of an owner were forgotten or not.
\textbf{\cite{MU_LinearFiltration}.}
Experimental work that introduce a linear filtration to remove a class from a predictive model. The authors consider models that can be decomposed as a logistic regression and a feature extraction function. The focus of this work is for a model to forget one of its class.
\textbf{\cite{ApproximateDataDeletion}.}
This work investigates MU for linear regression. The method proposed is based on the introduction of synthetic points for the deleted points. The method proposed by the authors is linear with the dimension of the model contrarily to current quadratic complexity.
Authors also verify their work on \cite{TowardsProbabilisticVerification} and not just on testing and training accuracy.
This work is not based on SGD approaches. These methods gives directly the close form of the new model.
\textbf{\cite{VariationalBayesianUnlearning}.}
This work considers the problem of unlearning a model trained with a variational Bayesian approach. They propose a new lower bound to obtain the resulting posterior when forgetting some samples. Experiments compare the posterior distance of the new posterior and the one obtained with retraining which enables to measure the forgetting guarantees of their method.
\textbf{\cite{CertifiedDataRemoval}.}
DP is used to ensure unlearning like in \cite{DescentToDelete}. The authors theoretical work assume bounded gradients and Lipschitz second derivative for samples loss function.
Authors assume that the gradient of the trained model contains too much information about the dataset used. They thus consider adding a linear perturbation to change the optimum and thus reduce the data information in the final model.
The authors measure the effectiveness of their method on testing accuracy and norm of the gradient.
\textbf{\cite{Cao2015TowardsMS}.}
This paper introduces 4 MU algorithms (See Table 1).
However no theoretical guarantees are given for any of them. Correctness of unlearning algorithms are measures by comparison of the unlearned model true and false positive with respect to the ones when retraining from scratch.
\textbf{\cite{Golatkar_2020_CVPR}.}
Definition of forgetting data based on KL divergence.
This paper has similar elements to \cite{VariationalBayesianUnlearning}.
Authors consider the problem of forgetting while keeping at heart a low training loss by proposing the forgetting Lagrangian. A loss function composed of the new loss function with the remaining samples regularized with the KL divergence between the current model posterior and the theoretical one.
The formalization of the problem is very interesting.
\textbf{\cite{MakingAIForgetYou}.}
This work proposed two MU algorithms for the problem of k-means. No theoretical guarantees are given for any of these algorithms. Experiments are conducted in term of standard metric and does not able to know if a client has been forgotten or not. Major speed-up has to be noticed for the proposed algorithms.
\textbf{\cite{WhenMachineUnlearning}.} This work considers the amount of information that can be recovered by an attacker by using the trained and modified models.
\cite{Golatkar_2021_CVPR}
\cite{golatkar2020forgetting}
\cite{shibata2021learning}
\cite{gong2021bayesian}
\cite{wang2021federated}
\cite{DeltaGrad}
\cite{WhenMachineUnlearning}
\cite{AdaptiveMachineUnlearning}
\cite{CertifiableMchineUnlearning}
\cite{FedEraser}
\cite{zeroshotMU}
\section{Noise data point to forget}
\section{Formalizing the Unlearning problem}
\subsection{Optimization}
We consider a dataset $\Dcal$ composed of $|\Dcal| = N$ data samples $({\bm{x}}_i, {\bm{y}}_i)$ where ${\bm{x}}_i$ is the feature vector of a data sample and ${\bm{y}}_i$ its prediction or label.
For unsupervised learning, it suffices to consider that data samples in $\Dcal$ are only made with ${\bm{x}}_i$. We denote by $\Scal$ the set of $S = |\Scal|$ indices we want to forget from $\Dcal$ and $\Dcal_{-\Scal} = \Dcal\backslash\{({\bm{x}}_i, {\bm{y}}_i) | i \in \Scal \}$ the dataset without the forgotten data samples.
We consider a function $f$ depending on one data sample and model parameters ${\bm{\theta}}$.
For example, $f$ can be the loss of one data sample or its activation vector. We define the associated normalized functions
\begin{equation}
F({\bm{\theta}}) \coloneqq \frac{1}{|\Dcal|} \sum_{i \in \Dcal} f({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
\text{ and }
F_{-\Scal}({\bm{\theta}}) \coloneqq \frac{1}{|\Dcal_{-\Scal}|} \sum_{i \in \Dcal_{-\Scal}} f({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
.
\end{equation}
We consider $F$ is a $K$ dimensional vector where $K=1$ if $f$ is the loss function of one data samples and $K$ is the amount of classes in a classification problem.
For ease of writing, we consider $F_{-i}$ to be the resulting function when forgetting data sample $i$.
\subsection{$(\epsilon, \delta)$-indistinguishable}
We first define $(\epsilon, \delta)$-indistinguishable in Definition \ref{def:DP_forget}.
\begin{definition}[$(\epsilon, \delta)$-indistinguishable] Let $\epsilon$ and $\delta$ be two positive real numbers. Two distributions $X$ and $Y$ are $(\epsilon, \delta)$-indistinguishable if for all subset $A \subseteq \Acal$, we have
\label{def:DP_forget}
\begin{equation}
\mathbb{P}(X \in A)
\le e^\epsilon \mathbb{P}(Y \in A) + \delta
\text{ and }
\mathbb{P}(Y \in A)
\le e^\epsilon \mathbb{P}(X \in A) + \delta
.
\end{equation}
\end{definition}
${\bm{\theta}}$ is the $d$ dimensional vector of the trained model parameters. We consider a stochastic perturbation ${\bm{\phi}} \sim N( {\bm{\mu}}, {\bm{\Sigma}})$, where ${\bm{\Sigma}}$ is positive definite and can thus be expressed as ${\bm{\Sigma}} = {\bm{S}}^T {\bm{S}}$, such that $F({\bm{\theta}} + {\bm{\phi}})$ and $F_{-\Scal}({\bm{\theta}} + {\bm{\phi}})$ are $(\epsilon, \delta)$-indistinguishable.
Definition \ref{def:DP_forget} is linked to differential privacy(DP). With DP, we are looking at the noise ${\bm{\phi}}$ such that for any sample $i$, $F({\bm{\theta}} + {\bm{\phi}})$ and $F_{-i}({\bm{\theta}} + {\bm{\phi}})$
are $(\epsilon, \delta)$-indistinguishable. In that case, no information specific to any data sample can be retrieve from $\hat{{\bm{\theta}}} \coloneqq {\bm{\theta}} + {\bm{\phi}}$. With Machine Unlearning, we are instead looking at ${\bm{\phi}}$ such that $\hat{{\bm{\theta}}}$ does not contain specific information from any forgotten data sample. As such, DP implies Unlearning but Unlearning is weaker than DP.
\begin{definition}[$(\epsilon, \delta, \Scal)$-unlearning]
If $F({\bm{\theta}} + {\bm{\phi}})$ and $F_{-\Scal}({\bm{\theta}} + {\bm{\phi}})$ are $(\epsilon, \delta)$-indistinguishable, $\hat{{\bm{\theta}}} \coloneqq {\bm{\theta}}+ {\bm{\phi}}$ contains no information from $\Scal$.
\end{definition}
\section{Complete Unlearning}
We assume that $f$ satisfies Assumption \ref{ass:linear_theta}.
\begin{assumption}
\label{ass:linear_theta}
For any data point $({\bm{x}}, {\bm{y}})$ and model with parameters ${\bm{\theta}}$,
we consider that $f$ can be linearly approximated in ${\bm{\theta}}$ for a perturbation ${\bm{\epsilon}}$, i.e.
\begin{equation}
f({\bm{x}}, {\bm{y}}, {\bm{\theta}} + \epsilon)
= f({\bm{x}}, {\bm{y}}, {\bm{\theta}}) + {\bm{J}}_{{\bm{\theta}}}({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
where ${\bm{J}}_{{\bm{\theta}}}$ is the Jacobian matrix of $f$ with respect to ${\bm{\theta}}$.
\end{assumption}
With Assumption \ref{ass:linear_theta} and the stochastic perturbation ${\bm{\phi}}$, we have
\begin{align}
F({\bm{\theta}} + {\bm{\phi}})
= F({\bm{\theta}})
+ \frac{1}{N}\sum_{ i = 1 }^{N} {\bm{J}}_{\bm{\theta}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}) {\bm{\phi}}
= F({\bm{\theta}})
+ {\bm{J}}_{\bm{\theta}} ( \Dcal, {\bm{\theta}}) {\bm{\phi}}
,
\end{align}
where we define $J_{\bm{\theta}} ( \Dcal, {\bm{\theta}}) = \frac{1}{N}\sum_{ i = 1 }^{N} {\bm{J}}_{\bm{\theta}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})$
with a similar derivation for $F_{-\Scal}$. We remind that ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{\Sigma}} = {\bm{S}}^T {\bm{S}})$. Hence, we have
\begin{equation}
F({\bm{\theta}} + {\bm{\phi}} )
\sim \mathcal{N}( F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}){\bm{\mu}}, {\bm{J}}_{\bm{\theta}} ( \Dcal, {\bm{\theta}})^T{\bm{\Sigma}} {\bm{J}}_{\bm{\theta}}(\Dcal, {\bm{\theta}}))
,
\end{equation}
with a similar derivation for $F_{-\Scal}$. With a non zero expected perturbation ${\bm{\mu}}$, a perfect unlearning can be obtained with $F({\bm{\theta}} + {\bm{\phi}}) = F_{-\Scal}({\bm{\theta}} + {\bm{\phi}})$ by choosing ${\bm{\mu}}$ and ${\bm{\Sigma}}$ such that the probability density functions of these two queries are identical. Considering the dimensions of the Jacobian, numerous ${\bm{\mu}}$ can satisfy this condition, we are thus interested in the ones with the smallest norm where Assumption \ref{ass:linear_theta} is the most valid.
Hence, we have
\begin{alignat}{2}
&\min_{{\bm{\mu}}} &\qquad& \norm{{\bm{\mu}}}^2
\\
&\text{subject to} & & \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]{\bm{\mu}} = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}),
\nonumber\\
& & & \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]^T{\bm{\Sigma}} \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right] = 0 \label{eq:constraint_sigma}.
\nonumber
\end{alignat}
We define ${\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal}) = \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]$.
When this optimization problem is feasible, then its optimum ${\bm{\mu}}^*$ satisfies
\begin{equation}
{\bm{\mu}}^* = {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal}) \left[ {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal})^T {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}})\right].
\end{equation}
Any $\Sigma$ such that equation(\ref{eq:constraint_sigma}) is satisfied works including ${\bm{\Sigma}} =0$. Yet, considering a positive ${\bm{\Sigma}}$ to mitigate the linear approximation would be helpful. We note that when $K=1$, we have ${\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) = \nabla_{\bm{\theta}} F({\bm{\theta}})$ and can simplify the close form of ${\bm{\mu}}^*$ with
\begin{equation}
{\bm{\mu}}^* = \frac{\nabla_{\bm{\theta}} F({\bm{\theta}}) - \nabla_{\bm{\theta}} F_{-\Scal}({\bm{\theta}})}{\norm{\nabla_{\bm{\theta}} F({\bm{\theta}}) - \nabla_{\bm{\theta}} F_{-\Scal}({\bm{\theta}})}^2} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}})\right].
\end{equation}
\subsubsection{Interpretability using Local Perturbation}
Perturbating the model ${\bm{\theta}}$ with ${\bm{\phi}}$ impacts the performance of all the data samples. As such, even with Definition \ref{def:DP_forget} only based on the data points to forget, all the samples are impacted. For interpretability concerns, we consider a local perturbation ${\bm{\psi}}$ to the parameters to forget and look at the resulting pdf.
\begin{assumption}
For any data point $({\bm{x}}, {\bm{y}})$ and model parameters ${\bm{\theta}}$,
we consider that $f$ can be linearly approximated in ${\bm{x}}$ such that for a feature perturbation $\epsilon$ we have
\begin{equation}
f({\bm{x}}+ {\bm{\epsilon}}, {\bm{y}}, {\bm{\theta}} )
= f({\bm{x}}, {\bm{y}}, {\bm{\theta}})
+ J_{\bm{x}} ({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
where ${\bm{J}}_{{\bm{x}}}$ is the Jacobian matrix of $f$ with respect to ${\bm{x}}$.
\end{assumption}
We consider clients are given the same Gaussian Noise ${\bm{\psi}} \sim N({\bm{\nu}}, {\bm{\Phi}})$ which gives the following optimization problem and results ${\bm{\nu}}^*$.
\begin{equation}
F(\Dcal, {\bm{\theta}})
= F({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} {\bm{J}}_{{\bm{x}}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}){\bm{\psi}}
= F({\bm{\theta}}) + \frac{|\Scal|}{|\Dcal|}{\bm{J}}_{{\bm{x}}} (\Dcal_\Scal, {\bm{\theta}}) {\bm{\psi}}
\end{equation}
and
\begin{alignat}{2}
&\min_{{\bm{\nu}}} &\qquad& \norm{{\bm{\nu}}}^2
\\
&\text{subject to} & & {\bm{J}}_{\bm{x}} (\Dcal_\Scal, {\bm{\theta}}){\bm{\nu}} = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^*,
\nonumbe
\end{alignat}
\begin{equation}
{\bm{\nu}}^*
= \frac{|\Dcal|}{|\Scal|}{\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})
\left[ {\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})^T {\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) - {\bm{J}}_{\bm{\theta}}(\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^* \right].
\end{equation}
We can also give every client their own perturbation ${\bm{\psi}}_i \sim N({\bm{\nu}}_i, {\bm{\Phi}}_i)$ which gives the following optimization problem and results ${\bm{\nu}}_i^*$.
\begin{equation}
F(\Dcal, {\bm{\theta}})
= F({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} {\bm{J}}_{{\bm{x}}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}){\bm{\psi}}_i
\end{equation}
which gives the following optimization problem
\begin{alignat}{2}
&\min_{{\bm{\nu}}_i} &\qquad& \frac{1}{|\Scal|}\sum_{ i \in S}\norm{{\bm{\nu}}_i}^2
\\
&\text{subject to} & & \frac{1}{|\Dcal|}\sum_{ i \in \Scal} {\bm{J}}_{\bm{x}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}) {\bm{\nu}}_i = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^*,
\nonumbe
\end{alignat}
\begin{equation}
{\bm{\nu}}_i^*
= |\Dcal| {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
\left[ \sum_{ i \in S} {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})^T {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) - {\bm{J}}_{\bm{\theta}}(\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^* \right].
\end{equation}
\section{Differential Privacy}
Let us consider
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
We provide sufficient conditions for ${\bm{\mu}}$ and ${\bm{S}}$ such that ${\bm{\theta}} + {\bm{\phi}}$ satisfies Definition \ref{def:DP_forget}.
${\bm{\phi}}$ is a multinomial Gaussian distribution ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{S}}^T {\bm{S}})$. Therefore,
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
We are investigating the probability, given that the database is $\mathcal{D}$, of observing an output that occurs with a very different probability under $\mathcal{D}$ than under and adjacent database $\Dcal_{-\Scal}$, where the probability space is $\mathbb{R}^d$.
We define $\Delta \Lcal ({\bm{\theta}}) = \Lcal({\bm{\theta}}) - \Lcal_{-\Scal}({\bm{\theta}})$ and the privacy loss $R$ as
\begin{equation}
R(\xi)
\coloneqq \ln \left[ \frac{\mathbb{P}\left(\Lcal({\bm{\theta}} + {\bm{\phi}}) = \xi \right)}{\mathbb{P}\left(\Lcal_{-N}({\bm{\theta}} + {\bm{\phi}}) = \xi \right)} \right]
= \ln \left[ \frac{ \mathbb{P} \left(\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}} = \xi - \Lcal({\bm{\theta}}) \right)}
{\mathbb{P} \left(\inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}}) }{{\bm{\phi}}} = \xi - \Lcal_{-N}({\bm{\theta}})\right)} \right]
.
\end{equation}
The numerator in the ratio describes the probability of seeing $\Lcal({\bm{\theta}}) + \xi$ when the database is $\mathcal{D}$, the denominator corresponds to the probability of seeing this same value when the database is $\Dcal_{-\Scal}$.
${\bm{\phi}}$ is a multinomial Gaussian distribution ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{S}}^T {\bm{S}})$. Therefore,
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
Therefore, we have
\begin{equation}
\mathbb{P}\left(\Lcal({\bm{\theta}} + {\bm{\phi}}) = \xi \right)
=
\left[2 \pi \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2 \right]^{-1/2} e^{ - \frac{1}{2} \left(\xi - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}} \right)^2 \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^{-2}}
.
\end{equation}
With similar results for $\Lcal_{-\Scal}$. We can thus rewrite the privacy loss $R$ as
\begin{align}
R
& = \ln \left[ \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}} \right]
+ \frac{1}{2} \left[ \left(\frac{\xi - \Lcal_{-N}({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal_{-N}({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}\right)^2
- \left(\frac{\xi - \Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}\right)^2 \right]
\end{align}
We define $Q \coloneqq \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}} $.
\subsubsection*{$Q = 1 $}
We define $\alpha(\Lcal, {\bm{\theta}}) \coloneqq \frac{\Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}$.
Therefore, we can simplify $R(\xi)$ as
\begin{align}
R (\xi)
& =
\left[\alpha(\Lcal, {\bm{\theta}}) - \alpha(\Lcal_{-\Scal}, {\bm{\theta}})\right]\xi
+
\frac{1}{2} \left[\alpha^2(\Lcal_{-\Scal}, {\bm{\theta}})
- \alpha^2(\Lcal, {\bm{\theta}}) \right]
\end{align}
Therefore, we have $R(\xi) \le \epsilon$ when
\begin{equation}
R(\xi) \le \epsilon
\Leftrightarrow
\xi \le \frac{\epsilon - \frac{1}{2} \left[\alpha^2(\Lcal_{-\Scal}, {\bm{\theta}})
- \alpha^2(\Lcal, {\bm{\theta}}) \right]}
{\alpha(\Lcal, {\bm{\theta}}) - \alpha(\Lcal_{-\Scal}, {\bm{\theta}})}
\eqqcolon \xi_1
\end{equation}
Let us assume that $\xi_1 > 0$.
To satisfy DP privacy, it suffices
\begin{equation}
\mathbb{P}(\Lcal({\bm{\theta}} + {\bm{\phi}}) > \xi_1)
\le \mathbb{P}(\inner{\nabla_{\bm{\theta}} \Lcal({\bm{\theta}})}{{\bm{\phi}}} > \xi_1 - \Lcal({\bm{\theta}}))
\end{equation}
We define $Q \coloneqq \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}} $, $U \coloneqq \Delta \Lcal ({\bm{\theta}})
+ \left[ \nabla_{\bm{\theta}} \Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal_{-N}({\bm{\theta}}) \right]^T {\bm{\mu}} $,
$
S_1 =
\{
\xi \ | \ R(\xi) \le \epsilon
\}$
, and
$
S_2 =
\{
\xi \ | \ R(\xi) \ge - \epsilon
\}$
.
\subsubsection*{1st case}
We change variable for ease of writing. We set $y = (\xi - \Lcal({\bm{\theta}}) - \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}) / \norm{{\bm{S}} \nabla \Lcal ({\bm{\theta}})}$ which gives $\xi = \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y + \Lcal({\bm{\theta}}) + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}} $. We also consider $U_{-N} = \frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})} } U$.
\begin{align}
R
& = - \ln Q
+ \frac{1}{2} \left[ \left(Q y + U_{-N} \right)^2
- y^2 \right]
\end{align}
which can be developed as
\begin{align}
R
& = \frac{1}{2}\left[Q^2 - 1\right] y^2
+ U_{-N} Q y
+ \frac{1}{2}U_{-N}^2
- \ln Q
\end{align}
To find $S_1$, we consider $R_\epsilon$ defined as
\begin{align}
R_{\epsilon}
& = \frac{1}{2}\left[Q^2 - 1\right] y^2
+ \Delta({\bm{\theta}}) Q y
+ \frac{1}{2}\Delta^2({\bm{\theta}})
- \ln Q
- \epsilon
.
\end{align}
We define $\Delta_{\epsilon}$ the associated characteristic polynomial of $R_\epsilon$, i.e.
\begin{equation}
\Delta_{ \epsilon}
= U_{-N}^2 Q^2 - 4 \frac{1}{2}\left[Q^2 - 1\right]\left[\frac{1}{2}U_{-N}^2 - \ln Q - \epsilon\right]
= U_{-N}^2 + 2\left[Q^2 - 1\right]\left[\ln Q + \epsilon\right]
\end{equation}
Let us call $y_1$ and $y_2$ the two roots of $\Delta_{\epsilon}$ such that $y_{1} < y_{2}$.
Therefore, we have
\begin{equation}
y_{1}
= \frac{- \Delta({\bm{\theta}}) Q - \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
\text{ and }
y_{2}
= \frac{- \Delta({\bm{\theta}}) Q + \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
.
\end{equation}
Therefore, we have
\begin{equation}
S_1
= \left[
\xi_1 \coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_1 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}},
\xi_2
\coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_2 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}}
\right]
\end{equation}
We consider that $\Delta$ is such that $y_1 < 0$ and $y_2>0$.
For DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\eta_1 \notin S_1
\right)
\le \delta
\Leftrightarrow
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \delta
\end{equation}
Let us consider $X \sim N(\mu, \sigma)$, then we have
\begin{equation}
p(X < x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x <0
\end{equation}
\begin{equation}
p(X > x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x >0
\end{equation}
We can bound the cumulative distribution function of a normal distribution using the complementary error function which gives
\begin{equation}
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )}
+ \sqrt{2}\sigma\frac{e^{-(x_2 - \mu)^2/2 \sigma^2}}{2 (x_2 - \mu_1 )}
\end{equation}
Finally, for DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\Lcal({\bm{\theta}} + {\bm{\phi}} ) - \Lcal({\bm{\theta}})\in S_1
\right)
\ge 1- \delta
\end{equation}
\subsubsection*{2nd case}
We change variable for ease of writing. We set
$z = (\xi - \Lcal_{-N}({\bm{\theta}}) - \inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}{{\bm{\mu}}}) / \norm{{\bm{S}} \nabla \Lcal_{-N} ({\bm{\theta}})}$
which gives
$\xi = \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N} ({\bm{\theta}})} z + \Lcal_{-N}({\bm{\theta}}) + \inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})} {{\bm{\mu}}} $.
We also consider $\frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})} } U
= \frac{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})} } \frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}U
= \frac{1}{Q}U_{-N}$.
\begin{align}
R
& = - \ln Q
+ \frac{1}{2} \left[ z^2
- \left( \frac{1}{Q} z - \frac{1}{Q} U_{-N} \right)^2 \right]
\end{align}
which we can develop as
\begin{align}
R
& = \frac{1}{2}\left[1 - \frac{1}{Q^2}\right] z^2
+ \frac{1}{Q^2} U_{-N} z
- \ln Q
- \frac{1}{2} \frac{1}{Q^2}U_{-N}^2
\end{align}
We define $\Delta_{\epsilon}$ the associated characteristic polynomial of $R_\epsilon$, i.e.
\begin{equation}
\Delta_{ \epsilon}
= \frac{1}{Q^4} U_{-N}^2
- 4 \frac{1}{2}\left[ 1 - \frac{1}{Q^2}\right]\left[- \frac{1}{2}\frac{1}{Q^2}U_{-N}^2 - \ln Q + \epsilon\right]
= \frac{1}{Q^2} U_{-N}^2 + 2\left[1 - \frac{1}{Q^2}\right]\left[\ln Q - \epsilon\right]
\end{equation}
Let us call $y_1$ and $y_2$ the two roots of $\Delta_{\epsilon}$ such that $y_{1} < y_{2}$.
Therefore, we have
\begin{equation}
y_{1}
= \frac{- \Delta({\bm{\theta}}) Q - \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
\text{ and }
y_{2}
= \frac{- \Delta({\bm{\theta}}) Q + \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
.
\end{equation}
Therefore, we have
\begin{equation}
S_1
= \left[
\xi_1 \coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_1 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}},
\xi_2
\coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_2 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}}
\right]
\end{equation}
We consider that $\Delta$ is such that $y_1 < 0$ and $y_2>0$.
For DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\eta_1 \notin S_1
\right)
\le \delta
\Leftrightarrow
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \delta
\end{equation}
Let us consider $X \sim N(\mu, \sigma)$, then we have
\begin{equation}
p(X < x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x <0
\end{equation}
\begin{equation}
p(X > x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x >0
\end{equation}
We cn bound the cumulative distribution function of a normal distribution using the complementary error function which gives
\begin{equation}
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )}
+ \sqrt{2}\sigma\frac{e^{-(x_2 - \mu)^2/2 \sigma^2}}{2 (x_2 - \mu_1 )}
\end{equation}
Finally, for DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\Lcal({\bm{\theta}} + {\bm{\phi}} ) - \Lcal({\bm{\theta}})\in S_1
\right)
\ge 1- \delta
\end{equation}
\subsubsection{Interpretability with local perturbation}
Perturbating the model ${\bm{\theta}}$ with ${\bm{\phi}}$ impacts the performance of all the data samples. As such, even with Definition \ref{def:DP_forget} only based on the data points to forget, all the samples are impacted. For interpretability concerns, we consider a local perturbation ${\bm{\psi}}$ to the parameters to forget and look at the resulting pdf.
We also consider the following linear approximation
\begin{assumption}
For any data point $({\bm{x}}, {\bm{y}})$ and model with parameters ${\bm{\theta}}$,
we consider that $l$ can be linearly approximated in ${\bm{x}}$, i.e.
\begin{equation}
l({\bm{x}}+ {\bm{\epsilon}}, {\bm{y}}, {\bm{\theta}} ) = l({\bm{x}}, {\bm{y}}, {\bm{\theta}}) + \nabla_{\bm{x}} l({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
with $\norm{{\bm{\epsilon}}} \ll \norm{{\bm{x}}}$.
\end{assumption}
Therefore, we consider
\begin{equation}
\Lcal(\Dcal, {\bm{\theta}})
= \Lcal({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} \inner{\nabla_{\bm{x}} l({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})}{{\bm{\psi}}_i}
\end{equation}
\subsection{Practical Implementation}
\section{Formalizing the Unlearning problem}
\subsection{Optimization}
We consider a dataset $\Dcal$ composed of $|\Dcal| = N$ data samples $({\bm{x}}_i, {\bm{y}}_i)$ where ${\bm{x}}_i$ is the feature vector of a data sample and ${\bm{y}}_i$ its prediction or label.
For unsupervised learning, it suffices to consider that data samples in $\Dcal$ are only made with ${\bm{x}}_i$. We denote by $\Scal$ the set of $S = |\Scal|$ indices we want to forget from $\Dcal$ and $\Dcal_{-\Scal} = \Dcal\backslash\{({\bm{x}}_i, {\bm{y}}_i) | i \in \Scal \}$ the dataset without the forgotten data samples.
We consider a function $f$ depending on one data sample and model parameters ${\bm{\theta}}$.
For example, $f$ can be the loss of one data sample or its activation vector. We define the associated normalized functions
\begin{equation}
F({\bm{\theta}}) \coloneqq \frac{1}{|\Dcal|} \sum_{i \in \Dcal} f({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
\text{ and }
F_{-\Scal}({\bm{\theta}}) \coloneqq \frac{1}{|\Dcal_{-\Scal}|} \sum_{i \in \Dcal_{-\Scal}} f({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
.
\end{equation}
We consider $F$ is a $K$ dimensional vector where $K=1$ if $f$ is the loss function of one data samples and $K$ is the amount of classes in a classification problem.
For ease of writing, we consider $F_{-i}$ to be the resulting function when forgetting data sample $i$.
\section{Scrubbing Methods}
We consider a quadratic approximation of every sample loss function.
Therefore, the scrubbing method $h$ satisfies
\begin{equation}
h(w) = w - H_w^{-1} \nabla f_{\Scal}(w)
\end{equation}
We can also analyze this transformation in function of the samples features on the scrubbed model
\begin{equation}
t(x_i) = x_i - H_x^{-1}(x_i, h(w)) \nabla_{x} f(x_i, h(w))
\end{equation}
The learnt model is not necessarily the best model to start the new learning process.
We can bound the distance between a given model $w^t$ and the optimum model $w^*$ as :
We consider $H$ to be symmetric positive definite matrix.
\begin{align}
\norm{w^t - w^*}_2^2
&= \norm{w^t - h(w^t)}_2^2
= \norm{H^{-1}\nabla f}_2^2
\le \norm{H^{-1}}_F^2 \norm{\nabla f}_2^2
\\
&= \Tr{\left(H^{-2}\right)} \norm{\nabla f}_2^2
\le \Tr{\left(H\right)}^{-2} \norm{\nabla f}_2^2
= \left(\sum_{ i = 1 }^M \Tr H_i \right)^{-2} \norm{\nabla f}_2^2
\end{align}
\section{Complete Unlearning}
We assume that $f$ satisfies Assumption \ref{ass:linear_theta}.
\begin{assumption}
\label{ass:linear_theta}
For any data point $({\bm{x}}, {\bm{y}})$ and model with parameters ${\bm{\theta}}$,
we consider that $f$ can be linearly approximated in ${\bm{\theta}}$ for a perturbation ${\bm{\epsilon}}$, i.e.
\begin{equation}
f({\bm{x}}, {\bm{y}}, {\bm{\theta}} + \epsilon)
= f({\bm{x}}, {\bm{y}}, {\bm{\theta}}) + {\bm{J}}_{{\bm{\theta}}}({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
where ${\bm{J}}_{{\bm{\theta}}}$ is the Jacobian matrix of $f$ with respect to ${\bm{\theta}}$.
\end{assumption}
With Assumption \ref{ass:linear_theta} and the stochastic perturbation ${\bm{\phi}}$, we have
\begin{align}
F({\bm{\theta}} + {\bm{\phi}})
= F({\bm{\theta}})
+ \frac{1}{N}\sum_{ i = 1 }^{N} {\bm{J}}_{\bm{\theta}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}) {\bm{\phi}}
= F({\bm{\theta}})
+ {\bm{J}}_{\bm{\theta}} ( \Dcal, {\bm{\theta}}) {\bm{\phi}}
,
\end{align}
where we define $J_{\bm{\theta}} ( \Dcal, {\bm{\theta}}) = \frac{1}{N}\sum_{ i = 1 }^{N} {\bm{J}}_{\bm{\theta}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})$
with a similar derivation for $F_{-\Scal}$. We remind that ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{\Sigma}} = {\bm{S}}^T {\bm{S}})$. Hence, we have
\begin{equation}
F({\bm{\theta}} + {\bm{\phi}} )
\sim \mathcal{N}( F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}){\bm{\mu}}, {\bm{J}}_{\bm{\theta}} ( \Dcal, {\bm{\theta}})^T{\bm{\Sigma}} {\bm{J}}_{\bm{\theta}}(\Dcal, {\bm{\theta}}))
,
\end{equation}
with a similar derivation for $F_{-\Scal}$. With a non zero expected perturbation ${\bm{\mu}}$, a perfect unlearning can be obtained with $F({\bm{\theta}} + {\bm{\phi}}) = F_{-\Scal}({\bm{\theta}} + {\bm{\phi}})$ by choosing ${\bm{\mu}}$ and ${\bm{\Sigma}}$ such that the probability density functions of these two queries are identical. Considering the dimensions of the Jacobian, numerous ${\bm{\mu}}$ can satisfy this condition, we are thus interested in the ones with the smallest norm where Assumption \ref{ass:linear_theta} is the most valid.
Hence, we have
\begin{alignat}{2}
&\min_{{\bm{\mu}}} &\qquad& \norm{{\bm{\mu}}}^2
\\
&\text{subject to} & & \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]{\bm{\mu}} = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}),
\nonumber\\
& & & \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]^T{\bm{\Sigma}} \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right] = 0 \label{eq:constraint_sigma}.
\nonumber
\end{alignat}
We define ${\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal}) = \left[{\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) - {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}})\right]$.
When this optimization problem is feasible, then its optimum ${\bm{\mu}}^*$ satisfies
\begin{equation}
{\bm{\mu}}^* = {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal}) \left[ {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal})^T {\bm{J}}_{\bm{\theta}}(\Dcal, \Dcal_{-\Scal})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}})\right].
\end{equation}
Any $\Sigma$ such that equation(\ref{eq:constraint_sigma}) is satisfied works including ${\bm{\Sigma}} =0$. Yet, considering a positive ${\bm{\Sigma}}$ to mitigate the linear approximation would be helpful. We note that when $K=1$, we have ${\bm{J}}_{\bm{\theta}} (\Dcal, {\bm{\theta}}) = \nabla_{\bm{\theta}} F({\bm{\theta}})$ and can simplify the close form of ${\bm{\mu}}^*$ with
\begin{equation}
{\bm{\mu}}^* = \frac{\nabla_{\bm{\theta}} F({\bm{\theta}}) - \nabla_{\bm{\theta}} F_{-\Scal}({\bm{\theta}})}{\norm{\nabla_{\bm{\theta}} F({\bm{\theta}}) - \nabla_{\bm{\theta}} F_{-\Scal}({\bm{\theta}})}^2} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}})\right].
\end{equation}
\subsubsection{Interpretability using Local Perturbation}
Perturbating the model ${\bm{\theta}}$ with ${\bm{\phi}}$ impacts the performance of all the data samples. As such, even with Definition \ref{def:DP_forget} only based on the data points to forget, all the samples are impacted. For interpretability concerns, we consider a local perturbation ${\bm{\psi}}$ to the parameters to forget and look at the resulting pdf.
\begin{assumption}
For any data point $({\bm{x}}, {\bm{y}})$ and model parameters ${\bm{\theta}}$,
we consider that $f$ can be linearly approximated in ${\bm{x}}$ such that for a feature perturbation $\epsilon$ we have
\begin{equation}
f({\bm{x}}+ {\bm{\epsilon}}, {\bm{y}}, {\bm{\theta}} )
= f({\bm{x}}, {\bm{y}}, {\bm{\theta}})
+ J_{\bm{x}} ({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
where ${\bm{J}}_{{\bm{x}}}$ is the Jacobian matrix of $f$ with respect to ${\bm{x}}$.
\end{assumption}
We consider clients are given the same Gaussian Noise ${\bm{\psi}} \sim N({\bm{\nu}}, {\bm{\Phi}})$ which gives the following optimization problem and results ${\bm{\nu}}^*$.
\begin{equation}
F(\Dcal, {\bm{\theta}})
= F({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} {\bm{J}}_{{\bm{x}}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}){\bm{\psi}}
= F({\bm{\theta}}) + \frac{|\Scal|}{|\Dcal|}{\bm{J}}_{{\bm{x}}} (\Dcal_\Scal, {\bm{\theta}}) {\bm{\psi}}
\end{equation}
and
\begin{alignat}{2}
&\min_{{\bm{\nu}}} &\qquad& \norm{{\bm{\nu}}}^2
\\
&\text{subject to} & & {\bm{J}}_{\bm{x}} (\Dcal_\Scal, {\bm{\theta}}){\bm{\nu}} = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^*,
\nonumbe
\end{alignat}
\begin{equation}
{\bm{\nu}}^*
= \frac{|\Dcal|}{|\Scal|}{\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})
\left[ {\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})^T {\bm{J}}_{\bm{x}}(\Dcal_\Scal, {\bm{\theta}})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) - {\bm{J}}_{\bm{\theta}}(\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^* \right].
\end{equation}
We can also give every client their own perturbation ${\bm{\psi}}_i \sim N({\bm{\nu}}_i, {\bm{\Phi}}_i)$ which gives the following optimization problem and results ${\bm{\nu}}_i^*$.
\begin{equation}
F(\Dcal, {\bm{\theta}})
= F({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} {\bm{J}}_{{\bm{x}}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}){\bm{\psi}}_i
\end{equation}
which gives the following optimization problem
\begin{alignat}{2}
&\min_{{\bm{\nu}}_i} &\qquad& \frac{1}{|\Scal|}\sum_{ i \in S}\norm{{\bm{\nu}}_i}^2
\\
&\text{subject to} & & \frac{1}{|\Dcal|}\sum_{ i \in \Scal} {\bm{J}}_{\bm{x}} ({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}}) {\bm{\nu}}_i = F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) + {\bm{J}}_{\bm{\theta}} (\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^*,
\nonumbe
\end{alignat}
\begin{equation}
{\bm{\nu}}_i^*
= |\Dcal| {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})
\left[ \sum_{ i \in S} {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})^T {\bm{J}}_{\bm{x}}({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})\right]^{-1} \left[F_{-\Scal}({\bm{\theta}}) - F({\bm{\theta}}) - {\bm{J}}_{\bm{\theta}}(\Dcal_{-\Scal}, {\bm{\theta}}){\bm{\mu}}^* \right].
\end{equation}
\section{Differential Privacy}
Let us consider
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
We provide sufficient conditions for ${\bm{\mu}}$ and ${\bm{S}}$ such that ${\bm{\theta}} + {\bm{\phi}}$ satisfies Definition \ref{def:DP_forget}.
${\bm{\phi}}$ is a multinomial Gaussian distribution ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{S}}^T {\bm{S}})$. Therefore,
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
We are investigating the probability, given that the database is $\mathcal{D}$, of observing an output that occurs with a very different probability under $\mathcal{D}$ than under and adjacent database $\Dcal_{-\Scal}$, where the probability space is $\mathbb{R}^d$.
We define $\Delta \Lcal ({\bm{\theta}}) = \Lcal({\bm{\theta}}) - \Lcal_{-\Scal}({\bm{\theta}})$ and the privacy loss $R$ as
\begin{equation}
R(\xi)
\coloneqq \ln \left[ \frac{\mathbb{P}\left(\Lcal({\bm{\theta}} + {\bm{\phi}}) = \xi \right)}{\mathbb{P}\left(\Lcal_{-N}({\bm{\theta}} + {\bm{\phi}}) = \xi \right)} \right]
= \ln \left[ \frac{ \mathbb{P} \left(\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}} = \xi - \Lcal({\bm{\theta}}) \right)}
{\mathbb{P} \left(\inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}}) }{{\bm{\phi}}} = \xi - \Lcal_{-N}({\bm{\theta}})\right)} \right]
.
\end{equation}
The numerator in the ratio describes the probability of seeing $\Lcal({\bm{\theta}}) + \xi$ when the database is $\mathcal{D}$, the denominator corresponds to the probability of seeing this same value when the database is $\Dcal_{-\Scal}$.
${\bm{\phi}}$ is a multinomial Gaussian distribution ${\bm{\phi}} \sim \mathcal{N}({\bm{\mu}}, {\bm{S}}^T {\bm{S}})$. Therefore,
\begin{equation}
\inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}) }{{\bm{\phi}}}
\sim \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T{\bm{\Sigma}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}}))
= \mathcal{N}( \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}, \norm{S \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2)
.
\end{equation}
Therefore, we have
\begin{equation}
\mathbb{P}\left(\Lcal({\bm{\theta}} + {\bm{\phi}}) = \xi \right)
=
\left[2 \pi \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^2 \right]^{-1/2} e^{ - \frac{1}{2} \left(\xi - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}} \right)^2 \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}^{-2}}
.
\end{equation}
With similar results for $\Lcal_{-\Scal}$. We can thus rewrite the privacy loss $R$ as
\begin{align}
R
& = \ln \left[ \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}} \right]
+ \frac{1}{2} \left[ \left(\frac{\xi - \Lcal_{-N}({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal_{-N}({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}\right)^2
- \left(\frac{\xi - \Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}\right)^2 \right]
\end{align}
We define $Q \coloneqq \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}} $.
\subsubsection*{$Q = 1 $}
We define $\alpha(\Lcal, {\bm{\theta}}) \coloneqq \frac{\Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal({\bm{\theta}})^T {\bm{\mu}}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}$.
Therefore, we can simplify $R(\xi)$ as
\begin{align}
R (\xi)
& =
\left[\alpha(\Lcal, {\bm{\theta}}) - \alpha(\Lcal_{-\Scal}, {\bm{\theta}})\right]\xi
+
\frac{1}{2} \left[\alpha^2(\Lcal_{-\Scal}, {\bm{\theta}})
- \alpha^2(\Lcal, {\bm{\theta}}) \right]
\end{align}
Therefore, we have $R(\xi) \le \epsilon$ when
\begin{equation}
R(\xi) \le \epsilon
\Leftrightarrow
\xi \le \frac{\epsilon - \frac{1}{2} \left[\alpha^2(\Lcal_{-\Scal}, {\bm{\theta}})
- \alpha^2(\Lcal, {\bm{\theta}}) \right]}
{\alpha(\Lcal, {\bm{\theta}}) - \alpha(\Lcal_{-\Scal}, {\bm{\theta}})}
\eqqcolon \xi_1
\end{equation}
Let us assume that $\xi_1 > 0$.
To satisfy DP privacy, it suffices
\begin{equation}
\mathbb{P}(\Lcal({\bm{\theta}} + {\bm{\phi}}) > \xi_1)
\le \mathbb{P}(\inner{\nabla_{\bm{\theta}} \Lcal({\bm{\theta}})}{{\bm{\phi}}} > \xi_1 - \Lcal({\bm{\theta}}))
\end{equation}
We define $Q \coloneqq \frac{ \norm{ {\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}}
{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}} $, $U \coloneqq \Delta \Lcal ({\bm{\theta}})
+ \left[ \nabla_{\bm{\theta}} \Lcal({\bm{\theta}}) - \nabla_{\bm{\theta}} \Lcal_{-N}({\bm{\theta}}) \right]^T {\bm{\mu}} $,
$
S_1 =
\{
\xi \ | \ R(\xi) \le \epsilon
\}$
, and
$
S_2 =
\{
\xi \ | \ R(\xi) \ge - \epsilon
\}$
.
\subsubsection*{1st case}
We change variable for ease of writing. We set $y = (\xi - \Lcal({\bm{\theta}}) - \inner{\nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})}{{\bm{\mu}}}) / \norm{{\bm{S}} \nabla \Lcal ({\bm{\theta}})}$ which gives $\xi = \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y + \Lcal({\bm{\theta}}) + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}} $. We also consider $U_{-N} = \frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})} } U$.
\begin{align}
R
& = - \ln Q
+ \frac{1}{2} \left[ \left(Q y + U_{-N} \right)^2
- y^2 \right]
\end{align}
which can be developed as
\begin{align}
R
& = \frac{1}{2}\left[Q^2 - 1\right] y^2
+ U_{-N} Q y
+ \frac{1}{2}U_{-N}^2
- \ln Q
\end{align}
To find $S_1$, we consider $R_\epsilon$ defined as
\begin{align}
R_{\epsilon}
& = \frac{1}{2}\left[Q^2 - 1\right] y^2
+ \Delta({\bm{\theta}}) Q y
+ \frac{1}{2}\Delta^2({\bm{\theta}})
- \ln Q
- \epsilon
.
\end{align}
We define $\Delta_{\epsilon}$ the associated characteristic polynomial of $R_\epsilon$, i.e.
\begin{equation}
\Delta_{ \epsilon}
= U_{-N}^2 Q^2 - 4 \frac{1}{2}\left[Q^2 - 1\right]\left[\frac{1}{2}U_{-N}^2 - \ln Q - \epsilon\right]
= U_{-N}^2 + 2\left[Q^2 - 1\right]\left[\ln Q + \epsilon\right]
\end{equation}
Let us call $y_1$ and $y_2$ the two roots of $\Delta_{\epsilon}$ such that $y_{1} < y_{2}$.
Therefore, we have
\begin{equation}
y_{1}
= \frac{- \Delta({\bm{\theta}}) Q - \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
\text{ and }
y_{2}
= \frac{- \Delta({\bm{\theta}}) Q + \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
.
\end{equation}
Therefore, we have
\begin{equation}
S_1
= \left[
\xi_1 \coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_1 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}},
\xi_2
\coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_2 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}}
\right]
\end{equation}
We consider that $\Delta$ is such that $y_1 < 0$ and $y_2>0$.
For DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\eta_1 \notin S_1
\right)
\le \delta
\Leftrightarrow
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \delta
\end{equation}
Let us consider $X \sim N(\mu, \sigma)$, then we have
\begin{equation}
p(X < x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x <0
\end{equation}
\begin{equation}
p(X > x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x >0
\end{equation}
We can bound the cumulative distribution function of a normal distribution using the complementary error function which gives
\begin{equation}
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )}
+ \sqrt{2}\sigma\frac{e^{-(x_2 - \mu)^2/2 \sigma^2}}{2 (x_2 - \mu_1 )}
\end{equation}
Finally, for DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\Lcal({\bm{\theta}} + {\bm{\phi}} ) - \Lcal({\bm{\theta}})\in S_1
\right)
\ge 1- \delta
\end{equation}
\subsubsection*{2nd case}
We change variable for ease of writing. We set
$z = (\xi - \Lcal_{-N}({\bm{\theta}}) - \inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}{{\bm{\mu}}}) / \norm{{\bm{S}} \nabla \Lcal_{-N} ({\bm{\theta}})}$
which gives
$\xi = \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N} ({\bm{\theta}})} z + \Lcal_{-N}({\bm{\theta}}) + \inner{\nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})} {{\bm{\mu}}} $.
We also consider $\frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})} } U
= \frac{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})} } \frac{1}{\norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal_{-N}( {\bm{\theta}})}}U
= \frac{1}{Q}U_{-N}$.
\begin{align}
R
& = - \ln Q
+ \frac{1}{2} \left[ z^2
- \left( \frac{1}{Q} z - \frac{1}{Q} U_{-N} \right)^2 \right]
\end{align}
which we can develop as
\begin{align}
R
& = \frac{1}{2}\left[1 - \frac{1}{Q^2}\right] z^2
+ \frac{1}{Q^2} U_{-N} z
- \ln Q
- \frac{1}{2} \frac{1}{Q^2}U_{-N}^2
\end{align}
We define $\Delta_{\epsilon}$ the associated characteristic polynomial of $R_\epsilon$, i.e.
\begin{equation}
\Delta_{ \epsilon}
= \frac{1}{Q^4} U_{-N}^2
- 4 \frac{1}{2}\left[ 1 - \frac{1}{Q^2}\right]\left[- \frac{1}{2}\frac{1}{Q^2}U_{-N}^2 - \ln Q + \epsilon\right]
= \frac{1}{Q^2} U_{-N}^2 + 2\left[1 - \frac{1}{Q^2}\right]\left[\ln Q - \epsilon\right]
\end{equation}
Let us call $y_1$ and $y_2$ the two roots of $\Delta_{\epsilon}$ such that $y_{1} < y_{2}$.
Therefore, we have
\begin{equation}
y_{1}
= \frac{- \Delta({\bm{\theta}}) Q - \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
\text{ and }
y_{2}
= \frac{- \Delta({\bm{\theta}}) Q + \sqrt{\Delta_{\epsilon}}}{Q^2 - 1}
.
\end{equation}
Therefore, we have
\begin{equation}
S_1
= \left[
\xi_1 \coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_1 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}},
\xi_2
\coloneqq \norm{{\bm{S}} \nabla_{\bm{\theta}} \Lcal ({\bm{\theta}})} y_2 + \nabla_{\bm{\theta}} \Lcal( {\bm{\theta}})^T {\bm{\mu}}
\right]
\end{equation}
We consider that $\Delta$ is such that $y_1 < 0$ and $y_2>0$.
For DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\eta_1 \notin S_1
\right)
\le \delta
\Leftrightarrow
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \delta
\end{equation}
Let us consider $X \sim N(\mu, \sigma)$, then we have
\begin{equation}
p(X < x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x <0
\end{equation}
\begin{equation}
p(X > x ) \le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )} \text{for} x >0
\end{equation}
We cn bound the cumulative distribution function of a normal distribution using the complementary error function which gives
\begin{equation}
\mathbb{P}
\left(
\eta_1 \le \xi_1
\right)
+
\mathbb{P}
\left(
\eta_1 \ge \xi_2
\right)
\le \sqrt{2}\sigma\frac{e^{-(x_1 - \mu)^2/2 \sigma^2}}{2 (\mu_1 - x_1 )}
+ \sqrt{2}\sigma\frac{e^{-(x_2 - \mu)^2/2 \sigma^2}}{2 (x_2 - \mu_1 )}
\end{equation}
Finally, for DP to be satisfied,we want
\begin{equation}
\mathbb{P}
\left(
\Lcal({\bm{\theta}} + {\bm{\phi}} ) - \Lcal({\bm{\theta}})\in S_1
\right)
\ge 1- \delta
\end{equation}
\subsubsection{Interpretability with local perturbation}
Perturbating the model ${\bm{\theta}}$ with ${\bm{\phi}}$ impacts the performance of all the data samples. As such, even with Definition \ref{def:DP_forget} only based on the data points to forget, all the samples are impacted. For interpretability concerns, we consider a local perturbation ${\bm{\psi}}$ to the parameters to forget and look at the resulting pdf.
We also consider the following linear approximation
\begin{assumption}
For any data point $({\bm{x}}, {\bm{y}})$ and model with parameters ${\bm{\theta}}$,
we consider that $l$ can be linearly approximated in ${\bm{x}}$, i.e.
\begin{equation}
l({\bm{x}}+ {\bm{\epsilon}}, {\bm{y}}, {\bm{\theta}} ) = l({\bm{x}}, {\bm{y}}, {\bm{\theta}}) + \nabla_{\bm{x}} l({\bm{x}}, {\bm{y}}, {\bm{\theta}}) {\bm{\epsilon}}
,
\end{equation}
with $\norm{{\bm{\epsilon}}} \ll \norm{{\bm{x}}}$.
\end{assumption}
Therefore, we consider
\begin{equation}
\Lcal(\Dcal, {\bm{\theta}})
= \Lcal({\bm{\theta}}) + \frac{1}{|\Dcal|}\sum_{ i \in S} \inner{\nabla_{\bm{x}} l({\bm{x}}_i, {\bm{y}}_i, {\bm{\theta}})}{{\bm{\psi}}_i}
\end{equation}
\subsection{Practical Implementation}
\subsection{Proof of Theorem \ref{theo:diff_bound} for $K=1$}\label{app:subsec:proof_DL}
\begin{proof}
We define by ${\bm{\theta}}^N = \textsc{FedAvg}(I, N)$ and ${\bm{\phi}}^N = \textsc{FedAvg}(I_{-c}, N)$ the models trained with $\textsc{FedAvg}$ on ${\bm{\theta}}_0$ with respectively all the clients, i.e. $I$, and all the clients but client $c$, i.e. $I_{-c}$, performing $K=1$ GD step.
When clients perform $K=1$ GD step, two consecutive global models can be related, when training with clients in $I$ as a simple GD step, i.e.
\begin{equation}
{\bm{\theta}}^{n+1}
= {\bm{\theta}}^n
- \eta \nabla f_I({\bm{\theta}}^n)
.
\end{equation}
By considering the same process for $I_{-c}$ and with Assumption \ref{ass:linear_approx}, we get
\begin{align}
{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}
& = {\bm{\phi}}^n - {\bm{\theta}}^n
- \eta \left[ \nabla f_{I_{-c}}({\bm{\phi}}^n) - \nabla f_I({\bm{\theta}}^n) \right]
\\
& = \left[I - \eta H_{I_{-c}}({\bm{\theta}}^n) \right] \left[{\bm{\phi}}^n - {\bm{\theta}}^n \right]
- \eta \left[ \nabla f_{I_{-c}}({\bm{\theta}}^n) - \nabla f_I({\bm{\theta}}^n) \right]
\label{eq:diff}
.
\end{align}
$H_{I_{-c}}({\bm{\theta}}^n)$ is semi-positive, Assumption \ref{ass:linear_approx}. Let us define $\sigma_\text{max} (H_{I_{-c}}({\bm{\theta}}^n))$ the highest eigenvalue of $H_{I_{-c}}({\bm{\theta}}^n)$. When consider that $\eta \le 1/\sigma_\text{max} (H_{I_{-c}}({\bm{\theta}}^n))$, and due to the subadditivity of the norm, we get the following recurrent inequality
\begin{align}
\norm{{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}}_2
&\le
\eta \norm{\nabla f_I({\bm{\theta}}^n) - \nabla f_{I_{-c}}({\bm{\theta}}^n)}_2
+ \norm{{\bm{\phi}}^n - {\bm{\theta}}^n}_2
,
\end{align}
which when developed completes the proof when clients perform $K=1$ GD.
\end{proof}
\subsection{Proof of Theorem \ref{theo:diff_bound} for $K\ge 1$}\label{app:subsec:proof_FL}
\begin{proof}
We maintain the definitions of ${\bm{\theta}}^n$ and ${\bm{\phi}}^n$ introduced in Section \ref{app:subsec:proof_DL}. To account for the amount of local work $K$, we introduce ${\bm{\theta}}_i^{n , k}$ the model of client $i$ after $k$ GD steps performed on global model ${\bm{\theta}}^n$. We apply a similar reasoning for ${\bm{\phi}}_i^{n , k}$.
With Assumption \ref{ass:linear_approx}, we have
\begin{equation}
\nabla f_i ({\bm{\phi}}_i^{n, k})
= \nabla f_i ({\bm{\theta}}_i^{n, k})
+ H_i ({\bm{\theta}}_i^{n, k}) \left({\bm{\phi}}_i^{n, k} - {\bm{\theta}}_i^{n, k}\right)
\label{eq:rewrite_linear}
,
\end{equation}
which gives
\begin{align}
{\bm{\phi}}_i^{n, k+1} - {\bm{\theta}}_i^{n, k+1}
& = \left( {\bm{\phi}}_i^{n, k+1} - {\bm{\phi}}_i^{n, k}\right)
- \left( {\bm{\theta}}_i^{n, k+1 } - {\bm{\theta}}_i^{n, k} \right)
+ \left({\bm{\phi}}_i^{n, k} - {\bm{\theta}}_i^{n, k}\right)
\\
& = - \eta \left[\nabla f_i \left({\bm{\phi}}_i^{n, k}\right)
- \nabla f_i \left({\bm{\theta}}_i^{n, k} \right) \right]
+ \left({\bm{\phi}}_i^{n, k} - {\bm{\theta}}_i^{n, k}\right)
\\
& = \left[I - \eta H_i ({\bm{\theta}}_i^{n, k}) \right]\left({\bm{\phi}}_i^{n, k} - {\bm{\theta}}_i^{n, k}\right)
\\
& = \left[\prod_{r=0}^{k}\left[I - \eta H_i ({\bm{\theta}}_i^{n, r}) \right]\right]\left({\bm{\phi}}^n - {\bm{\theta}}^n \right)
\label{eq:diff_two_locals}
,
\end{align}
where the third equality follows from equation (\ref{eq:rewrite_linear}), and the fourth from expanding the recurrent equation. For the rest of this work, we define $Q_i^n = \prod_{k=0}^{K-1}\left[I - \eta H_i ({\bm{\theta}}_i^{n, k}) \right]$.
Using equation (\ref{eq:diff_two_locals}), we relate the difference between two global models with every client in $I$ and in $I_c$. When removing client $c$ the clients' importance changes. We consider importance $p_i$ when training with $I$. Instead, when training with clients in $I_c$, we consider the regularized importance $q_i = p_i / (1 - p_c)$ for the remaining clients and $q_c = 0$. We have
\begin{align}
{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}
& = \sum_{i=1}^M q_i \left({\bm{\phi}}_i^{n+1} - {\bm{\phi}}^n \right)
- \sum_{i=1}^M p_i \left({\bm{\theta}}_i^{n+1} - {\bm{\theta}}^n\right)
\\
& = \sum_{i=1}^M q_i \left[ \left({\bm{\phi}}_i^{n+1} - {\bm{\theta}}_i^{n+1}\right) + \left({\bm{\theta}}_i^{n+1} - {\bm{\theta}}^n \right) \right]
- \sum_{i=1}^M p_i \left({\bm{\theta}}_i^{n+1} - {\bm{\theta}}^n\right)
\\
& = \left(\sum_{i=1}^M q_i Q_i^n \right) \left({\bm{\phi}}^n - {\bm{\theta}}^n\right)
+ \Delta(I_{-c}, {\bm{\theta}}^n)
- \Delta(I, {\bm{\theta}}^n)
.
\label{app:eq:diff_one}
\end{align}
We consider a learning rate $\eta$ such that $\eta \le 1/\sigma_\text{max} (H_i({\bm{\theta}}^{n , k}))$. Hence, $\norm{Q_i^n}_2 \le 1$. With equation (\ref{app:eq:diff_one}), we get the following inequality
\begin{align}
\norm{{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}}_2
& \le \norm{{\bm{\phi}}^n - {\bm{\theta}}^n}_2
+ \norm{ \Delta(I, {\bm{\theta}}^n)
- \Delta(I_{-c}, {\bm{\theta}}^n)}_2
,
\end{align}
which expansion completes the proof.
\end{proof}
\subsection{Local Loss Functions' Regularization and Strong Convexity, Proof of Corollary \ref{cor:tighter}}
\label{app:subsec:regularization}
\begin{proof}
Under L2 regularization, every client's regularized loss function $F_i$ is expressed as
\begin{equation}
F_i({\bm{\theta}})
= f_i({\bm{\theta}})
+ \frac{\lambda}{2}\norm{{\bm{\theta}}}^2
\text{ and }
\nabla F_i({\bm{\theta}})
= \nabla f_i({\bm{\theta}})
+ \lambda{\bm{\theta}}
.
\end{equation}
When clients perform $K=1$ GD step, equation (\ref{app:eq:diff_one}) reduces to
\begin{align}
{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}
& = \eta \left[\nabla f_I({\bm{\theta}}^n) - \nabla f_{I_{-c}}({\bm{\theta}}^n) \right]
+ \left[(1 - \eta \lambda) I -\eta H_{I_{-c}}(\theta^n) \right]({\bm{\phi}}^n - \theta^n)
,
\end{align}
which, if $\eta \le 1 / (\lambda + \sigma_{\max}(H_i({\bm{\theta}}^n))$, gives
\begin{align}
\norm{{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}}_2
&\le
\eta \norm{\nabla f_I({\bm{\theta}}^n) - \nabla f_{I_{-c}}({\bm{\theta}}^n)}_2
+ (1- \eta \lambda -\eta \mu)\norm{{\bm{\phi}}^n - {\bm{\theta}}^n}_2
.
\end{align}
When clients perform $K\ge 1$ GD steps, we have $ {\bm{\phi}}_i^{n + 1} - {\bm{\theta}}_i^{n + 1} = Q_i^n \left[{\bm{\phi}}^n - {\bm{\theta}}^n\right]$ with
\begin{equation}
Q_i^n
= \prod_{r=0}^{K-1}\left[(1 - \eta \lambda)I - \eta H_i ({\bm{\theta}}_i^{n, r}) \right]
.
\end{equation}
Hence, we retrieve equation (\ref{app:eq:diff_one}). We consider the local learning rate satisfy $\eta \le 1 /( \lambda + \sigma_{\max} (H_i ({\bm{\theta}}^n)))$. Hence, considering that $Q_i^n$ can be bounded with the $\mu$-strong convexity of the Hessian, we get
\begin{align}
\norm{{\bm{\phi}}^{n+1} - {\bm{\theta}}^{n+1}}_2
&\le
\eta \norm{\Delta(I, {\bm{\theta}}^n) - \Delta(I_{-c}, {\bm{\theta}}^n)}_2
+ (1- \eta \lambda -\eta \mu)^K \norm{{\bm{\phi}}^n - {\bm{\theta}}^n}_2
.
\end{align}
Developing this recurrent equation completes the proof.
\end{proof}
\subsection{Generalization}
The proof of Theorem \ref{theo:diff_bound} can be also extended to account for FL regularization methods \citep{FedProx, FedDane, FedDyn}, other SGD solvers \citep{Adam, AdaGrad,pmlr-v89-li19c, OnTheLienarSpeedUp, Yu_Yang_Zhu_2019, haddapour2019trading}, client sampling \citep{FedProx, OnTheConvergence, TheoryClientSampling} and/or gradient compression/quantization \citep{FedPaq, QSparse, Atomo}.
\subsection{Calculus simplification with uniform importance}
For computation purposes, we propose the following expression to estimate a client bounded sensitivity, equation (\ref{eq:def_Psi}.
When removing client $c$, each client has new importance $q_i = p_i /(1 - p_c)$ for the remaining clients and $q_c = 0$. Hence, we have
\begin{align}
\norm{\Delta(I, {\bm{\theta}}^n) - \Delta({\bm{\theta}}^n, \Dcal_{-c})}_2
& = \norm{\left[{\bm{\theta}}^{n+1} - {\bm{\theta}}^n\right] - \left[\sum_{i=1}^{M}q_i {\bm{\theta}}_i^{n+1} - {\bm{\theta}}^n\right]}_2
\\
& = \norm{{\bm{\theta}}^{n+1} - \frac{1}{1 - p_c}\left[{\bm{\theta}}^{n+1} - p_c {\bm{\theta}}_i^{n+1}\right]}_2
\\
& = \frac{p_c}{1 - p_c} \norm{ {\bm{\theta}}_i^{n+1} - {\bm{\theta}}^{n+1} }_2
\end{align}
In the special case where clients have identical importance, we have $p_c/(1 - p_c) = 1/ (M -1)$.
\section{Introduction}
With the emergence of new data regulations, such as the EU General Data Protection Regulation (GDPR) \citep{GDPR} and the California Consumer Privacy Act (CCPA) \citep{CCPA}, the storage and processing of sensitive personal data is often subject of strict constraints and restrictions. In particular, the “right to be forgotten” states that personal data must be erased upon request from the concerned individuals, with subsequent potential implications on machine learning models trained by using this data.
Machine Unlearning (MU) is an emerging discipline that studies methods to ideally remove the contribution of a given data instance used to train a machine learning model.
Current MU approaches are essentially based on routines that modify the model weights in order to guarantee the “forgetting" of a given data point, i.e. to obtain a model equivalent to an hypothetical one trained without this data point \citep{Cao2015TowardsMS, SISA}.
Motivated by data governance and confidentiality concerns,
Federated learning (FL) has gained popularity in the last years to allow data owners to collaboratively learn a model without sharing their respective data.
Among the different FL approaches, federated averaging (\textsc{FedAvg}) has emerged as the most popular optimization scheme \citep{FedAvg}. An optimization round of \textsc{FedAvg} requires data owners, also called clients, to receive from the server the current global model which they update
before sending it back to the server. The new global model is then created as the weighted average of the client updates, according to their data ratio.
FL communication design guarantees to clients that their data is solely used to compute their model update, while theoretical work guarantees FL convergence to a stationary point of the clients' joint optimization problem \citep{FedNova, OnTheConvergence}.
With the current deployments of FL in the real-world, it is of crucial importance to extend MU to guarantee the unlearning of clients wishing to opt-out from a collaborative training routine.
This is not straightforward, since current MU schemes have been proposed essentially in the centralized learning setting, and cannot be seamlessly applied to the federated one.
For example, several MU methods require the estimation of the Hessian of the loss function \citep{CertifiedDataRemoval, ApproximateDataDeletion, Golatkar_2020_CVPR, golatkar2020forgetting, Golatkar_2021_CVPR}, an operation which is notoriously computationally heavy and intractable for high dimensional models. Moreover, sharing the Hessian would require clients to share with the server additional information about their data, thus exposing the federated setting to information leakage and attacks, for example under the form of model inversion \citep{Fredrikson-MI-2015}.
Alternative MU methods draw from the concept of differential privacy \cite{DP_book} and are based on a Gaussian noise perturbation of the trained model \citep{DescentToDelete, CertifiedDataRemoval, AdaptiveMachineUnlearning}. The magnitude of the noise perturbation should be estimated directly from the clients data, which is by construction inaccessible to the server in the FL regime. We also note that while recent federated unlearning (FU) methods have been proposed to unlearn a client from the global FL model \citep{FedEraser, wang2021federated, halimi2022federated, wu2022federated}, these approaches do not come with theoretical guarantees on the effectiveness of the unlearning.
The main contribution of this work consists in Informed Federated Unlearning (IFU), a novel efficient FU approach to unlearn a client's contribution with quantifiable unlearning guarantees. IFU requires minimal additional computations to the server in a standard \textsc{FedAvg} procedure. Specifically, the server quantifies at every optimization round each client's contribution to the global model.
Upon receiving an unlearning request from a client, the server identifies in the FL training history the optimal FL iteration and associated intermediate global model from which re-initializing the unlearning procedure. Unlearning guarantees are provided by introducing a novel randomized mechanism to perturb the selected intermediate model with client-specific noise.
We also extend IFU to Sequential Informed Federated Unlearning (SIFU), to account for realistic unlearning scenarios where the server receives sequential unlearning requests from one or more clients at different times \citep{DescentToDelete, AdaptiveMachineUnlearning}.
This manuscript is structured as follows.
In Section \ref{sec:background}, we provide formal definitions for MU, FL, and FU, and introduce the randomized mechanism with associated unlearning guarantees.
In Section \ref{sec:theory}, we introduce sufficient conditions for IFU to unlearn a client from the FL routine (Theorem \ref{theo:noise_DP}). In Section \ref{sec:SIFU}, we extend IFU to the sequential unlearning setting with Sequential IFU (SIFU).
Finally, in Section \ref{sec:experiments}, we experimentally demonstrate on different tasks and datasets that SIFU leads to more efficient unlearning procedures as compared to basic re-training and state-of-the-art FU approaches.
\section{Conclusions}
In this work, we introduce informed federated unlearning (IFU), a novel federated unlearning scheme to unlearn a client's contribution from a model trained with federated learning. Upon receiving an unlearning request from a given client, IFU identifies the optimal FL iteration from which FL has to be reinitialised, with statistical unlearning guarantees defined by Definition \ref{def:DP_adpated}. We extend the theory of IFU to account for the practical scenario of sequential unlearning (SIFU), where the server receives a series of forgetting request of one or more clients. We prove that SIFU can unlearn a series of forgetting requests while satisfying our unlearning guarantees, and demonstrate the effectiveness of our methods on a variety of tasks and dataset.
An additional contribution of this work consists in a new theory for bounding the clients contribution in FL. The server can compute this bound for every client without asking for any additional computation and communication. The theoretical justification of this approach relies on the linear approximation of the clients' loss function, and its relevance is here demonstrated across several benchmarks. Future extensions of the work will focus on generalizing our unlearning framework to more general settings.
\section{Alternative SGD}
We first formalize the learning and unlearning problem. We consider $\Lcal_1$ the averaged loss over a dataset of $n$ samples. Without loss of generality we consider removing the sample with index $n$ from the dataset which gives the new loss function $\Lcal_2$. Hence, we have
\begin{equation}
\Lcal_1(\theta)
= \frac{1}{n}\sum_{i=1}^{n} l(\theta, X_i, y_i)
\text{ and }
\Lcal_2(\theta)
= \frac{1}{n-1}\sum_{i=1}^{n-1} l(\theta, X_i, y_i)
.
\end{equation}
We denote by $\theta_1^*$ and $\theta_2^*$ the respective minimum of $\Lcal_1$ and $\Lcal_2$. For simplicity, we denote $l_i(\theta) \coloneqq l(\theta, X_i, y_i)$. Retraining is done on amount $K$ of gradient descents. In practice for computation concerns, optimizations steps have to use GD instead of SGD. Let us consider $m$ to be the batch size. The variance between the gradient estimator and the true gradient is shown to slow down the learning process. We thus propose another gradient method enabling faster optimization and thus faster unlearning.
We can express the gradient of $\Lcal_2$ in function of the one of $\Lcal_1$ as
\begin{equation}
\nabla\Lcal_2(\theta)
= \frac{1}{n-1}\sum_{i=1}^{n-1} \nabla l_i(\theta)
= \frac{n}{n-1} \nabla\Lcal_1 (\theta) - \frac{1}{n-1}\nabla l_n(\theta)
.
\end{equation}
We note that if the training has been properly done then the learnt model is $\theta_1^*$ and therefore, the unlearning process starts with $\nabla\Lcal_2(\theta_1^*) = - \frac{1}{n-1}\nabla l_n(\theta_1^*)$ showing that the deleted data is the only sample with importance when forgetting this sample from the learnt model.
\subsection*{Expansion}
Let us consider $\theta$ the model obtained by training with algorithm $\Acal$, and the perturbation $\delta$ brought to $S$, the resulting trained model is
\begin{equation}
\Acal(S + \delta)
= \Acal(S)
+ \delta \nabla \Acal(S)
\end{equation}
$\delta_i$ is the modification of sampled $i$ and $S_i=(X_i, y_i)$ is sample $i$. We define $u$ the set of samples to forget. We consider there are $K$ classes. We define
\begin{equation}
\delta_i =
\begin{cases}
\EE{X \sim X_i | y_i =y_k, i \in S'}{X} - X_k
& \text{if } k\in u\\
0
& \text{otherwise}
\end{cases}
\end{equation}
\section{Experiments}
\label{sec:experiments}
\begin{figure*}[ht]
\begin{centering}
\includegraphics[width=\linewidth]{./plots/fig_SIFU_every_dataset.pdf}
\end{centering}
\caption{
Total amount of aggregation rounds (1\textsuperscript{st} row) and model accuracy of unlearned clients (2\textsuperscript{nd} row) for MNIST, FashionMNIST, CIFAR10, CIFAR100, and CelebA (the lower the better).
The server runs a federated routine with $M=100$ clients, and unlearns 10 of them at each unlearning request ($R=3$).
}
\label{fig:SIFU}
\end{figure*}
In this section, we experimentally demonstrate the effectiveness of SIFU on a series of benchmarks introduced in Section \ref{subsec:experimental_setup}. In Section \ref{subsec:experimental_results}, we illustrate and discuss our experimental results.
Results and related code are publicly available at URL.
\subsection{Experimental Setup}
\label{subsec:experimental_setup}
\textbf{Datasets.} We report experiments on reduced versions of MNIST \citep{MNIST}, FashionMNIST \citep{FashionMNIST}, CIFAR-10 \citep{CIFAR-10}, CIFAR-100 \citep{CIFAR-10}, and CelebA \citep{CelebA}. For each dataset, we consider $M=100$ clients, with 100 data points each. For MNIST and FashionMNIST, each client has data samples from only one class, so that each class is represented in 10 clients only. For CIFAR10 and CIFAR100, each client has data samples with ratio sampled from a Dirichlet distribution with parameter 0.1 \citep{FL_and_CIFAR_dir}.
Finally, in CelebA, clients own data samples representing the same celebrity. With these five datasets, we consider different level of heterogeneity based on label and feature distribution.
\textbf{Models.}
For MNIST, we train a logistic regression model to consider a convex classification problem, while, for the other datasets, we train a neural network with convolutional layers followed by fully connected ones. More details on the networks are available in Appendix \ref{app:sec:experiments}.
\textbf{Unlearning schemes.}
In addition to SIFU, we consider the following unlearning schemes from the state-of-the-art: \textsc{Scratch}, where retraining of a new initial model is performed on the remaining clients; \textsc{Fine-Tuning}, where retraining is performed on the current global model with the remaining clients; \textsc{Last} \citep{DescentToDelete}, where retraining is performed on the remaining clients via perturbation of the final FL global model; \textsc{DP} \citep{DP_book}, where training with every client is performed with differential privacy, and \textsc{FedAccum} \citep{FedEraser}, where retraining is performed on the current global model from which the server removes the updates of the clients to unlearn, by re-aggregating the parameter updates
of clients that were stored by the server
across FL iterations. We provide in Appendix \ref{app:sec:FedAccum} the pseudo-code of \textsc{FedAccum} with the notation of our paper. We remind that \textsc{FedAccum} does not provide quantitative guarantees of the unlearning procedure, and requires the server to store the full sequence of models during the FL procedure.
\textbf{Experimental scenario.}
We consider a sequential unlearning scenario in which the server performs the FL training procedure and then receives $R=3$ sequential unlearning requests to unlearn 10 random clients per request. In the special case of MNIST and FashionMNIST, the server must unlearn 10 clients owning the same class.
The server orchestrates each unlearning scheme through retraining until the global model accuracy on the remaining clients exceeds a fixed value specific to each dataset. We set the minimum number of 50 aggregation rounds, and a maximum budget of 10000 rounds when the stopping accuracy criterion is not met.
Each unlearning method is applied with the same hyperparameters, i.e. stopping accuracy, local learning rate $\eta$, and amount of local work $K$ (Appendix \ref{app:sec:experiments}). We define the set of clients requesting unlearning as:
\begin{equation}
F_r
= \cup_{s=1}^r W_s
.
\label{eq:F_r}
\end{equation}
In our experimental scenario, we have $|F_0|=0$ during training and $|F_1|=10$, $|F_2|=20$, and $|F_3|=30$ after each unlearning request.
\textbf{Unlearning quantification.}
We verify the success of an unlearning scheme with two metrics: (a) the amount of server aggregation rounds needed for retraining, and (b) the resulting model accuracy on the unlearned clients. we note that, by construction, \textsc{Scratch} perfectly unlearns the clients from a request $W_r$. Therefore, we consider an unlearning scheme successful if it reaches similar accuracy of \textsc{Scratch} with less aggregation rounds, when tested on the data samples of $F_r$.
\subsection{Experimental Results}
\label{subsec:experimental_results}
\begin{figure}
\begin{centering}
\includegraphics[width=\linewidth]{./plots/fig_SIFU_small_backdoored.pdf}
\end{centering}
\caption{Total amount of aggregation rounds (1\textsuperscript{st} row) and model accuracy of unlearned clients (2\textsuperscript{nd} row) for the unlearning of watermarked data from CIFAR100 and CelebA.
}
\label{fig:SIFU_small_backdoored}
\end{figure}
Figure \ref{fig:SIFU} shows that for every dataset and unlearning index, \textsc{Fine-Tuning}, \textsc{FedAccum}, and DP provide similar model accuracy for the unlearned clients in $F_r$ (Figure \ref{fig:SIFU}-2\textsuperscript{nd} row), albeit significantly higher than for \textsc{Scratch}, the unlearning standard. Noteworthy, unlearning with \textsc{Fine-Tuning}, \textsc{FedAccum}, and DP results in significantly less aggregation rounds than \textsc{Scratch} (Figure \ref{fig:SIFU}-1\textsuperscript{st} row).
We note that SIFU and \textsc{Scratch} lead to similar unlearning results, quantified by low accuracy on the unlearned clients $F_r$ (Figure \ref{fig:SIFU}-2\textsuperscript{nd} row), while SIFU unlearns these clients in roughly half the amount of aggregation rounds needed for \textsc{Scratch} (Figure \ref{fig:SIFU}-1\textsuperscript{st} row).
However, the model accuracy of SIFU is slightly higher than the one of \textsc{Scratch}, with perfect overlap only for FashionMNIST. This behavior is natural and can be explained by our privacy budget $(\epsilon, \delta)$, which trades unlearning capabilities for effectiveness of the retraining procedure. With highest unlearning budget, i.e. $\epsilon =0$ and $\delta=0$, SIFU would require to retrain from the initial model ${\bm{\theta}}_0^0$, thus reducing to \textsc{Scratch}.
Finally, we observed that when unlearning with \textsc{Last}, the retrained model always converged to a local optimum with accuracy inferior to our target after $10000$ aggregation rounds. This behavior is likely due to the difficulty of calibrating the noise perturbation due to the numerous heterogeneous contributions of the clients. For this reason, we decided to exclude \textsc{Last} from the plots of Figure \ref{fig:SIFU}.
\subsection{Verifying Unlearning through Watermarking}
\label{subsec:exp_watermark}
The work of \cite{TowardsProbabilisticVerification} proposes an adversarial approach to verify the efficiency of an unlearning scheme based on watermarking.
We apply here this method to our federated setting, in which watermarking is operated by each client by randomly assigning on all its data samples the maximum possible value to 10 given pixels.
To ensure that clients' data heterogeneity is only due to the modification of the pixels, we define heterogeneous data partitioning across clients by randomly assigning the data according to a Dirichlet distribution with parameter 1. Figure \ref{fig:SIFU_small_backdoored} shows our results for this experimental scenario on CIFAR100 and CelebA, while Appendix \ref{app:sec:experiments} provides similar results for MNIST, FashionMNIST and CIFAR10. We retrieve the same conclusions drawn from Figure \ref{fig:SIFU}. SIFU and \textsc{Scratch} have similar accuracies on the unlearned clients in $F_r$, to demonstrate the effectiveness of the unlearning. Moreover, SIFU unlearns these clients in significantly less aggregation rounds than \textsc{Scratch}.
\subsection{Impact of the noise perturbation on SIFU}
\label{subsec:exp_nosie_std}
Appendix \ref{app:sec:experiments} illustrates the impact of the perturbation amplitude $\sigma$ on convergence speed when unlearning with SIFU. We note that when unlearning with a small $\sigma$, SIFU has identical behavior to \textsc{Scratch} as the unlearning is applied to the initial random model ${\bm{\theta}}_0^0$. With large values of $\sigma$, SIFU performs instead identically to \textsc{Last} and applies the unlearning to the finale global model ${\bm{\theta}}_r^{N_r}$.
\section{Sequential FU with SIFU}
\label{sec:SIFU}
\input{./tex/SIFU_algo}
\input{./tex/example_SIFU}
In this section, we extend IFU to the sequential unlearning setting with Sequential IFU (SIFU). With Algorithm \ref{alg:SIFU}, SIFU is designed to satisfy a series of $R$ unlearning requests $\{W_r\}_{r=1}^R$, where $W_r$ is the set of clients to unlearn at request index $r$.
SIFU generalizes IFU for which $R=1$ and $W_1 = \{c\}$. We provide an illustration of SIFU with an example in Figure \ref{fig:example_with_R3}.
The notations introduced thus far need to be generalized to account for our series of unlearning requests $W_1, W_2, \ldots, W_R$. Global models are now referenced by their coordinates $(r, n)$, i.e. ${\bm{\theta}}_r^n$, which represent the unlearning request index $r$ and the amount of server aggregations $n$ performed during the retraining. Hence, ${\bm{\theta}}_r^0$ is the initialization of the model when unlearning the clients in $W_r$.
Also, we consider that the retraining at request index $r$ requires $N_r$ server aggregations on the remaining clients.
Therefore, by construction, ${\bm{\theta}}_r^{N_r}$ is the model obtained after using SIFU to $(\epsilon, \delta)$-unlearn the sequence of unlearning requests $\{W_s\}_{s=1}^r$. Finally, we define $I_r$ as the set of remaining clients after unlearning request $r$, i.e. $I_r \coloneqq I\setminus \cup_{s=1}^r W_s = I_{r-1} \setminus W_r$ with $I_0 = I$.
We extend the bounded sensitivity (\ref{eq:def_Psi}) with $\Psi_r(n, i)$ to compute the metric of client $i$ at unlearning index $r$ with
\begin{equation}
\Psi_r(n, i)
\coloneqq \sum_{s=0}^{n-1}\norm{\Delta(I_r, {\bm{\theta}}_r^s) - \Delta(I_r\setminus\{i\}, {\bm{\theta}}_r^s)}_2
\label{eq:def_Psi_extended}
.
\end{equation}
When unlearning client $c$ at $r=1$, the metric at $r=0$ is equivalent to the previous definition of $\Psi$. Also, when computing the metric on a client already unlearned, i.e. $i \notin I_r$, we retrieve $\Psi_r(n, i) = 0$.
Finally, for a set of clients $S$, we generalize the bounded sensitivity (\ref{eq:def_Psi_extended}) to
\begin{equation}
\Psi_r(n, S)
= \max_{i \in S}\Psi_r(n, i)
\label{eq:def_Psi_extended_set}
.
\end{equation}
With SIFU, the selection of the unlearning index $T$ for a request $r$ depends of the past history of unlearning requests.
To keep track of the unlearning history, we introduce the oracle $O(r)$ which returns at each request $r$ the coordinates of the history of global models where unlearning has been applied. These coordinates represent the nodes of the training history across unlearning requests (Figure \ref{fig:example_with_R3}). With reference to Figure \ref{fig:example_with_R3}, we start with the original sequence of global models obtained at each FL round, i.e. (${\bm{\theta}}_0^0, \ldots, {\bm{\theta}}_0^{N_0})$. Similarly to IFU, the first unlearning request requires to identify the unlearning index $T_1$ for which the corresponding global model ${\bm{\theta}}_0^{T_1}$ must be perturbed to obtain ${\bm{\theta}}_1^0$ and retrained until convergence, i.e. up to ${\bm{\theta}}_1^{N_1}$. The oracle is updated with the coordinates of the branching $O(1) = \{(0, T_1)\}$, and the current training history is now $({\bm{\theta}}_0^0, \ldots, {\bm{\theta}}_0^{T_1}, {\bm{\theta}}_1^0, \ldots, {\bm{\theta}}_1^{N_1})$. At the next unlearning request, the server needs to identify the coordinates $(\zeta_r, T_r)$ in the new training history for which unlearning must be applied on the model ${\bm{\theta}}_{\zeta_r}^{T_r}$ to obtain ${\bm{\theta}}_r^0 = {\bm{\theta}}_{\zeta_r}^{T_r} + \Ncal(0, \sigma^2{\bm{I}}_{\bm{\theta}})$. The oracle is subsequently updated with the new set of nodes describing the new branching in the training history.
By construction, we have $\zeta_r \le r-1$ and $T_r \le N_{\zeta_r}$.
More precisely, we define the index $\zeta_r$ associated to the first coordinate in $O(r-1)$ for which the bounded sensitivity (\ref{eq:def_Psi_extended}) of clients in $W_r$ exceeds $\Psi^*$.
Formally, we have
\begin{align}
\zeta_r
\coloneqq \min_s \{ &s : \Psi_s(n, W_r) > \Psi^* \text{ and } (s, n) \in O(r-1), \nonumber\\
&r-1 \}
.
\label{eq:zeta_r}
\end{align}
The definition of $T_r$ follows directly from the one of $\zeta_r$. Similarly as for IFU, the unlearning index $T_r$ quantifies the maximum amount of server aggregations starting from the unlearning request index $\zeta_r$ such that the bounded sensitivity $\Psi_{\zeta_r}(n, W_r)$ on this global model is inferior to $\Psi^*$, i.e.
\begin{equation}
T_r
\coloneqq \argmax_n \{\Psi_{\zeta_r}(n, W_r) \le \Psi^* \}
.\label{eq:T_SIFU}
\end{equation}
Finally, we update the oracle $O(r-1)$ to $O(r)$ with the following recurrent equation
\begin{equation}
O(r) = \{(s, n) \in O(r-1) \text{ s.t. } s < \zeta_r, (\zeta_r, T_r) \}
\label{eq:oracle_recurrent}
.
\end{equation}
Theorem \ref{theo:zeta} shows that for a model trained with SIFU after a given training request $r$, $(\epsilon, \delta)$-unlearning is guaranteed for every client belonging to the sets $W_s$, $s\leq r$.
\begin{theorem}
\label{theo:zeta}
The model ${\bm{\theta}}_r^{N_r}$ obtained with SIFU satisfies $(\epsilon, \delta)$-unlearning for every client in current and previous unlearning requests, i.e. clients in $\cup_{s=1}^r W_s$.
\end{theorem}
\begin{proof}
See Appendix \ref{app:sec:SIFU_convergence}.
\end{proof}
\subsection{A more general case}
Let us consider that clients loss functions are $\mu$ strongly convex. We denote by ${\bm{\theta}}^*$ and ${\bm{\theta}}_{-k}^*$ respectively the optimum of the loss function with all the clients and all but client $k$. We can obtain the following convergence guarantees
\begin{equation}
\norm{{\bm{\theta}}^N - {\bm{\theta}}^*}^2
\le ( 1 - \eta \mu)^N \norm{{\bm{\theta}}^0 - {\bm{\theta}}^*}^2
\end{equation}
As such, we have
\begin{equation}
\norm{{\bm{\theta}}_{-k}^N - {\bm{\theta}}^N}
\le
\norm{{\bm{\theta}}_{-k}^N - {\bm{\theta}}_{-k}^*}
+ \norm{{\bm{\theta}}^N - {\bm{\theta}}^*}
+ \norm{{\bm{\theta}}_{-k}^* - {\bm{\theta}}^*}
\end{equation}
which gives
\begin{align}
\norm{{\bm{\theta}}_{-k}^N - {\bm{\theta}}^N}
&\le
\sqrt{2} ( 1 - \eta \mu)^{N/2}
\max(\norm{{\bm{\theta}}^0 - {\bm{\theta}}^*}^2, \norm{{\bm{\theta}}^0 - {\bm{\theta}}_{-k}^*}^2)
\nonumber\\
&+ \norm{{\bm{\theta}}_{-k}^* - {\bm{\theta}}^*}
\end{align}
While this result holds for more functions the main issue lies in the provided bound. Indeed, it depends of the new optimum which we do not know.
\section{Background and Related Work}
\label{sec:background}
In Section \ref{subsec:unlearning_baselines}, we introduce the state-of-the art behind Machine Unlearning, while in Section \ref{subsec:FedAvg}, we introduce FL and \textsc{FedAvg}.
Finally, we introduce Federated Unlearning (FU) in Section \ref{subsec:federated_unlearning}.
\subsection{Machine Unlearning}
\label{subsec:unlearning_baselines}
\input{./tex/unlearning_baselines}
\subsection{Federated Optimization and \textsc{FedAvg}}
\label{subsec:FedAvg}
\input{./tex/formalization}
\subsection{Federated Unlearning}
\label{subsec:federated_unlearning}
\input{./tex/background_federated_unlearning}
\section{Unlearning a FL client with IFU}
\label{sec:theory}
In this section, we develop our theory for the scenario where a model is trained with \textsc{FedAvg} on the set of clients $I$, after which a client $c$ requests unlearning of its own data.
In Section \ref{subsec:bounding_drift}, we define the sensitivity of the global model with respect to a client's contribution, and provide a bound relating this sensitivity to the FL procedure.
In Section \ref{subsec:tighter_sensitivity}, we provide a tighter model sensitivity for some specific FL applications.
Using Theorem \ref{theo:diff_bound}, we introduce in Section \ref{subsec:unlearning_guarantees} the perturbation procedure to unlearn a client $c$ from the model trained with \textsc{FedAvg} (Theorem \ref{theo:noise_DP}).
Finally, using Theorem \ref{theo:noise_DP}, we introduce Informed Federated Unlearning (IFU) (Algorithm \ref{alg:unlearning_ours}).
\subsection{Theorem \ref{theo:diff_bound}, Bounding the Model Sensitivity}
\label{subsec:bounding_drift}
\input{./tex/diff_grad}
\subsection{Improving the Tightness of the Sensitivity Bound}
\label{subsec:tighter_sensitivity}
\input{./tex/corollary}
\subsection{Satisfying Definition \ref{def:DP_adpated}}
\label{subsec:unlearning_guarantees}
\input{./tex/unlearning_client}
\subsection{Informed Federated Unlearning (IFU)}
\label{subsec:unlearning_ours}
\input{./tex/unlearning_procedure}
\input{./tex/generalization}
|
2004.10492
|
\section{Introduction}
Source localization using measurements from spatially separated passive sensors has turned into a go-to scheme in many location-based services including target tracking \cite{FHoeflinger,JBordoy}, human-computer interaction \cite{VGReju}, and Internet of Things \cite{SLi}. Among plentiful measurement models, the time-of-arrival (TOA) and time-difference-of-arrival (TDOA), especially the latter that eliminates the need for synchronization between the source and sensors \cite{YHuang,NOno1,NOno2,LLin}, is perhaps the most widely used owing to its high accuracies. For an insight into the rationale of single source localization, the uninitiated readers are referred to \cite{So,Guvenc1} and the references therein.
One of the key issues in source localization is the so-called non-line-of-sight (NLOS) propagation, which commonly arises in real environments (e.g., urban canyons and indoor sites), and can adversely degrade the positioning performance if left untreated \cite{Guvenc1,STomic1,STomic2,GWang,HChen,WXiong,AProrok,GWang2,WWang,GWang3,ZSu,WXiong2}. Over the past decade, a vast variety of advanced NLOS mitigation methods have been developed for TOA-based localization: the worst-case least squares (LS) \cite{STomic1}, joint estimation of the source location and a balancing parameter \cite{STomic2,GWang,HChen}, and robust multidimensional similarity analysis \cite{WXiong}, to name a few. These approaches are practically more favorable than the straightforward maximum likelihood (ML) technique \cite{AProrok}, as their implementations rely on neither the path status nor the specified error distribution, but merely a few assumptions regarding the measurement noise and/or NLOS errors. Different from what one might expect, extension of the aforementioned TOA-based schemes to the TDOA case is not at all a trivial task. This is mainly because the possible NLOS error in a TDOA measurement is essentially the difference of those occurred in two related TOA measurements, and hence may not necessarily be a positive outlier anymore. To settle this matter, the authors of \cite{GWang2} follow again the worst-case rule, but this time the upper bound is imposed on the magnitude of NLOS errors. As a modification to \cite{GWang2} which treats each measurement equally, additional path status information is utilized in \cite{WWang} for placing less reliance on the error-prone measurements. More recently, the authors of \cite{GWang3} point out that the formulations in \cite{GWang2} and \cite{WWang} may not perform well due to the loose upper bound and inexact triangle inequality, whereupon they put forward several refinements to alleviate the impacts. Despite considerable resistance of the worst-case criterion to NLOS errors, solving the resultant robust LS problems in \cite{GWang2,WWang,GWang3}, however, involves the use of convex optimization such as second-order cone programming (SOCP) and semidefinite programming (SDP), which will bring in heavy computational burdens. On the other hand, whereas the TDOA model with a structure more complex than the TOA counterpart can impede the formulation derivation \cite{TLe}, the idea of model transformation is suggested in \cite{ZSu,EXu}. Such a tactic is well-motivated to the extent that the metrics of TOA and TDOA differ by only one degree of freedom, i.e., the time at which the signal departs from the source. Moreover, the selection of a proper reference sensor is no longer a prerequisite after the model transformation. Nevertheless, the constrained LS estimator with NLOS mitigation in \cite{ZSu} still ends up with solving a complicated SDP problem.
Conventional numerical methods for optimization are often realized and run on digital computers. Consequently, the computing time can grow dramatically with the increase of problem size, implying less effectiveness in time-varying scenarios. To overcome this drawback, employing physically implementable recurrent neural networks by which distributed, parallel, and real-time computation is enabled has become a promising alternative for tackling various classes of mathematical programming problems \cite{SZhang,DTank,MPKennedy,ANazemi,XHu,ZShi2,HChe}. The mechanism is to build a dynamical system that will ultimately settle down to an equilibrium point, at which the optimal solution to the problem is obtained from the outputs, given suitable inputs as the initial point. In particular, the Lagrange programming neural networks (LPNN) \cite{SZhang} developed based on the gradient model \cite{MPKennedy} and Lagrange multiplier theory has provided a general framework for coping with the nonlinear constrained optimization problems. With the use of an augmented Lagrangian function, the LPNN model can further be empowered to handle nonconvex optimization, and recent studies have successfully utilized the augmented LPNN to solve a mass of source localization problems \cite{HWang,ZHan,JLiang,ZFHan,CJia}. However, the standard LPNN framework is unfriendly towards the presence of inequality constraints, since it requires introducing slack variables to convert them into the equality ones as a preprocessing step. This is apparently not a fine option if a large number of inequality constraints are involved, in view of the fact that promptness and real-time responses are the main purposes of applying the recurrent neural networks. Unfortunately, for the sake of binding the additional nuisance variable, there do exist many inequality constraints in the model transformation approaches \cite{ZSu}, which indicates that a more efficient means of neurodynamic optimization is still a yearning in our application.
In this paper, we formulate TDOA-based source localization in NLOS environments as a nonconvex constrained optimization problem by robust model transformation, and then devise an effective and efficient neurodynamic solution to it. To start with, the least absolute deviation (LAD) (also known as (a.k.a.) the $\ell_1$-norm) criterion is adopted to achieve robustness against the bias-like NLOS error in the reconstructed TOA measurement model. For the higher-order properties in the design of dynamical system, certain smoothed approximations are made to the LAD objective function to yield a twice differentiable surrogate. Unlike most of the neurodynamic source localization approaches adapting their formulations to the standard LPNN setting (e.g., by either discarding the inequality constraints \cite{HWang,JLiang} or transforming them into the equalities \cite{ZHan,ZFHan,CJia}), we follow \cite{XHu} to redefine the augmented Lagrangian and establish a different projection-type neural network (PNN) model which can directly take the inequality constraints into account. It is worth noting that although the LPNN and PNN share the same terminology ``neural network'' with the booming deep neural networks in machine learning, they refer to totally distinct approaches and should not be mixed up with each other. The presented scheme obviates the need for acquiring any information (e.g., an upper bound \cite{GWang2,WWang,GWang3}) concerning NLOS errors or tuning the hyperparameters \cite{ZSu} beforehand, thereby resulting in a lower prior knowledge demand compared to the methods in \cite{GWang2,WWang,GWang3,ZSu}. It should be noted that though bearing some resemblance to \cite{ZSu} which also remodels the problem into a TOA framework, our work should be distinguished from it, as neither the $\ell_2$-space-based objective function nor the time-consuming SDP is counted on any longer. In addition, our neurodynamic solution is shown to be computationally more efficient even when it is executed on the general purpose digital computers.
The remainder of the paper is organized as follows. Section \ref{PF} states the localization problem and introduces the robust model transformation formulation. Section \ref{NN} reviews the classical LPNN framework and defines the neural dynamics of the presented PNN, whose stability and convergence properties are then briefly discussed in Section \ref{CSCA}. To ensure a fair comparison between the proposed neurodynamic method and the state-of-the-art convex optimization counterparts in terms of computational expense, its algorithmic complexity when implementing in a numerical fashion is also analyzed in Section \ref{CSCA}. Section \ref{SR} evaluates the performance of our approach through computer simulations. Finally, conclusions are drawn in Section \ref{CC}.
\section{Problem formulation}
\label{PF}
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{xiong1}
\caption{Signal timestamp diagram of TDOA-based localization system.}
\label{FIG1}
\end{figure}
Consider a TDOA-based localization system in $k$-dimensional space ($k = 2$ or $3$) with $L \geq k+1$ sensors and a single source. The known sensor positions and unknown source location are denoted by $\bm{x}_i \in \mathbb{R}^k$ (for $i = 1,2,...,L$) and $\bm{x} \in \mathbb{R}^k$, respectively. As demonstrated in Fig. \ref{FIG1}, the local clocks of the sensors are well synchronized such that the received signal timestamp $t_i$ (for $i = 1,2,...,L$) can be collected from the $i$th sensor, whereas the time at which the signal is emitted from the source, $t_0$, is unknown because there is no synchronization between the source and sensors. Without loss of generality, the first sensor is designated as the reference and the TDOA measurements are modeled as
\begin{align}{\label{TDOA}}
t_{i,1} = \frac{1}{c} ({\| \bm{x} - \bm{x}_i \|}_2 - {\| \bm{x} - \bm{x}_1 \|}_2 + n_{i,1} + b_{i,1}) = t_i - t_1,~~i = 2,3,...,L,
\end{align}
where $c$ denotes the signal propagation speed, ${\| \cdot \|}_2$ represents the $\ell_2$-norm of a vector, $n_{i,1} = n_i - n_1$ and $b_{i,1} = q_i - q_1$ (both for $i=2,3,...,L$) are the measurement noise and possible NLOS error in the corresponding range difference measurement, respectively, $n_i$ (for $i = 1,2,...,L$) is assumed to be zero-mean Gaussian noise with variance $\sigma_i^2$, and $q_i$ (for $i = 1,2,...,L$) equals either $0$ or a positive bias error $e_i$, contingent on whether the path between the $i$th sensor and source is line-of-sight (LOS) or NLOS. Before proceeding with the formulation derivation, we decompose the TDOA measurements in (\ref{TDOA}) into the related TOA components
\begin{equation}
t_i - t_0 = \frac{1}{c}({\| \bm{x} - \bm{x}_i \|}_2 + n_{i} + q_{i}),~~i = 1,2,...,L\label{TOA}
\end{equation}
by making as if the synchronization between the source and sensors is established, namely, including $t_0$ as a variable of interest.
Exhibiting less sensitivity to outliers than the conventional $\ell_2$-norm criterion, the $\ell_1$-norm has been widely utilized in robust signal processing, with low-rank matrix completion under impulsive noise circumstances \cite{AEriksson}, robust principal component analysis \cite{ECandes}, and sensor network localization under Laplacian noise assumption \cite{POE} being a few representative applications of it. Borrowing the similar idea, we employ the LAD cost function as the objective of minimization to mitigate the positive bias errors in (\ref{TOA}):
\begin{equation*}
\min_{t_0, \bm{x}}~\sum_{i=1}^{L} \left| (t_i - t_0) c - {\| \bm{x} - \bm{x}_i \|}_2 \right|.
\end{equation*}
To bind the nuisance variable $t_0$, the temporal constraints\footnote{The temporal constraints are premised on $t_i > 0$ so as to be meaningful.}
\begin{equation}
0 \leq t_0 \leq t_i,~~i = 1,2,...,L,\label{TEMC}
\end{equation}
geometrical constraints by the triangle inequalities \cite{OJean}
\begin{equation}
(t_i - t_0)c + (t_j - t_0)c \geq {\| \bm{x}_i - \bm{x}_j \|}_2,~~i \not= j,~i,j = 1,2,...,L,\label{TRIC}
\end{equation}
and general consensus that $e_i$ is much greater than $|n_i|$ are thereupon incorporated into the formulation, yielding:
\begin{subequations}{\label{CLAD}}
\begin{align}
\min_{t_0, \bm{x}, \bm{d}} &~\sum_{i=1}^{L} \left| (t_i - t_0) c - d_i \right|\nonumber\\
\textup{s.t.}~~&d_i^2 = {\| \bm{x} - \bm{x}_i \|}_2^2,~~i = 1,2,...,L,\\
&d_i \geq 0,~~i = 1,2,...,L,\\
&\text{(\ref{TEMC})},~\text{(\ref{TRIC})},\nonumber\\
&(t_i - t_0) c \geq d_i,~~i = 1,2,...,L,
\end{align}
\end{subequations}
where $\bm{d} = \left[ d_1,d_2,...,d_L \right]^T \in \mathbb{R}^L$ is a vector containing the auxiliary variables for source-sensor distances, and the constraint $d_i = {\| \bm{x} - \bm{x}_i \|}_2$ (for $i = 1, 2, ..., L$) is replaced by (\ref{CLAD}a) and (\ref{CLAD}b) in the quadratic form to avoid ill-posing \cite{ZHan}. Falling into the category of nonlinear and nonconvex constrained optimization problems, (\ref{CLAD}) is appropriately tackled in the next section by constructing a dynamical system whose equilibrium state is reached at a Karush-Kuhn-Tucker (KKT) point of the underlying problem.
\section{Preliminaries and proposed neurodynamic method}
\label{NN}
Assume that we have a nonlinear programming problem with equality constraints:
\begin{align}{\label{ENP}}
\min_{\bm{z}}~f(\bm{z}),\quad\textup{s.t.}~~\bm{h}(\bm{z}) = \bm{0}_M,
\end{align}
where $\bm{z}\in \mathbb{R}^N$, $f:\mathbb{R}^N \rightarrow \mathbb{R}$, $\bm{h}(\bm{z}) = \left[ h_1 (\bm{z}),h_2 (\bm{z}),...,h_M (\bm{z}) \right]^T \in \mathbb{R}^M$ is an $M$-dimensional vector-valued function of $N$ variables with $M \leq N$, the functions $f(\bm{z})$ and $h_i (\bm{z})$ (for $i = 1,2,...,M$) are supposed to be twice differentiable, and $\bm{0}_M \in \mathbb{R}^M$ denotes an all-zero vector of length $M$. In a nutshell, the widely used LPNN approach \cite{SZhang} deals with (\ref{ENP}) by invoking the Lagrange multiplier theory and designing a neurodynamic model whose time-domain transient behavior is defined as
\begin{subequations}{\label{dynamicsENP}}
\begin{align}
\frac{d\bm{z}}{dt} &= -\bm{\nabla}_{\bm{z}} \mathcal{L}_{\star}(\bm{z},\bm{\lambda}),\\
\frac{d{\bm{\lambda}}}{dt} &= \bm{\nabla}_{\bm{\lambda}} \mathcal{L}_{\star}(\bm{z},\bm{\lambda}),
\end{align}
\end{subequations}
where $\bm{\nabla}_{\bm{z}} (\cdot) \in \mathbb{R}^N$ denotes the gradient of a function at $\bm{z}$, $\bm{\lambda} \in \mathbb{R}^M$ is a vector containing the Lagrange multipliers for the constraints in (\ref{ENP}), $\bm{z}$ and $\bm{\lambda}$ are assigned physical meanings as the activities of the variable and Lagrangian neurons, respectively, and $\mathcal{L}_{\star}(\bm{z},\bm{\lambda})$ can be either the Lagrangian or augmented Lagrangian of (\ref{ENP}), differing in the stability of the built system under nonconvexity. In the dynamic process of the LPNN, (\ref{dynamicsENP}a) ensures that the value of $\mathcal{L}_{\star}(\bm{z},\bm{\lambda})$ decreases over time, whereas (\ref{dynamicsENP}b) plays a role in leading the solution into the feasible region. After performing appropriate initialization of the variable and Lagrangian neurons, the network governed by (\ref{dynamicsENP}) is expected to approach an equilibrium point satisfying the first-order necessary conditions of optimality (a.k.a. the KKT conditions).
It is obvious that (\ref{CLAD}) does not conform to the paradigm shown in (\ref{ENP}) owing to the existence of numerous inequality constraints. Instead of introducing slack variables \cite{ZHan,ZFHan} to fit in with (\ref{ENP}), in the following we seek for a simpler way to directly handle the general constrained optimization problem (GCOP)
\begin{align}{\label{GCOP}}
\min_{\bm{z}}~f(\bm{z}),\quad\textup{s.t.}~~\bm{g}(\bm{z}) \leqq \bm{0}_K,~~\bm{h}(\bm{z}) = \bm{0}_M,
\end{align}
where the definitions pertaining to $\bm{z}$, $\bm{\lambda}$, $f$, and $\bm{h}$ remain the same as those in (\ref{ENP}) except that $M \leq N$ is no longer requested, the $K$-dimensional vector-valued function $\bm{g}(\bm{z}) = \left[ g_1 (\bm{z}),g_2 (\bm{z}),...,g_K (\bm{z}) \right]^T \in \mathbb{R}^K$ is assumed to be twice differentiable, and the vector inequality $\bm{a} \leqq \bm{b}$ means each component of $\bm{a}$ is less than or equal to each corresponding component of $\bm{b}$.
The Lagrangian of (\ref{GCOP}) is $\mathcal{L}(\bm{z},\bm{\nu}) = f(\bm{z}) + \bm{\mu}^T \bm{g}(\bm{z}) + \bm{\lambda}^T \bm{h}(\bm{z})$. Here, we have $\bm{\nu} = \left[ \bm{\mu}^T, \bm{\lambda}^T \right]^T \in \mathbb{R}^{K+M}$, where $\bm{\mu} = \left[ \mu_1,\mu_2,...,\mu_K \right]^T \in \mathbb{R}^K$ and $\bm{\lambda} = \left[ \lambda_1,\lambda_2,...,\lambda_M \right]^T \in \mathbb{R}^M$ are the vectors containing Lagrange multipliers for the inequality and equality constraints in (\ref{GCOP}), respectively. The KKT conditions \cite{JNocedal} for (\ref{GCOP}) that a pair $(\bm{z}^{*},\bm{\nu}^{*})$ satisfies\footnote{In this paper, we stipulate that the asterisk in the superscript of a vector is by default applied to each component of the vector.}, namely, the first-order necessary conditions for $\bm{z}^{*}$ to be a local minimizer of (\ref{GCOP}), are
\begin{subequations}{\label{KKT}}
\begin{numcases}{}
\bm{\nabla}_{\bm{z}} \mathcal{L}(\bm{z}^{*},\bm{\nu}^{*}) = \bm{0}_N,\\
g_i (\bm{z}^{*}) \leq 0, \mu_i^{*} \geq 0, \mu_i^{*} g_i (\bm{z}^{*}) = 0,~~i=1,2,...,K,\\
\bm{h}(\bm{z}^{*}) = \bm{0}_M.
\end{numcases}
\end{subequations}
Analogous to the strategy taken by \cite{XHu}, we point out that the KKT conditions in (\ref{KKT}) actually share the same solution set with
\begin{subequations}{\label{KKT2}}
\begin{numcases}{}
\bm{\nabla}_{\bm{z}} \mathcal{L}_{\rho} (\bm{z}^{*},\bm{\nu}^{*}) = \bm{0}_N,\\
\left[ \mu_{i}^{*} + \alpha g_i (\bm{z}^{*}) \right]^{+} = \mu_{i}^{*},~~i=1,2,...,K,\\
\bm{h}(\bm{z}^{*}) = \bm{0}_M,
\end{numcases}
\end{subequations}
where $\mathcal{L}_{\rho}(\bm{z},\bm{\nu}) = \mathcal{L}(\bm{z},\bm{\nu}) + \frac{\rho}{2} \Big\{ \sum_{i = 1}^{K} \left[ \mu_{i} g_i (\bm{z}) \right]^2 + \sum_{i = 1}^{M} \left[ \lambda_{i} h_i (\bm{z}) \right]^2 \Big\}$ is a redefined augmented Lagrangian of (\ref{GCOP}), the scale factor $\alpha > 0$ indicates the convergence rate of the neural network and we let $\alpha = 1$ in this paper for simplicity, $\rho > 0$ is the augmented Lagrangian parameter, and the operator
\begin{equation}{\label{PO}}
\left[ \cdot \right]^{+} = \max(\cdot,0)
\end{equation}
is introduced to re-express the primal feasibility, dual feasibility, and complementarity conditions for the inequality constraints in a projection form. The equivalence between the solution sets of (\ref{KKT}) and (\ref{KKT2}) is illustrated in the proposition below.
\textbf{Proposition 1.} Denote the solution sets of equations in (\ref{KKT}) and (\ref{KKT2}) by $\Omega_1$ and $\Omega_2$, respectively, then $\Omega_1 = \Omega_2$.
\textbf{Proof.} We begin with proving that (\ref{KKT}b) is true if and only if (\ref{KKT2}b) is true.
\textit{Sufficiency:}
The conditions in (\ref{KKT}b) are partitioned into two cases as: (i) $g_i (\bm{z}^{*}) = 0, \mu_i^{*} \geq 0$, and (ii) $g_i (\bm{z}^{*}) < 0, \mu_i^{*} = 0$. The equalities in (\ref{KKT2}b) can be trivially deduced in both two cases, thus the sufficiency holds.
\textit{Necessity:}
\textit{Case 1:} $\mu_{i}^{*} + \alpha g_i (\bm{z}^{*}) \geq 0$.
It follows from (\ref{KKT2}b) and (\ref{PO}) that $\left[ \mu_{i}^{*} + \alpha g_i (\bm{z}^{*}) \right]^{+} = \mu_{i}^{*} + \alpha g_i (\bm{z}^{*}) = \mu_{i}^{*}$, which subsequently implies $g_i (\bm{z}^{*}) = 0$ and $\mu_{i}^{*} \geq 0$.
\textit{Case 2:} $\mu_{i}^{*} + \alpha g_i (\bm{z}^{*}) < 0$.
Likewise, we arrive at $\left[ \mu_{i}^{*} + \alpha g_i (\bm{z}^{*}) \right]^{+} = 0 = \mu_{i}^{*}$ and $g_i (\bm{z}^{*}) < 0$.
It is evident that the conditions in (\ref{KKT}b) are formed by merging the two cases together. Therefore, the necessity is satisfied.
In this way, we now only need to prove that (\ref{KKT}a) and (\ref{KKT2}a) are equivalent to each other under the conditions in (\ref{KKT}b) and (\ref{KKT}c). The gradient of $\mathcal{L}_{\rho}(\bm{z},\bm{\nu})$ at $\bm{z}$ is calculated as
\begin{align}{\label{gradLrho}}
\bm{\nabla}_{\bm{z}} \mathcal{L}_{\rho} (\bm{z},\bm{\nu}) = \bm{\nabla}_{\bm{z}} \mathcal{L} (\bm{z},\bm{\nu}) + \rho \Bigg[ \sum_{i = 1}^{K} \mu_{i}^2 g_i (\bm{z}) \bm{\nabla}_{\bm{z}} g_i (\bm{z}) + \sum_{i = 1}^{M} \lambda_{i}^2 h_i (\bm{z}) \bm{\nabla}_{\bm{z}} h_i (\bm{z}) \Bigg].
\end{align}
Substituting the conditions in (\ref{KKT}b) and (\ref{KKT}c) into (\ref{gradLrho}) at $(\bm{z}^{*},\bm{\nu}^{*})$ produces $\bm{\nabla}_{\bm{z}} \mathcal{L}_{\rho} (\bm{z}^{*},\bm{\nu}^{*}) = \bm{\nabla}_{\bm{z}} \mathcal{L} (\bm{z}^{*},\bm{\nu}^{*})$, which verifies the equivalence between (\ref{KKT}a) and (\ref{KKT2}a). The proof is complete.
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{xiong2}
\caption{Sketch for neural network defined by (\ref{dynamicsGCOP}).}
\label{xiong2}
\end{figure}
Based on (\ref{KKT2}), a KKT point of the GCOP (\ref{GCOP}) is to be searched by employing a three-layer PNN, with its dynamical equations being given by
\begin{subequations}{\label{dynamicsGCOP}}
\begin{align}
\frac{d\bm{z}}{dt} &= -\bm{\nabla}_{\bm{z}} \mathcal{L}_{\rho} (\bm{z},\bm{\nu}),\\
\frac{d{\mu_{i}}}{dt} &= -\mu_{i} + \left[ \mu_{i} + g_i (\bm{z}) \right]^{+},~~i=1,2,...,K,\\
\frac{d{\bm{\lambda}}}{dt} &= \bm{h} (\bm{z}).
\end{align}
\end{subequations}
A simplified block diagram of how such a neural network can be implemented on hardware is sketched in Fig. \ref{xiong2}. What may be noteworthy is that (\ref{dynamicsGCOP}) can be viewed as either a projection-type extension of the standard LPNN \cite{SZhang}, a GCOP-treatable augmentation of the neurodynamic model in \cite{XHu}, or a simplification leaving out the bound constraints of that in \cite{HChe}. On this account, several existing analyses in the literature will be referenced for the property discussion on (\ref{dynamicsGCOP}) in the related sections.
In what follows, the neurodynamic system described by (\ref{dynamicsGCOP}) is exploited for working out the solution to (\ref{CLAD}). To meet the higher-order (more precisely, twice in our scenario) differentiability condition for the neural network implementation \cite{SZhang,HChe}, the absolute value function in (\ref{CLAD}) is replaced by the following smoothed robust loss function with arbitrary-order derivatives\footnote{Note that the celebrated Huber loss function which is a trade-off between the $\ell_1$- and $\ell_2$-norm \cite{RZhang} also suffers from the differentiability issues, i.e., it is only first-order differentiable \cite{KFount}.} \cite{LZhao}:
\begin{equation*}
f_1 (z) = \frac{\ln \left( \left( e^{\gamma z} + e^{-\gamma z} \right) / 2 \right)}{\gamma},
\end{equation*}
where $\gamma > 0$ is a predefined parameter and $\log(\cdot)$ denotes the logarithm operation with base $e$. For illustrative purpose, the comparison between the absolute value function $|z|$ and $f_1 (z)$ is provided in Fig. \ref{xiong3}, from which it is clearly seen that acceptable approximation can be achieved if a sufficiently large $\gamma$ is chosen.
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{xiong3}
\caption{Comparison of the functions $|z|$ and $f(z)$.}
\label{xiong3}
\end{figure}
Accordingly, the problem (\ref{CLAD}) is approximated by
\begin{align*}
\min_{t_0, \bm{x}, \bm{d}} ~\sum_{i=1}^{L} f_1 ((t_i - t_0) c - d_i), \quad \textup{s.t.}~~\text{(\ref{TEMC})},~\text{(\ref{TRIC})},~\text{(\ref{CLAD}a)--(\ref{CLAD}c)},
\end{align*}
which can then be cast into the standard GCOP form shown in (\ref{GCOP}) by letting
\begin{subequations}
\begin{align*}
&\bm{z} = \left[ t_0, \bm{x}^T, \bm{d}^T \right]^T \in \mathbb{R}^{L+k+1},\\
&N = L+k+1,\\
&K = \frac{L^2 + 5L + 2}{2},\\
&M = L,\\
&f(\bm{z}) = \sum_{i=1}^{L} f_1 ((t_i - t_0) c - d_i),\\
&g_1 (\bm{z}) = - t_0,\\
&g_{i+1} (\bm{z}) = t_0 - t_i,~~i = 1,2,...,L,
\end{align*}
\end{subequations}
\begin{subequations}
\begin{align*}
&g_{i+L+1} (\bm{z}) = -d_i,~~i = 1,2,...,L,\\
&g_{i+2L+1} (\bm{z}) = d_i - (t_i - t_0)c,~~i = 1,2,...,L,\\
&\left[ \bm{g}(\bm{z}) \right]_{3L+2:K} =\left[g_{3L+2} (\bm{z}),..., g_{\frac{(2L-i)(i-1)}{2}+j-i+3L+1} (\bm{z}),...,g_{K} (\bm{z})\right]^T\\
&~~=\left[ g_{1,2} (\bm{z}),...,g_{1,L} (\bm{z}),g_{2,3} (\bm{z}),...,g_{L-1,L} (\bm{z}) \right]^T \in \mathbb{R}^{\frac{L(L-1)}{2}}\\
&h_i (\bm{z}) = d_i^2 - {\| \bm{x} - \bm{x}_i \|}_2^2,~~i = 1,2,...,L,
\end{align*}
\end{subequations}
where
\begin{align*}
g_{i,j} (\bm{z}) = (2t_0 - t_i - t_j)c + {\| \bm{x}_i - \bm{x}_j \|}_2,~~i = 1,2,...,L-1,~j = i+1, i+2,...,L.
\end{align*}
While the dynamical equations are readily constructed pursuant to the rules in (\ref{dynamicsGCOP}), a more detailed description of the most crucial step (\ref{dynamicsGCOP}a) is presented as follows:
\begin{subequations}
\begin{align*}
\frac{d\bm{z}}{dt} &= \left[ \frac{d t_0}{dt}, \left(\frac{d\bm{x}}{dt}\right)^T, \left(\frac{d\bm{d}}{dt}\right)^T \right]^T = -\bm{\nabla}_{\bm{z}} \mathcal{L}_{\rho} (\bm{z},\bm{\nu}) = -\frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial \bm{z}}\\
&= -\left[ \frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial t_0}, \left(\frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial \bm{x}}\right)^T, \left(\frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial \bm{d}}\right)^T \right]^T,
\end{align*}
\end{subequations}
where
\begin{subequations}
\begin{align*}
&\frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial t_0} = c \sum_{i=1}^{L} \frac{e^{2 \gamma \left[ d_i + (t_0 - t_i)c \right]} - 1}{e^{2 \gamma \left[ d_i + (t_0 - t_i)c \right]} + 1} - \mu_1 + \sum_{i=1}^{L} \mu_{i+1} + c \sum_{i=1}^{L} \mu_{i+2L+1}\\
&~~+ 2c \sum_{i=1}^{L-1} \sum_{j=i+1}^{L} \mu_{\frac{(2L-i)(i-1)}{2}+j-i+3L+1} + \rho \Bigg\{ \mu_1^2 t_0 + \sum_{i=1}^{L} \mu_{i+1}^2 (t_0 - t_i)\\
&~~+ c \sum_{i=1}^{L} \mu_{i+2L+1}^2 \left[ d_i - (t_i - t_0) c \right] + 2c \sum_{i=1}^{L-1} \sum_{j=i+1}^{L} \mu_{\frac{(2L-i)(i-1)}{2}+j-i+3L+1}^2\\
&~~\left[ (2t_0 - t_i - t_j)c + {\| \bm{x}_i - \bm{x}_j \|}_2 \right] \Bigg\},\\
&\frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial \bm{x}} = 2 \sum_{i=1}^{L} \left[ \lambda_{i} + \rho \lambda_{i}^2 \left(d_i^2 - {\| \bm{x} - \bm{x}_i \|}_2^2 \right) \right] \left( \bm{x}_i - \bm{x} \right),\\
&\frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial \bm{d}} = \left[ \frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial d_1}, \frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial d_2},..., \frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial d_L} \right]^T,
\end{align*}
\end{subequations}
and
\begin{align*}
&\frac{\partial \mathcal{L}_{\rho} (\bm{z},\bm{\nu})}{\partial d_i} = \frac{e^{2 \gamma \left[ d_i + (t_0 - t_i)c \right]} - 1}{e^{2 \gamma \left[ d_i + (t_0 - t_i)c \right]} + 1} - \mu_{i+L+1} +\mu_{i+2L+1} + 2 \lambda_{i} d_i + \rho \Big\{ \mu_{i+L+1}^2 d_i\\
&~~+ \mu_{i+2L+1}^2 \left[ d_i - (t_i - t_0)c \right] + 2 \lambda_{i}^2 d_i \left( d_i^2 - {\| \bm{x} - \bm{x}_i \|}_2^2 \right) \Big\},~~i = 1,2,...,L.
\end{align*}
\section{Stability, convergence, and complexity analyses}
\label{CSCA}
Since the original objective function of our formulation lies in the $\ell_1$-space, we succinctly term the proposed projection-type recurrent neural network method $\ell_1$-PNN. In this section, several important aspects including the stability, convergence, and complexity properties of $\ell_1$-PNN are discussed.
\subsection{Local stability and convergence analysis}
As a preparation for the formal statements and by taking the GCOP (\ref{GCOP}) as an example, we define three concepts which frequently appear in the optimization literature:
\textbf{Definition 1.} (\texttt{Feasible region}). A feasible region is the set of all possible solutions to an optimization problem (namely, the GCOP (\ref{GCOP})) that satisfy the problem's constraints ($\bm{g}(\tilde{\bm{z}}) \leqq \bm{0}_K$ and $\bm{h}(\tilde{\bm{z}}) = \bm{0}_M$).
\textbf{Definition 2.} (\texttt{Regularity condition}). A feasible point $\tilde{\bm{z}}$ is said to be a regular point if the gradients of the active inequality constraints (i.e., $\bm{\nabla}_{\bm{z}} g_i (\tilde{\bm{z}}), \forall i \in \mathcal{I} = \left\{ i | g_i (\tilde{\bm{z}}) = 0 \right\}$) and those of the equality constraints (i.e., $\bm{\nabla}_{\bm{z}} h_i (\tilde{\bm{z}})$ for $i = 1,2,...,M$) are linearly independent at $\tilde{\bm{z}}$. This is a.k.a. the linear independence constraint qualification (LICQ).
\textbf{Definition 3.} (\texttt{Strict local minimum}). A point $\bm{z}^{*}$ is said to be a strict local minimum if $f(\bm{z}^{*}) < f(\bm{z}), \forall \bm{z} \in \mathcal{N}(\bm{z}^{*}, \delta) \cap \rm{S}$, where $\mathcal{N}(\bm{z}^{*}, \delta)$ represents the neighborhood of the point $\bm{z}^{*}$ with radius $\delta > 0$ and $\rm{S}$ denotes the feasible region.
A lemma presenting the second-order sufficient conditions (SOSC) \cite{MSBazaraa} is then introduced as:
\textbf{Lemma 1.} (\texttt{SOSC} \cite{MSBazaraa}). Let $\bm{z}^{*}$ be a feasible and regular point of the GCOP (\ref{GCOP}). If there exists a $\bm{\nu}^{*} = \left[ {\bm{\mu}^{*}}^T, {\bm{\lambda}^{*}}^T \right]^T \in \mathbb{R}^{K+M}$, such that $(\bm{z}^{*},\bm{\nu}^{*})$ is a KKT pair and $\bm{\nabla}_{\bm{z} \bm{z}}^{2} \bar{\mathcal{L}}(\bm{z}^{*},\bm{\nu}^{*})$ is positive definite on the cone
\begin{align*}
\mathcal{C} = \Big\{\bm{y} \in \mathbb{R}^N \Big| \left[ \bm{\nabla}_{\bm{z}} g_i (\bm{z}^{*}) \right]^T \bm{y} &= 0, \forall i \in \mathcal{I}_{+},~\left[ \bm{\nabla}_{\bm{z}} g_i (\bm{z}^{*}) \right]^T \bm{y} \leq 0, \forall i \in \mathcal{I}_{0},\\
\left[ \bm{\nabla}_{\bm{z}} h_i (\bm{z}^{*}) \right]^T \bm{y} &= 0, \forall i = 1,2,...,M,~\bm{y} \neq \bm{0}_N \Big\},
\end{align*}
where $\bar{\mathcal{L}}(\bm{z}^{*},\bm{\nu}^{*}) = f(\bm{z}^{*}) + \sum_{i \in \mathcal{I}} \mu_{i}^{*} g_i (\bm{z}^{*}) + \sum_{i = 1}^{M} \lambda_{i}^{*} h_i (\bm{z}^{*})$ is the restricted Lagrangian function at $(\bm{z}^{*},\bm{\nu}^{*})$, and $\mathcal{I}_{+} = \left\{ i \in \mathcal{I}|\mu_{i}^{*} > 0 \right\}$ and $\mathcal{I}_{0} = \left\{ i \in \mathcal{I}|\mu_{i}^{*} = 0 \right\}$ are often referred to as the sets of strongly active and weakly active constraints, respectively.
We now finally arrive at the following lemma in which the analytical results concerning the behaviors of iterative sequences produced by (\ref{dynamicsGCOP}) are established.
\textbf{Lemma 2.} (\texttt{Local stability} \cite{HChe}). Suppose that $(\bm{z}^{*},\bm{\nu}^{*})$ is a KKT point of the GCOP (\ref{GCOP}) satisfying the SOSC in Lemma 1. There exists a sufficiently large $\rho > 0$, such that the neurodynamic system described by (\ref{dynamicsGCOP}) is asymptotically stable at $(\bm{z}^{*},\bm{\nu}^{*})$, where $\bm{z}^{*}$ is a strict local minimum of the GCOP (\ref{GCOP}).
The detailed proof of Lemma 2 is omitted, because it constitutes a special case of the analysis of Theorem 2 in \cite{HChe} if we set the lower and upper bounds therein as negative and positive infinities, respectively. Based on Lemma 2, we embark on a careful examination of the local stability of $\ell_1$-PNN below. In general, the source onset time should be a proper value at least greater than $0$ \cite{ZSu}, the positions of the sensors are different from that of the source (otherwise there is no need for localization), and the positive bias error is much larger than the magnitude of the measurement noise in a TOA measurement under NLOS conditions \cite{GWang}. Therefore, the inequality constraints in (\ref{CLAD}) are actually all inactive (viz. $\mathcal{I} = \emptyset$), which means that the LICQ in our case is subject to only the equality constraints. The gradients of the equality constraints in (\ref{CLAD}) at a KKT point $(\bm{z}^{*},\bm{\nu}^{*})$ are calculated as
\begin{align}{\label{Gradh}}
\bm{\nabla}_{\bm{z}} \bm{h}(\bm{z}^{*}) &= \frac{\partial \bm{h}(\bm{z})}{\partial \bm{z}}\bigg|_{\bm{z} = \bm{z}^{*}} = \left[ \frac{\partial h_1 (\bm{z}^{*})}{\partial \bm{z}}, \frac{\partial h_2 (\bm{z}^{*})}{\partial \bm{z}},..., \frac{\partial h_{L} (\bm{z}^{*})}{\partial \bm{z}} \right]^T\nonumber\\
&= \begin{bmatrix}
\begin{array}{c : c : c}
\bm{0}_L & 2 \left( \bm{X}^T - \bm{1}_L {\bm{x}^{*}}^T \right) & 2\mathrm{diag}(\bm{d}^*)
\end{array}
\end{bmatrix},
\end{align}
where $\bm{1}_L \in \mathbb{R}^L$ is an all-one vector of length $L$, $\mathrm{diag}(\bm{a})$ stands for a diagonal matrix with vector $\bm{a}$ being its main diagonal, and $\bm{X} = \left[ \bm{x}_1, \bm{x}_2,..., \bm{x}_L \right] \in \mathbb{R}^{k \times L}$ represents a matrix including the positions of all sensors. Given the aforementioned practical considerations, we can easily deduce that the row vectors of the matrix in (\ref{Gradh}) are linearly independent and therewith $\mathcal{C} = \emptyset$. As a result, the SOSC hold trivially, and from Lemma 2 our $\ell_1$-PNN is assured locally stable as long as the Lagrangian parameter takes a large enough value. It is worth mentioning that due to the nonconvexity of the problem being solved, we investigate only the local stability of $\ell_1$-PNN here, but refer the interested readers to \cite{HChe,ZYan,HChe2} for the very recent developments of global convergence guaranteed neurodynamic optimization. Nevertheless, it is shown in Section \ref{SR} through extensive simulations that even local minimization can yield satisfactory performance in terms of positioning accuracy.
\subsection{Complexity analysis}
Since $\ell_1$-PNN is intended to be implemented analogously by designated hardware (e.g., application specific integrated circuits), it may not be meaningful to compare its complexity with those of the numerical approaches. Yet, we still manage to analyze the computational complexity of the neural network framework (\ref{dynamicsGCOP}) when it is realized in a discrete and numerical manner \cite{JLiang}:
\begin{equation}{\label{discrete}}
\left\{
\begin{aligned}
&\bm{z}_{(\kappa+1)} = \bm{z}_{(\kappa)} + \tau \frac{d\bm{z}}{dt},\\
&\bm{\mu}_{(\kappa+1)} = \bm{\mu}_{(\kappa)} + \tau \frac{d\bm{\mu}}{dt},\\
&\bm{\lambda}_{(\kappa+1)} = \bm{\lambda}_{(\kappa)} + \tau \frac{d\bm{\lambda}}{dt},
\end{aligned}
\right.
\end{equation}
where the subscript $(\cdot)_{(\kappa)}$ denotes the iteration index, $\tau$ is the step size, and the derivatives $\frac{d\bm{z}}{dt}, \frac{d\bm{\mu}}{dt}, \frac{d\bm{\lambda}}{dt}$ follow the definitions in (\ref{dynamicsGCOP}). With the help of Horner's scheme \cite{EHildebrand}, the evaluation of a polynomial of degree $n$ with fixed-size coefficients can be computed in $\mathcal{O}(n)$ time. Then, by considering polynomial evaluation as the operation in each step of (\ref{discrete}) governing the computational complexity, it is not hard to conclude that the dominant complexity of $\ell_1$-PNN is $\mathcal{O}\Big(N_{\text{PNN}}\Big(\max(\zeta,3)+5k+L\max(\zeta,5)+\frac{L^2 + 5L + 2}{2}+2L\Big)\Big) = \mathcal{O}\left(N_{\text{PNN}}L^2\right)$, where $\zeta$ is the degree of the Maclaurin polynomial for the hyperbolic tangent function $\frac{e^{2 \gamma \left[ d_i + (t_0 - t_i)c \right]} - 1}{e^{2 \gamma \left[ d_i + (t_0 - t_i)c \right]} + 1}$ and $N_{\text{PNN}}$ is the iteration number of the PNN using discrete realization. Table \ref{table} presents a comparison of complexity of the proposed neurodynamic method for solving (\ref{CLAD}) (termed $\ell_1$-PNN), SDP-based robust method for solving Formulation 1 in \cite{GWang3} (termed SDP-Robust-Refinement-1), SDP-based robust method for solving Formulation 2 in \cite{GWang3} (termed SDP-Robust-Refinement-2), and SDP-based model transformation method in \cite{ZSu} (termed SDP-TOA) as the function of $L$. Note that the naming of SDP-TOA is consistent with that in \cite{GWang3}, and the computational costs of dealing with the mixed SDP/SOCP problems are determined by following the calculation rule in \cite{ABenTal}. It can be concluded that $\ell_1$-PNN has a significantly lower complexity than those convex optimization approaches in \cite{GWang3,ZSu}.
\begin{table*}[!t]
\renewcommand{\arraystretch}{0.6}
\caption{Complexity of considered NLOS mitigation algorithms}
\label{table}
\centering
\begin{tabular}{c|c}
\hline\hline
\bfseries Algorithm & \bfseries Complexity\\
\hline
$\ell_1$-PNN & $\mathcal{O}\left(N_{\text{PNN}}L^2\right)$\\
\hline
SDP-Robust-Refinement-1 & $\mathcal{O}\left(L^{6.5}\right)$\\
\hline
SDP-Robust-Refinement-2 & $\mathcal{O}\left(L^{6.5}\right)$\\
\hline
SDP-TOA & $\mathcal{O}\left(L^{4}\right)$\\
\hline\hline
\end{tabular}
\end{table*}
\section{Simulation results}
\label{SR}
This section substantiates the efficacy of our proposed neurodynamic approach through simulation studies. To be specific, $\ell_1$-PNN is compared with representative NLOS mitigation algorithms including SDP-Robust-Refinement-1 in \cite{GWang3}, SDP-Robust-Refinement-2 in \cite{GWang3}, and SDP-TOA in \cite{ZSu} just as what have been provided in Table \ref{table}\footnote{It is remarkable that the definition of matrix $\bm{E}$ in \cite{GWang3} is incorrect and should be amended before putting the involved algorithms into use.}, and additionally the separated constrained weighted LS (SCWLS) approach in \cite{LLin}. Furthermore, the Cram\'{e}r-Rao lower bounds (CRLBs) for positioning with TDOA measurements in the LOS \cite{LLin} and NLOS \cite{YQi} scenarios are also included as the benchmark (when applicable). It should be mentioned that the invocation of $\ell_1$-PNN and SDP-TOA needs only the sensor positions and known signal timestamps as the inputs, whereas additional prior knowledge of the error bound/noise variance is a must for SDP-Robust-Refinement-1, SDP-Robust-Refinement-2, and SCWLS. In the following numerical examples, a perfect upper bound of the NLOS error is always ensured and passed into SDP-Robust-Refinement-1 and SDP-Robust-Refinement-2. The CVX package \cite{MGrant} and MATLAB$^\circledR$ ODE solver are utilized for realizing the convex programs and solving the systems of equations, respectively. All hyperparameters involved in SDP-TOA are assigned the same values as those in the demonstration program\footnote{\url{https://github.com/xmuszq/Semidefinite-Programming-SDP-optimization}} coded by the authors of \cite{ZSu}. As a global setup of $\ell_1$-PNN, the values held in the variable and Lagrangian neurons are initialized with 0s. For the selection of the augmented Lagrangian parameter $\rho$, the existing numerical results \cite{HChe} demonstrate that a relatively large $\rho$ can reduce transient oscillation of the neurodynamic model and speed up the convergence. In our simulations, we simply set $\rho = 5$ and it is observed that such a value always makes $\ell_1$-PNN settle down within several tens of time constants. Another predefined parameter associated with the quality of approximation to the original $\ell_1$-norm is fixed as $\gamma = 100$, based on which the resultant estimator is robust enough (see Fig. \ref{xiong3}). All simulations are carried out using a laptop with Intel$^\circledR$ Core$^\text{TM}$ i7-10710U processor and 16 GB memory.
Basically, two representative configurations with $k = 2$ are covered. The first configuration considers source localization in a 20 m $\times$ 20 m square region with $L=8$ sensors being evenly placed on the perimeter of the area and a single source being deployed at $\bm{x}=[2, 3]^T$ m. On the other hand, a typical setting in \cite{WXiong} with multiple sensors and a single source, whose locations are all randomly selected from the 20 m $\times$ 20 m square region in each Monte Carlo (MC) run, is adopted as the second configuration. The true value of the unknown source onset time is fixed as $t_0 = 0.1$ s, while the known signal timestamps received at the sensors and the TDOA measurements in the simulated system are obtained in accordance with (\ref{TOA}) and (\ref{TDOA}), respectively. Particularly, the signal propagation speed is set as $c=1$ m/s to keep things simple, the zero-mean Gaussian distributed noise $n_i$ is assumed to be of identical variance $\sigma^2$ for all $i$s, and the possible NLOS error in the TOA measurement between the source and $i$th sensor, namely $q_i$, is generated from the uniform distribution\footnote{Unquestionably, the corresponding source-sensor path is LOS if $\omega_i$ is assigned $0$.} $\mathcal{U}(0,\omega_i)$.
\begin{figure*}[!t]
\centering
\subfigure[]{\includegraphics[width=2.36in]{xiong5.eps}}%
\subfigure[]{\includegraphics[width=2.36in]{xiong6.eps}}
\subfigure[]{\includegraphics[width=2.36in]{xiong7.eps}}
\subfigure[]{\includegraphics[width=2.36in]{xiong8.eps}}
\centering
\caption{Dynamic behaviors of estimated source position versus time constant number for deterministic deployment in LOS and mild NLOS environments. (a) Outputs of 100 independent trials when $\sigma^2 = 0.1$ and $\omega_i = 0$ for all $i$s. (b) Mean of 100 outputs when $\sigma^2 = 0.1$ and $\omega_i = 0$ for all $i$s. (c) Outputs of 100 independent trials when $\sigma^2 = 0.1$, $\omega_1 = 5$, $\omega_5 = 5$, and $\omega_i = 0$ for other $i$s. (d) Mean of 100 outputs when $\sigma^2 = 0.1$, $\omega_1 = 5$, $\omega_5 = 5$, and $\omega_i = 0$ for other $i$s.}
\label{xiong5678}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=3.5in]{CDF_NLOS.eps}
\centering
\caption{Empirical CDF of Euclidean distance between source location and its estimate for deterministic deployment in mild NLOS environment based on 100 MC runs when $\sigma^2 = 0.1$, $\omega_1 = 5$, $\omega_5 = 5$, and $\omega_i = 0$ for other $i$s.}
\label{ECDF}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=3.5in]{NLOS_CRLB.eps}
\centering
\caption{RMSE versus $\sigma$ for deterministic deployment in mild NLOS scenario when $\omega_1 = 5$, $\omega_2 = 5$ and $\omega_i = 0$ for other $i$s.}
\label{NLOSCRLB}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=3.5in]{xiong9.eps}
\caption{RMSE versus $\sigma$ in LOS scenario (viz. $L_{\textup{NLOS}} = 0$).}
\label{xiong9}
\end{figure*}
\begin{figure*}[!t]
\centering
\subfigure[]{\includegraphics[width=2.36in]{xiong10.eps}}
\subfigure[]{\includegraphics[width=2.36in]{xiong11.eps}}
\subfigure[]{\includegraphics[width=2.36in]{xiong12.eps}}
\subfigure[]{\includegraphics[width=2.36in]{xiong13.eps}}
\subfigure[]{\includegraphics[width=2.36in]{xiong14.eps}}
\subfigure[]{\includegraphics[width=2.36in]{xiong15.eps}}
\centering
\caption{RMSE versus parameter of uniform distribution $b$ in different NLOS scenarios when $\sigma^2 = 0.1$. (a) $L_{\textup{NLOS}} = 2$, $\omega_1 = b$. (b) $L_{\textup{NLOS}} = 5$, $\omega_1 = b$. (c) $L_{\textup{NLOS}} = 8$, $\omega_1 = b$. (d) $L_{\textup{NLOS}} = 2$, $\omega_1 = 0$. (e) $L_{\textup{NLOS}} = 5$, $\omega_1 = 0$. (f) $L_{\textup{NLOS}} = 8$, $\omega_1 = 0$.}
\label{xiong10to15}
\end{figure*}
In the first test, the dynamic behaviors of the estimated source position using $\ell_1$-PNN in the deterministic deployment scenario are investigated. Taking the LOS and a mild NLOS environments for instance, Fig. \ref{xiong5678} plots the dynamics of the second and third variable neurons (i.e., those holding the variable $\bm{x}$) based on 100 MC runs. It is seen that $\ell_1$-PNN settles down and converges to a point close to the true source location within 20 to 40 time constants. For this reason, in the following we simply take the corresponding neuron output right after 40 time constants as the final position estimate produced by $\ell_1$-PNN. As a preliminary evaluation of $\ell_1$-PNN in comparison with other considered methods, Fig. \ref{ECDF} shows the empirical cumulative distribution function (CDF) of the Euclidean distance between source location and its estimate in the above-defined mild NLOS environment, from which we see that $\ell_1$-PNN and SDP-TOA demonstrate superior positioning performance. Next, the root mean square error (RMSE) criterion with 500 ensemble trials, defined as $\textup{RMSE} = \sqrt{\frac{1}{500}\sum_{i=1}^{500}{{\left\|\hat{\bm{x}}^{\{i\}} - \bm{x}^{\{i\}}\right\|}^2}}$ where $\hat{\bm{x}}^{\{i\}}$ represents the estimate of source position in the $i$th MC run (namely ${\bm{x}}^{\{i\}}$), is utilized as a measure to further compare the location estimation performance of diverse approaches. Fig. \ref{NLOSCRLB} depicts the RMSE versus $\sigma$ for the deterministic deployment scenario when the number of NLOS connections is fixed as $L_{\textup{NLOS}} = 2$ and the parameter of uniform distribution is set to 5. Especially, comparison with the CRLB when no prior NLOS statistics are available (namely, the one depending only on LOS signals \cite{YQi}) is also made. The results reveal that: (i) $\ell_1$-PNN exhibits the best robustness to NLOS propagation in such circumstances as long as $\sigma$ is not large enough, and (ii) taking advantage of rather than simply discarding the NLOS links results in an improvement in performance.
The random deployment scenario with $L=10$ is now considered to assess the localization performance of $\ell_1$-PNN together with other state-of-the-art algorithms under LOS and NLOS conditions. It must be pointed out that the setup is quite different from and in one sense more general than those in \cite{GWang2} and \cite{GWang3}, as the sensors here are neither fixed nor placed on a certain circle but all randomly drawn from the square region. Fig. \ref{xiong9} illustrates the RMSE as a function of $\sigma$ in the scenario where LOS transmissions are guaranteed for all source-sensor paths, i.e., $L_{\textup{NLOS}} = 0$. Clearly, only the SCWLS algorithm at sufficiently lower-level measurement disturbances (e.g., when $\sigma = 0.2$ m) can attain the CRLB \cite{LLin}. On the other side, there is always a performance gap between $\ell_1$-PNN and SDP-TOA/CRLB (i.e., SDP-TOA and CRLB are superior to $\ell_1$-PNN by about $0.25$ m and $0.5$ m across the whole range of $\sigma$). This is owing to the fact that SDP-TOA tightly approximates the ML estimator for small noise of the same level, whereas $\ell_1$-PNN derived in $\ell_1$-space is inherently suboptimal under the Gaussian noise assumption. It is also observed that the two worst-case robust methods SDP-Robust-Refinement-1 and SDP-Robust-Refinement-2 in general perform badly in the LOS scenario. We divide the test conditions in scenarios where NLOS propagation exists into two separate groups: (i) the path between the source and reference sensor is NLOS, and (ii) the path between the source and reference sensor is LOS. In each group, three diverse cases with $L_{\textup{NLOS}}=2,5,8$ are included, standing for the mild, moderate, and severe NLOS environments, respectively. Fig. \ref{xiong10to15} shows the comparison results with the detailed parameter settings being given in the caption. We can see that the location estimation accuracy of the non-robust SCWLS scheme deteriorates considerably as $b$ increases. $\ell_1$-PNN and SDP-TOA have comparable RMSEs, and they both outperform SDP-Robust-Refinement-1 and SDP-Robust-Refinement-2. Note that although $\ell_1$-PNN is slightly inferior to SDP-TOA in most cases (e.g., for $b < 4$ in Figs. \ref{xiong10to15}(a), \ref{xiong10to15}(b), and \ref{xiong10to15}(d) and all $b$s in Figs. \ref{xiong10to15}(c), \ref{xiong10to15}(e), and \ref{xiong10to15}(f)), the former is computationally more efficient and gets rid of the cumbersome hyperparameter tuning problems.
\section{Conclusion}
\label{CC}
In this paper, we proposed a robust model transformation formulation for TDOA-based source localization and devised a novel neurodynamic optimization solution to it. The new scheme does not require any \textit{a priori} information except the positions of the sensors, received signal timestamps thereat, and signal propagation speed, as the mitigation of NLOS biases in the reconstructed TOA measurements are achieved via the $\ell_1$-norm criterion. To address the problem of non-differentiability of the $\ell_1$-norm, certain approximations were applied to the original objective function for yielding a differentiable surrogate. Benefiting from the use of a projection-type recurrent neural network approach, the biggest advantage of the presented algorithm over the existing ones is its quadratic computational complexity in $L$. Through the theoretical analysis and extensive simulation investigations, we verified that the dynamics of the proposed $\ell_1$-norm-based PNN are locally stable, and confirmed its superiority over several existing TDOA-based localization schemes in terms of the estimation accuracy.
\section*{Acknowledgment}
This work was supported by the German Federal Ministry of Education and Research (BMBF) in the framework of the ASSIST~ALL project under Grant FKZ:16SV8162.
The first author would also like to thank Ms Ge Cheng at China Railway Major Bridge Reconnaissance \& Design Institute Co., Ltd. (BRDI), the team of Telocate GmbH, and the Laboratory for Electrical Instrumentation of the University of Freiburg for their assistance with the manuscript preparation.
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0808.3818
|
\section{Introduction}
\label{sec:intro}
The center of our Galaxy harbors not only a supermassive black hole
\citep[Sgr A*, $M_\bullet \sim 4 \times 10^6$ M$_{\odot}$\,;][]{
eckart96,genzel96,ghez98pm,ghez00nat,ghez03spec,ghez05orbits,
schodel02,schodel03,eisenhauer06}, but also
a population of massive (10-120 M$_{\odot}$\,), young ($\lesssim$10-100 Myr) stars
whose existence is a puzzle.
The origin of such young stars has been difficult to explain since
the gas densities observed today are orders of magnitude too low
for a gas clump to overcome the extreme tidal forces and
collapse to form stars
\citep[e.g.][for reviews]{sanders92,morris93,ghez05orbits,alexander05review}.
And yet, within the central parsec of our Galaxy, nearly 100 stars have
been classified as OB main-sequence stars,
more luminous OB giants and supergiants, and post-main-sequence
Wolf-Rayet stars \citep{allen90,krabbe91,blum95heI,krabbe95,tamblyn96,
najarro97,ghez03spec,paumard06}, with the more evolved massive stars having
ages as young as 6$\pm$2 Myr \citep{paumard06}.
Populations of young stars have also been observed in the nuclei of other
galaxies, such as M31 \citep{bender05}, suggesting that star formation
near a supermassive black hole may be a common, but not understood,
phenomenon in galaxy evolution. The close proximity
of the black hole at the center of the Milky Way provides a unique laboratory
for studying this ''paradox of youth''
\citep[e.g.][]{ghez03spec,ghez05orbits,schodel03,eisenhauer06}.
Proposed resolutions to the paradox of youth
can be grouped into several broad categories, including
(1) rejuvenation of an older population such that older stars appear young,
(2) dynamical migration from larger radii, and
(3) {\it in situ} formation.
Rejuvenation scenarios include stripping \citep{davies98,davies05}
or tidal heating of the atmospheres of old stars \citep{alexander03},
or combining multiple low mass stars via collisional mergers
to form a higher-mass hot star akin to a ``blue straggler''
\citep{lee96gcmergers,morris93,genzel03cusp}.
Although these processes may be candidates for explaining the
closest young stars within the central arcsecond,
they cannot account for the OB giants, OB supergiants, and
Wolf-Rayet stars that are located at larger radii (1\arcsec-14\arcsec),
since the rate of collisions is too low to produce
the observed total numbers.
Thus, it appears that these massive young stars must have formed,
or were deposited, in the central region within the last 4-8 Myr.
Dynamical migration scenarios attempt to resolve the paradox of youth
with the formation of a massive star cluster at larger distances from
the black hole (3-30 pc).
Such a cluster would spiral in due to dynamical friction
and deposit stars at smaller radii where they
are observed today \citep{gerhard01}. However,
for a cluster to reach the central parsec in only a few million
years, it must be very massive and centrally concentrated
\citep{kim03,pzwart03irs16,mcmillan03,gurkan05},
and it may even require the existence of an intermediate-mass
black hole (IMBH) as an anchor in the cluster core \citep{hansen03,kim04}.
{\it In situ} star formation scenarios can resolve the paradox
of youth if a massive, self-gravitating gas disk
was once present around the black hole \citep{levin03}.
Such a disk would be sufficiently dense to overcome the strong tidal
forces, and gravitational instabilities would then lead to
fragmentation and the formation of stars,
as has been suggested in the context of both the Galactic Center
circumnuclear disk and AGN accretion disks in other galaxies
\citep[e.g.][]{kolykhalov80,shlosman89,morris96,sanders98,goodman03,nayakshinCuadra05}.
Insight into the origins of the massive, young stars may be obtained
through observations of the spatial distribution and stellar dynamics of
this population.
Already, high-resolution infrared imaging and spectroscopy have shown
that the young stars between 0\farcs5 and 14\arcsec (0.02-0.6 pc)
exhibit coherent rotation \citep{genzel00}.
Analyses of the statistical properties of the
three-dimensional velocity vectors for these stars
suggest that they may reside in two disks.
The first proposed disk has a clockwise sense of rotation, as
projected onto the plane of the sky
\citep[][hereafter: clockwise-rotating or CW disk]{levin03},
while the second proposed disk is
counter-clockwise-rotating \citep[CCW][]{genzel03cusp} and is nearly
perpendicular to the first. The proposed disks extend
from $\sim$0\farcs8 to at least 7\arcsec \citep{paumard06}.
Other velocity vector analyses show that there are possible
co-moving groups or clusters of stars, including the IRS 13
cluster, which is proposed to lie within the putative CCW
disk \citep{maillard04irs13,schodel05}, and the IRS 16SW co-moving
group, which are also consistent with the proposed CW disk \citep{lu05irs16sw}.
The two proposed disks are inferred
to be oriented with an inclination and angle to the ascending node of
[$i_{CW}$=127$^\circ \pm$ 2$^\circ$, $\Omega_{CW}$=99$^\circ \pm$ 2$^\circ$] and
[$i_{CCW}$=24$^\circ \pm$ 4$^\circ$, $\Omega_{CCW}$=167$^\circ \pm$ 7$^\circ$] and
to have a finite angular thickness of
$\Delta\theta_{CW} \sim 14^\circ$ and
$\Delta\theta_{CW} \sim 19^\circ$ where $\Delta\theta$ is the
standard deviation of the orbital inclinations distributed
normally about the disk plane \citep{paumard06}.
The thickness of the stellar disks has been attributed to
thickening as a result of gravitational interactions between the two disks,
which provides an estimate of the disk masses \citep{nayakshin06thick}.
The derived mass is smaller than the mass inferred from the number of
observed young stars, assuming a Salpeter initial mass function (IMF);
accordingly, \citet{nayakshin06thick}
suggest that the disks have a top-heavy mass function.
Both {\it in situ} gas disk and in-spiraling star cluster
formation scenarios have been used to explain
the kinematics of this young star population and to
predict that the stars should lie in a common orbital plane.
However, the presence of two stellar disks with similarly aged
populations requires either two nearly concurrent gas disks or two
infalling star clusters; and both of these scenarios are difficult to produce.
Therefore, to understand the recent star formation history,
it is critical to measure the orbital planes of individual
stars in order to confirm the existence of the two stellar disks previously
derived from a statistical analysis of velocity vectors alone.
The {\it in situ} gas disk and inspiraling star cluster
formation scenarios predict different structures and evolutions
for the resulting stellar disk, particularly with respect to the
eccentricities and radial distribution of stars within the disk.
Early models of a self-gravitating gas disk around the supermassive
black hole at the center of the Milky Way produce stars with
a steep radial profile in the disk surface density,
$\Sigma \propto r^{\alpha}$, with $\alpha \sim -2$ \citep{linPringle87,levin06}.
These models typically result in stars on
circular orbits as would be the case for the slow build up of a
gas disk that is circularized before there is sufficient mass for
gravitational instabilities to set in
\citep{milosav04,nayakshinCuadra05,levin06}.
The stellar eccentricities of an initially circular disk can relax
to higher eccentricities up to $e_{rms} = \sqrt{<e^2>} \sim $0.15 for a normal
IMF or $e_{rms} \sim $0.3 for a top-heavy IMF \citep{alexander07imf,cuadra08}.
More recent models have also shown that star formation can occur rapidly
before circularization in an initially eccentric
disk as might result from the infall of a single massive molecular cloud
or a cloud-cloud collision \citep{sanders98,nayakshin07sims,alexander08}.
These eccentric self-gravitating accretion disk models typically produce
a more top-heavy IMF than initially circular disks.
On the other hand, an inspiraling star cluster would dissolve into a
disk of stars with a flatter radial profile
\citep[$\Sigma \propto r^{-0.75}$;][]{berukoff06}
whose orbital eccentricities would reflect the
eccentricity of the cluster's orbit, which could be either circular
or eccentric \citep{pzwart03irs16,mcmillan03,kim03,kim04,gurkan05,berukoff06}.
Previous measurements of the radial distribution of young stars
yields a steep radial profile consistent with {\it in situ} formation
\citep{paumard06}.
Also, the eccentricities of the stars have previously been estimated from
observations by assuming that the stars orbit in a disk;
however, there are conflicting
results claiming that the stars in the clockwise-rotating disk are on
nearly circular orbits \citep{paumard06} or on eccentric orbits
\citep{beloborodov06}.
Determining the radial profile and stellar eccentricities of stars in a
disk may provide observational constraints on the origin of the young stars.
We present an improved proper motion study that yields an order
of magnitude more precise proper motions and
the first measurement of accelerations in the
plane of the sky for stars outside the central arcsecond.
By combining the stellar positions, proper motions, radial velocities,
and accelerations, we estimate stellar orbital parameters
and test whether the young stars reside on one or two stellar disks in a more
direct manner than previous methods using only velocity information.
This provides a {\it direct} test of the existence, membership, and properties
of these disks. The observations are described in
\S\ref{sec:obs} and the astrometric analysis procedure and results are detailed
in \S\ref{sec:astrometry}. Orbit analysis and results are presented in
\S\ref{sec:orbitAnalysis} and \S\ref{sec:orbitResults} and a discussion
of the implications for the origin of the massive, young stars
at the Galactic Center is presented in \S\ref{sec:discussion}.
\section{Observations}
\label{sec:obs}
This study utilizes 29 epochs of high-resolution, infrared images of the
Galaxy's central stellar cluster, which were taken from 1995 to 2005
using both speckle and laser guide star adaptive optics (LGS AO) observing
techniques on the W.~M. Keck 10 m telescopes.
These data sets are listed in Table \ref{tab:obs} and all but the
additional LGS AO observation from 2005 are described in detail in earlier
papers \citep{ghez98pm,ghez00nat,ghez05orbits,lu05irs16sw,rafelski07}.
Columns 3 and 4 list the individual exposure times and the
total number of frames for each epoch of data.
All 27 speckle imaging observations were taken using the
facility near-infrared camera, NIRC \citep{NIRC,NIRCs}, which has a
plate scale of $\sim$20 mas per pixel, and
a 5\farcs22 $\times$ 5\farcs22 field of view.
The two adaptive optics imaging observations used the facility LGS AO
system \citep{wizinowich06,vanDam06} and the near-infrared camera,
NIRC2 (PI: K. Matthews) with a plate scale of
9.963 $\pm$ 0.006 mas per pixel \citep{ghez08}
and a 10\farcs2 $\times$ 10\farcs2 field of view.
While the laser guide star is used to correct most of the
atmospheric aberrations, the low-order, tip-tilt terms were corrected
using visible observations of USNO 0600-28577051 (R = 13.7 mag and
$\Delta r_{SgrA*}$ = 19\arcsec).
In addition to the 27 speckle observation and the 2004 LGS AO observations
described in previous works, a new LGS AO data set was obtained in
2005 June. This data set was taken using two different
narrow-band filters,
K$_{CO}$ ($\lambda_{o}$=2.289 \micron, $\Delta\lambda$=0.027 \micron) and
K$_{cont}$ ($\lambda_{o}$=2.270 \micron, $\Delta\lambda$=0.030 \micron),
rather than the K' broadband filter used for the 2004 LGS AO observations.
For each filter, images were taken in a 5 position pattern around a
4\farcs0 box with exposure times of 36 s
(t$_{exp}$ = 7.2 s, 5 coadds) and 59.5 s (t$_{exp}$ = 11.9 s, 5 coadds)
for the K$_{CO}$ and K$_{cont}$ filters, respectively.
The choice of narrow-band filters was driven by a different project
and the data sets from the two filters were combined together
for the present study (see \S\ref{sec:images}).
Resulting Strehl ratios were $\sim$0.25-0.35 in the individual frames.
\section{Astrometric Data Analysis and Results}
\label{sec:astrometry}
The goal of this analysis is to obtain high precision astrometry for
a sample of young stars that are candidate disk members and
have existing radial velocity measurements.
Based on spectroscopic identification, there are currently 90 known
young stars with radial velocity measurements listed in \citet{paumard06}
based on high quality (``quality 1 or 2'') spectral classifications.
We define a {\it primary sample} that includes those known young stars found
in our astrometric data sets that
have projected radii between 0\farcs8 and 3\farcs5. The inner
radius is set by the proposed inner edge of the clockwise disk of young
stars and young stars interior to this radius are on more randomly
oriented orbits \citep{ghez05orbits,eisenhauer06}.
The outer radius is set by the field of view of the speckle data sets.
Over this region, \citet{paumard06} note that all young
stars brighter than K=13.5 should be identified, which includes
OB giants and supergiants.
A total of 32 such young stars are in our 11 year
astrometric data set and comprise the sample for this study. Of the
32 stars in our sample, 23 are among the 36 stars thought to be part
of the clockwise disk, 2 are among the 12 candidate members of the
counter-clockwise disk, and the remaining 7 are among the 42
stars not assigned to either disk by \citet{paumard06}.
We also define an {\it extended sample} that includes both the primary
sample of 32 stars and an additional 41 young stars found by
\citet{paumard06} at larger radii that are outside the field of view
of our astrometric measurements. The astrometry for the additional
41 stars is taken from \citet{paumard06}\footnote{We note that there
are 4 additional young stars at larger radii that are not included
in our extended sample since they do not have proper motions
listed in \citet{paumard06}.},
which has an order of magnitude lower precision and lacks any constraints
on the accelerations.
However, we use the extended sample to explore the kinematics of the
young stars at larger radii with the same analysis techniques used
on the primary sample. We also note that the spectroscopic observations
used to identify the young stars at larger radii were taken in a different
setup than in the central regions, with lower spectral resolution
and lower Strehl; thus the
completeness limit may be somewhat brighter in this region.
However, any difference is statistically insignificant given that
a two-sample KS-test yields a 50\% probability that
the primary sample and those additional stars added to the extended
sample have the same K-band luminosity function. The extended sample
is used only to supplement our analysis; therefore, to avoid confusion,
all analysis and results are reported for the
primary sample, which has more precise proper motions and accelerations,
unless specifically noted otherwise.
Astrometric positions for the young stars in the primary sample
are extracted from the imaging
data sets listed in Table \ref{tab:obs} using similar techniques to those
described in \citet{ghez98pm,ghez00nat}, \citet{lu05irs16sw},
and \citet{ghez05orbits}, with the
following key changes:
(1) geometric distortion is corrected in the speckle images using
an improved distortion solution (see \S\ref{sec:images},
Appendix \ref{app:speckDistort}),
(2) speckle images are combined with an improved algorithm developed
and implemented by \citet{sethThesis}, and
(3) image coordinates are transformed between data sets with more
degrees of freedom (see \S\ref{sec:coords}).
Sections \ref{sec:images} and \ref{sec:coords} describe the analysis in
detail and Section \ref{sec:astroResults} presents the astrometric results.
\subsection{Image Processing}
\label{sec:images}
To achieve precise astrometry, the basic image reduction steps,
particularly geometric distortion correction, must be carefully implemented.
First, both speckle and LGS AO individual exposures are processed using
standard techniques of sky subtraction, flat-fielding, and bad pixel correction.
Next, the images are transformed to correct for optical distortion.
For the LGS AO/NIRC2 images, optical distortions are well characterized at the
$\sim$2 milli-arcsecond level over 2''
\citep[][Appendix \ref{app:speckDistort}]{ghez08}
by the pre-ship review distortion
coefficients\footnote{http://www2.keck.hawaii.edu/inst/nirc2/}
and the distortions are removed from the images using the IRAF routine,
{\it Drizzle} \citep{drizzle}.
The speckle images, obtained with NIRC, have a known off-axis
distortion that can be corrected as described in \citet{ghez98pm}.
However, this distortion solution
does not account for any distortion introduced by the
additional optics in the NIRC reimager, which magnifies the image scale
by a factor of $\sim$7 from
seeing limited sampling to diffraction limited sampling.
Speckle data sets were acquired in such a way as to minimize
the effects of this residual distortion in the center of the field
of view and have resulting residual distortion errors that are
smaller than the typical centroiding error, which is $\sim$2 mas,
for stars at radii $<$ 0\farcs5.
However, astrometric uncertainties for stars outside this region
are dominated by the uncorrected distortion, which grows to
$\sim$6 mas near the field edge at a radius of 2\farcs5 \citep{ghez05orbits}.
In order to characterize the residual distortion in NIRC, simultaneous images
of the Galactic Center were obtained with both NIRC and NIRC2 with the NIRC2
images serving as a reference coordinate system
(see Appendix \ref{app:speckDistort}). The speckle image distortion
is mapped by comparing stars' positions in both NIRC and NIRC2 images.
As shown in Appendix \ref{app:speckDistort}, the resulting NIRC to NIRC2
transformation is characterized at the $\sim$2 mas level over the entire
field of view.
After distortion correction, individual exposures are combined into a final
diffraction-limited image using different methods for speckle and LGS AO data
sets.
Speckle images are produced by first rejecting the low Strehl ratio frames
(typically 75\% of frames are rejected)
and then stacking the remaining frames using a
weighted shift-and-add (SAA) routine \citep{sethThesis}.
The resulting combined images have a point-spread function (PSF)
composed of a diffraction-limited core
(FWHM$\sim$0\farcs055) on top of a broad seeing halo (FWHM$\sim$0\farcs4).
The improved image combination algorithm attempts to maximize the
signal-to-noise ratio (SNR) of the final image while preserving
the highest spatial resolution. Quantitatively, the weighted SAA method
doubles the fraction of light contained in the diffraction-limited
core (from 3.5\% to 7.0\%) over the standard SAA scheme with no weighting
and no frame rejection \citep{sethThesis}.
The LGS AO individual exposures are all of similar
quality and are thus all averaged together, without weighting, in order
to produce the final high-resolution image for each data set.
Although the 2005 June data were taken in two different filters
(K$_{CO}$ and K$_{cont}$), all the images
were combined together to increase the final SNR.
While photometry from this epoch is marginally impacted,
the astrometry is comparable to other epochs.
Each data set was also sub-divided to produce three equivalent
quality (randomized in time) subsets to make three images used for determining
photometric and astrometric uncertainties.
The resulting images are summarized in Table \ref{tab:obs},
including the achieved spatial resolution (FWHM) and
the Strehl ratio.
\subsection{Stellar Positions and Coordinate Transformations}
\label{sec:coords}
In order to extract astrometric information for the sample of young stars, the
coordinate system from each data set is transformed into a common
reference frame using the stars in each image to determine the
transformation parameters. Since the
accuracy of this transformation relies on the assumption that there
is no net rotation of the sample, we use all stars
detected in each data set, not just the young stars, in this analysis.
The steps for (1) measuring stars' positions in each epoch,
(2) transforming to a common (relative) reference frame, and
(3) determining the absolute coordinate system are
described below and utilize all stars detected in the data sets;
then as a final step, the young star sample is extracted.
In each data set, stars are identified and their positions measured
using the IDL point-spread function fitting routine ``StarFinder''
\citep{starfinder}.
StarFinder generates a PSF from several bright stars in the field
and cross-correlates the resulting PSF with the image.
The PSF was iteratively constructed using
IRS 16C and IRS 16NW for the speckle maps and
IRS 16C, 16NW, 16NE, 16SW, 33E, 33W, 7, 29N, and GEN+2.33+4.60
for the LGS AO images.
Candidate stars are those for which StarFinder correlation
peaks have a correlation value higher than 0.8 and positions
and fluxes are extracted by fitting the PSF to each correlation peak.
From the candidate star list, spurious detections are then eliminated by
requiring that each star be detected in all three of the subset-images
with a correlation of higher than 0.6.
The positional centroiding uncertainties for each candidate star are estimated
from the rms of their locations in the three subset-images, and an additional
systematic error term of 0.88 mas is added in quadrature to all stars in
LGS AO epochs to account for residual distortion in the central 5'' of
NIRC2 \citep{ghez08}.
The candidate stars are flux calibrated using the apparent
magnitudes of the non-variable stars,
IRS 16C, IRS 16SW-E, S2-17, S1-23, S1-3, S1-4, S2-22, S2-5, S1-68, S0-13,
and S1-25, as measured by \citet{rafelski07}.
The star detections from each epoch are cross-identified
with stars from all other epochs and those stars that are detected in
at least 16 out of 29 epochs are used to create a master star list.
The threshold of 16 or more epochs is used in order to insure
high astrometric precision; for a threshold of less than 16 epochs,
the number of detected stars rises dramatically as does the number of
sources showing significant ($\gtrsim$3$\sigma$) accelerations in
non-physical directions, indicating a high frequency of false detections
(see \S\ref{sec:astroResults} for further discussion).
Stars in the master list are also examined for source confusion, which may occur
when two stars pass close enough to each other such that StarFinder
only detects a single source with biased astrometry rather than detecting
both stars. Source measurements from individual epochs are rejected if two
stars pass within 55 mas ($\sim$1 spatial resolution element) of each other and
only one source is detected by StarFinder.
The results of this stage of the analysis are summarized in
Table \ref{tab:obs}, which provides for
each data set the average centroiding error for the brightest stars
(K$<$13; also see Figure \ref{fig:posError})
and the sensitivity as estimated
by the peak in a histogram of the K-band magnitudes (bins = 0.1 mag)
of all the stars in the data set.
Averaged over all stars in all maps, the centroiding uncertainties have a
mean value of 1.6 mas for the brightest stars (K $\leq$ 13 mag) and
3.4 mas for the fainter stars (13 $<$ K $<$ 16 mag).
The coordinate system for each image is transformed to a common
local reference frame defined by the 2004 July LGS AO/NIRC2 image's
coordinates and pixel scale.
This particular LGS AO epoch was chosen as the reference because
the NIRC speckle distortion solution is tied to this epoch, thus
providing a smooth transition between speckle and LGS AO data sets.
The procedure for deriving the coordinate transformation for
all of the data sets is non-trivial, since the stars in the
images have detectable motions.
Optimal alignment is achieved by minimizing the error-weighted, net
displacement for all the stars as described by \citet{ghez98pm}
while allowing for translation, rotation, and two magnifications in
arbitrary, but perpendicular, directions. This is a higher order
transformation than was used in our earlier astrometric works, which
only allowed for translation and rotation.
The new transformation equations have the form
\begin{eqnarray}
x_{pix} = a_0 + a_1 x'_{pix} + a_2 y'_{pix} \\
y_{pix} = b_0 + b_1 y'_{pix} + b_2 x'_{pix}
\end{eqnarray}
where $x'_{pix}$ and $y'_{pix}$ are the input detector coordinates in pixels
and $x_{pix}$ and $y_{pix}$ are the output coordinates for each star,
and all other variables are free parameters that are common across all stars
in the alignment fit.
As in \citet{ghez05orbits}, stars within 0\farcs5 of Sgr A* are excluded
from the transformation as they exhibit large non-linear motions.
Additionally, all spectroscopically identified young stars are excluded
from the transformation as they have a known net rotation \citep{genzel00}.
Initially, each image is aligned to the reference image by assuming the
stars have no proper motions and finding the best-fit values for the
free parameters of the transformation, $a_0, a_1, a_2, b_0, b_1, b_2$, for
that image. However, after a first pass at the alignment of all the images,
proper motions are estimated and
used to refine the alignment solutions in a second pass.
Sources with estimated proper motions higher than
1.5 mas yr$^{-1}$ (600 km s$^{-1}$)
are excluded from the transformation resulting in the elimination of
2 sources that are near the edge of the speckle field-of-view and suffer
from edge effects.
Alignment uncertainties are estimated by a half-sample bootstrap method
\citep{astrostats,ghez05orbits}
and are small ($\sim$0.2 mas for
stars at $r<$2\arcsec) compared to the centroiding uncertainties
(see Figure \ref{fig:posError}).
Alignment and centroiding
uncertainties are added in quadrature to produce a final relative
positional uncertainty for each star at each epoch.
The resulting astrometric data set contains stellar positions
and uncertainties for all epochs, transformed into the 2004 July NIRC2 pixel
coordinate system ($x_{pix}, y_{pix}$).
The relative positions and uncertainties are transformed into J2000
absolute astrometric coordinates defined by radio observations of
SiO masers and Sgr A*. Using observations of the SiO masers in the infrared,
a set of {\it infrared absolute astrometric standards} are defined in
a process described in detail in an
appendix of \citet{ghez08}.
These astrometric standards are used to derive the transformation from
2004 July NIRC2 pixel coordinates into absolute coordinates.
A statistically insignificant adjustment is made to place the origin at
the dynamical center of S0-2's orbit, which is known to high precision,
by offseting from the radio position of Sgr A* by 1 mas to the East
and 5 mas to the South. This offset is well within the absolute astrometric
uncertainty of $\sim$6 mas for Sgr A* \citep{ghez08}.
The stellar positions in all epochs are thus expressed in arcseconds
offset from the dynamical center with $+x$ increasing East and $+y$
increasing North and can be converted
into celestial coordinates using
(x, y) = ($\cos{\delta}\; \Delta\alpha$, $\Delta\delta$)
\footnote{When converting from (x, y) to ($\Delta\alpha$, $\Delta\delta$),
higher order terms are negligible (0.06 mas over 5\arcsec) because the
celestial sphere is sufficiently flat over our field of view.
}.
Positional uncertainties are taken as the quadratic sum of the relative
errors, which dominate, and the absolute error from uncertainties in the
plate scale and position angle.
Errors in the relative position of Sgr A* ($\sim$2 mas) are incorporated later
during the orbit analysis stage as a parameter of the potential of the
supermassive black hole
(see \S\ref{sec:orbitAnalysis}).
From the resulting absolute astrometric data set, the sample of young
stars is extracted.
\subsection{Proper Motions and Acceleration Results}
\label{sec:astroResults}
For each of the young stars in the sample, positions, velocities, and
accelerations in the plane of the sky are derived by fitting second-order
polynomials to the star's
position as a function of time, weighted by the positional uncertainties.
The polynomials are fit independently in $x$ and $y$
coordinates and have the form
\begin{eqnarray}
x(t) = x_{ref} + v_{x,ref} (t - t_{ref}) + \frac{1}{2} a_{x,ref} (t - t_{ref})^2 \\
y(t) = y_{ref} + v_{y,ref} (t - t_{ref}) + \frac{1}{2} a_{y,ref} (t - t_{ref})^2
\end{eqnarray}
where $t$ is the time in years, $t_{ref}$ is a reference time
taken to be the mean of the time of all epochs weighted by the positional
uncertainties for each star,
$x_{ref}$ and $y_{ref}$ are the positions at the reference time, $v_{ref}$ is the
velocity at the reference time, and $a_{ref}$ is the acceleration at the
reference time. Uncertainties in the fit
parameters are determined from the covariance matrix.
Figures \ref{fig:posTime_S0-15} and \ref{fig:posTime_irs16NW} show the
polynomial fits for two example stars and
the resulting values for the kinematic variables for all stars are reported
in Table \ref{tab:yng_pm_table}.
Since the stars' motions
are assumed to be dominated by the central force from the black hole,
we convert $a_{x,ref}$ and $a_{y,ref}$ into radial and tangential
accelerations \footnote{This assumption
may not hold for stars in a gravitationally bound cluster, such as may be the
case for the 4 stars in the extended sample that make up the IRS 13
co-moving group; however, the deviations from the potential assumed above
should result in only $5-10$\% changes in the velocity vectors.}.
All tangential accelerations and
positive radial accelerations are non-physical and therefore provide
a check on the systematic errors of the acceleration measurements.
Figure \ref{fig:histAccel} shows a histogram of the
significance of the acceleration measurements both in the radial and
tangential directions for the young stars in our primary sample.
While the tangential and positive radial distributions are
slightly offset (0.6$\sigma$) from zero and
broader (1.5$\sigma$ vs. 1$\sigma$) than is expected
for a normal distribution,
any systematic errors appear to impact the results at the
$\lesssim 1\sigma$ level.
The resulting velocity measurements for the young star sample outside
the central arcsecond are improved by at least a factor of 7 when compared
with our previous work \citep{ghez98pm,lu05irs16sw} and other recently
reported Galactic Center proper motions \citep[e.g.][]{genzel00,ott03thesis}.
The absolute uncertainties in our proper motions
are typically $\sim$0.06 mas yr$^{-1}$ ($\sim$2 km s$^{-1}$),
although stars detected in fewer epochs have somewhat higher values
(0.1 - 0.5 mas yr$^{-1}$; 4 - 20 km s$^{-1}$).
Figures \ref{fig:posTime_S0-15} and \ref{fig:posTime_irs16NW}
show examples of the measurements for two stars in our sample (S0-15
and IRS 16NW), and their corresponding proper motion fits
with 1$\sigma$ errorbars.
In the young star sample, significant ($>$3$\sigma$) acceleration,
or curvature, in the plane of the sky is detected only for S0-15
(Figure \ref{fig:posTime_S0-15}).
This star has the second smallest projected separation from Sgr A*
in our sample,
at $\rho$ = 1\farcs0 (0.04 pc), and has a projected radial acceleration of
-0.21 $\pm$ 0.05 mas yr$^{-2}$ or, equivalently,
-9.6 $\pm$ 2.0 km s$^{-1}$ yr$^{-1}$ (see Figure \ref{fig:PMimage}).
S0-15 is more than twice as far from Sgr A*, in projection, than the seven
stars with previously detected accelerations,
which were all within a projected radius of less than 0\farcs4
\citep[0.016 pc, ][]{ghez00nat,eckart02,ghez05orbits,eisenhauer06}.
The detection of acceleration is important in that it allows us to
solve for the line-of-sight distance, and thus the
three-dimensional position of a star relative to the black hole.
For a star in the gravitational potential well of a supermassive black hole,
the plane-of-the-sky acceleration, at a three-dimensional distance r,
in cylindrical coordinates is
\begin{equation}
\label{eqn:a2z}
a_{\rho} = \frac{-GM \rho}{r^3} = \frac{-GM \rho}{(\rho^2 + z^2)^{3/2}}
\end{equation}
where $\rho$ is the plane-of-the-sky radial coordinate and z is the
coordinate along the line of sight relative to Sgr A*.
The magnitude of the line-of-sight distance from Sgr A*, z,
can be solved for by adopting a black hole mass of
$M_{\bullet} = 4.4 \times 10^6$ M$_{\odot}$\, and a distance of
$R_{\circ} = 8.0$ kpc \citep[see \S\ref{sec:orbitAnalysis};][]{ghez08};
it is important to note that there is a remaining sign ambiguity for z.
The resulting line-of-sight distance from Sgr A*
for S0-15 is $|0.045 \pm 0.004|$ pc bringing the total separation between
S0-15 and Sgr A* to 0.060 pc.
The remaining stars in our sample have acceleration measurements that
constrain the line-of-sight distance.
While the lower limits of these acceleration magnitudes are not
significantly different
from zero at the 3$\sigma$ level, their upper limits are
smaller than the maximum allowed acceleration.
The maximum possible magnitude of the acceleration for a star at a
given $\rho$ occurs when z = 0. When the measured acceleration limits are
below this value, they provide a
lower limit on the star's line-of-sight distance to the SMBH.
Figure \ref{fig:accSignificance2} compares the measured acceleration limits
with the maximum possible acceleration for each star.
Any 3$\sigma$ acceleration limits below the maximum allowed value
gives useful constraints on the line-of-sight distances.
In addition to our explicit measurement for S0-15,
our high precision astrometric measurements are now yielding
3$\sigma$ acceleration limits with a median value of
-0.19 mas yr$^{-2}$ (-7.3 km s$^{-1}$ yr$^{-1}$)
that can significantly constrain the line-of-sight
distance for nine stars in our sample that are located as far as
1\farcs7 (0.07 pc), in projection, away from the black hole.
\section{Orbit Analysis}
\label{sec:orbitAnalysis}
For a known point-mass Newtonian gravitational potential,
a star's orbital elements can be fully determined from
the measurement of only six kinematic variables.
For this analysis, we assume that the central point mass is a
black hole with characteristics determined by analysis of the
orbit of the star S0-2, which has been observed for nearly one
complete revolution
\citep{eisenhauer06,ghez08}.
Our proper motion analysis (\S\ref{sec:astroResults}) yields information
on five kinematic variables, including two positions, two velocities, and one
acceleration.
The sixth kinematic variable comes from radial velocities measured by
\citet{paumard06}. The reported radial velocities are averaged
over several years of observations; however, we adopt the same
reference epoch, $t_{ref}$, as for the proper motion analysis since
any change in the radial velocity due to acceleration along the
line-of-sight should be well
within the large measurement uncertainties in radial velocity
($\sigma_{v_{z,ref}}\sim$20-100 km s$^{-1}$).
As described in \S\ref{sec:astroResults}, the plane-of-the-sky acceleration
can be converted into a line-of-sight distance that, when combined with the
projected distance, gives the full three-dimensional position for a star.
Although most of the stars in our sample have plane-of-the-sky accelerations
that are consistent with zero, the upper limits on the magnitude of the
acceleration provide valuable information by ruling out small
line-of-sight distances. We therefore use our best-fit accelerations
and uncertainties as formal measures of the acceleration when converting
to a line-of-sight distance.
Therefore the six measured quantities can be expressed as a
three-dimensional position and three-dimensional velocity at a certain
epoch ($t_{ref}$).
Given the properties of the black hole, these kinematic quantities can be
translated directly into 6 standard orbital elements
(see Appendix \ref{app:orbits}).
A Monte Carlo simulation is carried out to transform
each star's six measured kinematic variables
($x_{ref}$, $y_{ref}$, $v_{x,ref}$, $v_{y,ref}$, $v_{z,ref}$, $a_{\rho,ref}$)
into six orbital parameters ($i$, $\Omega$, $\omega$, $e$, $P$, $T_o$)
and their uncertainties.
A total of 10$^5$ Monte Carlo trials are run and, in each trial,
$4 + (6 \cdot 32)$ variables are randomly generated; four for the potential
parameters and six for each of the 32 stars' measured kinematic variables.
The four potential parameters are pulled from a four-dimensional probability
density function, PDF(M$_\bullet$, R$_o$, x$_o$, y$_o$),
based on the orbit of S0-2 derived by \citet{ghez08},
where the black hole's mass and line-of-sight distance are centered on
$M_{\bullet}=4.4 \times 10^6$ M$_{\odot}$\, and
$R_{\circ}=8.0$ kpc \footnote{These values correspond to a 12-parameter
orbit model for S0-2 (i.e. $v_z=0$ case) from an early version of
\citet{ghez08}. In this version, local distortions were
not corrected \citep[][Appendix B]{ghez08}; but the
resulting black hole mass and distance differ by $<1\sigma$ from the final
reported values.},
the dynamical center is adopted as the origin
with x$_o$ and y$_o$ defined as zero, and the projected one-dimensional
probability distributions' RMS errors are [1.0, 1.6] mas for [$x_o$, $y_o$],
$0.3 \times 10^6$ M$_{\odot}$\, for $M_{\bullet}$, and $0.3$ kpc for $R_\circ$
\footnote{Simulations were also performed using the lower black hole mass and
distance reported by \citep{eisenhauer06}. Our results on the detection of
only one stellar disk and on the properties of the disk are all consistent
within 1$\sigma$ error bars.}.
For each trial, all the stars' orbits are calculated using the same
potential parameters in order to preserve correlations between the
potential parameters and the orbital parameters such as eccentricity.
The kinematic variables for each star
are sampled from independent gaussian distributions,
each of which is centered at the best-fit value from Table
\ref{tab:yng_pm_table} and has a 1$\sigma$ width set to the measurement
uncertainty.
Any correlations between the measured kinematic variables are negligible
given the small uncertainties in the stars' relative angular positions
($\lesssim$0.2\%) and velocities ($\lesssim$3\%) in the plane-of-the-sky
as compared to the uncertainties in the black hole mass ($\sim$10\%)
and the accelerations ($\sim$60\%).
The distribution for the acceleration, $a_\rho$, is truncated such that only
accelerations of bound orbits are allowed\footnote{The assumption that the
orbits are bound does not effect the results presented in this paper
discussed in \S\ref{sec:orbitResults} since
all unbound orbital solutions yield high inclination (edge-on) orbits
and large eccentricities (e > 1). Considering only bound orbits
simplifies the orbit analysis as we need only consider equations for
elliptical orbits rather than hyperbolic or parabolic orbits.
}, which follows from requiring a negative specific orbital energy,
\begin{equation}
E = \frac{v^2}{2} - \frac{GM}{r} < 0
\end{equation}
and substituting from Eq. \ref{eqn:a2z} to give the acceleration constraint
\begin{equation}
|a_\rho| > \frac{\rho v^6}{8(GM)^2}.
\end{equation}
For each trial and each star, the orbital parameters are computed and
the results of all the trials are combined into a
six-dimensional probability density function (PDF) by dividing up parameter
space into bins, summing the number of trials in each bin, and then
normalizing by the total number of trials.
This Monte Carlo method is a straight-forward way to combine
a star's six measurement PDFs and the
four-dimensional PDF for the central point mass, which shows
strong correlations between $M_\bullet$ and $R_o$,
to produce a six-dimensional PDF for each star's orbital elements,
PDF($i$, $\Omega$, $\omega$, $e$, $P$, $T_o$), which has strong correlations
between the orbital parameters.
The results of these simulations are plotted for an example star, IRS 16SW,
in Figure \ref{fig:pdfParams_irs16SW}
to show that $i$ and $\Omega$ are generally well determined and that
$e$, in some cases, can be usefully constrained.
Similar figures of the orbital parameters for every star are
shown in Figure Set 7, which is available online in the electronic edition
of this manuscript.
The resulting stellar orbital parameters are constrained by several different
factors.
First, a measured acceleration that is significantly different from zero,
such as for S0-15, yields the best
determined orbit since the line-of-sight distance is confined to a small
range of values (Figure \ref{fig:pdfParams_example}, {\it top}).
Secondly, each star has a maximum allowed acceleration,
$a_{{\rho}, max} = |-GM/\rho^2|$, at the closest
possible distance set by the observed projected radius.
Stars with measured accelerations more than 3$\sigma$ below
the maximum allowed acceleration, such as IRS 16NW,
have strong lower limits on their line-of-sight distances, which translate
into significant constraints on the direction of the angular momentum
vector, $\vec{L}$, and can be equivalently expressed as constraints on
inclination, $i$, and on the angle to the ascending node, $\Omega$
(Figure \ref{fig:pdfParams_example}, {\it middle}).
Finally, even stars without significant limits on their
line-of-sight distance from accelerations have some well
constrained orbital elements. In particular, $i$ and $\Omega$
are well constrained as a result of the precise measurements for the
stellar velocities and potential parameters.
Furthermore, if the star's total velocity is higher than the circular
velocity at the two-dimensional projected radius, then it is higher than the
circular velocity at {\it all} distances and only non-zero eccentricity
orbits are allowed (Figure \ref{fig:pdfParams_example}, {\it bottom}).
The Monte Carlo analysis described above assumes that, in the absence
of an acceleration measurement, the acceleration should be drawn from a
uniform probability distribution; or, in other words, we adopt a
uniform acceleration prior. For those stars that are
only in the extended sample, the Monte Carlo
orbit analysis samples from this uniform acceleration prior
ranging from the largest allowed acceleration by the
projected radius to the smallest allowed for the orbit to remain bound.
For these stars and for stars in the primary sample with acceleration
limits that are not significantly
smaller than the maximum physically allowed acceleration,
the uniform-$a_\rho$ prior is an important assumption.
To test how sensitive our results are to this assumption, we performed
the same Monte Carlo analysis as detailed above using an
alternative assumption that the prior acceleration distribution is
uniform in z,
which shifts the line-of-sight distance PDF to larger values when
compared with a uniform-$a_\rho$ prior. On a star-by-star basis, the
resulting orbital parameters are consistent within 1$\sigma$
for both priors, with one exception. The young star S0-14 has an
eccentricity that is constrained to be higher than 0.93 (3$\sigma$)
with a uniform-$a_\rho$ prior, while with a uniform-z prior, all
eccentricities are allowed within 3$\sigma$. S0-14 is distinguishable
from all other stars in our sample in that it has a total velocity of
only 50 km s$^{-1}$, as compared to 160-640 km s$^{-1}$ for the rest of
the sample.
Such a small velocity translates into a very large range of allowed
line-of-sight distances which are not well sampled by a uniform-$a_\rho$
prior. S0-14's range of $i$ and $\Omega$ are not largely affected by the
choice of prior; therefore, we exclude S0-14 from our eccentricity analysis,
but we keep it in all other orbital analyses.
To distinguish between these two possible priors, we examine the resulting
distribution of orbital phases.
For a set of stars whose motion is dominated by the supermassive black hole
and that have been orbiting for more than a few
orbital time scales, the distribution of orbital phases should be
uniform.
The distribution of orbital phases for our sample is constructed by
summing the orbital phase PDFs for all the stars.
Figure \ref{fig:comparePriorsPhase} shows that
while the uniform-$a_\rho$ prior produces a population that is
uniformly distributed in orbital phase, the uniform-z prior produces
a distribution that is strongly peaked at 0 (periapse) due to the
higher occurence of large line-of-sight distances that, for a given
velocity, creates an artificial bias towards periapse.
Such a strong bias towards periapse is unlikely to occur even if
some of the young stars reside in a gravitationally bound cluster,
such as IRS 13, where all the cluster members would have a similar
orbital phase.
Based on our assumption that the distribution of orbital phases
should be roughly uniform, we adopt a uniform-$a_\rho$ prior instead of
the uniform-z prior in the following sections.
\section{Orbit Results}
\label{sec:orbitResults}
\subsection{Detection of the Clockwise Disk}
\label{sec:diskDetect}
A large number of stars appear to share a common orbital plane
based on our analysis, which has no prior assumption about the
existence of a disk.
The orientation of a star's orbital plane can be described by
a unit vector originating at Sgr A*'s position and pointing
normal to the orbital plane ($\vec{n}$); and, this normal vector's
direction can be expressed by the inclination angle ($i$) and the angle to the
ascending node ($\Omega$) using
\begin{equation}
\vec{n} = \left( \begin{array}{c} n_x \\ n_y \\ n_z \end{array} \right) =
\left( \begin{array}{c}
\sin\, i\; \cos\, \Omega \\
-\sin\, i\; \sin\, \Omega \\
-\cos\, i\; \end{array} \right).
\end{equation}
The direction of each star's orbital plane normal vector is
determined from the joint two-dimensional probability density
function of $i$ and $\Omega$,
PDF($i$, $\Omega$),
which is constructed by
binning the resulting $i$ and $\Omega$ values from the Monte Carlo
simulation in a two-dimensional histogram with equal solid angle bins using
the HEALpix framework \citep{healpix}.
Figure \ref{fig:iomap} shows PDF($i$, $\Omega$)
projected onto the sky as viewed from Sgr A*
for the same three example stars shown in Figure \ref{fig:pdfParams_example}.
Figure \ref{fig:allPlanePDF} shows, for all stars,
the contours for the 68\% confidence region, which,
on average, covers a solid angle of $SA_{\vec{n}} \sim$0.2 steradian (sr)
for the primary sample and 0.6 sr for stars found only in the extended sample,
which have larger proper motion uncertainties.
Table \ref{tab:eccDisk} \& \ref{tab:eccDiskExtended}
list this solid angle, $SA_{\vec{n}}$, for each star in the primary and
extended samples.
The bound orbit assumption does not greatly impact the size of the
$SA_{\vec{n}}$ because the orbital
parameters $i$ and $\Omega$ asymptote at large line-of-sight distances
as can be seen in Figure \ref{fig:pdfParams_example}.
Stars with acceleration limits significantly smaller than $a_{{\rho}, max}$
have two isolated solutions because small line-of-sight distances (z)
are not permitted and at large line-of-sight distances the positive-z
and negative-z solutions asymptote to
two different values of $\Omega$ (see Figure \ref{fig:pdfParams_example}).
Despite this degeneracy, the clockwise ($i$=90$^\circ$-180$^\circ$)
stars' normal vectors appear to cluster around a common point indicating
that many of these stars lie on a common orbital plane.
The directions of the stars' normal vectors show a
statistically significant clustering as measured by the
the density of normal vectors in the sky as viewed from Sgr A*.
To quantify the density of normal vector directions, we use
a nearest neighbor density estimate, which is commonly used to identify
galaxy clusters \citep[e.g.,][]{nearestNeighbor}, and take the density
at each point on the sky to be
\begin{equation}
\Sigma = \frac{k}{2\pi (1-\cos{\theta_k})} \textrm{stars sr}^{-1}
\end{equation}
where $\theta_k$ is the angle to the $k^{th}$ nearest star
and $k$ is taken to be 6.
We calculate the expectation value for the density of normal vectors at
each point on the sky using the Monte Carlo simulation discussed earlier.
For each Monte Carlo trial, the sky is divided into 12288 equal area pixels
(0.001 sr) using a HEALpix grid and the
density of normal vectors is calculated for each pixel.
These estimates are then averaged together over all the trials to provide
an average density per pixel on the sky.
The resulting average density of normal vectors is nearly the same
for a choice of 4th, 5th, or 7th nearest neighbor.
Additionally, a similar analysis using a fixed
aperture to calculate the density of normal vectors at each point on the
sky produced similar, but less smooth, results as the nearest neighbor
approach we adopt here.
A peak in the density of normal vectors is detected at
$i = 115^\circ \pm 3^\circ$ and $\Omega = 100^\circ \pm 3^\circ$,
which provides direct evidence of a common orbital plane without any
prior assumptions (see Figure \ref{fig:orbitPlane}).
The uncertainty on the peak position is taken as the
half-width at half-maximum of the peak divided by the square-root of the
number of stars that are candidate disk members,
$\sqrt{N_{disk-stars}}$ (see below).
We also note that an analysis of the entire extended
sample produces a peak at the exact same position.
The mean density of normal vectors at the peak is
0.016 stars deg$^{-2}$ with a negligible uncertainty on the mean value
($< 10^{-4}$ stars deg$^{-2}$).
The significance of the peak is determined by comparing the
background density of normal vectors, which is defined by the
average (0.001 stars deg$^{-2}$) and standard deviation (0.0008 stars deg$^{-2}$)
of all other pixels on the sky
after first rejecting those pixels ($\sim$0.25 sr)
that are high outliers (more than three standard deviations).
The density peak is $\sim$19$\sigma$ above the observed
background density.
A second comparison can be made to the density expected
if the 32 stars in our sample were isotropically distributed over
4$\pi$ steradians.
The observed peak in the density is
$\gtrsim$20 times higher than this isotropic density. Thus we conclude
that there is a statistically significant common orbital plane of young stars.
The majority of the young stars that are orbiting in the clockwise direction
are likely to be orbiting in this common plane.
A comparison of each star's normal vector to the common plane's normal vector
allows us to determine which stars are {\it not} on the common plane with
high statistical significance.
All other stars are then considered {\it candidate} members.
First, a preliminary estimate of the thickness of the common plane is
determined by defining the solid angle extent of the
plane, $SA_{plane}$, encompassed by the contour at which the
density drops to half of the peak value.
This corresponds to a region
with a solid angle of $SA_{plane} \sim$0.1 sr, which gives a half-opening
angle of 0.2 radians (10$^\circ$) for a cone with the same $SA_{plane}$.
Then, each star's probability density function, PDF($i$, $\Omega$),
is integrated over this region to determine the probability that the
star is a disk member.
The orientation of the stars' normal vectors have a wide range of
uncertainties as expressed by the total solid angle covered by each star,
so it is
necessary to distinguish between those stars that have a low probability
due to a large $\vec{n}$-uncertainty (i.e. large solid angle)
vs. those stars that have a low probability
because they are significantly offset from the common plane.
Therefore, we normalize the above integrated probability by
the probability at the peak of the star's PDF integrated over a region
that has the same total area as the common plane
\begin{eqnarray}
\mathrm{L}(\mathrm{not\;on\;plane}) & = & 1 -
\frac{\int_{\mathrm{plane}} \mathrm{PDF}(i, \Omega)\;\; d\mathrm{SA}}
{\int_{\mathrm{peak}} \mathrm{PDF}(i, \Omega)\;\; d\mathrm{SA}} \\
\int_{\mathrm{plane}} d\mathrm{SA} & = & \int_{\mathrm{peak}} d\mathrm{SA}
\end{eqnarray}
where SA is the solid angle and L(not on plane)
is the likelihood that the star is {\it not} on the common plane.
Those stars with likelihoods, L(not on plane),
of greater than 0.9973 (equivalent to 3$\sigma$ for a gaussian distribution)
are flagged as non-members of the common plane.
The remaining set of stars are considered candidate members of the common plane.
Table \ref{tab:eccDisk} \& \ref{tab:eccDiskExtended}
list [1 - L(not on disk)] for each
star and Figure \ref{fig:PMimage} shows the positions of
candidate members of the common plane in red and non-members in blue.
Of the primary sample of 32 stars, 26 of which are orbiting in a clockwise
sense on the plane-of-the-sky, we find that 22 are possible members of the
common plane ($N_{disk-stars} = 22$).
The clockwise common plane that we measure is slightly offset from the
clockwise planes proposed in earlier works.
Over-plotted in black on Figure \ref{fig:orbitPlane} is the candidate
orbital plane proposed by \citet{levin03} with
updated values from \citet{paumard06} for the candidate plane normal
vector (solid black) and thickness (dashed black).
The previously proposed plane was derived by minimizing a statistical metric,
K, in order to find the best-fit common orbital plane from the velocity
vectors of a sample of stars (see Appendix \ref{app:kmetric}).
However, some stars are not members of the common plane and including
them in the fit biases the result since they have extremely well
measured velocities (S0-15, IRS 16C, S3-19).
For example, using the K metric
approach of \citet{levin03}, fitting all 26 clockwise stars in our
primary sample gives i = 128$^\circ$ and $\Omega$ = 102$^\circ$ with K = 0.7,
which is closer to the disk found by \citet{paumard06} at
i = 127$^\circ$ and $\Omega$ = 99$^\circ$.
While fitting only the 22 stars that are consistent with the clockwise disk
based on our orbit analysis gives i = 117$^\circ$ and $\Omega$ = 98$^\circ$ with
K = 0.2. Therefore, using the K metric to determine the common plane can
produce biased results due to the inclusion of non-members.
By combining position, velocity, and acceleration information in order to
determine the orbital plane for each star, the direction of a common
orbital plane can be estimated more robustly.
The detected common orbital plane is composed of stars dispersed in a
disk rather than in a single cluster as can be seen from the
stars' positions within the common plane shown in
Figure \ref{fig:diskVelRadius}.
In this figure, the stars' positions have been
converted into a disk coordinate system defined as
[$\hat{p}$, $\hat{q}$, $\hat{n}$] where $\hat{n}$ is perpendicular
to the disk plane,
$\hat{p}$ is along the line of ascending nodes (where the plane of the sky
intersects the disk plane), and $\hat{q} = \hat{n}\times\hat{p}$.
For each star, all orbital solutions that fall within 10$^\circ$ of the
common orbital plane are combined to create
a probability distribution for the star's position in the disk,
PDF($p$, $q$), which is shown in Figure \ref{fig:diskVelRadius} ({\it left}).
Each stars probability distribution is elongated in the $q$-direction
due to the large range of line-of-sight distances, $z$, that
are possible within the small range of possible disk inclinations for this
nearly edge-on plane of the disk.
The thickness in the $p$ direction is largely set by the uncertainties
in the potential parameters ($M_\bullet$, R$_o$) and velocities.
The distribution of young stars within the plane shows a range of
position angles on the plane, consistent with a stellar disk rather
than a stellar cluster.
The CW stellar disk is detected both in our analysis of the primary sample
and in a similar analysis of the entire extended sample.
The additional young stars in the extended sample have larger velocity
uncertainties and no acceleration information, therefore the Monte Carlo
orbit analysis samples from a prior probability distribution that is
uniform in acceleration ranging from the largest allowed by the
projected radius to the smallest allowed for the orbit to
remain bound.
We note that even if we ignore the acceleration measurements for our primary
sample analysis, the CW stellar disk is still detected, although the
significance is lowered from $\sim$19$\sigma$ to $\sim$8$\sigma$ above the
background density of normal vectors.
Thus the additional stars' orbits in the extended
sample are still constrained (see Figure \ref{fig:histSolidAngle}),
even though they have larger uncertainties as compared to the stars in
just the primary sample.
The density of normal vectors from the extended sample analysis shows a
peak within 1$^\circ$ of the disk's position from the primary sample.
The analysis of the extended sample shows that $\sim$50\% of the young
stars reside on the CW disk and there is no statistically significant
change ($>3\sigma$) in the fractional number of disk stars at different radii.
For reference, the 73 young stars in the extended sample are
distributed on the plane of the sky with a surface density
that decreases with radius as $\rho^{-2.1 \pm 0.4}$.
Within a projected radius of 3'', the fraction of candidate
disk members is 72\% $\pm$ 9\% (18 out of 25) and
at projected radii larger than 3'', the fraction of candidate disk members
is 42\% $\pm$ 7\% (20 out of 48).
Given the small number of known young stars, Poisson statistics indicate
that this change in the fraction of candidate disk members is only marginally
statistically significant at the 2.6$\sigma$ level.
Likewise, the projected surface density for the on-disk and off-disk
populations shown no significant difference from each other or from that of
the total population.
Thus the number of candidate disk members does not change with radius and
roughly half of the young stars reside on the CW disk.
The K-band luminosity function (KLF) of the young stars does not change
significantly with radius or when considering stars on and off the
disk. To compare the KLF as a function of radius, the entire extended
sample of young stars is divided into a near sample (r $<$ 3\farcs5)
and a far sample (r $\geq$ 3\farcs5) and the KLF is constructed for
each. A two-sample KS test yields a probability of 46\% that the
near and far samples have the same KLF. Similarly, the KLF is constructed
for stars on and off the disk and a two-sample KS test yields
a probability of 74\% that the on-disk and off-disk samples have the
same KLF. Finding more young stars will allow for a more detailed
comparison of the KLF for different subsets within the young stars
population.
\subsection{Limits on Additional Stellar Disks}
In our primary sample, no common orbital plane is detected for the
counter-clockwise population of stars; however, our sample is limited to
six counter-clockwise
orbiting stars, only two of which (IRS 16NE, IRS 16NW) are claimed by
\citet{paumard06} to
reside on the counter-clockwise disk.
Out of the 6 counter-clockwise stars in our primary sample, we find that only
IRS 16NE and S2-66 could be consistent with the previously proposed
counter-clockwise disk.
The proposed counter-clockwise disk may have a larger
radial extent than is covered by our observations,
so in order to fully explore whether
our lack of detection of a 2nd disk is due to our limited field-of-view,
it is necessary to analyze the extended sample.
As discussed in \S\ref{sec:orbitAnalysis},
the uniform acceleration prior adopted for this analysis tends to
overemphasize face-on orbital planes, making it easier to detect the
proposed CCW disk, as \citet{paumard06} suggest it has an inclination
of 24$^\circ$.
Using the extended sample, our analysis of the density of normal vectors,
in the region of the proposed counter-clockwise disk, reveals no
significant over-density.
Of the 73 stars in the extended sample, at least 34 are not on the clockwise
disk and thus we compare the density observed in the region of the
proposed counter-clockwise
disk to that expected for an isotropic distribution of 34 stars.
The observed density of normal vectors in the region of the
counter-clockwise disk
is 2.4$\times$10$^{-3}$ stars deg$^{-2}$, which is only a factor of 3 above
what is expected for an isotropic distribution and is less than
1$\sigma$ above the background over the rest of the sky (excluding
the clockwise peak).
This density of normal vectors corresponds to only 3 stars
within 19$^\circ$ of the putative CCW disk,
where 19$^\circ$ is the disk thickness proposed
by \citet{paumard06}, and is consistent
with random fluctuations of an isotropic distribution having the
$\vec{n}$-uncertainties shown in Figure \ref{fig:histSolidAngle}.
We estimate that this analysis is capable of revealing, at the 3$\sigma$ level,
a stellar disk with more than 7 stars within a solid angle cone of radius =
19$^\circ$ at the location of the proposed CCW disk;
thus the proposed CCW disk containing 17 stars as suggested by
\citet{paumard06} should have been detected with this approach.
There are several principle differences between our analysis and
that in earlier works. First, previous works make the {\it a priori}
assumption that a disk exists through the use of the statistical metric,
K, and the results were not compared to a null hypothesis (i.e. no disk)
to establish the statistical significance of a disk detection.
Furthermore, the K metric used in previous works suffers from a bias which
is described in Appendix \ref{app:kmetric}.
The primary goal of our methodology
is to minimize the number of {\it a priori} assumptions and to
fully quantify the significance of any disk detected as compared to the null
hypothesis that there is no disk. Therefore, we choose to search for disks
using all the young stars rather than first trimming out stars based on
projected angular momentum criteria or radii.
Also, we determine the range of allowed orbital orientations for each
star individually rather than searching for a disk from a statistical
sample of young stars. In this fashion, we utilize not only the
direction information for a velocity vector, as has been used previously,
but also the physical relationships between the magnitude of the velocity
and the positional information. This method allows for no disk to be
detected, while the previously used statistical tests assumed a disk
model and, therefore, must be compared
to the no-disk hypothesis using simulations of isotropic populations.
Without the simulations, the significance of any disk detection via
the K metric cannot be fully quantified. The resulting distribution
of orbits from our analysis is consistent with the
hypothesis of a single, clockwise disk plus a more randomly
distributed population.
\subsection{Properties of the Clockwise Disk}
\label{sec:diskProperties}
We now examine, in detail, the properties of the detected clockwise disk.
With the identification of a single stellar disk and a
candidate list of disk members, we investigate the following:
(1) the thickness of the disk,
(2) the radial profile of the disk,
(3) the azimuthal isotropy of the disk,
(4) the eccentricities of stars in the disk,
and (5) the luminosity function of the stars in the disk.
These properties are critical for distinguishing between {\it in situ}
and infalling cluster formation scenarios, as well as for understanding the
dynamical evolution of the young stars both on and off the disk.
The observed disk of young stars has a significant intrinsic thickness;
however, the vertical velocity dispersion is less than previously determined.
To measure the thickness of the disk,
the dispersion of the velocities out of the plane (along the $\vec{n}$
direction) is calculated from all candidate disk members by projecting
each star's three-dimensional velocity vector along the
disk's normal vector to give $v_{\vec{n}}$.
The measurement uncertainties in both $\vec{v}$ and $\vec{n}$
are propagated through this coordinate transformation.
The intrinsic velocity dispersion is calculated using
\begin{eqnarray}
\sigma^2_{\vec{n},intrinsic} & \;=\; &
\sigma^2_{\vec{n},measured} \;-\; \sigma^2_{\vec{n},bias} \\
\sigma^2_{\vec{n},intrinsic} & \;=\; &
\left( \frac{1}{N_{disk-stars} - 1} \right) \left(
\displaystyle\sum_{i=0}^{N_{disk-stars}} v^2_{\vec{n},i} \;-\;
\displaystyle\sum_{i=0}^{N_{disk-stars}} error^2(v_{\vec{n},i})
\right)
\end{eqnarray}
where the bias term, $\sigma_{\vec{v},bias}$, is 19 km s$^{-1}$ and accounts for
added dispersion as a result of uncertainties in the measurements.
The resulting intrinsic velocity dispersion is 28 $\pm$ 6 km s$^{-1}$, which
is significantly different from zero, thus a finite thickness is required.
However, this velocity dispersion is a factor of 2 smaller than
that found using the previously proposed disk solution of \citet{paumard06}
and is slightly smaller than the value reported in \citet{beloborodov06}
due to our improved identification of candidate disk members.
The disk's
thickness can be expressed as the ratio of the vertical scale height to
radius, $h/r = \sigma_{\vec{n},intrinsic} / <|\vec{v}|>$, and is 0.08 $\pm$ 0.02.
Following a similar analysis to \citet{beloborodov06}, but with the above
relationship between $h/r$ and the velocity dispersion, the disk thickness
can also be described using a gaussian distribution of inclination angles
about the disk plane with a standard deviation of
$\Delta\theta$ and is related to the scale height of the disk by
$h/r \sim \sqrt{1/2} \Delta\theta$. This yields a dispersion angle of
$\Delta\theta = 7^\circ \pm 2^\circ$ for the young stellar disk. This
more rigorous determination of the disk thickness is consistent with the
thickness we derived in \S\ref{sec:diskDetect} from the half-width at
half-maximum
of the peak in the density of normal vectors; thus the selection of the
candidate disk members is likely robust.
In comparison, the previously proposed disk solutions
yield a disk thickness of $h/r = 0.2$ ($\Delta\theta = 14^\circ$)
and $h/r = 0.1$ ($\Delta\theta = 9^\circ$)
for \citet{paumard06} and \citet{beloborodov06}, respectively.
We caution that all of these conversions from velocity dispersion to disk
scale height and dispersion angle assume circular orbits and an isothermal
disk structure.
From our analysis, we note that the out-of-the-plane velocity dispersion
shows no statistically significant variation with radius in the disk both
for the primary (difference is $1\sigma \sim$ 7 km s$^{-1}$) and
the full extended samples (difference is $1\sigma \sim$ 14 km s$^{-1}$).
Therefore, the observations are consistent with a thin disk of
uniform velocity dispersion at all radii.
The surface density of stars in the disk falls off rapidly as a
function of radius. In order to extend the radial coverage, we
consider the entire extended sample in this analysis.
The young stars that are candidate disk members have constraints
on their three-dimensional radii if we limit their orbital solutions
to those close to the disk plane. Thus the disk's surface
density can be determined as a function of three-dimensional radius
rather than just the projected two-dimensional radius as discussed at the
end of \S\ref{sec:diskDetect}.
The distribution for each star's position within the disk plane,
PDF($p$, $q$), is constructed from orbits that are within 10$^\circ$
of the disk and is shown in Figure \ref{fig:diskVelRadius}.
Then the disk's surface density at each radius is computed
numerically by sampling the PDF($p$, $q$) 10$^5$ times for all the candidate
disk members and constructing a radial histogram for each trial.
The radial histograms are combined for all the trials to find the peak
and 68\% confidence bounds for the expected number of stars at each radius.
This is converted into an azimuthally integrated
surface density by dividing by the area of a ring
at each radius. This method of constructing the surface density captures
both the measurement error in the individual stars and the finite
thickness of the disk, which has not been incorporated into previous
estimates. The resulting azimuthally averaged surface density on the
disk is shown for the extended sample in Figure \ref{fig:diskRadialDist}
and has a best-fit power-law profile of $r^{-2.3\pm0.7}$.
This is consistent with the previous results \citep{paumard06},
but our analysis accounts for the uncertainty in each stars
line-of-sight distance due to the finite disk thickness and, therefore
yields a larger uncertainty on the power-law index.
Visual examination of the stars' positions in the disk plane
(Figure \ref{fig:diskVelRadius}) suggests there may be
some anisotropy as evidenced by the clustering of stars on the lower
part of the disk; however, this over-density is only marginally
statistically significant based on the following analysis.
In order to search for non-uniformities,
we compare the observed stellar surface density of the
extended sample within the disk plane
with the surface density expected for an azimuthally symmetric disk.
The observed stellar surface density is measured by
sampling from all stars' PDF($p$, $q$) for 10$^5$ trials and calculating the
stellar surface density over a grid of points in the disk plane
for each trial. For each point on the
disk plane, the surface densities from all trials are combined, yielding
the most probable surface density with uncertainties.
The resulting two-dimensional map of observed surface densities is then
compared to the expected surface densities for an azimuthally symmetric
disk by subtracting the two values and dividing by the uncertainties.
This produces a surface density excess map that shows the significance
of any excess.
The disk shows a marginally significant ($\sim 3\sigma$) over-density
on the front side (q $<$ 0) of the disk and a corresponding under-density on
the back side (q $>$ 0).
A few candidate disk stars show evidence for eccentric orbits.
To determine whether any of the stars' eccentricities are consistent
with a circular orbit, the six-dimensional probability density function for
the orbital parameters is marginalized and re-expressed as a PDF for
the eccentricity vector (see Appendix \ref{app:orbits}),
PDF($e_x$, $e_y$, $e_z$). The magnitude of this
vector is the orbital eccentricity and the direction
points along the semi-major axis towards the periapse position.
The PDF for the eccentricity vector cannot be further marginalized to
produce a PDF of the eccentricity magnitude without introducing a bias
due to the positive, definite nature of a vector magnitude. This is the
same bias term as described in the velocity dispersion analysis; however,
unlike the velocities, the eccentricity distributions are strongly
non-gaussian and the bias term cannot be easily accounted for in the
marginalization.
The peak of PDF($e_x$, $e_y$, $e_z$) gives the unbiased orbital
eccentricity and the 99.7\% confidence interval of the three-dimensional
distribution is used to determine the range for the
one-dimensional eccentricity.
Tables \ref{tab:eccDisk} and \ref{tab:eccDiskExtended}
show the 99.7\% confidence range of the eccentricities for all stars
in the primary and extended samples. Also,
Figure \ref{fig:eccentricity} shows the eccentricity 99.7\% confidence
lower limit
for the candidate disk members in red, non-disk members in blue, and excludes
S0-14 (see \S\ref{sec:orbitAnalysis}).
When considering all possible orbital solutions, the resulting eccentricity
ranges show that 2 candidate disk members from the primary sample
have 99.7\% confidence eccentricity lower limits of greater than 0.2.
Restricting the possible orbital solutions to only those
having normal vectors oriented within 10$^\circ$ of the disk normal
vector increases the number to 8 candidate disk members
with 99.7\% confidence eccentricity lower limits larger than 0.2.
We find high-eccentricity
stars in the disk, similar to the analysis of \citet{beloborodov06} in
which they assumed an infinitely thin disk. However, our analysis
incorporates the finite thickness of the disk and places statistical
errors on the eccentricities for individual stars.
The average eccentricity of the entire population is not yet well
constrained.
The eccentricity for the stellar disk is determined using the
eccentricity vector.
For each candidate disk member, orbital solutions are selected
whose normal vectors point within 10$^\circ$ of the disk normal vector.
These orbital solutions are combined for all the disk stars by
averaging their PDFs to create
a combined probability distribution for all stars' eccentricity vectors,
which is then projected into the disk plane and plotted in two dimensions
(Figure \ref{fig:eccVector}).
This two-dimensional probability distribution gives an unbiased estimate
of the eccentricity magnitude
and shows that while the characteristic disk eccentricity peaks
at e=0.22, it is consistent with e=0.0 $-$ 0.8 at the 1$\sigma$ level,
reflecting the large eccentricity uncertainties for the majority of the
candidate disk members.
\section{Discussion}
\label{sec:discussion}
The kinematic analysis of the young stars in the central parsec
around our Galaxy's supermassive black hole has implications for the recent
star formation history in this region.
Our first attempt at determining individual orbits for young stars that
reside outside the central arcsecond shows definitive evidence for
the clockwise-rotating disk that was suggested
by \citet{levin03} and was subsequently refined by
\citet{genzel03cusp} and \citet{paumard06}.
Our results do not show a statistically significant second disk.
The presence of a single stellar disk eliminates the need to
invoke two distinct starburst events occuring roughly 6 Myr ago and
greatly simplifies the demands on both {\it in situ} and infalling
cluster scenarios. For instance, in the self-gravitating gas disk scenario,
the detection of only a single stellar disk lifts the requirement
for a second disk to rapidly build up gas, fragment,
and form stars within 1-2 Myr of the formation of the first disk.
Likewise, for the infalling cluster
scenario, the presence of only one stellar disk means that the
frequency of such infall events is half that required for
the existence of two disks.
On the strength of our confirming only one stellar disk, we consider
whether all of the young stars within the central parsec may have formed
in a single burst of star formation.
Such a scenario must explain not only the observed
clockwise stellar disk, oriented at
$i\sim$115$^\circ$ and $\Omega\sim$100$^\circ$, but also the
presence of roughly half of the young stars from our extended
sample on more isotropically distributed orbits out of the disk.
In the single starburst scenario, the out-of-the-disk stars could either
be generated during the formation process or could intially be in
the disk and then perturbed through subsequent dynamical evolution.
Self-relaxation of the disk has not had sufficient time to produce
the out-of-the-plane population \citep{alexander07imf,cuadra08},
but other mechanisms have been
proposed such as scattering by an inward-migrating IMBH \citep{yu07}.
Currently, our results show that the
on-disk and off-disk populations of young stars look very similar
outside the central arcsecond (0.04 pc)
both in terms of the K-band luminosity function and the surface density
profiles that decreases at larger projected radii as $\propto r^{-2}$.
However, the number of young stars in the disk drops at radii smaller
than 0.08 pc; and at radii of $\lesssim$0.04 pc, none of the observed young
S-stars are in the disk \citep{ghez05orbits,eisenhauer06}.
This drop in the number of disk stars at
small radii may be the result of resonant relaxation or other dynamical
processes if the central arcsecond S-stars are a continuation of the
disk population \citep{hopman06}.
Thus, if dynamical evolution produced the off-disk population, then
the dynamical process must not be a strong function of radius
beyond 0.08 pc.
Our distributions show that a potential problem with the single starburst
scenario is the presence of the apparent massive star cluster, IRS 13, located
$\sim$4'' from the supermassive black hole \citep{maillard04irs13,schodel05}.
The cluster's orbit is not in the disk plane
and, given the proposed mass of IRS 13 ($>$10$^3$ M$_{\odot}$\,), it is
unlikely that it could have been ejected from the disk.
However, the definition of IRS 13 as a cluster and the derived
mass is based on observations of only 3-4 bright stars and
is complicated by enhanced dust and gas emission in the vicinity.
More data are needed to determine the total mass of IRS 13 and its
relationship to the disk stars.
Our results also have implications for the star formation mechanism.
For both infalling cluster and {\it in situ} formation
scenarios, we consider whether the observed characteristics
of the young stellar disk can be explained.
We observe a stellar disk with an out-of-the-disk velocity dispersion
of 28 $\pm$ 6 km s$^{-1}$. Additionally, if we consider only orbital
solutions within the disk (disk prior), we find that at least 8 of the 22
candidate disk stars have 99.7\% confidence lower limits on the
eccentricity of greater than 0.2.
Therefore, any formation scenario should explain not only a single thin
stellar disk but also allow for non-circular stellar orbits of
some stars in the disk.
First, for the infalling star cluster formation scenario,
some of the disk properties we observe are well explained
and others appear difficult to reconcile with this model.
For instance, eccentric orbits are easily produced.
Stars that are stripped from a cluster as
it spirals in should have a similar inclination and
eccentricity as the cluster itself. Therefore, an infalling cluster
with an initially eccentric orbit will produce a disk of stars with
similarly eccentric orbits \citep{berukoff06}.
Previous studies have observed co-moving clumps of stars,
such as IRS 16SW \citep{lu05irs16sw} and IRS 13 \citep{schodel05},
that appeared to support the infalling cluster formation scenario as
they could be the remnant core of the dissipated cluster.
We tentatively observe evidence for an over-density of stars on the
front half of the disk at the position of the IRS 16SW co-moving group.
However this over-density may be explained by the effects of
extinction that reduces the number of young stars identified
on the back half of the disk at a given magnitude.
The extinction is highly variable throughout the region
and the back half of the disk is behind a patch of higher exctinction
\citep[$\Delta$A$_K$ = 0.3 - 1.4; ][]{scoville03,schodel07}.
Thus the apparent overdensity on the front half of the disk,
corresponding to the IRS 16SW co-moving group, can perhaps be ascribed to
differential extinction.
More data are needed to confirm the observed disk asymmetry and to determine
whether the cause is extinction.
Our results yield a steep radial profile for the young stars in
the disk, as also found by \citet{paumard06}, which appears to be
inconsistent with the flatter profile expected for an infalling cluster
\citep[$r^{-0.75}$, ][]{berukoff06}.
We note that mass segregation is observed in massive star
clusters that are only a few million years old
\citep{hillenbrand98,fischer98,stolte06}. Any mass segregation that existed
prior to the cluster's dissolution may impact the observed radial profile
as the massive stars would have resided preferentially in the cluster core
and would therefore have been deposited at the smallest radii.
Thus, the massive stars O stars that we observe today
may have a steeper radial profile than the entire young star population.
Additionally, the lack of X-ray emission from pre-main-sequence stars
\citep{nayakshinSunyaev06} is not well explained by an infalling cluster
model.
A larger and deeper survey for young stars over the central $\sim$5 pc could
definitively rule out this scenario if the tidal tails of the disrupted
clusters are not detected.
Some theories of {\it in situ} star formation
take place in a circular gas disk. Such a gas disk
can be built up from a steady inward migration of
material or from many small cloud-infall events and the disk
would circularize prior to becoming massive enough to form stars from
self-gravity ($>10^4$ M$_{\odot}$\,). Such a formation scenario would most
likely produce a steep radial profile in agreement with our observations.
Our observations of over 30\% of the
candidate disk members with eccentricities greater than 0.2 appears to be
inconsistent with an initially circular disk of stars and a normal
initial mass function.
A disk of stars on initially circular orbits and with a normal IMF
will relax over 6 Myr and produce a thermal distribution of
eccentricities with an rms eccentricity of 0.15 or less \citep{alexander07imf}.
For such a disk, only 4 out of 22 stars should have eccentricities higher than
0.2, compared with the 8 out of 22 observed when a disk prior
is imposed on the primary sample.
Therefore, in order for the disk to have been initially circular with
a normal-IMF, some additional dynamical processing other than self-relaxation
is needed. Other possibilities are that the initial mass function may
have been top-heavy, the binary fraction may have been extremely high,
or IMBHs could have formed, all resulting in faster relaxation
to higher eccentricities, but these are not
sufficient to explain the out-of-the-disk population of young stars
\citep{alexander07imf,cuadra08}. The gas disk formation scenario may be
modified \citep{alexander07imf,cuadra08} to accommodate the observed high
stellar eccentricities and out-of-the plane population by
building up a massive gas disk in a single cloud infall or a cloud-cloud
collision event, in which the clouds are on eccentric orbits
\citep{sanders98,vollmerDuschl01,nayakshin07sims}.
The gas disk would then have a high eccentricity for a short period
of time during which stars might form \citep{alexander08,bonnell08}.
The cloud-cloud collision scenario may yield both a thin stellar disk and
a more distributed population of stars at larger radii
with a range of angular momenta as a result of the complex interactions
and shocks during the collision.
It is also conceivable that a cloud-cloud
collision scenario might give rise to out-of-the-disk clumps of gas
that could form a cluster such as IRS 13.
Refined estimates of the eccentricity and inclination distributions
of the young stars and more detailed theoretical
analysis are needed to investigate the viability of this scenario.
\section{Conclusions}
In summary, the advent of laser guide star adaptive optics has allowed
us to retroactively improve our 11 year astrometric data set used for
monitoring stars orbiting our Galactic Center. This has increased our
proper motion precision, with resulting uncertainties of $\sim$3 km s$^{-1}$,
and allowed us, for the first time, to make measurements of and place
limits on accelerations for stars outside the central arcsecond
out to a radius of 3\farcs5, with typical 3$\sigma$ acceleration
limits of -0.19 mas yr$^{-2}$.
By combining our improved stellar positions and proper motions with radial
velocity information from the literature, we compute orbits
for individual young stars proposed to lie in stellar disks orbiting
the supermassive black hole.
The orbits for the young stars confirm only a single disk
of young stars at a high inclination rotating in a clockwise sense and
there is no statistically significant evidence for a second disk.
Stars within the well-defined, clockwise disk
have an out-of-the-disk velocity dispersion of 28 $\pm$ 6 km s$^{-1}$
and several stars have high eccentricities. These disk properties
suggest that star formation may have occurred in a single event, rather
than the two events previously needed to explain two stellar disks; however,
there are open questions as to how $\sim$50\% of all young stars can be
perturbed out of the disk plane and whether
the apparent compact cluster, IRS 13, which is not part of the stellar
disk, requires a separate star formation or dynamical event.
Future directions include (1) obtaining new LGSAO
data sets with improved astrometry to measure accelerations for the
young stars at all radii and (2) identifying new young stars within
the central parsec in order to better constrain the orbital
properties of these stars and to study in detail the
distribution of eccentricities and semi-major axes for stars both in and
out of the disk.
\acknowledgements
Support for this work was provided by NSF grant AST-0406816, and
the NSF Science \& Technology Center for AO, managed by UCSC
(AST-9876783).
Additional support for J.R.L. was provided by
a NSF Graduate Research Fellowship.
We would like to thank Brad Hansen
and the anonymous referee for helpful comments.
The W.M.~Keck Observatory is operated as a scientific
partnership among the California Institute of Technology, the University
of California and the National Aeronautics and Space Administration.
The Observatory was made possible by the generous financial support of
the W.M.~Keck Foundation.
{\it Facilities:} \facility{Keck:II (NIRC2)}, \facility{Keck:I (NIRC)}
\begin{appendix}
\section{NIRC Speckle Distortion}
\label{app:speckDistort}
In the speckle data sets, optical distortions, introduced by the NIRC
reimager, are small near the center of the field-of-view where
Sgr A* was positioned, but grow to dominate the positional
uncertainties for stars located more than $\sim$0\farcs5 from Sgr A*
(see Figure \ref{fig:distort} and \S\ref{sec:images}).
Now, utilizing images of the Galactic Center obtained with NIRC2,
which has optical distortions characterized at the $\sim$2 mas level
\citep{ghez08},
we can, for the first time, similarly quantify and correct
the optical distortions in the NIRC reimager speckle data sets.
Images of the Galactic Center were obtained with both NIRC and NIRC2 on
consecutive nights during July 2004
and the NIRC2 images were used as a reference coordinate system.
The individual NIRC speckle exposure times are only
0.1 seconds and have insufficient signal-to-noise to detect more than the
brightest 5 stars. Exposures were obtained in sets of 100 and each set is
combined to produce a single image in which approximatly 100 stars are
detected. It is assumed that the images are mostly stationary on the
NIRC detector during each set of exposures. For each stacked image, the stars'
positions are compared to those in the NIRC2 image and the offsets are
mapped into NIRC detector coordinates
(see Figure \ref{fig:distort2d}, {\it left}).
In this fashion, a distortion map is built up from many stacks of images
which are dithered and rotated such that stars fall on many different
positions on the detector.
The distortion solution was obtained by fitting the distortion map with
polynomials of the form
\begin{equation}
(x^\prime + 128) = a_0 + a_1(x - 128) + a_2(y - 128)
\end{equation}
\begin{equation}
(y^\prime + 128) = b_0 + b_1(x - 128) + b_2(y - 128)
\end{equation}
where the best-fit distortion parameters are listed in Table
\ref{tab:distort}.
The new distortion solution improves the RMS residual errors
per stack by a factof of 3 to $\sim$3 mas
(Figure \ref{fig:speck_distort}), which is further reduced in the
final image by averaging the dithered stacks.
Higher-order polynomial terms did not sufficiently improve the
fit to warrant inclusion. The above solution is applied
after the initial application of the standard NIRC distortion correction.
The map of positional differences between stars in the NIRC and NIRC2
images before and after the NIRC-reimager distortion correction is shown
in Figure \ref{fig:distort2d} ({\it right}).
The resulting radial dependence on the RMS
positional uncertainty is greatly improved and is shown in
Figure \ref{fig:distort}, which plots many stars' RMS residual offset from
their best-fit proper motions across all epochs. In the final analysis
of the speckle data, the relative astrometric uncertainty is $\sim$2 mas.
\section{Analytic Orbit Equations}
\label{app:orbits}
The orbit of a star in a known point source potential can be derived from a
single measurement of a star's orbital state vector.
At epoch t$_{ref}$, the orbital state vector is usually
described by the star's position, $\vec{r}$, and velocity, $\vec{v}$,
relative to the central mass. For the analysis in this paper,
the state vector is estimated using measurements of the three-dimensional
velocity, $\vec{v} = [v_x, v_y, v_z]$, and the
projected position, $\vec{r}_{2D} = [x, y]$, and $z$ is derived from
the radial acceleration on the plane of the sky. For brevity, we have
removed the $ref$ subscript notation and all of the above variables are
measured at t$_{ref}$.
Orbital trajectories are then inferred from conservation of
energy, specific angular momentum, and eccentricity
($\epsilon$, $\vec{h}$, $\vec{e}$),
which are related by $\vec{e} \cdot \vec{h} = 0$ and
$|e|^2 - 1 = 2\,\epsilon\,h^2 / GM$ giving 5 constants of motion
plus an undetermined reference time.
Equivalently, the orbital trajectory can
be expressed using the standard Keplerian orbital elements:
period ($P$), eccentricity ($e$), time of periapse passage (T$_{\circ}$),
inclination ($i$), position angle of the ascending node ($\Omega$),
and the longitude of periapse
\citep[$\omega$; see Equations \ref{eqn:i}, \ref{eqn:e}, \ref{eqn:w},
\ref{eqn:o}, \ref{eqn:p}, \ref{eqn:t0} and][for detailed descriptions of these orbital parameters]{ghez05orbits}.
The 3D position and velocity state vectors can be used to calculate
the orbit of the star around the black hole (by algebraic manipulation of
Kepler's Laws).
Here we present the analytic expressions used to compute the orbital elements
from the state vectors.
Orbit determination for the young stars in our sample is tractable because
the mass and position of the black hole are determined by independent means,
namely the well determined orbits of stars much closer to the black hole.
The coordinate system is set such that Sgr A* resides at the origin,
$\hat{x}$ and $\hat{y}$ increase with right ascension and declination,
and $\hat{z}$ increases with the line-of-sight distance from the Earth
to Sgr A* with z=0 at the location of the black hole.
Combining the two state vectors, $\vec{r}$ and $\vec{v}$, and the
black hole mass, there are three intermediate vectors that describe the
geometry of the orbit both in three-dimensions and projected onto the plane
of the sky. These are (1) the specific angular momentum vector,
$\vec{h}$, which points normal to the plane of the orbit, (2) the
eccentricity vector, $\vec{e}$, which points in the direction of periapse,
and (3) the ascending node vector, $\vec{\Omega}$, which points to where
the star passes through the plane of the sky moving away from us,
and are given by
\begin{eqnarray}
\vec{h} & = & \vec{r} \times \vec{v} \\
\vec{e} & = & \frac{\vec{v} \times \vec{h}}{GM} - \frac{\vec{r}}{|\vec{r}|} \\
\vec{\Omega} & = & \vec{h} \times \hat{z}.
\end{eqnarray}
The semi-major axis can also be calculated as an intermediate quantity
\begin{eqnarray}
a & = & \left ( \frac{2}{|\vec{r}|} - \frac{|\vec{v}|^2}{GM} \right )^{-1}.
\end{eqnarray}
Then the five standard orbital parameters that describe the shape and
period of the orbit are then
\begin{eqnarray}
i & = & \arccos{ \left ( \frac{-\vec{h} \cdot \hat{z}}{|\vec{h}|} \right )}
\label{eqn:i} \\
e & = & |\vec{e}|
\label{eqn:e} \\
\omega & = & \arccos \left ( \frac{(\hat{z} \times \vec{h}) \cdot \vec{e}}
{|\hat{z} \times \vec{h}||\vec{e}|} \right ) \quad
(\textrm{if } \vec{e} \cdot \hat{z} < 0 \textrm{ then }
\omega = 2\pi - \omega)
\label{eqn:w} \\
\Omega & = & \arctan \left ( \frac{\vec{\Omega} \cdot \hat{x}}
{\vec{\Omega} \cdot \hat{y}} \right )
\label{eqn:o} \\
\left ( \frac{P}{[yr]} \right ) & = & \sqrt{ \left ( \frac{a}{[AU]} \right )^3
\left ( \frac{[M_\sun]}{M} \right ) }
\label{eqn:p} \\
\end{eqnarray}
where i = 0 if the orbit is in the plane of the sky and $\Omega$ is
measured East ($\hat{x}$) of North ($\hat{y}$).
The remaining orbital parameter is the epoch of
periapse passage and can be computed in a number of different ways.
We first compute several intermediate quantities of interest such as the
Thiele-Innes constants (A,B,C,F,G,H), and the eccentric anomaly as shown below:
\begin{eqnarray}
A & = & a(\cos \omega \cos \Omega - \sin \omega \sin \Omega \cos i) \\
B & = & a(\cos \omega \sin \Omega + \sin \omega \cos \Omega \cos i) \\
F & = & a(-\sin \omega \cos \Omega - \cos \omega \sin \Omega \cos i) \\
G & = & a (-\sin \omega \sin \Omega + \cos \omega \cos \Omega \cos i) \\
\cos E & = & \frac{Gr_y - Fr_x}{AG - BF} + e \\
\sin E & = & \frac{Ar_x - Br_y}{AG - BF} \frac{1}{\sqrt{ 1 - e^2 }} \\
E & = & \arctan \left( \frac{\sin E}{\cos E} \right ).
\end{eqnarray}
And finally the epoch of periapse passage are calculated from
these intermediate quantities using
\begin{eqnarray}
T_o & = & t_{ref} - \frac{P}{2\pi} (E - e \sin E).
\label{eqn:t0}
\end{eqnarray}
\section{K Metric}
\label{app:kmetric}
The previously proposed planes were derived by minimizing a metric that
\citet{levin03} call $\chi^2$, but we call K, and which is defined as
\begin{equation}
K = \frac{1}{N-1}\displaystyle\sum^N_{i=1}
\frac{(\vec{n}\cdot\vec{v_i})^2}{(n_x\sigma_{v_{x,i}})^2 +
(n_y\sigma_{v_{y,i}})^2 + (n_z\sigma_{v_{z,i}})^2}
\end{equation}
where $N$ is the number of stars, $\vec{v}_i$ is the velocity of each star,
$\sigma_{v_{x,i}}$, $\sigma_{v_{y,i}}$, $\sigma_{v_{z,i}}$ are the velocity
uncertainties for each star, and $\vec{n}$ is the normal vector to the disk
plane that is found in the fitting process.
This metric is used to find, statistically, the best-fit common orbital
plane from the velocity vectors of a sample of stars.
The K metric suffers from several shortcomings.
First, the K metric is described as a $\chi^2$ metric; however,
standard $\chi^2$ minimization takes the form of
(data - model)$^2$/(data errors)$^2$ where the data errors have no
dependency on the model parameters. The K metric includes the model
parameters in the data-error term and does not necessarily have an
expectation value of 1 for normal errors. The appropriate function
to minimize in order to find the best-fit common orbital plane
can be derived from maximum likelihood theory if we assume that the
likelihood function is given by
\begin{equation}
L = \prod_{i=1}^N \frac{1}{\sqrt{2 \pi \sigma_i^2}}
\exp{ \left [ - \frac{(\vec{n} \cdot \vec{v}_i)^2}{2 \sigma_i^2} \right ]}
\end{equation}
where $\sigma_i$ depends on the disk model parameters that are
being sought by
\begin{equation}
\sigma_i^2 = (n_x \sigma_{v_{x,i}})^2 +
(n_y\sigma_{v_{y,i}})^2 + (n_z\sigma_{v_{z,i}})^2.
\end{equation}
Standard practice is then to take the logarithm of the likelihood, $L$, and
minimize the resulting function in Equation \ref{eq:minfunc} in order to find
the best fit disk model parameters. The above likelihood function then becomes
\begin{eqnarray}
\ln L & = & -\frac{N}{2} \ln (2\pi) - \sum_{i=1}^N \ln \sigma_i
+ \sum_{i=1}^N -\frac{ (\vec{n} \cdot \vec{v}_i)^2 }{ 2 \sigma_i^2 } \\
-2 \ln L & = & N \ln (2 \pi) + 2 \sum_{i=1}^N \ln \sigma_i +
\sum_{i=1}^N \frac{ (\vec{n} \cdot \vec{v}_i)^2 }{\sigma_i^2} \label{eq:minfunc}
\end{eqnarray}
and the first two terms are constant and do not factor into finding an
extremum in the above equation. The third term on the right-hand side is the
K metric previously used to determing the disk parameters. However, the
second term on the right-hand side also depends on the free parameters
in $\vec{n}$ and must be included in the minimization process.
This extra term that has not
previously been included in the disk fitting process has the full form
\begin{equation}
\ln{ \sqrt{(n_x\sigma_{v_{x,i}})^2 + (n_y\sigma_{v_{y,i}})^2 +
(n_z\sigma_{v_{z,i}})^2}}
\end{equation}
and standard chi-squared probability functions cannot be applied.
Second, even when accounting for the extra term, the metric can still
introduce substantial bias. In particular, radial
velocity uncertainties, $\sigma_{v_{z,i}}$, are larger than the proper motion
errors by a factor of 2 on average in previous publications. During
K-minimization, this over-weights solutions with a larger $n_z$ resulting
in a bias against edge-on planes. Finally, in order to properly
evaluate the probability of obtaining a given
value of the K-metric by random chance, one must perform simulations of
an isotropic distribution of stars. However, such simulations are extremely
sensitive to the input distribution of semi-major axes and eccentricities
which are not yet well constrained by observations.
Thus, when utilizing such statistical tests for finding a common orbital
plane, it is difficult to compare to the null hypothesis -- an isotropic
distribution of stars -- and to quantify the significance of a disk.
\end{appendix}
|
0808.3141
|
\section{Introduction}
With its first observation, INTEGRAL \citep{Winkler03} began a collection of highly absorbed
sources, some of which may represent a new class of binary (see \cite{Kuulkers05} for a review; K05 hereafter).
IGRJ16318-4848 was a bright X-ray source with extreme absorption that would later be shown to have
bright X-ray emission lines. Since then, the INTEGRAL Galactic plane survey has
discovered several more similar sources, suggesting that these objects form a new class
of X-ray binary, characterized by high absorption columns, slow spin periods and occasional bright X-ray
outbursts. The first INTEGRAL catalog \citep{Bird04} notes 28 unidentified sources from the INTEGRAL
Galactic plane survey, ten of which have been followed up at X-ray and optical wavelengths and have been
shown to belong to this new source class.
Nearly simultaneously to the discovery of this potentially new class of source by INTEGRAL, the {\it Swift}
BAT Galactic plane survey began to uncover many new sources from its own survey. It was
speculated, due to their absence from previous soft X-ray surveys, that some of these sources may be
highly absorbed binaries, possibly members of the new class of INTEGRAL binaries.
Several objects identified in the first INTEGRAL catalog, as of Fall 2006, remained without significant
follow-up in the X-ray or at other wavelengths. During the Suzaku cycle 1 call for proposals, we
proposed to follow-up three INTEGRAL detected sources that were suspected to be members of the new class but
had yet to be studied in detail. We proposed, furthermore, to observe two newly identified sources from
the BAT Galactic plane survey to determine whether they were also members of this new source class.
In this paper, we detail the analysis of the Suzaku data collected on these sources, discuss the
likelihood that these sources are similar to the originally discovered IGR sources and briefly discuss the
possible nature of the new class. The paper is organized as follows: in \S2 we describe the
observations and data analysis; in \S3 we present our results; in \S4 we discuss the
similarity of each source and the propriety of calling each a member of the IGR source class and discuss
the potential nature of these sources; in \S5 we summarize our results and conclusions.
\section{Observations and Data Reduction}
Observations of the five targets, three INTEGRAL sources and two additional sources from the BAT Galactic plane
survey, were conducted between April 12 and October 31, 2006 (see Table 1). On four of the five targets,
a single observation was collected while on one source two observations
were collected, separated by $\sim$6 months.
All sources were observed using the HXD aimpoint and with the XIS instruments in normal
imaging mode. One of the sources (SWJ1010.1-5747) was found to be a symbiotic star and these observations
were published in \cite{Smith08}; this source will not be discussed further here.
Data from the four other targets were reduced using the standard Suzaku processing software, xisrmfgen
version 2007-05-14 and xissimarfgen version 2007-09-22 and the other standard analysis tools contained
in HEADAS version 6.4. Point source processing was carried out as described in the Suzaku ABC
Data Reduction Guide \footnotemark\footnotetext{http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/abc}.
In processing the data from the HXD instrument, which is a non-focused instrument, it was discovered that
the observation of IGRJ16465-4507 was contaminated by the nearby bright X-ray source GX340+0.
As a result the HXD instrument data for this source have not been used. The HXD data of
the other sources were checked for contamination by nearby bright sources but none was found.
Data from the XIS0, XIS2 and XIS3 front illuminated instruments were combined together while data from
the XIS1 back illuminated instrument were kept separate. Data from the combined front illuminated
instruments, from the back illuminated instrument and from the HXD PIN (except in the case of
IGRJ16465-4507) were then fit simultaneously in XSPEC (version 12.4.0). Data from the XIS instruments
were binned to have a minimum of 50 events per bin. Data from the HXD instruments were binned to have a
minimum of 100 events per bin.
The Xronos timing analysis software version 5.21 was used to search for periodicity in the data. In
cases where significant temporal structure was found in the data (either periodicity or transient
outbursts), time-separated spectral analysis was done to search for spectral variation.
\begin{deluxetable}{cccccc}
\tablecaption{Observational Parameters}
\tabletypesize{\normalsize} \tablecolumns{6} \tablewidth{0pt}
\tablehead{ \colhead{Obsnum} & \colhead{Source} & \colhead{observation start} & \colhead{observation stop} & \colhead{XIS $\Delta$t} & \colhead{HXD PIN $\Delta$t}\\
\colhead{} & \colhead{} & \colhead{} & \colhead{} & \colhead{(s)} & \colhead{(s)}} \startdata
\hline
\hline
401052010 & IGRJ16465-4507 & 2006-09-09-09:12:56 & 2006-09-09-22:05:14 & 14536 & 24645\\
401053010 & SWJ2000.6+3210 & 2006-04-12-15:53:10 & 2006-04-12-21:56:04 & 12444 & 9877\\
401053020 & SWJ2000.6+3210 & 2006-10-31-00:29:37 & 2006-10-31-07:16:19 & 10146 & 11727\\
401054010 & IGRJ16493-4348 & 2006-10-05-21:10:30 & 2006-10-06-10:05:24 & 18975 & 20220\\
401055010 & SWJ1010.1-5747 & 2006-06-05-05:13:12 & 2006-06-05-18:25:25 & 19171 & 20000\\
401056010 & IGRJ16195-4945& 2006-09-20-20:25:12 & 2006-09-21-17:21:20 & 27908 & 42265\\
\hline
\enddata
\end{deluxetable}
\section{Results}
Two of the sources in our sample show only random temporal variability while two of the
sources show important temporal and spectral variations that warrant further discussion. We begin this
section with an overview of the characteristics of each source, considering the observation as a whole,
and the global characteristics of the sample. We will then focus, in turn, on each of the two sources
whose data are worthy of more detailed analysis.
\subsection{Global fitting}
In some cases, fitting a simple absorbed powerlaw model produces a fit with $\chi^2_{\nu}\sim$1, but in
all cases a partial covering absorber model is strongly preferred (see Table 2).
The partial covering model is invoked here for its ability to quantify interesting
parameters from a wide range of geometries using a minimum of fit components.
Assuming a dust halo subtending 4$\pi$ steradians, for example, the partial covering
model provides information about the scattering fraction and thus about the density of the
halo. Assuming, instead, a geometry similar to accretion disk corona (ADC) systems, the partial
covering model provides information about the vertical extent of the ADC and the scale height
of the neutral disk material. While the precision of the observations discussed in this
paper is too low to confidently distinguish between these different possible geometries,
we adopt the use of the partial covering model here to lend our results to future
discussion in this context.
The photon index ($\Gamma$)
in the partial covering model, ranges from 1.8 to 2.4, the partial covering fraction (PCF) ranges from 0.5
to 0.8 and the \NH column ranges from $\sim$1$\times$10$^{23}$~cm$^{-2}$ to $\sim$1$\times$10$^{24}$~cm$^{-2}$. These $\Gamma$
are similar to those seen previously in highly absorbed HMXBs \citep{Kuulkers05}. In one case, the \NH
column is slightly below 1$\times$10$^{23}$~cm$^{-2}$ (though within errors; 9$\times$10$^{22}$~cm$^{-2}$).
We note that the HXD data for this
target, IGRJ16465-4507, are
contaminated by a field source and cannot be used. As a result, the high energy spectral slope
for this source is not well constrained as it is for the others, probably leading to an unusually
high value of the PCF compared to the other sources. The effect of the higher PCF
is to suppress the soft X-ray flux which helps to explain the lower \NH column in this source.
If we assume a PCF and $\Gamma$
for this source similar to the other three sources in our sample (defined by the mean values of
PCF=0.6 and $\Gamma$=2.15 respectively) we find \NH column values of 3.2$\times$10$^{22}$~cm$^{-2}$ and 1.0$\times$10$^{23}$~cm$^{-2}$
for the Galactic and local components respectively, similar to what is seen in the other three sources.
\begin{deluxetable}{cccccccccc}
\tablecaption{All sources Partial Covering Absorber Fits\label{T7}}
\tabletypesize{\scriptsize} \tablecolumns{10} \tablewidth{0pt}
\tablehead{\colhead{Source} & \colhead{\NH} & \colhead{\NH$_{\rm{part}}$} & \colhead{PCF} & \colhead{$\Gamma$} & \colhead{Flux} & \colhead{Fe EW} & \colhead{$\chi^2_{\nu_{PC}}$} & \colhead{$\chi^2_{\nu_{PL}}$} & \colhead{dof}\\
\colhead{} & \colhead{(10$^{22}$cm$^{-2}$)\tablenotemark{a}} & \colhead{(10$^{22}$cm$^{-2}$)\tablenotemark{b}} & \colhead{\tablenotemark{c}} & \colhead{\tablenotemark{d}} & \colhead{(10$^{-12}\frac{\rm{ergs}}{\rm{cm^{2}~s}}$)\tablenotemark{e}} & \colhead{eV} & \colhead{\tablenotemark{f}} & \colhead{\tablenotemark{g}} & \colhead{\tablenotemark{h}}} \startdata
\hline
\hline
IGRJ16465-4507 & 2.0$_{-0.7}^{+0.7}$ & 7.3$_{-2.1}^{+2.8}$ & 0.82$_{-0.09}^{+0.06}$ & 2.19$_{-0.31}^{+0.35}$ & 8.95$_{-3.1}^{+0.2}$ & $<$135 & 0.859 & 0.994 & 270\\
IGRJ16493-4348 & 8.6$_{-1.0}^{+0.9}$ & 26$_{-7.9}^{+9.4}$ & 0.62$_{-0.07}^{+0.06}$ & 2.37$_{-0.17}^{+0.18}$ & 13.5$_{-2.0}^{+0.3}$ & $<$84 & 0.902 & 1.09 & 389\\
IGRJ16195-4945 & 11$_{-1}^{+1}$ & 78$_{-17}^{+17}$ & 0.53$_{-0.10}^{+0.09}$ & 1.80$_{-0.13}^{+0.14}$ & 16.1$_{-2.5}^{+0.2}$ & $<$43 & 0.934 & 0.999 & 614\\
SWJ2000.6+3210(1) & 2.3$_{-0.2}^{+0.2}$ & 9.3$_{-1.0}^{+1.2}$ & 0.68$_{-0.02}^{+0.03}$ & 2.21$_{-0.09}^{+0.08}$ & 32.0$_{-0.9}^{+0.5}$ & 51$^{+34}_{-37}$ & 0.918 & 1.58 & 417\\
SWJ2000.6+3210(2) & 2.1$_{-0.2}^{+0.2}$ & 8.2$_{-1.0}^{+1.1}$ & 0.70$_{-0.03}^{+0.02}$ & 2.01$_{-0.05}^{+0.05}$ & 55.0$_{-0.6}^{+0.8}$ & 71$^{+32}_{-29}$ & 0.897 & 1.66 & 725\\
\hline
\hline
\enddata
\tablenotetext{a}{\begin{footnotesize}Fully covered neutral hydrogen column\end{footnotesize}}
\tablenotetext{b}{\begin{footnotesize}Partially covered neutral hydrogen column\end{footnotesize}}
\tablenotetext{c}{\begin{footnotesize}Partial covering fraction\end{footnotesize}}
\tablenotetext{d}{\begin{footnotesize}Photon index\end{footnotesize}}
\tablenotetext{e}{\begin{footnotesize}Fit range set to 0.2keV to 10 keV\end{footnotesize}}
\tablenotetext{f}{\begin{footnotesize}Reduced $\chi^2$ of partial covering fit\end{footnotesize}}
\tablenotetext{g}{\begin{footnotesize}Reduced $\chi^2$ of simple absorbed powerlaw fit\end{footnotesize}}
\tablenotetext{h}{\begin{footnotesize}degrees of freedom in partial covering fit\end{footnotesize}}
\end{deluxetable}
All sources show variability on timescales of hundreds of seconds. Furthermore, IGRJ16195-4945
shows a bright outburst lasting $\sim$5000~s and SWJ2000.6+3210 shows periodic variations. We will return
to discuss these two objects in greater detail shortly. Overall flux levels range from 1$\times$10$^{-11}$
to 5$\times$10$^{-11}$ ergs~cm$^{-2}$~s$^{-1}$ (absorbed) in the 0.2-10 keV energy band.
As a check of consistency with previous work, we have compared the results of absorbed
cutoff powerlaw fits of the three sources in our sample that were also fit to cutoff powerlaws by
K05. Our best fit results for IGRJ16195-4945 match well. Our best fit results for
IGRJ16493-4348 do not match those of K05, but if we fix $\Gamma$ to be similar to that
found by K05, we find a column density similar to theirs and can additionally note a cutoff
energy of 17.5 keV. Our best fit results of IGRJ16465-4507 do not match that of K05. Even
after fixing $\Gamma$ and the cutoff energy to match that of K05, the \NH column that we
find is still more than an order of magnitude lower than that found by K05. As
noted earlier, the observation of IGRJ16465-4507 suffered from high energy contamination, leaving us with
data only form 0.2-10.0 keV. While this may account for some of the discrepancy between our results and
K05, we also note that we see no evidence in our data of the $\sim$4 minute period noted by
K05. This suggests that our data may be taken during the apoastron phase of an elliptical binary orbit when
the flux level, spectrum and absorption column have different values than they do
near periastron and where the higher wind density is likely to produce periodic variations as are reported in the
K05 observation. Thus, it is not surprising that our observations of this source appear
different. In general, however, our overall results appear consistent with those of K05.
{\it Swift} BAT data from the forthcoming 22-month survey \citep{Tueller08} have been analyzed for each of the
sources in our sample except for IGRJ16465-4507 which is contaminated by the same nearby source which hampers
the Suzaku HXD analysis. No evidence of outbursts on daylong timescales (similar to those seen in the
Suzaku observation of IGRJ16915-4945, e.g.) are found. The BAT daily survey detection threshold
is $\sim$5-10 mCrab \citep{Markwardt05},
however, so the non-detection of the $\sim$100-500 $\mu$Crab outbursts seen in
our Suzaku data is not surprising.
We have, furthermore, examined the BAT data binned over longer timescales (2, 4, 8, 16, 32 and 64 days)
to improve the sensitivity of our variability search.
While all sources show stochastic variability on these timescales, no evidence of periodicity is seen.
For the 3 sources with uncontaminated BAT data, we have made simultaneous fits with the Suzaku and BAT
data. In the cases of IGRJ16493-4348 and SWJ2000.6+3210, the
BAT data are well fit by the same model derived from fitting the Suzaku data alone. This implies that the
snapshot captured in the Suzaku observations of these 2 sources is representative of the overall source
behavior and shows that the partial covering scenario is supported to energies as high as 200keV. In the
case of IGRJ16195-4945, the BAT data are not well fit to the same spectrum as the total Suzaku observation, nor to
any of the segmented spectra shown in Table 5. Since the total Suzaku observation does not match the
BAT data, we can infer that flares such as that seen in the Suzaku data do not dominate the flux from the
source. Since
the BAT data are also not well fit by the spectrum seen in the Suzaku quiescent data (the BAT data fit a softer
spectrum than any segment of the Suzaku data and have a higher intensity than the quiescent segment but lower intensity than
the Post-Flare segment), we can also infer that the quiescent state is not best representative of this source.
Since the post-flare state is also a poor fit to the BAT data, we infer that the source spends more time
in the quiescent state than is seen in the Suzaku observation, but that flares such as that seen in the Suzaku
data are somewhat common (since the quiescent state does not match the BAT data). We will discuss this in
further detail in \S4.
Spectra and lightcurves (in cases where the lightcurve shows noteworthy behavior)
of each source are shown in Figures 1-5. In the spectral plots, black datapoints show Suzaku data while
green datapoints show the average 22-month survey data for comparison. We note that the spectral
fits detailed in Tables 2, 4 and 5 are derived using Suzaku data only.
\subsection{IGRJ16195-4945}
The lightcurve of IGRJ16195-4945 is marked by a short, bright flare, seen in both the XIS and HXD
instruments. The flare lasts for $\sim$5000~s and reaches a peak flux level $\sim$10$\times$ brighter than the
prior emission level. Such flaring behavior is characteristic of the subset of absorbed HMXBs known as
Supergiant Fast X-ray Transients (SFXTs, \cite{Negueruela06, Smith06}).
To better understand the nature of the flare and how it compares
to the source during quiescence, we have separated the data as shown in Figure 3 and Table 3.
The data segmentation separates the period prior to the flare (quiescence) from the flare itself (flare)
and from the period after the giant flare has decayed to near the original level (post-flare).
The definition of the
quiescent period is clear due to an extended gap in the data (20ks) prior to the onset of the giant
flare. The definition of the end of the flare is less clear, but we have chosen to define the end of the
flare with the end of the orbit during which the flare decays below $\sim$2$\times$ the original level.
The subsequent data to the end of the observation are defined as post-flare. This definition ensures that
the flare segment is dominated by flare emission even if
some small amount of ''post-flare'' emission is included. It also ensures that the post-flare data are
sufficient to produce an accurate spectrum. We have further subdivided the giant
flare into the rising leg of the flare, the decaying leg of the flare (which are conveniently
separated by an orbital gap) and a segment that encompasses all data from
the onset of the flare to the end of the observation (''on'').
\begin{table}[h]
\centering
\caption{IGRJ16195-4945 Segmentation - Data from IGRJ16195-4945 are segmented to separate the period
prior to the flare (quiescence) from the flare itself (flare) and from the period after the giant
flare has decayed to near the original level (post-flare).}
\begin{tabular}{c c c c}
\hline\hline
Segment & start & stop & duration\\
& (s) & (s) & (s)\\
\hline
quiescence & 0 & 12000 & 12000\\
total flare & 40068 & 46684 & 6616\\
flare onset & 40068 & 41769 & 1701\\
flare decay & 44669 & 46684 & 2015\\
post-flare & 49000 & 72000 & 23000\\
on & 40068 & 72000 & 31932\\
\hline
\end{tabular}
\label{table:Segs}
\end{table}
The results of absorbed powerlaw and partial covering model spectral fits to each of the six segments
(quiescence, total flare, flare onset, flare decay, post-flare, on) are shown in Table 4.
To limit the number of free parameters in the fit we have fixed the global \NH column density,
the partial covering column and the PCF to the values that they have when fitting the complete
observation, allowing only $\Gamma$ to vary. We will return momentarily to the
possibility that the spectral variation is due to these other parameters.
When allowing only $\Gamma$ to vary, we find that the ''on'' segment is
well fit by a model with $\Gamma\sim$1.8. The quiescent data, in
contrast, are fit by a much softer $\Gamma\sim2.5$. No significant Fe
lines are seen, although upper limits are given in Table 4.
It is interesting to note that $\Gamma$ remains nearly unchanged (perhaps even
becoming harder) as the flare subsides and the ''post-flare'' phase begins. Considering the modest flux
level of the ''post-flare'' phase, one would expect that as the flare decays the spectrum would soften
as the quiescent component once again becomes comparable to the flaring component. That this is
not seen suggests
that, during the ''post-flare'' phase, the emission component that was seen during the quiescent phase is
not merely being overwhelmed by the component responsible for the flare emission, but rather that the
quiescent component is absent altogether.
\begin{deluxetable}{cccccccccc}
\tablecaption{J16195-4945 Time Resolved Partial Covering Absorber fits; Fixed \NH and covering fraction\label{T7}}
\tabletypesize{\normalsize} \tablecolumns{10} \tablewidth{0pt}
\tablehead{\colhead{Seg\#}& \colhead{Source} & \colhead{\NH} & \colhead{\NH$_{\rm{part}}$} & \colhead{PCF} & \colhead{$\Gamma$} & \colhead{Flux} & \colhead{$\chi^2$} & \colhead{dof} & \colhead{Fe EW}\\
\colhead{} & \colhead{} & \colhead{(10$^{22}$cm$^{-2}$)} & \colhead{(10$^{22}$cm$^{-2}$)} & \colhead{} & \colhead{} & \colhead{(10$^{-12}\frac{\rm{ergs}}{\rm{cm^{2}~s}}$)\tablenotemark{a}} & \colhead{} & \colhead{} & \colhead{(eV)}} \startdata
\hline
\hline
1-6 & Total & 11 & 76 & 0.55 & 1.81$_{-0.03}^{+0.03}$ & 16.1$_{-1.0}^{+0.1}$ & 565.0 & 591 & $<$43\\
1 & quiescent & 11 & 76 & 0.55 & 2.46$_{-0.44}^{+0.48}$ & 2.68$_{-1.13}^{+0.25}$ & 64.5 & 88 & $<$320\\
2 & total flare & 11 & 76 & 0.55 & 1.78$_{-0.03}^{+0.03}$ & 48.0$_{-0.8}^{+0.7}$ & 290.7 & 340 & $<$19\\
3 & flare onset & 11 & 76 & 0.55 & 1.81$_{-0.12}^{+0.10}$ & 58.5$_{-1.9}^{+2.0}$ & 71.3 & 123 & $<$65\\
4 & flare decay & 11 & 76 & 0.55 & 1.82$_{-0.08}^{+0.08}$ & 45.8$_{-1.4}^{+1.5}$ & 99.4 & 122 & $<$37\\
5 & post-flare & 11 & 76 & 0.55 & 1.73$_{-0.06}^{+0.06}$ & 9.68$_{-0.25}^{+0.34}$ & 242.2 & 269 & $<$81\\
6 & on & 11 & 76 & 0.55 & 1.75$_{-0.03}^{+0.03}$ & 20.6$_{-0.3}^{+0.3}$ & 508.5 & 554 & $<$28\\
\hline
\hline
\enddata
\tablenotetext{a}{\begin{footnotesize}Fit range set to 0.2keV to 10 keV\end{footnotesize}}
\end{deluxetable}
If we leave all components of the partial covering model free to vary we find the results shown in Table 5.
Here we see the quiescent phase described by a high PCF, moderately hard powerlaw
and low column density. In contrast, the flare shows a low PCF and softer powerlaw with
much higher column density. Finally, the ''post-flare'' phase shows a PCF
between the other two, the softest $\Gamma$ and the highest column density of all the segments.
This suggests that the flare signals the onset of emission from a region of much
greater local column density, perhaps from interaction with a disk. We will return to
discuss this possibility in \S4.
Previous observations of this source by both ASCA \citep{Sidoli05} and INTEGRAL \citep{Sguera06} have also shown
evidence of outbursts lasting $\sim$1-2 hours. Translated into a common
energy range the flux of these previous outbursts is lower than that seen in our Suzaku observations
by a factor of a few, but differences in the measured spectral parameters and the lower signal to noise of these
earlier observations limit the precision of the comparison. Nevertheless, the similarity of the outbursts
in these three observations seems suggestive either of a characteristic timescale of the accretion flow of this
source (perhaps indicative of an associated clump size in the flow) or of a periodic interaction of the neutron star
with an over-dense region in the wind, possibly a disk. We note that a short ($<$5ks) Chandra
observation has been made of this source but no variability has previously been reported, though
irregularities in the data complicate the analysis \citep{Tomsick06}.
\begin{deluxetable}{cccccccccc}
\tablecaption{J16195-4945 Time Resolved Partial Covering Absorber fits\label{T7}}
\tabletypesize{\normalsize} \tablecolumns{10} \tablewidth{0pt}
\tablehead{\colhead{Seg\#} & \colhead{Source} & \colhead{\NH} & \colhead{\NH$_{\rm{part}}$} & \colhead{PCF} & \colhead{$\Gamma$} & \colhead{Flux} & \colhead{$\chi^2$} & \colhead{dof} & \colhead{Fe EW}\\
\colhead{} & \colhead{} & \colhead{(10$^{22}$cm$^{-2}$)} & \colhead{(10$^{22}$cm$^{-2}$)} & \colhead{} & \colhead{} & \colhead{(10$^{-12}\frac{\rm{ergs}}{\rm{cm^{2}~s}}$)\tablenotemark{a}} & \colhead{} & \colhead{} & \colhead{(eV)}} \startdata
\hline
\hline
1-6 & Total & 11$_{-1}^{+1}$ & 77$_{-18}^{+17}$ & 0.55$_{-0.10}^{+0.09}$ & 1.82$_{-0.13}^{+0.14}$ & 16.1$_{-2.4}^{+0.1}$ & 573.8 & 615 & $<$43\\
1 & quiescent & 3.5$_{-3.0}^{+6.0}$ & $<$10 & 0.95\tablenotemark{b} & 1.58$_{-0.5}^{+1.1}$ & 2.85$_{-2.85}^{+0.21}$ & 64.5 & 85 & $<$356\\
2 & total flare & 11$_{-1}^{+1}$ & 72$_{-18}^{+17}$ & 0.57$_{-0.10}^{+0.08}$ & 1.82$_{-0.15}^{+0.15}$ & 48.2$_{-8.2}^{+0.7}$ & 289.0 & 337 & $<$15\\
3 & flare onset & 10$_{-2}^{+2}$ & 48$_{-24}^{+23}$ & 0.52$_{-0.22}^{+0.16}$ & 1.73$_{-0.26}^{+0.30}$ & 59.4$_{-27.1}^{+2.4}$ & 69.0 & 120 & $<$49\\
4 & flare decay & 11$_{-2}^{+2}$ & 65$_{-28}^{+36}$ & 0.58$_{-0.21}^{+0.14}$ & 1.89$_{-0.30}^{+0.33}$ & 46.4$_{-28.6}^{+1.4}$ & 95.7 & 119 & $<$25\\
5 & post-flare & 12.4$_{-1.7}^{+1.8}$ & 121$_{-31}^{+30}$ & 0.75$_{-0.18}^{+0.11}$ & 2.07$_{-0.32}^{+0.33}$ & 9.73$_{-2.54}^{+2.18}$ & 237.0 & 266 & $<$68\\
6 & on & 11$_{-1}^{+1}$ & 83$_{-16}^{+16}$ & 0.55$_{-0.10}^{+0.08}$ & 1.75$_{-0.14}^{+0.14}$ & 20.4$_{-2.7}^{+0.3}$ & 507.2 & 551 & $<$43\\
\hline
\hline
\enddata
\tablenotetext{a}{\begin{footnotesize}Fit range set to 0.2keV to 10 keV\end{footnotesize}}
\tablenotetext{b}{\begin{footnotesize}PCF is poorly constrained in the quiescent fit due to the low number of counts\end{footnotesize}}
\end{deluxetable}
\subsection{SW2000.6+3210}
Data were collected on SWJ2000.6+3210 during two epochs, separated by six months (see Figures 4-5).
We have analyzed each of these observations separately. Neither observation is
adequately fit by a simple absorbed powerlaw (see Table 2). Both observations are well fit by a
partial covering model and the parameters of both fits are quite similar. The
later observation has a slightly harder $\Gamma$ and significantly larger flux. Both observations
show evidence of a weak Fe fluorescence line at 6.4 keV with apparently constant equivalent width.
Interestingly, while the first observation shows only random variability, the second observation shows
regular variations with a period of 1056~s. We interpret this as the spin period of the
neutron star companion in a binary, similar to long periodic variations noted in several other
sources \citep{Kuulkers05}. There is the suggestion of secular variation beneath the 1056~s period,
but the data are insufficient to confidently determine any further periodic components.
This may suggest that the system is a HMXB with an elliptical orbit. During the first observation, the compact object is far from the donor star where the wind density is low. In this case,
the X-ray flux is likely to be approximately constant, possibly associated with an accretion
disk or corona around the compact object. During the second observation, the compact object is
nearer to the donor star where the stellar wind is more dense, channeling more material onto
the neutron star and producing the observed periodicity through accretion.
\section{Discussion}
All of the sources in our sample are generally consistent with being members of the new INTEGRAL highly
absorbed binary source class. The most likely candidate among our sources to be a member of this new
class is SWJ2000.6-3210, which displays moderately high \NH ($\sim$1$\times$10$^{23}$~cm$^{-2}$), $\Gamma$ and
cutoff energy similar to other members of the class and periodic variations with a
period of 1056~s.
IGRJ16195-4945 shows a high absorbing column and a brief but bright
X-ray outburst, identifying it as a new SFXT. We note, however, that \NH variations are not strictly required
during the flare.
If N$_{\rm{H}}$ variations are not invoked, the outbursts can be
associated with variable accretion and the duration of the flare may be indicative of the clump size in the companion
star wind.
A recent alternative model \citep{Sidoli07} suggests that SFXT outbursts such as this
are due to periodic interaction with an equatorial disk wind from the donor star. A dramatic increase in \NH
column is seen when all parameters are left free to
vary during fitting, supporting this scenario. Sidoli et al. suggest that such outbursts are actually
longer lived than the previously generally observed duration of hours, lasting instead for several days
\citep{Romano07}.
Since our data coverage ends $\sim$30~ks after the flare onset, we cannot rule out that an elevated level of
emission continues for several days after the initial flare, though the brightest emission appears to last for
only $\sim$1 hour in our data.
If we assume the equatorial disk wind model to be applicable to this source, however, we can estimate the orbital
period of the binary as follows. We assume that the three
broad states seen in the Suzaku observation (quiescent, post-flare and total flare) approximately represent the
full range of states of IGRJ16195-495 and thus we can write the following relationship between the flux of the 3 Suzaku states
and the orbit-averaged flux level seen by BAT:
\begin{equation}
a\times~a_{f}+b\times~b_{f}+c\times~c_{f}=d
\end{equation}
where a is the flux level of the Suzaku quiescent state observation and a$_{f}$ is the fraction of the orbit
spent in this state, b is the flux level of the Suzaku post-flare state observation and b$_{f}$ is the fraction
of the orbit spent in this state, c is the flux level of the Suzaku total flare state observation and c$_{f}$ is the
fraction of the orbit spent in this state and d is the (orbit averaged) flux of the BAT observation.
If we now make the assumption that the bright flare seen in the Suzaku data (the total flare state) represents
the entry of the compact object into an equatorial disk wind from the donor star,
we must interpret the subsequent post-flare state as the longer lived period of moderate flaring behavior that
accompanies the remainder of the journey of the compact object through the disk wind. According to the \cite{Romano07}
observation of SwiftJ11215-5952, this longer lived period of activity can last for up to $\sim$5 days. Assuming a
similar duration for this phase in IGRJ16195-4945, we have b$_{f}=$4.3e5/P and c$_{f}=$5e3/P and a$_{f}=(1-\frac{4.35e5}{P})$
where P is the period of the binary in seconds. Thus equation (1) becomes:
\begin{equation}
P=\frac{-a\times4.35e5+b\times4.3e5+c\times5e3}{d-a}
\end{equation}
To estimate the flux levels a, b, c and d, we fit each corresponding dataset using the average spectral fit to the
complete dataset (including the BAT data) and determined the associated flux level in the 15-150~keV band. We find
a=5.13e-12, b=8.61e-11, c=3.93e-10 and d=3.11e-11~ergs/cm$^2$/s. Plugging these values into equation (2)
returns a predicted period P$\sim$16 days. This period is unusually short compared to other measured
SFXT periods \citep{Heras07, Romano07}. Even if we alter our calculation to interpret the on
state (see Tables 4-5) as the measure of the flux during the passage through the disk wind rather
than weighting it heavily to the post-flare state as shown above, the predicted period only increases by a factor
of $\sim$2, which is still well below the more typically reported SFXT orbital periods of $\gtrsim$200 days. An orbital
period of $\sim$16 days, however, is consistent with previous measurements of orbital periods associated
with highly absorbed IGR sources \citep{Corbet04, Corbet05}.
We are pursuing further observations to followup this predicted orbital
period. If the predicted period is shown to be accurate, IGRJ16195-4945 may represent an important cross-over source between
the known SFXTs with orbital periods of hundreds of days with periodic flaring behavior and the new class of
highly absorbed INTEGRAL sources, which have much shorter orbital periods but are without clear observations of periodic
flaring behavior. Such a cross-over object would suggest a common mechanism at work for the emission seen from
HMXBs of widely varying orbital parameters. It would also suggest a useful method of refining
the period search for such sources in instances where a single flare has been observed but large amounts of
observatory time are not available for a dedicated follow-up monitoring campaign.
A long-term monitoring campaign of this source would be very useful to determine
the true duration of the associated outbursts.
IGRJ16493-4843 is a variable source with
a high column density and non-periodic variations, also potentially qualifying it with
this new source class. Finally, IGRJ16465-4507 presents the
fewest similarities to previous observations of the class, but part of the discrepancy may be due to the
narrow energy window in which our observations are made for this source. We note that we do not
see periodic variability in this source as previously reported by others \citep{Lutovinov05, Kudryavtsev06}.
One of the interesting results of this work is that two of the four sources in
our sample (IGRJ16465-4507 and SwiftJ2000.6+3210) appear to be transient pulsars.
It would be very interesting
to know if this is a common characteristic of all sources in this new class. If so, it would imply that
elliptical orbits are common in these sources. Since it has been shown, however, that binaries will
circularize rapidly (10$^4$-10$^6$ years) through tidal dissipation following the
supernova detonation of the more massive star \citep{Savonije83, Lecar76}, the occurrence of highly elliptical orbits,
if found to be common in these sources, will require explanation. As a rough estimate of the fraction of sources
that may be expected to be in highly elliptical orbits,
we may proceed from work showing that the companion source in several of these systems is a supergiant O or B star
\citep{Nespoli08b, Nespoli08a, Halpern06, Tomsick06, Negueruela05}. This implies that the HMXB
phase of the lifetime of the system (the time after the detonation of the more massive star and prior to the
detonation of the less massive star) will be typically 5-10$\times$10$^6$ years. Therefore, at most 10-20\%
of the observed sources should be found in strongly elliptical orbits. Finding
a significantly higher fraction of highly elliptical orbits would suggest another mechanism at work during the
HMXB evolution. Possible explanations for the lower circularization rate include
generally wide orbital separation of the binaries, unusually large initial supernova kicks, and capture events or
other gravitational interactions that may amplify eccentricity of the system.
It has been shown by \cite{Abt05} that the ellipticity of visual binaries is roughly a function of the mass
of the primary and the orbital period of the system, with tidal dissipation responsible for rapid circularization
of systems with orbital periods less than about 10 days. Since the same mechanism of tidal dissipation is thought to
be responsible for the circularization of X-ray binary orbits, it seems reasonable to assume that a similar
function applies to HMXBs. Measuring both the ellipticity of the orbit and the period for a sample of
these systems through a monitoring campaign will allow us to test whether this hypothesis is true, and thus will
indicate whether tidal dissipation is the dominant effect in HMXB circularization.
Monitoring would also
determine orbital periods of each system, a necessary step toward determining the orbital characteristics
of the system and ultimately the mass of the compact object, and would allow a more in-depth analysis of
spectral variability as a function of orbital phase. Since the neutron stars in these binaries likely orbit only a
few stellar radii from the companion and are believed to have
orbital periods on the order of ten days, daily or perhaps bi-weekly observations for about a month
seem sufficient to accurately determine all of these characteristics. The {\it Swift} satellite offers the
capability to perform such monitoring and, with its simultaneous X-ray and optical observations, would
provide an ideal platform for future observations. Finally, monitoring on long timescales (covering a
baseline of several years) will allow investigation of the spin-up/spin-down of the pulsar which
can shed light on the magnetic field strength of the pulsar as well as the wind environment in which it is
embedded (see, e.g., \cite{Patel07}).
Three of the four sources in our sample are confidently associated with OB stars from optical spectroscopy
\citep{Negueruela05, Halpern06, Nespoli08a}.
In the case of IGRJ16915-4945, a foreground dwarf star has thwarted attempts to identify
the secondary. IGRJ16915-4945 has a history of X-ray outbursts, however, which are a trait of SFXTs that
contain OB secondary stars \citep{Negueruela06}. Thus association with a massive star seems likely for this system as well.
Since the companion stars in these sources are most likely O or B stars, an interesting potential application
of these data is to study the porosity or clump size of the absorbing material in the winds of
massive stars. Though the periodic variability and extremely bright outbursts seen in some of the data
cannot be explained by variations in the column density along the line of the sight to the observer in classic HMXBs,
stochastic variations of lower contrast, which are also seen in the data, are a potential signature of
variable obscuration. We have searched for variations in column density (assuming all other spectral parameters
remain constant as a simplifying approximation) during the aperiodic pulses seen in some of the data
but do not find any to within the uncertainty of the data. We point out, however, that typical fluences
in the pulses analyzed are $<$1000 photons, leading to weak constraints on the column density. While some
of these sources have prior observations with XMM-Newton, whose greater effective area will reduce
the uncertainty in these measurements, the exposures are extremely short (3-4 ks) and thus are not able to
constrain column density variations on timescales of 5-10 ks as seen in these data.
Longer observations of these sources with XMM-Newton would be useful in this regard and would offer the
additional benefit of high resolution soft X-ray spectroscopy to probe the ionization
state of the wind through metal line strengths. \cite{Walter07} have investigated
the size of clumps in the accreting wind of SFXTs and determined typical masses of 10$^{22}$-10$^{23}$~g and mass loss
rates of 10$^{-5}$-10$^{-6}$ M$_\odot$/yr from analysis of INTEGRAL data.
INTEGRAL data do not constrain N$_{\rm{H}}$, however, and thus
cannot distinguish between variability due to changing column density levels and that due to changing accretion
levels. XMM-Newton observations at 0.2-10.0~keV will be able to distinguish between these two variability mechanisms.
The limits on the Fe line equivalent widths (EWs) found in these sources are low for sources of such high N$_{\rm{H}}$ column density
\citep{Makishima86}. Such low Fe line EWs imply either an intrinsically low Fe abundance relative to N$_{\rm{H}}$
(as has been seen in extragalactic sources, e.g., Centaurus A \citep{Markowitz07}) or that the Fe fluorescence
covering fraction is less than 1. If the high N$_{\rm{H}}$ column density absorber is local to the compact object,
as has been suggested through comparisons between X-ray and optical absorption measures (e.g., \cite{Revnivtsev03, Walter03, Chaty04}),
and if the Fe fluorescence originates in a shell structure surrounding the compact object,
it is difficult to envision a geometry in which the coverage would be less than 4$\pi$. The Fe emission
may originate in the inner region of the accretion disk, however, in which case a lower covering
fraction may be reasonable. Furthermore, there are reasons to believe that the assumptions inherent in
relating X-ray and optical absorption measures may not be valid for all sources \citep{Maiolino01}, and thus the disagreement
between X-ray and optical absorption measures may not necessarily imply that the high column absorber is
local to the compact object. Alternatively, such low Fe EWs might also be considered as evidence of the presence of ADC
emission. The presence of an ADC seems unlikely, however, given the lack
of other emission lines from 1-5 keV that are usually produced in ADCs. Still other explanations
for low Fe EWs are similar to those discussed by \cite{Markowitz07} in reference to Cen A and include
the Fe fluorescence source being displaced from the X-ray continuum source or
the presence of an attenuating obstruction between the X-ray source and the site of the Fe fluorescence. While our
data argue against (but do not rule out) the low covering fraction scenario, the intrinsically low Fe EWs
scenario requires non-trivial assumptions about the geometry of the system. Further observations, particularly
phase-resolved spectroscopy, will be very helpful in disentangling these many potential explanations and are
encouraged.
\section{Summary and Conclusions}
Our primary goal in this study is to determine whether these four sources exhibit similar characteristics to
the emerging new class of highly absorbed IGR sgHMXB or SFXT sources.
The defining characteristics of this new class are i) high absorption column ($>$1$\times$10$^{23}$~cm$^{-2}$), ii)
periodicity on timescales of a few to $\sim$100 minutes, generally
interpreted as a neutron star spin period, iii) periodicity on timescales of $\sim$10 days, generally
interpreted as a binary orbital period and iv) occasionally strong X-ray emission lines.
\noindent {\bf i)} In all four sources the total absorbing column
is $\gtrsim$1$\times$10$^{23}$~cm$^{-2}$. Generally,
we do not see significant changes in the absorption over the duration of the observations. While this is
not surprising during a single day-long observation, which probably samples a small segment of
the orbital phase, it is somewhat surprising that two observations of SWJ20006.+3210, separated by more
than six months (and furthermore apparently at different parts of the orbital phase) also show no
significant difference in \NH. A dramatic increase in absorbing column is seen in IGR16195-4945 for a
duration of $\sim$30~ks, during a bright outburst (when all parameters are left free to vary). Since this dramatic increase in \NH is
associated with a dramatic increase in flux, one might interpret this as an
increase in emission due to interaction of the neutron star with a density enhancement. Such a scenario might be
expected due to the passage of the neutron star through a thick disk associated with the donor star.
Since an interpretation in which \NH is approximately constant and only $\Gamma$
varies also produces adequate spectral fits, an alternative model in which the increased emission is due to
variable accretion is also possible.
\noindent {\bf ii)} In SWJ2000.6+3210, we
see a period of 1056~s which is only observed during one of two observations of the
source. This is a newly identified transient X-ray pulsar. Given the low
magnetic field implied by the relatively slow rotation period of 1056~s, it is not surprising that the
same mechanism that produces periodic observations near periastron is too weak to produce an observable
period near apoastron where the wind density will be lower.
\noindent {\bf iii)} Due to the short nature of the observations (generally $\sim$1 day), we do not expect to
directly measure orbital periods, previously reported to be on the order of ten days in other sgHMXBs and, indeed,
we do not. During
observations of the one source for which we have well separated observations we also do not see evidence of
an orbital period. We have, however, used our observations to calculate a predicted orbital period for
IGRJ16195-4945. Assuming that IGRJ16195-4945 is a SFXT and that the bright flare we see signals the
interaction of the compact object with an equatorial wind of the donor star, we have combined our Suzaku
observations with long-term {\it Swift}-BAT observations to predict the orbital period for this source as P$\sim$16 days.
This is unusually short compared to previously measured SFXT orbital periods, but is consistent with orbital periods
previously measured for the highly absorbed IGR sources. More consistent monitoring, perhaps using the {\it Swift} satellite,
would be very useful in confirming or refuting this predicted period and in characterizing the orbital periods
of these highly absorbed binaries in general.
\noindent {\bf iv)} We find only weak evidence of Fe fluorescence emission in one source, and only upper limits
to Fe lines in the other three sources. This is similar to the behavior reported in the original INTEGRAL highly
absorbed X-ray binary sources
\citep{Kuulkers05} in which only 1 of 10 sources showed strong emission lines while the others showed only
weak lines or upper limits. The Fe line measurements that we report here
are either the first for the source in the literature (IGRJ16465-4507, IGRJ16493-4348, SwiftJ2000.6+3210) or several times
more restrictive than previous measurements (IGRJ16195-4945). Moreover, our measurements are similarly or more restrictive
than those reported on similar sources in the literature (see, e.g., \cite{Sidoli05, Patel04, Rodriguez03}).
While none of these sources is ruled out from the IGR class based on these observations, further
observations would be helpful in refining their nature. Of particular value would be periodic
monitoring of all four systems.
\acknowledgements
We are grateful to Hans Krimm and Gerry Skinner for providing access to and help in analyzing the BAT survey data and to Fotis Gavril for help in the timing analysis studies. We also thank Maurice Leutenegger for useful discussion and comments. D.~C.~M. acknowledges support from the Center for Nuclear Studies (CNS) through the Research Enhancement Fund at the George Washington University and from NASA grant NNH05ZDA001N-SUZ/11132.
|
0808.3712
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\section*{Extended english abstract}
\noindent{\bf Introduction}
\noindent The symbol to be transmitted is modulating a carrier,
which is assumed to be a solution of a linear differential equation
with polynomial coefficients. Most signals utilized in practice,
like $\sum_{\tiny{\mbox{\rm finite}}} A_\iota \sin (\omega_\iota t +
\varphi_\iota)$, $A_\iota, \omega_\iota, \varphi_\iota \in
\mathbb{R}$, ${\mbox {\rm sinc}} ~t = \frac{\sin t}{t}$, $\frac{\cos
t}{1 + t^2}$, do satisfy this property. New algebraic estimation
techniques \cite{esaim,garnier} permit to achieve demodulation even
with very ``strong'' corrupting additive noises and therefore to
question the importance of the signal to noise ratio, which is
playing such a crucial r\^ole in information theory (see
\cite{bell,ire} and \cite{battail,blahut,brillouin,cover,proakis}).
Operational calculus, differential algebra and nonstandard analysis
are the main mathematical tools.
\\
\noindent{\bf Identifiability}
\noindent Let $k_0 (\Theta)$ be the field generated by a finite set
$\Theta = \{\theta_1, \dots, \theta_\varrho \}$ of unknown {\em
parameters}, where $k_0$ is a ground field of characteristic $0$. We
are utilizing the classical notations of operational calculus
\cite{yosida}. Introduce the differential field
\cite{chambert,singer} $\bar{k}(s)$ of rational functions in the
indeterminate $s$ over the algebraic closure $\bar{k}$ of $k_0
(\Theta)$, with derivation $\frac{d}{ds}$. Any {\em signal} $x$, $x
\not\equiv 0$, is assumed to satisfy a homogeneous linear
differential equation with coefficients in $\bar{k}(s)$ and
therefore to belong to a Picard-Vessiot extension
\cite{chambert,singer} of $\bar{k}(s)$. Write $\bar{k}(s)
[\frac{d}{ds}]$ the noncommutative ring of linear differential
operators with coefficients in $\bar{k}(s)$. The left $\bar{k}(s)
[\frac{d}{ds}]$-module spanned by $x$ and $1$ is a
$\bar{k}(s)$-vector space of finite dimension $n + 1$, $n \geq 0$.
It yields the {\em minimal} non necessarily homogeneous linear
differential equation (\ref{nh}) where the polynomials $p, q_0,
\dots, q_{n}$ in $\bar{k}[s]$ are coprime. Introduce the square
matrix $\mathfrak{M}$ of order $N + M + 1$, where the
$\xi^{\tiny\mbox{\rm th}}$ line, $0 \leq \xi \leq N + M$, is
(\ref{line}). Techniques stemming from Wronskian determinants
\cite{chambert,singer} demonstrate that the rank of $\mathfrak{M}$
is $N + M$. It follows that the coefficients of the polynomials $p,
q_0, \dots, q_{n}$ are {\em projectively linearly identifiable}
\cite{esaim,garnier}. Consider as a particular case $x =
\frac{p(s)}{q(s)}$ where the polynomials $p, q \in \bar{k} [s]$ are
coprime. Then the coefficients of $p$ and $q$ are also projectively
linearly identifiable.
\\
\noindent{\bf Perturbations and estimators}
\noindent Assume that the unknown parameters $\Theta$ are {\em
linearly identifiable} \cite{esaim,garnier}. With and additive
perturbation $w$, we obtain the estimator (\ref{estim}) where
$\mathfrak{A}$, and $\mathfrak{B}$, $\mathfrak{C}$ are respectvely
$\varrho \times \varrho$ and $\varrho \times 1$ matrices, such that
the entries of $\mathfrak{A}$ and $\mathfrak{B}$ belong to
$\mbox{\rm span}_{k_0 (s) [\frac{d}{ds}]} (1, x)$ and those of
$\mathfrak{C}$ to $\mbox{\rm span}_{R [\frac{d}{ds}]} (w)$, where
$R$ is the localized ring \cite{lang} $k_0(\Theta)[s] (k_0
[s])^{-1}$. Moreover $\det ( \mathfrak{A}) \neq 0$. It is always
possible to obtain an estimator which is {\em strictly polynomial}
with respect to $\frac{1}{s}$, i.e., where the rational functions in
$s$ are polynomials in $\frac{1}{s}$ without constant terms. As in
\cite{ans} it yields in the time domain, if $x$ is an analytic
function, Formula (\ref{estimat}) where $c$ is a constant, $[0, t]$
is the {\em estimation time window}, the {\em divisor} $\delta (t)$
is an analytic function such that $\delta (0) = 0$, $[ \theta_\iota
]_e (t)$ is the estimated value of $\theta_\iota$ at time $t$.
\\
\noindent{\bf Noises}
\noindent We are considering two types of perturbations, which are
{\em noises} in the sense of \cite{ans}:
\begin{itemize}
\item The first noise, which is zero-mean, is a
finite sum $\sum_{\tiny{\mbox{\rm finite}}} A_i \sin (\Omega_i t +
\varphi_i)$ where the frequencies $\Omega_i
>0$ are unlimited.
\item Let $^*\mathbb{N}$, $^*\mathbb{R}$ be the nonstandard
extensions \cite{robinson} of $\mathbb{N}$, $\mathbb{R}$. Replace
$[0, 1] \subset {\mathbb{R}}$ by the hyperfinite set \cite{robinson}
${\mathrm{I}} = \{0, \frac{1}{\bar{N}}, \dots, \frac{\bar{N} -
1}{\bar{N}}, 1 \}$, where $\bar{N} \in {^*\mathbb{N}}$ is unlimited.
A {\em zero-mean white noise} is a function $w: {\mathrm{I}}
\rightarrow {^*\mathbb{R}}$, $\iota \mapsto w(\iota) = A n(\iota)$,
where
\begin{itemize}
\item $A \in {^*\mathbb{R}}$ is a constant, such that
$\frac{A^2}{\bar{N}}$ is limited,
\item the $n(\iota)$ are independent
zero-mean random variables with a normalized covariance $1$.
\end{itemize}
\end{itemize}
The estimator (\ref{estimat}) yields ``good'' values for the unknown
parameters for any limited values of the amplitudes $A_i$, $A$, and
even for some unlimited values of them.
\newpage
\selectlanguage{francais}
\section{Introduction}
\label{intro} Le {\em rapport signal \`{a} bruit}\footnote{Rappelons au
lecteur peu au fait que toute transmission physique de signal est
perturb\'{e}e, no\~tamment par du \og bruit \fg. L'extraction des
informations utiles malgr\'{e} ces alt\'{e}rations est un but essentiel en
traitement du signal et en th\'{e}orie des communications. Que l'on
pense par exemple aux {\em codes correcteurs d'erreurs}, o\`{u}, comme
les math\'{e}maticiens le savent, la th\'{e}orie des {\em codes en blocs}
est d'une grande richesse alg\'{e}brique.}, que l'on retrouve dans les
formules de la th\'{e}orie de l'information, telle qu'elle s'est impos\'{e}e
depuis Shannon (voir \cite{bell,ire} et, par exemple, dans la vaste
litt\'{e}rature sur le sujet,
\cite{battail,blahut,brillouin,cover,proakis}), est un ingr\'{e}dient
fondamental pour d\'{e}finir la qualit\'{e} des communications. Le but de ce
travail\footnote{Voir \cite{arxiv} pour une version pr\'{e}liminaire.}
est de d\'{e}montrer qu'une nouvelle approche de l'estimation rapide et
du bruit (voir \cite{ans} et sa bibliographie) rend ce rapport sans
objet dans un certain cadre num\'{e}rique. Revoyons, donc, le \og
paradigme de Shannon \fg. Le {\em symbole} \`{a} transmettre (voir, par
exemple, \cite{glavieux,proakis}) {\em module} une {\em porteuse}
$z(t)$, solution d'une \'{e}quation diff\'{e}rentielle lin\'{e}aire \`{a}
coefficients polynomiaux:
$$
\sum_{\tiny{\mbox{\rm finie}}} a_\nu (t)
z^{(\nu)} (t) = 0, \quad \quad ~ ~ a_\nu \in \mathbb{C}[t]
$$
La plupart des signaux utilis\'{e}s en pratique, comme une somme
trigonom\'{e}trique finie $\sum_{\tiny{\mbox{\rm finie}}} A_\iota \sin
(\omega_\iota t + \varphi_\iota)$, un sinus cardinal $\frac{\sin
(\omega t)}{t}$ ou un cosinus sur\'{e}lev\'{e} $\frac{\cos (\omega t)}{1 +
t^2}$, $A_\iota$, $\omega_\iota$, $\varphi_\iota$, $\omega \in
\mathbb{R}$, v\'{e}rifient une telle \'{e}quation, qui se traduit dans le
domaine op\'{e}rationnel (cf. \cite{yosida}), pour $t \geq 0$, par
\begin{equation}\label{nh0} \sum_{\tiny{\mbox{\rm finie}}} a_\nu (-
\frac{d}{ds}) s^\nu \hat{z} = I(s) \end{equation} o\`{u} $I \in
\mathbb{C}[s]$ est un polyn\^{o}me dont les coefficients d\'{e}pendent des
conditions initiales en $t = 0$. La {\em d\'{e}modulation} revient,
alors, \`{a} estimer certains des coefficients de (\ref{nh0}). On y
parvient, ici, gr\^{a}ce \`{a} des techniques alg\'{e}briques r\'{e}centes (cf.
\cite{esaim,garnier}).
Un bruit, selon \cite{ans}, est une fluctuation rapide, que l'on
d\'{e}finit de fa\c{c}on efficace et \'{e}l\'{e}gante gr\^{a}ce \`{a} l'analyse non
standard\footnote{Voir aussi \cite{lobry}.}. Les calculs du {\S}
\ref{bruit} sont effectu\'{e}s avec une somme finie de sinuso\"{\i}des \`{a} tr\`{e}s
hautes fr\'{e}quences et un bruit blanc, dont la d\'{e}finition non standard
clarifie l'approche usuelle des manuels de traitement du signal. Ils
d\'{e}montrent la possibilit\'{e} d'obtenir de \og bonnes \fg ~ estimations
avec des bruits \og tr\`{e}s forts \fg, c'est-\`{a}-dire de \og grandes \fg
~ puissances, fait confirm\'{e} par des simulations num\'{e}riques et des
expriences de laboratoire (voir
\cite{fmmsr,liu,mboup,ajaccio,neves,trapero,trapero-bis,trapero-ter}).
Les imperfections, in\'{e}vitables en pratique, proviennent de
l'implantation num\'{e}rique des calculs, notamment de celui des
int\'{e}grales (voir \cite{liu,mboup}), des interf\'{e}rences entre symboles
(voir \cite{battail,glavieux,proakis0,proakis} et leurs
bibliographies), et du fait que les bruits ne sont pas
n\'{e}cessairement centr\'{e}s (voir \`{a} ce propos le {\S} 3.2.2 de \cite{ans}).
Calcul op\'{e}rationnel et alg\`{e}bre diff\'{e}rentielle aux {\S} \ref{algebre} et
\ref{perturb}, analyse non standard au {\S} \ref{bruit} sont les
principaux outils math\'{e}matiques.
\begin{remarque}
Ajoutons pour le lecteur \'{e}tranger, voire hostile, \`{a} l'analyse non
standard\footnote{Lobry \cite{N} a \'{e}crit un pamphlet \'{e}difiant sur
l'histoire \og agit\'{e}e \fg ~ et la r\'{e}ception \og houleuse \fg ~ de
cette analyse, en d\'{e}pit (\`{a} cause?) de sa beaut\'{e} et de sa puissance
inconstestables. Les propres d\'{e}boires de l'auteur lui ont prouv\'{e} que
ce t\'{e}moignage n'est pas exag\'{e}r\'{e}!} qu'il est loisible de la remplacer
par des consid\'{e}rations \og classiques \fg. On y perdrait en
concision et, \`{a} notre avis, en intuition.
\end{remarque}
\begin{remarque}
Avec des signaux analytiques par morceaux (le sens du mot {\em
analytique} est celui de la th\'{e}orie des fonctions et non pas, ici,
celui usuel en traitement du signal (cf.
\cite{battail,proakis0,proakis})), qui ne satisfont pas d'\'{e}quations
diff\'{e}rentielles connues \`{a} l'avance, on utilise, selon les m\^{e}mes
principes alg\'{e}briques, des d\'{e}rivateurs num\'{e}riques \`{a} fen\^{e}tres
glissantes pour obtenir les estimations (voir \cite{ath} et
\cite{compression,nl,join}, leurs exemples et leurs bibliographies).
On ne peut, alors, esp\'{e}rer les m\^{e}mes r\'{e}sultats que pr\'{e}c\'{e}demment.
\end{remarque}
\begin{remarque}
La possibilit\'{e} de liens entre th\'{e}orie de l'information et m\'{e}canique
quantique a \'{e}t\'{e} examin\'{e}e par divers auteurs (voir, par exemple,
\cite{brillouin,austria,green}). Rappelons \`{a} ce propos que
l'approche du bruit en \cite{ans} a d\'{e}j\`{a} conduit \`{a} une tentative
nouvelle de formalisation du quantique \cite{mecaqua}, qui sera
compl\'{e}t\'{e}e gr\^{a}ce \`{a} un r\'{e}sultat remarquable et tout r\'{e}cent, d\^{u} \`{a}
Charreton \cite{charreton}.
\end{remarque}
\begin{remarque}
Les techniques d'estimation \'{e}voqu\'{e}es plus haut ont permis des
avanc\'{e}es notables en automatique\footnote{Voir, par exemple,
\cite{linz,diag,nl,esaim,recons,garnier} et leurs bibliographies.
Des questions classiques sur l'identification param\'{e}trique, les
observateurs, le diagnostic et l'att\'{e}nuation de perturbations y
re\c{c}oivent des solutions d'une grande simplicit\'{e} conceptuelle et
faciles \`{a} mettre en {\oe}uvre en temps r\'{e}el. Mentionnons aussi la {\em
commande sans mod\`{e}le} \cite{sm}, particuli\`{e}rement prometteuse.},
lin\'{e}aire ou non.
\end{remarque}
\begin{remarque}
Des travaux en cours portant sur l'ing\'{e}ni\'{e}rie financi\`{e}re devraient
\'{e}galement d\'{e}montrer l'applicabilit\'{e} de notre point de vue \`{a} cette
discipline\footnote{Voir \cite{finance} pour une premi\`{e}re \'{e}bauche.}.
\end{remarque}
\vspace{0.4cm} \noindent{\bf Remerciements}. L'auteur exprime sa
reconnaissace \`{a} O. Gibaru (Lille), M. Mboup (Paris) et \`{a} tous les
membres du projet INRIA--ALIEN pour des \'{e}changes fructueux.
\vspace{0.4cm}
\section{Identifiabilit\'{e}}\label{algebre}
\subsection{\'Equations diff\'{e}rentielles}
Renvoyons \`{a} \cite{chambert} pour des rappels sur les corps,
diff\'{e}rentiels ou non.
Soit $k_0$ le corps de base de caract\'{e}ristique nulle, $\mathbb{Q}$
par exemple. Soit $k_0 (\Theta)$ le corps engendr\'{e} par un ensemble
fini $\Theta = \{\theta_1, \dots, \theta_\varrho \}$ de {\em
param\`{e}tres} inconnus. Soit $\bar{k}$ la cl\^{o}ture alg\'{e}brique de $k_0
(\Theta)$. Introduisons le corps $\bar{k}(s)$ des fractions
rationnelles en l'ind\'{e}termin\'{e}e $s$, que l'on munit d'une structure
de corps diff\'{e}rentiel gr\^{a}ce \`{a} la d\'{e}rivation $\frac{d}{ds}$ (les
\'{e}l\'{e}ments de $k_0$, de $\Theta$ et, donc, de $\bar{k}$, sont des
constantes). Tout {\em signal} $x$, $x \not\equiv 0$, est suppos\'{e}
satisfaire une \'{e}quation diff\'{e}rentielle lin\'{e}aire homog\`{e}ne, \`{a}
coefficients dans $\bar{k}(s)$, et donc appartenir \`{a} une extension
de Picard-Vessiot de $\bar{k}(s)$.
\begin{remarque}
Il suffit pour se convaincre de l'existence d'une telle \'{e}quation
homog\`{e}ne de d\'{e}river les deux membres de (\ref{nh0}) suffisamment de
fois par rapport \`{a} $s$.
\end{remarque}
L'anneau non commutatif $\bar{k}(s) [\frac{d}{ds}]$ des op\'{e}rateurs
diff\'{e}rentiels lin\'{e}aires
$$
\sum_{\tiny{\mbox{\rm finie}}} \varpi_\alpha (s)
\frac{d^\alpha}{ds^\alpha}, \quad \quad ~ \varpi_\alpha (s) \in
\bar{k}(s)
$$
est principal \`{a} droite et \`{a} gauche (cf. \cite{McC}). Le $\bar{k}(s)
[\frac{d}{ds}]$-module \`{a} gauche engendr\'{e} par $x$ et $1$ est un
module de torsion (cf. \cite{McC}), et, donc, un $\bar{k}(s)$-espace
vectoriel de dimension finie, $n + 1$, $n \geq 0$. D'o\`{u} le r\'{e}sultat
suivant qui semble nouveau (cf. \cite{chambert,singer}):
\begin{proposition}
Il existe un entier minimal $n \geq 0$, tel que $x$ satisfait
l'\'{e}quation diff\'{e}rentielle lin\'{e}aire, d'ordre $n$, non n\'{e}cessairement
homog\`{e}ne,
\begin{equation}\label{nh}
\left(\sum_{\iota = 0}^{n} q_\iota \frac{d^\iota}{ds^\iota}\right) x
- p = 0
\end{equation}
o\`{u} les polyn\^{o}mes $p, q_0, \dots, q_{n} \in \bar{k}[s]$ sont premiers
entre eux. Cette \'{e}quation, dite {\em minimale}, est unique \`{a} un
coefficient multiplicatif constant non nul pr\`{e}s.
\end{proposition}
\subsection{Identifiabilit\'{e} lin\'{e}aire projective}
Rappelons que l'ensemble $\Theta = \{\theta_1, \dots, \theta_\varrho
\}$ de param\`{e}tres est dit (cf. \cite{esaim,garnier})
\begin{itemize}
\item {\em lin\'{e}airement identifiable} si, et selement si,
\begin{equation}\label{li}
\mathfrak{A} \left(\begin{array}{c} \theta_1 \\ \vdots \\
\theta_\varrho \end{array} \right) = \mathfrak{B}
\end{equation}
o\`{u}
\begin{itemize}
\item les entr\'{e}es des matrices $\mathfrak{A}$, carr\'{e}e $\varrho
\times \varrho$, et $\mathfrak{B}$, colonne $\varrho \times 1$,
appartiennent \`{a} $\mbox{\rm span}_{k_0 (s) [\frac{d}{ds}]} (1, x)$;
\item $\det ( \mathfrak{A}) \neq 0$.
\end{itemize}
\item {\em projectivement lin\'{e}airement identifiable} si, et
seulement si,
\begin{itemize}
\item il existe un param\`{e}tre, $\theta_1$ par exemple, non nul,
\item l'ensemble $\{ \frac{\theta_2}{\theta_1}, \dots,
\frac{\theta_\varrho}{\theta_1} \}$ est lin\'{e}airement identifiable.
\end{itemize}
\end{itemize}
R\'{e}\'{e}crivons (\ref{nh}) sous la forme suivante:
\begin{equation}\label{nhcoef}
\left( \sum_{\tiny{\mbox{\rm finie}}} a_{\mu \nu} s^\mu
\frac{d^\nu}{ds^\nu} \right) x - \sum_{\tiny{\mbox{\rm finie}}}
b_\kappa s^\kappa = 0
\end{equation}
o\`{u} les $N + 1$ coefficients $a_{\mu \nu}$ et les $M$ coefficients
$b_\kappa$ appartiennent \`{a} $\bar{k}$. La matrice carr\'{e}e
$\mathfrak{M}$ d'ordre $N + M + 1$, dont la $\xi^{\tiny\mbox{\rm
\`{e}me}}$ ligne, $0 \leq \xi \leq N + M$, est
\begin{equation}\label{line} \dots,
\frac{d^\xi}{ds^\xi} \left( s^\mu \frac{d^\nu x}{ds^\nu}\right),
\dots, \frac{d^\xi s^\kappa}{ds^\xi}, \dots
\end{equation}
est singuli\`{e}re d'apr\`{e}s (\ref{nh}) et (\ref{nhcoef}). La minimalit\'{e}
de (\ref{nh}) permet de d\'{e}montrer selon des techniques bien connues
sur le rang du wronskien (cf. \cite{chambert,singer}) que le rang de
$\mathfrak{M}$ est $N + M$. Il en d\'{e}coule:
\begin{theoreme}
Les coefficients $a_{\mu \nu}$ et $b_\kappa$ de (\ref{nhcoef}) sont
projectivement lin\'{e}airement identifiables.
\end{theoreme}
\begin{corollaire}
Posons $x = \frac{p(s)}{q(s)}$, o\`{u} les polyn\^{o}mes $p, q \in \bar{k}
[s]$ sont premiers entre eux. Alors, les coefficients de $p$ et $q$
sont projectivement lin\'{e}airement identifiables.
\end{corollaire}
Il est loisible de supposer l'ensemble des param\`{e}tres inconnus
$\Theta = \{\theta_1, \dots, \theta_\varrho \}$ strictement inclus
dans celui des coefficients $a_{\mu \nu}$ et $b_\kappa$ de
(\ref{nhcoef}), et donc lin\'{e}airement identifiable.
\section{Perturbations et estimateurs}\label{perturb}
Avec une perturbation additive $w$ le capteur fournit non pas $x$
mais $x + w$. Soient
\begin{itemize}
\item $R = k_0(\Theta)[s] (k_0 [s])^{-1}$ l'anneau {\em localis\'{e}}
(cf. \cite{lang}) des fractions rationnelles \`{a} num\'{e}rateurs dans
$k_0(\Theta)[s]$ et d\'{e}nominteurs dans $k_0 [s]$,
\item $R[\frac{d}{ds}]$ l'anneau non commutatif des op\'{e}rateurs
diff\'{e}rentiels lin\'{e}aires \`{a} coefficients dans $R$.
\end{itemize}
On obtient, \`{a} partir de (\ref{li}), la
\begin{proposition}\label{estimperturb}
Les param\`{e}tres inconnus v\'{e}rifient
\begin{equation}\label{estim}
\mathfrak{A} \left(\begin{array}{c} \theta_1 \\ \vdots \\
\theta_\varrho \end{array} \right) = \mathfrak{B} + \mathfrak{C}
\end{equation}
o\`{u} les entr\'{e}es de $\mathfrak{C}$, matrice colonne $\varrho \times
1$, appartiennent \`{a} $\mbox{\rm span}_{R [\frac{d}{ds}]} (w)$.
\end{proposition}
On appelle (\ref{estim}) un {\em estimateur}. Il est dit {\em
strictement polynomial en $\frac{1}{s}$} si, et seulement si, toutes
les fractions rationnelles en $s$, rencontr\'{e}es dans les coefficients
des matrices $\mathfrak{A}$, $\mathfrak{B}$, $\mathfrak{C}$ de
(\ref{estim}), sont des polyn\^{o}mes en $\frac{1}{s}$ sans termes
constants. On peut toujours s'y ramener en multipliant les deux
membres de (\ref{estim}) par une fraction rationnelle de $k_0(s)$
convenable. On aboutit, alors, dans le domaine temporel, aux
estimateurs consid\'{e}r\'{e}s en \cite{ans}, si l'on suppose l'analyticit\'{e}
du signal:
\begin{equation}\label{estimat}
\delta (t) \left( [ \theta_\iota ]_e (t) - \theta_\iota \right) =
\sum_{\tiny{\mbox{\rm finie}}} c \int_{0}^{t} \dots
\int_{0}^{\tau_2} \int_{0}^{\tau_1} \tau_{1}^{\nu} w(\tau_1)d\tau_1
d\tau_2 \dots d\tau_k \quad \quad ~ ~\iota = 1, \dots, \varrho
\end{equation}
o\`u
\begin{itemize}
\item $c$ est une constante,
\item $[0, t]$ est la {\em fen\^etre d'estimation}, de {\em largeur}
$t$,
\item $\delta (t)$ est une fonction analytique,
appel\'{e}e {\em diviseur}, nulle en $0$,
\item $[ \theta ]_e (t)$ est l'estim\'{e}e de $\theta$ en $t$.
\end{itemize}
\section{Bruits}\label{bruit}
Renvoyons \`{a} \cite{robinson} et \cite{diener} pour la terminologie de
l'analyse non standard, d\'{e}j\`{a} utilis\'{e}e en \cite{ans}. Les
propositions \ref{propsin} et \ref{propbr} ci-dessous affinent la
proposition 3.2 de \cite{ans}, o\`{u} les estimations sont obtenues en
temps limit\'{e}, \og court \fg ~ en pratique.
\subsection{Sinuso\"{\i}des hautes fr\'{e}quences}\label{sinus}
La perturbation du {\S} \ref{perturb} est de la forme
$$\sum_{\iota
= 1}^{M} A_\iota \sin (\Omega_\iota t + \varphi_\iota)$$
o\`{u}
\begin{itemize}
\item $M$ est un entier limit\'{e} standard,
\item les fr\'{e}quences $\Omega_\iota
>0$ sont des constantes illimit\'{e}es,
\item les amplitudes $A_\iota$ sont des constantes, limit\'{e}es ou non,
\item les phases $\varphi_\iota$, $0 \leq \varphi_\iota < 2 \pi$, sont des constantes.
\end{itemize}
Si les quotients $\frac{A_\iota}{\Omega_\iota}$ sont infinit\'{e}simaux,
c'est un bruit centr\'{e}, c'est-\`{a}-dire de moyenne nulle, au sens de
\cite{ans}. Des manipulations \'{e}l\'{e}mentaires des int\'{e}grales it\'{e}r\'{e}es
(\ref{estimat}) conduisent \`{a} la
\begin{proposition}\label{propsin} Si \begin{itemize}
\item les quotients $\frac{A_\iota}{\Omega_\iota}$ sont
infinit\'{e}simaux, et, en particulier, si les $A_\iota$ sont limit\'{e}s,
\item la largeur de la fen\^{e}tre d'estimation est limit\'{e}e et n'appartient
pas au halo d'un z\'{e}ro du diviseur,
\end{itemize}
les estim\'{e}es des param\`{e}tres inconnus, obtenues gr\^{a}ce \`{a}
(\ref{estimat}), appartiennent aux halos de leurs vraies valeurs. Il
n'en va plus de m\^{e}me si l'un des quotients
$\frac{A_\iota}{\Omega_\iota}$ est appr\'{e}ciable.
\end{proposition}
\begin{remarque}
Il existe des valeurs illimit\'{e}es des amplitudes $A_\iota$,
$\sqrt{\Omega_\iota}$ par exemple, telles que les estim\'{e}es
pr\'{e}c\'{e}dentes appartiennent aux halos des vraies valeurs.
\end{remarque}
\subsection{Bruits blancs}
D\'{e}signons par $^*\mathbb{N}$, $^*\mathbb{R}$ les extensions non
standard de $\mathbb{N}$, $\mathbb{R}$. Rempla\c{c}ons l'intervalle $[0,
1] \subset {\mathbb{R}}$ par l'ensemble hyperfini ${\mathrm{I}} =
\{0, \frac{1}{\bar{N}}, \dots, \frac{\bar{N} - 1}{\bar{N}}, 1 \}$,
o\`u $\bar{N} \in {^*\mathbb{N}}$ est illimit\'{e}. Un {\em bruit blanc
centr\'{e}} est une fonction $w: {\mathrm{I}} \rightarrow
{^*\mathbb{R}}$, $\iota \mapsto w(\iota) = A n(\iota)$, o\`{u}
\begin{itemize}
\item l'amplitude $A \in {^*\mathbb{R}}$ est constante,
\item le quotient $\frac{A^2}{\bar{N}}$ est limit\'{e},
\item les $n(\iota)$ sont des variables al\'{e}atoires
r\'{e}elles, suppos\'{e}es centr\'{e}es, de m\^{e}me \'{e}cart-type $1$ normalis\'{e}, et
deux \`{a} deux ind\'{e}pendantes.
\end{itemize}
\begin{remarque}
Cette d\'{e}finition non restreinte au cas gaussien, qu'il convient de
comparer \`{a} celle de \cite{al}, pr\'{e}cise \cite{ans}; elle est inspir\'{e}e
de publications d'ing\'{e}nieurs sur le bruit blanc en temps discret
(voir, par exemple, \cite{proakis0}). Elle clarifie, \`{a} la mani\`{e}re de
\cite{nelson}, l'approche en temps continu usuelle dans les manuels
de traitement du signal (voir, \`{a} ce sujet,
\cite{battail,cover,proakis0,proakis} et leurs bibliographies).
Rappelons que cette approche continue est bas\'{e}e, en g\'{e}n\'{e}ral, sur
l'analyse de Fourier et renvoyons, \`{a} ce sujet, \`{a} \cite{fourier}.
Mentionnons, enfin, les travaux de \cite{gelfand,hida}, bas\'{e}s sur
l'analyse fonctionnelle.
\end{remarque}
\begin{remarque}
Un pas suppl\'{e}mentaire, inutile ici pour nos besoins, consisterait \`{a}
remplacer, comme en \cite{nelson}, les variables al\'{e}atoires
$n(\iota)$ par des analogues \og discrets \fg.
\end{remarque}
Comme au {\S} \ref{sinus}, il vient:
\begin{proposition}\label{propbr} Si
\begin{itemize}
\item le quotient $\frac{A^2}{\bar{N}}$ est
infinit\'{e}simal, et, en particulier, si $A$ est limit\'{e},
\item la largeur $t$, $t \in {\mathrm{I}}$, de la fen\^{e}tre d'estimation
n'appartient pas au halo d'un z\'{e}ro du diviseur,
\end{itemize}
les estim\'{e}es des param\`{e}tres inconnus, obtenues gr\^{a}ce \`{a}
(\ref{estimat}), appartiennent presque s\^{u}rement aux halos de leurs
vraies valeurs. Il n'en va plus de m\^{e}me si le quotient
$\frac{A^2}{\bar{N}}$ est appr\'{e}ciable.
\end{proposition}
\begin{remarque}
Il existe des valeurs illimit\'{e}es de $A$, $\sqrt[3]{\bar{N}}$ par
exemple, telles que les estim\'{e}es pr\'{e}c\'{e}dentes appartiennent presque
s\^{u}rement aux halos des vraies valeurs.
\end{remarque}
\begin{remarque}
Il est loisible de remplacer l'ind\'{e}pendance de $n(\iota)$ et
$n(\iota^\prime)$, $\iota \neq \iota^\prime$, par le fait que
l'esp\'{e}rance du produit $n(\iota) n(\iota^\prime)$ est
infinit\'{e}simale.
\end{remarque}
|
0808.3895
|
\section{Introduction}
High angular resolution observations of molecular gas have revealed the presence of dense equatorial discs and tori towards several late asymptotic giant branch (AGB) stars and young planetary nebulae (PNe): for instance, \citet{bie88}, \citet{for98}, \citet{sah98b}, \citet{buj03}.
The interaction between the post-AGB wind and such equatorial structures
has been proposed as one of the possible physical mechanisms in shaping the bipolar and multipolar morphologies seen in PNe and proto-PNe (PPNe) \citep{bal87,mel95,sok00,ick03}.
The origin of molecular discs and tori around late AGB stars is not completely clear, but a possible explanation is the presence of a binary system \citep{mor87,liv88,taa89,sok06}. In this case, when one of the stars enters the AGB phase, some of the ejected material can be retained, generating an extended torus. However, in the case of the high velocity jets observed in some late AGB stars, post-AGB stars, and young PNe \citep{fei85,sah98a,ima02,rie03} it is believed that a more effective collimation mechanism(s) should be present in the innermost region of the object, such as an accretion disc \citep{mor87,sok94,sok00} or a stellar magnetic field produced by a rotating star \citep{pas85,che94,gar97}.
Furthermore, recent observations reveal compelling evidence for magnetic fields in post-AGB stars and PNe, associated with equatorial discs and/or jets (e.g. \citealt{sab07,vle06}).
The process for the formation of accretion discs in PPNe has been investigated in detail by \citet{rey99}. They find that a disc forms when a close binary system (with a substellar companion) undergoes common envelope evolution.
For the case in which a low-mass secondary is disrupted during a dynamically unstable mass transfer process, an accretion disc, with a radius of $\sim$10~AU and a mass of $\sim$2$\times10^{-3}$~M$_{\sun}$, forms within $\sim$100~yr.
Recently, \citet{rij05} from their 3D simulations based on a two-wind model with a warped disc, suggested that, in order to explain the observed multipolar and point-symmetric shape of PNe, the required discs are quite small ($\sim$10--100~AU). Furthermore, these disc-like structures should be dense ($10^7-10^8$~cm$^{-3}$) and in Keplerian rotation.
Interferometric CO observations show larger toroidal molecular structures with sizes in the range of $\simeq$1000--6000 AU in PPNe and young PNe.
These structures seem to be systematically
in expansion with a mean velocity of $\sim$7~km~s$^{-1}$, such as in M~1-92 \citep{buj98}, M~2-9 \citep{zwe97}, \mbox{M~2-56} \citep{cas02}, or KjPn~8 \citep{for98}. These velocities are comparable to or below those found in expanding AGB envelopes \citep{hug07}.
Importantly, rotation has been observed in the Red Rectangle as well as slower expansion ($\sim$0.8~km~s$^{-1}$), superimposed on rotation, according to \citet{buj05}.
Note that the sizes of the tori are about one or two orders of magnitude more than the sizes of the disc-like structures proposed by \citet{rey99} and \citet{rij05}.
Maps of water maser emission allow the identification of a disc-like structure with a radius of 0\farcs12 in IRAS 17347-3139 \citep{deg04}, which corresponds to $\simeq$100--750~AU at the source distance \citep{gom05a}. In this case, the gas kinematics suggests the presence of both rotation and expansion in the disc traced by the water masers.
In this context, it is very important to study in detail the kinematics of much smaller disc-like structures that might be related to the collimation of the observed bipolar outflows in some PNe.
K3-35 is a young PN that shows a bipolar outflow in optical images \citep{mir00}.
At radio wavelengths, K3-35 exhibits a bright core and two bipolar lobes with an S-shape \citep{mir01}.
The distance to this object has been estimated to be $\sim$5~kpc \citep{zha95}, using an statistical method. However, we note the large uncertainty of this type of estimate, since the application of different statistical methods could give distances varying by factors of $\sim$3 \citep{phi04}.
The characteristic S-shape morphology of the radio lobes can be successfully reproduced by a precessing jet evolving in a dense circumstellar medium \citep{vel07}. Water maser emission has been found in three regions: two regions located at the tips of the bipolar radio jet about $\simeq$1$''$
from the centre (regions N and S), and another region towards the core of the nebula within $\simeq$0\farcs02 (region C), suggesting the presence of a torus \citep{mir01}.
In addition, OH maser emission has been detected towards the centre of K3-35 (within $\simeq$0\farcs04), showing circular polarisation that suggests the presence of a magnetic field \citep{mir01,gom05b}.
We decided to study the spatio-kinematical distribution of the water masers reported by \citet{mir01} towards the centre of K3-35 in order to identify possible expansion and/or rotation motions.
The paper is organised as follows.
In Section 2, we present a simple kinematical model of a ring including both expansion and rotation, and we calculate the pattern delineated in the position-velocity diagrams. We also describe the least-squares fit procedure that we used. In Section 3, we present the current observational data of H$_2$O masers in the PN K3-35. We then apply our model to the H$_2$O masers located towards the centre of the PN K3-35, making a comparison between the results and the observations. Finally, in Section 4, we discuss the implications of our results.
\section[]{Rotating and Expanding Ring Model}
\subsection{Model}
We assume a narrow, uniform, rotating, and expanding ring of radius $R$, arbitrarily oriented with respect to the line of sight. Its projection on the plane of the sky is an ellipse with semimajor and semiminor axes $a$ and $b$, respectively.
We define the two frames of reference shown in Fig.~1. Both coincide with the plane of the sky, one of them has its origin at the centre of the ellipse and is oriented such that the $x'$-axis is along the major axis of the projected ellipse, and the other one has the axes parallel to the RA and Dec axes.
The semimajor and semiminor axes are related to the ring radius by $a=R$ and $b=R\cos i$, where $i$ is the inclination angle between the line of sight and the normal to the ring plane as shown in Fig.~2.
The equation of the ellipse is given by
\begin{equation}
\frac{x'^2}{a^2}+\frac{y'^2}{b^2}=1,
\end{equation}
and the transformation equations between the coordinate systems are
\begin{equation}
x'=(x-x_0)\cos\theta+(y-y_0)\sin\theta,
\end{equation}
\begin{equation}
y'=-(x-x_0)\sin\theta+(y-y_0)\cos\theta,
\end{equation}
where
($x_0,y_0$) is the position of the centre of the ellipse, and
$\theta$ is the angle between the $x$-axes of the two frames of reference, and is defined as positive clockwise. The
angle $\theta$ is related to the position angle (PA) of the major axis of the ellipse by $\rmn{PA}=90\degr-\theta$.
\begin{figure}
\includegraphics[width=72mm]{fig1.eps}
\caption{Reference systems. Both the $x'y'$- and the $xy$-coordinate systems are in the plane of the sky. The $x'y'$-system has its origin at the centre of the ellipse ($x_0,~y_0$), and $\theta$ is the angle between the $x$-axis and the $x'$-axis. The $x$ and $y$ axes are parallel to the RA and Dec axes.}
\end{figure}
\begin{figure*}
\includegraphics[width=150mm]{fig2.eps}
\caption{Position-velocity diagrams for a rotating and expanding ring model. The top panels correspond to a positive inclination angle and the bottom panels correspond to a negative inclination angle.
The sense of rotation (clockwise or counterclockwise) as seen from the observer is indicated. The $x'$ and $y'$ axes are those defined in Fig.~1.}
\end{figure*}
Let $v_s$, $v_{\rmn{rot}}$, and $v_{\rmn{exp}}$ be the local standard of rest (LSR) systemic velocity, the rotation velocity, and the expansion velocity of the ring, respectively. Then the observed LSR velocity of a point in the ring can be expressed as
\begin{equation}
V_{\rmn{LSR}}=v_s+\frac{x'}{a}v_{\rmn{rot}}\sin i+\frac{y'}{a}v_{\rmn{exp}}\tan i.
\end{equation}
Hence the observed $V_{\rmn{LSR}}$ will be a linear function of either $x'$ or $y'$, if only one type of motion (rotation or expansion, respectively) is present in the ring \citep{usc05}.
Using equation (1), equation (4) can be written either in terms of the $x'$ or $y'$ coordinate as
\begin{equation}
\frac{[V_{\rmn{LSR}}-v_s-(x'/a)v_{\rmn{rot}}\sin i]^2}{(v_{\rmn{exp}}\sin i)^2}+\frac{x'^2}{a^2}=1,
\end{equation}
\begin{equation}
\frac{[V_{\rmn{LSR}}-v_s-(y'/a)v_{\rmn{exp}}\tan i]^2}{(v_{\rmn{rot}}\sin i)^2}+\frac{y'^2}{(a\cos i)^2}=1.
\end{equation}
Therefore, equations (5) and (6) indicate that the observed $V_{\rmn{LSR}}$ has a quadratic form (ellipse) expressed in terms of $x'$ or $y'$, when both motions are present.
In this model, we do not solve the radiative transfer through the ring, but we assume that the emission at a given $V_{\rmn{LSR}}$ comes from the point of the ring having this line-of-sight velocity component. We then use this information to construct position-velocity diagrams.
The orientation of the major axis of the ellipse in the position-velocity ($x'$-$V_{\rmn{LSR}}$ or $y'$-$V_{\rmn{LSR}}$) diagrams changes depending on the value of the inclination angle and on whether the sense of rotation is clockwise or counterclockwise as seen from the observer's point of view, as shown in Fig.~2. Accordingly, we are able to distinguish between a positive or negative value of the inclination angle and the sense of the rotation by doing a comparison between the position-velocity diagram delineated by the maser emission and the position-velocity diagrams expected for a rotating and expanding ring.
It is important to note that similar position-velocity diagrams can be obtained considering contraction instead of expansion in the ring, but changing the sign of the inclination angle and the sense of rotation. This ambiguity can be solved by constraining the value of the inclination angle (positive or negative) with additional information (see Section 3.2).
\subsection[]{Least-squares fit}
We carried out a least-squares fit of an ellipse to the observed emission, to estimate its spatial distribution on the sky.
In order to do this, we considered the curve on the
$xy$-plane given by equations (1)--(3) for a given set of values
of the parameters $(x_0,y_0)$ (the centre of the ellipse on the
plane of the sky), $a$ and $b$ (the semimajor and semiminor axes, respectively)
and $\theta$ (the angle between the $x$-axes of the two frames of reference).
We then compute the minimum distances
$d_j$ between the position $(x_j,y_j)$ of each of the masers and the
ellipse (these distances are measured along straight lines that pass through the maser and intersect the ellipse at right angles to
the curve). With these distances, we define the $\chi^2$ as
\begin{equation}
\chi^2={1\over {N-5}}\sum_{j=1}^N \left({d_j\over \sigma_j}\right)^2\,,
\label{chi2}
\end{equation}
where $N$ is the number of data points (i.e. the factor $N-5$ is the number of degrees of freedom), and $\sigma_j$ is the error associated with the position of the
maser spots. We then find the set of parameters $(x_0,y_0,a,b,\theta)$ (see
above) which give the minimum of $\chi^2$.
As a first approximation, we began by setting the values of $\sigma_j$ equal to the observational uncertainties $\Delta_j$ for the measured positions of the masers, and then finding the ellipse that gives the minimum value of ${\chi^2}$
(see equation \ref{chi2}). The actual structure of the maser-emitting region could be a ring of finite width, for which a broad ellipse is a rough approximation of its projection on the plane of the sky. Therefore, we do not expect maser spots to exactly trace an ellipse. We can characterise the width of the ring, assuming that the fitted ellipse traces the mean projected angular distance of the maser emission to the central star, and the actual emission will be distributed around this ellipse, with a dispersion $\Delta_e$. We treat $\Delta_e$ as a source of error for the ellipse fit, additional to the measured error, so that ${\sigma_j}^2={\Delta_j}^2+{\Delta_e}^2$. We then try different values of the width parameter ($\Delta_e$), until we obtain a fit with a minimum ${\chi^2}(\Delta_e)=1$. We consider $2 \Delta_e$ as the characteristic width of the maser ring.
The $(x_0,y_0,a,b,\theta)$ parameters obtained from the
minimization of ${\chi^2}$
give the best elliptical fit to the observed positions of the masers.
With these spatial parameters, we carried out a kinematical fit, using the LSR velocities of the maser components
in order to define a ${\chi^2}_v$ of the form
\[
{\chi^2}_v={1\over {N-3}}\sum_{j=1}^N {1\over {{\sigma_v}^2}}
\]
\begin{equation}
\times\left(V_{\rmn{LSR}},j-v_s-{{x_j}'\over a}v_{\rmn{rot}}\sin i-
{{y_j}'\over a}v_{\rmn{exp}}\tan i\right)^2\,,
\end{equation}
where $N-3$ is the number of degrees of freedom, and $({x'}_j,{y'}_j)$ are given by equations (2)--(3).
Here $\sigma_v$ is the uncertainty in the observed LSR velocity that we adopt as the spectral resolution of the observations.
The minimization of ${\chi^2}_v$ yielded the best values
for the systemic ($v_s$), rotation ($v_{\rmn{rot}}$), and expansion
($v_{\rmn{exp}}$) velocities.
\begin{table*}
\centering
\begin{minipage}{112mm}
\caption{H$_2$O Masers in K3-35 (Region C)}
\label{symbols}
\begin{tabular}{@{}ccrcllccl}
\hline
$V_{\mathrm{LSR}}$ & \multicolumn{3}{c}{Flux Density} &
\multicolumn{2}{c}{Position\footnote{ Units of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcseconds. Data from \citet{mir01} and \citet{gom03}.}} &
\multicolumn{3}{c}{Position Uncertainty\footnote{Relative position uncertainties (2$\sigma$) between maser spots.
The position of the 1.3~cm continuum emission peak is $\alpha(\rmn{J2000})=19^{\rmn{h}}27^{\rmn{m}}$44\fs0233, $\delta(\rmn{J2000})=21\degr30'$03\farcs441. The relative position uncertainty between the continuum and the H$_2$O masers is 0\farcs002.
The accuracy of the absolute positions is 0\farcs05.}} \\
(km~s$^{-1})$ & \multicolumn{3}{c}{(mJy)} & $\alpha$(J2000) & $\delta$(J2000) &
\multicolumn{3}{c}{(arcsec)} \\
\hline
24.6 & & 23 & & 19 27 44.0243 & 21 30 03.438 & & & 0.010 \\
24.0 & & 61 & & 19 27 44.0242 & 21 30 03.428 & & & 0.004 \\
23.3 & & 218 & & 19 27 44.02246 & 21 30 03.4460 & & & 0.0012 \\
22.6 & & 1010 & & 19 27 44.022254 & 21 30 03.45146 & & & 0.00024 \\
22.0 & & 1572 & & 19 27 44.022364 & 21 30 03.45335 & & & 0.00014 \\
21.3 & & 945 & & 19 27 44.02247 & 21 30 03.4564 & & & 0.0003 \\
20.7 & & 201 & & 19 27 44.02257 & 21 30 03.4625 & & & 0.0011 \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\section{H$_2$O masers in K3-35 (region C)}
\subsection{Observational Data}
There are only two sets of VLA water maser observations towards the PN K3-35. The VLA 1999.7 epoch observations reported by \citet{mir01}, and the VLA 2002.3 epoch observations reported by \citet{deg04}.
In the latter observations, only a group of four maser spots was detected towards the central region of this source. No maser emission was detected at the tips of the bipolar lobes of the PN.
Note that in the \citet{deg04} paper the position of the continuum peak used to align the positions of the masers at the two epochs was not used with enough precision, resulting in a spurious shift of the maser spots from one epoch to another (see their Fig. 4). Using the position with the adequate precision, we find the positions of the masers at the two epochs to be consistent within the uncertainties, 0\farcs01 (2$\sigma$).
\subsection{Model Application and Results}
In our analysis, we have used the water maser data from the VLA 1999.7 epoch observations towards the central region of K3-35 reported by \citet{mir01}.
At this epoch, the number of maser spots detected was larger than during the other epoch, allowing us a better identification of possible expansion and/or rotation motions at the centre of K3-35.
The velocity resolution of the VLA observations was 1.2~km~s$^{-1}$ and the accuracy in the relative positions of the water maser spots was of the order of milliarcseconds.
The positions of the observed water maser spots towards the core (region C) are listed in Table 1. We adopt as the origin of the $xy$-coordinate system the position of the 1.3 cm continuum emission peak.
Based on a least-squares fit to the positions of the maser spots (see Section 2.2) and using the observational uncertainties given in Table~1, we have found that the H$_2$O masers located towards the core of K3-35 can be fitted by a circular ring of $R\simeq0$\farcs021 radius ($\simeq$100~AU at the estimated distance of $\sim$5~kpc) with an angular width of 2$\Delta_e=0$\farcs003, observed at an inclination angle of $\vert i\vert\simeq\,55\degr$ (see Table 2 and Fig.~3).
\begin{figure}
\includegraphics[width=84mm]{fig3_light_cyan.eps}
\caption{
Positions of the K3-35 water maser spots in offsets relative to the position of the 1.3~cm continuum emission peak \citep{mir01}.
Each spot is labelled with its corresponding LSR velocity (in km s$^{-1}$).
The dashed ellipse corresponds to the least-squares fit to the maser spots positions whose parameters are indicated in Table 2.
The open circle indicates the nominal position of the 1.3~cm continuum emission peak (see Table 1), its size is equal to the uncertainty of this position. The diamond indicates the position of the centre of the ellipse that was obtained from the fit.
The straight line shows the direction of the bipolar outflow traced by the innermost region of the jet. The arrows show the sense of rotation of the proposed ring (see Section 2.1).
The broad cyan ellipse has a width $2\Delta_e$, with $\Delta_e$ fulfilling $\chi^2(\Delta_e)=1$ (see Section 2.2).
The positional error bars indicate the uncertainties in the relative positions between maser spots given in Table 1.}
\end{figure}
Spectroscopic observations show that the northeastern lobe of the outflow is blueshifted, and the southwestern one is redshifted \citep{mir00}. If we assume that the ring traced by the water masers is perpendicular to the bipolar lobes then the inclination angle should be positive ($i\simeq+55\degr$). This means that the western half of the ring is closer to the observer.
The calculated position of the centre of the ellipse relative to the position of the 1.3 cm continuum emission peak $(x_0,y_0)$ is given in Table 2. Both positions are in agreement within the uncertainties (see the overlap of these positions in Fig.~3).
The kinematic trend is shown in Fig.~4. Since we have determined that the inclination angle of the ring is positive, the ambiguity between expansion and contraction can be solved when we compare the position-velocity diagrams delineated by the water masers (see the top panel of Fig. 4) and the position-velocity diagrams of the model (see Fig. 2). We have found that the ring traced by the masers rotates clockwise as seen from the observer at a velocity $v_{\rmn{rot}}\simeq 3.1$~km~s$^{-1}$ and expands at a velocity $v_{\rmn{exp}}\simeq 1.4$~km~s$^{-1}$ (see Table 2).
The rotation and expansion velocities are estimated from a purely kinematical fit to the LSR velocities of the maser spots (see equation 8).
The kinematical fit yields a ${\chi^2}_v$ value of 1.94, which means that the fit is good, assuming a conservative confidence level of 90\%.
Although not all maser spots may be completely independent, given the limited angular and spectral resolution, the observed velocity trend and the good kinematical fit suggest that these motions are real and systematic. We note that we have considered a single value of $v_{\rmn{rot}}$ instead a Keplerian rotation law. This is reasonable, since the velocity gradient over the ring width would be only $\sim$7\%. Tracing more subtle velocity variations would require more data points than the ones available.
The expansion velocity we found for the ring is close to that of thermal motions or subsonic turbulence. However we do not expect that thermal or turbulent motions could produce the systematic motions described in the position-velocity diagrams. The water masers seem to be tracing a spatio-kinematical structure with organised motions (at macroscopic scales).
\begin{figure}
\includegraphics[width=84mm]{fig4.eps}
\caption{Position-velocity diagrams. \textit{Top}: The ordinate axis corresponds to the observed LSR velocity and the abscissa axis corresponds to the coordinate $x'$ or $y'$. \textit{Bottom}: Same as Fig.~4 (\textit{top}), but the abscissa axis corresponds to right ascension offset or declination offset relative to the position of the 1.3~cm continuum emission peak.
The points correspond to the observed maser spots towards the centre of K3-35 and the dashed ellipses correspond to the kinematical model using the parameters listed in Table 2. The error bars in position are those shown in Fig.~3. The uncertainty in velocity is $\simeq$0.6~km~s$^{-1}$.}
\end{figure}
\section{Discussion}
From our model, we conclude that a ring is a likely explanation for the distribution and kinematics shown by the water masers located towards the centre of the PN K3-35.
This ring may be arising from the innermost region of a disc or torus probably formed at the end of the AGB phase.
Since masers trace regions with very stringent physical conditions, it is not possible to know whether the ring is tracing part of a toroidal or a disc-like structure.
The kinematics of the ring in K3-35 suggests the presence of both rotating and expanding motions, as was also proposed in the young PN IRAS 17347-3139 \citep{deg04}. The estimated expansion and rotation velocity values for K3-35 are similar (a few km~s$^{-1}$) to those obtained from water maser observations of IRAS 17347-3139.
The calculated expansion velocity of the ring ($\simeq$1.4~km~s$^{-1}$) in K3-35 is comparable to, but below, the expansion velocities of the tori inferred from interferometric CO observations in some PPNe and young PNe. For instance, the expansion velocities in M~1-92, M 2-9, M 2-56, and KjPn 8, have values in the range of $\simeq$5--8~km~s$^{-1}$ \citep{buj98,zwe97,cas02,for98}. It is noteworthy that the estimated expansion velocity in K3-35 obtained from water maser observations is quite similar to the slow expansion velocity ($\simeq$0.8~km~s$^{-1}$) deduced from CO observations in the PPN Red Rectangle \citep{buj05}.
However, the estimated sizes of the tori detected in CO observations in those sources are about one order of magnitude more than the size of structure traced by the water masers in K3-35, suggesting that the structures observed in CO are probably related to the outermost equatorial region, while the water masers could be arising from the innermost one, such as a circumstellar disc, closer to the central star.
\begin{table}
\caption{Results of the rotating and expanding ring model
fit to the H$_2$O masers towards the centre of K3-35.}
\label{symbols}
\begin{tabular}{@{}ccrcl}
\hline
Parameter & \multicolumn{4}{c}{Value$^a$} \\
\hline
$a$ & & 0\farcs021 & $\pm$ & 0\farcs003 \\
$b$ & & 0\farcs012 & $\pm$ & 0\farcs002 \\
$x_0\,^*$ & & 0\farcs001 & $\pm$ & 0\farcs001 \\
$y_0\,^*$ & & 0\farcs004 & $\pm$ & 0\farcs004 \\
PA & & 158$\degr$ & $\pm$ & 10$\degr$ \\
$i$ & & 55$\degr$ & $\pm$ & 7$\degr$ \\
$v_s$ & & 22.8 & $\pm$ & 0.5~km~s$^{-1}$ \\
$v_{\rmn{exp}}$ & & 1.4 & $\pm$ & 0.9~km~s$^{-1}$ \\
$v_{\rmn{rot}}$ & & 3.1 & $\pm$ & 0.8~km~s$^{-1}$ \\
\hline
\end{tabular}
\medskip
$^a$ Note: Uncertainties are 2$\sigma$.\\
$^*$ Coordinates of the centre of the ellipse relative to the position of the 1.3~cm continuum emission peak.
\end{table}
In the case of K3-35, the water masers may be delineating an ellipse on the plane of the sky with a semimajor axis of $\simeq$100 AU and a PA$\simeq$158$\pm$10\degr, suggesting the presence of a disc or torus projected on the sky. The major axis of the ellipse is almost perpendicular (within the uncertainties) to the direction of the bipolar emission traced by the innermost region of the jet, with a PA$\simeq$65\degr observed by \citet{mir01}, suggesting that the disc or torus could be physically related to the collimation of the bipolar outflow.
Recently, \citet{hug07} found that jets typically appear a few hundred years after the torus formation in PPNe and young PNe.
This time sequence provides evidence that jets and tori are physically related. In the case of K3-35, \citet{vel07} modelled its radio continuum emission as a precessing dense jet with an age of $\sim$40~yrs. On the other hand,
our estimate of the dynamical age of the ring traced by the maser emission is about $\sim$350 years, using its radius ($\sim$100~AU) and assuming a uniform expansion velocity of $\sim$1.4~km~s$^{-1}$.
These values would, in principle, suggest a lag in time similar to the value found by \citet{hug07}. However, the dynamical age we derive is not reliable enough as an age estimate. Proper motion studies of the water maser emission (e.g. using e-MERLIN and VLBA) could provide better estimates.
The kinematics of the ring in K3-35 suggests the presence of both expansion and rotation.
A rough estimate for the central stellar mass can be obtained by assuming that the total energy (kinetic plus gravitational)
of the masing gas is close to zero.
In this case, $M\simeq (R/2G)(v_{\rmn{exp}}^2+v_{\rmn{rot}}^2)$, where $R\simeq100~(D/5~\rmn{kpc})$~AU, is the radius of the ring.
Hence, $M\simeq[0.7~(D/5~\rmn{kpc})\pm 0.3]~\rmn{M}_{\sun}$, where $D$ is the source distance. The error in the mass only includes the errors in the fitted parameters. This estimate of the central mass is in agreement with the core mass required for a PNe, according to evolutionary models \citep{kwo03}.
\section*{Acknowledgments}
LU and YG acknowledge support from DGAPA-UNAM grant IN100407 and CONACyT grant 49947.
LU is supported by Secretar\'{\i}a de Estado de Universidades e Investigaci\'on of MEC.
AR and JC are supported by CONACyT grants 46828-F and 61547.
LFM acknowledges support from MEC AYA2005-01495 grant (co-funded with FEDER funds).
GA, JFG, and JMT are supported by the MEC AYA2005-05823-C03 grant (co-funded with FEDER funds). GA, JFG, LFM, JMT and LU are also supported by Consejer\'{\i}a de Innovaci\'on, Ciencia y Empresa of Junta de Andaluc\'{\i}a.
We thank L. F. Rodr\'{\i}guez for valuable comments.
We are thankful to our referee, Anita Richards, for her useful comments on the manuscript.
|
1606.00426
|
\section{Introduction}
Throughout this note,
$f: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$
is a $\mathbb{C}$-algebra endomorphism
that satisfies
$\Jac(p,q) \in \mathbb{C}^*$,
where
$p:=f(x)$ and $q:=f(y)$.
Formanek's field of fractions theorem
~\cite[Theorem 2]{formanek field of fractions thm}
in dimension two says that
$\mathbb{C}(p,q,x)=\mathbb{C}(x,y)$.
{}From this it is not difficult to obtain that
$y = u/v$, for some
$u \in \mathbb{C}[p,q,x]-0$
and
$v \in \mathbb{C}[p,q]-0$.
We show in Theorem \ref{my keller} that
for such $f$,
if, in addition,
the group of invertible elements of
$\mathbb{C}[p,q,x][1/v]$
is contained in
$\mathbb{C}(p,q)-0$,
then $f$ is an automorphism.
Our proof of Theorem \ref{my keller} is almost identical to the proof of
Formanek's automorphism theorem ~\cite[Theorem 1]{formanek automorphism thm};
we did only some slight changes in his proof,
and also used Formanek's field of fractions theorem
and Wang's intersection theorem ~\cite[Theorem 41 (i)]{wang}.
Keller's theorem in dimension two
follows immediately from our theorem:
Assume that
$\mathbb{C}(p,q)=\mathbb{C}(x,y)$.
Then our condition of Theorem \ref{my keller} is satisfied,
because the group of invertible elements of
$\mathbb{C}[p,q,x][1/v] \subset \mathbb{C}(x,y)$
is contained in
$\mathbb{C}(x,y)-0 = \mathbb{C}(p,q)-0$.
\section{Preliminaries}
Our Theorem \ref{my keller}
deals with the two-dimensional case only.
However, the results we rely on are valid in any dimension $n$,
so we add the following notation:
$F: \mathbb{C}[x_1,\ldots,x_n] \to \mathbb{C}[x_1,\ldots,x_n]$
is a
$\mathbb{C}$-algebra endomorphism
that satisfies
$\Jac(F_1,\ldots,F_n) \in \mathbb{C}^*$,
where
$F_1:=F(x_1),\ldots,F_n:=F(x_n)$.
When $n=2$ we will keep the above notation, namely,
$x_1=x, x_2=y, F_1=p, F_2=q$.
\begin{theorem}[Formanek's automorphism theorem]
Suppose that there is a polynomial
$W$ in $\mathbb{C}[x_1,\ldots,x_n]$
such that
$\mathbb{C}[F_1,\ldots,F_n,W]=
\mathbb{C}[x_1,\ldots,x_n]$.
Then
$\mathbb{C}[F_1,\ldots,F_n]=
\mathbb{C}[x_1,\ldots,x_n]$,
namely, $F$ is an automorphism.
\end{theorem}
\begin{proof}
See ~\cite[Theorem 1]{formanek automorphism thm}
and
~\cite[page 13, Exercise 9]{essen book}.
\end{proof}
\begin{itemize}
\item If there exists
$w \in \mathbb{C}[x,y]$
such that
$\mathbb{C}[p,q,w]=\mathbb{C}[x,y]$,
then
$\mathbb{C}[p,q]=\mathbb{C}[x,y]$,
namely, $f$ is an automorphism.
\end{itemize}
\begin{theorem}[Formanek's field of fractions theorem]
$$
\mathbb{C}(F_1,\ldots,F_n,x_1,\ldots,x_{n-1})=
\mathbb{C}(x_1,\ldots,x_n).
$$
\end{theorem}
\begin{proof}
See ~\cite[Theorem 2]{formanek field of fractions thm}.
\end{proof}
\begin{itemize}
\item
$\mathbb{C}(p,q,x)=\mathbb{C}(x,y)$
and
$\mathbb{C}(p,q,y)=\mathbb{C}(x,y)$.
\end{itemize}
Formanek remarks that when $n=2$,
$\mathbb{C}(p,q,w)=\mathbb{C}(x,y)$,
where $w$ is the image of $x$ under any automorphism of
$\mathbb{C}[x,y]$;
see
~\cite[page 370, just before Theorem 6]{formanek field of fractions thm}.
The two-dimensional case was already proved by Moh ~\cite[page 151]{moh} and by
Hamann ~\cite[Lemma 2.1, Proposition 2.1(2)]{hamann}.
Moh and Hamann assumed that $p$ is monic in $y$,
but this is really not a restriction.
It is easy to see that:
\begin{corollary}\label{cor fractions}
There exist
$u \in
\mathbb{C}[p,q,x]-0$
and
$v \in
\mathbb{C}[p,q]-0$
such that
$y = u/v$.
\end{corollary}
\begin{proof}
$y \in
\mathbb{C}(x,y)=\mathbb{C}(p,q,x)=\mathbb{C}(p,q)(x)$.
Since $x$ is algebraic over $\mathbb{C}(p,q)$,
we have
$\mathbb{C}(p,q)(x)=\mathbb{C}(p,q)[x]$
(see ~\cite[Remark 4.7]{rowen}).
Hence,
$y \in \mathbb{C}(p,q)[x]$.
Therefore, there exist
$a_i, b_i \in \mathbb{C}[p,q]$
($b_i \neq 0$)
such that
$y= \sum (a_i/b_i)x^i$.
Then if we denote
$B =\prod{b_i}$ and
$B_i$ the product of the $b_j$'s except $b_i$,
we get
$y= (1/B) \sum B_i a_i x^i$.
Just take $v:=B$ and
$u:=\sum B_i a_i x^i$.
\end{proof}
\begin{theorem}[Wang's intersection theorem]
$\mathbb{C}(F_1,\ldots,F_n) \cap \mathbb{C}[x_1,\ldots,x_n] =
\mathbb{C}[F_1,\ldots,F_n]$.
\end{theorem}
\begin{proof}
See ~\cite[Theorem 41 (i)]{wang}
and
~\cite[Corollary 1.1.34 (ii)]{essen book}.
\end{proof}
Wang's intersection theorem has a more general version due to Bass
~\cite[Remark after Corollary 1.3, page 74]{bass},
~\cite[Proposition D.1.7]{essen book};
we will not need the more general version here.
\begin{itemize}
\item $\mathbb{C}(p,q) \cap \mathbb{C}[x,y] = \mathbb{C}[p,q]$.
\end{itemize}
The following is immediate:
\begin{corollary}\label{cor intersection}
$\mathbb{C}(p,q) \cap R = \mathbb{C}[p,q]$,
for any
$\mathbb{C}[p,q] \subseteq R \subseteq \mathbb{C}[x,y]$.
In particular,
$\mathbb{C}(p,q) \cap \mathbb{C}[p,q,x] = \mathbb{C}[p,q]$.
\end{corollary}
\begin{proof}
$\mathbb{C}(p,q) \cap R \subseteq
\mathbb{C}(p,q) \cap \mathbb{C}[x,y] = \mathbb{C}[p,q]$.
The other inclusion,
$\mathbb{C}(p,q) \cap R \supseteq \mathbb{C}[p,q]$,
is trivial.
\end{proof}
\begin{theorem}[Keller's theorem]
If
$\mathbb{C}(F_1,\ldots,F_n)=\mathbb{C}(x_1,\ldots,x_n)$,
then $F$ is an automorphism.
\end{theorem}
$F$ as in Keller's theorem is called birational
($F$ has an inverse formed of rational functions).
\begin{proof}
See ~\cite{keller}, ~\cite[Corollary 1.1.35]{essen book}
and ~\cite[Theorem 2.1]{bcw}.
\end{proof}
\begin{itemize}
\item If $\mathbb{C}(p,q)=\mathbb{C}(x,y)$,
then $f$ is an automorphism.
\end{itemize}
\begin{remark}
Notice that the above results are dealing with
$k[x_1,\ldots,x_n]$, where $k$ is:
\begin{itemize}
\item $\mathbb{C}$: Formanek's field of fractions theorem.
\item a field of characteristic zero: Formanek's automorphism theorem.
\item any field: Keller's theorem.
\item a UFD: Wang's intersection theorem.
\end{itemize}
We have not checked if Formanek's field of fractions theorem
is valid over a more general field than $\mathbb{C}$;
if, for example, it is valid over any algebraic closed field
of characteristic zero, then our Theorem \ref{my keller}
is valid over any algebraic closed field
of characteristic zero, not just over $\mathbb{C}$.
Anyway, working over $\mathbb{C}$ is good enough in view of
~\cite[Lemma 1.1.14]{essen book}.
\end{remark}
\section{A new proof of Keller's theorem in dimension two}
Our proof of Theorem \ref{my keller} relies heavily on the proof of Formanek's automorphism theorem;
we did only some slight changes in his proof,
changes that seem quite natural in view of
Corollary \ref{cor fractions}:
Although we do not know if
$\mathbb{C}[p,q,x]=\mathbb{C}[x,y]$
(if so, then $f$ is an automorphism
by Formanek's automorphism theorem),
we do know that $\mathbb{C}(p,q,x)=\mathbb{C}(x,y)$
(by Formanek's field of fractions theorem),
so by Corollary \ref{cor fractions},
$y=u/v$ for some
$u \in \mathbb{C}[p,q,x]-0$
and
$v \in \mathbb{C}[p,q]-0$.
Therefore, it seems natural to consider
$\beta: \mathbb{C}[U_1,U_2,U_3][1/V] \to \mathbb{C}[p,q,x][1/v]$,
where $V=v(U_1,U_2)$.
This $\beta$ has $x$ and $y$ in its image,
so most of Formanek's proof can be adjusted here,
except that the group of invertible elements
of $\mathbb{C}[p,q,x][1/v]$
is not as easily described as
the group of invertible elements
of $\mathbb{C}[x,y]$,
which is obviously $\mathbb{C}^*$.
Only after adding a condition on the group of invertible elements
of $\mathbb{C}[p,q,x][1/v]$,
we are able to show that $f$ is an automorphism.
Now we are ready to bring our theorem;
we recommend the reader to first read the proof of
Formanek's automorphism theorem, and then read our proof,
with $p,q,x$ in our proof instead of $F_1,F_2,F_3$ in his proof.
\begin{theorem}[Main Theorem]\label{my keller}
If the group of invertible elements of
$\mathbb{C}[p,q,x][1/v]$
is contained in
$\mathbb{C}(p,q)-0$,
then $f$ is an automorphism.
\end{theorem}
\begin{proof}
By Corollary \ref{cor fractions},
there exist
$u \in \mathbb{C}[p,q,x]-0$
and
$v \in \mathbb{C}[p,q]-0$
such that
$y= u/v$.
Let $U_1,U_2,U_3$ be independent variables over $\mathbb{C}$.
Define
$\alpha: \mathbb{C}[U_1,U_2,U_3] \to \mathbb{C}[p,q,x]$
by
$\alpha(U_1):=p$,
$\alpha(U_2):=q$,
$\alpha(U_3):=x$.
Clearly, $\alpha$ is surjective.
Claim: The kernel of $\alpha$ is a principal prime ideal
of $\mathbb{C}[U_1,U_2,U_3]$.
Proof of claim: $\mathbb{C}(U_1,U_2,U_3)$ has transcendence degree $3$
over $\mathbb{C}$,
and $\mathbb{C}(p,q,x)=\mathbb{C}(x,y)$ has transcendence degree $2$
over $\mathbb{C}$.
{}From ~\cite[Theorem 5.6]{crt},
$\mathbb{C}[U_1,U_2,U_3]$ is of Krull dimension $3$
and
$\mathbb{C}[p,q,x]$ is of Krull dimension $2$.
Hence, the kernel of $\alpha$ is of height $1$,
and in a Noetherian UFD a height one prime ideal is principal,
see ~\cite[Theorem 15.9]{pete}.
Denote by $H$ a generator of the kernel of $\alpha$:
$H= H_r U_3^r+\ldots+H_1U_3+H_0$,
where $H_j \in \mathbb{C}[U_1,U_2]$ and $r \geq 1$.
$H$ is a product of the minimal polynomial for $x$ over
$\mathbb{C}(p,q)$ by some element $H_r$ of $\mathbb{C}[U_1,U_2]$
which clears the denominators of the minimal polynomial
for $x$ over $\mathbb{C}(p,q)$.
Notice that $r=0$ is impossible,
since then
$H= H_0(U_1,U_2)$:
\begin{itemize}
\item
If $H_0(U_1,U_2) \equiv 0$,
then $H(U_1,U_2,U_3) \equiv 0$,
so the kernel of $\alpha$ is zero,
but then we have
$\mathbb{C}[U_1,U_2,U_3] \cong \mathbb{C}[p,q,x]$,
which is impossible from considerations of Krull dimensions.
\item
If $H_0(U_1,U_2) \neq 0$,
then
$0=\alpha(H)=\alpha(H_0(U_1,U_2))=H_0(p,q)$
is a non-trivial algebraic dependence
of $p$ and $q$ over $\mathbb{C}$.
But $p$ and $q$ are algebraically independent over
$\mathbb{C}$,
because $\Jac(p,q) \neq 0$;
see ~\cite[pages 19-20]{makar} or
~\cite[Proposition 6A.4]{rowen}.
\end{itemize}
Since we do not know if $y$ is in the image of $\alpha$,
we define the following (surjective)
$\beta: \mathbb{C}[U_1,U_2,U_3][1/V] \to \mathbb{C}[p,q,x][1/v]$
by
$\beta(U_1):=p$,
$\beta(U_2):=q$,
$\beta(U_3):=x$,
$\beta(1/V):= 1/(\beta(V))$,
where
$V:=v(U_1,U_2)$, namely,
in $v \in \mathbb{C}[p,q]-0$ replace
$p$ by $U_1$ and $q$ by $U_2$
and get $V$.
It is clear that
$\beta(V)= v$,
so
$\beta(1/V)= 1/v$.
Notice that $V \in \mathbb{C}[U_1,U_2]$;
the fact that the $U_3$-degree of $V$
is zero will be crucial in what follows.
Now, $y$ is in the image of $\beta$;
indeed, let $U:=u(U_1,U_2,U_3)$, namely,
in $u \in \mathbb{C}[p,q,x]-0$ replace
$p$ by $U_1$, $q$ by $U_2$ and $x$ by $U_3$,
and get $U$.
Then clearly $\beta(U/V)=u/v=y$.
Take:
$T_1:=U_3$ and $T_2:=U/V$.
Then, $\beta(T_1)=\beta(U_3)=x$,
and
$\beta(T_2)=\beta(U/V)=u/v=y$.
Each of the following three elements lie in the kernel of $\beta$:
$U_1-p(T_1,T_2)$,
$U_2-q(T_1,T_2)$
and
$U_3-x(T_1,T_2)=U_3-T_1=0$.
Indeed,
$\beta(U_1-p(T_1,T_2))=\beta(U_1)-\beta(p(T_1,T_2))=p-p=0$
and
$\beta(U_2-q(T_1,T_2))=\beta(U_2)-\beta(q(T_1,T_2))=q-q=0$.
Claim: The kernel of $\beta$ is a principal prime ideal of
$\mathbb{C}[U_1,U_2,U_3][1/V]$, generated by exactly the same
$H \in \mathbb{C}[U_1,U_2,U_3]$ that generates the kernel of
$\alpha$.
Proof of claim: Assume that $R/V^j$ is in the kernel of $\beta$,
where $R \in \mathbb{C}[U_1,U_2,U_3]$.
We have
$0=\beta(R/V^j)=\beta(R)/\beta(V)^j=\beta(R)/v^j$,
hence $0=\beta(R)$.
Since $\beta$ restricted to $\mathbb{C}[U_1,U_2,U_3]$
is $\alpha$, we get that $R$ belongs to the kernel of
$\alpha$, hence $R = \tilde{R} H$,
for some $\tilde{R} \in \mathbb{C}[U_1,U_2,U_3]$.
So,
$R/V^j= \tilde{R} H/ V^j= (\tilde{R}/ V^j) H$,
as claimed.
Therefore, there exist $R_1,R_2 \in \mathbb{C}[U_1,U_2,U_3]$
($R_3=0$) and $n,m \geq 0$
such that
$U_1-p(T_1,T_2) = (R_1/V^n) H$
and
$U_2-q(T_1,T_2)= (R_2/V^m) H$.
So,
$U_1= p(T_1,T_2) + (R_1/V^n) H$
and
$U_2= q(T_1,T_2) + (R_2/V^m) H$
(and
$U_3= T_1$).
Differentiating these three equations with respect to $U_1,U_2,U_3$
and using the Chain Rule, we get similar matrices to those in Formanek's proof;
the difference is that instead of $R_1, R_2, R_3$ of Formanek's proof,
we have here $R_1/V^n, R_2/V^m, 0$.
Applying $\beta$ gives a matrix equation over
$\mathbb{C}[p,q,x][1/v]$, similar to the matrix equation
$(2)$ of Formanek's proof.
Cramer's Rule shows that
$\beta(\partial H/\partial U_3) = \lambda/ d$,
where $\lambda = \Jac(p,q) \in \mathbb{C}^*$
and
$d \in \mathbb{C}[p,q,x][1/v]-0$
is the determinant of the matrix on the left.
$d$ belongs to the group of invertible elements
of $\mathbb{C}[p,q,x][1/v]$,
hence, by our assumption,
$d$ belongs to $\mathbb{C}(p,q)-0$.
On the one hand,
$d \in \mathbb{C}[p,q,x][1/v]-0$,
hence $d= \tilde{d}/v^l$
for some
$\tilde{d} \in \mathbb{C}[p,q,x]-0$
and $l \geq 0$.
On the other hand,
$d \in \mathbb{C}(p,q)-0$,
hence $d=a/b$
for some $a,b \in \mathbb{C}[p,q]-0$.
Combining the two we get,
$\tilde{d}/v^l = a/b$,
so
$\mathbb{C}[p,q,x]-0 \ni \tilde{d}=
v^l(a/b) \in \mathbb{C}(p,q)-0$.
{}From Corollary \ref{cor intersection}
we get that
$\tilde{d} \in \mathbb{C}[p,q]-0$.
(Remark: Actually, one can use Wang's intersection theorem directly,
without Corollary \ref{cor intersection},
and still get
$\tilde{d} \in \mathbb{C}[p,q]-0$,
as long as one observes that
$\mathbb{C}[p,q,x][1/v]= \mathbb{C}[x,y][1/v]$.
Indeed,
$d \in \mathbb{C}[p,q,x][1/v] = \mathbb{C}[x,y][1/v]$,
hence $d= \tilde{d}/v^l$
for some $\tilde{d} \in \mathbb{C}[x,y]-0$
and $l \geq 0$, etc.).
So $d= \tilde{d}/v^l$,
with $\tilde{d} \in \mathbb{C}[p,q]-0$.
Let
$D=d(U_1,U_2)=\tilde{d}(U_1,U_2)/v^l(U_1,U_2)=
\tilde{d}(U_1,U_2)/V^l$.
Clearly,
$\beta(D) = d$.
For convenience, multiply the above equation
$\beta(\partial H/\partial U_3) = \lambda/ d$
by $d$ and get
$d \beta(\partial H/\partial U_3) = \lambda$.
Then
$\beta(D) \beta(\partial H/\partial U_3) = \lambda$,
so
$\beta(D \partial H/\partial U_3) = \beta(\lambda)$.
Therefore,
$D \partial H/\partial U_3 - \lambda$
is in the kernel of $\beta$.
We have seen that the kernel of $\beta$ is a principal ideal of
$\mathbb{C}[U_1,U_2,U_3][1/V]$, generated by
$H \in \mathbb{C}[U_1,U_2,U_3]$,
hence there exist
$S \in \mathbb{C}[U_1,U_2,U_3]$
and $t \geq 0$
such that
$D \partial H/\partial U_3 - \lambda = (S/V^t)H$.
Replace $D$ by $\tilde{d}(U_1,U_2)/V^l$
and get,
$(\tilde{d}(U_1,U_2)/V^l) \partial H/\partial U_3 - \lambda = (SH)/V^t$.
Multiply both sides by
$V^{l+t}$ and get,
$V^t \tilde{d}(U_1,U_2) \partial H/\partial U_3 - \lambda V^{l+t} = V^l(SH)$.
Now, as promised above, we use the fact that the
$U_3$-degree of $V$ is zero:
The $U_3$-degree of the right side is at least $r$
(= that of $H$, which is exactly $r$, plus that of $S$, which is $\geq 0$),
while the $U_3$-degree of the left side is exactly $r-1$
(= that of $\partial H/\partial U_3$).
It follows that $S=0$ and $r-1=0$,
so $r=1$
and
$H= H_1(U_1,U_2) U_3 + H_0(U_1,U_2)$.
Apply $\beta$ and get
$0 = H_1(p,q) x + H_0(p,q)$,
so
$x = -H_0(p,q)/H_1(p,q) \in \mathbb{C}(p,q)$.
By Wang's intersection theorem,
$x \in \mathbb{C}[p,q]$.
Then obviously,
$\mathbb{C}[p,q][y]= \mathbb{C}[x,y]$.
Finally, Formanek's automorphism theorem
implies that
$\mathbb{C}[p,q]= \mathbb{C}[x,y]$,
namely $f$ is an automorphism.
\end{proof}
All the arguments and known results we use do not depend on
Keller's theorem, hence we have a new proof of Keller's theorem in dimension two:
\begin{theorem}[Keller's theorem]
Let
$f:\mathbb{C}[x,y] \to \mathbb{C}[x,y]$
be a $\mathbb{C}$-algebra endomorphism
that satisfies
$\Jac(p,q) \in \mathbb{C}^*$.
If
$\mathbb{C}(p,q)=\mathbb{C}(x,y)$,
then $f$ is an automorphism.
\end{theorem}
\begin{proof}
The group of invertible elements of
$\mathbb{C}[p,q,x][1/v] \subset \mathbb{C}(x,y)$
is contained in
$\mathbb{C}(x,y)-0 = \mathbb{C}(p,q)-0$.
Now apply Theorem \ref{my keller}.
\end{proof}
Notice that the converse of Theorem \ref{my keller} is trivially true:
If $f$ is an automorphism,
then
$\mathbb{C}[p,q] = \mathbb{C}[x,y]$,
so
$\mathbb{C}(p,q) = \mathbb{C}(x,y)$,
hence the group of invertible elements of
$\mathbb{C}[p,q,x][1/v] \subset \mathbb{C}(x,y)$
is contained in
$\mathbb{C}(x,y)-0 = \mathbb{C}(p,q)-0$.
Another argument:
If $f$ is an automorphism,
then we can take
$u=y$ and $v=1$.
Then
$\mathbb{C}[p,q,x][1/v]= \mathbb{C}[x,y]$,
and its group of invertible elements
is $\mathbb{C}^*$,
which is contained in $\mathbb{C}(p,q)-0$.
Therefore, the condition in Keller's theorem
is equivalent to our condition, not just implies our condition:
\begin{proposition}
TFAE:
\begin{itemize}
\item [(i)] $f$ is an automorphism, i.e. $\mathbb{C}[p,q]=\mathbb{C}[x,y]$.
\item [(ii)] $f$ is birational, i.e. $\mathbb{C}(p,q)=\mathbb{C}(x,y)$.
\item [(iii)] The group of invertible elements of
$\mathbb{C}[p,q,x][1/v]$
is contained in
$\mathbb{C}(p,q)-0$.
\end{itemize}
\end{proposition}
We do not know how to show directly that $(iii)$ implies $(ii)$.
\section{Further discussion}
We wish to bring some related ideas.
\textbf{First idea:}
We have already mentioned in the Preliminaries
that Formanek remarks that
$\mathbb{C}(p,q,w)=\mathbb{C}(x,y)$,
where $w$ is the image of $x$ under any automorphism of
$\mathbb{C}[x,y]$.
Therefore, we can obtain similar theorems
to Theorem \ref{my keller} with $x$
replaced by any image of $x$ under an
automorphism of $\mathbb{C}[x,y]$.
More elaborately, take any automorphism
$g: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$
and denote
$g_1:=g(x)$ and $g_2:=g(y)$.
We have
$\mathbb{C}(p,q,g_1)=\mathbb{C}(x,y)=
\mathbb{C}(g_1,g_2)$;
the first equality follows from Formanek's remark,
while the second equality trivially follows from
$\mathbb{C}[x,y]=\mathbb{C}[g_1,g_2]$.
Then,
$g_2 \in \mathbb{C}(p,q)(g_1) = \mathbb{C}(p,q)[g_1]$,
because $g_1$ is algebraic over $\mathbb{C}(p,q)$.
It is easy to obtain
$g_2 = u_g/v_g$,
where
$u_g \in \mathbb{C}[p,q,g_1]-0$
and
$v_g \in \mathbb{C}[p,q]-0$.
\begin{theorem}
If the group of invertible elements of
$\mathbb{C}[p,q,g_1][1/v_g]$
is contained in
$\mathbb{C}(p,q)-0$,
then $f$ is an automorphism.
\end{theorem}
\begin{proof}
In the proof of Theorem \ref{my keller}
replace $x$ and $y$ by $g_1$ and $g_2$,
do the appropriate adjustments,
and get a proof for the new theorem.
Notice that now, instead of considering
$p$ and $q$ as functions of $x$ and $y$,
one has to consider
$p$ and $q$ as functions of $g_1$ and $g_2$.
\end{proof}
\textbf{Second idea:}
For $v$ as in Corollary \ref{cor fractions}
write
$v= v_1 \cdots v_m$,
where $v_1,\ldots,v_m \in \mathbb{C}[p,q]$
are irreducible elements of
$\mathbb{C}[p,q]$.
There are two options, either
one (or more) of the $v_j$'s
becomes reducible in $\mathbb{C}[x,y]$
or all the $v_j$'s remain irreducible
in $\mathbb{C}[x,y]$.
If one (or more) of the $v_j$'s
becomes reducible in $\mathbb{C}[x,y]$,
then it is possible to show that
our condition of Theorem \ref{my keller}
is not satisfied,
and hence $f$ is not an automorphism:
Assume that
$v_1=w_1 \cdots w_l$, where
$w_1, \ldots, w_l \in \mathbb{C}[x,y]$
are irreducible in
$\mathbb{C}[x,y]$,
$l > 1$.
It is not difficult to see
(use Wang's intersection theorem)
that at least two factors are in
$\mathbb{C}[x,y] - \mathbb{C}[p,q]$,
w.l.o.g $w_1$ and $w_2$.
We claim that $w_1$ is invertible in
$\mathbb{C}[p,q,x][1/v] = \mathbb{C}[x,y][1/v]$.
Indeed,
$1=v/v= v_1 \cdots v_m / v =
w_1 w_2 \cdots w_l v_2 \cdots v_m / v =
w_1 (w_2 \cdots w_l v_2 \cdots v_m / v)$.
Clearly,
$w_1 \notin \mathbb{C}(p,q)$,
because otherwise,
$w_1 \in \mathbb{C}(p,q) \cap \mathbb{C}[x,y] = \mathbb{C}[p,q]$,
but
$w_1 \in \mathbb{C}[x,y] - \mathbb{C}[p,q]$.
Actually, if one (or more) of the $v_j$'s
becomes reducible in $\mathbb{C}[x,y]$,
then it is immediate that
$f$ is not an automorphism,
since an automorphism satisfies
$\mathbb{C}[p,q]=\mathbb{C}[x,y]$,
so trivially every irreducible element of
$\mathbb{C}[p,q]$
is an irreducible element of
$\mathbb{C}[x,y]$.
Next, if all the $v_j$'s remain irreducible
in $\mathbb{C}[x,y]$,
then our condition of Theorem \ref{my keller}
is satisfied:
\begin{theorem}[A special case of the main theorem]\label{my keller special case}
If $v_1,\ldots,v_m$ remain irreducible in
$\mathbb{C}[x,y]$,
then $f$ is an automorphism.
\end{theorem}
Of course, since $\mathbb{C}[p,q]$ ($\mathbb{C}[x,y]$)
is a UFD,
every irreducible element of
$\mathbb{C}[p,q]$ ($\mathbb{C}[x,y]$) is prime.
\begin{proof}
By assumption,
$v_1,\ldots,v_m \in \mathbb{C}[p,q]$
are irreducible elements of
$\mathbb{C}[x,y]$,
hence,
$v_1,\ldots,v_m$
are prime elements of
$\mathbb{C}[x,y]$.
Claim: The condition of Theorem \ref{my keller}
is satisfied.
Proof of claim:
Let
$a \in \mathbb{C}[p,q,x][1/v]= \mathbb{C}[x,y][1/v]$
be an invertible element,
so there exists
$b \in \mathbb{C}[p,q,x][1/v]= \mathbb{C}[x,y][1/v]$
such that
$ab = 1$.
We can write
$a= r/v^k$
and
$b= s/v^l$,
for some
$r,s \in \mathbb{C}[x,y]-0$
and
$k,l \geq 0$.
Then $ab=1$
becomes
$rs = v^{k+l} = (v_1 \cdots v_m)^{k+l}$.
Since
$v_1,\ldots,v_m$ are prime elements of
$\mathbb{C}[x,y]$,
we obtain that
$r= v_1^{\alpha_1} \cdots v_m^{\alpha_m}$
and
$s= v_1^{\beta_1} \cdots v_m^{\beta_m}$,
where
$\alpha_j+\beta_j = k+l$,
$1 \leq j \leq m$.
Therefore,
$r,s \in \mathbb{C}[p,q]-0$,
so
$a= r/v^k \in \mathbb{C}(p,q)-0$,
and we are done.
\end{proof}
Notice that in Theorem \ref{my keller special case}
we demand that each of the irreducible factors
$v_1,\ldots,v_m \in \mathbb{C}[p,q]$
of $v$
remain irreducible in $\mathbb{C}[x,y]$,
but we do not demand that other irreducible elements of
$\mathbb{C}[p,q]$
remain irreducible in
$\mathbb{C}[x,y]$.
If one demands that every irreducible element of
$\mathbb{C}[p,q]$
remains irreducible in
$\mathbb{C}[x,y]$,
then, without relying on Theorem \ref{my keller},
one can get that $f$ is an automorphism,
thanks to the result ~\cite[Lemma 3.2]{J}
of Jedrzejewicz and Zieli\'{n}ski.
Their result says the following:
Let $A$ be a UFD.
Let $R$ be a subring of $A$ such that
$R^* = A^*$.
The following conditions are equivalent:
\begin{itemize}
\item [(i)] Every irreducible element of $R$
remains irreducible in $A$.
\item [(ii)] $R$ is factorially closed in $A$.
\end{itemize}
(Recall that a sub-ring $R$ of a ring $A$
is called factorially closed in $A$
if whenever $a_1,a_2 \in A$
satisfy $a_1 a_2 \in R-0$,
then $a_1,a_2 \in R$).
In ~\cite[Lemma 3.2]{J} take
$A=\mathbb{C}[x,y], R=\mathbb{C}[p,q]$;
since we now assume that every irreducible element of
$\mathbb{C}[p,q]$
remains irreducible in
$\mathbb{C}[x,y]$,
we obtain that
$\mathbb{C}[p,q]$ is factorially closed in
$\mathbb{C}[x,y]$,
and we are done by the following easy lemma:
\begin{lemma}\label{lemma}
If $\mathbb{C}[p,q]$ is factorially closed in
$\mathbb{C}[x,y]$,
then $f$ is an automorphism.
\end{lemma}
\begin{proof}
Let $H$ be as in the proof of Theorem \ref{my keller},
and denote
$h_j := H_j(p,q)$, $0 \leq j \leq r$.
Obviously, $h_0 \neq 0$ by the minimality of $r$.
We have
$x(h_rx^{r-1}+h_{r-1}x^{r-2}+\ldots+h_1)=
-h_0 \in \mathbb{C}[p,q]-0$.
By assumption
$\mathbb{C}[p,q]$ is factorially closed in
$\mathbb{C}[x,y]$,
hence
$x \in \mathbb{C}[p,q]$
(and
$h_rx^{r-1}+h_{r-1}x^{r-2}+\ldots+h_1 \in \mathbb{C}[p,q]$).
Then
$\mathbb{C}[p,q,y]=\mathbb{C}[x,y]$,
and $f$ is an automorphism
by Formanek's automorphism theorem.
\end{proof}
Notice that in the proof of Lemma \ref{lemma},
$h_rx^{r-1}+h_{r-1}x^{r-2}+\ldots+h_1 \in \mathbb{C}[p,q]$
also yields that $f$ is an automorphism,
because by the minimality of $r$,
we must have $r=1$,
so $h_1 x + h_0 =0$.
Then $x = -h_0 / h_1 \in \mathbb{C}(p,q)$,
and by Wang's intersection theorem,
$x \in \mathbb{C}[p,q]$, etc.
\textbf{Third idea:} Notations as in the second idea,
another special case is when all the $v_j$'s are primes
in
$\mathbb{C}[p,q,x]$;
this special case is dealt with in
~\cite{vered}:
It is shown in ~\cite[Theorem 2.2]{vered}
that if all the $v_j$'s are primes
in $\mathbb{C}[p,q,x]$, then
$\mathbb{C}[p,q,x]$ is a UFD,
and it is shown in ~\cite[Theorem 2.1]{vered}
that if
$\mathbb{C}[p,q,x]$ is a UFD,
then $f$ is an automorphism.
It is not yet clear to us what happens in
the more general case when
all the $v_j$'s are irreducibles
in $\mathbb{C}[p,q,x]$.
It may happen that some (or all)
of the $v_j$'s are not primes in
$\mathbb{C}[p,q,x]$,
since we just know that
$\mathbb{C}[p,q,x]$ is an integral domain
(if we knew it is a UFD, then $f$ is an automorphism
by ~\cite[Theorem 2.1]{vered}).
\textbf{Fourth idea:}
We do not know if a similar result to Theorem \ref{my keller}
holds in higher dimensions.
Even if the answer is positive, the proof should be somewhat different
from the proof of the two-dimensional case.
For example,
already in the three-dimensional case some problems may arise
when trying to generalize the proof of the two-dimensional case:
Let
$f: \mathbb{C}[x,y,z] \to \mathbb{C}[x,y,z]$
be a $\mathbb{C}$-algebra endomorphism
having an invertible Jacobian.
Denote
$p:= f(x), q:=f(y), r:=f(z)$.
It is not difficult to generalize Corollary \ref{cor fractions}:
\begin{corollary}
There exist
$u \in
\mathbb{C}[p,q,r,x,y]-0$
and
$v \in
\mathbb{C}[p,q,r]-0$
such that
$z = u/v$.
\end{corollary}
\begin{proof}
By Formanek's field of fractions theorem,
$\mathbb{C}(p,q,r,x,y)= \mathbb{C}(x,y,z)$.
Since $x$ and $y$ are algebraic over
$\mathbb{C}(p,q,r)$,
a generalization of ~\cite[Remark 4.7]{rowen}
implies that
$\mathbb{C}(p,q,r)[x,y]= \mathbb{C}(p,q,r)(x,y)$.
Then,
$\mathbb{C}(p,q,r)[x,y]= \mathbb{C}(x,y,z) \ni z$.
{}From this it is not difficult to obtain that
$z =u/v$,
where
$u \in
\mathbb{C}[p,q,r,x,y]-0$
and
$v \in
\mathbb{C}[p,q,r]-0$.
\end{proof}
Define:
$\alpha:
\mathbb{C}[U_1,U_2,U_3,U_4,U_5] \to \mathbb{C}[p,q,r,x,y]$
by
$\alpha(U_1):=p$,
$\alpha(U_2):=q$,
$\alpha(U_3):=r$,
$\alpha(U_4):=x$,
$\alpha(U_5):=y$.
Clearly, $\alpha$ is surjective.
We can define
$\beta: \mathbb{C}[U_1,U_2,U_3,U_4,U_5][1/V]
\to \mathbb{C}[p,q,r,x,y][1/v]$.
It is clear that
$z \in \mathbb{C}[p,q,r,x,y][1/v]$.
The kernel of $\alpha$ is a height two prime ideal;
indeed,
$\mathbb{C}[U_1,U_2,U_3,U_4,U_5]$ is of Krull dimension $5$
and $\mathbb{C}[p,q,r,x,y]$ is of Krull dimension $3$,
hence the kernel of $\alpha$ is of height two.
{}From Krull's principal ideal theorem
~\cite[Theorem 8.42]{pete},
the kernel of $\alpha$ is generated by at least two elements.
Assume for the moment that the kernel of $\alpha$
is generated by exactly two elements,
hence the kernel of $\beta$
is generated by the same two elements.
The matrix equation $(2)$ in Formanek's proof
will involve the product of a
$5 \times 5$ matrix with a $5 \times 5$ matrix,
but Cramer's Rule seems not to help here.
We do not know yet if it is possible to overcome this problem.
\bibliographystyle{plain}
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\section{Introduction}
\label{Intro}
We consider the min-max optimization problem
\begin{equation}
\label{MainDeterministicProblem}
\min_{x_1 \in\mathbb{R}^{d_1}} \max_{x_2 \in\mathbb{R}^{d_2}} g(x_1,x_2)
\end{equation}
where $g:\mathbb{R}^{d_1} \times \mathbb{R}^{d_2}\rightarrow \mathbb{R}$ is a smooth objective.
Our goal is to find $x^*=(x_1^*, x_2^*)^\top \in \mathbb{R}^{d}$ where $d=d_1+ d_2$ such that
\begin{equation}
g(x_1^*, x_2) \leq g(x_1^*, x_2^*) \leq g(x_1, x_2^*),
\end{equation}
for every $x_1 \in \mathbb{R}^{d_1}$ and $x_2 \in \mathbb{R}^{d_2}$.
We call point, $x^*$, a \emph{ saddle point}, \emph{min-max solution} or {\em Nash equilibrium} of \eqref{MainDeterministicProblem}.
In its general form, this problem is hard.
In this work we focus on the simplest family of problems where some important questions are still open:
the case where all stationary points are global min-max solutions.
Motivated by recent applications in machine learning, we are particularly interested in cases where the objective, $g$, is naturally expressed as a finite sum
\begin{equation}
\label{MainStochasticProblem}
\min_{x_1 \in\mathbb{R}^{d_1}} \max_{x_2 \in\mathbb{R}^{d_2}} g(x_1, x_2) = \frac{1}{n} \sum_{i=1}^n g_i(x_1,x_2)
\end{equation}
where each component function $g_i:\mathbb{R}^{d_1} \times \mathbb{R}^{d_2}\rightarrow \mathbb{R}$ is assumed to be smooth.
Indeed, in problems like domain generalization \cite{albuquerque2019adversarial},
generative adversarial networks \cite{goodfellow2014generative},
and some formulations in reinforcement learning \cite{pfau2016connecting},
empirical risk minimization yields finite sums of the form of \eqref{MainStochasticProblem}.
We refer to this formulation as a \textbf{{\em stochastic smooth game}}.\footnote{We note that all of our results except the one on variance reduction do not require the finite-sum assumption and can be easily adapted to the stochastic setting (see Appendix~\ref{BeyondFiniteSum}).} We call problem \eqref{MainDeterministicProblem} a {\em deterministic game}.
The deterministic version of the problem has been studied in a number of classic \cite{korpelevich1976extragradient,nemirovski2004prox} and recent results \cite{mescheder2017numerics,ibrahim2019linear,gidel2018variational,daskalakis2017training,gidel2018negative,mokhtari2020unified,azizian2019tight,azizian2020accelerating} in various settings.
Importantly, the majority of these results provide
\textbf{last-iterate convergence} guarantees.
In contrast, for the stochastic setting, guarantees on the classic extragradient method and its variants rely on iterate averaging over compact domains \cite{nemirovski2004prox}.
However, \citet{chavdarova2019reducing} highlighted a possibility of pathological behavior where the iterates \emph{diverge towards} and then rotate near the boundary of the domain, far from the solution, while their average is shown to converge to the solution (by convexity).\footnote{This is qualitatively very different to stochastic minimization where the iterates converge towards a neighborhood of the solution and averaging is only used to stabilize the method.}
This behavior is also problematic in the context of applying the method on non-convex problems, where averaging do not necessarily yield a solution \cite{daskalakis2017training,abernethy2019last}.
It is only very recently that last-iterate convergence guarantees over a \textbf{non-compact domain} appeared in literature for the stochastic problem \cite{palaniappan2016stochastic,chavdarova2019reducing,hsieh2019convergence,mishchenko2020revisiting} under the assumption of strong monotonicity.
Strong monotonicity, a generalization of strong convexity for general operators, seems to be an essential condition for fast convergence in optimization.
Here, we make \textbf{no strong monotonicity assumption}.
The algorithms we consider belong to a recently introduced family of computationally-light second order methods which in each step require the computation of a Jacobian-vector product. Methods that belong to this family are the consensus optimization (CO) method \cite{mescheder2017numerics} and Hamiltonian gradient descent \cite{balduzzi2018mechanics,abernethy2019last}. Even though some convergence results for these methods are known for the deterministic problem, there is no available analysis for the stochastic problem. We close this gap.
We study {\em stochastic Hamiltonian gradient descent} (SHGD), and propose the first stochastic variance reduced Hamiltonian method, named L-SVRHG. Our contributions are summarized as follows:
\vspace{-0.1in}
\begin{itemize}[leftmargin=*]
\setlength{\itemsep}{0pt}
\item
Our results provide the first set of global non-asymptotic last-iterate convergence guarantees for a stochastic game over a non-compact domain, in the absence of strong monotonicity assumptions.
\item The proposed stochastic Hamiltonian methods use \emph{novel unbiased estimators} of the gradient of the Hamiltonian function. This is an essential point for providing convergence guarantees. Existing practical variants of SHGD use biased estimators \cite{mescheder2017numerics}.
\item We provide the first efficient convergence analysis of stochastic Hamiltonian methods. In particular, we focus on solving two classes of stochastic smooth games:
\begin{itemize}
\item \emph{Stochastic Bilinear Games}.
\item Stochastic games satisfying the ``sufficiently bilinear" condition or simply \emph{Stochastic Sufficiently Bilinear Games}. The deterministic variant of this class of games was firstly introduced by \citet{abernethy2019last} to study the deterministic problem and notably includes some \textbf{non-monotone problems}.
\end{itemize}
\item For the above two classes of games, we provide convergence guarantees for SHGD with a constant step-size (linear convergence to a neighborhood of stationary point), SHGD with a variable step-size (sub-linear convergence to a stationary point) and L-SVRHG. For the latter, we guarantee a linear rate.
\item We show the benefits of the proposed methods by performing numerical experiments on simple stochastic bilinear and sufficiently bilinear problems, as well as toy GAN problems for which the optimal solution is known. Our numerical findings corroborate our theoretical results.
\end{itemize}
\section{Further Related work}
\label{sec:related}
In recent years, several second-order methods have been proposed for solving the min-max optimization problem~\eqref{MainDeterministicProblem}. Some of them require the computation or inversion of a Jacobian which is a highly inefficient operation \cite{wang2019solving,mazumdar2019finding}. In contrast, second-order methods like the ones presented in
\citet{mescheder2017numerics,balduzzi2018mechanics,abernethy2019last} and in this work are more efficient as they only rely on the computation of a Jacobian-vector product in each step.
\citet{abernethy2019last} provide the first last-iterate convergence rates for the deterministic Hamiltonian gradient descent (HGD) for several classes of games including games satisfying the sufficiently bilinear condition. The authors briefly touch upon the stochastic setting and by using the convergence results of \citet{karimi2016linear}, explain how a stochastic variant of HGD with decreasing stepsize behaves. Their approach was purely theoretical and they did not provide an efficient way of selecting the unbiased estimators of the gradient of the Hamiltonian. In addition, they assumed bounded gradient of the Hamiltonian function which is restrictive for functions satisfying the Polyak-Lojasiewicz (PL) condition \cite{gower2020sgd}. In this work we provide the first efficient variants and analysis of SHGD. We did that by choosing practical unbiased estimator of the full gradient and by using the recently proposed assumptions of expected smoothness \cite{gower2019sgd} and expected residual \cite{gower2020sgd} in our analysis. The proposed theory of SHGD allow us to obtain as a corollary tight convergence guarantees for the deterministic HGD recovering the result of \citet{abernethy2019last} for the sufficiently bilinear games.
In another line of work, \citet{carmon2019variance} analyze variance reduction methods for constrained finite-sum problems and \citet{ryu2019ode} provide an ODE-based analysis and guarantees in the monotone but potentially non-smooth case.
\citet{chavdarova2019reducing} show that both alternate stochastic descent-ascent and stochastic extragradient diverge on an unconstrained stochastic bilinear problem. In the same paper, \citet{chavdarova2019reducing} propose the stochastic variance reduced extragradient (SVRE) algorithm with restart, which empirically achieves last-iterate convergence on this problem. However, it came with no theoretical guarantees. In Section~\ref{sec:experiments}, we observe in our experiments that SVRE is slower than the proposed L-SVRHG for both the stochastic bilinear and sufficiency bilinear games that we tested.
In concurrent work, \citet{yang2020global} provide global convergence guarantees for stochastic alternate gradient descent-ascent (and its variance reduction variant) for a subclass of nonconvex-nonconcave objectives satisfying a so-called two-sided Polyak-Lojasiewicz inequality, but this does not include the stochastic bilinear problem that we cover.
\section{Technical Preliminaries}
\label{sec:preliminaries}
In this section, we present the necessary background and the basic notation used in the paper. We also describe the update rule of the deterministic Hamiltonian method.
\subsection{Optimization Background: Basic Definitions}
\label{optBackground}
We start by presenting some definitions that we will later use in the analysis of the proposed methods.
\begin{definition}
\label{QSCdefinition}
Function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ is $\mu$--quasi-strongly convex if there exists a constant
$\mu > 0$ such that $\forall x \in \mathbb{R}^d$:
$f^* \geq f(x)+ \dotprod{\nabla f(x) , x^*-x} + \tfrac{\mu}{2} \norm{x^*-x}^2,$
where $f^*$ is the minimum value of $f$ and $x^*$ is the projection of $x$ onto the solution set $\mathcal{X}^*$ minimizing~$f$.
\end{definition}
\begin{definition}
\label{Polyak}
We say that a function satisfies the Polyak-Lojasiewicz (PL) condition if there exists $\mu >0$ such that
\begin{equation}
\label{PLcondition}
\frac{1}{2}\|\nabla f(x)\|^2 \geq \mu \left[f(x)-f^*\right] \quad \forall x \in \mathbb{R}^d \, ,
\end{equation}
where $f^*$ is the minimum value of $f$.
\end{definition}
An analysis of several stochastic optimization methods under the assumption of PL condition \citep{polyak1987introduction} was recently proposed in \citet{karimi2016linear}. A function can satisfy the PL condition and not be strongly convex, or even convex.
However, if the function is $\mu-$quasi strongly convex then it satisfies the PL condition with the same $\mu$ \cite{karimi2016linear}.
\begin{definition}
\label{Lsmooth}
Function $f:\mathbb{R}^d\rightarrow \mathbb{R}$ is $L$-smooth if there exists $L >0$ such that:\\ ${\|\nabla f(x) -\nabla f(y)\| \leq L \|x-y\|} \quad \forall x, y \in \mathbb{R}^d$.
\end{definition}
If $f=\frac{1}{n} \sum_{i=1}^n f_i(x)$, then a more refined analysis of stochastic gradient methods has been proposed under new notions of smoothness. In particular, the notions of \emph{expected smoothness (ES)} and \emph{expected residual (ER)} have been introduced and used in the analysis of SGD in \citet{gower2019sgd} and \citet{gower2020sgd} respectively. ES and ER are generic and remarkably weak assumptions. In Section~\ref{sec:analysis} and Appendix~\ref{ESandER}, we provide more details on their generality. We state their definitions below.
\begin{definition}[Expected smoothness, \citep{gower2019sgd}]
\label{ass:Expsmooth} We say that the function $f=\frac{1}{n} \sum_{i=1}^n f_i(x)$ satisfies the \emph{expected smoothness} condition if there exists ${\cal L}>0$ such that for all $x\in\mathbb{R}^d$,
\begin{equation}
\label{eq:expsmooth}
\EE{i}{\norm{\nabla f_i(x)-\nabla f_i(x^*)}^2} \leq 2{\cal L} (f(x)-f(x^*)),
\end{equation}
\end{definition}
\begin{definition}[Expected residual, \citep{gower2020sgd}]\label{ass:expresidual}
We say that the function $f=\frac{1}{n} \sum_{i=1}^n f_i(x)$ satisfies the \emph{expected residual} condition if there exists $\rho >0$ such that for all $x\in\mathbb{R}^d$,
\begin{multline}
\label{eq:expresidual}
\EE{i}{\norm{\nabla f_i(x)-\nabla f_i(x^*) - ( \nabla f(x)-\nabla f(x^*))}^2} \\
\leq 2\rho\left(f(x)-f(x^*) \right).
\end{multline}
\end{definition}
\subsection{Smooth Min-Max Optimization}
We use standard notation used previously in \citet{mescheder2017numerics,balduzzi2018mechanics,abernethy2019last,letcher2019differentiable}.
Let $x=(x_1, x_2)^\top \in \mathbb{R}^{d}$ be the column vector obtained by stacking $x_1 $ and $ x_2 $ one on top of the other. With $\xi(x):=\left(\nabla_{x_1} g, -\nabla_{x_2} g \right)^\top$, we denote the signed vector of partial derivatives evaluated at point $x$.
Thus, $\xi(x):\mathbb{R}^d \rightarrow \mathbb{R}^d$ is a vector function.
We use
$$\mathbf{J}=\nabla \xi=\begin{pmatrix}
\nabla^2_{x_1,x_1} g & \nabla^2_{x_1,x_2} g \\
-\nabla^2_{x_2,x_1} g & -\nabla^2_{x_2,x_2} g \\
\end{pmatrix} \in \mathbb{R}^{d \times d}
$$
to denote the Jacobian of the vector function $\xi$.
Note that using the above notation, the simultaneous gradient descent/ascent (SGDA) update can be written simply as: $x^{k+1}=x^k - \eta_k \xi(x_k)$.
\begin{definition}
The objective function $g$ of problem \eqref{MainDeterministicProblem} is $L_g$-smooth if there exist $L_g>0$ such that: \\ $\|\xi(x)-\xi(y)\| \leq L_g \|x-y\| \quad \forall x, y \in \mathbb{R}^d$.
We also say that $g$ is $L$-smooth in $x_1$ (in $x_2$) if $\|\nabla_{x_1} g(x_1,x_2)-\nabla_{x_1} g(x_1',x_2)\| \leq L \|x_1-x_1'\|$ (if $\|\nabla_{x_2} g(x_1,x_2)-\nabla_{x_2} g(x_1,x_2')\| \leq L \|x_2-x_2'\|$) $\quad \text{for all } x_1, x_1' \in \mathbb{R}^{d_1}$ ($ \text{for all } x_2, x_2' \in \mathbb{R}^{d_2}$).
\end{definition}
\begin{definition}
A stationary point of function $f:\mathbb{R}^{d} \rightarrow \mathbb{R}$ is a point $x^* \in \mathbb{R}^d$ such that $\nabla f(x^*)=0$. Using the above notation, in min-max problem \eqref{MainDeterministicProblem}, point $x^* \in \mathbb{R}^d$ is a stationary point when $\xi(x^*)=0$.
\end{definition}
As mentioned in the introduction, in this work we focus on smooth games satisfying the following assumption.
\begin{assumption}
\label{criticalminmax}
The objective function $g$ of problem~\eqref{MainStochasticProblem} has at least one stationary point and all of its stationary points are global min-max solutions.
\end{assumption}
With Assumption~\ref{criticalminmax}, we can guarantee convergence to a min-max solution of problem \eqref{MainStochasticProblem} by proving convergence to a stationary point. This assumption is true for several classes of games including strongly convex-strongly concave and convex-concave games. However, it can also be true for some classes of non-convex non-concave games~\cite{abernethy2019last}. In Section~\ref{sec:hamiltonian}, we describe in more details the two classes of games that we study. Both satisfy this assumption.
\subsection{Deterministic Hamiltonian Gradient Descent}
Hamiltonian gradient descent (HGD) has been proposed as an efficient method for solving min-max problems in \citet{balduzzi2018mechanics}. To the best of our knowledge, the first convergence analysis of the method is presented in \citet{abernethy2019last} where the authors prove non-asymptotic linear last-iterate convergence rates for several classes of games.
In particular, HGD converges to saddle points of problem \eqref{MainDeterministicProblem} by performing gradient descent on a particular objective function ${\cal H}$, which is called the Hamiltonian function \citep{balduzzi2018mechanics}, and has the following form:
\begin{equation}
\label{HamiltonianProblem}
\min_x \quad {\cal H}(x)= \frac{1}{2} \|\xi(x)\|^2.
\end{equation}
That is, HGD is a gradient descent method that minimizes the square norm of the gradient $\xi(x)$.
Note that under Assumption~\ref{criticalminmax}, solving problem \eqref{HamiltonianProblem} is equivalent to solving problem \eqref{MainDeterministicProblem}.
The equivalence comes from the fact that
${\cal H}$ only achieves its minimum at stationary points.
The update rule of HGD can be expressed using a Jacobian-vector product \citep{balduzzi2018mechanics,abernethy2019last}:
\begin{equation}
\label{DetHamiltonian}
x^{k+1}=x^k - \eta_k \nabla {\cal H}(x)= x^k - \eta_k \left[ \mathbf{J}^\top \xi \right],
\end{equation}
making HGD a second-order method.
However, as discussed in~\citet{balduzzi2018mechanics}, the Jacobian-vector product can be efficiently evaluated in tasks like training neural networks and the computation time of the gradient and the Jacobian-vector product is comparable~\cite{pearlmutter1994fast}.
\section{Stochastic Smooth Games and Stochastic Hamiltonian Function}
\label{sec:hamiltonian}
In this section, we provide the two classes of stochastic games that we study. We define the stochastic counterpart to the Hamiltonian function as a step towards solving problem~\eqref{MainStochasticProblem} and present its main properties.
Let us start by presenting the basic notation for the stochastic setting.
Let $\xi(x)=\frac{1}{n}\sum_{i=1}^n \xi_i(x),$
where $\xi_i(x):=\left(\nabla_{x_1} g_i, -\nabla_{x_2} g_i \right)^\top$, for all $i \in [n]$ and
let
$$\mathbf{J}=\frac{1}{n}\sum_{i=1}^n \mathbf{J}_i,
\quad\textrm{where\ }
\mathbf{J}_i=\begin{pmatrix}
\nabla^2_{x_1,x_1} g_i & \nabla^2_{x_1,x_2} g_i \\
-\nabla^2_{x_2,x_1} g_i & -\nabla^2_{x_2,x_2} g_i \\
\end{pmatrix}.$$
Using the above notation, the stochastic variant of SGDA can be written as $x^{k+1}=x^k - \eta_k \xi_i(x_k)$
where $\mathbb{E}_i[ \xi_i(x_k)]= \xi(x_k)$.\footnote{Here the expectation is over the uniform distribution. That is, $\mathbb{E}_i[ \xi_i(x)]=\frac{1}{n}\sum_{i=1}^n \xi_i(x)$.}
In this work, we focus on stochastic smooth games of the form~\eqref{MainStochasticProblem} that satisfy the following assumption.
\begin{assumption}
\label{AssumptionOnGi}
Functions $g_i:\mathbb{R}^{d_1} \times \mathbb{R}^{d_2}\rightarrow \mathbb{R}$ of problem \eqref{MainStochasticProblem} are twice differentiable, $L_i$-smooth with $S_i$-Lipschitz Jacobian. That is, for each $i \in[n]$ there are constants $L_i >0$ and $S_i >0$ such that $\|\xi_i(x)-\xi_i(y)\| \leq L_i \|x-y\|$ and $\|\mathbf{J}_i(x)-\mathbf{J}_i(y)\| \leq S_i \|x-y\|$ for all $x ,y \in R^d$.
\end{assumption}
\subsection{Classes of Stochastic Games}
\label{theclasses}
Here we formalize the two families of stochastic smooth games under study: (i) stochastic bilinear, and (ii) stochastic sufficiently bilinear. Both families satisfy Assumption~\ref{criticalminmax}. Interestingly, the latter family includes some non-convex non-concave games, i.e. non-monotone problems.
\paragraph{Stochastic Bilinear Games.}
A stochastic bilinear game is the stochastic smooth game \eqref{MainStochasticProblem} in which function $g$ has the following structure:
\begin{equation}
\label{bilinearGame1}
g(x_1,x_2)=\frac{1}{n} \sum_{i=1}^n \left( x_1^\top b_i+x_1^\top \mathbf{A}_i x_2 +c_i^\top x_2 \right) \, .
\end{equation}
While this game appears simple, standard methods diverge on it~\citep{chavdarova2019reducing} and L-SVRHG gives the first stochastic method with last-iterate convergence guarantees.
\paragraph{Stochastic sufficiently bilinear games.}
A game of the form~\eqref{MainStochasticProblem} is called \emph{stochastic sufficiently bilinear} if it satisfies the following definition.
\begin{definition}
\label{SuffBilinear}
Let Assumption~\ref{AssumptionOnGi} be satisfied and let the objective function $g$ of problem~\eqref{MainStochasticProblem} be $L$-smooth in $x_1$ and $L$-smooth in $x_2$. Assume that a constant $C>0$ exists, such that $\mathbb{E}_i\|\xi_i(x)\|<C$. Assume the cross derivative $\nabla^2_{x_1,x_2} g$ be full rank with $0 < \delta \leq \sigma_i \left(\nabla^2_{x_1,x_2} g\right) \leq \Delta$ for all $x \in \mathbb{R}^d$ and for all singular values $\sigma_i $. Let $\rho^2 = \min_{x_1,x_2} \lambda_{\min} \left[ \nabla^2_{x_1,x_1} g(x_1,x_2)\right]^2$ and $\beta^2 = \min_{x_1,x_2} \lambda_{\min} \left[ \nabla^2_{x_2,x_2} g(x_1,x_2)\right]^2$. Finally let the following condition to be true:
\begin{equation}
\label{SufficientBilinear}
(\delta^2 +\rho^2)(\delta^2 +\beta^2) -4L^2 \Delta^2 >0.
\end{equation}
\end{definition}
Note that the definition of the stochastic sufficiently bilinear game has no restriction on the convexity of functions $g_i(x)$ and $g(x)$. The most important condition that needs to be satisfied is the expression in equation~\eqref{SufficientBilinear}. Following the terminology of \citet{abernethy2019last}, we call the condition~\eqref{SufficientBilinear}: ``\emph{sufficiently bilinear}" condition. Later in our numerical evaluation, we present stochastic non convex-non concave min-max problems that satisfy condition~\eqref{SufficientBilinear}.
We highlight that the deterministic counterpart of the above game was first proposed in~\citet{abernethy2019last}. The deterministic variant of \citet{abernethy2019last} can be obtained as special case of the above class of games when $n=1$ in problem~\eqref{MainStochasticProblem}.
\subsection{Stochastic Hamiltonian Function}
\label{stoHamilFunction}
Having presented the two main classes of stochastic smooth games, in this section we focus on the structure of the stochastic Hamiltonian function and highlight some of its properties.
\paragraph{Finite-Sum Structure Hamiltonian Function.}
Having the objective function $g$ of problem~\eqref{MainStochasticProblem} to be stochastic and in particular to be a finite-sum function, leads to the following expression for the Hamiltonian function:
\begin{eqnarray}
\label{StochHamiltonianFunction}
{\cal H}(x)=\frac{1}{n^2} \sum_{i,j=1}^n \underbrace{\frac{1}{2} \langle \xi_i(x), \xi_j(x)\rangle}_{{\cal H}_{i,j}(x)} \, .
\end{eqnarray}
That is, the Hamiltonian function ${\cal H}(x)$ can be expressed as a finite-sum with $n^2$ components.
\paragraph{Properties of the Hamiltonian Function.}
As we will see in the following sections, the finite-sum structure of the stochastic Hamiltonian function \eqref{StochHamiltonianFunction} allows us to use popular stochastic optimization problems for solving problem~\eqref{HamiltonianProblem}. However in order to be able to provide convergence guarantees of the proposed stochastic Hamiltonian methods, we need to show that the stochastic Hamiltonian function \eqref{StochHamiltonianFunction} satisfies specific properties for the two classes of games we study. This is what we do in the following two propositions.
\begin{proposition}
\label{BilinearGameProposition}
For stochastic bilinear games of the form~\eqref{bilinearGame1}, the stochastic Hamiltonian function~\eqref{StochHamiltonianFunction} is a smooth quadratic $\mu_{{\cal H}}$--quasi-strongly convex function with constants $L_{{\cal H}}=\sigma_{\max}^2 (\mathbf{A})$ and $\mu_{{\cal H}} = \sigma_{\min}^2(\mathbf{A})$ where $\mathbf{A}=\frac{1}{n} \sum_{i=1}^n \mathbf{A}_i$ and $\sigma_{\max}$ and $\sigma_{\min}$ are the maximum and minimum non-zero singular values of $\mathbf{A}$.
\end{proposition}
\begin{proposition}
\label{SufficientlyBilinearGameProposition}
For stochastic sufficiently bilinear games, the stochastic Hamiltonian function~\eqref{StochHamiltonianFunction} is a $L_{{\cal H}}= \bar{S} C+\bar{L}^2$ smooth function and satisfies the PL condition~\eqref{PLcondition} with $\mu_{{\cal H}}=\frac{(\delta^2 +\rho^2)(\delta^2 +\beta^2) -4L^2 \Delta^2 }{2\delta^2+\rho^2+\beta^2}$. Here $\bar{S}=\mathbb{E}_i[S_i]$ and $\bar{L}=\mathbb{E}_i[L_i]$.
\end{proposition}
\section{Stochastic Hamiltonian Gradient Methods}
\label{sec:methods}
In this section we present the proposed stochastic Hamiltonian methods for solving the stochastic min-max problem~\eqref{MainStochasticProblem}. Our methods could be seen as extensions of popular stochastic optimization methods into the Hamiltonian setting. In particular, the two algorithms that we build upon are the popular stochastic gradient descent (SGD) and the recently introduced loopless stochastic variance reduced gradient (L-SVRG). For completeness, we present their form for solving finite-sum optimization problems in Appendix~\ref{AppendixTechnical}.
\subsection{Unbiased Estimator}
\label{unbiased-estimator}
One of the most important elements of stochastic gradient-based optimization algorithms for solving finite-sum problems of the form~\eqref{StochHamiltonianFunction} is the selection of unbiased estimators of the full gradient $\nabla {\cal H}(x)$ in each step. In our proposed optimization algorithms for solving~\eqref{StochHamiltonianFunction}, at each step we use the gradient of only one component function ${\cal H}_{i,j}(x)$:
\begin{eqnarray}
\nabla {\cal H}_{i,j}(x)=\frac{1}{2} \left[ \mathbf{J}_i^\top \xi_j + \mathbf{J}_j^\top \xi_i \right].
\end{eqnarray}
It can easily be shown that this selection is an unbiased estimator of $\nabla {\cal H}(x)$. That is, $\mathbb{E}_{i,j}\left[\nabla {\cal H}_{i,j}(x)\right]= \nabla {\cal H}(x).$
\subsection{Stochastic Hamiltonian Gradient Descent (SHGD)}
Stochastic gradient descent (SGD) \cite{robbins1951stochastic, NemYudin1978, NemYudin1983book, Nemirovski-Juditsky-Lan-Shapiro-2009, HardtRechtSinger-stability_of_SGD, gower2019sgd, gower2020sgd, loizou2020stochastic} is the workhorse for training supervised machine learning problems. In Algorithm~\ref{SHGD_Algorithm}, we apply SGD to~\eqref{StochHamiltonianFunction}, yielding stochastic Hamiltonian gradient descent (SHGD) for solving problem~\eqref{MainStochasticProblem}. Note that at each step, $i \sim {\cal D}$ and $j \sim {\cal D}$ are sampled from a given well-defined distribution ${\cal D}$ and then are used to evaluate $\nabla {\cal H}_{i,j}(x^k)$ (unbiased estimator of the full gradient). In our analysis, we provide rates for two selections of step-sizes for SHGD. These are the constant step-size $\gamma^k=\gamma$ and the decreasing step-size (switching rule which
describe when one should switch from a constant to a decreasing stepsize regime).
\begin{algorithm}[tb]
\caption{Stochastic Hamiltonian Gradient Descent (SHGD)}
\label{SHGD_Algorithm}
\begin{algorithmic}
\STATE {\bfseries Input:} Starting stepsize $\gamma^0>0$. Choose initial points $x^0 \in \mathbb{R}^d$. Distribution ${\cal D}$ of samples.
\FOR{$k=0,1,2,\cdots, K$}
\STATE Generate fresh samples $i \sim {\cal D}$ and $j \sim {\cal D}$ and evaluate $\nabla {\cal H}_{i,j}(x^k)$.
\STATE Set step-size $\gamma^k$ following one of the selected choices (constant, decreasing)
\STATE Set $x^{k+1}=x^k -\gamma^k \nabla {\cal H}_{i,j}(x^k)$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsection{Loopless Stochastic Variance Reduced Hamiltonian Gradient (L-SVRHG)}
One of the main disadvantage of Algorithm~\ref{SHGD_Algorithm} with constant step-size selection is that it guarantees linear convergence only to a neighborhood of the min-max solution $x^*$. As we will present in Section~\ref{sec:analysis}, the decreasing step-size selection allow us to obtain exact convergence to the min-max but at the expense of slower rate (sublinear).
One of the most remarkable algorithmic breakthroughs in recent years was the development of variance-reduced stochastic gradient algorithms for solving finite-sum optimization problems. These algorithms, by reducing the variance of the stochastic gradients, are able to guarantee convergence to the exact solution of the optimization problem with faster convergence than classical SGD. For example, for smooth strongly convex functions, variance reduced methods can guarantee linear convergence to the optimum. This is a vast improvement on the sub-linear convergence of SGD with decreasing step-size. In the past several years, many efficient variance-reduced methods have been proposed. Some popular examples of variance reduced algorithms are SAG \cite{schmidt2017minimizing}, SAGA \cite{defazio2014saga}, SVRG \cite{johnson2013accelerating} and SARAH \cite{nguyen2017sarah}. For more examples of variance reduced methods in different settings, see \citet{defazio2016simple, mS2GD, GowerRichBach2018, sebbouh2019towards}.
In our second method Algorithm~\ref{LSVRHG_Algorithm}, we propose a variance reduced Hamiltonian method for solving~\eqref{MainStochasticProblem}. Our method is inspired by the recently introduced and well behaved variance reduced algorithm, Loopless-SVRG (L-SVRG) first proposed in \citet{hofmann2015variance, kovalev2019don} and further analyzed under different settings in \citet{qian2019svrg, gorbunov2020unified, khaled2020unified}. We name our method loopless stochastic variance reduced Hamiltonian gradient (L-SVRHG). The method works by selecting at each step the unbiased estimator $g^k= \nabla {\cal H}_{i,j}(x^k)-\nabla {\cal H}_{i,j}(w^k)+\nabla {\cal H}(w^k)$ of the full gradient. As we will prove in the next section, this method guarantees linear convergence to the min-max solution of the stochastic bilinear game~\eqref{bilinearGame1}.
\begin{algorithm}[tb]
\caption{Loopless Stochastic Variance Reduced Hamiltonian Gradient (L-SVRHG)}
\label{LSVRHG_Algorithm}
\begin{algorithmic}
\STATE {\bfseries Input:} Starting stepsize $\gamma>0$. Choose initial points $x^0=w^0 \in \mathbb{R}^d$. Distribution ${\cal D}$ of samples. Probability $p \in (0,1]$
\FOR{$k=0,1,2,\cdots, K-1$}
\STATE Generate fresh samples $i \sim {\cal D}$ and $j \sim {\cal D}$ and evaluate $\nabla {\cal H}_{i,j}(x^k)$.
\STATE Evaluate $g^k= \nabla {\cal H}_{i,j}(x^k)-\nabla {\cal H}_{i,j}(w^k)+\nabla {\cal H}(w^k)$.
\STATE Set $x^{k+1}=x^k -\gamma g^k$
\STATE Set $$ w^{k+1} = \begin{cases} x^k \quad \text{with probability} \quad p\\ w^k \quad \text{with probability} \quad 1-p \end{cases}$$
\ENDFOR
\STATE {\bf Output:} \\
Option I: The last iterate $x=x^k$.\\
Option II: $x$ is chosen uniformly at random from $\{x^i\}^K_{i=0}$.
\end{algorithmic}
\end{algorithm}
To get a linearly convergent algorithm in the more general setup of sufficiently bilinear games~\ref{SuffBilinear}, we had to propose a restarted variant of Alg.~\ref{LSVRHG_Algorithm}, presented in Alg.~\ref{PL-LSVRHG_Algorithm}, which calls at each step Alg.~\ref{LSVRHG_Algorithm} with the second option of output, that is L-SVRHG$_{II}$. Using the property from Proposition~\ref{SufficientlyBilinearGameProposition} that the Hamiltonian function~\eqref{StochHamiltonianFunction} satisfy the PL condition~\ref{Polyak}, we show that Alg.~\ref{PL-LSVRHG_Algorithm} converges linearly to the solution of the sufficiently bilinear game (Theorem~\ref{TheoremLSVRHGforPL}).
\begin{algorithm}[tb]
\caption{L-SVRHG (with Restart)}
\label{PL-LSVRHG_Algorithm}
\begin{algorithmic}
\STATE {\bfseries Input:} Starting stepsize $\gamma>0$. Choose initial points $x^0=w^0 \in \mathbb{R}^d$. Distribution ${\cal D}$ of samples. Probability $p \in (0,1]$, $T$
\FOR{$t=0,1,2,\cdots, T$}
\STATE Set $x^{t+1} $ = L-SVRHG$_{II}(x^{t}, K, \gamma, p \in (0,1]$)
\ENDFOR
\STATE {\bf Output:} The last iterate $x^T$.
\end{algorithmic}
\end{algorithm}
\section{Convergence Analysis}
\label{sec:analysis}
\label{ConvergenceAnalysis}
We provide theorems giving the performance of the previously described stochastic Hamiltonian methods for solving the two classes of stochastic smooth games: stochastic bilinear and stochastic sufficiently bilinear. In particular, we present three main theorems for each one of these classes describing the convergence rates for (i) SHGD with constant step-size, (ii) SHGD with decreasing step-size and (iii) L-SVRHG and its restart variant (Algorithm~\ref{PL-LSVRHG_Algorithm}).
The proposed results depend on the two main parameters $\mu_{{\cal H}}$, $L_{{\cal H}}$ evaluated in Propositions~\ref{BilinearGameProposition} and~\ref{SufficientlyBilinearGameProposition}. In addition, the theorems related to the bilinear games (the Hamiltonian function is quasi-strongly convex) use the expected smoothness constant ${\cal L}$~\eqref{eq:expsmooth}, while the theorems related to the sufficiently bilinear games (the Hamiltonian function satisfied the PL condition) use the expected residual constant $\rho$~\eqref{eq:expresidual}.
We note that the expected smoothness and expected residual constants can take several values according to the well-defined distributions ${\cal D}$ selected in our algorithms and the proposed theory will still hold \cite{gower2019sgd, gower2020sgd}.
As a concrete example, in the case of $\tau$-minibatch sampling,\footnote{In each step we draw uniformly at random $\tau$ components of the $n^2$ possible choices of the stochastic Hamiltonian function~\eqref{StochHamiltonianFunction}. For more details on the $\tau$-minibatch sampling see Appendix~\ref{ESandER}.} the expected smoothness and expected residual parameters take the following values:
\begin{gather}
\label{nic1}
{\cal L}(\tau) = \tfrac{n^2(\tau-1)}{\tau(n^2-1)}L_{{\cal H}} + \tfrac{n^2-\tau}{\tau(n^2-1)}L_{\max}\\
\label{nic2}
\rho(\tau)=L_{\max} \tfrac{n^2-\tau}{(n^2-1)\tau}
\end{gather}
where $L_{\max}=\max_{\{1,\dots,n^2\}} \{L_{{\cal H}_{i,j}}\}$ is the maximum smoothness constant of the functions ${\cal H}_{i,j}$.
By using the expressions \eqref{nic1} and \eqref{nic2}, it is easy to see that for single element sampling where $\tau=1$ (the one we use in our experiments) ${\cal L}=\rho=L_{\max}$. On the other limit case where a full-batch is used ($\tau=n^2$), that is we run the deterministic Hamiltonian gradient descent, these values become ${\cal L}=L_{{\cal H}}$ and $\rho=0$ and as we explain below, the proposed theorems include the convergence of the deterministic method as special case.
\subsection{Stochastic Bilinear Games}
We start by presenting the convergence of SHGD with constant step-size and explain how we can also obtain an analysis of the HGD \eqref{DetHamiltonian} as special case. Then we move to the convergence of SHGD with decreasing step-size and the L-SVRHG where we are able to guarantee convergence to a min-max solution $x^*$. In the results related to SHGD we use $\sigma^2 := \mathbb{E}_{i,j}[\norm{\nabla {\cal H}_{i,j}(x^*)}^2]$ to denote the finite gradient noise at the solution.
\begin{theorem}[Constant stepsize]
\label{theo:strcnvlin}
Let us have the stochastic bilinear game \eqref{bilinearGame1}. Then iterates of SHGD with constant step-size $\gamma^k=\gamma \in (0, \frac{1}{2{\cal L}}]$ satisfy:
\begin{equation}\label{eq:convsgd}
\mathbb{E} \| x^k - x^* \|^2 \leq \left( 1 - \gamma \mu_{{\cal H}} \right)^k \| x^0 - x^* \|^2 + \frac{2 \gamma \sigma^2}{\mu}.
\end{equation}
\end{theorem}
That is, Theorem~\ref{theo:strcnvlin} shows linear convergence to a neighborhood of the min-max solution.
Using Theorem~\ref{theo:strcnvlin} and following the approach of~\citet{gower2019sgd}, we can obtain the following corollary on the convergence of deterministic Hamiltonian gradient descent (HGD)~\eqref{DetHamiltonian}. Note that for the deterministic case $\sigma=0$ and ${\cal L}=L$ \eqref{nic1}.
\begin{corollary}
\label{aoskjnda}
Let us have a deterministic bilinear game. Then the iterates of HGD with step-size $\gamma=\frac{1}{2L}$ satisfy:
\begin{equation}
\| x^k - x^* \|^2 \leq \left( 1 - \gamma \mu_{{\cal H}} \right)^k \| x^0 - x^* \|^2
\end{equation}
\end{corollary}
To the best of our knowledge, Corollary~\ref{aoskjnda} provides the first linear convergence guarantees for HGD in terms of $\| x^k - x^* \|^2$ (\citet{abernethy2019last} gave guarantees only on ${\cal H}(x^k)$).
Let us now select a decreasing step-size rule (switching strategy) that guarantees a sublinear convergence to the exact min-max solution for the SHGD.
\begin{theorem}[Decreasing stepsizes/switching strategy]
\label{theo:decreasingstep}
Let us have the stochastic bilinear game \eqref{bilinearGame1}. Let $\mathcal{K} := \left.{\cal L}\right/\mu_{{\cal H}}$. Let
\begin{equation}\label{eq:gammakdef}
\gamma^k=
\begin{cases}
\displaystyle \tfrac{1}{2{\cal L}} & \mbox{for}\quad k \leq 4\lceil\mathcal{K} \rceil \\[0.3cm]
\displaystyle \tfrac{2k+1}{(k+1)^2 \mu_{{\cal H}}} & \mbox{for}\quad k > 4\lceil\mathcal{K} \rceil.
\end{cases}
\end{equation}
If $k \geq 4 \lceil\mathcal{K} \rceil$, then SHGD given in Algorithm~\ref{SHGD_Algorithm} satisfy:
\begin{equation}\label{eq:rateofdecreasing}
\mathbb{E}\| x^{k} - x^*\|^2 \le \tfrac{\sigma^2 }{\mu_{{\cal H}}^2 }\tfrac{8 }{k} + \tfrac{16 \lceil\mathcal{K} \rceil^2}{e^2 k^2 } \|x^0 - x^*\|^2.\end{equation}
\end{theorem}
Lastly, in the following theorem, we show under what selection of step-size L-SVRHG convergences linearly to a min-max solution.
\begin{theorem}[L-SVRHG]
\label{TheoremLSVRHGBilinear}
Let us have the stochastic bilinear game \eqref{bilinearGame1}.
Let step-size $\gamma= 1/6L_{{\cal H}}$ and $p \in (0,1]$. Then L-SVRHG with Option I for output as given in Algorithm~\ref{LSVRHG_Algorithm} convergences linearly to the min-max solution $x^*$ and satisfies:
$$\mathbb{E}[\Phi^k]\leq \max \left\{ 1-\frac{\mu}{6L_{{\cal H}}} , 1-\frac{p}{2} \right\}^k \Phi^0$$
where $\Phi^k :=\|x^k-x^*\|^2+\frac{4\gamma^2}{p n^2}\sum_{i,j=1}^{n}\|\nabla {\cal H}_{i,j}(w^k)-\nabla {\cal H}_{i,j}(x^*)\|^2$.
\end{theorem}
\subsection{Stochastic Sufficiently-Bilinear Games}
\label{sufficiently-bilinear-games}
As in the previous section, we start by presenting the convergence of SHGD with constant step-size and explain how we can obtain an analysis of the HGD \eqref{DetHamiltonian} as special case. Then we move to the convergence of SHGD with decreasing step-size and the L-SVRHG (with restart) where we are able to guarantee linear convergence to a min-max solution $x^*$.
In contrast to the results on bilinear games, the convergence guarantees of the following theorems are given in terms of the Hamiltonian function $\mathbb{E}[{\cal H}(x^{k})]$. In all theorems we call ``sufficiently-bilinear game" the game described in Definition~\ref{SuffBilinear}. With $\sigma^2 := \mathbb{E}_{i,j}[\norm{\nabla {\cal H}_{i,j}(x^*)}^2]$, we denote the finite gradient noise at the solution.
\begin{theorem}
\label{SGDforPolyak}
Let us have a stochastic sufficiently-bilinear game. Then the iterates of SHGD with constant steps-size $\gamma^k=\gamma \leq \frac{\mu}{L (\mu +2\rho)}$ satisfy:
\begin{equation}\label{functionTheorem}
\mathbb{E}[{\cal H}(x^{k})] \leq \left(1- \gamma \mu_{{\cal H}} \right)^k [{\cal H}(x^{0})] + \frac{L_{{\cal H}} \gamma \sigma^2} {\mu_{{\cal H}}}.
\end{equation}
\end{theorem}
Using the above Theorem and by following the approach of \citet{gower2020sgd}, we can obtain the following corollary on the convergence of deterministic Hamiltonian gradient descent (HGD) \eqref{DetHamiltonian}. It shows linear convergence of HGD to the min-max solution. Note that for the deterministic case $\sigma=0$ and $\rho=0$ \eqref{nic2}.
\begin{corollary}
\label{ncaoskla}
Let us have a deterministic sufficiently-bilinear game. Then the iterates of HGD with step-size $\gamma=\frac{1}{L_{{\cal H}}}$ satisfy:
\begin{equation}
{\cal H}(x^{k}) \leq \left( 1 - \gamma \mu_{{\cal H}} \right)^k {\cal H}(x^{0})
\end{equation}
The result of Corollary~\ref{ncaoskla} is equivalent to the convergence of HGD as proposed in \citet{abernethy2019last}.
\end{corollary}
Let us now show that with decreasing step-size (switching strategy), SHGD can converge (with sub-linear rate) to the min-max solution.
\begin{theorem}[Decreasing stepsizes/switching strategy]
\label{theo:decreasingstepPL}
Let us have a stochastic sufficiently-bilinear game. Let $k^* := 2\tfrac{L}{\mu} \left(1+2\tfrac{\rho}{\mu}\right)$ and
\begin{equation}
\gamma^k=
\begin{cases}
\displaystyle \tfrac{\mu_{{\cal H}}}{L_{{\cal H}} (\mu_{{\cal H}} +2\rho)} & \mbox{for}\quad k \leq \lceil k^*\rceil\\[0.3cm]
\displaystyle \tfrac{2k+1}{(k+1)^2 \mu_{{\cal H}}} & \mbox{for} \quad k > \lceil k^* \rceil.
\end{cases}
\end{equation}
If $k \geq \lceil k^* \rceil$, then SHGD given in Algorithm~\ref{SHGD_Algorithm} satisfy:
\begin{equation*}
\mathbb{E}[{\cal H}(x^{k})] \le \tfrac{4 L_{{\cal H}} \sigma^2 }{\mu_{{\cal H}}^2 }\tfrac{1}{k} + \tfrac{( k^*)^2}{k^2 e^2} [{\cal H}(x^{0})] .
\end{equation*}
\end{theorem}
In the next Theorem we show how the updates of L-SVRHG with Restart (Algorithm~\ref{PL-LSVRHG_Algorithm}) converges linearly to the min-max solution. We highlight that each step $t$ of Alg.~\ref{PL-LSVRHG_Algorithm} requires $K=\frac{4}{\mu_{{\cal H}} \gamma}$ updates of the L-SVRHG.
\begin{theorem}[L-SVRHG with Restart]
\label{TheoremLSVRHGforPL}
Let us have a stochastic sufficiently-bilinear game. Let $p \in (0,1]$ and $
\gamma \leq \min \left\{ \frac{1}{4L_{{\cal H}}}, \frac{p^{2/3}}{36^{1/3}(L_{{\cal H}} \rho)^{1/3}},\frac{\sqrt{p}}{\sqrt{6\rho}} \right\}
$
and let $K=\frac{4}{\mu_{{\cal H}} \gamma}$. Then the iterates of L-SVRHG (with Restart) given in Algorithm~\ref{PL-LSVRHG_Algorithm} satisfies
$$\mathbb{E}[{\cal H}(x^{t})] \le \left(1/2\right)^t [{\cal H}(x^{0})].$$
\end{theorem}
\section{Numerical Evaluation}
\label{sec:experiments}
In this section, we compare the algorithms proposed in this paper to existing methods in the literature. Our goal is to illustrate the good convergence properties of the proposed algorithms as well as to explore how these algorithms behave in settings not covered by the theory.
We propose to compare the following algorithms: \textbf{SHGD} with constant step-size and decreasing step-size, a biased version of SHGD \citep{mescheder2017numerics}, \textbf{L-SVRHG} with and without restart, consensus optimization (\textbf{CO})\footnote{\textbf{CO} is a mix between SGDA and SHGD, with the following update rule $x^{k+1}=x^k - \eta_k (\xi_i(x^k) + \lambda \nabla {\cal H}_{i,j}(x^k))$ (See Appendix~\ref{app:CO})} \citep{mescheder2017numerics}, the stochastic variant of \textbf{SGDA}, and finally the stochastic variance-reduced extragradient with restart \textbf{SVRE} proposed in \citep{chavdarova2019reducing}.
For all our experiments, we ran the different algorithms with 10 different seeds and plot the mean and 95\% confidence intervals.
We provide further details about the experiments and choice of hyperparameters for the different methods in Appendix~\ref{app:experiments-details}.
\subsection{Bilinear Games}
\label{exp:bilinear-games}
First we compare the different methods on the stochastic bilinear problem \eqref{bilinearGame1}. Similarly to \citet{chavdarova2019reducing}, we choose $n=d_1=d_2=100$, $[\mathbf{A}_i]_{kl} = 1$ if $i = k = l$ and 0 otherwise, and $[b_i]_k, [c_i]_k \sim \mathcal{N}(0, 1/n)$.
We show the convergence of the different algorithms in Fig.~\ref{fig:bilinear-game}. As predicted by theory, \textbf{SHGD} with decreasing step-size converges at a sublinear rate while \textbf{L-SVRHG} converges at a linear rate. Among all the methods we compared to, \textbf{L-SVRHG} is the fastest to converge; however, the speed of convergence depends a lot on parameter $p$. We observe that setting $p=1/n$ yields the best performance.
To further illustrate the behavior of the Hamiltonian methods,
we look at the trajectory of the methods on a simple 2D version of the bilinear game, where we choose $x_1$ and $x_2$ to be scalars. We observe that while previously proposed methods such as \textbf{SGDA} and \textbf{SVRE} suffer from rotations which slow down their convergence and can even make them diverge, the Hamiltonian methods converge much faster by removing rotation and converging ``straight" to the solution.
\subsection{Sufficiently-Bilinear Games}
\label{exp:nonlinear-games}
In section~\ref{sufficiently-bilinear-games}, we showed that Hamiltonian methods are also guaranteed to converge when the problem is non-convex non-concave but satisfies the sufficiently-bilinear condition~\eqref{SufficientBilinear}.
To illustrate these results, we propose to look at the
following game inspired by \citet{abernethy2019last}:
\begin{multline}
\label{eq:exp-sufficiently-bilinear}
\min_{x_1 \in\mathbb{R}^{d}}\max_{x_2 \in\mathbb{R}^{d}}\frac{1}{n} \sum_{i=1}^n \big( F(x_1) + \delta \,\, x_1^\top \mathbf{A}_i x_2 \, + \\
b_i^\top x_1 + c_i^\top x_2 - F(x_2) \big) ,
\end{multline}
where $F(x)$ is a non-linear function (see details in Appendix~\ref{app:nonlinear-games}).
This game is non-convex non-concave and satisfies the sufficiently-bilinear condition if $\delta > 2L$, where $L$ is the smoothness of $F(x)$. Thus, the results and theorems from Section~\ref{sufficiently-bilinear-games} hold.
Results are shown in Fig.\ref{fig:nonlinear-game}. Similarly to the bilinear case, the methods follow very closely the theory.
We highlight that while the proposed theory for this setting only guarantees
convergence for \textbf{L-SVRHG} with restart, in practice using restart is not strictly necessary: \textbf{L-SVRHG} with the correct choice of stepsize also converges in our experiment.
Finally we show the trajectories of the different methods on a 2D version of the problem. We observe that contrary to the bilinear case, stochastic SGDA converges but still suffers from rotation compared to Hamiltonian methods.
\begin{figure*}[h]
\captionsetup[subfigure]{justification=centering}
\begin{subfigure}[t]{0.5\textwidth}
\begin{center}
\hspace{-5mm}
\includegraphics[width=.53\columnwidth]{bilinear_game.pdf}
\hspace{-5mm}
\includegraphics[width=.53\columnwidth]{bilinear2d.pdf}
\caption{Bilinear game}
\label{fig:bilinear-game}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.5\textwidth}
\begin{center}
\hspace{-4mm}
\includegraphics[width=.53\columnwidth]{nonlinear_game.pdf}
\hspace{-5mm}
\includegraphics[width=.53\columnwidth]{non-linear2d.pdf}
\caption{Sufficiently-bilinear game}
\label{fig:nonlinear-game}
\end{center}
\end{subfigure}
\vspace{-2mm}
\caption{\textbf{a)} Comparison of different methods on the stochastic bilinear game~\eqref{bilinearGame1}. Left: Distance to optimality $\frac{||x_k - x^*||^2}{||x_0 - x^*||^2}$ as a function of the number of samples seen during training. Right: The trajectory of the different methods on a 2D version of the problem.\\
\textbf{b)} Comparison of different methods on the sufficiently bilinear games~\eqref{eq:exp-sufficiently-bilinear}. Left: The Hamiltonian $\frac{H(x_k)}{H(x_0)}$ as a function of the number of samples seen during training. Right: The trajectory of the different methods on a 2D version of the problem.}
\vspace{-3mm}
\end{figure*}
\subsection{GANs}
\label{exp:GANs}
In previous experiments, we verify the proposed theory for the stochastic bilinear and sufficiently-bilinear games. Although we do not have theoretical results for more complex games, we wanted to test our algorithms on a simple GAN setting, which we call \em GaussianGAN\em.
In \em GaussianGAN\em, we have a dataset of real data $x_{real}$ and latent variable $z$ from a normal distribution with mean 0 and standard deviation 1. The generator is defined as $G(z)=\mu + \sigma z$ and the discriminator as $D(x_{data})=\phi_0 + \phi_1 x_{data} + \phi_2 x_{data}^2$, where $x_{data}$ is either real data ($x_{real}$) or fake generated data ($G(z)$). In this setting, the parameters are $x=(x_1,x_2)=([\mu, \sigma], [\phi_0,\phi_1,\phi_2])$.
In GaussianGAN, we can directly measure the $L^2$ distance between the generator's parameters and the true optimal parameters: $||\hat{\mu}-\mu|| + || \hat{\sigma}-\sigma ||$,
where $\hat{\mu}$ and $\hat{\sigma}$ are the sample's mean and standard deviation.
We consider three possible minmax games: Wasserstein GAN (WGAN) \citep{arjovsky2017wasserstein}, saturating GAN (satGAN) \citep{goodfellow2014generative}, and non-saturating GAN (nsGAN) \citep{goodfellow2014generative}. We present the results for WGAN and satGAN in Figure \ref{figgan1}. We provide the nsGAN results in Appendix~\ref{app:gans-other} and details for the different experiments in Appendix~\ref{app:gans}.
\begin{figure}[h]
\captionsetup[subfigure]{justification=centering}
\begin{subfigure}[t]{.23\textwidth}
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/GAN_fixed/wgan_Ham_batch100.pdf}}
\caption{Hamiltonian for WGAN}
\end{center}
\vspace{-6mm}
\end{subfigure}
\begin{subfigure}[t]{.23\textwidth}
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/GAN_fixed/wgan_batch100.pdf}}
\caption{Distance to optimum for WGAN}
\end{center}
\vspace{-6mm}
\end{subfigure}\\
\begin{subfigure}[t]{.23\textwidth}
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/GAN_fixed/satgan_Ham_batch100.pdf}}
\caption{Hamiltonian for satGAN}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{.23\textwidth}
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/GAN_fixed/satgan_batch100.pdf}}
\caption{Distance to optimum for satGAN}
\end{center}
\end{subfigure}
\vspace{-4mm}
\caption{The Hamiltonian $\frac{H(x_k)}{H(x_0)}$ (\textbf{left}) and the distance to the optimal generator (\textbf{right}) as a function of the number of samples seen during training for WGAN and satGAN. The distance to the optimal generator corresponds to $\frac{||\hat{\mu}-\mu_k|| + ||\hat{\sigma}-\sigma_k||}{||\hat{\mu}-\mu_0|| + ||\hat{\sigma}-\sigma_0||}$.}
\label{figgan1}
\end{figure}
For WGAN, we see that stochastic \textbf{SGDA} fails to converge and that \textbf{L-SVRHG} is the only method to converge linearly on the Hamiltonian.
For satGAN, \textbf{SGDA} seems to perform best. Algorithms that take into account the Hamiltonian have high variance. We looked at individual runs and found that, in 3 out of 10 runs, the algorithms other than stochastic \textbf{SGDA} fail to converge, and the Hamiltonian does not significantly decrease over time. While WGAN is guaranteed to have a unique critical point, which is the solution of the game, this is not the case for satGAN and nsGAN due to the non-linear component. Thus, as expected, Assumption~\ref{criticalminmax} is very important in order for the proposed stochastic Hamiltonian methods to perform well.
\section{Conclusion and Extensions}
We introduce new variants of SHGD (through novel unbiased estimator and step-size selection) and present the first variance reduced Hamiltonian method L-SVRHG. Using tools from optimization literature, we provide convergence guarantees for the two methods and we show how they can efficiently solve stochastic unconstrained bilinear games and the more general class of games that satisfy the ``sufficiently bilinear” condition. An important result of our analysis is the first set of global non-asymptotic last-iterate convergence guarantees for a stochastic game over a non-compact domain, in the absence of strong monotonicity assumptions.
We believe that our results and the Hamiltonian viewpoint could work as a first step in closing the gap between the stochastic optimization algorithms and methods for solving stochastic games and can open up many avenues for further development and research in both areas.
A natural extension of our results will be the proposal of accelerated Hamiltonian methods that use momentum \cite{loizou2017momentum, assran2020convergence} on top of the Hamiltonian gradient update. We speculate that similar ideas to the ones presented in this work can be used for the development of efficient decentralized methods \cite{assran2018stochastic,koloskova2020unified} for solving problem~\eqref{MainStochasticProblem}.
\newpage
\section*{Acknowledgements}
The authors would like to thank Reyhane Askari, Gauthier Gidel and Lewis Liu for useful discussions and feedback.
Nicolas Loizou acknowledges support by the IVADO postdoctoral funding program.
This work was partially supported by the FRQNT new researcher program (2019-NC-257943), the NSERC Discovery grants (RGPIN-2017-06936 and RGPIN-2019-06512) and the Canada CIFAR AI chairs program.
Ioannis Mitliagkas acknowledges support by an IVADO startup grant and a Microsoft Research collaborative grant.
Simon Lacoste-Julien acknowledges support by a Google Focused Research award.
Simon Lacoste-Julien and Pascal Vincent are CIFAR Associate Fellows in the Learning in Machines \& Brains program.
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1011.6448
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\section*{THE PROBLEM}
Let us first explain our problem more formally. Consider two dits $y_0$ and $y_1 \in \{0,\ldots,d-1\}$, where the string $y = y_0y_1$ plays the role of the whole, and $y_0$, $y_1$ are the individual parts. Let $\rho_y$ denote an encoding of the string $y$ into a classical or quantum state. In quantum theory,
$\rho_y$ is simply a density operator, and in a NC-HV model it is a preparation $\ensuremath{\mathcal{P}}_y$ described by a probability distribution over hidden variables $\lambda \in \Lambda$. Let $P_Y$ be a probability distribution over $\{0,\ldots,d-1\}^{2}$, and imagine that with probability $P_Y(y)$ we are given the state $\rho_y$.
The optimum probability of guessing $y$ given its encoding $\rho_y$, which lies in a register $E$, can be written as
\begin{align}\label{eq:guessingProb}
P_{\rm guess}(Y|E) = \max_{\{\ensuremath{\mathcal{M}}\}} \sum_{y \in \{0,\ldots,d-1\}^{ 2}} P_{Y}(y)\, p(y|\ensuremath{\mathcal{M}},\ensuremath{\mathcal{P}}_y)\ ,
\end{align}
where
$p(y|\ensuremath{\mathcal{M}},\ensuremath{\mathcal{P}}_y)$ is the probability of obtaining outcome $y$ when measuring the preparation $\ensuremath{\mathcal{P}}_y$ with $\ensuremath{\mathcal{M}}$,
and the maximization is taken over all $d^2$-outcome measurements allowed in the theory.
In the case of quantum theory, for example,
the maximization is taken over POVMs $\ensuremath{\mathcal{M}} = \{M_y\}_y$ and $p(y|\ensuremath{\mathcal{M}},\ensuremath{\mathcal{P}}_y) = \mathop{\mathrm{tr}}\nolimits(M_y \rho_y)$.
The guessing probability is directly related to the conditional min-entropy
${\ensuremath{{\rm H}}_{\infty}}(Y|E)$ through the equation~\cite{krs:entropy}
\begin{align}
{\ensuremath{{\rm H}}_{\infty}}(Y|E) := - \log P_{\rm guess}(Y|E)\ .
\end{align}
This measure plays an important role in quantum cryptography and is the relevant measure of information
in the single shot setting corresponding to our everyday experience, as opposed to the asymptotic setting captured by the von Neumann entropy.
A closely related variant is the smooth min-entropy ${\ensuremath{{\rm H}}_{\infty}^\eps}(Y|E)$ which can be thought of as being like ${\ensuremath{{\rm H}}_{\infty}}(Y|E)$ except with some
small error probability $\varepsilon$.
The main question we are interested in can then be loosely phrased as:
\begin{quote}
How does ${\ensuremath{{\rm H}}_{\infty}}(Y=Y_0Y_1|E)$ (ignorance about the whole)
relate to ${\ensuremath{{\rm H}}_{\infty}}(Y_C|EC)$, for $C \in \{0,1\}$ (ignorance about the parts)?
\end{quote}
Here the introduction of the additional random variable $C$ is crucial, and it can be understood as a pointer to the part of $Y$ about which there is large ignorance (given a large ignorance of the whole string $Y$); see Figure~\ref{fig:game} for an illustration of this role.
It is important to note that the choice of $C$ should be consistent with the encoding prior to its definition. That is, whereas $C$
may of course depend on $Y_0,Y_1$ and the encoding $E$, the reduced state on registers holding $Y_0,Y_1$ and $E$ after tracing out $C$ should remain the same.
In particular, this condition states that $C$ cannot be the result of a measurement causing disturbance to the encoding register; if we were allowed to destroy information
in the encoding we would effectively alter the original situation.
\section*{RESULTS}
{\bf An inequality valid in any NC-HV model.}
We first show that classically, or more generally in any non-contextual hidden variable model~\cite{endnote23},
ignorance about the whole really \emph{does} imply ignorance about a part.
More specifically, we show that for any random variable $Y=Y_0Y_1$ and side information $E$, there exists a random variable $C \in \{0,1\}$ such
that
\begin{align}\label{eq:classicalSplitting}
{\ensuremath{{\rm H}}_{\infty}}(Y_C|EC) \gtrsim\frac{{\ensuremath{{\rm H}}_{\infty}}(Y_0Y_1|E)}{2}\ .
\end{align}
This inequality can be understood as an information-theoretic analogue of Bell inequalities to the question of non-contextuality.
Classically, this inequality is known as the \emph{min-entropy splitting inequality}, and
plays an important role in the proof of security of some (classical) cryptographic primitives~\cite{juerg:splitting, serge:new}.
The proof of~\eqref{eq:classicalSplitting} is a straightforward extension to the case of standard
NC-HV models~\cite{klyachko:nc,andreas:nc} of a
classical technique known as min-entropy splitting first introduced by Wullschleger~\cite{juerg:splitting}, and we defer details to the appendix.
The fact that $C$ is a random variable, rather than being deterministically chosen, is important, and an example will help clarify its role. Consider $Y$ uniformly distributed over $\{0,\ldots,d-1\}^{2}$ and $E = Y_0$ with probability $1/2$, and $Y_1$ with probability $1/2$. In this case it is easy to see that \emph{both} $Y_0$ and $Y_1$ can be guessed from $E$ with average success probability $1/2+1/(2d)$, so that ${\ensuremath{{\rm H}}_{\infty}}(Y_0|E) = {\ensuremath{{\rm H}}_{\infty}}(Y_1|E) \approx 1$, which is much less than ${\ensuremath{{\rm H}}_{\infty}}(Y|E)\approx \log d$. However, define $C$ as $0$ if $E=Y_1$ and $1$ if $E=Y_0$. Then it is clear that ${\ensuremath{{\rm H}}_{\infty}}(Y_C|EC) = \log d$, as we are always asked to predict the variable about which we have no side information at all! In this case the random variable $C$ ``points to the unknown'' by being correlated with the side information $E$, but is entirely consistent with our knowledge about the world: by tracing out $C$ we recover the initial joint distribution on $(Y,E)$. This also highlights the important difference between the task we are considering and the well-studied random access codes~\cite{nayak:rac,nayak:original}, in which the requirement is to be able to predict one of $Y_0,Y_1$ (adversarially chosen) from their encoding; for this task it has been demonstrated that there is virtually no asymptotic difference between classical and quantum encodings (see below for a discussion).
It is interesting to note that~\eqref{eq:classicalSplitting} still holds if we consider a somewhat ``helpful'' physical model
in which in addition to the encoding one might learn a small number of ``leaked'' bits of information about $Y$. More specifically, if the NC-HV discloses $m$ extra bits of information
then it follows from the chain rule for the min-entropy (see appendix) that
\begin{align}
{\ensuremath{{\rm H}}_{\infty}}(Y_C|EC) \gtrsim\frac{{\ensuremath{{\rm H}}_{\infty}}(Y_0Y_1|E)}{2} - m\ .
\end{align}
{\bf Violation in quantum theory.} Our main result shows that~\eqref{eq:classicalSplitting} is violated in the strongest possible sense by quantum theory. More specifically, we provide an explicit construction that demonstrates this violation:
Let $Y = Y_0Y_1$ be uniformly distributed over $\{0,\ldots,d-1\}^{2}$. Given $y=y_0y_1\in\{0,\ldots,d-1\}^2$, define its encoding $\rho_{y_0y_1}^E = \proj{\Psi_y}$ as
\begin{align}\label{eq:encoding}
\ket{\Psi_y} := X^{y_0}_d Z^{y_1}_d \ket{\Psi}\ ,
\end{align}
where $X_d$ and $Z_d$ are the generalized Pauli matrices and
\begin{align}
\ket{\Psi} :=
\frac{1}{\sqrt{2\left(1+\frac{1}{\sqrt{d}}\right)}}
(\ket{0} + F\ket{0})\ ,
\end{align}
with $F$ being the matrix of the Fourier transform over $\mathbb{Z}_d$. Since we are only interested in showing a quantum violation,
we will for simplicity always assume that $d$ is prime~\cite{endnoteAdd}. The system $YE$ is then described by the ccq-state
\begin{align}
\rho_{Y_0Y_1E} = \frac{1}{d^2} \sum_{y_0,y_1} \proj{y_0} \otimes \proj{y_1} \otimes \rho_{y_0y_1}^E\ .
\end{align}
We first prove that ${\ensuremath{{\rm H}}_{\infty}}(Y|E) = \log d$ for our choice of encoding.
We then show the striking fact that, even though the encoding we defined gives very little information about the whole string $Y$, for any adversarially chosen random variable $C$ (possibly correlated with our encoding) one can guess $Y_C$ from its encoding $\rho_E$ with essentially
constant probability.
More precisely, for any ccqc-state $\rho_{Y_0Y_1EC}$, with $C \in \{0,1\}$, that satisfies the consistency relation $\mathop{\mathrm{tr}}\nolimits_C(\rho_{Y_0Y_1EC}) = \rho_{Y_0Y_1E}$, we have
\begin{align}
{\ensuremath{{\rm H}}_{\infty}}(Y_C|EC) \approx 1
\end{align}
for \emph{any} sufficiently large $d$.
This shows that the inequality~\eqref{eq:classicalSplitting} can be violated arbitrarily (with $d$), giving a striking example of the malleability
of quantum information.
What's more, it is not hard to show that this effect still holds even for ${\ensuremath{{\rm H}}_{\infty}^\eps}$, for constant error $\varepsilon$,
and a ``helpful'' physical model leaking $m \approx c \log d$ bits of information with $c < 1/2$.
Hence, the violation of
the inequality~\eqref{eq:classicalSplitting} has the appealing feature of being very robust. Indeed, for any number of bits $m$ a NC-HV might leak in
addition, we could find a $d$ to ensure a violation.
\begin{center}
\begin{figure}
\includegraphics[scale=0.5]{gameDoodle.png}
\caption{{\scriptsize Intuitively, one can also understand our result in terms of a game between Bob and a malicious challenger, the Owl. Imagine Bob is taking a philosophy class teaching him knowledge about $Y$, clearly chosen uniformly at random. Unfortunately, he never actually attended and had insufficient time to prepare for his exam. Luckily, however, he has been given some encoding $E$ of the possible answers $Y_0Y_1$, hastily prepared by his old friend Alice. When entering the room, he had to submit $E$ for inspection to the challenger who knows $Y_0$, $Y_1$ as well as the encoding Alice might use. After inspection, the challenger may secretly keep a
system $C$, possibly correlated with $E$, but such that the reduced system on $Y_0$, $Y_1$ and $E$ looks untampered with.
It is immediately obvious to the challenger that Bob must be ignorant about the whole of $Y_0Y_1$. But can it always measure and point to a $C=c$
such that Bob is ignorant about $Y_C$? That is, can it always detect Bob's ignorance by challenging him to output a single $Y_C$?
Classically, this is indeed possible: ignorance about the whole of $Y_0Y_1$ implies significant ignorance about one of the parts, $Y_C$.
However, a quantum Bob could beat the Owl.}}
\label{fig:game}
\end{figure}
\end{center}
\section*{PROOF OF THE QUANTUM VIOLATION}
We now provide an outline of the proof that the encoding specified in~\eqref{eq:encoding} leads to a quantum violation of the splitting
inequality~\eqref{eq:classicalSplitting}; for completeness, we provide a more detailed derivation in the appendix.
Our proof proceeds in three steps: first, by computing ${\ensuremath{{\rm H}}_{\infty}}(Y|E)$ we show that the encoding does indeed not reveal much information about the whole.
Second, we compute the optimal measurements for extracting $Y_0$ and $Y_1$ on average, and show that these measurements perform equally well
for any other prior distribution on $Y$.
Finally, we show that even introducing an additional system $C$ does not change one's ability to extract $Y_C$ from the encoding.
\smallskip
\noindent
{\bf Step 1: }
Very intuitively, ignorance about the whole string already follows from Holevo's theorem and the fact
that we are trying to encode 2 dits into a $d$-dimensional quantum system.
To see this more explicitly, recall that ${\ensuremath{{\rm H}}_{\infty}}(Y|E) = \log d$ is equivalent to showing that
$P_{\rm guess}(Y|E) = 1/d$.
From~\eqref{eq:guessingProb} we have that this guessing probability is given by the solution to the following semidefinite program (SDP)
\begin{sdp}{maximize}{$\frac{1}{d^2} \sum_{y_0,y_1} \mathop{\mathrm{tr}}\nolimits\left(M_{y_0y_1}\proj{\Psi_{y_0y_1}}\right)$}
& $M_{y_0y_1} \geq 0 \mbox{ for all } y_0,y_1$\ ,\\
& $\sum_{y_0,y_1} M_{y_0,y_1} = \mathbb{I}$\ .
\end{sdp}
The dual SDP is easily found to be
\begin{sdp}{minimize}{$\mathop{\mathrm{Tr}}\nolimits(Q)$}
& $Q \geq \frac{1}{d^2} \proj{\Psi_{y_0y_1}} \mbox{ for all } y_0,y_1$\ .
\end{sdp}
Let $v_{\rm primal}$ and $v_{\rm dual}$ be the optimal values of the primal and dual respectively. By the property of weak duality, $v_{\rm dual} \geq v_{\rm primal}$ always holds. Hence, to prove our result, we only need to find a primal and dual solutions for which $v_{\rm primal} = v_{\rm dual} = 1/d$.
It is easy to check that $\hat{Q} = \mathbb{I}/d^2$ is a dual
solution with value $v_{\rm dual} = \mathop{\mathrm{tr}}\nolimits(\hat{Q}) = 1/d$.
Similarly, consider the measurement $M_{y_0y_1} = \proj{\Psi_{y_0y_1}}/d$. Using Schur's lemma, one can directly verify that
$\sum_{y_0,y_1} M_{y_0y_1} = \mathbb{I}$, giving $v_{\rm primal} = 1/d$. The claimed value of the conditional min-entropy follows.
\smallskip
\noindent
{\bf Step 2: }
A similar argument, exploiting the symmetries in the encoding, can be used to show that
\begin{align}\label{eq:racRecover}
P_{\rm guess}(Y_0|E) = P_{\rm guess}(Y_1|E) = \frac{1}{2} + \frac{1}{2\sqrt{d}}\ .
\end{align}
The measurements that attain these values are given by the eigenbases of $Z_d$ and $X_d$ respectively.
As a remark to quantum information theorists, note that this means that our encoding doubles
as a random access encoding of the string $y$ into a $d$-dimensional quantum state $\rho_y$ with
probability~\eqref{eq:racRecover} to recover $y_0$ or $y_1$.
For $d=2$, such encodings have previously been considered in the realm of contextuality
as a reinterpretation of the CHSH inequality~\cite{chsh,Spekkens2009}.
However, we note that this is \emph{not} what is surprising here, as
there exists an obvious classical random access encoding for 2 dits into a single dit (see discussion on $C$ above), with recovery probability
$1/2 + 1/(2d)$.
Simply computing~\eqref{eq:racRecover} is hence insufficient for our purposes. Let us write $\{\ket{y_0},\,y_0 \in \{0,\ldots,d-1\}\}$ for
the eigenbasis of $Z_d$, and note that its Fourier transform $\{F\ket{y_1},\,y_1 \in \{0,\ldots,d-1\}\}$ is then the eigenbasis of $X_d$.
Exploiting the symmetries in our problem, it is straightforward to verify that for all $y_0,y_1 \in \{0,\ldots,d-1\}$
\begin{align}
|\langle y_0|\Psi_{y_0y_1}\rangle|^2 =
|\langle y_1|F^\dagger|\Psi_{y_0y_1}\rangle|^2 =
\frac{1}{2} + \frac{1}{2 \sqrt{d}}\ .
\end{align}
An important consequence of this is that for \emph{any} other prior distribution $P_{y_0y_1}$,
measurement in the $Z_d$ eigenbasis
distinguishes the states
\begin{align}
\sigma_{y_0} = \sum_{y_1} P_{y_0y_1}(y_0,y_1) \proj{\Psi_{y_0y_1}}\ ,
\end{align}
with probability at least $1/2 + 1/(2\sqrt{d})$, even when the distribution is unknown.
A similar argument can be made for the marginal states $\sigma_{y_1}$ and measurement in the $X_d$ eigenbasis.
\smallskip
\noindent
{\bf Step 3: }
It now remains to show that, for any possible choice of an additional classical system
$C$~\cite{endnote25},
one can still guess $Y_C$ from the encoding with a good success probability: one cannot construct a $C$ which would ``point to the unknown''.
Note that we may express the joint state with any other system $C$ as
\begin{align}
\rho_{Y_0Y_1 E C} = \frac{1}{d^2} \sum_{y_0 y_1} \proj{y_0} \otimes \proj{y_1} \otimes \rho_{y_0y_1c}^{EC}\ ,
\end{align}
for some states $\rho_{y_0y_1c}^{EC}$ on registers $E$ and $C$.
Since the reduced state on $Y_0$,$Y_1$ and $E$ should be the same for any $C$
we have by the fact that $Y_0$ and $Y_1$ are classical that $tr_{C}(\rho_{y_0y_1c}^{EC}) = \proj{\Psi_{y_0y_1}}$. Since $\proj{\Psi_{y_0y_1}}$ is a pure state, this implies
that $\rho_{y_0y_1c}^{EC} = \proj{\Psi_{y_0y_1}} \otimes \sigma_{y_0 y_1}^C$.
Now imagine that we were to perform some arbitrary measurement on $C$, whose outcome would supposedly point to an unknown substring. But this merely creates a different distribution $P_{y_0y_1}$
over encoded strings, and we already know from the above that we can still succeed in retrieving either $y_0$ or $y_1$
with probability at least $1/2 + 1/(2\sqrt{d})$ by making a measurement in the $X_d$ or $Z_d$ basis respectively.
Hence for large $d$ we have a recovery probability of roughly $1/2$, implying
\begin{align}
{\ensuremath{{\rm H}}_{\infty}}(Y_0|EC=0) \approx {\ensuremath{{\rm H}}_{\infty}}(Y_1|EC=1) \approx 1\ ,
\end{align}
which is our main claim.
Note that the consistency condition, which states that our choice of $C$ should be compatible
with the original situation and should not affect the reduced state, is important, and
makes our task non-trivial. As an example, consider our construction for $d=2$. In that case
the encoding states lie in the XZ-plane of the Bloch sphere.
Imagine now that we measured the encoding register $E$ in the eigenbasis of $\sigma_y$, and let the outcome be $C$. But for
any measurement in the eigenbasis of $\sigma_y$ we observe entirely random outcomes, and the post-measurement states trivially no longer carry any information about the
encoded string. Indeed, any choice of $C$ would do if we are allowed to destroy information in such a manner.
\section*{IMPLICATIONS FOR CRYPTOGRAPHY}
Our result answers an interesting open question in quantum cryptography~\cite{chris:talk},
namely whether min-entropy splitting can still be performed
when conditioned on quantum instead of classical knowledge. This technique was used to deal with classical side information $E$
in~\cite{serge:bounded,serge:new}.
Our example shows that quantum min-entropy splitting is impossible, even when we would
be willing to accept subtracting a large error term on the r.h.s. of~\eqref{eq:classicalSplitting}.
This tells us that classical protocols that rely on such statements may become insecure in the presence of quantum side information,
and highlights the importance of so-called min-entropy sampling results of~\cite{kr:sampling}
used in quantum cryptography~\cite{KoeWehWul09} instead. It also indicates that contextuality may play a more important role in our understanding
of the possibilities and limits of quantum cryptography than previously thought.
\section*{DISCUSSION}
The first indication that something may be amiss when looking at knowledge from a quantum perspective was given by Schr{\"o}dinger~\cite{schroedinger:eprGerman}, who
pointed out that one can have \emph{knowledge} (not ignorance) about the whole, while still being ignorant about the parts~\cite{endnote26}.
Here, we tackled this problem from
a very different direction, starting with the premise that one has ignorance about the whole.
Our results show that contextuality is responsible for much more significant effects than have previously been noted.
In particular, it leads to arbitrarily large quantum violations of~\eqref{eq:classicalSplitting}, which can be understood
as a Bell-type inequality for non-contextuality. This is still true even for a somewhat ``helpful'' physical model, leaking
additional bits of information. To our knowledge, this is the first \emph{information-theoretic} inequality distinguishing
NC-HV models from quantum theory. Our question and perspective are completely novel, and we hope that our observations will lead to an increased understanding
of the role of contextuality. In this work, we have considered standard NC-HVs in which all HVs can be decomposed as convex combinations of extremal HVs which give deterministic outcomes for effects (see appendix). It is an interesting open question whether our results can be generalized to very general models that distinguish between measurement and preparation contextuality~\cite{Spekkens2005}.
At the heart of our result lies the fact that contextuality allows for strong forms of complementarity in quantum mechanics (often conflated with
uncertainty~\cite{js:urvsnl}),
which intuitively is responsible for allowing the violation of~\eqref{eq:classicalSplitting}.
Typically, complementarity is discussed by considering examples of properties of a physical system that one may be able to determine individually, but which cannot all be learned at once. In spirit, this is similar to the notion of a random access encoding where we could determine either property
$Y_0$ or $Y_1$ quite well, but not all of $Y$. However, as discussed above this can also be true classically, in a probabilistic sense.
We would thus like to emphasize the novelty of our perspective, as we approach the problem from the other end, and first demonstrate
the general result that in an NC-HV ignorance about the whole always implies ignorance about a part. We then show that
in a quantum world, this principle is violated in the strongest possible sense, even with respect to an additional system $C$.
One could think of this as a much more robust way of capturing the intuitive notion of complementarity~\cite{js:inprep}.
Finally, it is an interesting open question whether our inequality can be experimentally verified. Note that this made difficult by the fact that our aim would be to test \emph{ignorance} rather than knowledge. However, it is conceivable that such an experiment can be performed by building a larger cryptographic protocol whose security relies on being ignorant about one of the parts of a string $Y$ created during that protocol~\cite{endnote27}.
A quantum violation could then be observed by breaking the security of the protocol, and exhibiting \emph{knowledge} (rather than ignorance) about some information that could not have been obtained if the protocol was secure.
\acknowledgments
We thank Jonathan Oppenheim, Christian Schaffner, Tony Short, Robert Spekkens and CQT's ''non-local club'' for useful comments.
We are particularly grateful to Tony Short for pointing out that our problem could more easily
be explained by means of the game depicted in Figure~\ref{fig:game}.
TV was supported by ARO Grant W911NF-09-1-0440 and NSF Grant CCF-0905626.
SW was supported by the National Research Foundation, and the Ministry of Education, Singapore.
TV is grateful to CQT, Singapore, for hosting him while part of this work was done. SW is grateful for an invitation from the Mittag-Leffler Institute, Sweden, where part of this work was performed.
|
1011.6198
|
\section{The first result: the nonlinear integral equation connected with the Bessel's functions}
\subsection{}
In this paper we obtain some new properties of the signal
\bdis
Z(t)=e^{i\vartheta(t)}\zf
\edis
that is generated by the Riemann zeta-function, where
\bdis
\vartheta(t)=-\frac t2\ln\pi+\text{Im}\ln\Gamma\left(\frac 14+i\frac t2\right)=\frac t2\ln\frac{t}{2\pi}-\frac t2-\frac{\pi}{8}+
\mcal{O}\left(\frac{1}{t}\right) .
\edis
Let us remind that
\bdis
\tilde{Z}^2(t)=\frac{{\rm d}\vp_1(t)}{{\rm d}t},\ \vp_1(t)=\frac 12\vp(t)
\edis
where
\be \label{1.1}
\tilde{Z}^2(t)=\frac{Z^2(t)}{2\Phi^\prime_\vp[\vp(t)]}=\frac{\left|\zf\right|^2}{\left\{ 1+\mcal{O}\left(\frac{\ln\ln t}{\ln t}\right)\right\}\ln t}
\ee
(see \cite{1}, (3.9); \cite{2}, (1.3); \cite{7}, (1.1), (3.1), (3.2)), and $\vp(t)$ is the Jacob's ladder, i.e. a solution of the nonlinear integral
equation (see \cite{1})
\bdis
\int_0^{\mu[x(T)]}Z^2(t)e^{-\frac{2}{x(T)}t}{\rm d}t=\int_0^TZ^2(t){\rm d}t .
\edis
\subsection{}
The Gram's sequence $\{ t_\nu\}$ is defined by the equation
\bdis
\vartheta(t_\nu)=\pi\nu,\ \nu=1,2,\dots
\edis
where (see \cite{20}, p. 102)
\be \label{1.2}
t_{\nu+1}-t_\nu=\frac{2\pi}{\ln t_\nu}+\frac{2\pi\ln 2\pi}{\ln^2 t_\nu}+\mcal{O}\left(\frac{1}{\ln^3t_\nu}\right) .
\ee
The following theorem holds true.
\begin{theorem}
Every Jacob's ladder $\vp_1(t)=\frac{1}{2}\vp(t)$ where $\vp(t)$ is the exact solution to the nonlinear integral equation
\bdis
\int_0^{\mu[x(T)]}Z^2(t)e^{-\frac{2}{x(T)}t}{\rm d}t=\int_0^TZ^2(t){\rm d}t
\edis
is the asymptotic solution of the following nonlinear integral equation
\be \label{1.3}
\int_{x^{-1}(t_\nu)}^{x^{-1}(t_{\nu+1})}J_1[x(t)]\left|\zf\right|^2{\rm d}t=\frac{2\sqrt{2\pi}}{\sqrt{t_\nu}}\sin\left(t_\nu-\frac{\pi}{4}\right)
\ee
for every sufficiently big $t_\nu$ that fulfils the conditions
\be \label{1.4}
\begin{split}
& [t_\nu,t_{\nu+1}]\subset [\mu_n^{(1)},\mu_{n+1}^{(1)}], \\
& [t_\nu,t_{\nu+1}]\cap [k\pi-\epsilon,k\pi+\epsilon]=\emptyset,\ \nu,k\in\mbb{N},\ \nu\to\infty
\end{split}
\ee
where $J_1(\mu_n^{(1)})=0,\ n=1,2,\dots $, i.e. the following asymptotic formula
\be \label{1.5}
\int_{\vp_1^{-1}(t_\nu)}^{\vp_1^{-1}(t_{\nu+1})}J_1[\vp_1(t)]\left|\zf\right|^2{\rm d}t\sim\frac{2\sqrt{2\pi}}{\sqrt{t_\nu}}
\sin\left(t_\nu-\frac{\pi}{4}\right)
\ee
holds true.
\end{theorem}
\begin{remark}
Since
\be \label{1.6}
\mu_{n+1}^{(1)}-\mu_n^{(1)}\sim\pi,\ n\to\infty
\ee
then the number $N_{\nu,n}$ of the intervals $[t_\nu,t_{\nu+1}]$, for which the first condition in (\ref{1.4}) is fulfilled, is given by
the asymptotic formula
\bdis
N_{\nu,n}\sim\frac 12\ln t_\nu,\ t_\nu\to\infty ,
\edis
((\ref{1.2}), (\ref{1.6})).
\end{remark}
This paper is a continuation of the series \cite{1} - \cite{19}.
\section{The second result: some nonlinear integral equation connected with the function $|\zf|^4$}
Let us remind that the Jacob's ladder $\vp_2(T)$ of the second order is a solution to the nonlinear integral equation
\be \label{2.1}
\int_0^{\mu[x(T)]} Z^4(t)e^{-\frac{t}{x(T)}}{\rm d}t=\int_0^T Z^4(t){\rm d}t
\ee
(see \cite{8}). In this case the following asymptotic formula (see \cite{8}, (1.5))
\be \label{2.2}
\begin{split}
& \int_{\vp_1^{-1}(T)}^{\vp_1^{-1}(T+U)}\left|\zeta\left(\frac 12+i\vp_2(t)\right)\right|^4\left|\zf\right|^4{\rm d}t\sim \\
& \sim\frac{1}{4\pi^4}U\ln^8T,\ U=T^{13/14+2\epsilon},\ T\to\infty
\end{split}
\ee
holds true.
\begin{remark}
The small improvements of the exponent $\frac{13}{14}$ that are of the type $\frac{13}{14}\rightarrow \frac{8}{9}\rightarrow \dots$ are
irrelevant in this question.
\end{remark}
Next, similarly to the Theorem 1, the following theorem holds true.
\begin{theorem}
Every Jacob's ladder of the second order $\vp_2(t)$, i.e. the (exact) solution to the nonlinear integral equation (\ref{2.1}) is the asymptotic
solution of the nonlinear integral equation
\be \label{2.3}
\int_{x^{-1}(T)}^{x^{-1}(T+U)}\left|\zeta\left(\frac 12+ix(t)\right)\right|^4\left|\zf\right|^4{\rm d}t=\frac{1}{4\pi^4}U\ln^8T,\ T\to\infty ,
\ee
(comp. (\ref{2.2})).
\end{theorem}
\begin{remark}
There are the fixed-point methods and other methods of the functional analysis used to study the nonlinear equations. What can be obtained by using
these methods in the case of the nonlinear integral equations (\ref{1.3}), (\ref{2.3})?
\end{remark}
\section{Proof of the Theorem 1}
\subsection{}
Let us remind that the following lemma holds true (see \cite{6}, (2.5); \cite{7}, (3.3)): for every integrable function (in the Lebesgue sense)
$f(x),\ x\in [\vp_1(T),\vp_1(T+U)]$ we have
\be \label{3.1}
\int_T^{T+U}f[\vp_1(t)]\tilde{Z}^2(t){\rm d}t=\int_{\vp_1(T)}^{\vp_1(T+U)}f(x){\rm d}x,\ U\in \left(\left. 0,\frac{T}{\ln T}\right]\right.
\ee
where
\bdis
t-\vp_1(t)\sim (1-c)\pi(t) ,
\edis
$c$ is the Euler's constant and $\pi(t)$ is the prime-counting function. In the case $\mT=\vp_1^{-1}(T),\ \widering{T+U}=\vp_1^{-1}(T+U)$ we
obtain from (\ref{2.1})
\be \label{3.2}
\int_{\vp_1^{-1}(T)}^{\vp_1^{-1}(T+U)}f[\vp_1(t)]\tilde{Z}^2(t){\rm d}t=\int_T^{T+U}f(x){\rm d}x .
\ee
\subsection{}
By the simple formula
\bdis
\int_0^a J_1(x){\rm d}x=1-J_0(a) ,
\edis
known from the theory of the Bessel's functions, we obtain
\be \label{3.3}
\int_{t_\nu}^{t_{\nu+1}}J_1(x){\rm d}x=J_0(t_\nu)-J_0(t_{\nu+1}) .
\ee
Hence, from (\ref{3.3}) by (\ref{3.2}) the formula
\be \label{3.4}
\int_{\vp_1^{-1}(t_\nu)}^{\vp_1^{-1}(t_{\nu+1})}J_1[\vp_1(t)]\tilde{Z}^2(t){\rm d}t=J_0(t_\nu)-J_0(t_{\nu+1})
\ee
is obtained.
\subsection{}
It is also well-know that
\be \label{3.5}
J_\nu(x)=\sqrt{\frac{2}{\pi x}}\cos\left( x-\nu\frac{\pi}{2}-\frac{\pi}{4}\right)+\mcal{O}\left(\frac{1}{x^{3/2}}\right),\ x\to\infty
\ee
(the asymptotic formula for $J_\nu(x)$). Since, by the Titchmarsh' formula (\ref{1.2})
\be \label{3.6}
\frac{1}{\sqrt{t_{\nu+1}}}=\frac{1}{\sqrt{t_\nu}}+\mcal{O}\left(\frac{1}{t_\nu^{3/2}\ln t_\nu}\right),
\ee
it follows (see (\ref{3.5}), (\ref{3.6})) that
\be \label{3.7}
\begin{split}
& J_0(t_\nu)-J_0(t_{\nu+1})= \\
& =\sqrt{\frac{2}{\pi t_\nu}}\left\{\cos\left( t_\nu-\frac{\pi}{4}\right)-\cos\left( t_{\nu+1}-\frac{\pi}{4}\right)\right\}+
\mcal{O}\left(\frac{1}{t_\nu^{3/2}}\right) .
\end{split}
\ee
Next, (see (\ref{1.2}))
\be \label{3.8}
\begin{split}
& \cos\left(t_\nu-\frac{\pi}{4}\right)-\cos\left( t_{\nu+1}-\frac{\pi}{4}\right)=2\sin\frac{t_{\nu+1}-t_\nu}{2}
\sin\left(\frac{t_{\nu+1}+t_\nu}{2}-\frac{\pi}{4}\right)= \\
& =2\sin\frac{t_{\nu+1}-t_\nu}{2}\sin\left(\frac{t_{\nu+1}-t_\nu}{2}+t_\nu-\frac{\pi}{4}\right) = \\
& =2\sin^2\frac{t_{\nu+1}-t_\nu}{2}\cos\left(t_\nu-\frac{\pi}{4}\right)+\sin(t_{\nu+1}-t_\nu)\sin\left( t_\nu-\frac{\pi}{4}\right) = \\
& =\frac{2\pi}{\ln t_\nu}\sin\left( t_\nu-\frac{\pi}{4}\right)+\mcal{O}\left(\frac{1}{\ln^2 t_\nu}\right) .
\end{split}
\ee
Hence, from (\ref{3.4}) by (\ref{3.7}), (\ref{3.8}) the asymptotic formula
\be \label{3.9}
\begin{split}
& \int_{\vp_1^{-1}(t_\nu)}^{\vp_1^{-1}(t_{\nu+1})}J_1[\vp_1(t)]\tilde{Z}^2(t){\rm d}t= \\
& =\frac{2\sqrt{2\pi}}{\sqrt{t_\nu}\ln t_\nu}\sin\left( t_\nu-\frac{\pi}{4}\right)+\mcal{O}\left(\frac{1}{\sqrt{t_\nu}\ln^2 t_\nu}\right)
\end{split}
\ee
follows if the second condition in (\ref{1.4}) is fulfilled. Then from (\ref{3.9}) by the mean-value theorem (see (\ref{1.1}), (\ref{1.4}) and
\cite{19}, (3.3)) we obtain
\bdis
\begin{split}
& \int_{\vp_1^{-1}(t_\nu)}^{\vp_1^{-1}(t_{\nu+1})}J_1[\vp_1(t)]\left|\zf\right|^2{\rm d}t= \\
& =\frac{2\sqrt{2\pi}}{\sqrt{t_\nu}}\sin\left( t_\nu-\frac{\pi}{4}\right)+\mcal{O}\left(\frac{\ln\ln t_\nu}{\sqrt{t_\nu}\ln t_\nu}\right) ,
\end{split}
\edis
i.e. the formula (\ref{1.5}) holds true.
\thanks{I would like to thank Michal Demetrian for helping me with the electronic version of this work.}
|
2106.13688
|
\section{Introduction}
Neutron scattering is one of the few experimental techniques that allow one to probe both the structure and the dynamics of physical systems at the {\r A}ngstr{\"o}m scale\cite{Sivia:2011,Squires:2012}. Typically, structural information is obtained through the elastic scattering of cold or thermal neutrons (SANS). The dynamic information is obtained through inelastic and quasi-elastic scattering, as the neutrons gain or lose energy when they interact with moving phases in the system. As for most scattering techniques, however, converting experimental data to real-space and time-dependent structures can be challenging. This is particularly the case for complex and disordered structures that cannot be described in simple geometrical terms.
When studying disordered systems, stochastic models often provide a practical compromise between geometrical realism and mathematical simplicity. The former is necessary to account for as many geometrical features as possible, and the latter improves the robustness of the analysis by avoiding unnecessarily large number of parameters \cite{Serra:1982,Torquato:2002,Lantuejoul:2002}. In that spirit, stochastic models have often been used to analyze small-angle scattering data from a variety of physical systems and reconstruct their structure \cite{Sonntag:1981,Roberts:1997,Gille:2011,Gommes:2018,Prehal:2020}. In the present paper, we generalize this type of approach to analyze and model time-dependent structures investigated by inelastic neutron scattering.
The paper focuses specifically on a family of descriptive models based on clipped Gaussian random fields. These models originate in the work of Cahn on spinodal decomposition \cite{Cahn:1965}, but they have since been used as general geometrical models of disordered structures in a variety of contexts, including porous materials \cite{Quiblier:1984,Roberts:1995,Gommes:2018}, polymers \cite{Chen:1996,DHollander:2010}, emulsions \cite{Berk:1987,Teubner:1991}, gels \cite{Gommes:2008}, confined liquids \cite{Gommes:2013,Gommes:2018B}, nanoparticles \cite{Gommes:2020}, etc. Gaussian random fields are comprehensively characterized by their correlation function, which makes them particularly useful in the context of scattering studies.
The theoretical developments of the present paper are illustrated on previously-published elastic and inelastic neutron scattering data measured on water/oil microemulsion, which are presented shortly in Section II, together with some general results pertaining to elastic and inelastic neutron scattering. Section III covers some classical results of static Gaussian-field models, which are generalized to time-dependent structures in Section IV. Three families of dynamic models are proposed, which are applicable to any static Gaussian field and endow it with qualitatively different time-dependence. In Section V, some aspects of the models relevant to inelastic scattering are discussed, and the models are used to analyze the microemulsion data.
\section{Neutron small-angle scattering and spin-echo data}
\label{sec:experimental}
The methods and models developed in the paper are illustrated with published neutron small-angle scattering (SANS) patterns and neutron spin-echo (NSE) data measured on a water/oil microemulsion stabilised with a surfactant\cite{Holderer:2005,Holderer:2007}. The relevant data are available on the authors institutional repository\cite{Holderer:2021} and they are displayed in Fig. \ref{fig:data}.
The bicontinuous phases of the microemulsion consisted in water and decane with decyl-polyglycol-ether (C$_{10}$E$_{4}$) as a surfactant. The volume fractions of the three phases were $\phi_o \simeq 0.4075$, $\phi_s \simeq 0.185$ and $\phi_a \simeq 0.4075$ for oil, surfactant and aqueous phases, respectively. Small amounts (0.25 wt.\%) of homopolymers were dispersed in the continuous phases in order to slightly modify their viscosity and the efficiency of the surfactant (see Refs. \cite{Holderer:2005,Holderer:2007}), namely polyethylene oxide (PEO) in water and polyethylene propylene (PEP) in decane. The molecular weights slightly differed in SANS (10 kg/mol) and NSE (5 kg/mol) experiments, which has only minor effects for the purposes of this study on the relaxation rate in the NSE experiments ($<10$ \%), but gave a complete set of SANS and NSE data. The microemulsion was prepared in two different neutron scattering contrasts, by exchanging hydrogen with deuterium. In the so-called bulk contrast, deuterated water (D$_2$O) was used with protonated surfactant and oil, which results in a contrast between the water domains and the oil-surfactant-domains. In film contrast, the decane was deuterated as well, leaving only the protonated surfactant film visible in the deuterated water/oil surrounding. The SANS experiments were conducted on the KWS-2 small angle scattering instrument at the DIDO reactor of Forschungszentrum J\"ulich, the NSE experiments were conducted on the IN15 instrument at the Institut Laue-Langevin in Grenoble. The resolutions the SANS and NSE data in Fig. \ref{fig:data} are $\sigma^{SANS}_q=0.0034$ \AA$^{-1}$ and $\sigma^{NSE}_q=0.0085$ \AA$^{-1}$, respectively.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig1_data_with_realization_N50.eps}
\caption{Neutron Small-Angle Scattering data (SANS, $a$) measured on a microemulsion in bulk (grey) and film (red) contrasts, together with structure reconstructed from it as a clipped Gaussian field model ($b_1$: field, $b_2$ clipped structure with oil in grey and surfactant in red). The neutron spin-echo (NSE) data measured in the same conditions are shown in $c_1$ and $c_2$. In the SANS patterns (a) the dots are the experimental values, and the solid lines are the fitted model. The values in the Gaussian field shown in $b_1$ range from -2.5 (blue) to + 2.5 (yellow). The error bars are $\pm 2 \sigma$ for both SANS and NSE.}
\label{fig:data}
\end{center}
\end{figure}
Microemulsions are strong coherent scatterers, so that incoherent scattering from individual atoms (mainly hydrogen) does not play a significant role at the length scales discussed in this paper. Therefore, the central structural characteristic of the microemulsion relevant to both the SANS and NSE data is the scattering-length correlation function\cite{VanHove:1954,Squires:2012}
\begin{equation} \label{eq:C_rho_def}
C_{\rho}(r,\tau) = \langle \rho( \mathbf{x},t) \rho(\mathbf{x}+ \mathbf{r},t+\tau) \rangle - \langle \rho \rangle^2
\end{equation}
which characterises the statistical correlation between the scattering length density $\rho$ at two points at a distance $r$ apart, and time lag $\tau$. Throughout the paper we assume statistical isotropy, so that correlation functions depend only on the modulus of the distance $r=|\mathbf{r}|$. In Eq. (\ref{eq:C_rho_def}) the brackets $\langle \rangle$ stand for the average value, evaluated over all accessible positions $\mathbf{x}$ and times $t$. For the type of ergodic models considered later in the paper, they can also be thought of as ensemble averages.\cite{Torquato:2002,Lantuejoul:2002,Gommes:2018}
When working with stochastic models it is convenient to introduce the concept of covariance \cite{Serra:1982,Lantuejoul:2002}, which is occasionally also referred to as 2-point probability functions\cite{Torquato:2002} or stick-probability functions\cite{Ciccariello:1981}. The covariance of, say the oil phase $o$ of the microemulsion, is defined as the probability for two points at distance $\mathbf{r}$ from one another to belong to that phase at two moments separated with time lag $\tau$, namely
\begin{equation}
C_{oo}(r,\tau ) =\textrm{Prob} \left[ \left(\mathbf{x} \in o \ \textrm{at time} \ t \right) \& \left( \mathbf{x} + \mathbf{r} \in o \ \textrm{at time} \ t+\tau \right) \right]
\end{equation}
As this generalizes to {\it cross-covariances} for two points belonging to two distinct phases, the name {\em self-covariance} is occasionally used to insist that the two points belong to the same phase. Because each of the three phases of the microemulsion - oily, aqueous and surfactant - has a specific scattering-length density, the correlation function $C_\rho(r)$ is a linear combination of the covariances of the phases. Out of the six self- and cross-covariances that are defined for a three-phase system, only three are linearly independent.\cite{Torquato:2002} A convenient expression for $C_\rho $ is therefore \cite{Gommes:2013}
\begin{align} \label{eq:C_rho}
C_{\rho}(r,\tau) &= (\rho_o - \rho_s)(\rho_o-\rho_a) [C_{oo}(r,\tau) - \phi_o^2]+ (\rho_s - \rho_o)(\rho_s-\rho_a) [C_{ss}(r,\tau) - \phi_s^2]\cr
&+(\rho_a - \rho_o)(\rho_a-\rho_s) [C_{aa}(r,\tau) - \phi_a^2]
\end{align}
where $\rho_o$, $\rho_s$ and $\rho_a$ are the scattering-length densities of the oil, surfactant, and aqueous phases, respectively; $C_{oo}$, $C_{ss}$ and $C_{aa}$ are the corresponding self-covariances. A derivation of Eq. (\ref{eq:C_rho}) is provided in the Supplementary Material (Sec. SM-1). The bulk contrast relevant to Fig. \ref{fig:data} correspond to $\rho_o=\rho_s \neq \rho_a$, in which case $C_\rho$ is proportional to $C_{aa}$. The film contrast corresponds to $\rho_o=\rho_a \neq \rho_s$, and in that case $C_\rho$ is proportional to $C_{ss}$.
The coherent inelastic neutron scattering data is expressed in terms of the intermediate scattering function $I(q,\tau)$. The latter is defined as the Fourier transform of the correlation function $C_\rho(r,\tau)$, namely \cite{Sivia:2011,Squires:2012}
\begin{equation} \label{eq:I_Fourier}
I(q,\tau) = \int_0^\infty \frac{\sin(qr)}{qr} C_{\rho}(r,\tau) 4 \pi r^2 \textrm{d}r
\end{equation}
and the instrument resolution is accounted by multiplying $C_\rho$ by a spread function with width $\sigma_q$, prior to Fourier transform. The situation relevant to SANS is elastic scattering corresponding to $I(q,0)$, to which we refer simply as $I(q)$ when there is no ambiguity. The data measured in NSE instruments is $I(q,\tau)/I(q)$, as given in Fig. \ref{fig:data}c1 and \ref{fig:data}c2 for the microemulsion.
In Fig. \ref{fig:data} the SANS data in both film and bulk contrasts were fitted jointly with a clipped Gaussian field model, adapting a procedure developed elsewhere\cite{Gommes:2018}. For the sake of completeness, the detailed procedure is described in the Supplementary Material (Sec. SM-3.3). A realisation of the model is shown in Fig. \ref{fig:data}b.
\section{Clipped Gaussian-field models}
\label{sec:static}
\subsection{Static Gaussian random fields}
\label{sec:GRF}
We focus here on static, that is time-independent, Gaussian random fields (GRF) and we introduce time-dependence in Sec. \ref{sec:timedependentmodels}. A convenient and classical way to think of GRFs is as a superposition of random sine waves \cite{Berk:1991,Levitz:1998}
\begin{equation} \label{eq:W_def}
W(\mathbf{x}) = \sqrt{\frac{2}{N}} \sum_{n=1}^N \sin\left[ \mathbf{q}_n \cdot \mathbf{x} - \varphi_n \right]
\end{equation}
where the phases are uniformly distributed over $[0, 2 \pi)$ and the wavevectors $\mathbf{q}$ are drawn from a user-specified density distribution over reciprocal space $f_W(\mathbf{q}) \textrm{d}V_q$, referred to as the spectral density of the field. For asymptotically large values of $N$, the central limit theorem ensures that $W(\mathbf{x})$ is Gaussian-distributed at any point $\mathbf{x}$ with average equal to zero, and the factor in Eq. (\ref{eq:W_def}) ensures that the variance is equal to one.
A central characteristic of the GRF in the context of elastic scattering is its correlation function $g_W(r)$, defined as the statistical correlation between the values of $W(\mathbf{x})$ at two points at distance $r$ apart
\begin{equation} \label{eq:gW}
g_W(r) = \langle W(\mathbf{x}) W(\mathbf{x}+\mathbf{r}) \rangle
\end{equation}
where the brackets have the same meaning as in Eq. (\ref{eq:C_rho_def}), and the dependence is only on the modulus $r =|\mathbf{r}|$ for isotropic fields. The field correlation function is obtained as the Fourier transform of the spectral density, namely\cite{Berk:1987,Berk:1991}
\begin{equation} \label{eq:gW_f}
g_W(r) = \int_0^\infty \frac{\sin(qr)}{qr} f_W(q) 4 \pi q^2 \textrm{d}q
\end{equation}
In principle any integrable and positive function can be used as a spectral density. In practice, in order to ensure that the structures modelled by clipping the field have finite surface areas\cite{Berk:1991,Teubner:1991}, it is necessary to impose that the second moment of $f_W(q)$ be finite. This enables one to define $l_W$ as
\begin{equation} \label{eq:lW}
\frac{1}{l_W^2} = \frac{1}{6} \int_0^\infty q^2 f_W(q) 4 \pi q^2 \textrm{d}q
\end{equation}
which we refer to as the field characteristic length. The finiteness of $l_W$ corresponds to a quadratic behavior of the correlation function $g_W \simeq 1 - (r/l_W)^2 + \ldots $ for small distances, and is a condition for the modelled structures to have finite surface areas\cite{Teubner:1991,Berk:1991}.
\begin{sidewaystable}
\caption{Examples of static Gaussian random fields (GRFs) with their spectral densities $f_W(q)$, field correlation functions $g_W(r)$, and characteristic lengths $l_W$. The function $w(|\mathbf{x}|)$ is the corresponding elementary wave relevant to a dilution approach (see text, $l$ and $\mu$ are model parameters). These functions are plotted in Figs. SM-1 to SM-8 of the Supplementary Material.}
\begin{center}
\begin{tabular}{c|c|c|c|c|r}
GRF\# & $f_W(q)$ & $g_W(r)$ & $l_W$ & $w(\mathbf{x})$$^a$ & Ref. \cr
\hline
1
& $ \frac{l^2}{16 \pi^3} \delta[q - \frac{2 \pi}{l}] $
& $\frac{\sin[2\pi r/l]}{2\pi r /l} $ & $\sqrt{6} l/(2 \pi) $
& -$^{b}$
& \cite{Berk:1987} \cr
2
& $\left( \frac{l}{2 \sqrt{\pi}} \right)^3 e^{-[q l]^2/4} $
& $e^{-[r/l]^2}$
& $l$
& $ e^{- 2 \left( \frac{|\mathbf{x}|}{l} \right)^2} $
& \cite{Lantuejoul:2002,Gelfand:2010} \cr
3
& $ \frac{l^2}{4q} \frac{\sinh[\pi ql/2]}{1 + \cosh[\pi q l]} $
& $\frac{1}{\cosh[r/l]} $ & $\sqrt{2} l $
& -$^{b}$
& \cite{Gommes:2008} \cr
4
& $\left(\frac{l}{\sqrt{\pi}} \right)^3 \frac{1}{480} (ql)^4 e^{-[ql]^2/4} $
& $\left[1 - \frac{4}{3} \left( \frac{r}{l} \right)^2 + \frac{4}{15} \left( \frac{r}{l} \right)^4 \right] e^{ -\left( \frac{r}{l} \right)^2 } $
& $\sqrt{3/7} l $
& $ \left[ \left(\frac{ |\mathbf{x}|}{l}\right)^2 - \frac{3}{4} \right] e^{- 2 \left(\frac{|\mathbf{x}|}{l} \right)^2} $
& \cr
5
& $ \frac{l^3}{4\pi^2 \mu} \frac{ \sinh[\pi^2/\mu] \sinh[\pi ql/(2\mu)]/(ql)}{\cosh[2 \pi^2 /\mu] + \cosh[\pi ql/ \mu]} $
& $\frac{\sin[2\pi r/l]}{(2\pi r /l)\cosh[\mu r/l]} $
& $ l /\sqrt{ \frac{2 \pi^2}{3} + \frac{\mu^2}{2} } $
& -$^{b}$
& \cite{Gommes:2008} \cr
6
& $ \left(\frac{l}{\sqrt{\pi}}\right)^3 \frac{\Gamma(\mu+\frac{3}{2})}{\Gamma(\mu) [1 + (ql)^2]^{\frac{3}{2}+\mu}} $
& $2^{1-\mu} \frac{(r/l)^\mu K_\mu(r/l)}{ \Gamma(\mu + 1)}$
& $ 2l \sqrt{\mu - 1} $
& $\left(\frac{|\mathbf{x}|}{l} \right)^{\frac{\mu}{2}-\frac{3}{4}} K_{\frac{\mu}{2}-\frac{3}{4}} \left(\frac{|\mathbf{x}|}{l} \right)$
& \cite{Gelfand:2010,Lantuejoul:2002} \cr
7
& $ \left(\frac{l}{\sqrt{\pi}}\right)^3 \frac{\Gamma(\mu+1) }{\Gamma(\mu-\frac{1}{2})} [1 - (ql)^2]^{\mu-\frac{3}{2}} $
& $2^\mu \Gamma(\mu + 1) \frac{J_\mu(r/l)}{(r/l)^\mu}$
& $ 2l \sqrt{\mu+1}$
& $\left(\frac{|\mathbf{x}|}{l} \right)^{-\frac{\mu}{2}-\frac{3}{4}} J_{\frac{\mu}{2}+\frac{3}{4}} \left(\frac{|\mathbf{x}|}{l} \right)$
& \cite{Lantuejoul:2002} -$^c$ \cr
8
& Piecewise Linear
& Cf. Sup. Info.
& Cf. Sup. Info.
& -$^b$
& \cite{Gommes:2018} \cr
\end{tabular}
\end{center}
\label{tab:fields}
$^a$: within unspecified normalizing factor; $^b$: not available; $^c$: a typo in the formula provided for $f_W(q)$ has been corrected in the present table.
\end{sidewaystable}%
The methods developed in the paper apply to any static Gaussian field. A few examples are given in Tab. \ref{tab:fields} with explicit spectral densities, correlation functions, and characteristic lengths $l_W$, which are also plotted in Figs. SM-1 to SM-8 of the Supplementary Material. These fields are referred to in the rest of the paper by the number in the first column. GRF-1 contains a single spectral component, and is arguably the simplest possible Gaussian field. By contrast, GRF-2 is extremely polydispersed and is referred to in geostatistics as the squared-exponential correlation function. GRF-3 is also polydispersed: its correlation function is exponential for asymptotically large distances, but the $1/\cosh$ function ensures quadratic shape at the origin and hence finite $l_W$. GRF-4 is introduced in Sec. \ref{sec:data_analysis}, and leads to structures with a scattering peak. GRF-5 is obtained by multiplying the correlation functions of GRF-1 and GRF-3, and provides one with a parameter to control the polydispersity of the structure, which makes it convenient for SAS data fitting\cite{Gommes:2008,Prehal:2017}. Other examples discussed in the scattering literature can be found {\it e.g.} in Refs. \cite{Teubner:1987,Chen:1996,Roberts:1997}. The following two entries in Tab. \ref{tab:fields} are classical in geostatistics but are seldom used in scattering studies. GRF-6 is the Mat\'ern model where $K_\mu$ is a modified Bessel function. The parameter $\mu$ controls the smoothness of the field, which is $\mu-1$ times differentiable\cite{Rasmussen:2006}, and GRF-2 is obtained as a particular case for $\mu \to \infty$. By contrast to GRF-6, field GRF-7 introduces strong correlations through Bessel function $J_\mu$. It leads to peaked scattering functions (see Fig. SM-7) and coincides with GRF-1 in the limit $\mu \to 1/2$.
When it comes to analyzing experimental scattering patterns, the simple analytical expressions in Tab. \ref{tab:fields} seldom provide sufficient flexibility for data fitting. Therefore, a convenient approach consists in linearly combining independent Gaussian fields $W_i(\mathbf{x})$, with spectral densities $f_W^{(i)}(q)$, so as to create a composite field
\begin{equation} \label{eq:combine_W}
W(\mathbf{x}) =\sum_i \sigma_i W_i(\mathbf{x})
\end{equation}
where $\sigma_i$ are constants. The spectral density of the resulting field is
\begin{equation}
f_W(q) =\sum_i \sigma^2_i f_W^{(i)}(q)
\end{equation}
and a similar relation holds for $g_W(r)$. Because the integral of $f_W(q)$ over the entire reciprocal space is the variance of the field, the parameter $\sigma_i^2$ can be thought of as the contribution of $W_i(\mathbf{x})$ to the total variance of the composite field $W(\mathbf{x})$. In that spirit, a possible approach to data fitting would consist in combining a large number of monodispersed fields ({\it e.g.} GRF-1 in Tab. \ref{tab:fields}), so as to approximate an experimental spectral density as a sum of Dirac peaks. As an unpractically large number of peaks might be needed to approximate a continuous function, a more practical approach consists in replacing the Dirac peaks by broader functions. The piecewise-linear model (GRF-8 in Tab. \ref{tab:fields}) corresponds to such an approach, which was developed in earlier work\cite{Gommes:2018}. As this approach was used here to fit the SANS data in Fig. \ref{fig:data}a, it is described in detail in the Supplementary Material (Sec. SM-3). In particular, the influence of the number of nodes for the SANS fit shown in Fig. \ref{fig:data}a is illustrated in Fig. SM-10.
When generalising the Gaussian-field modelling to time-dependent structures it will prove useful to use another construction of Gaussian fields, which is mathematically equivalent to Eq. (\ref{eq:W_def}). In so-called dilution random functions, \cite{Serra:1982,Lantuejoul:1991,Lantuejoul:2002} a field is created as a sum of localised elementary waves $w(\mathbf{x})$, randomly positioned in space, namely
\begin{equation} \label{eq:W_def_dilution}
W(\mathbf{x}) = \sum_s A_s w(\mathbf{x} - \mathbf{x}_s)
\end{equation}
where the sum is on all the seeds $\mathbf{x}_s$ of a Poisson point process with density $\theta$, and $A_s$ is any random amplitude satisfying $\langle A \rangle = 0$ and $\langle A_s A_{s'} \rangle = \langle A^2 \rangle \delta_{ss'}$. The latter condition corresponds to uncorrelated wave amplitudes. In the limit of a large density of the Poisson process, many elementary waves overlap at any given point of space so that the values of the field defined in Eq. (\ref{eq:W_def_dilution}) become Gaussian distributed.
In the context of a dilution approach, the correlation function of the field is calculated as\cite{Serra:1982,Lantuejoul:1991,Lantuejoul:2002}
\begin{equation} \label{eq:gW_dilution}
g_W(\mathbf{r})= \theta \langle A^2 \rangle K(\mathbf{r})
\end{equation}
where
\begin{equation}
K(\mathbf{r})= \int \textrm{d}V_x \ w(\mathbf{x}) w(\mathbf{x} - \mathbf{r})
\end{equation}
is the self-convolution of the elementary wave. In order to ensure that the variance of the field is equal to one, one has to impose $g_W(0)=1$, which requires adjusting the amplitudes so that $\theta \langle A^2 \rangle K(0)=1$. Although Eqs. (\ref{eq:W_def}) and (\ref{eq:W_def_dilution}) are conceptually different constructions, the two approaches are mathematically equivalent. The spectral density of the dilution model is indeed obtained as
\begin{equation} \label{eq:fW_w}
f_W(\mathbf{q}) = \theta \langle A^2 \rangle \left| \int \textrm{d}V_x \ w(\mathbf{x}) \exp ( i \mathbf{q} \cdot \mathbf{x} ) \right|^2
\end{equation}
which results from evaluating the Fourier transform of Eq. (\ref{eq:gW_dilution}). Among the static Gaussian fields presented in Tab. \ref{tab:fields}, the shape of the elementary wave is know for GRF-2, GRF-4, GRF-6 and GRF-7. Conceptually, however, any field such that $\sqrt{f_W(q)}$ is integrable can be thought of as resulting from a superposition of a large number of randomly positioned elementary waves.
\subsection{Clipping procedure}
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig2_demo_GRF_2D.eps}
\caption{Microemulsion modelling as a clipped Gaussian field, with the underlying field shown in (a) and the structure in (b). The clipping thresholds are $\alpha = -0.234$ and $\beta = +0.234$, resulting in aqueous (white) and oil (grey) phases with volume fractions $\phi_a=\phi_o=0.4075$, and $\phi_s=0.185$ for the surfactant (red). The figure is a 2D cut out of a 3D realization obtained from GRF-2 of Tab. \ref{tab:fields}, with distances normalized to $l_W$.}
\label{fig:GRF2D}
\end{center}
\end{figure}
According to a classical approach, the phases of disordered systems can be modelled as excursion sets of a Gaussian field $W(\mathbf{x})$, which is also referred to as clipped-Gaussian-field models.\cite{Quiblier:1984,Berk:1987} In the particular case of emulsions\cite{Teubner:1991} a convenient clipping procedure is based on two thresholds $\alpha \leq \beta$, as sketched in Fig. \ref{fig:GRF2D}. The oil phase is modelled as the points of space where $\beta \le W(\mathbf{x})$, the surfactant film-like phase as $\alpha \le W(\mathbf{x}) < \beta$, and the aqueous phase as $W(\mathbf{x})< \alpha$.
Because the values of $W(\mathbf{x})$ are Gaussian distributed, the values of the thresholds control the volume fractions of the phases. The volume fraction of the oil $\phi_o$, is obtained as
\begin{equation} \label{eq:phi_o}
\phi_o = \Lambda_1[\beta]
\end{equation}
where the function $\Lambda_1[x]$ is the probability for a univariate Gaussian variable to take values larger than $x$, which can be calculated as
\begin{equation}
\Lambda_1[x] = \frac{1}{2} \left( 1 - \textrm{erf}[x/\sqrt{2}] \right)
\end{equation}
where erf is the error function. With the same notation, the volume fraction of the surfactant phase is
\begin{equation} \label{eq:phi_s}
\phi_s = \Lambda_1[\alpha] - \Lambda_1[\beta]
\end{equation}
and the volume fraction of the remaining aqueous phase is $\phi_a = 1 - \phi_o - \phi_s$. Relevant values for the microemulsion of Sec. \ref{sec:experimental} are $\alpha=-0.234$ and $\beta \simeq +0.234$, corresponding to $\phi_f \simeq 0.185$ and $\phi_a = \phi_o= 0.4075$, as used in Fig. \ref{fig:GRF2D}.
The scattering functions are obtained from the covariances of the various phases of the microemulsion, which are calculated from the field correlation function $g_W(r)$ and the clipping thresholds $\alpha$ and $\beta$. In line with Eq. (\ref{eq:C_rho}), we consider here the covariances $C_{oo}(r)$, $C_{ss}(r)$ and $C_{aa}(r)$, defined as the probabilities for two randomly chosen points are distance $r$ from one another to belong both to the oil, surfactant, and aqueous phases, respectively. The covariances are expressed in terms of the bivariate error function $\Lambda_2[\alpha,\beta,g]$, defined as the probability for two correlated Gaussian variables, with correlation $g$, to take values larger than $\alpha$ and $\beta$, respectively \cite{Roberts:1995}. Explicitly the expressions are\cite{Teubner:1991,Levitz:1998}
\begin{equation} \label{eq:Coo}
C_{oo}(r) = \Lambda_2[\beta, \beta, g_W(r)]
\end{equation}
for the oil phase,
\begin{equation} \label{eq:Css}
C_{ss}(r) = \Lambda_2[\alpha, \alpha, g_W(r)] + \Lambda_2[\beta, \beta,g_W(r)] - 2 \Lambda_2[\alpha, \beta, g_W(r)]
\end{equation}
for the surfactant phase, and
\begin{equation} \label{eq:Caa}
C_{aa}(r) = 1 - 2 \Lambda_1[\alpha]+ \Lambda_2[\alpha, \alpha, g_W(r)]
\end{equation}
for the aqueous phase. The values used to calculate the scattering functions through Eqs. (\ref{eq:C_rho}) are the centred covariance $\bar C_{oo}(r)$, $\bar C_{ss}(r)$ and $\bar C_{aa}(r)$, obtained by subtracting the corresponding squared volume fraction, so as to enable their Fourier transformation through Eq. (\ref{eq:I_Fourier}).
In principle the function $\Lambda_2[\alpha,\beta,g]$ can be calculated as two-dimensional integral of a bivariate Gaussian distribution. Based on Dirichlet's representation of Heaviside's step function, it can be calculated in the following simpler way\cite{Berk:1991,Teubner:1991,Roberts:1995,Levitz:1998}
\begin{eqnarray} \label{eq:Lambda2}
\Lambda_2[\alpha, \beta, g] &=& \Lambda_1[\alpha] \Lambda_1[\beta]
+ \frac{1}{2 \pi} \int_0^{\textrm{asin}[g]} \exp\left[ - \frac{\alpha^2 + \beta ^2 - 2 \alpha \beta \sin(\theta)}{2 \cos^2(\theta)} \right] \ \textrm{d}\theta
\end{eqnarray}
which requires numerically evaluating only a one-dimensional integral. For any given volume fraction of the phases, that is for given $\alpha$ and $\beta$, Eqs. (\ref{eq:Coo}), (\ref{eq:Css}) and (\ref{eq:Caa}) define non-linear relations between the field correlation $g_W$ and the corresponding covariances. These relations are illustrated in Fig. \ref{fig:clipping} for {the centred covariances} $\bar C_{oo}$, $\bar C_{aa}$ and $\bar C_{ss}$, for the values of $\alpha$ and $\beta$ relevant to the microemulsion data. Note that the values satisfy $\alpha=-\beta$, corresponding to $\phi_o=\phi_a$ and $\bar C_{oo}(r)= \bar C_{aa}(r)$.
\begin{figure}
\begin{center}
\includegraphics[width=5cm]{fig3_clipping.eps}
\caption{Effect of clipping: non-linear relation between the field correlation $g_W$ and the centered covariance $\bar C = C - \phi^2$ of the aqueous, oil, and surfactant phases, calculated from Eqs. (\ref{eq:Coo}-\ref{eq:Caa}) with clipping constants $\alpha \simeq -0.234$ and $\beta \simeq +0.234$. The relations are highly non-linear for $g_W \simeq 1$ and linear for asymptotically small $g_W$, which limit is relevant for asymptotically large values of $r$ or $\tau$. The dashed lines are the asymptotic approximations for $g_W \to 1$ calculated through Eq. (\ref{eq:Lambda2_large_g}), relevant to vanishingly small $r$ and $\tau$.}
\label{fig:clipping}
\end{center}
\end{figure}
As visible in Fig. \ref{fig:clipping}, for small field correlations $g_W$ the centred covariances $\bar C_{oo}$ and $\bar C_{aa}$ are proportional to $g_W$. In that region - corresponding to large $r$ or $\tau$ - the bivariate error function $\Lambda_2$ is approximated by
\begin{eqnarray} \label{eq:Lambda2_small_g}
\Lambda_2[\alpha,\beta,g] &\simeq & \Lambda_1[\alpha] \Lambda_1[\beta]
+ \frac{g}{2 \pi} \exp\left(-\frac{\alpha^2 + \beta^2}{2} \right) + \ldots
\end{eqnarray}
which is the first term of a general development in terms of Hermite polynomials\cite{Lantuejoul:2002,Gommes:2013}. However, in general $\Lambda_2$ is a non-linear function. In particular the relation between $g_W$ and the covariances of any clipped structure is vertical when $g_W$ approaches 1 (see Fig. \ref{fig:clipping}). The following asymptotic relation is useful for further purposes
\begin{eqnarray} \label{eq:Lambda2_large_g}
\Lambda_2[\alpha,\beta,1 - \epsilon^2 ] = \Lambda_1\left[ \textrm{max}\{\alpha,\beta \} \right] &-&
\frac{\epsilon}{\pi \sqrt{2}} e^{-\frac{\alpha \beta}{2}} e^{-\frac{(\alpha - \beta)^2}{4 \epsilon^2}} \cr
+ \frac{|\alpha - \beta|}{2 \sqrt{2 \pi}} e^{-\frac{\alpha \beta}{2}} \Big( 1 &-& \textrm{erf} \left[ \frac{|\alpha - \beta|}{2 \epsilon} \right] \Big)
\end{eqnarray}
It is obtained by setting $g=1-\epsilon^2$ in Eq. (\ref{eq:Lambda2}), through a first-order expansion in $\epsilon$. This equation controls the shape of the covariances for asymptotically small $r$ and $\tau$, and therefore the asymptotic shape of the scattering functions for large $q$ and small $\tau$, as we discuss in detail later. The centred covariances approximated through Eq. (\ref{eq:Lambda2_large_g}), are shown as dashed lines in Fig. \ref{fig:clipping}.
\section{Time-dependent clipped Gaussian-field models}
\label{sec:timedependentmodels}
We now introduce three qualitatively different dynamic models to construct time-dependent Gaussian fields, starting from any static Gaussian field. This is achieved by adapting Eqs. (\ref{eq:W_def}) or (\ref{eq:W_def_dilution}), by which the fields are constructed. Although the models are quite general, the discussion is centred on static field GRF-2 of Tab. \ref{tab:fields}, which has the following spectral density
\begin{equation} \label{eq:squaredexponential_f}
f_W(q) = \left( \frac{l}{2 \sqrt{\pi}} \right)^3 \exp \left[-\frac{(q l)^2}{4} \right]
\end{equation}
and field correlation function
\begin{equation} \label{eq:squaredexponential_g}
g_W(r) = \exp\left[ -\left(\frac{r}{l} \right)^2 \right]
\end{equation}
where $l$ is model parameter that coincides with the characteristic length $l_W$. With this specific field the main results can be expressed in analytical form. Moreover the corresponding elementary wave $w(\mathbf{x})$ is also known analytically (see Tab. \ref{tab:fields}), which enables one to construct realizations and visually illustrate all considered dynamic models.
All results of Sec. \ref{sec:static} remain valid for time-dependent Gaussian fields. This is notably the case for the clipping relations between the field correlation and the covariances of the water, oil and surfactant phases. However, the field correlation function describes here the statistical correlation between the values of $W$ at two points at distance $r$ apart, with a time lag $\tau$, namely
\begin{equation} \label{eq:gW_rtau}
g_W(r,\tau) = \langle W(\mathbf{x},t) W(\mathbf{x}+\mathbf{r}, t + \tau) \rangle
\end{equation}
In this case, the covariances obtained through Eqs. (\ref{eq:Coo}-\ref{eq:Caa}) are Van-Hove correlation functions,\cite{Sivia:2011,Squires:2012} and the intensity obtained subsequently through Eqs. (\ref{eq:C_rho} - \ref{eq:I_Fourier}) is the coherent intermediate scattering function $I(q,\tau)$.
Throughout this section, the qualitative geometrical properties of the Gaussian fields are illustrated by clipping them at the value $\alpha=0$. This yields two-phase morphologies with volume fractions $\phi=1/2$, different from the emulsion in Fig. \ref{fig:data}. All the mathematical results, however, are quite general and remain valid for any clipping procedure. The specific case of the three-phase emulsion, with finite surfactant volume, is considered again in the discussion section.
\subsection{Dynamic model 1: independent time and space fluctuations}
\label{sec:model1}
In the first approach, a field is created with statistically-independent space and time fluctuations. This is achieved by starting from a series of independent static fields $W_n(\mathbf{x})$, with $n=1, \ldots, N$, and combining them linearly with time-dependent coefficients. The fields $W_n$ can be thought of as independent realisations of Eq. (\ref{eq:W_def}) each with different random numbers but the same spectral density $f_W(q)$. The statistical independence is expressed as
\begin{equation} \label{eq:W_def_model1}
\langle W_n(\mathbf{x}) W_m(\mathbf{x}+\mathbf{r}) \rangle = g_W(r) \delta_{mn}
\end{equation}
where $\delta_{mn} = 1$ for $m=n$ and 0 otherwise. Based on the set of $W_n(\mathbf{x})$ the time-dependent field is built as
\begin{equation} \label{eq:W_model1}
W(\mathbf{x},t) = \sqrt{\frac{2}{N}} \sum_{n=1}^N W_n(\mathbf{x}) \cos(\omega_n t - \varphi_n)
\end{equation}
where the phases $\varphi_n$ are random and uniform over $[0,2\pi)$, and the frequencies $\omega_n$ are drawn from a temporal spectral density $f'(\omega) \textrm{d}\omega$. With this first dynamic model, the space and time field correlation function in Eq. (\ref{eq:gW_rtau}) is found to be
\begin{equation} \label{eq:gW_model1}
g_W(r,\tau) = g_W(r) g'(\tau)
\end{equation}
in the limit of large $N$, with
\begin{equation} \label{eq:gW_time}
g'(\tau) = \int_0^\infty \cos[\omega \tau] f'(\omega) \textrm{d}\omega
\end{equation}
In geostatistics, models satisfying Eq. (\ref{eq:gW_model1}) are referred to as being separable.\cite{Gelfand:2010}
A physical interpretation of separable models is obtained by noting that they can be constructed in mathematically-equivalent way through a dilution approach, as in Eq. (\ref{eq:W_def_dilution}). Indeed, the field constructed as
\begin{equation}
W(\mathbf{x},t) = \sqrt{2} \sum_s A_s w(\mathbf{x}-\mathbf{x}_s) \cos(\omega_s t -\varphi_s)
\end{equation}
with $\varphi_s$ uniformly distributed in $[0, 2\pi)$, and $\omega_s$ distributed according to temporal spectral density $f'(\omega)$, has the same correlation function as in Eq. (\ref{eq:gW_model1}). Therefore, dynamic model 1 can be interpreted as resulting from incoherently fluctuating elementary waves.
\begin{figure}
\begin{center}
\includegraphics[width=6cm]{fig4_model1_realization.eps}
\caption{Two-dimensional $(x,t)$ cuts through four-dimensional $(x,y,z,t)$ realizations of dynamic model 1, with static field GRF-2 from Tab. \ref{tab:fields} and exponential (a) and hyperbolic secant (b) temporal correlation functions. The threshold assumed in the figure is $\alpha = 0$, corresponding to $\phi=0.5$ for the phases shown in white and grey.}
\label{fig:model1_realizations}
\end{center}
\end{figure}
For the purpose of data modelling, a natural choice for the temporal correlation function is the exponential $g'(\tau) = \exp[-\tau/\tau_c]$, where the correlation time $\tau_c$ is a model parameter. This choice corresponds to the following spectral density
\begin{equation}
f'(\omega) = \frac{1}{\pi} \frac{2 \tau_c}{1 + (\omega \tau_c)^2}
\end{equation}
A realization of the clipped Gaussian field obtained from this specific time-correlation function and static field GRF-2 is shown in Fig. \ref{fig:model1_realizations}a. The corresponding correlation function $g_W(r,\tau)$ is plotted in Fig. \ref{fig:model1}, together with the covariance assuming a single clipping threshold $\alpha=0$, and the corresponding intermediate scattering function in the form of $I(q,\tau)/I(q)$.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{fig5_model1_gw_c_i_dull.eps}
\caption{Correlation function $g_W(r,\tau)$ for static field GRF-2 and independent space and time fluctuations, with exponential (a1) and hyperbolic-secant (a2) time-correlation functions. The corresponding covariance (clipping threshold $\alpha=0$) and intermediate scattering function are shown in b1/b2 and c1/c2. The two solid lines in b1/b2 and c1/c2 highlight the values for $\tau = 0$ and for small yet finite $\tau$.}
\label{fig:model1}
\end{center}
\end{figure}
The type of dynamics obtained from the exponential time correlation function in Fig. \ref{fig:model1_realizations}a is extremely rugged. Smoother dynamics is obtained by modelling the temporal correlation function as a hyperbolic secant $g'(\tau) = 1/\cosh[\tau/\tau_c]$, which behaves like an exponential for asymptotically large times but differs for short times. Its temporal spectral density is
\begin{equation}
f'(\omega) = \frac{2 \tau_c \cosh[\pi \omega \tau_c/2]}{1 + \cosh[\pi \omega \tau_c]}
\end{equation}
A realization of the clipped Gaussian field obtained with this expression is shown in Fig. \ref{fig:model1_realizations}b. The corresponding correlation function $g_W(r,\tau)$, covariance (for $\alpha=0$), and intermediate scattering function are plotted in Fig. \ref{fig:model1}a2 to \ref{fig:model1}c2.
\subsection{Dynamic model 2: dispersion relation}
\label{sec:model2}
The second dynamic model introduces correlations between space and time fluctuations, and belongs to the class of non-separable models.\cite{Cressie:1999,Gneiting:2002} This is achieved through a dispersion relation that deterministically assigns a specific temporal frequency $\omega$ to any spatial frequency $q$ of the Gaussian field, and leads to the following generalization of Eq. (\ref{eq:W_def})
\begin{equation} \label{eq:W_model2}
W(\mathbf{x},t) = \sqrt{ \frac{2}{N}} \sum_{n=1}^N \sin\left[ \mathbf{q}_n \cdot \mathbf{x} + \omega(|\mathbf{q}_n|) t - \varphi_n \right]
\end{equation}
where $\omega(|\mathbf{q}|)$ is the dispersion relation, which we assume to be isotropic. Based on the statistical independence of the various components of the field in Eq. (\ref{eq:W_model2}), the field correlation function is calculated as
\begin{equation} \label{eq:gW_model2}
g_W(r,\tau) = \int_0^\infty f_W(q) \cos[\omega(q) \tau] \frac{\sin(qr)}{qr} 4 \pi q^2 \textrm{d}q
\end{equation}
in the limit of asymptotically large $N$.
In principle any suitable function can be used to model a dispersion relation. We consider here two simple analytical forms $\omega = c q$ and $\omega = D q^2$, where $c$ and $D$ are constants with dimensions of velocity and diffusion coefficient, respectively. Realizations obtained with static field GRF-2 and these two dispersion relations are given in Fig. \ref{fig:model2_realizations}a and \ref{fig:model2_realizations}b. In the case of the linear dispersion relation the structures propagate at constant velocity $c$, which appears as slanted features with slopes $\pm 1$ on the scales of the figure. In the case of quadratic dispersion the structures propagate with size-dependent velocity, which leads to more complicated temporal evolution.
\begin{figure}
\begin{center}
\includegraphics[width=6cm]{fig6_model2_realization.eps}
\caption{Two-dimensional $(x,t)$ cuts through four-dimensional $(x,y,z,t)$ realizations of model 2 of time-dependent Gaussian field, with GRF-2 from Tab. \ref{tab:fields}, and linear (a) and quadratic (b) dispersion relations. The threshold assumed in the figure is $\alpha = 0$, corresponding to $\phi=0.5$ for the phases shown in white and grey.}
\label{fig:model2_realizations}
\end{center}
\end{figure}
In the particular case of static field GRF-2, analytical expressions are obtained for the field correlation function through Eq. (\ref{eq:gW_model2}). For the linear dispersion relation, one finds
\begin{eqnarray} \label{eq:demo_g_linear}
g_W(r,\tau) = \exp\left[ -\frac{r^2 + (c \tau)^2 }{l_W^2} \right] \times
\Big\{
\cosh\left[ \frac{2 r c \tau }{l_W^2} \right] &-& 2 \left( \frac{c \tau}{l_W} \right)^2 \sinh\left[ \frac{2 r c \tau }{l_W^2} \right]/ \left[ \frac{2 r c \tau }{l_W^2} \right]
\Big\}
\end{eqnarray}
and for the quadratic dispersion, the relation is
\begin{equation} \label{eq:demo_g_quadratic}
g_W(r,\tau) = \frac{\exp\left[ - \frac{(r/l_W)^2}{1 + \left[ 4 D \tau/l_W^2\right]^2} \right]}{\left(1+\left[ 4 D \tau/l_W^2\right]^2 \right)^{3/4}}
\cos \left[ \frac{(r/l_W)^2 4D \tau/l_W^2}{1 + \left[ 4 D \tau/l_W^2\right]^2} - \frac{3}{2} \tan^{-1}\left[ \frac{4 D \tau}{l_W^2} \right] \right]
\end{equation}
Detailed derivations of these equations are given in the Supplementary Material (Sec. SM-4). The correlation functions and corresponding intermediate scattering functions are plotted in Fig. \ref{fig:model2}.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig7_model2_gw_c_i_dull.eps}
\caption{Correlation function $g_W(r,\tau)$ for GRF-2 from Tab. \ref{tab:fields} with linear (a1) and quadratic (a2) dispersion relations, together with corresponding intermediate scattering functions (b1 and b2) for clipping threshold $\alpha=0$.}
\label{fig:model2}
\end{center}
\end{figure}
An interesting characteristic of the time-dependent structure in Fig. \ref{fig:model2_realizations}a is that it displays temporal order in spite of being spatially disordered. The absence of any feature in $g_W(r,0)$ testifies to spatial disorder (see Fig. \ref{fig:model2}a1 and Fig. SM-3). By contrast, the correlation function $g_W(0,\tau)$ displays a sharp minimum at $\tau = \sqrt{3/2} \times l_W/c$. This corresponds to situation where a fixed point of space is visited alternatively by one phase and the other with quasi periodicity.
The temporal order is not obvious in the correlation function of the quadratic dispersion (Fig. \ref{fig:model2}a2), but the intermediate scattering function $I(q,\tau)$ exhibits marked oscillations for both dispersion relations (Figs. \ref{fig:model2}b1 and b2). This can be understood by noting that for asymptotically large values of $\tau$ the covariance is proportional to $g_W(r,\tau)$ (see the discussion of Fig. \ref{fig:clipping}), so that $I(q,\tau)$ is approximately the Fourier transform of $g_W(r,\tau)$. It therefore results from the Fourier inversion of Eq. (\ref{eq:gW_model2}) that $I(q,\tau)$ is proportional to $f_W(q) \cos[\omega(q) \tau]$ for large values of $\tau$. It is the cosine in this expression that is responsible for the observed oscillations in Fig. \ref{fig:model2}b1 and \ref{fig:model2}b2. For the particular representation as $I(q,\tau)/I(q)$ the oscillations are further amplified by the small value of $I(q)$ for large $q$.
\subsection{Dynamic model 3: moving waves}
\label{sec:model3}
Experimental scattering functions seldom display the type of marked oscillations obtained with dynamic model 2 and displayed in Fig. \ref{fig:model2}. In dynamic model 3, the temporal correlations and corresponding oscillations are naturally damped through a dilution approach, {\it i.e.} by using Eq. (\ref{eq:W_def_dilution}) instead of Eq. (\ref{eq:W_def}) to describe the Gaussian field. Explicitely, the Gaussian field is made time-dependent by allowing the elementary waves to propagate
\begin{equation} \label{eq:GRF_model3}
W(\mathbf{x},t) = \sum_s A_s w(\mathbf{x} - \mathbf{x}_s - \mathbf{j}_s(t))
\end{equation}
where $\mathbf{x}_s$ is the initial position of wave $s$, and $\mathbf{j}_s(t)$ is the vectorial distance it has travelled at time $t$. We consider two qualitatively different cases: the ballistic or diffusive motions of waves with velocity $c$ or diffusion coefficient $D$. For static field GRF-2 the shape of the elementary wave $w(\mathbf{x})$ is known mathematically (see Tab. \ref{tab:fields}), which enables one to construct realizations as shown in Fig. \ref{fig:model3_realizations_cD}a and b. The diffusive model displays the same type of rugged dynamics as in Fig. \ref{fig:model1_realizations}a, which we analyze in detail in the discussion section.
\begin{figure}
\begin{center}
\includegraphics[width=6cm]{fig8_model3_realization_cD.eps}
\caption{Two-dimensional $(x,t)$ cuts through four-dimensional $(x,y,z,t)$ realizations of dynamic model 3 with static field GRF-2 from Tab. \ref{tab:fields}, and (a) ballistic and (b) diffusive propagation of elementary waves. The threshold assumed in the figure is $\alpha = 0$, corresponding to $\phi=0.5$ for the phases shown in white and grey.}
\label{fig:model3_realizations_cD}
\end{center}
\end{figure}
The field correlation function corresponding to Eq. (\ref{eq:GRF_model3}) is calculated from the time-dependent distributions $f_t(\mathbf{j}) \textrm{d}V_j$ of the wave position $\mathbf{j}$ as follows
\begin{equation} \label{eq:integral_j}
g_W(\mathbf{r}, \tau )= \theta \langle A^2 \rangle \int K(\mathbf{r} - \mathbf{j}) f_{\tau}(\mathbf{j}) \ \textrm{d}V_j
\end{equation}
which generalizes Eq. (\ref{eq:gW_dilution}) to time-dependent dilution processes. If the elementary waves are compact in real space, then $K(\mathbf{r})$ has a range comparable to the characteristic length $l_W$ of the field. It therefore results from Eq. (\ref{eq:integral_j}) that all correlations disappear as soon as the elementary waves have travelled a distance comparable to $l_W$, which happens in a time $l_W/c$ or $l_W^2/D$ for the ballistic or diffusive case, respectively.
The case of ballistic propagation corresponds to density distribution
\begin{equation} \label{eq:fj_ballistic}
f_\tau(\mathbf{j}) = \frac{\delta(j - c \tau)}{4 \pi j^2}
\end{equation}
where $j =|\mathbf{j}|$, $\delta(.)$ is Dirac's function, and the denominator accounts for the normalization of the probabilities. With such distribution, Eq. (\ref{eq:integral_j}) reduces to an integration on the unit sphere, and leads to
\begin{equation} \label{eq:gW_model3_c}
g_W(r,\tau)= \theta \langle A^2 \rangle \frac{1}{2} \int_{-1}^{+1} K\left( \sqrt{r^2 + (c \tau)^2 - 2rc\tau \mu}\right) \ \textrm{d}\mu
\end{equation}
In the case of static field GRF-2, this can be calculated explicitly as
\begin{equation}
g_W(r, \tau)= \exp\left[ - \left(\frac{r - c\tau}{l_W} \right)^2 \right] \frac{1 - \exp\left[ -4 r c \tau /l_W^2 \right]}{4 r c \tau /l_W^2}
\end{equation}
which is plotted in Fig. \ref{fig:model3_cD}a1. The corresponding intermediate scattering function (assuming $\alpha=0$) is plotted in Fig. \ref{fig:model3_cD}b1.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig9_model3_gw_i_cD_dull.eps}
\caption{Correlation function $g_W(r,\tau)$ of model 3 for Gaussian fields GRF-2 with (a1) ballistic and (a2) diffusive motion of elementary waves, together with corresponding intermediate scattering functions (b1 and b2) for clipping threshold $\alpha=0$.}
\label{fig:model3_cD}
\end{center}
\end{figure}
In the diffusive case, the probability distribution of $\mathbf{j}$ is given by the classical expression for the position of a random walker\cite{Berg:1993,Cussler:2009}
\begin{equation} \label{eq:fj_diffusive}
f_t (\mathbf{j}) = \left( 4 \pi Dt \right)^{-3/2} \exp\left[ -\frac{|\mathbf{j}|^2}{4Dt} \right]
\end{equation}
For the case of static field GRF-2, Eq. (\ref{eq:integral_j}) then leads to the following field correlation function
\begin{equation} \label{eq:model3_D}
g_W(r,\tau) = \left( 1 + \frac{4 D \tau}{l_W^2} \right)^{-3/2} \exp\left[ -\frac{r^2}{l_W^2+4D\tau} \right]
\end{equation}
This function is plotted in Fig. \ref{fig:model3_cD}a2, with the corresponding intermediate scattering function (assuming $\alpha=0$) in Fig. \ref{fig:model3_cD}b2.
The asymptotic behaviour of the intermediate scattering function $I(q,\tau)$ for large $\tau$ can be understood by expressing the field correlation function in Eq. (\ref{eq:integral_j}) in terms of the spectral density as follows
\begin{equation} \label{eq:gW_model3_Fj}
g_W(r,\tau) = \int_0^\infty f_W(q) F_\tau(q) \frac{\sin[q r]}{qr} 4\pi q^2 \textrm{d}q
\end{equation}
where $F_\tau(q)$ is the Fourier transform of $f_\tau(\mathbf{j})$. This expression results directly from Eq. (\ref{eq:integral_j}) by equating $\theta \langle A^2 \rangle K(r)$ to the Fourier transform of $f_W(q)$. For the same reason as for model 2, the intermediate scattering function is proportional to the Fourier transform of $g_W(r,\tau)$ in the limit of asymptotically large $\tau$. In the case of model 3, this implies $I(q,\tau) \simeq f_W(q) F_\tau(q)$. In the case of the ballistic motion described by Eq. (\ref{eq:fj_ballistic}), the relevant value of $F_\tau(q)$ is
\begin{equation} \label{eq:Fj_c}
F_\tau(q) = \frac{\sin[qc \tau]}{q c \tau}
\end{equation}
which explains the mild oscillations in Fig. \ref{fig:model3_cD}b1. In the case of the diffusive motion described by Eq. (\ref{eq:fj_diffusive}), the relevant function is
\begin{equation} \label{eq:Fj_D}
F_\tau(q) = \exp\left[ - q^2 D \tau \right]
\end{equation}
which explains why no oscillations are observed at all in Fig. \ref{fig:model3_cD}b2. Finally, note that the expression of the field correlation function in Eq. (\ref{eq:gW_model3_Fj}) relies on the spectral density and makes no explicit reference to the elementary wave used to build the model. Therefore the usability of dynamic model 3 is not limited to static Gaussian fields for which the form of the elementary wave is known explicitly.
\section{Discussion}
\subsection{Temporal crossing rate}
\label{sec:crossing_rates}
An interesting characteristic of the Neutron Spin-Echo (NSE) data in Fig. \ref{fig:data}c1 and \ref{fig:data}c2 is the very steep $\tau$-dependence for small $\tau$ and large $q$. Among the three dynamic models discussed in Sec. \ref{sec:timedependentmodels}, this type of behavior was observed for model 1 with exponential time correlation function (Fig. \ref{fig:model1}c1), and for model 3 with diffusive motion of elementary waves (Fig. \ref{fig:model3_cD}b2). In both cases, the realisations testify to extremely rugged dynamics as illustrated in Figs. \ref{fig:model1_realizations}a and \ref{fig:model3_realizations_cD}b.
A useful mathematical concept to describe the two types of dynamics in both Fig. \ref{fig:model1_realizations} and Fig. \ref{fig:model3_realizations_cD} is the temporal crossing rate $n_t$, which characterizes how often a fixed point in space is crossed by moving interfaces. As discussed shortly, this concept is the temporal equivalent of the spatial notion of specific surface area. The surface area $a_V$ is defined as the total area of an interface per unit volume of the system. For an isotropic system it is mathematically related to the notion of average chord length, which characterizes how frequently one crosses the interface when traveling along any straight line crossing the system.\cite{Gommes:2020B} The significance of the surface area for scattering was first acknowledged by Debye, who related $a_V$ to the small-$r$ behaviour of the covariance as\cite{Debye:1957}
\begin{equation} \label{eq:Debye}
C(r,0) \simeq \phi - \frac{a_V}{4} r + \ldots
\end{equation}
which converts in reciprocal space to the well-known Porod's law\cite{Guinier:1963,Ciccariello:1988,Ciccariello:1995}
\begin{equation} \label{eq:Porod}
I(q,0) \simeq \frac{2\pi a_V}{q^4}
\end{equation}
In the particular case of clipped Gaussian-field structures, the surface area of an isosurface, say at $W(\mathbf{x})=\alpha$, is calculated as\cite{Teubner:1991,Berk:1991}
\begin{equation} \label{eq:surface_area}
a_V= \frac{2\sqrt{2}}{\pi l_W} e^{-\alpha^2/2}
\end{equation}
where $l_W$ is the characteristic length of the field defined in Eq. (\ref{eq:lW}).
The crossing rate $n_t$ is related to the covariance via
\begin{equation} \label{eq:C_nt}
C(0,\tau) = \phi - \frac{n_t}{2} \tau + \ldots
\end{equation}
which illustrates further the similarity between $n_t$ and the surface area $a_V$ in Eq. (\ref{eq:Debye}). The factor
$1/2$ differs from the factor $1/4$ in Eq. (\ref{eq:Debye}) because the temporal process considered here is one-dimensional.\cite{Torquato:2002} From Eq. (\ref{eq:C_nt}) the crossing rate can be calculated as the limit of $-2 (\partial C/\partial \tau)$ for vanishingly small $r$ and $\tau$. In the case of a clipped Gaussian field model, this limit corresponds to values of the field correlation function $g_W(r,\tau)$ asymptotically close to 1. The derivative can then be calculated using the asymptotic result in Eq. (\ref{eq:Lambda2_large_g}). The following expression is then obtained for the crossing rate
\begin{equation} \label{eq:nt}
n_t = \frac{\sqrt{2}}{\pi} e^{-\alpha^2/2} \lim_{\tau \to 0} \frac{\sqrt{1-g_W(0,\tau)}}{\tau}
\end{equation}
which provides a physical interpretation to the small-$\tau$ behavior of $g_W(0,\tau)$. The condition for having finite $n_t$ is that the field correlation function should be quadratic at the origin
\begin{equation}
g_W(0,\tau) \simeq 1 - (\tau/\tau_W)^2 + \ldots
\end{equation}
which defines a natural characteristic time $\tau_W$. Note that the two dynamic models of Sec. \ref{sec:timedependentmodels} displaying rugged dynamics both have correlation functions that are linear at the origin. In the case of model 1 with exponential correlation function, Eq. (\ref{eq:gW_model1}) leads to
\begin{equation}
g_W(0,\tau) = 1 - \tau/\tau_c + \ldots
\end{equation}
and in the case of model 3 with diffusive wave motion, Eq. (\ref{eq:model3_D}) leads to
\begin{equation} \label{eq:gW_tau_D}
g_W(0,\tau) = 1 - \frac{6 D \tau}{l_W^2} + \ldots
\end{equation}
In both cases $\tau_W$ is not defined and Eq. (\ref{eq:nt}) predicts infinite crossing rate. The effect of linear versus quadratic correlation at short times is illustrated further in Fig. \ref{fig:excursion_1D}.
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig10_excursions_1D.eps}
\caption{Field correlation functions at a fixed point in space $g_W(0,\tau)$ for dynamic model 1 with exponential (blue) and hyperbolic-secant (black) temporal correlation functions. The insets are realizations of the time-dependent Gaussian fields at a fixed point in space, together with clipped structure with $\alpha=0$, for the exponential (a) and hyperbolic-secant (b) models. The dashed line highlights the quadratic shape of the hyperbolic-secant model at the origin.}
\label{fig:excursion_1D}
\end{center}
\end{figure}
Whether the crossing rate $n_t$ is finite or infinite controls the shape of the intermediate scattering function $I(q,\tau)$ for asymptotically large $q$ and small $\tau$ (see Figs. \ref{fig:model1} and \ref{fig:model3_cD}). The analysis builds on the following two mathematical facts. (i) First, the non-linearity of the clipping function at $g_W=1$ is of the type $C \simeq 1 - \sqrt{1-g_W}$ (see Fig. \ref{fig:clipping} and Eq. \ref{eq:Lambda2_large_g}). In particular, this converts a quadratic field correlation $g_W(r,\tau) \simeq 1-r^2$ into a linear covariance $C(r,\tau) \simeq 1-r$. This also converts a linear field correlation $g_W(r,\tau) \simeq 1-\tau$ into a singular covariance $C(r,\tau) \simeq 1-\sqrt{\tau}$. (ii) Second, the asymptotic behavior of $I(q,\tau)$ for large $q$ is controlled by the small-$r$ behavior of $C(r,\tau)$. This results from a generalization of the Riemann-Lebesgue lemma by Lighthill\cite{Lighthill:1958,Berk:1991}, and is shortly discussed in the Supplementary Material (Sec. SM-6). In particular, a covariance that is linear at the origin $C(r,0) = 1 - r$ leads to a $1/q^4$ scattering, in line with Porod's law in Eq. (\ref{eq:Porod}). By contrast, a covariance whose derivatives all vanish at $r=0$ leads to a scattering that decreases faster than any power law.
Consider now the case where $\tau_W$ exists, {\it i.e.} where $g_W(r,\tau)$ is quadratic in $\tau$ at the origin, and $C(r,\tau)$ is linear in $\tau$ ({\it e.g.} Figs. \ref{fig:model1}b2). In that case Porod's law holds not only for $I(q)$ but also for $I(q,\tau)$ for small $\tau$s, because $C(r,\tau)$ is a smooth function of $\tau$. As a consequence $I(q,\tau)/I(q)$ approaches the value 1 horizontally for $\tau \to 0$ (Fig. \ref{fig:model1}c2). By contrast, if $\tau_W$ is not defined the covariance $C(r,\tau)$ varies like $\sqrt{\tau}$, which has infinite slope for $\tau \to 0$. Accordingly $C(r,\tau)$ passes from being linear in $r$ to having vanishing derivative over infinitesimally short interval of $\tau$ (Fig. \ref{fig:model1}b1). The intermediate scattering function $I(q,\tau)$ passes discontinuously from Porod's law for $\tau = 0$ to decreasing faster than $q^{-4}$ for arbitrarily small $\tau > 0$. This explains the very steep intermediate scattering function $I(q,\tau)$ for large $q$ and small $\tau$ in Fig. \ref{fig:model1}c1. The same explanation holds for Fig. \ref{fig:model3_cD}b2.
In the case of model 1 the existence $\tau_W$ and the finiteness of $n_t$ can be ascertained by direct examination of $g'_W(\tau)$. In the case of model 2, one has to examine both the dispersion relation and the spectral density. Assuming a dispersion relation of the type
\begin{equation}
\omega(q) = a_n q^n
\end{equation}
where $a_n$ and $n$ are constants, a truncated expansion of the cosine factor in Eq. (\ref{eq:gW_model2}) leads to
\begin{equation} \label{eq:tauW_dispersion}
\frac{1}{\tau_W^2} = \frac{a_n^2}{2} \int_0^\infty f_W(q) q^{2n} 4 \pi q^2 \textrm{d}q
\end{equation}
In the particular case of a linear dispersion relation, with $a_1 = c$, the characteristic time is proportional to the characteristic length $\tau_W = l_W/(c \sqrt{3})$. However, for exponents $n$ larger than one the conditions are more stringent for $\tau_W$ than for $l_W$. The condition for non-vanishing $\tau_W$ is that the spectral density $f_W(q)$ should decrease faster than $q^{-\nu}$ with $\nu = 2 n +3$ (see also Sec. SM-VII in the supporting information) . Finally, in the case of model 3, it results from Eq. (\ref{eq:gW_model3_Fj}) that the ballistic case always leads to finite $n_t$, provided $l_W$ is finite. Replacing Eq. (\ref{eq:Fj_c}) by a truncated expansion for small $\tau$, the following expression is indeed obtained
\begin{equation}
g_W(0,\tau) \simeq 1 - \left( \frac{c \tau}{l_W} \right)^2 + \ldots
\end{equation}
which shows that $\tau_W = l_W/c$. By contrast, in the diffusive case, $g_W(0,\tau)$ is linear in $\tau$ and $n_t$ is always infinite, as already shown in Eq. (\ref{eq:gW_tau_D}). The latter equation holds for all spectral densities, and it is not limited to static field GRF-2.
\subsection{NSE data analysis}
\label{sec:data_analysis}
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig11_SANS_twowaves.eps}
\caption{(a) Fitting of the microemulsion SANS patterns of Fig. \ref{fig:data} as a sum of two static GRF contributions, with bulk and film contrasts in red and grey, respectively. The dots are the data, and solid lines are the fits. The two contributions to the spectral density and the correlation functions are shown in b and c, with background contribution (GRF-2, dotted), correlated contribution (GRF-4,dashed) and their sum shown as a solid line.}
\label{fig:SANS_twowaves}
\end{center}
\end{figure}
The developed approach offers the possibility of decomposing a given static structure into distinct contributions, and building composite time-dependent models whereby each structural contribution is animated according to different dynamic models. In the particular case of the microemulsion, the static SANS data hint at two types of structures as illustrated in Fig. \ref{fig:SANS_twowaves}. A distinctive feature of the SANS is the presence of a sharp scattering peak in the bulk contrast data around 0.03 \AA$^{-1}$, corresponding to strongly correlated structures. That peak alone, however, does not describe the entire SANS as it is superimposed with a more diffuse and featureless background scattering that extends over the entire $q$ range. We now endeavour to model these two contributions with suitable Gaussian fields, and use these SANS-born models to explore the NSE data.
A natural choice for modelling the background-like contribution in the SANS is the static field GRF-2, which was used for illustrative purposes throughout Sec. \ref{sec:timedependentmodels}. As the spectral density of GRF-2 (see Eq. \ref{eq:squaredexponential_f}) displays no peak, it is not suitable to model the correlated part of the structure. For the latter contribution, we introduce a new static field, the elementary wave of which is built as the Laplacian of GRF-2, namely
\begin{equation} \label{eq:w_Laplacian}
w(\mathbf{x}) = \left[ \left(\frac{ |\mathbf{x}|}{l}\right)^2 - \frac{3}{4} \right] \exp\left[- 2 \left( \frac{|\mathbf{x}|}{l} \right)^2 \right]
\end{equation}
The static properties of this field are given in Tab. \ref{tab:fields} under the name GRF-4. By construction the elementary wave in Eq. (\ref{eq:w_Laplacian}) satisfies
\begin{equation} \label{eq:int_0}
\int w(\mathbf{x}) \textrm{d}V_x = 0
\end{equation}
This is equivalent to $f_W(0)=0$ by virtue of Eq. (\ref{eq:fW_w}), which leads to the desired scattering peak in the SAS patterns (See also Fig. SM-4). Interestingly, in the context of moving-wave dynamic models (model 3) when a given volume is crossed by any wave satisfying Eq. (\ref{eq:int_0}) the local average value of the Gaussian field $W(\mathbf{x})$ remains unchanged, in a neighbourhood of size larger than $l_W$. Therefore, the propagation of the elementary wave in Eq. (\ref{eq:w_Laplacian}) preserves locally the volumes of the phases. We therefore refer to GRF-4 as a {\em deformation} mode. The volume-preservation property can also be understood by noting that the low-$q$ limit of the scattered intensity is proportional to the compressibility of the phases \cite{Glatter:1982} so that a spectral density that vanishes for $q \to 0$ corresponds to incompressible phases. By contrast, GRF-2 leads to local modifications of the the volumes, and we refer to it as a {\em breathing} mode. The spectral densities of these two modes are illustrated in Fig. \ref{fig:SANS_twowaves}b.
To analyze the microemulsion SANS data the breathing and deformation modes were combined into a single Gaussian field, following Eq. (\ref{eq:combine_W}). This leads to a three-parameter static-field model, with the characteristic lengths of each mode and their relative contribution to the field variance. The clipping thresholds $\alpha=-0.234$ and $\beta=+0.234$ are imposed by the volume fractions. The least-square fit of both bulk- and film-contrast data is illustrated in Fig. \ref{fig:SANS_twowaves}. The breathing mode contributes 70 \% of the variance ($\sigma_b^2 \simeq 0.7$) with lengths $l_d \simeq $ 98 \AA \ and $l_b \simeq$ 65 \AA \ for the deformation and breathing modes. These numerical values of $l_d$ and $l_b$ coincidentally correspond to the same characteristic lengths $l_W \simeq 65$ \AA (see Tab. \ref{tab:fields}).
\begin{figure}
\begin{center}
\includegraphics[width=14cm]{fig12_NSE_composite_models.eps}
\caption{Fitting of the microemulsion NSE data with the composite breathing-and-deformation model (GRF-2 and GRF-4), with the dynamics of the modes modelled either as (i) fluctuations, (ii) ballistic waves, or (iii) diffusive waves. From top to bottom: the breathing mode is assigned fluctuation (a), ballistic (b), or diffusive dynamics (c). From left to right: the deformation mode is assigned fluctuation (1), ballistic (2), or diffusive dynamics (3). In each case the dots are the data with bulk (grey) and film (red) contrasts, and the surface is the model. The corresponding parameters are in Tab. \ref{tab:parameters}.}
\label{fig:NSE_composite}
\end{center}
\end{figure}
In order to analyze the NSE data, a dynamic model has to be assumed for each mode. As the notion of breathing and deformation is inspired by an elementary-wave interpretation we restrict the analysis to model 1 (fluctuating waves) and model 3 (ballistically or diffusively propagating waves). Moreover, as the NSE data exhibit steep slope for large $q$ and small $\tau$ (Fig. \ref{fig:data}c1-c2), we consider only the exponential correlation function for model 1 with infinite crossing rate $n_t$. In the following, we explore systematically all combinations of the three types of dynamics for the two modes, which leads to nine composite time-dependent Gaussian fields. The least-square fits of the NSE data are illustrated in Fig. \ref{fig:NSE_composite}, and the values of the corresponding parameters and $\chi^2$ are reported in Tab. \ref{tab:parameters}. The fitting required the correlation function $g_W(r,\tau)$ to be known for the deformation mode (GRF-4), for both ballistic and diffusive wave propagation. All details are provided in Sec. SM-V of the Supplementary Material.
\begin{table}
\begin{tabular}{c|c|p{4cm}|p{4cm}|p{4cm}| }
\multicolumn{2}{r}{}
& \multicolumn{3}{c}{Deformation mode} \\
\cline{3-5}
\multicolumn{2}{r}{}
& \multicolumn{1}{c}{Fluct. ($\tau_d$)}
& \multicolumn{1}{c}{Ball. ($c_d$)}
& \multicolumn{1}{c}{Diff. ($D_d$)} \\
\cline{3-5}
\parbox[t]{10mm}{\multirow{7}{*}{\rotatebox[origin=c]{90}{Breathing mode}}}
& \parbox[t]{8mm}{\rotatebox[origin=c]{90}{Fluct. ($\tau_b$)}} & \makecell{$\tau_b = 1000$ ns$^*$ \\ $\tau_d = 519$ ns \\ ($\chi^2=10$) } &
\makecell{$\tau_b = 130$ ns \\ $c_d = 0.088$ \AA/ns \\ ($\chi^2=11$) } &
\makecell{$\tau_b = 1000$ ns$^*$ \\ $D_d = 1.3$ \AA$^2$/ns \\ ($\chi^2=8.2$) } \\
\cline{3-5}
& \parbox[t]{8mm}{\rotatebox[origin=c]{90}{Ball. ($c_b$)} }& \makecell{$c_b = 0.001$ \AA/ns$^*$ \\ $\tau_d = 429$ ns \\ ($\chi^2=9.6$) } &
\makecell{$c_b = 0.82$ \AA/ns \\ $c_d = 0.003$ \AA/ns \\ ($\chi^2=14$) } &
\makecell{$c_b = 0.001$ \AA/ns$^*$ \\ $D_d = 1.7$ \AA$^2$/ns \\ ($\chi^2=7.4$) } \\
\cline{3-5}
& \parbox[t]{8mm}{\rotatebox[origin=c]{90}{Diff. ($D_b$)}} & \makecell{$D_b = 4.0$ \AA$^2$/ns \\ $\tau_d = 1000$ ns$^*$ \\ ($\chi^2=8.4$) } &
\makecell{$D_b = 8.6$ \AA$^2$/ns \\ $c_d = 0.081$ \AA/ns \\ ($\chi^2=7.6$) } &
\makecell{$D_b = 0.001$ \AA$^2$/ns$^*$ \\ $D_d = 1.7$ \AA$^2$/ns \\ ($\chi^2=7.4$) } \\
\cline{3-5}
\end{tabular}
\caption{Parameters of the composite model, whereby each mode (breathing or deformation, both with $l_W \simeq 65$ \AA) is assigned either a fluctuating, ballistic or diffusive dynamics. The values were obtained from the least-square fits in Fig. \ref{fig:NSE_composite}, and the $\chi^2$ values are also reported. The stars highlight values that have converged to the lower bound allowed for the fit.}
\label{tab:parameters}
\end{table}
Globally, none of the composite models in Fig. \ref{fig:NSE_composite} is able to quantitatively account for both the bulk- and film-contrast NSE data. The model that fares worst is the one that assumes ballistic wave propagation for both breathing and deformation modes. This model has finite crossing rate $n_t$, and is unable to capture the steep experimental intermediate scattering function. Based on the $\chi^2$ values in Tab. \ref{tab:parameters}, the data is best described when both modes have diffusive dynamics. More generally, it is interesting to note that the calculated intermediate scattering functions in Fig. \ref{fig:NSE_composite} are generally smaller than the data, particularly for the film-contrast scattering, so that the breathing and deformation-mode approach overestimates the dynamics. This is also manifest in the values of Tab. \ref{tab:parameters}, which often converge towards the lower limits allowed on the parameters. This means that one of the two modes is practically static and does not contribute to the dynamics.
As an alternative to the composite model based on breathing and deformation modes, the stochastic models offer the possibility of a more general approach based on a dispersion relation (dynamic model 2). This enables one to better match experimental data by tuning the dynamics in a scale-dependent way through the value of $q$. Moreover, the dispersion-relation approach does not require one to explicitly decompose the SANS into substructures, so that the same piecewise-linear spectral density (GRF-8) as in Fig. \ref{fig:data}a and SM-10 can be used to describe the underlying static Gaussian field.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig13_NSE_dispersion.eps}
\caption{Least-square fit of the NSE data from Fig. \ref{fig:data} with a dispersion relation (dynamic model 2) and a static field with piecewise-linear spectral density (GRF-8, same as Fig. \ref{fig:data}a), for both bulk contrast (a) and film contrast (b). The dispersion relation is Eq. (\ref{eq:dispersion}) with parameter $a \simeq 2053\ \AA^3$/ns and $q_c \simeq 0.047\ \AA^{-1}$, resulting in $\chi^2 = 6.3$. The surfaces are the model and the dots are the data, with errorbar $\pm 2 \sigma$.}
\label{fig:NSE_dispersion}
\end{center}
\end{figure}
Although any dispersion relation can in principle be chosen to model the NSE data, it is desirable that it should lead to an infinite crossing rate $n_t$ so as to capture the steep intermediate scattering function. As discussed in Sec. SM-III of the Supplementary Material, the spectral density of the piecewise-linear model decreases asymptotically as $q^{-6}$ so that any dispersion relation with order $n>3/2$ would lead to infinite $n_t$. In practice, the following dispersion relation is found to describe fairly the NSE data
\begin{equation} \label{eq:dispersion}
\omega = a (q-q_c)^3 H[q-q_c]
\end{equation}
where $H()$ is Heaviside's step function, $q_c$ is a cutoff frequency, and the parameter $a$ sets the value of $\omega$. The least-square fit is illustrated in Fig.\ref{fig:NSE_dispersion} and in Fig. SM-12 as 2D plots. The values of the fitted parameters are $a \simeq 2050 \pm 60 \ \AA^3$/ns and $q_c \simeq 0.047 \pm 0.0002 \ \AA^{-1}$, resulting in $\chi^2 = 6.3$, which is a significant improvement compared to the values in Tab. \ref{tab:parameters}. The reported uncertainties on $a$ and $q_c$ were obtained from a Monte-Carlo estimation, with a $\pm 2 \sigma $ normal-distributed error on each NSE data point. The small errors on the parameters results from the fact that all the NSE data - over the entire $q$ and $\tau$ ranges, and for the two contrasts - are fitted jointly with only two adjustable parameters which makes it quite robust. This also offers the prospect of reducing the number of experimental data points needed to reliably adjust a model (see Fig. SM-15).
As an alternative to Eq. (\ref{eq:dispersion}) the NSE data were also fitted with a dispersion relation modelled as a sum of power laws (from $n=1$ to $n=4$) with adjustable factors. Such fit converges to a situation where factors with alternating signs contribute to keeping $\omega$ close to zero for small $q$, leading to an overall shape similar to the cutoff frequency used in Eq. (\ref{eq:dispersion}) (see Fig. SM-13).
The $q^3$ dependence assumed in Eq. (\ref{eq:dispersion}) appears in a variety of dynamic structure factors involving hydrodynamic interactions, although the specifics of the relaxation curve are system-dependent. The exponent $3$ notably appears in the Zimm model of polymers in solvents to describe the thermally driven fluctuations of a polymer chain \cite{Zimm:1956,Dubois:1967}. It also appears in the Zilman-Granek analysis of membrane fluctuations with bending rigidity \cite{Zilman:1996}. In the present context this can be understood from the following general scaling argument. The very observation of infinite $n_t$ hints at thermal random motion. By itself, this would lead to a quadratic dispersion relation $\omega = D q^2$, where the diffusion coefficient $D$ is related to the characteristic size $L$ by Stokes-Einstein relation $D \simeq k_B T /(\eta L) $, where $k_BT$ is the thermal energy and $\eta$ is the viscosity of the medium. In the case of microemulsion deforming over a variety of length scales, inversely related to the scattering vector $L \sim q^{-1}$, this suggests the following cubic dispersion relation
\begin{equation}
\omega \sim \frac{k_B T}{\eta} q^3
\end{equation}
Using the value $\eta \simeq 10^{-3}$ Pa.s for the effective viscosity of the water/decane microemulsion as reported in ref.\cite{Holderer:2005} the factor in this scaling law takes the value 4000 \AA$^{3}$/ns, which is of the same order of magnitude as the value inferred from the NSE data fitting.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{fig14_realizations_fluctuations_M100_twolines.eps}
\caption{(a) Time-dependent realizations of the model of Fig. \ref{fig:NSE_dispersion} over 50 ns, with oil in grey, surfactant in red, and water in white; (b) realization from the static components of the field only, corresponding to spectral density limited to $q < q_c$; (c) average oil density calculated over the duration of the simulation; (d) average oil density profile as a function of distance to the interface of the static components ($d$ is positive into the oil). The blue area highlight the 80 \% confidence interval, with width $\simeq 60$ \AA. The four curves in d were obtained from independent realizations.}
\label{fig:realization_versus_time}
\end{center}
\end{figure}
The time-dependent structure of the microemulsion is illustrated in Fig. \ref{fig:realization_versus_time} with a particular realization of the dispersion-relation model over a time interval of 50 ns. One notes in particular the stability in time of large-scale structures, larger than approximately $2 \pi/q_c \simeq 130$ \AA, where $q_c$ is the cutoff frequency from Eq. (\ref{eq:dispersion}). The average position of the oil and water phases does not change over the timescale of the figure. However, the interfaces deform in random and very rugged way at smaller scale, as typically expected from a system with infinite crossing rate $n_t$. This picture matches the physical intuition as the redistribution of oil and water over long distances occurs through slow hydrodynamic flow, while no such obstacle exists for the local fluctuations. This scale-dependence is captured by the Stokes-Einstein relation, and is responsible for the observed $q^3$ dynamics. On the other hand, the physical origin of the cutoff size $q_c$ remains unclear, although its phenomenology is reminiscent of de-Gennes narrowing, whereby relaxation times are often observed to scale with the scattering structure factors\cite{Myint:2021}.
In order to better understand the realizations of the fitted model, the structure was further decomposed into its static and time-dependent components. The realisation in Fig. \ref{fig:realization_versus_time}b was obtained by setting to zero all components of the spectral density $f_W(q)$ with $q> q_c$, which results in much smoother interface. This is manifest in the characteristic lengths of the field (Eq. \ref{eq:lW}), which pass from $l_W \simeq 52$ \AA \ to $l_W \simeq 73$ \AA. Based on Eq. (\ref{eq:surface_area}) this corresponds to surface areas $a_V \simeq 170$ m$^2$/cm$^3$ and $a_V \simeq 120$ m$^2$/cm$^3$, respectively. The thermal fluctuations of the interface therefore contribute to as much as 30 \% of the area of the interfaces. Due to the symmetry of the model (with clipping constants $\alpha=-\beta$) the surfactant/oil and surfactant/water interfaces have identical areas. Another interesting aspect of the fluctuations is their amplitude, which can be estimated by evaluating first the average density, say, of oil over the entire duration of a simulation. This is illustrated in Fig. \ref{fig:realization_versus_time}c, where the smooth transition between the white and black areas correspond to all the successive positions of the interface over time. The extent of the transition in the direction locally orthogonal to the interface is given in Fig. \ref{fig:realization_versus_time}d. During 80 \% of the time, the interface fluctuates within a 60 \AA-thick layer that extends on both sides of the average position. It is interesting to compare that value to the size of the oil and water phases, estimated as an average chord length\cite{Gommes:2020B} as $4 \phi/a_V \simeq 130 \ \textrm{\AA}$ for the average structure in Fig. \ref{fig:realization_versus_time}b. In other words, the interface fluctuates over distances as large as half the size of the phases.
The dispersion-relation analysis hints at reasons why the breathing and deformation-mode analysis was unable to account for the NSE data of the emulsion. The dispersion relation points indeed at two dynamic regimes but they are separated by a cutoff frequency, which is much sharper a transition than between GRF-2 and GRF-4. It has also to be noted that the very concept of independent modes contributing additively to the dynamics, is strictly justified only as a linear approximation. Given the observed large amplitude of the interface fluctuations, non-linear effects could be expected which would rule out any possibility of linear-mode decomposition.
\section{Conclusion}
Clipped Gaussian field models have been extensively used to analyze the elastic small-angle scattering data of disordered systems. When applied to dynamical systems, such classical approach provides static snapshots of a structure. In the paper, the models are generalised to make them time-dependent, which enable one to analyze consistently both the instantaneous spatial structures and their dynamics within a single statistical description. General expressions are derived for all the space- and time-correlation functions relevant to coherent inelastic neutron scattering, for multiphase systems and arbitrary scattering contrasts between the phases.
With the proposed approach, for any given static structure inferred {\it e.g.} from small-angle scattering, a variety of distinctly different dynamics can be modelled. In a first family of models, the Gaussian field underlying the structure is decomposed into a large number of localised elementary waves. Qualitatively different dynamics are obtained by letting the waves randomly fluctuate, or propagate ballistically or diffusively through the system. In another family of models, the spectral components of the Gaussian field are assigned any desired dynamics through a suitable dispersion relation. The various types of dynamics lead to qualitatively different intermediate scattering functions, which enables one to discriminate them through neutron scattering. Moreover, all these approaches can be combined to yield models with composite and possibly realistic dynamics.
A central characteristic of the dynamic models, which controls the shape of the intermediate scattering functions, is their temporal crossing rate. This is defined by considering a fixed point in space, and evaluating how often it is passed through by a moving interface of the time-dependent structure. Systems undergoing Brownian-like thermal fluctuations have infinite crossing rate, which converts to infinitely steep intermediate scattering functions for asymptotically large $q$ and small $\tau$.
The methods of the paper were illustrated with the analysis of neutron small-angle scattering and spin-echo data measured on oil/water microemulsions. The methodology consisted in analyzing first the SANS data in order to determine the spectral density of the Gaussian field underlying the static structure, corresponding to snapshots of the time-dependent structure. As a second step, the NSE data was analyzed by complementing the so-determined static spectral density with few dynamic parameters. This enabled us to analyze jointly the entire SANS and NSE data, in both film and bulk contrasts and over the entire range of $q$ and $\tau$, with a single coherent model. The small number of adjusted parameter contributes to the robustness of the NSE analysis, and offers the prospect of reducing the number of experimental points required to reliably adjust a model.
From a physical perspective, the SANS and NSE data of the emulsion point to a static large-scale structure of the oil and water domains, with thermal fluctuations of the interfaces. The interface fluctuations take place over distances as large as 60 \AA , corresponding to half the domain size, and contribute to 30\% of the total interface area. In future work the stochastic approach will be explored further to analyze the wavelike dynamics observed by neutron spin-echo in lipid membranes\cite{Jaksch:2017}.
\section*{Supplementary Material}
See supplementary material for the mathematical derivation of some equations, for numerical data-analysis procedures, as well as for additional figures.
\section*{Data Availability}
All SANS and NSE data discussed in the paper can be downloaded from the authors' institutional repository at \url{https://doi.org/10.26165/JUELICH-DATA/DJ3LIN}.
\nocite{*}
|
2107.05590
|
\section{Introduction}
Relativistic heavy-ion collisions produce small systems of strongly interacting matter of extremely high energy densities in which possibly a new state of deconfined matter, consisting of free quarks and gluons, called the Quark Gluon Plasma (QGP) is created \cite{Stoecker:1986ci}. The created hot and dense system expands rapidly under its own pressure and gradually cools down back to a dilute gas of hadrons which can be detected as final state particles in experiments.
In QCD thermodynamics, the transition from a gas of hadrons to a QGP is likely a smooth crossover at high temperatures and very small baryon densities as established by lattice QCD \cite{Aoki:2006we,Borsanyi:2013bia,Bazavov:2014pvz}. A first order phase transition is conjectured at lower temperatures and moderate baryon densities \cite{Fukushima:2010bq}. Re-constructing the complete QCD phase diagram and identifying the regions of these transitions and thereby identifying the possible critical point, by means of experimental observations, is the major goal for the heavy-ion collision programs at Relativistic Heavy Ion Collider (RHIC), Large Hadron Collider (LHC) and the future Facility for Antiproton and Ion research (FAIR).
The Compressed Baryonic Matter (CBM) experiment at FAIR is a fixed target experiment that will study the phase structure of dense QCD matter with nucleus-nucleus collisions of energies up to $45 A$GeV in the lab frame \cite{Friese:2006dj, Senger:2006wd, Staszel:2010zza}. The physics program of the CBM experiment includes the exploration of high density equation of states such as in neutron star cores and the search for phase transitions at finite baryon densities \cite{Ablyazimov:2017guv,Senger:2020pzs}. The experiment will run at an unprecedented interaction rate of up to 10 MHz and the CBM detector will measure up to 1000 charged particles per collision. An online event selection algorithm \cite{deCuveland:2011zz} that performs ultra fast event reconstruction will be used to select interesting events for permanent storage from about 1 TBytes/s of collected data. Extracting the physics hidden in the vast amounts of data generated in this ambitious experiment requires the development of new techniques that can perform fast, accurate and real time physics analyses on raw experimental output.
The incoming data stream from the detector is processed by different algorithms to perform event reconstruction \cite{Kisel:2006yu}, particle identification and event selection \cite{deCuveland:2011zz} before different physics analyses can be performed. Events reconstructed and selected by these algorithms are used to calculate observables such as anisotropic flow and particle multiplicity fluctuations which are sensitive to a phase transition \cite{Rischke:1995pe}. Multi-parameter fits of the model simulations to the experimental data for these observables are currently used in experiments to search for phase transitions and to calculate the bulk properties of QCD medium. Bayesian analysis methods have been proposed as a method to fit the parameters to these observables\cite{Pratt:2015zsa, Bernhard:2016tnd,Bernhard:2019bmu}. An alternate approach to identify the appearance of a phase transition in QCD matter is based on Deep Learning \cite{lecun2015deep} techniques. Such DL techniques are considered so-called end-to-end approaches, where the DL model themselves determine the interesting features of the data and perform a classification task on these features. In \cite{Pang:2016vdc}, Convolutional Neural Networks \cite{gu2018recent} were trained on pion spectra (p$_t$, $\phi$) from hydrodynamic simulations to classify the EoS of a possible QCD transition. The study performed on the hydrodynamic output showed an average prediction accuracy greater than 95$\%$. A follow up study was presented in \cite{Du:2019civ} where a hadronic cascade model was employed after hydrodynamic evolution in the simulations to achieve a realistic freeze-out as well as including the effect of having a finite number of measurable particles in single events. The hadronic cascade "after-burner" introduces uncertainties in the final state spectra due to resonance decays and hadron rescatterings. This results in discrete particle spectra with predominant event-by-event fluctuations unlike the smooth spectra produced by pure hydrodynamic simulations.
DL methods are reliable and accurate in identifying QCD transitions in heavy-ion collisions. However, as reported in \cite{Du:2019civ}, the performance depends largely on the fluctuations in the final state spectra. Therefore, if such a DL based EoS-meter is to be used on the direct output of a heavy ion experiment, an extensive analysis on the response of the DL model on additional uncertainties introduced by e.g. the detector resolution, acceptance region and efficiency of the reconstruction algorithms is necessary. The model should not only be robust against these constraints but also meet the performance in terms of accuracy and speed as demanded by the experiment.
In this study, the effects of experimental uncertainties and detector effects on the predictions of DL models for classifying QCD transitions at CBM experiment are investigated. The DL models were trained on a data similar to an experimental output by the use of a comprehensive data preparation pipeline that includes detector simulation and reconstruction algorithms. We demonstrate a novel DL model that can identify the EoS of QCD transition from raw experimental output and its performance on different situations of detector resolution and acceptance. We also studied its dependence on collision centrality and the model parameters for hydrodynamic evolution. This simulation study thereby shows for the first time how DL models can be employed in heavy-ion collision experiments to identify phase transitions directly from experimental output.
\section{The CBM detector}\label{cbmdet}
The CBM detector is designed to make fast and precise measurements of the hadrons, muons and electrons produced in nucleus-nucleus collisions. The experiment will exploit modern radiation hard detectors with self triggered read out electronics to achieve the desired performance. Among the key components of the CBM experiment are the Silicon Tracking System (STS)\cite{heuser2013technical} and Micro Vertex Detector (MVD)\cite{Deveaux:2014cda} which are placed inside a superconducting dipole magnet with a magnetic field integral of 1 Tm. The MVD consists of 4 layers of Monolithic Active Pixel Sensors (MAPS) placed 5-20 cm downstream the target. The main purpose of the MVD is to reconstruct open charm decay vertices and has an excellent position resolution of 3.5 - 6 $\mu m$ and secondary vertex resolution of about 50 $\mu m$. The STS comprises of 8 layers of silicon microstrip sensors placed 30 - 100 cm downstream the target. The task of STS is to reconstruct the tracks and momenta of charged particles. The STS has an excellent momentum resolution of about 1\%. Other sub detector systems of CBM include Ring Imaging Cherenkov Detector (RICH), a MUon CHamber system (MUCH), Transition Radiation Detector (TRD), Multi Gap Resistive Plate Chambers (MRPC) based Time of Flight (TOF) system, Electromagnetic CALorimeter (ECAL) and Projectile Spectator Detector (PSD). However, in this study we consider the data only from STS and MVD for the analyses.
\section{Microscopic and macroscopic dynamical models used to generate the data}\label{data prep}
To generate the training data for the DL analysis, this study uses the hybrid mode \cite{Petersen:2008dd} of the Ultra-relativistic Quantum Molecular Dynamics model (UrQMD 3.4) \cite{Bass:1998ca,Bleicher:1999xi} to simulate heavy-ion collision events with and without a phase transition. In this hybrid approach, a combination of microscopic and macroscopic description of collisions is used where the microscopic UrQMD model is used to generate realistic initial states of the collision at high baryon density. The consecutive hydrodynamic evolution models the intermediate hot and dense stage during which the system may undergo a phase transition \cite{Steinheimer:2007iy}. The hydrodynamic evolution starts once the Lorentz-contracted nuclei have passed through each other. This time ($t_{start}$) is given in natural units by
\begin{equation}
t_{start}=2R\sqrt{\frac{2m}{E_{lab}}}
\end{equation}
where $R$ is the radius of the nuclei, $m$ is the mass of the nucleon and $E_{lab}$ is the kinetic energy of beam. At this time the particle list of UrQMD is transformed into an initial distribution of the energy-momentum and net baryon number density required for the subsequent hydrodynamic evolution. The required smoothing of the density is achieved by treating each hadron from UrQMD as a three dimensional Gaussian distribution of its energy-momentum as well as baryon number. One should note that this initial state will give reasonable event-by-event fluctuations for the initial eccentricities and is also independent of the equation of state that is employed for the fluid dynamical evolution. Any effect of the EoS will therefore be confined only to the expansion phase. The SHASTA \cite{Rischke:1995ir,Rischke:1995mt} algorithm is then used for the 3+1D ideal fluid dynamic evolution on a Cartesian grid with a spacing of $\Delta x= 0.2$ fm and a grid size of $200^3$ cells.
The equation of state of the medium is an essential input that is required to solve the fluid dynamic equations. The EoS combines the microscopic and macroscopic properties of the system created and provides the pressure of the medium for any given energy and net baryon number densities. The EoS incorporates the QCD transition, as the evolution of the medium is driven by pressure gradients. In this study, we use two distinctly different equations of state for training and validation. One based on a Maxwell construction between a bag model quark gluon EoS and a gas of pions and nucleons \cite{Rischke:1995ir,Rischke:1995mt} to simulate the first order phase transitions scenario. The second EoS is dubbed the Chiral Mean Field hadron-quark EoS \cite{Steinheimer:2010ib} which describes a smooth crossover transitions as predicted by lattice QCD. To investigate the models output for an unknown EoS we also employ a hadron resonance gas equation of state which is based on a free gas of hadrons according to the known hadronic resonances from the particle data group \cite{Zyla:2020zbs}. The three equations of state, along trajectories of constant entropy per baryon, as expected for heavy ion collisions at $\mathrm{E_{lab}}=10 \ A$GeV, are visualised in figure \ref{0}. While the crossover EoS is the stiffest and the phase transition the softest equation of state, the HRG lies in between these two extreme cases.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{eosfig.pdf}
\caption{(Color online) The Equations of State, along an isentropic trajectory, for first order phase transition, crossover transition and hadron resonance gas as incorporated in the simulation. The pressure of the medium for a central cell is plotted as a function of its energy density in central Au-Au collision at $10 \ A$GeV in lab frame. The phase transition is associated with a plateau region where the pressure remains almost constant as the system evolves while the pressure is a continuous function for a crossover transition. The phase transition and crossover EoS are used to train the models while the hadron gas EoS is used only to test the performance of the models on an unseen EoS. }
\label{0}
\end{figure}
The fluid dynamical evolution proceeds until the energy density in all cells falls below a freezout energy density ($\epsilon$) after which the evolution is stopped. The default value for $\epsilon$ is five times nuclear ground state energy density $(\epsilon_0)$ but it can be adjusted freely. More details on the motivations behind the chosen values for $t_{start}$ and $\epsilon$ are discussed in \cite{Petersen:2008dd,Steinheimer:2007iy}. Particles are then generated from an iso-energy density hypersurface which has been created throughout the whole time evolution. The density that defines this particlization hypersurface is the above defined value of $n \epsilon_0$. The sampling of particles is done using the well known Cooper-Frye formula
\begin{equation}
E\frac{dN}{d^{3}p}=\int_{\sigma}f(x,p)p^{\mu}d\sigma_{\mu }
\end{equation}
where $f(x,p)$ is the boosted Fermi or Bose distribution and $d\sigma_{\mu }$is the freeze out hypersurface element. Here, global conservation of baryon number, charge, strangeness is exactly observed.
The particles are then transferred to UrQMD where hadronic cascade calculations happen. Important final state effects such as hadronic rescattering and resonance decays are performed at this stage.
The output of the UrQMD-hybrid model is then an event-wise list of particles with their four momenta and positions.
The main objective of the study is to develop a DL model that uses information similar to the experimental output of the CBM experiment, without any significant analysis chain. Furthermore our study will analyse the effects of experimental constraints on the performance of this model. Therefore, an accurate modelling of the experimental condition is necessary. The CbmRoot \cite{root_url} package is used to transport the final state particles from UrQMD through the CBM detector simulation. CbmRoot uses GEANT3 \cite{Brun:1987ma} to simulate the electromagnetic and weak interactions as well as decays of particles traversing the detector. The hits in the detector are then digitised to mimic the detector resolution and finally these digitised hit positions are used to reconstruct the tracks using a Kalman filter based algorithm \cite{Kisel:2006yu}. The standard CbmRoot macros are used for the transport simulation, digitisation and track reconstruction. As a result we obtain realistic event-wise output from the detector simulation which now can be used as input for the DL analysis.
It is also important to note that CbmRoot can perform the full detector simulation according to the experimental specifications. However the default setup does not include a realistic simulation of different backgrounds which may lead to additional noise and could potentially weaken the discrimination performance.
In the actual experimental data taking, quasi real-time processing of free-streaming detector data requires an extra stage of event building, i. e. the identification of clusters of detector hits sufficiently close in time and space. After the step of event building separate events are technically defined and can be processed, also in the approach of this analysis.
It is interesting to note that the process of event building might also be improved by DL-based methods, similar to the PointNet recently developed in \cite{Kuttan:2020kha}.
\section{PointNet for classifying the EoS}
Deep Learning is a well established Machine Learning method inspired by the way information is processed in biological systems. It employs multiple layered Artificial Neural Networks to learn higher dimensional correlations in the data. Machine learning and Deep Learning methods have been widely used both in theory \cite{Zhou:2018ill,Fujimoto:2019hxv, Steinheimer:2019iso, Thaprasop:2020mzp,Pang:2019aqb,Wang:2020hji,Jiang:2021gsw,Shi:2021qri,Song:2021rmm, Li:2020qqn, Wang:2020tgb, Kvasiuk:2020izb, Boyda:2020hsi, Liu:2020omw,Pang:2021vwl} and in experimental high energy physics \cite{Bourilkov:2019yoi,Radovic:2018dip,Guest:2018yhq,Larkoski:2017jix,deOliveira:2015xxd,Baldi:2016fql,Komiske:2016rsd,Almeida:2015jua,Kasieczka:2017nvn,Kasieczka:2019dbj,Qu:2019gqs,Moreno:2019bmu,Samuel:2018xci,Samuel:2019crc,Kasieczka:2020nyd,Sirunyan:2020lcu,Esmail:2019ypk,Haake:2017dpr,Samuel:2021jho,Banerjee:2020iab}. Previous studies \cite{Pang:2016vdc,Du:2019civ} on identifying the QCD phase transitions have shown that Convolutional Neural Network (CNN) based models can accurately classify the underlying equation of state from a hydrodynamic evolution using the p$_t$- $\phi$ spectra of pions (differential transverse and angular distributions in the transverse plane). In \cite{Sergeev:2020fir}, CNN was used to detect the formation of QGP in CBM experiment. CNNs are a good choice of algorithm for extracting correlations from image like data, i.e. data which is provided in the form of equally spaced multi dimensional histograms. However, the purpose of this study was to train DL models directly on experimental outputs such as the information of discrete reconstructed tracks of particles in a collision event. The state vector which represents a reconstructed track in a CBM detector plane comprises of transverse x, y coordinates, tangential directions to the track and the charge to momentum ratio (q/P) of the particle. This data can be fed to a neural network as a 3D voxel array (trajectories in 3D) or as two separate 2D pixel arrays (trajectories in x-z and y-z planes). However, this would render the data to be highly voluminous causing large memory requirements. Moreover, processing the data into these images and combing through it with CNNs would be computationally inefficient and slow. Considering the potential use of a DL based EoS meter for fast online data analysis at CBM, this conversion of the data to images could slow down the whole analysis chain. A solution to these issues is to use a point cloud representation of the data. Point clouds are collections of disordered points in space. A track can be considered as a point in the point cloud in a N-dimensional space where "N" is the number of attributes describing the track. The data therefore becomes an order invariant list of tracks where each entry of the list is the state vector of the track.
DL models can be trained on point cloud data using the PointNet \cite{qi2017pointnet} architecture. PointNet based DL models have been shown to learn from heavy ion collision data to reconstruct the impact parameter of collisions in \cite{Kuttan:2020kha, Kuttan:2021zcu}. In this study, we used a similar network architecture but less complex (i.e.; lesser number of trainable parameters) than the one described in the above paper. The PointNet based models accept the point cloud in the form of a 2D array where each row is a point (i.e. a track information in the event) in the point cloud and each column is an attribute of the point/track. This array is then processed with symmetric, order invariant operations to extract global features which finally pass through a fully connected deep neural network to identify the EoS that created the given point cloud.
\section{Training and testing PointNet models}
The present study was conducted on a set of Au+Au collisions at a beam energy of $10$ $A$GeV in the lab frame. CBM will also study other heavy ions at similar energies. However, as the underlying physics of the collisions remains the same, the models developed in this study can be easily extended for application to other nuclei. The dataset for this study was generated using the UrQMD-hybrid model and CbmRoot package as described in section \ref{data prep}. It consists of 30000 training events and 10000 validation events each for the crossover and first order phase transition equation of states with an uniform impact parameter (b) distribution from 0 to 7 fm. To study the effects of experimental uncertainties and constraints on the performance of the DL models, the PointNet model was trained on different outputs:
\begin{enumerate}
\item Firstly, the final state output (\emph{Dataset 1}), i.e. the particle information directly from the UrQMD model without any acceptance cuts. This dataset contains essentially the full event information and has not been transported through the detector simulation.
\item Secondly, the final state output within CBM detector acceptance (\emph{Dataset 2}). The dataset contains final state particles from UrQMD model within the CBM acceptance region of 2-25$\degree$ polar angles. This corresponds to a hypothetical, ideal detector output which detects all particles within its acceptance with infinite resolution.
\item Lastly, the CbmRoot simulated data (\emph{Dataset 3}), i.e. the final state output from UrQMD is passed through CbmRoot. This dataset comprises of the reconstructed tracks from the digitised hits of particles in the simulated CBM detector.
\end{enumerate}
The network structure and other training parameters were fine tuned through trial and error to achieve the best performance on the final state output (\emph{Dataset 1}). The same network architecture and hyperparameters (however, with different input dimensions depending on the dataset) were then used for training the model with experimental effects (\emph{Dataset 2,3}). In this way, it was possible to study the response of the same DL network to different experimental constraints.
\subsection{Network architecture}
The input point cloud passes through 3 1D-convolution layers to extract 128, 256 and 512 feature maps respectively. Batch normalisation layers are present between every convolution layer. An average pooling layer then extracts one global feature of the point cloud from each of the 512 feature map generated by the final convolution layer. The 512 global features are the input to a 3 layer fully connected Deep Neural Network (DNN) with 256, 128 and 2 neurons respectively. Batch normalisation and dropout layers (with drop out probability 0.5) are present between every DNN layer. All layers except the final layer use the ReLU activation function. A softmax activation is used on the final layer to classify the EoS. The models use the Adam optimiser with a learning rate of $10 ^{-5}$ and categorical cross entropy as the loss function. The models were trained until the network started overfitting the data and the best model in terms of validation accuracy and loss was chosen for further analyses.
\subsection{Training results}
As discussed above, three different scenarios for the input data were investigated in this study. In the first case (\emph{Dataset 1}), the input for training was the event-by-event list of four-momenta of all particles from UrQMD. The input data has dimensions N$\times 4$ where N is the maximum number of particles present in an event. Events with less number of particles are filled with zeros to maintain the same input dimensions. In this scenario, the trained PointNet model achieved a validation accuracy of 77.2\% for the correct event-wise classification between crossover and phase transition EoS. This accuracy can be improved if multiple events are combined to create the input. This was done by randomly selecting K events, i.e. all rows (without replacement) in that event, from the event-by-event lists (along with rows filled with zeros) and concatenating them to create a longer list with dimensions (K*N)$\times 4$. It must be noted that the combined events are randomly chosen from b=0-7 fm. A validation accuracy of 99.7\% was achieved by the model when the input was the combined data from 15 events as can be seen in figure \ref{1}. The model learns a set of unique observables for classifying the underlying EoS and the boundaries of these observables for either classes are accurately learned with a combined dataset.
In the second case, the input for training was the four momentum of particles from UrQMD which were within the CBM detector acceptance. Particles beyond the CBM acceptance range of a 2-25$\degree$ polar angles
were removed from the events. The validation accuracy in this case was decreased to about 72.2\% for the event by event input and the model was able to achieve an accuracy of 99.5\% by combining 20 events for the input.
The decrease in accuracy can be understood. Supplying the PointNet with only a shortened or partial list of particles increases the difficulty of learning the observables capable of classifying the EoS. The DL model therefore requires a few more events to achieve a classificaton accuracy similar to the first case. The models cannot distinguish particles belonging to one event from another. Therefore, it is likely that the unique DL constructed observables are some aggregate quantities, probably within certain region of the phase space. An acceptance cut could remove part of the information which was otherwise available (in first case) and calculating these observables accurately would naturally require more statistics.
In the third dataset, more realistic experimental constraints of acceptance and resolution were introduced. The UrQMD output was passed through the CBM detector simulation and the model was trained on the tracks reconstructed from the hits of the particles in MVD and STS detectors of the detector simulation. In this case, the average classification accuracy for single event inputs was only 62.4\%. However, after combining 40 events for an input, the accuracy increased again to 96.6\%. For this model to achieve a performance similar to the second dataset, the number of events that were combined to create the input had to be doubled. This model, based on dataset 3, that uses 40 events of reconstructed tracks as input is henceforth referred to as \emph{Model-1}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{neveacc.pdf}
\caption{(Color online) Validation accuracy of the PointNet models as a function of number of events combined to create the input. The number of training and validation samples were fixed to be 60000 and 20000 respectively for all the models although multiple events were combined to create a sample. While randomly combining events, it was ensured that an event in training sample was never present in a validation sample. The DL models achieved greater than 99\% accuracy with the combination of 15 events in an ideal case while 20 events had to be combined in the presence of an acceptance cut. However, the DL model required a combination of 40 events to achieve about 96\% accuracy in a more realistic experimental condition. }
\label{1}
\end{figure}
The accuracy of PointNet models in the three cases as a function of number of events combined is plotted in figure \ref{1}. It is evident from the plot that the performance of DL model is only marginally decreased in the presence of a simple acceptance cut but there is a large drop in the accuracy when a more realistic experimental scenario is considered. This shows that the final state particles have strong features that are characteristic of the macroscopic properties that governed the evolution of QCD medium. However, in an experiment these distinct features become weaker and difficult to identify. Uncertainties in measurements due to the detector resolution and randomness in the detected particle spectra arising from interactions of particles in the detector diminish the relevant signals in the data. Inefficiencies of reconstruction algorithms and selection cuts also introduce errors in the final data. Nevertheless, the DL model is able circumvent these issues by combining more events for decision making. A similar behaviour was also reported in \cite{Du:2019civ}. Increasing the statistics reduces the stochasticity in the data thereby improving the predictive power of DL. For instance, the global feature used by the PointNet models for classifying EoS are the average (given by average pooling layer) values of each feature extracted by the convolution kernels. These averages could be more accurate determined when more sample points are used. In this way, the PointNet models could improve in performance when more events are used.
However, this does not mean that conventional mean observables such as mean transverse momentum ($<p_{T}>$), collective flow ($v2$) etc. can be used for classifying the EoS as accurately as PointNet models. The above mentioned DL models do not require any event selection based on centrality while the traditional observables have strong centrality dependence. Without a centrality selection and high statistics, the traditional observables will not have well separated boundaries that can aid an accurate classification. The $<p_{T}>$ and $v2$ distributions for 15 events averaged data from UrQMD are plotted in figure \ref{ptv}. It is evident from the plots that the distributions of these observables, after averaging over only 15 events, overlap significantly and cannot be used to classify the two classes of data as accurately as the PointNet model does. We have also checked that simply calculating averages of the different components of the input features in the PointNet will also not lead to easily distinguishable distributions. A more in depth discussion on the interpretability of the network is given in appendix \ref{appendix}. There, we describe a method to interpret the decision making process of our PointNet model which helps to understand why the model outperforms conventional observables.
In other words, the PointNet model is able to learn unique observables that produce a close to perfect classification accuracy from only combining the input of 15 random events. The PointNet model is able to learn such observables even from an "experiment like data" in which the reconstructed tracks are the input (\emph{Model-1}).
\begin{figure}
\includegraphics[width=0.5\textwidth]{15evev2pt-2000all.pdf}
\caption{(Color online) Distributions of Mean transverse momentum (left) and elliptical flow (right) for crossover and first order phase transitions. The values are averaged over all particles from 15 UrQMD events with b=0-7 fm (\emph{Dataset-1}). The distributions have significant overlap that it is not possible to classify the EoS using these observables while the DL model with 15 event combined input achieved an accuracy of 99.7\%.}
\label{ptv}
\end{figure}
In order for the CBM experiment to make complete utilisation of the high event rates, accurate online event selection and analysis techniques are necessary. The DL models require a maximum of just 40 events to achieve a classification accuracy greater than 96\%. The PointNet based EoS meter can serve this purpose and can be coupled with other DL based algorithms (eg. centrality meter \cite{Kuttan:2020kha}) for a comprehensive online event analysis.
It is well known that conventional observables are very sensitive on model parameters such as the centrality selection, initial state, freeze-out condition etc. Therefore, we investigate in detail the generalisation ability of the PointNet models on these parameters in the following sections.
\subsection{Centrality dependence}
\emph{Model-1} which had an accuracy of 96.6\% was trained on events with impact parameters 0-7 fm. Although the accuracy is already good enough, the model showed slightly better performance on central events which hints to a centrality dependence. To examine if the accuracy of the model can be increased with a different centrality selection, a model (\emph{Model-2}) was trained explicitly on events with an impact parameter of 0-3 fm. This model also used the tracks reconstructed from the detector and combined the data from 40 events to form an input. The \emph{Model-2} achieved a prediction accuracy of 99.8\% on events with impact parameters 0-3 fm: Choosing a smaller centrality bin therefore improved the performance of the model. However, most of the events collected in the experiment will be unusable if we choose only central collisions. To tackle this issue, a model (\emph{Model-3}) was trained which combined only events with impact parameters 0-3 fm and 3-7 fm separately. The input for this model was a combinations of 40 events (reconstructed tracks) either from the impact parameter bin of 0-3 fm or from 3-7 fm. In addition to this selection of events, the network had 1 extra input to feed in the impact parameter bin of the given sample (i.e; 0 if b=0-3 fm and 1 if b=3-7 fm). This input is concatenated with other extracted global features and is fed into the DNN. The \emph{Model-3} achieved a validation accuracy of about 99.65\% for events with impact parameter 0-3 fm and 81.27\% for impact parameter 3-7 fm. The PointNet models can achieve the best performance for central events, assuming they can be accurately identified \cite{Kuttan:2020kha}. However, significant accuracies can also be achieved on peripheral events if they are separated from central events for training.
\subsection{Dependence on model parameters}
In the previous section, it was shown how PointNet models can be employed to correctly classify the nature of the QCD transition with large accuracy in a wide range of centralities or in a small centrality range depending on the experimental requirement. However, the physical and model parameters have been kept constant, i.e. they where assumed to be known exactly. In reality this is not the case and thus, to ensure the reliability of DL model in an experiment, the models must be robust against reasonable changes of the physical parameters of the hydrodynamic event generator. Two such parameters are the starting time for hydrodynamic evolution ($t_{start}$), which essentially determines the time at which one can assume local equilibration to be reached and the particlization energy density ($\epsilon$), which determines at which point the system starts to fall out of local equilibrium. Since at that energy density particles are emitted from the hydro to the non-equilibrium hadronic rescattering phase, matter below this criterion will effectively not be influenced by the EoS. To evaluate the dependency of the DL models on these parameters, the trained PointNet models were tested on events where $t_{start}$ is varied by $\pm$ 10\% and $\epsilon$ by $\pm$ 40\% from the training value. The performance of the DL models are illustrated and compared in figure \ref{2}. The models in general seem to achieve an accuracy similar to the validation accuracy when $t_{start}$ or $\epsilon$ is decreased. However, the accuracies decrease considerably when the $t_{start}$ or $\epsilon$ is increased. This effect can be understood if one studies the fraction of the matter which is below the particlization criterion, and therefore does not carry any information on the EoS, for the different initial and freeze out conditions . This fraction varies also as a function of the impact parameter as shown in figure \ref{fr}.
The decrease in performance with decrease in the duration of hydrodynamic evolution and centrality is therefore nicely illustrated with figure \ref{fr}. It can be seen that a smaller portion of the emitted hadrons experiences the dynamics of the phase transition as the impact parameter is increased. This explains the higher validation accuracy for \emph{Model-2} compared to \emph{Model-1} and the decreased validation accuracy for \emph{Model-3} on events with impact parameter 3-7 fm.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{modelacc.pdf}
\caption{(Color online) Variation in the testing accuracy of the PointNet models with change in $t_{start}$ and $\epsilon$. The blue bars show the validation accuracies of the models while the other colours represent the testing accuracy on datasets different from the training data. Each testing dataset comprised of 2000 events for each EoS. The \emph{Model-3a} and \emph{Model-3b} are the testing results of the \emph{Model-3} on impact parameters 0-3 fm and 3-7 fm respectively.}
\label{2}
\end{figure}
A delayed starting time of the hydrodynamic evolution or an increased freeze out energy density reduces the contribution of the hydrodynamic evolution of the system to the emitted particles and therefore the EoS will have less influence on the final particle spectra. While an increase of the duration of the hydrodynamic evolution leads to a prolonged influence of the EoS on the evolution of the medium and thus a higher accuracy, the performance drop can be related to a limitation imposed by physics which may not be avoidable.
Similarly, an increase in the freeze out energy density by 40\%, for b=0 fm, causes about 50 \% of the final particles being already emitted before the hydrodynamic evolution even begins. The DL-models have to rely on the artefacts left by the EoS in the remaining 50\% of the emitted particles to make a decision. This is why the accuracy decreases considerably with an increase in freeze out energy density. However, the decrease of the portion of the emitted particles that undergo the hydrodynamic evolution from an increase in the freeze out energy density by 40\% is larger than when $t_{start}$ is increased by 10\%. This is why the drop in the accuracy is comparatively lower when the $t_{start}$ is increased by 10\%. In short, hadrons from central events with early starting of the hydrodynamic evolution or a decreased freeze out energy density carry more information on the EoS as they are, on average, emitted after a longer hydrodynamic evolution.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{fracmatter.pdf}
\caption{(Color online) Fraction of the medium which is below the freezout energy density at the beginning of hydrodynamics as a function of impact parameter. This is simply the fraction of the medium which does not undergo the hydrodynamic evolution. The blue curve represents the initial conditions used while training the model. Curves above the blue curve corresponds to the initial conditions which reduces the duration of hydrodynamic evolution and vice versa.}
\label{fr}
\end{figure}
\subsection{Testing on an unseen EoS}
We have shown that the PointNet models can accurately classify the data into one of the two training EoS. However, the actual EoS of the fluid dynamic evolution can be different from the ones used during the training. To understand how the DL model would perform in such a scenario, we tested \emph{Model-1} on an EoS which it was not trained on. On the hadron resonance gas EoS, the \emph{Model-1} classified 68\% of the samples as crossover and the remaining as a first order phase transition. As evident from figure \ref{0}, the hadron gas EoS is similar to a crossover EoS. At low energy densities, the hadron gas EoS traces the crossover equation of state and at high densities, the pressure is in between the phase transition and crossover equation of states. The hadron gas EoS also doesn't have a plateau like region of constant pressure which is characteristic of the phase transition EoS. This explains why the model prefers to predict the hadron gas EoS as a crossover equation of state. The PointNet based binary classifier of EoS can therefore provide reliable predictions not just on the trained EoS but also on other similar equation of states for crossover and phase transition.
\subsection{Comparison to a single event classifier}
We have shown that the performance of the model can be improved by combining multiple events to train the PointNet models. A recent study \cite{Nachman:2021yvi} pointed out, that a single event classifier, when applied on N events could outperform a classifier trained on combinations of N events if these events are statistically independent. This raises the question if an event-by-event EoS classifier, combined over N events, would outperform the combined events models developed in this study. To check this, 20000 validation events of \emph{Dataset-3} were tested using a model trained to classify the EoS of individual events. The final prediction is then defined as the predicted EoS of the majority of the events for groups of 40 random events. This procedure achieved an accuracy of 92.01\%. At the same time, the \emph{Model-1} which was trained on combinations of 40 events to make predictions had an accuracy of 96.6\%. The single event classifier therefore doesn't achieve the accuracies of the combined events classifier. An accuracy of about 92\% can be achieved by training the model on combinations of about 25 events while the single event classifier required 40 events to achieve the same accuracy. The superior performance of the PointNet models trained on combinations of multiple events is due to the centrality dependent influence of the EoS on the system. As shown in figure \ref{fr}, a significantly larger fraction of the system is influenced by the supplied EoS for a central event while most part of the system is not influenced by the EoS for peripheral events. Therefore central events contain more information on the EoS which governed its evolution than a peripheral event. When the PointNet is trained on combinations of random events with all centralities, the model can learn to make decisions using the signals from the central events present in the data. A single event classifier on the other hand would struggle to correctly classify the peripheral events which would often contain only very weak signatures of the EoS. This centrality dependent performance bias would further worsen the performance of single event classifiers when a realistic impact parameter distribution $(P(b) \propto b)$ is considered where the central events are rare compared to peripheral events. Another practical advantage of using combinations of events is that such models can potentially work with a continuous datastream without event building or event separation. This can be extremely useful to the CBM experiment which will require extremely fast analysis methods for the data collected at rates upto 10 MHz.
\section{Conclusion and Discussion}\label{conc}
In this study, we have developed PointNet based DL models that can extract very complex universal event features from basic event information of heavy ion collisions at the CBM experiment. This model is even able to classify events by very abstract event features like the EoS present during the hot and dense stage of the collision, i.e. whether a phase transition was present or not. The prediction accuracy was found to be improving when more events were combined to make the predictions. This shows that with increased statistics, PointNet models learn the global features that can classify the EoS despite the uncertainties in the data arising from a discrete particle spectra with final state effects, detector effects and inefficiencies of reconstruction algorithms. It is noteworthy that the PointNet models can achieve a classification accuracy of up to 96.6\% from the reconstructed tracks of particles from just 40 collision events. The PointNet models can work on a wide range of impact parameters but they achieve the best performance by choosing only central collisions for analysis. However it is also possible to include non central collisions for analysis if central collision events are not mixed with non central collisions. The predictions of the DL models were also robust to some changes in the physical parameters like the initial condition. The performance of the models was consistent when $t_{start}$ or $\epsilon$ was decreased from the training value while a decrease in the performance is observed when these parameters are increased. This is interpreted as a physical consequence of a decreased influence of the hydrodynamic evolution, and the EoS, on emitted particles. Nevertheless, the DL models show good performance in comparison to conventional averaged event features like $<p_{T}>$ or $v_2$ which have similar values for both the classes. The values of these features also differ widely for different model parameters.
The use of experimental output such as the tracks of particles can eliminate any possible biases in the data that might appear in later stages of data processing. The point cloud representation of data requires minimal pre-processing before being fed to the DL model. This enables the model to be deployed in the experiment for fast, online analysis of experimental data. Moreover, these models can be easily translated to any other heavy-ion collision experiment for similar tasks. The capability of these models to work on large range of impact parameters make it an ideal tool to search for phase transitions in heavy-ion programmes. Due to their ability to find out global features in the input, the PointNet based models can also be easily adapted for analysing any other global event feature of heavy-ion collisions. Future studies in this direction can be focused on incorporating more equation of states making it a multi class classification problem and testing the performance of the models for other FAIR energies. It would also be interesting to study the performance of DL models in a continuous datastream and in the presence of detector noise, event pileup etc. Studies on training the DL models on low level detector data such as the signals from readout channels and deploying them directly on the detectors using FPGAs is another interesting direction which could be extremely beneficial to the CBM experiment. Such methods can be exploited for ultra fast event selection and analysis based on yet unachievable, complex event features.
\begin{acknowledgments}
The authors thank Benjamin Nachman and Jesse Thaler for their insightful comments and discussions. M.O.K. thanks the GSI and HFHF as well as the SAMSON AG for their support.
K.Z. and J.S. thank the Samson AG and the BMBF through the ErUM Data project for funding. H.S. acknowledges the Walter Greiner Gesellschaft zur F\"orderung der physikalischen Grundlagenforschung e.V. through the Judah M. Eisenberg Laureatus Chair at Goethe Universit\"at Frankfurt am Main.
\end{acknowledgments}
\bibliographystyle{apsrev}
|
1211.2794
|
\section{Introduction}
The search for Higgs and supersymmetry (SUSY) are the main focus of ATLAS and CMS experiments. Although before the start of the LHC the expectation for an early discovery of supersymmetric partners of the Standard Model (SM) particles was very high (mainly driven by the studies in the constrained SUSY scenarios), no SUSY particle has been observed yet. On the other hand, ATLAS and CMS collaborations have reported the discovery of a new bosonic state with a mass of around 126 GeV, compatible with the SM-Higgs \cite{ATLAS:2012gk,CMS:2012gu}. These results have significant implications for the Minimal Supersymmetric Standard Model (MSSM).
In the following, we discuss the consequences of the latest SUSY and Higgs search results in the context of the MSSM.
\section{Implication of SUSY searches}
To study the implication of the LHC SUSY searches we consider the unconstrained phenomenological MSSM (pMSSM) with 19 parameters \cite{Djouadi:1998di}. Most of the previous studies considered the highly constrained models with a small number of free parameters. However, these models are not representative of a generic MSSM scenario where the particle mass parameters are independent. As we will see below, the results can be very different in such generic scenarios.
To explore the pMSSM, we perform a flat scan over the parameters in the ranges given in Table~\ref{tab:paramSUSY}. %
The particle spectra are generated for more than 100M points using {\tt SOFTSUSY} \cite{softsusy} and {\tt SUSPECT} \cite{suspect}. We impose the SUSY and Higgs mass limits from LEP and Tevatron as described in \cite{Arbey:2011aa}. The flavour observables, muon anomalous magnetic moment and relic density are computed with {\tt SuperIso Relic} \cite{superiso}, and we apply the constraints given in Table~\ref{tab:constraints}. We do not discuss here the consequences of the dark matter direct detection results which are discussed thoroughly in \cite{Arbey:2012na}.
\begin{table}[b!]
\begin{center}
\begin{tabular}{|c|c||c|c|}
\hline
~~~~Parameter~~~~ & ~~~~~~~~~~Range~~~~~~~~~~&~~~~Parameter~~~~ & ~~~~~~~~~~Range~~~~~~~~~~\\
\hline\hline
$\tan\beta$ & [1, 60]&$M_{\tilde{e}_L}=M_{\tilde{\mu}_L}$ & [50, 2500]\\
\hline
$M_A$ & [50, 2000]&$M_{\tilde{e}_R}=M_{\tilde{\mu}_R}$ & [50, 2500]\\
\hline
$M_1$ & [-2500, 2500]&$M_{\tilde{\tau}_L}$ & [50, 2500]\\
\hline
$M_2$ & [-2500, 2500]&$M_{\tilde{\tau}_R}$ & [50, 2500]\\
\hline
$M_3$ & [50, 2500]&$M_{\tilde{q}_{1L}}=M_{\tilde{q}_{2L}}$ & [50, 2500]\\
\hline
$A_d=A_s=A_b$ & [-10000, 10000]&$M_{\tilde{q}_{3L}}$ & [50, 2500]\\
\hline
$A_u=A_c=A_t$ & [-10000, 10000]&$M_{\tilde{u}_R}=M_{\tilde{c}_R}$ & [50, 2500]\\
\hline
$A_e=A_\mu=A_\tau$ & [-10000, 10000]&$M_{\tilde{t}_R}$ & [50, 2500]\\
\hline
$\mu$ & [-1000, 2000]&$M_{\tilde{d}_R}=M_{\tilde{s}_R}$ & [50, 2500]\\
\hline
&&$M_{\tilde{b}_R}$ & [50, 2500]\\
\hline
\end{tabular}
\end{center}
\caption{SUSY parameter ranges (in GeV when applicable).\label{tab:paramSUSY}}
\end{table}%
To evaluate the consequences of the SUSY searches, we compute the supersymmetric particle decay rates with {\tt SDECAY} \cite{Muhlleitner:2003vg} and we use {\tt PYTHIA 6} \cite{Sjostrand:2006za} for event generation of inclusive SUSY production in $pp$ interactions. The generated events are then passed through fast detector simulation using {\tt Delphes} \cite{Ovyn:2009tx}.
The Higgs decay rates are computed with {\tt HDECAY} \cite{Djouadi:1997yw} and the gluon fusion and VBF cross sections of the lightest CP-even Higgs with {\tt HIGLU} \cite{Spira:1995mt} and {\tt FeynHiggs} \cite{Heinemeyer:1998yj}. More details can be found in \cite{Arbey:2011aa,Arbey:2011un}.
\begin{table}
\begin{center}
\begin{tabular}{|c|}
\hline
$2.16 \times 10^{-4} < \mbox{BR}(B \to X_s \gamma) < 4.93 \times 10^{-4}$\\
\hline
$\mbox{BR}(B_s \to \mu^+ \mu^-) < 5.0 \times 10^{-9}$\\
\hline
$0.56 < \mbox{R}(B \to \tau \nu) < 2.70$\\
\hline
$4.7 \times 10^{-2} < \mbox{BR}(D_s \to \tau \nu ) < 6.1 \times 10^{-2}$\\
\hline
$2.9 \times 10^{-3} < \mbox{BR}(B \to D^0 \tau \nu) < 14.2 \times 10^{-3}$\\
\hline
$0.985 < \mbox{R}_{\mu23}(K \to \mu \nu) < 1.013$\\
\hline
$-2.4 \times 10^{-9} < \delta a_\mu < 4.5 \times 10^{-9}$\\
\hline
$10^{-4} < \Omega_\chi h^2 < 0.155$\\
\hline
\end{tabular}
\end{center}
\caption{Constraints applied in our pMSSM analysis. The points passing all the constrained are called ``accepted points''.\label{tab:constraints}}
\end{table}%
We consider the consequences of the SUSY searches in all hadronic events with $\alpha_T$ \cite{11-003}, in same-sign isolated dilepton events with jets \cite{11-010} and missing energy and in opposite-sign dilepton events with missing transverse energy \cite{11-011} in the CMS detector at 7~TeV with 1~fb$^{-1}$ of data, and extrapolate to the 8 TeV run with 15~fb$^{-1}$ of data.
In Fig.~\ref{fig:susysearches} we show the consequences on the masses of the lightest squark of the first two generations, the lightest neutralino and $\tan \beta$, where the distribution of the points compatible with SUSY searches with 1~fb$^{-1}$ at 7~TeV and the projections for 15~fb$^{-1}$ at 7 and 8~TeV data are displayed.
\begin{figure}[t!]
\begin{center}
\vspace*{0.5cm}
\hspace*{-0.3cm}\includegraphics[width=0.35\textwidth]{fig1a.png}%
\includegraphics[width=0.35\textwidth]{fig1b.png}%
\includegraphics[width=0.35\textwidth]{fig1c.png}%
\end{center}
\caption{Fraction of accepted pMSSM points with $M_{h}>$ 111~GeV, not excluded by the SUSY searches with 1~fb$^{-1}$ of 7~TeV data (red), and by a projection of 15~fb$^{-1}$ at 7~TeV (blue) and 15~fb$^{-1}$ at 8~TeV (green), as functions of the masses of the lightest squark of the first two generations (left panel), the lightest neutralino $\tilde \chi^0_1$ (central panel) and $\tan \beta$ (right panel).}
\label{fig:susysearches}
\end{figure}
We notice that with 15~fb$^{-1}$ of data at 8~TeV, more than 30\% of the points with squark masses below 1~TeV will still be allowed. The spectrum of the weakly interacting particles will be even less affected as there is basically no sensitivity for neutralino masses above 700~GeV. The spectrum of
$\tan\beta$ is also rather flat. The full set of the results can be found in \cite{Arbey:2011un}. These results in the unconstrained pMSSM are very different from those obtained in highly constrained scenarios such as mSUGRA.
\section{Implication of Higgs searches}
An alternative way to efficiently constrain SUSY is using the information from the Higgs sector.
In the following, we consider that the new boson discovered at the LHC corresponds to the lightest CP-even Higgs boson. The combination of the Higgs search results presented by ATLAS and CMS are given in Table~\ref{tab:input}.
\begin{table}[t!]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Parameter & Value & Experiment \\ \hline \hline
$M_H$ & 125.9$\pm$2.1 GeV & ATLAS \cite{ATLAS:2012gk} + CMS \cite{CMS:2012gu} \\
$\mu_{\gamma \gamma}$ & 1.71$\pm$0.33 & ATLAS \cite{ATLAS-2012-091} + CMS \cite{CMS-12-015} \\
$\mu_{Z Z}$ & 0.95$\pm$0.40 & ATLAS \cite{ATLAS-2012-092} + CMS \cite{CMS-12-016} \\
$\mu_{b \bar b}$ & $<$1.64 (95\% C.L.) & CMS \cite{CMS-12-019}\\ \hline
$\mu_{\tau \tau}$ & $<$1.06 (95\% C.L.) & CMS \cite{CMS-12-018}\\ \hline
\end{tabular}
\end{center}
\caption{Input parameters used for the pMSSM study.}
\label{tab:input}
\end{table}
In \cite{Arbey:2011ab,Arbey:2012dq}, we have shown that the Higgs mass measurement has strong implications on the constrained MSSM scenarios. This is demonstrated in Fig.~\ref{fig:cMSSM}, where the maximal value of the light Higgs mass is given in mAMSB, mGMSB, mSUGRA and some of its variants, as a function of $\tan\beta$ and the SUSY scale $M_S$. %
\begin{figure}[!t]
\begin{center}
\vspace*{0.5cm}
\includegraphics[width=6.7cm]{fig2a.png}~\raisebox{0.5cm}{\includegraphics[width=2.cm]{Mh_legend.png}}~\includegraphics[width=6.7cm]{fig2b.png}
\end{center}
\caption{The maximal $h$ mass value $M_h^{\rm max}$ as functions of
$\tan\beta$ (left) and $M_S$ (right) in the mASMB, mGMSB as well as in mSUGRA and
some of its variants.}
\label{fig:cMSSM}
\end{figure}
The parameters of the models are varied within the ranges given in \cite{Arbey:2012dq}, and the top quark mass is taken to be $m_t=173$ GeV, and $M_S$ is limited to 3 TeV. While mSUGRA and NUHM provide solutions compatible with a Higgs mass $\sim$126 GeV, it is clear that the minimal versions of GMSB and AMSB, and the even more constrained mSUGRA scenarios (VCMSSM, no-scale) are disfavoured.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=7.cm]{fig3.png}
\end{center}
\caption{Maximal Higgs mass in mSUGRA, mAMSB and mGMSB, as a function of $M_S$ for the top quark mass varied in the range $m_t = 170-176$ GeV.}
\label{fig:top}
\end{figure}
It should be noted that the value of top mass has a significant impact on the maximal Higgs mass, in particular in constrained scenarios, where $m_t$ also enters in the evaluation of
the soft SUSY breaking parameters and the minimisation of the scalar potential. This effect is demonstrated in Fig.~\ref{fig:top} for the minimal SUGRA, AMSB and GMSB models.
We turn now to the pMSSM and in Fig.~\ref{fig:msq125} we show the distribution of points compatible with SUSY searches with 15~fb$^{-1}$ of 7~TeV data as well as the Higgs mass constraints and the $\mu_{\gamma\gamma}$ and $\mu_{ZZ}$ signal strengths given in Table~\ref{tab:input}, as functions of the masses of the lightest squark of the first two generations, the lightest neutralino $\tilde \chi^0_1$ and $\tan \beta$.
\begin{figure}[t!]
\begin{center}
\vspace*{0.5cm}
\hspace*{-0.3cm}\includegraphics[width=0.35\textwidth]{fig4a.png}%
\includegraphics[width=0.35\textwidth]{fig4b.png}%
\includegraphics[width=0.35\textwidth]{fig4c.png}%
\end{center}
\caption{Fraction of accepted pMSSM points, with 123 $< M_{h}<$ 127~GeV (filled squares),
not excluded by the SUSY searches with 15~fb$^{-1}$ of 7~TeV data as functions of the masses of the lightest squark of the first two generations (left panel), the lightest neutralino $\tilde \chi^0_1$ (central panel) and $\tan \beta$ (right panel). The open square points show the fraction of pMSSM points after imposing the additional requirements on the Higgs rates $\mu_{\gamma\gamma}$ and $\mu_{ZZ}$.\label{fig:msq125}
}
\end{figure}%
A comparison with Fig.~\ref{fig:susysearches} reveals first that the Higgs constraints strongly reduce the statistics. The squark and neutralino distribution shapes are quite unaffected, but the small $\tan\beta$ region (below 15) is now more constrained.
In Fig.~\ref{fig:pdf2D} we present the distribution of pMSSM points compatible with the $h$ boson mass and the observed yields given, in the $(X_t, m_{\tilde t_1})$, $(X_b, m_{\tilde b_1})$, $(X_\tau, m_{\tilde \tau_1})$ and $(M_A$, $\tan \beta)$ parameter planes. To do so, we combined all the constraints in Table~\ref{tab:input} with a $\chi^2$ combination. We notice first that small values of $|X_t|$ are clearly disfavoured, and that stop masses as low as 400 GeV are still compatible with the data. This is mainly the results of the Higgs mass measurement which calls for non minimal mixing in the stop sector. Second, the negative $X_b$ region is favoured by the rate constraints. This can be explained by the fact that a decrease in the $h\to b\bar{b}$ rate, generated in particular by negative $X_b$, would result in an increase of $\mu_{\gamma\gamma}$, which is favoured by present data. Similarly, negative $X_\tau$ are favoured since a reduced $h\to\tau\tau$ rate is more likely to be consistent with the current $\mu_{\tau\tau}$ limit. Finally, $M_A<400$ GeV values are strongly disfavoured by the Higgs mass and rate measurements for any value of $\tan\beta$, and therefore the decoupling regime seems to be favoured by the data.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=7.5cm]{fig5a.png}\includegraphics[width=7.5cm]{fig5b.png}\\
\includegraphics[width=7.5cm]{fig5c.png}\includegraphics[width=7.5cm]{fig5d.png}
\end{center}
\caption{Distributions of the pMSSM points in the $(X_t, m_{\tilde t_1})$ (upper left), $(X_b, m_{\tilde b_1})$ (upper right), $(X_\tau, m_{\tilde \tau_1})$ (lower left) and $(M_A$, $\tan \beta)$ (lower right) parameter planes. The black dots show the accepted pMSSM points, those in light (dark) grey the same points compatible at 68\% (90\%) C.L. with the Higgs constraints of Table~\protect\ref{tab:input}.\label{fig:pdf2D}}
\end{figure}%
\section{Conclusion}
We have considered the results from SUSY and Higgs searches at the LHC in the context of the MSSM. Contrary to the constrained MSSM scenarios, in the pMSSM the current LHC limits from the SUSY searches still leave a substantial room for low energy SUSY. This conclusion does not change when we include in addition the information from the Higgs sector, namely the mass and signal yields of the new boson. However, the current Higgs data already point to specific regions of the MSSM, in particular to the decoupling regime with large stop and negative sbottom and stau mixings. With more data becoming available, and more precise experimental results on the Higgs boson properties, more important impacts on the parameter regions of the SUSY scenarios are to be expected.
|
1211.2424
|
\section{Introduction}\label{sec:intro}
Resonance phenomena appear in many fields of quantum physics: from unstable elementary particles, to resonances in atomic or molecular systems and to collective excitations in the condensed phase. They are described as long-lived states of a system that have enough energy to undergo a decay process. Wave functions of resonant configurations resemble bound-states over a period of time, called the decaying lifetime, when they are captured in a small area of space. Likewise the bound states, they may be treated as the eigenstates of the Hamiltonian that are associated with the natural frequencies of the system. The difference lies in the complex character of the resonance eigenvalues which is related to the purely outgoing boundary condition they fulfil. Such an approach to resonances has been started by Gamow in the paper on $\alpha$-decay of radioactive nuclei \cite{Gamow} and further developed by Siegert \cite{Siegert}. It has been shown that the problem of non-square integrability of their eigenfunctions can be rigorously overcome by enlarging the functional space to a rigged Hilbert space \cite{riggedGelfand, riggedMaurin}. On the other hand, the complex scaling idea~\cite{BC} has enabled development of practical approaches where the resonant states are treated on the same footing as the bound-states~\cite{Reinhardt,Moiseyev}. One of the most practical methods is the Rayleigh-Ritz (RR) determination of the eigenstates of a complex-scaled Hamiltonian.
In this work, we discuss the determination of the resonances by the optimized RR scheme~\cite{ao} which proved successful for bound-states~\cite{Amore,RRopty,coll}. The method uses a basis of functions with adjustable nonlinear parameters, the values of which are fixed so as to make the trace of the RR matrix stationary. Generalization to resonances proceeds straightforwardly by allowing nonlinear parameters to be complex numbers. Through the study of several systems with different potentials we demonstrate the efficiency of our method in finding both the energy and lifetime of the resonant states.
The plan of our work is as follows. In section~\ref{sec:RR}, the optimized RR method is described and its extension to resonances is discussed.
The calculations of the resonance parameters are presented in section~\ref{sec:one} for one-dimensional potentials, and in section~\ref{sec:radial} for the spherically symmetric case in $D$ dimensions. Section~\ref{sec:con} is devoted to conclusion.
\section{The method}\label{sec:RR}
The Schr\"{o}dinger equation
\begin{equation} \hat{H}\psi(x)= \varepsilon \psi(x),\label{eq:prob1}\end{equation}
in the vast majority of cases cannot be solved exactly and has to be dealt with approximately. One of the classical methods for calculating numerically accurate solutions is the linear Rayleigh-Ritz procedure.
\subsection{Rayleigh-Ritz determination of bound-states}
The RR method for solving the Schr\"{o}dinger equation (\ref{eq:prob1}) is based on the variational principle which says that the Rayleigh quotient \begin{equation} R[\Phi]=\frac{<\Phi(x)|\hat{H}|\Phi(x)>}{<\Phi(x)|\Phi(x)>},\label{eq:roqq}\end{equation}
achieves minimum when $\Phi(x)$ fulfils Eq.(\ref{eq:prob1}) with boundary conditions $\psi(x \to \pm \infty)\to 0$. This allows determination of bound-state wave-functions by approximating them by finite linear combinations
\begin{equation} \Phi(x)=\sum ^{M-1}_{j=0} c_j \phi^{A}_j(x),\label{eq:baza}\end{equation} where the functions $\phi^{A}_j(x)$are taken from an orthonormal basis in the function space.
The variational principle yields the matrix equation for the linear parameters $c_j$ in the form
\begin{equation}
\sum ^{M-1}_{j=0} (H^{A}_{jm} - \varepsilon \delta_{jm})c_j = 0,~m=0,1,...,M-1,
\label{eq:new}
\end{equation}
where the matrix elements of the Hamiltonian are \begin{equation}H^{A}_{jm}= \langle \phi^A_j |\hat{H}| \phi^A_m \rangle=\int_{-\infty}^{\infty}\phi^A_j(x)\hat{H}\phi^A_m(x)dx. \label{eq:real}
\end{equation}
Diagonalisation of the RR matrix provides $M$th order approximations to the $M$ of the lowest energy states. Systematically increasing the matrix dimension $M$, we increase the number of determined bound-states, wherein their approximate energies approach the exact results from above. The convergence of the method depends heavily on the choice of the basis set $\{\phi^{A}_j(x), j=0,1,...\}$. It appears advantageous to make the functions of the basis adaptable to the problem under study by allowing their dependence on nonlinear parameters ($A$), the values of which can be conveniently adjusted in each order approximation. To ensure a fast convergence of the particular eigenvalue (usually the ground-state energy), the values of nonlinear parameters are chosen by the trial and error or determined in numerically demanding optimization procedures \cite{rych}. Another option, that does not need any starting values or iterations, is to fix the nonlinear parameters according to the principle of minimal sensitivity \cite{PMS}, i.e. so that the approximation to a physical quantity would depend as weakly as possible on infinitesimal changes of their values. In the optimized RR method, proposed by one of us~\cite{ao}, the sum of $M$ bound-state energies is chosen as the physical quantity, the $M$th order approximation to which is given by the trace of the RR matrix
$TrH_{A}^{(M)}=\sum^{M-1}_{j=0}<\phi^{A}_j|\hat{H}|\phi^{A}_j>$. The stationarity of the trace requirement
\begin{equation} \frac{\delta}{\delta A} TrH_{A}^{(M)}=0,\label{eq:sta}\end{equation}
is used to fix the values of nonlinear parameters, and a diagonalization of the so optimized matrix determines a set of $M$ aproximate eigenstates which are mutually orthogonal. The method is computationally less demanding although its convergence for a particular state may be slower than that achieved with nonlinear parameters iteratively optimized for that state. In the case of bound-states, the effectiveness of our method has been demonstrated for various potentials and various basis sets \cite{ao,Amore,RRopty,coll}. Here we extend its application to resonant states.
\subsection{Rayleigh-Ritz determination of resonant states}
Experimentally, the resonances manifest themselves as sharp peaks in the collision cross sections which are well described by the two parameter Breit-Wigner formula. The resonance energy $E$ and the half-width of the peak $\Gamma$ may be related to the complex eigenvalues
\begin{equation}\varepsilon=E-i\Gamma/2, \label{eq:enercms}\end{equation}
of the Schr\"{o}dinger equation
(\ref{eq:prob1}), allowing $\Gamma$ to be interpretated as the inverse of the resonance lifetime. In finite range potentials, the wave function of a resonant state exhibits an asymptotic behavior of the form
\begin{equation}\psi_{rez}(x\to \pm \infty ) \approx e^{\pm ik_{rez}x} , \label{eq:asy1}\end{equation}
where
\begin{equation}k_{rez}= |k_{rez}|e^{-i\alpha_{rez}}, \label{eq:krez}\end{equation}
with $0<\alpha_{rez}<\pi /2$, as corresponds to the position of wave vector $k_{rez}$ in the fourth quarter of the complex plane. Since the wave function $\psi_{rez}(x)$ diverges exponentially, the Hamiltonian is not hermitian and its complex eigenvalues are hidden on a higher Riemann sheet of the complex energy plane.
\subsubsection{Complex Scaling}
The complex scaling transformation
\begin{equation}U(\theta): x \mapsto x e^{i\theta}, \label{eq:tran}\end{equation}
allows the treatment of resonant states in analogy to bound-states.
The corresponding complex-rotated Hamiltonian
\begin{equation} \hat{H}_{\theta}=U(\theta) \hat{H}U^{-1}(\theta),\label{eq:hamt}\end{equation}
satisfies the eigenequation
\begin{equation} \hat{H}_{\theta} \psi_{\theta}(x)=\varepsilon_{\theta} \psi_{\theta}(x),\label{eq:rsr}\end{equation}
and its spectrum is described by the Basley-Combes theorem. For dilatation analytic potentials~\cite{BC}, among which are the Coulomb and Yukawa potentials in addition to the finite range ones, the theorem states that the real bound-state eigenvalues, the complex resonance eigenvalues and the thresholds are the same as those of the original Hamiltonian, but the eigenvalues of the continuous spectrum are rotated about the thresholds by an angle $2\theta$ into the lower energy half-plane, exposing complex resonance eigenvalues. The complex scaling transformation (\ref{eq:tran}) turns the function $\psi_{rez}(x)$ into a normalizable one, if the real parameter $\theta$ is such that $0<\theta-\alpha_{rez}<\pi /2$. In this case, the resonances can be determined as the eigenstates of the non-hermitian Hamiltonian $\hat{H}_{\theta}$ by using bound-state-like strategies. Here we use the optimized RR method with a complex basis to this end.
\subsubsection{Complex Basis}
Since the resonance eigenvalues are complex numbers, their spectrum is determined by stationarity rather than minimization condition. This requires that the Rayleigh quotient
\begin{equation} R[\Phi]=\frac{<\Phi(x)|\hat{H}_{\theta}|\Phi(x)>}{<\Phi(x)|\Phi(x)>},\label{eq:roqq}\end{equation}
be stationary at the square integrable solutions of the complex rotated Hamiltonian~(\ref{eq:hamt}). With the solution $\Phi(x)$ approximated by a finite linear combination of the real functions (\ref{eq:baza}), the same secular equation is obtained as for bound states (\ref{eq:new}) but
with the matrix element $H^A_{jm}$ replaced by
\begin{equation}
H^{A,\theta}_{jm}=\langle \phi^A_j |\hat{H}_{\theta}| \phi^A_m \rangle=\int_{-\infty}^{\infty}\phi^A_j(x)\hat{H}_{\theta}\phi^A_m(x)dx.
\end{equation}
It has been observed~\cite{ComplexBasis} that changing the variable $x$ to $x e^{-i\theta}$ and using Cauchy's theorem to distort the integration contour back to the real axis, the matrix elements turn into
\begin{equation}
H^{A,\theta}_{jm}=\int_{-\infty}^{\infty}\phi^A_j(xe^{-i\theta})\hat{H}\phi^A_m(xe^{-i\theta})dx.\label{scale}
\end{equation}
The complex scaling is thus equivalent to working with original Hamiltonian and using the basis functions with coordinates rescaled with $e^{-i\theta}$ factor~\cite{Reinhardt,ComplexBasis}. Note, however, that instead of the ordinary scalar product of the Hilbert space $<f|g>= \int_{-\infty}^{\infty} f^{*}(x)g(x) dx$, the c-scalar product
$(f|g)= \int_{-\infty}^{\infty} f(x)g(x) dx$ is to be used in the complex basis approach~\cite{nm}. The advantage of the complex basis approach is that it applies also for non-dilatation analitic potentials. Moreover, generalization to many-body systems allows introducing different scaling parameter for each degree-of-freedom, which makes the method more flexible than that of the complex scaled Hamiltonian. In the case when the nonlinear parameter $A$ is the scale parameter, so that \be \phi^{A}_j(x)={1\over \sqrt{A}}\phi_{j}\left({x\over A}\right),\ee the RR matrix element (\ref{scale}) may be written as
\begin{equation}
H^{\alpha}_{jm}=\int_{-\infty}^{\infty}\phi^\alpha_j(x)\hat{H}\phi^\alpha_m(x)dx,\label{Hcomplex}
\end{equation}
where $\alpha=Ae^{i\theta}$ and may be simply obtained by replacing the real parameter $A$ in (\ref{eq:real}) by a complex parameter $\alpha$.
\subsubsection{Optimized RR method for resonances}
In this work we generalize the optimized RR method~\cite{ao} to the case of resonant states by using the complex basis approach. Dealing with the unscaled Hamiltonian $\hat{H}$, we choose a basis set of real functions $\{\phi^{\alpha}_j(x), j=0,1,...\}$ so that the matrix elements $<\phi^{\alpha}_m|\hat{H}|\phi^{\alpha}_n>$ are given by analitic expressions. In the $M$-th order calculation, we fix the value of the parameter $\alpha$ to be equal to the complex solution of the stationarity of the trace condition (\ref{eq:sta}) and after a single diagonalization of the $M$-dimensional symmetric complex matrix we obtain a set of $M$ approximate eigenvalues. The number of calculated eigenvalues and their accuracy may be increased by increasing $M$, which permits quantification of the precision of the obtained results. In order to demonstrate the effectiveness of our method, we consider several very different systems described by one-dimensional and radial Schr\"{o}dinger equations, using various basis sets of functions with adjustable scale parameters. In some cases it turns out to be advantageous to introduce additional parameters that are not the scaling parameters.
\section{One-dimensional Schr\"{o}dinger equations}\label{sec:one}
\subsection{Harmonic oscillator basis: $RR_{opt}^{\Omega}$ method}
First, we consider the case of reflection symmetric potentials $V(x)=V(-x)$. We use the $RR_{opt}^{\Omega}$ method with the basis of the harmonic oscillator (HO) eigenfunctions
\begin{equation} \phi_j^{\Omega}(x)=\left(\frac{\sqrt{\Omega}}{\sqrt{\pi}2^j j!}\right)^{1/2} H_j (\sqrt{\Omega}x) e^{-\frac{\Omega x^2}{2}}, \label{eq:one}\end{equation}
where the square root of the oscillator frequency $\Omega$ plays a role of an inverse scaling parameter. Due to reflection symmetry, the even- (odd-) parity states may be obtained by diagonalization of the Hamiltonian matrix in the basis of the first $M$ even (odd) functions, which substantially reduces the computational cost.
\subsubsection{Quartic resonance potential}
A simple example of a resonant system is the inverse quartic anharmonic oscillator with a Hamiltonian
\begin{equation}
\hat{H}=-\frac{1}{2}\frac{d^2}{dx^2}+ V(x)=-\frac{1}{2}\frac{d^2}{dx^2}+ \frac{1}{2}x^2 -\frac{\lambda}{2} x^4,
\label{eq:hl1}
\end{equation}
where the units $\hbar=1$ and $m=1$ are used. In the case of $\lambda>0$, the potential $V(x)$ is not bounded from below and the system possess only resonant states, the lowest energies of which are marked on Fig.~\ref{fig:quartic} for $\lambda = 0.02$. The lifetime of the resonances increases with decreasing $\lambda$, and the $RR_{opt}^{\Omega}$ calculations require using larger basis sets to obtain a satisfactory accuracy. Our numerical approximations to $\epsilon_0$ and $\epsilon_2$ for $\lambda = 0.02$, which is the most demanding case considered in the literature \cite{ferSiegert}, are presented in table \ref{eq:tab1} as a function of the dimension of the RR matrix $M$. Here and in the following tables the results are accurate to the number of figures shown. The best accuracy is obtained for the lowest resonance, and for all states it quickly improves with increasing $M$. We can see that the literature results of the Riccati-Pad\'e method \cite{ferSiegert} are reproduced with RR matrix of dimension $M=25$; for larger $M$, more accurate values are easily obtained. The imaginary part of $\epsilon_2$ is larger than that of $\epsilon_0$, and we observed its further increase for higher resonances, which confirms that the lifetime decreases with increasing resonance energy.
\begin{figure} [h!]
\begin{center}
\scalebox{0.9}{\includegraphics{rr.eps}}
\caption{The first five resonance's energies $Re\varepsilon$ for the potential $V(x)=\frac{1}{2}x^2-0.01 x^4$.}
\label{fig:quartic}
\end{center}
\end{figure}
\begin{table} [h!]
\begin{center}
\scriptsize
\begin{tabular}{ccll}
\hline
$M$&$\Omega_{opt}$& $E_0$ & $\Gamma_0$\\
&&$E_2$&$\Gamma_2$\\
\hline
20&0.723 - 0.754 I&0.4922138348826277005 &$5.109*10^{-14}$\\
&&2.393167523963263&$2.842626*10^{-9}$\\
25&0.759 - 0.853 I&0.49221383488262770042136&$5.109394888*10^{-14}$\\
&&2.39316752396326281772&$2.84262607840*10^{-9} $\\
30&0.791 - 0.937 I&0.4922138348826277004213624190&$5.10939488839463*10^{-14}$\\
&&2.393167523963262817721830&$2.8426260783914863*10^{-9}$ \\
35&0.821 - 1.010 I&0.49221383488262770042136241897612033&$5.109394888394627276*10^{-14}$\\
&&2.39316752396326281772183026343&$2.842626078391486200557*10^{-9}$\\
\hline
\end{tabular}
\caption{Optimal values of the parameter $\Omega$ and the energies and widths of the two lowest even resonances for the Hamiltonian (\ref{eq:hl1}) with $\lambda =0.02$ calculated by the $RR_{opt}^{\Omega}$ method.}
\label{eq:tab1}
\end{center}
\end{table}
\subsubsection{Triple-well oscillator}
The sextic oscillator Hamiltonian
\begin{equation}
\hat{H}=-\frac{1}{2}\frac{d^2}{dx^2}+ \frac{1}{2}x^2 -g^2x^4+\frac{g^4}{2} x^6,
\label{eq:hx6}
\end{equation}
where the triple-well potential $V(x)=\frac{1}{2}x^2 -g^2x^4+\frac{g^4}{2} x^6$ is bounded from below
and increases to infinity at $|x|\to \infty$, describes an interesting system which supports only bound states. It has been shown~\cite{BGG,irregular} that the complex scaling transformation turns the asymptotically divergent solutions of this problem into square integrable ones that are associated with complex eigenvalues which describe the rates of tunneling between the potential wells. The eigenvalues $\epsilon_0$ and $\epsilon_4$ determined with the $RR_{opt}^{\Omega}$ method for various matrix dimensions $M$ are presented in table \ref{eq:tabg1} for $g=0.08$ and in table \ref{eq:tabg2} for $g=0.3$. We observe that with increasing $g$, the imaginary part of the eigenvalue grows, i.e. the resonance lifetime decreases, and the accuracy of the method improves. Comparison with the best published results for $\varepsilon_0$, obtained by the Ricatti-Pad\'e method \cite{ferSiegert}, shows that in the most unfavorable case of $g=0.08$ we attain the same level of accuracy with the matrix of dimension $M=60$, while $M=30$ is sufficient in the case of $g=0.3$.
\begin{table} [h!]
\begin{center}
\scriptsize
\begin{tabular}{ccll}
\hline
$M$ &$\Omega_{opt}$& $E_0$& $\Gamma_0$\\
&& $E_4$& $\Gamma_4$\\
\hline
40&0.5161 - 0.8673 I&0.4951282297707530015692954656&$10^{-28}$\\
&&4.28962031383623330069490&$4*10^{-20}$\\
50&0.5162 - 1.0001 I&0.4951282297707530015692954656874446&$1.2*10^{-32}$\\
&&4.289620313836233300694904739&$4.180046*10^{-20}$\\
60&0.5163 - 1.1171 I&0.4951282297707530015692954656874446179669&$1.16994174*10^{-32}$\\
&&4.2896203138362333006949047388180616&$4.1800456118133*10^{-20}$\\
70&0.5163 - 1.2230 I&0.495128229770753001569295465687444617966881040&$1.1699417439855*10^{-32}$\\
&&4.2896203138362333006949047388180616237064&$4.1800456118132521255*10^{-20}$\\
\hline
\end{tabular}
\caption{Optimal values of the parameter $\Omega$ and the energies and widths of the even resonances for the Hamiltonian (\ref{eq:hx6}) with $g =0.08$ calculated by the $RR_{opt}^{\Omega}$ method.}
\label{eq:tabg1}
\end{center}
\end{table}
\begin{table} [h!]
\begin{center}
\scriptsize
\begin{tabular}{ccll}
\hline
$M$ &$\Omega_{opt}$& $E_0$& $\Gamma_0$\\
&& $E_4$& $\Gamma_4$\\
\hline
10&0.5159 - 1.7799 I&0.40780&0.0294002\\
&&2.6095&4.7968\\
20&0.5163 - 2.5792 I&0.4078039790737&0.02940021689214\\
&&2.6094307234&4.79672853029\\
30&0.5163 - 3.1840 I&0.40780397907366957146&0.029400216892153485663\\
&&2.60943072337167570&4.79672853029023136\\
40&0.5164 - 3.6909 I&0.40780397907366957146548001&0.029400216892153485662928541\\
&&2.60943072337167569736879988&4.796728530290231359778820\\
50&0.5164 - 4.1363 I&0.40780397907366957146548000596165423&0.02940021689215348566292854082324361\\
&&2.60943072337167569736879987097172&4.7967285302902313597788200335659\\
\hline
\end{tabular}
\caption{Same as in Table \ref{eq:tabg1} but for $g =0.3$.}
\label{eq:tabg2}
\end{center}
\end{table}
\subsection{Shifted harmonic oscillator basis: $RR_{opt}^{\Omega,t}$ method}
Discussing problems with the resonant potential that is not symmeric about the origin, it is advantegeous to use $RR_{opt}^{\Omega, t}$ method with additional complex parameter $t$ that shifts the argument of the HO basis functions
\begin{equation}
\phi_j^{\Omega, t}(x)=\left(\frac{\sqrt{\Omega}}{\sqrt{\pi}2^j j!}\right)^{1/2} H_j \left(\sqrt{\Omega}(x-t)\right) e^{-\frac{\Omega (x-t)^2}{2}}.
\label{eq:bazat}
\end{equation}
We apply the above basis to determine the spectrum of the cubic anharmonic Hamiltonian
\begin{equation}
\hat{H}=-\frac{1}{2} \frac{d^2}{dx^2}+\frac{1}{2} x^2 + \gamma x^3,
\label{eq:hxc2}
\end{equation}
for an exemplary value of $\gamma=0.1$. In table \ref{eq:tabxc1}, the complex energies of the lowest resonant state are presented with the corresponding optimal values of the nonlinear parameters $\Omega_{opt}$ and $t_{opt}$ for various dimensions $M$ of the RR matrix. The best previously published results \cite{k3} are reproduced with the RR matrix of dimension $M=40$. By increasing $M$, we easily obtain more accurate eigenvalues. It is interesting to note that although the shift parameter is not a scale parameter, its value determined by the stationarity of the trace condition turns out to be complex and this appears crucial for a fast convergence of the optimized RR scheme.
\begin{table} [h!]
\begin{center}
\scriptsize
\begin{tabular}{cccll}
\hline
M & $t_{opt}$ &$\Omega_{opt}$ &$E_0$& $\Gamma_0$ \\
\hline
20 &-0.67 - 2.26 I&1.02 - 0.66 I&0.48432&0.0000161\\
30 &-0.54 - 2.78 I&1.11 - 0.75 I&0.48431599700&0.0000161204\\
40 &-0.43 - 3.20 I&1.18 - 0.81 I&0.484315997004117&0.000016120419000\\
50 &-0.34 - 3.55 I&1.24 - 0.86 I&0.4843159970041175430&0.00001612041900013357\\
60 &-0.25 - 3.86 I&1.29 - 0.90 I&0.484315997004117543023&0.0000161204190001335639\\
\hline
\end{tabular}
\caption{Optimal values of nonlinear parameters and the energy and width of the lowest resonance state for the Hamiltonian (\ref{eq:hxc2}) with $\gamma=0.1$ calculated by the $RR_{opt}^{\Omega,t}$ method.}
\label{eq:tabxc1}
\end{center}
\end{table}
\subsection{Trigonometric basis: $RR_{opt}^{L}$ method}
Another convenient basis is provided by the set of trigonometric (TRIG) functions that satisfy the Dirichlet boundary condition at $x=\pm L$, where the width of the box $L$ serves as a scaling parameter. The even functions are of the form
\begin{equation} \phi_{j}^L(x)={1\over\sqrt{L}}\cos\left[\left(j+\frac{1}{2}\right)\frac{\pi x}{L}\right], \label{eq:te} \end{equation}
and the odd ones are given by
\begin{equation} \phi_{j}^L(x)={1\over\sqrt{L}}\sin\left[(j+1)\frac{\pi x}{L}\right]. \label{eq:to} \end{equation}
We use the TRIG basis to find the eigenvalues of the Hamiltonian with an inverted Gaussian potential with quartic perturbation
\begin{equation}
\hat{H}=-\frac{1}{2}\frac{d^2}{dx^2} -5e^{-0.1 x^2}-\frac{\lambda}{2} x^4.
\label{eq:hex3}
\end{equation}
\begin{table} [h!]
\begin{center}
\scriptsize
\begin{tabular}{ccll}
\hline
$M$&$L_{opt}$& $E_0$ & $\Gamma_0$\\
&&$E_1$&$\Gamma_1$\\
\hline
20&5.114 + 2.888 I&-4.5665655093777188&0.0177068941054286\\
&5.155 + 2.915 I&-3.8381035089108826&0.2743215904678811\\
30&5.840 + 3.348 I&-4.5665655093777188168702314&0.0177068941054286198302607\\
&5.872 + 3.367 I&-3.8381035089108826586063027&0.27432159046788112432336\\
40&6.427 + 3.699 I&-4.5665655093777188168702313367733584&0.01770689410542861983026074495278\\
&6.454 + 3.715 I&-3.838103508910882658606302626356831&0.274321590467881124323361759999012\\
50&6.924 + 3.989 I&-4.5665655093777188168702313367733583615406723&0.017706894105428619830260744952784024460349\\
&6.947 + 4.003 I&-3.83810350891088265860630262635683043046341&0.27432159046788112432336175999901239136805\\
\hline
\end{tabular}
\caption{Optimal values of the parameter $L$ and the energies and widths of the two lowest resonances for the Hamiltonian (\ref{eq:hex3}) with $\lambda =0.08$ calculated by the $RR_{opt}^{L}$ method.}
\label{eq:tabl1}
\end{center}
\end{table}
\begin{table} [h!]
\begin{center}
\scriptsize
\begin{tabular}{ccll}
\hline
$M$&$L_{opt}$& $E_0$ & $\Gamma_0$\\
&&$E_1$&$\Gamma_1$\\
\hline
20&7.202 + 4.098 I&-4.52348287219&$2.20682*10^{-7} $\\
&7.264 + 4.135 I&-3.62331693435&0.00006769098\\
30&8.261 + 4.726 I&-4.5234828721860057186&$2.20682214291*10^{-7} $\\
&8.306 + 4.753 I&-3.6233169343531338&0.0000676909876076569\\
40&9.089 + 5.218 I&-4.52348287218600571860826015&$2.20682214290912985423*10^{-7} $\\
&9.126 + 5.240 I&-3.62331693435313378989538384&0.0000676909876076569149953\\
50&9.788 + 5.630 I&-4.5234828721860057186082601443634245&$2.2068221429091298542221588*10^{-7}$\\
&9.821 + 5.649 I&-3.62331693435313378989538384275228&0.00006769098760765691499530503897\\
\hline
\end{tabular}
\caption{Same as Table \ref{eq:tabl1}, but for $\lambda =0.01$}
\label{eq:tabl1a}
\end{center}
\end{table}
\noindent The energies and widths presented in Tables \ref{eq:tabl1} and \ref{eq:tabl1a} show that already with matrices of dimension $M=20$ we reach the accuarcy of the literature results \cite{gauss}. We observe that the energy of the resonant state does not change much with decreasing $\lambda$, while its width decreases rapidly to zero, switching to the bound state at $\lambda=0$.
\section{Radial Schr\"{o}dinger equation in $D$-dimensional space}\label{sec:radial}
For spherically symmetric problems in $D$-dimensional space, the solution of the Schr\"{o}dinger equation factorizes into the angular part given by hyperspherical harmonics and the radial part $R(r)$ that fulfils
\begin{equation}
\left[-\frac{1}{2}\frac{d^2}{dr^2}-\frac{(D-1)}{2r}\frac{d}{dr}+\frac{l(l+D-2)}{2r^2}+V(r)\right]R(r)=ER(r).
\label{eq:hradial}
\end{equation}
By substituting $R(r)=r^{(1-D)/2}u(r)$, the radial equation is brought to the one-dimensional Schr\"{o}dinger form
\begin{equation}
\hat{H}_r u(r)=\left[-\frac{1}{2}\frac{d^2}{dr^2}+\frac{\Lambda(\Lambda+1)}{2r^2}+V(r)\right]u(r)=Eu(r),
\label{eq:hr}
\end{equation}
where $\Lambda=l+D/2-3/2$.
\subsection{Radial harmonic oscillator basis: $RR_{opt}^{\Omega}$ method}
For determining the resonant spectrum of radial anharmonic oscillators, it is convenient to use the basis of the spherically symmetric HO eigenfunctions that are given by
\begin{equation}
\phi_{j\Lambda}^{\Omega}(r)=\sqrt{\frac{2 j! \Omega^{\frac{3}{2}+\Lambda}}{\Gamma(j+\Lambda+\frac{3}{2})}}r^{\Lambda+1} e^{-\frac{r^2\Omega}{2}} L_j^{\Lambda+\frac{1}{2}}(r^2\Omega).\label{eq:baza2}
\end{equation}
As an example, we consider the case of two dimensions, where $\Lambda=l-1/2$. We determine the resonances in the inverted Mexican hat potential with the radial Hamiltonian given by
\begin{equation}
\hat{H}=-\frac{1}{2}\frac{d^2}{dr^2} +\frac{l^2-\frac{1}{4}}{2r^2}+ \frac{1}{2}r^2 - \frac{g}{2} r^4,
\label{eq:hr}
\end{equation}
where $g>0$. The above Hamiltonian has been studied before in Ref.\cite{k4} by the RR method with the same basis (\ref{eq:baza2}), but with iteratively adjusting the value of $\Omega$ in each order so as to obtain the best convergence for a single selected state. Our results obtained with $\Omega$ being fixed a priori from the trace condition~(\ref{eq:sta}) are presented in Table \ref{eq:tabr1}. The comparison shows that our method automatically provides a fast convergence in determining a number of resonances with different quantum number $n$ in one run.
\begin{table} [!ht]
\begin{center}
\scriptsize
\begin{tabular}{ccll}
\hline
$M$&$\Omega_{opt}$&$E_0^0$&$\Gamma_0^0$\\
&&$E_0^1$&$\Gamma_0^1$\\
\hline
10&0.8982 - 1.1917 I&0.856745041& 0.04543667 \\
&0.9095 - 1.2172 I&1.56818293&0.34682442\\
15&1.0000 - 1.4142 I&0.85674504114583&0.0454366707026\\
&1.0090 - 1.4333 I&1.568182929694&0.34682442355763\\
20&1.0832 - 1.5874 I&0.8567450411458273172&0.0454366707025907851 \\
&1.0907 - 1.6029 I&1.5681829296936992519&0.346824423557637908916\\
25&1.1545 - 1.7316 I&0.8567450411458273172142177&0.0454366707025907850723740 \\
&1.1611 - 1.7448 I&1.568182929693699251881382&0.3468244235576379089157114\\
30&1.2174 - 1.8564 I&0.8567450411458273172142177179279& 0.04543667070259078507237390548 \\
&1.2233 - 1.8681 I&1.568182929693699251881381495503&0.346824423557637908915711315930\\
\hline
\end{tabular}
\caption{Optimal values of the parameter $\Omega$ and the energies and widths of the lowest resonances with $l=0$ and $l=1$ for the Hamiltonian (\ref{eq:hr}) with $g=0.1$, calculated by the $RR_{opt}^{\Omega}$ method for increasing dimension $M$.}
\label{eq:tabr1}
\end{center}
\end{table}
\subsection{Radial trigonometric basis: $RR_{opt}^{L}$ method}
For studying problems with $\Lambda=0$, it appears convenient to use the basis of antisymmetric trigonometric functions
\begin{equation} \phi_{j}^L(x)=\sqrt\frac{2}{L}\sin\left[(j+1)\frac{\pi x}{L}\right],\label{eq:to1} \end{equation}
where $L$ represents the confinement radius. We apply the above basis to calculate the resonant states of the Bardsley Hamiltonian~\cite{Bar}
\begin{equation}
\hat{H}=-\frac{1}{2}\frac{d^2}{dr^2} + V_{0}r^2 e^{-r}.
\label{eq:ht1}
\end{equation}
\begin{figure} [h!]
\begin{center}
\scalebox{0.8}{\includegraphics{rys1.eps}}
\caption{Bardsley potential $V(r)=7.5r^2 e^{-r}$.}
\label{eq:p_nr12}
\end{center}
\end{figure}
In Ref.\cite{rak}, the very accurate results obtained using the Jost-function method are presentes for nine lowest resonances. We show our results for the first and ninth state in Table \ref{eq:tabt1}. The convergence rate decreases with increasing resonance energy but with matrices of dimension $N=160$ we automatically reproduce all the results of Ref.\cite{rak}. Our results for the next resonance ($E_9$, $\Gamma_9$ are also presented in the Table.
\begin{table} [!ht]
\begin{center}
\scriptsize
\begin{tabular}{ccll}
\hline
$M$&$L_{opt}$&$E_0$&$\Gamma_0$\\
&&$E_8$&$\Gamma_8$\\
&&$E_9$&$\Gamma_9$\\
\hline
100&-0.841+6.661I&3.4264&0.025549\\
&&-3.7 &40 \\
&& -6.9& 45\\
120&-1.099+6.804I&3.4263903&0.02554896\\
&& -3.75& 40.01\\
&& -6.801& 45.527\\
140&-1.319+6.920I&3.42639031015&0.02554896118\\
&& -3.754144 & 40.0090149\\
&& -6.80030& 45.5263\\
160&-1.511+7.018I&3.4263903101482&0.02554896118580\\
&& -3.754144122 &40.00901499 \\
&&-6.80030389 & 45.52631015\\
180&-1.680+7.102I&3.4263903101482505&0.025548961185791\\
&&-3.754144122607 &40.0090149933 \\
&&-6.800303886379 &45.52631015000\\
\hline
\end{tabular}
\caption{Optimal values of the parameter $L$ and the energies and widths of a few resonances for the Hamiltonian (\ref{eq:hr}) with $V_{0}=7.5$, calculated by the $RR_{opt}^{\Omega}$ method for increasing dimension $M$.}
\label{eq:tabt1}
\end{center}
\end{table}\newpage
\section{Conclusion}\label{sec:con}
We applied the optimized RR method with complex nonlinear parameters to determine the resonance states in various one-dimensional potentials. The expansion basis were adapted to the considered problems so that the RR matrix elements were given by analytic expressions. The values of nonlinear parameters were fixed by requiring that the trace of the truncated matrix be stationary. We have shown that the basis of the HO eigenfunctions with frequency $\Omega$ optimized by the stationarity of the trace condition is efficient in determining the resonance spectrum in anharmonic potentials. In the case of nonsymmetric about the origin potentials, an additional complex shift parameter $t$ has proved to be useful to obtain quick convergence. The trigonometric basis with optimized boundary period parameter $L$ appears convenient, especially in the case of potentials described by exponential functions. Effectiveness of a similar approach in determining resonances of the radial Schr\"odinger equation has been also shown. The advantage of the optimized RR scheme is that a set of resonances is determined automatically in one run without the necessity of specifying any starting value. The computational cost of our method is much lower than in the case of iterative optimization of nonlinear parameters. The method appears especially effective in the cases where only resonant modes exist. For the class of resonant potentials considered in the present work, it is highly competitive with existing methods, the results of which are easily recovered in our approach.
\section{Acknowledgements}\label{ack} This work was partially supported by the ESF Human Capital Operational Programme grant 6/1/8.2.1./POKL/ 2009. One of the authors (AO) is deeply indebted to Jacek Karwowski for inspiring questions.\\
|
1003.5641
|
\section{Introduction}
\label{introduction}
\noindent
A peculiar feature distinguishing strong (QCD) and electroweak (EW)
effects in higher orders
is that the latter are enhanced by (Sudakov) double logarithmic factors,
$\ln^2(\frac{s}{M^2_{{W}}})$,
which, unlike in the former, do not cancel for `infrared-safe'
observables \cite{Kuroda:1991wn,Beenakker:1993tt,Ciafaloni:1999xg,Denner:2000jv}.
The origin of these `double logs' is well understood.
It is due to a lack of the Kinoshita-Lee-Nauenberg (KLN)
\cite{KLN} type
cancellations of Infra-Red (IR) -- both soft
and collinear -- virtual and real emission in
higher order contributions originating from $W^\pm$ (and, possibly,
$Z$) exchange.
This is in turn a consequence of the
violation of the Bloch-Nordsieck theorem \cite{BN} in non-Abelian theories
\cite{Ciafaloni:2000df}.
The problem is in principle present also in QCD. In practice, however,
it has no observable consequences, because of the final averaging of the
colour degrees of freedom of partons.
This does not occur in the EW case,
where the initial state has a non-Abelian charge,
dictated by the given collider beam configuration, such as in $e^+e^-$
collisions.
These logarithmic corrections are finite (unlike in
QCD), as the masses of the weak gauge bosons provide a physical
cut-off for $W^\pm$ and $Z$ emission. Hence, for typical experimental
resolutions, softly and collinearly emitted weak bosons need not be included
in the production cross-section and one can restrict oneself to the
calculation of weak effects originating from virtual corrections and
affecting a purely hadronic final state.
Besides, these contributions can be
isolated in a gauge-invariant manner from electromagnetic (EM) effects
\cite{Ciafaloni:1999xg},
at least in some specific cases, and
therefore may or may not
be included in the calculation, depending on the observable being studied.
As for purely EM effects,
since
the (infinite) IR real photon emission cannot be resolved experimentally,
this ought to be combined with the (also infinite) virtual one, through the
same order, to recover a finite result, which is however not
doubly logarithmically enhanced (as QED is an Abelian theory).
In view of all this, it becomes of crucial importance to assess
the quantitative relevance of such EW corrections
affecting, in particular, key QCD processes studied at past, present and
future
colliders, such as $e^+e^-\to3$~jets.
\section{Calculation}
\label{calculation}
\noindent
In Ref.~\cite{ee3jets}, we calculated the full
one-loop EW effects entering three-jet production in $e^+e^-$
annihilation at any collider energy
via the subprocesses $e^+e^-\to\gamma^*,Z\to \bar
qqg$. Ref.~\cite{oldpapers} tackled part of these, restricted to the
case of $W^\pm$ and $Z$ (but not $\gamma$) exchange and
when the higher order effects arise only from initial or final state
interactions
(the so-called `factorisable' corrections).
The remainder, `non-factorisable' corrections,
while being typically small at $\sqrt s=M_{Z}$,
are expected to play a quantitatively relevant role as $\sqrt s$ grows
larger.
We improved on the
results of Ref.~\cite{oldpapers} in two respects: (i) we include now all
the non-factorisable terms; (ii) we also incorporate previously
neglected genuine QED corrections, including photon bremsstrahlung.
A more complete account of the corrections discussed here has recently
appeared in Ref.~\cite{ee3jetsgermans}.
Combining the enhancement associated with the weak Sudakov logarithms
to the decrease of $\alpha_{\mathrm{S}}$ with energy,
in general, one expects one-loop EW effects to become comparable to QCD ones
at future Linear Colliders (LCs) \cite{LCs} running at TeV energy scales,
like those available at an International Linear Collider (ILC) or the
Compact LInear Collider (CLIC).
In contrast, at the $Z$ mass peak, where logarithmic enhancements are
not effective, one-loop EW corrections are expected to appear
at the percent level, hence being of limited relevance at
LEP1 and SLC, where the final error on $\alpha_{\mathrm{S}}$
is of the same order or larger,
but of crucial importance at a GigaZ stage of a future
LC \cite{oldpapers}, where the relative accuracy
of $\alpha_{\mathrm{S}}$ measurements is expected to be at the
$0.1\%$ level or better.
Concerning higher order QCD
effects, a great deal of effort has
recently been devoted to evaluate two-loop contributions
to the three-jet process \cite{QCD2Loops}
while the one-loop QCD results have been known for quite some time \cite{ERT}.
In $e^+e^-$ annihilations, the most important QCD quantity to be
extracted from multi-jet events is $\alpha_{\mathrm{S}}$.
The confrontation of the measured value of the strong coupling
constant with that predicted by the theory through the
renormalisation group evolution is an important test of the Standard Model
(SM). Alternatively, it may be an indication of new physics, when its
typical mass scale is larger than the collider energy, so that
the new particles cannot be
produced as `real' detectable states
but may manifest themselves through `virtual' effects.
Not only jet rates,
but also
jet shape observables would be affected.
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{diagrams/pentagons}
\end{center}
\caption{Pentagon graphs}
\label{fig:pentagons}
\end{figure}
The detailed discussion of the calculation can be found in
Ref.~\cite{ee3jets}. Here, for the sake of completeness, we mention
that the calculation of virtual corrections is performed in the
't~Hooft-Feynmann gauge.
It is also worth mentioning that initial state
electron-positron polarisations are retained and it is possible to
study EW effects in presence of polarised incoming beams. For genuinely weak interaction
corrections, this is of particular interest, since such corrections violate
parity conservation.
In the calculation IR divergences are regulated by means of a small
photon mass $\lambda$, both in virtual and real QED corrections. The
independence of the final results from the photon mass has been successfully checked.
A new feature of this calculation
is the
occurrence of pentagon graphs, as those shown in Fig.~\ref{fig:pentagons}.
We have handled these in two
separate ways (with two independently developed codes), in order to check
for possible numerical instabilities, finding good agreement.
The collinear QED divergence
gives rise to a large logarithm ($\ln(s/m_f^2)$),
which is associated with the Initial State Radiation (ISR) induced by
the incoming electrons and positrons.
In the case of
electron-positron colliders this large correction is always present
and it is universal to all processes.
For sensible numerical
results, it has to be accounted for to all orders of perturbation
theory, e.g., within the so-called electron/positron structure function formalism~\cite{sf},
which automatically resums in QED all Leading Logarithmic (LL) terms.
In Ref.~\cite{MNP} a method of combining consistently
resummed LL calculations with exact ${\cal O}(\alpha_{\mathrm{EM}})$ ones
has been devised both in additive and factorisable form.
Here, we adopted the additive approach.
In order to integrate over the phase-space,
the width, $\Gamma_Z$, of the $Z$ boson has been included in the propagator.
For consistency, this means that the same width has to
be included in the $Z$ propagator for the virtual corrections.
The essential ingredient for the
evaluation of virtual corrections is the ability to compute
one-loop integrals with complex internal masses.
We implemented the general expression
for the scalar four-point function of Ref.~\cite{tHV},
valid also for complex masses.
Particular attention has been
devoted to the occurrence of numerical instabilities in
certain regions of phase space because of strong cancellations.
We have neglected the masses of light quarks throughout. However,
in the case in which the final state
contains a $b\bar b$ pair, whenever there is a $W^\pm$ boson
in the virtual loops, account had to be taken of the mass of the
top (anti)quark.
We are therefore in a position to present the results for such
`$b$-jets' separately, as reported in~\cite{eebjets}.
\section{Numerical results}
\label{numbers}
\noindent
The numerical results presented in this section are obtained
considering a realistic experimental setup. The input parameters and
the setup of the cuts is described in Ref.~\cite{ee3jets}. A Cambridge
jet algorithm is used to cluster parton momenta into jets.
Finally, we sum over the final-state quarks, if not stated otherwise.
\begin{figure}\begin{center}{
\includegraphics[width=6cm]{figures/energyscan}~\includegraphics[width=6cm]{bjets-figures/energyscan}
\caption{Relative effect on the integrated cross section due to
different contributions to the order $\alpha\equiv
\alpha_{\mathrm{EM}}$ correction, as a function of the CM energy.
On the left the sample inclusive over the quark flavours is shown, on the right
the $b$-jets subsample is considered.
}
\label{energyscan}}
\end{center}
\end{figure}
On the left of Fig.~\ref{energyscan}, the relative effects
on the cross
section
induced by different contributions to the order $\alpha_{\mathrm{S}}
\alpha_{\mathrm{EM}}^3$
correction are plotted as a function of the CM energy, in
the range from 150 GeV to 1 TeV, when considering a sample summed over
the quark flavours. The curves represent the effect of
the QED
corrections only, the effect of the gauge
bosons self-energy corrections,
the
effect of the non-factorisable graphs
with $WW$
exchange,
the effect of
the weak corrections with the non-factorizing $WW$ graphs removed
(labelled as ``full weak - non-fact $WW$ graphs'')
and the total effect as the sum of the previous
ones: the total effect is increasingly negative, reaching the $-13\%$ level at
1 TeV.
It is worth mentioning that, as far as the non-factorisable
$WW$ corrections
are concerned, in the case of $d$,
$s$ and $b$ final-state quarks, only the direct diagrams
are present due to charge conservation, while, for $u$ and $c$ quarks, only
crossed diagrams are present, if the sum over initial- and final-state
helicities is taken. In the case of $ZZ$ exchange, all the graphs survive,
giving rise to a cancellation at the leading-log level between direct
and crossed diagrams, which does not occurr for $WW$ exchange. Hence,
the big negative correction is due to the presence of the
$WW$ non-factorisable graphs, which develop the aforementioned
large Sudakov double logarithms in the high energy regime. In the
right panel of Fig.~\ref{energyscan}, the corrections to the process
$e^+e^-\to b\bar{b}g$ are shown, assuming that an efficient
$b$-tagging is present.
We then show the impact of the EW corrections on some
differential distributions of phenomenological interest.
The plots show the tree-level contributions and the higher order
corrections in three different
contributions: the purely weak-interaction contribution
(labelled ``weak ${\cal O}(\alpha)$"), purely weak plus QED corrections,
which are dominated by the above-mentioned ISR (labelled ``exact ${\cal
O}(\alpha)$"), and the weak plus electromagnetic
correction in which the LL have been summed (labelled ``exact
${\cal O}(\alpha)$ + h.o. LL"). The figures show in the upper panel
the absolute distributions and in the lower panel the relative differences with
respect to the tree-level rates.
\begin{figure}
\begin{center}
\includegraphics[width=6cm]{figures/thrust-peak}~\includegraphics[width=6cm]{figures/thrust-1000}
\caption{$(1-T)\frac{d\sigma}{dT}$ distribution at the $Z$ peak (left)
and at 1 TeV (right).}
\label{thrust-1000}
\end{center}\end{figure}
In Fig.~\ref{thrust-1000}, the {\em thrust} event shape distribution
is shown, in the form $(1-T)\frac{d\sigma}{dT}$.
The $T$ distribution is one of the key observables used for the
measurement of $\alpha_{\mathrm{S}}$ in $e^+e^-$
collisions~\cite{kunsztnason}. It is worth noticing that while the
purely weak corrections give an almost constant effect on the whole
$T$ range, the presence of the real bremsstrahlung gives a non trivial
effect in the region $T>0.92$. In view of a precise measurement of
$\alpha_{\mathrm{S}}$ at a future LC, EW corrections can
play an important role.
The ability to efficiently tag $b$-quark jets enables one to define
observables in $b\bar bg$ final states
which are not (easily) reconstructable in the case
of the full three-jet sample. One example is the invariant mass
of the $b\bar b$ pair, $M_{b\bar b}$, which we plot in
Fig.~\ref{minbbj-peak}.
Here, the largest contribution to the total correction comes
from QED ISR, primarily because of the radiative return phenomenon.
\begin{figure}\begin{center}
\includegraphics[width=6cm]{bjets-figures/minbbj-peak}~\includegraphics[width=6cm]{bjets-figures/minbbj-1000}
\caption{$b\bar b$ invariant mass distribution at the $Z$ peak (left)
and at 1 TeV (right).}
\label{minbbj-peak}\end{center}\end{figure}
\section{Conclusions}
\noindent
In summary, we have shown the phenomenological relevance that the calculation
up to
${\cal O}(\alpha_{\mathrm{S}}\alpha_{\mathrm{EM}}^3)$
can have in the study of (unflavoured) three-jet samples
in $e^+e^-$ annihilation, for all energies ranging from $\sqrt s=M_Z$ to
1 TeV. Not only inclusive jet rates are affected, but also more exclusive
distributions, both global (like the event shape variables)
and individual (like invariant mass) ones. Effects range from a few percent to several tens
of percent, depending on the energy and the observable being studied,
and we have shown cases where such higher-order contributions would impinge
on the experimental measurements of jet quantities.
Finally, notice that, depending on experimental procedures, a
different normalisation of the distributions,
like, e.g., the one adopted in Ref.~\cite{ee3jetsgermans}, would lead
to somewhat different corrections in general.
\acknowledgments
\noindent
C.M. Carloni Calame would like to warmly thank the Local Organizing Committee
for the pleasant and stimulating atmosphere during the LC09 conference.
|
1407.3800
|
\section{Introduction}
A \emph{causal structure} for a set of classical variables is a graph,
where every variable is associated with a node and a directed edge
denotes functional dependence.
Such a causal model offers a means of \emph{explaining} dependencies
between variables, by specifying the process that gave rise to them.
More formally, variables $X_1, \dots, X_n$ form a
\emph{Bayesian network} with respect to a directed,
acyclic graph,
if every variable $X_i$
depends only on its graph-theoretic parents
$\operatorname{pa_i}$. This is the case
\cite{Pearlbook,Spirtesbook}
if and only if
the distribution factorizes as in
\begin{equation}\label{eqn:factorized}
p(x_1, \dots, x_n) = \prod_{i=1}^n p (x_i | x_{\mathrm{PA}_{i}} ).
\end{equation}
One can ask the following fundamental question: \emph{Given a subset of
of variables, which correlations between them are compatible with a
given causal structure?}
In this work, we measure ``correlations'' in terms of the collection
of joint entropies of the the variables (a precise definition will be
given below).
This problem appears in several contexts. In the young field of \emph{causal
inference}, the goal is to learn causal dependencies from empirical
data
\cite{Pearlbook,Spirtesbook}.
If observed correlations are incompatible with a presumed causal
structure, it can be discarded as a possible model.
This is close to the reasoning employed in \emph{Bell's Theorem} \cite{Bell1964} -- a
connection which is increasingly appreciated among quantum physicists \cite{Spekkens2012,Fritz2012,FritzChaves2013,Chaves2014,Fritz2014,Henson2014,Pienaar2014}.
In the context of \emph{communication theory},
these joint entropies describe the capacities that can be achieved in
network coding protocols \cite{Yeung2008}.
In this work, we are interested in quantum generalizations of causal
structures. Nodes are now allowed to represent either quantum or
classical systems, and edges are quantum operations.
An important conceptual difference to the purely classical setup
is rooted in the fact that quantum operations disturb their input. Put
differently, quantum mechanics does not assign a joint state to the
input and the output of an operation. Therefore, there is in general
no analogue to (\ref{eqn:factorized}), i.e., a global density
operator for all nodes in a quantum causal structure cannot be
defined. However, if we pick a set of nodes that do coexist (e.g.\
because they are classical, or because they are created at the same
instance of time), then we can again ask: \emph{Which joint entropies
of coexisting nodes can result from a given quantum causal structure?}
The main contribution of this work is to describe a systematic
algorithm for answering this question, generalizing previous results on
the classical case \cite{Chaves2012,FritzChaves2013,Chaves2013entropic,Chaves2014,Chaves2014b}.
We illustrate the versatility and practical relevance with two
examples. The details, along with more examples including, e.g.\ dense
coding schemes \cite{Bennett1992}, are presented in the main text.
\emph{Distributed architectures}.---Consider a scenario where, in a
first step,
several few-body quantum states are distributed among a number of
parties.
In a second step, each party processes those parts of the states it
has access to (e.g.\ by performing a coherent operation or a joint measurement).
Such setups are studied e.g.\ in distributed quantum computing
\cite{van2014quantum,buhrman2003distributed},
quantum networks
\cite{Acin2007entanglement}, quantum non-locality \cite{Brunner2014review}, and quantum
repeaters \cite{Sangouard2011}.
Which limits on the resulting correlations are implied by the network
topology alone? Our framework can be used to compute these
systematically. We will, e.g., prove certain \emph{monogamy relations}
between the correlations that can result from measurements
on distributed quantum states.
\emph{Information causality}.---The ``no-signalling principle'' alone
is insufficient to explain the ``degree of non-locality'' exhibited by
quantum mechanics \cite{Popescu1994}. This has motivated the search for stronger,
operationally motivated principles, that may single out quantum
mechanical correlations
\cite{Van1999nonlocality,Brassard2006limit,Pawlowski2009,
gross2010all,de2012deriving,Navascues2010glance,Fritz2013local,Sainz2014,Navascues2014almost}.
One of these is \emph{information causality} (IC)
\cite{Pawlowski2009,Barnum2010entropy}
which posits that an $m$ bit message from Alice to Bob must not allow
Bob to learn more than $m$ bits about a string held by Alice. A
precise formulation of the protocol involves a relatively complicated
quantum causal structure (Fig.~\ref{fig:triangle_and_IC}b). It implies
an information-theoretic bound on the mutual information between bits
$X_i$ held by Alice and guesses $Y_i$ of these by Bob
\cite{Pawlowski2009}. Here, we note that the IC setup falls into our
framework and we put the machinery to use to generalize and strengthen
it. We will show below that by taking additional information into
account, our strengthened IC principle can identify super-quantum
correlations that could not have been detected in the original
formulation.
\begin{figure} [!t]
\centering
\includegraphics[width=0.49\textwidth]{triangle_and_IC}
\caption{
\textbf{(a)} An example of distributed architecture involving bipartite entangled states. Each of the underlying quantum states can connect at most two of the observable variables, what implies a non-trivial monogamy of correlations as captured in \eqref{ineq_mn}. \textbf{b} The quantum causal structure associated with the information causality principle.}
\label{fig:triangle_and_IC}
\end{figure}
\section{Quantum Causal Structures}
Informally, a quantum causal structure
specifies the functional dependency between a collection of quantum
and classical variables.
We find it helpful to employ a graphical notation, where we aim to
closely follow the conventions of classical graphical models \cite{Pearlbook,Spirtesbook}.
There are two basic building blocks: Root nodes are labeled by
a set of quantum systems and represent a density operator for these
systems
\begin{equation*}
\raisebox{-.10cm}{\includegraphics[width=.6cm]{rootnode}}
\>
\triangleq \rho_{AB}.
\end{equation*}
The second type is given by
nodes with incoming edges. Again, both the edges and
the node carry the labels of quantum systems. Such symbols represent a
quantum operation (completely positive, trace-preserving map) from the
systems associated with the edges to the ones
associated with the node:
\begin{equation*}
\raisebox{-.23cm}{\includegraphics[width=1.0cm]{incoming}}
\>
\triangleq
\Phi_{AB \to CD}: A\otimes B \to C \otimes D.
\end{equation*}
These blocks may be combined: a node containing a system $X$
can be connected to an edge with the same label. The interpretation
is, of course, that $X$ serves as the input to the associated
operation. For example,
\begin{equation*}
\raisebox{-0.60cm}{\includegraphics[width=2.0cm]{connect}}
\>
\triangleq
\rho_C = \Phi_{AB \to C} (\rho_A \otimes \rho_B)
\end{equation*}
says that the state of system $C$ is the result of applying an
operation $\Phi_{AB \to C}$ to a product state on $AB$.
To avoid ambiguities, we will never use the same label in two
different nodes (in particular, we always assume that the output
systems of an operation are distinct from the input systems).
For a more involved example,
note that Fig.~\ref{fig:triangle_and_IC}(a)
gives a fairly readable representation of the following cumbersome
algebraic statement:
\begin{eqnarray}
\label{eqn:cumbersome}
\rho_{ABC} &=&
\big[
\Phi_{A_1 A_2 \to A}
\otimes
\Phi_{B_1 B_2 \to B}
\otimes
\Phi_{C_1 C_2 \to C}
\big] \\
&&
(
\rho_{A_1 B_1} \otimes
\rho_{A_2 C_2} \otimes
\rho_{B_2 C_2}
) \nonumber
\end{eqnarray}
(where the operation defined in the first line is acting on the state
defined in the second line).
The graphical representation does not indicate \emph{which} input
state or \emph{which} operation to employ. We suppress this
information, because we will be interested only in constraints on the
resulting correlations that are implied by the topology of the
interactions alone, regardless of the choice of states and maps.
We will use round edges to denote classical variables (equivalently,
quantum systems described by states which are diagonal in a given
basis).
In principle, classical variables could have more than one outgoing
edge, though this does not happen in the examples considered here. Of
course, the no-cloning principle precludes a quantum system being used
as the input to two different operations.
Only graphs that are free of cyclic dependencies can be interpreted as
specifying a causal structure. Thus, as is the case in classical
Bayesian networks, every quantum causal structure is associated with a
directed, acyclic graph (commonly abbreviated \emph{DAG}).
We note that graphical notations for quantum processes have been used
frequently before. The most popular graphical calculus is probably the
gate model of quantum computation \cite{Nielsen2010quantum}, where,
directly opposite to our conventions, operations are nodes and systems
are edges. Quantum communication scenarios are often visualized the
same way we employ here \cite{wilde2013quantum}. The recently introduced
\emph{generalized Bayesian networks} of \cite{Henson2014} are closely related
to our system. There, the authors even allow for post-quantum
resources.
We have noted in the introduction that a classical Bayesian network
not only defines the functional dependencies between random variables,
but also provides a structural formula (\ref{eqn:factorized}) for
the joint distribution of all variables in the graph. Again, such a
joint state for all systems that appear in a quantum causal structure
is not in general defined. However, other authors have considered
quantum versions of distributions that factor as in
(\ref{eqn:factorized}) and have developed graphical notations to this
end. Well-known examples include the related constructions that go by
the name of finitely correlated states,
matrix-product states, tree-tensor networks, or projected entangled pairs
states (a highly incomplete set of starting points to the literature
is given by
\cite{fannes1992finitely,perez2007matrix,shi2006classical}).
Also, certain definitions of quantum Bayesian
networks \cite{tucci1995quantum} fall into that class.
\section{Entropic description of quantum causal structures}
The entropic description of classical-quantum DAGs can be seen as a generalization of the framework for case of purely classical variables \cite{Chaves2012,FritzChaves2013,Chaves2013entropic,Chaves2014,Chaves2014b} that consists of three main steps. In the first, we describe the constraints (given in terms of linear inequalities) over the entropies of the $n$ variables describing a DAG. In the second step one needs to add to this basic set of inequalities, the causal entropic constraints as encoded in the conditional independencies implied by the DAG. In the last step, we need to eliminate from our description all terms involving variables that are not observable. The final result of this three steps program is the description of the marginal entropic constraints implied by the model under test.
We denote the set of indices of the random variables by $[n]=\{1, \dots, n\}$ and its power set (i.e., the set of subsets) by $2^{[n]}$. For every subset $S\in 2^{[n]}$ of indices, let $X_S$ be
the random vector $(X_i)_{i\in S}$ and denote by $H(S):=H(X_S)$ the associated entropy vector (for some, still unspecified entropy function $H$). Entropy is then a function $H: 2^{[n]} \to \mathbbm{R}, \qquad S \mapsto H(S)$ on the power set.
Note that as entropies must fulfill some constraints, not all entropy vectors are possible. That is, given the linear space of all set functions denoted by $R_n$ and a function $h\in R_n$ the region of vectors in $R_n$ that correspond to entropies is given by
\begin{eqnarray*}
\left\{ h \in R_n \,|\, h(S) = H(S) \text{ for some entropy function
} H \right\}.
\end{eqnarray*}
Clearly, this region will depend on the chosen entropy function.
For purely classical variables, $H$ is chosen to be the Shannon entropy given by $H(X_S)=-\sum_{x_s}p(x_s)\log_2 p(x_s)$. In this case an outer approximation to the associated entropy region has been studied extensively in information theory, the so called \emph{Shannon cone} $\Gamma_n$ \cite{Yeung2008}, which is the basis of the entropic approach in classical causal inference \cite{Chaves2014b}. The Shannon cone is the polyhedral closed convex cone of set functions $h$ that respect two elementary inequalities, known as polymatroidal axioms: The first relation is the \emph{sub-modularity} (also known as strong subadditivity) condition which is equivalent to the positivity of the conditional mutual information, e.g. $I(A:B\vert C)= H(A,C)+H(B,C)-H(A,B,C)-H(C) \geq 0$. The second inequality -- known as \emph{monotonicity} -- is equivalent to the positivity of the conditional entropy, e.g. $H(A \vert B) = H(A,B)- H(B) \geq 0 $.
Here lies the first difference between the classical and quantum variables, the latter being described in terms of the quantum analog of the Shannon entropy, the von Neumann entropy $H(\rho_{A,B})=-\mathrm{Tr} \left(\rho_{A,B} \log \rho_{A,B} \right) $. While quantum variables respect sub-modularity, the von Neumann entropy fails to commit with monotonicity. Note, however, that for sets consisting of both classical and quantum variables, monotonicity may still hold. That is because the uncertainty about a classical variable $A$ cannot be negative, even if we condition on an arbitrary quantum variable $\rho$, following then that $H(A \vert \rho) \geq 0$ \cite{Barnum2010entropy}. Furthermore, for a classical variable $A$, the entropy $H(A)$ reduce to the Shannon entropy \cite{Safi2011}.
Another important difference in the quantum case is the fact that
measurements (or more generally complete positive and trace preserving
(CPTP) maps) on a quantum state will generally destroy/disturb the
state. To illustrate that consider the classical-quantum DAG in Fig.
\ref{fig:triangle_and_IC}. Consider the classical and observable
variable $A$. It can without loss of generality be considered a
deterministic function of its parents $\rho_{A_1}$ and $\rho_{A_2}$,
as any additional local parent can be absorbed in the latter. For the
variable $A$ to assume a definite outcome, a joint CPTP map is applied
to both parents $\rho_{A_1}$ and $\rho_{A_2}$ that will in general
disturb these variables. The variable $A$ does not coexist with
variables $A_1$ and $A_2$. Therefore, no entropy can be
associated to these variables simultaneously, that is,
$H(A,A_1,A_2)$ cannot be part of the entropic
description of the classical-quantum DAG. Classically, this problem
does not arise as the underlying classical hidden variables could be
accessed without disturbing them.
The elementary inequalities discussed above encode the constraints that the entropies of \emph{any} set of classical or quantum random variables are subject to. Classically, the causal relationships between the variables are encoded in the conditional independencies (CI) implied by the graph. These can be algorithmically enumerated using the so-called \emph{$d$-separation criterion} \cite{Pearlbook}. Therefore, if one further demands that classical random variables are a Bayesian network with respect to some given DAG, their entropies will also ensue the additional CI relations implied by the graph. The CIs, relations of the type $p(x,y \vert z)=p(x\vert z)p(y\vert z)$, defining non-linear constraints in terms of probabilities are faithfully translate to homogeneous linear constraints on the level of entropies, e.g. $H(X,Y \vert Z)=0$. The CIs involving jointly coexisting variables also hold for the quantum causal structures considered here \cite{Henson2014}. However, some classically valid CIs may, in the quantum case, involve non coexisting variables and therefore are not valid for quantum variables. An example of that is illustrated below for the information causality scenario.
Furthermore, because terms like $H(A,A_1,A_2)$ are not part of our description, we need, together with the CIs implied by the quantum causal structure, a rule telling us how to map the underlying quantum variables in their classical descendants, for example, how to map $H(A_1,A_2) \rightarrow H(A)$. This is achieved by the data processing (DP) inequality, another basic property that is valid both for the classical and quantum cases \cite{Nielsen2010quantum}. The DP inequality basically states that the information content of a system cannot be increased by acting locally on it. To exemplify, one DP inequality implied by the DAG in Fig. \ref{fig:triangle_and_IC} is given by $I(A:B) \leq I(A_1,A_2:B_1,B_2)$, that is, the mutual information between the classical variables cannot be larger then the information shared by their underlying quantum parents.
Finally, we are interested in situations where not all joint distributions are accessible. Most commonly, this is because the variables of a DAG can be divided into observable and not directly observable ones (e.g. the underlying quantum states in Fig. \ref{fig:triangle_and_IC}). Given the set of observable variables, in the classical case, it is natural to assume that any subset of them can be \emph{jointly} observed. However, in quantum mechanics that situation is more subtle. For example, position $Q$ and momentum $P$ of a particle are individually measurable, however, there is no way to consistently assign a joint distribution to both position and momentum of the same particle \cite{Bell1964}. That is while $H(Q)$ and $H(P)$ are part of the entropic description of classical-quantum DAGs, joint terms like $H(Q,P)$ cannot be part of it. This motivates the following definition: Given a set of variables $X_{1}, \dots, X_{n}$ contained in a DAG, a \emph{marginal scenario} $\mathcal{M}$ is the collection of those subsets of $X_1, \dots, X_n$ that are assumed to be jointly measurable.
Given the inequality description of the DAG and the marginal scenario $\mathcal{M}$ under consideration, the last step consists of eliminating from this inequality description, the variables that are not directly observable, that is the variables that are not contained in $\mathcal{M}$. This is achieved, for example, via a Fourier-Motzkin (FM) elimination (see appendix for further details). In two of the examples below (information causality and quantum networks), all the observable quantities correspond to classical variables, corresponding, for example, to the outcomes of measurements performed on quantum states. Therefore, the marginal description will be given in terms of linear inequalities involving Shannon entropies only. For the super dense coding case, the final description involves a quantum variable, therefore implying a mixed inequality with Shannon as well von Neumann entropy terms.
\section{Information Causality}
The IC principle can be understood as a kind of game: Alice receives a bit string $x$ of length $n$, while Bob receives a random number $s$ ($1 \leq s \leq n$). Bob's task is to make a guess $Y_s$ about the $s$th bit of the bit string $x$ using as resources i) a $m$-bit message $M$ sent to him by Alice and ii) some correlations shared between them. It would be expected that the amount of information available to Bob about $x$ should be bounded by the amount of information contained in the message, that is, $H(M)$. IC makes this notion precise, stating that the following inequality is valid in quantum theory \cite{Pawlowski2009}
\begin{equation}
\label{IC1}
\sum_{s=1}^{n} I(X_s:Y_s) \leq H(M)
\end{equation}
where $I(X:Y)$ is the classical mutual information between the variables $X$ and $Y$ and the input bits of Alice are assumed to be independent. This inequality is valid for quantum correlations but is violated by all nonlocal correlations beyond Tsirelson's bound \cite{Pawlowski2009,Barnum2010entropy,Dahlsten2012tsirelson}.
Consider the case where $X=(X_1,X_2)$ is a 2-bit string. The corresponding causal structure to the IC game is then the one shown in Fig. \ref{fig:triangle_and_IC} b). The only relevant CI is given by $I(X_1,X_2: AB)=0$. Note that classically the CI $I(X_1,X_2:Y_s \vert M,B)=0$ (with $s=1,2$) would also be part of our entropic description. However, because we cannot assign a joint entropy to $Y_s$ and $\rho_B$, that is not possible in quantum case anymore. We can now proceed with the general framework. But before doing that we first need to specify in which marginal scenario we are interested. In Ref. \cite{Pawlowski2009} the authors implicitly restricted their attention to the marginal scenario defined by $\left\{ X_1,Y_1 \right\},\left\{ X_2,Y_2 \right\},\left\{M \right\}$. Proceeding with this marginal scenario we find that the only non-trivial inequality characterizing this marginal entropic cone is given by
\begin{equation}
\label{IC2}
I(X_1:B_1)+I(X_2:B_2) \leq H(M)+I(X_1:X_2),
\end{equation}
that corresponds exactly to the IC inequality obtained in \cite{Safi2011} where the input bits are not assumed to be independent.
Note, however, that using the aforementioned marginal scenario, available information is being discarded. The most general possible marginal scenario is given by $\left\{ X_1,X_2,Y_s,M \right\}$ (with $s=0,1$). That is, in this case we are also interested in how much information the guess $Y_1$ of the bit $X_1$ together with the message $M$ may contain about the bit $X_2$ (similarly for $B_2$ and $X_1$). Proceeding with this marginal scenario we find different classes of non-trivial tight inequalities describing the marginal information causality cone. Of particular relevance is the following tighter version of the original IC inequality
\begin{eqnarray}
\label{ICtighter}
\nonumber
& I(X_1:Y_1,M)+I(X_2:Y_2,M) +I(X_1:X_2 \vert Y_2,M) \\
& \leq H(M)+I(X_1:X_2).
\end{eqnarray}
Two different interpretations can be given to this inequality: as a monogamy of correlations or as a classical quantification of causal influence.
For the first interpretation, consider for simplicity the case where the input bits are independent, that is, $I(X_1:X_2)=0$. These independent variables may, however, become correlated given we know the values of other variables that depend on them. That is, in general $I(X_1:X_2 \vert Y_2,M) \neq 0$. However, the underlying causal relationships between the variables impose constraints on how much we can correlate these variables. In fact, as we can see from \eqref{ICtighter}, the more information the message $M$ and the guess $Y_i$ contain about about the input bit $X_i$, the smaller is the correlation we can generate between the input bits. As an extreme example suppose Alice decides to send $M=X_1\oplus X_2$. Then $X_1$ and $X_2$ are fully correlated given $M$, but $M$ doesn't contain any information about the individual inputs $X_1$ and $X_2$.
As for the second interpretation, we need to rely on the classical concept of how to quantify causal influence between two sets of variables $X$ and $Y$. As shown in \cite{Janzing2013}, a good measure $\mathcal{C}_{X \rightarrow Y}$ of the causal influence of a variable $X$ over a variable $Y$ should be lower bounded as $\mathcal{C}_{X \rightarrow Y} \geq I(X:Y \vert Pa^X_Y)$, where $Pa^X_Y$ stands for all the parents of $Y$ but $X$. That is, excluded the correlations between $X$ and $Y$ that are mediated via $Pa^X_Y$, the remaining correlations give a lower bound to the direct causal influence between the variables. Consider for instance that we allow for an arrow between the input bits $X$ and the guess $Y$. Therefore, the classical CI $I(X_1,X_2:Y_1,Y_2 \vert M,B)=0$ that is valid for the DAG in Fig. \ref{fig:triangle_and_IC} b), does not hold any longer. In this case $I(X:Y \vert Pa^X_Y)=I(X_1,X_2:Y_1,Y_2 \vert M,B)$, an object that is part of the entropic description in the classical case. Proceeding with the general framework one can prove that
\begin{eqnarray}
\label{causal_interpretation}
\nonumber
& \mathcal{C}_{X \rightarrow Y} \geq I(X_1:Y_1,M)+I(X_2:Y_2,M) \\
& +I(X_1:X_2 \vert Y_2,M) - H(M)- I(X_1:X_2).
\end{eqnarray}
That is, the degree of violation of \eqref{ICtighter} (for example, via a PR-box) gives exactly the minimum amount of direct causal influence required to obtain the same level of correlations within a classical model.
Inequality \eqref{ICtighter} refers to the particular case of two input bits for Alice. As we prove in the appendix the following generalization for any number of input bits is valid within quantum theory:
\begin{eqnarray}
\label{ICtighter2}
\nonumber
\sum^{n}_{i=1} I(X_i:Y_i,M) + \sum^{n}_{i=2} I(X_1:X_i \vert Y_i,M) \\
\leq H(M)+ \sum^{n}_{i=1}H(X_i)-H(X_1,\dots,X_{n}).
\end{eqnarray}
We further notice that the IC scenario is quite similar to the super dense coding scenario \cite{Bennett1992}, the only difference being on the fact that for the latter the message $M$ is a quantum state. On the level of the entropies this difference is translated in the fact that the monotonicity $H(M\vert X_0,X_1,B) \geq 0$ must be replaced by a the weak monotonicity $H(M\vert X_0,X_1,B) +H(M) \geq 0$. As proved in the appendix this implies that a similar inequality \eqref{ICtighter2} is a also valid for the super dense coding scenario if one replaces $H(M)$ by $2H(M)$, that is, a quantum message (combined with the shared entangled state) may allow for the double of information to be transmitted.
Finally, to understand how much more powerful inequality \eqref{ICtighter} may be in order to witness postquantum correlations we perform a similar analysis to the one in Ref. \cite{Allcock2009recovering}. We consider the following section of the nonsignalling polytope
\begin{equation}
\label{sec_poly}
p(a,b \vert x,y)=\gamma P_{PR}+\epsilon P_{det}+(1-\gamma-\epsilon) P_{white}
\end{equation}
with $P_{PR}(a,b \vert x,y)=(1/2)\delta_{a\oplus b, xy}$, $P_{white}(a,b \vert x,y)=1/4$ and $P_{det}(a,b \vert x,y)=\delta_{a,0}\delta_{b,0}$ corresponding, respectively to the PR-box, white noise and a deterministic box. The results are shown in Fig. \ref{fig:ICplot} where it can be seen that the new inequality is considerably more powerful then the original one. It can for instance witness the postquantumness of distributions that could not be detected before even in the limit of many copies.
\begin{figure} [!t]
\centering
\includegraphics[width=0.45\textwidth]{ICplot}
\caption{A slice of the non-signalling polytope corresponding to the distributions \eqref{sec_poly}. The lower black dashed line is an upper limit on quantum correlations obtained via the criterion in Ref. \cite{Navascues2007} while the upper solid black line bounds the set of non-signalling correlations. The solid red, blue and orange curves correspond, respectively, to the boundaries obtained with the IC inequalities \eqref{ICtighter}, \eqref{IC1} and \eqref{IC2}. Above each of this curves, the corresponding inequalities are violated. See appendix for details of how this curves are computed.
}
\label{fig:ICplot}
\end{figure}
\section{Quantum networks} Quantum networks are ubiquitous in quantum
information. The basic scenario consists of a collection of entangled
states that are distributed among several spatially separated parties
in order to perform some informational task, e.g., entanglement
percolation \cite{Acin2007entanglement}, entanglement swapping
\cite{Zukowski1993} or distributed computing
\cite{van2014quantum,buhrman2003distributed}.
A similar setup
is of relevance in classical causal inference, namely the inference of
latent common ancestors \cite{Steudel2010,Chaves2014b}. As we will
show next, the topology alone of these quantum networks imply
non-trivial constraints on the correlations that can be obtained
between the different parties. We will consider the particular case
where all the parties can be connected by at most bipartite states. We
note, however, that our framework applies as well to the most general
case and results along this line are presented in the appendix.
The problem can be restated as follows. Consider $n$ observable variables that may be assumed to have no direct causal influence on each other (as they are space-like separated). Given some observed correlations between them, the basic question is then: Can the correlations between these $n$ variables be explained by (hidden) common ancestors connecting at most $2$ of them? The simplest of such common ancestors scenarios ($n=3$), the so called triangle scenario \cite{Steudel2010,Branciard2012,Fritz2012}, is illustrated in Fig. \ref{fig:triangle_and_IC}.
In the case where the underlying hidden variables are classical (for example, separable states), the entropic marginal cone associated to this DAG has been completely characterized in Ref. \cite{Chaves2014}. Following the framework delineated before, we can prove that the same cone is obtained if we replace the underlying classical variables by quantum states (see appendix). This implies that \emph{entropically there are no quantum correlations in the triangle scenario}.
The natural question is how to generalize this result to more general common ancestor structures for arbitrary $n$. With this aim, we prove in the appendix that the monogamy relation
\begin{equation}
\label{ineq_mn}
\sum_{\substack{
i=1,\cdots,n \\
i \neq j
}}I(V_{i}:V_{j})\leq H(V_{j}),
\end{equation}
recently derived in \cite{Chaves2014b} is also valid for quantum theory. We also prove in the appendix that this inequality is valid for general non-signalling theories, generalizing the result obtained in \cite{Henson2014} for $n=3$. In addition we exhibit that for any nontrivial common ancestor structure there are entropic corollaries even if we allow for general non-signalling parents.
The inequality \eqref{ineq_mn} can be seen as a kind of monogamy of correlations. Consider for instance the case $n=3$ and label the commons ancestor (any nonsignalling resource) connecting variables $V_i$ and $V_j$ by $\rho_{i,j}$. If the dependency between $V_1$ and $V_2$ is large, that means that $V_1$ has a strong causal dependence on their common mutual ancestor $\rho_{1,2}$. That implies that $V_1$ should depend only mildly on its common ancestor $\rho_{1,3}$ and therefore its correlation with $V_3$ should also be small. The inequality \eqref{ineq_mn} makes this intuition precise.
\section{Discussion}
\label{sec:discussion}
In this work, we have introduced a systematic algorithm for computing information-theoretic constraints arising from quantum causal structures. Moreover, we have demonstrated the versatility of the framework by applying it to a set of diverse examples from quantum foundations, quantum communication, and the analysis of distributed architectures. In particular, our framework readily allows to obtain a much stronger version of information causality.
These examples aside, we believe that the main contribution of this work is to highlight the power of systematically analyzing entropic marginals. A number of future directions for research immediately suggest themselves. In particular, it will likely be fruitful to consider multi-partite versions of information causality or other information theoretical principles and to further look into the operational meaning of entropy inequality violations.
\bigskip
\begin{acknowledgements}
We acknowledge support by the Excellence Initiative of the German
Federal and State Governments (Grant ZUK 43), the Research Innovation
Fund from the University of Freiburg. DG's research is supported by
the US Army Research Office under contracts W911NF-14-1-0098 and
W911NF-14-1-0133 (Quantum Characterization, Verification, and
Validation). CM acknowledges support by the German National Academic
Foundation.
\end{acknowledgements}
\section{Appendix}
\subsection{A linear program framework to entropic inequalities}
Given the inequality description of the entropic cone describing a causal structure, to obtain the description of an associated marginal scenario $\mathcal{M}$ we need to eliminate from the set of inequalities all variables not contained in $\mathcal{M}$. After this elimination procedure, we obtain a new set of linear inequalities, constraints that correspond to facets of a convex cone, more precisely the marginal entropic cone characterizing the compatibility region of a certain causal structure \cite{Chaves2014}. This can be achieved via a Fourier-Motzkin (FM) elimination, a standard linear programming algorithm for eliminating variables from systems of inequalities \cite{Williams1986}. The problem with the FM elimination is that it is a double exponential algorithm in the number of variables to be eliminated. As the number of variables in the causal structure of interest increases, typically this elimination becomes computationally intractable.
While it can be computationally very demanding to obtain the full description of a marginal cone, to check if a given candidate inequality is respected by a causal structure is relatively easy. Consider that a given causal structure leads to a number $N$ of possible entropies. These are organized in a $n$-dimensional vector $\boldsymbol{h}$. In the purely classical case, the graph consisting of $n$ nodes ($X_1,\dots, X_n$) will lead to a $N=2^n$ dimensional entropy vector that can be organized as $\boldsymbol{h}=(H(\emptyset), H(X_n), H(X_{n-1}),H(X_{n-1}X_n),\dots,H(X_1,\dots,X_{n}))$. In the quantum case, since not all subsets of variables may jointly coexist we will have typically that $N$ is strictly smaller than $2^n$.
As explained in details in the main text, for this entropy vector to be compatible with a given causal structure, a set of linear constraints must be fulfilled. These linear constraints can be casted as a system of inequalities of the form $M\boldsymbol{h} \geq \boldsymbol{0}$, where $M$ is a $m \times N$ matrix with $m$ being the number of inequalities characterizing the causal structure.
Given the entropy vector $\boldsymbol{h}$, any entropic linear inequality can be written simply as the inner product $\langle \boldsymbol{\mathcal{I}}, \boldsymbol{h} \rangle \geq 0$, where $\boldsymbol{\mathcal{I}}$ is the associated vector to the inequality. A sufficient condition for a given inequality to be valid for a given causal structure is that the associated set of inequalities $M\boldsymbol{h} \geq \boldsymbol{0}$ to be true for any entropy vector $\boldsymbol{h}$. That is, to check the validity of a test inequality, one simply needs to solve the following linear program:
\begin{eqnarray}
\underset{\boldsymbol{h} \in \mathbbm{R}^N}{\textrm{minimize}} & & \langle \boldsymbol{\mathcal{I}}, \boldsymbol{h} \rangle \label{LP} \\
\textrm{subject to } & & M\boldsymbol{h} \geq \boldsymbol{0} \nonumber
\end{eqnarray}
In general, this linear program only provides a sufficient but not necessary condition for the validity of a inequality. The reason for that is the existence of non-Shannon type inequalities, that are briefly discussed below.
\subsection{Details about the new IC inequality}
In the following we will discuss how to characterize the most general marginal scenario in the information causality scenario. We will start discussing the purely classical case (i.e Alice and Bob share classical correlations) and afterwards apply the linear program framework to prove that all inequalities characterizing the classical Shannon cone are also valid for quantum mechanical correlations.
The classical causal structure associated with information causality contains six classical variables $S=\left\{X_1,X_2,Y_1,Y_2,M,\lambda \right\}$. The variable $\lambda$ stands here for the classical analog of the quantum state $\rho_{AB}$. The most general marginal scenario that is compatible with the information causality game and thus with protocols using more general resources such as nonlocal boxes is given by $\mathcal{M}=\left\{ X_1,X_2,Y_i,M \right\}$ (with $i=1,2$). The relevant conditional independencies implied by the graph are given by $I(X_1,X_2:\lambda)=0$ and $I(X_1,X_2:Y_1,Y_2 \vert M, \lambda)=0$. CIs like $I(X_1:Y_1 \vert M, \lambda)=0$ are implied by the relevant ones together with the polymatroidal axioms for the set $S$ of variables, and in this sense are thus redundant. Given this inequality description (basic inequalities plus CIs) we need to eliminate from our description, via a FM elimination, all the variable not contained in $\mathcal{M}$.
Our first step was to eliminate from the system of inequalities the variable $\lambda$. Doing that one obtains a new set of inequalities for the five variables $S=\left\{X_1,X_2,Y_1,Y_2,M \right\}$. These set of inequalities is simply given by the basic inequalities plus one single non-trivial inequality, implied by the CIs:
\begin{equation}
H(Y_1,Y_2,M)+H(X_1,X_2) \leq H(M) +H(X_1,X_2,Y_1,Y_2,M)
\end{equation}
We then proceed eliminating all variables not contained in $\mathcal{M}$. The final inequality description of the marginal cone of $\mathcal{M}$ can be organized in two groups. The first group contains all inequalities that are valid for the collection of variables in $\mathcal{M}$ independently of the underlying causal relationships between them, that is, they follow from the basic inequalities alone. The second group contains the inequalities that follow from the basic inequalities plus the conditional independencies implied by the causal structure. These are the inequalities capturing the causal relations implied by information causality and there are $54$ of them. Among these $54$ inequalities, one of particular relevance is the tighter IC inequality \eqref{ICtighter} given in the main text.
One can prove, using the linear program framework delineated before, that this inequality is also valid for the corresponding quantum causal structure shown in Fig. \ref{fig:triangle_and_IC} b). Following the discussion in the main text, the sets of jointly existing variables in the quantum case are given by $S_0=\left\{X_1,X_2,A,B \right\}$, $S_1=\left\{X_1,X_2,M,B \right\}$ and $S_2=\left\{X_1,X_2,M,Y_i \right\}$ (with $i=1,2$). One can think about these sets of variables in a time ordered manner. At time $t=0$ the joint existing variables are the inputs $X_1$ and $X_2$ of Alice, together with the shared quantum state $\rho_{A,B}$. At time $t=1$ Alice encodes the input bits into the message $M$ also using her correlations with Bob obtained through the shared quantum state. Doing that, Alice disturbs her part $A$ of the quantum system that therefore does not coexist anymore with the variables defined in $S_1$. In the final step of the protocol at time $t=2$, Bob uses the received message $M$ and its part $B$ of the quantum state in order to make a guess $Y_1$ or $Y_2$ about Alice's inputs. Once more, by doing that $B$ ceases to coexist with the variables contained in $S_2$.
Following the general idea, we write down all the basic inequalities for the sets $S_0$ and $S_1$ and $S_2$, together with the conditional independencies and the data processing inequalities. As discussed before, because the quantum analogous of $I(X_1,X_2:Y_1,Y_2 \vert M, \lambda)=0$ has no description in the quantum case, the only CI implied here will be $I(X_1,X_2 : A,B)=0$. The causal relations encoded in the other CIs are taken care by the data processing inequalities. Below we list all used data processing inequalities:
\begin{eqnarray*}
I(X_1,X_2:Y_i) & & \leq I(X_1,X_2:M,B) \\ \nonumber
I(X_i:Y_j) & & \leq I(X_i:M,B) \\ \nonumber
I(X_i: X_{i \oplus 1}, Y_j) & & \leq I(X_i: X_{i \oplus 1}, B) \\ \nonumber
I(X_1,X_2:Y_i,M) & & \leq I(X_1,X_2:B,M) \\ \nonumber
I(X_i:Y_j,M) & & \leq I(X_i:B,M) \\ \nonumber
I(X_i:X_{i \oplus 1}, Y_j, M) \leq & & I(X_i:X_{i \oplus 1}, B, M)
\end{eqnarray*}
Note that some of these DP inequalities may be redundant, that is, they may be implied by other DP inequalities together with the basic inequalities.
We organize all the above constraints into a matrix $M$ and given a certain candidate inequality $\mathcal{I}$ we run the linear program discussed before. Doing that one can easily prove that inequality \eqref{ICtighter} is also valid in the quantum case.
Note that this computational analysis will in general be restricted by the number of variables involved in the causal structure. To circumvent that we provide in the following an analytical proof of the validity of the generalized IC inequality \eqref{ICtighter2} for the quantum causal structure in Fig. \ref{fig:triangle_and_IC} b).
\begin{proof}
First rewrite the following conditional mutual information as
\begin{equation}
I(X_1:X_i \vert Y_i,M) = I(X_1:X_i, Y_i,M)-I(X_i: Y_i,M).
\end{equation}
The LHS of the inequality \eqref{ICtighter2} can then be rewritten as
\begin{equation}
I(X_1:Y_1,M)+\sum^{n}_{i=2}I(X_i:X_1,Y_i,M).
\end{equation}
This quantity can be upper bounded as
\begin{widetext}
\begin{eqnarray}
& \leq I(X_1:B,M) +\sum^{n}_{i=2}I(X_i:X_1,B,M) \\
& = \sum^{n}_{i=1} H(X_i)+H(B,M)+(n-2)H(X_1,B,M) - \sum^{n}_{i=2} H(X_1,X_i,B,M) \\
& \leq \sum^{n}_{i=1} H(X_i)+H(B,M) - H(X_1,\dots,X_{n},B,M) \\
& \leq \sum^{n}_{i=1} H(X_i)+H(B,M) - H(X_1,\dots,X_{n},B) \\
& = \sum^{n}_{i=1} H(X_i)+H(B,M) - H(X_1,\dots,X_{n})-H(B) \\
& \leq \sum^{n}_{i=1} H(X_i)+H(M) - H(X_1,\dots,X_{n})
\end{eqnarray}
\end{widetext}
leading exactly to the inequality \eqref{ICtighter2}. In the proof above we have used consecutively i) the data processing inequalities $I(X_1:Y_1,M) \leq I(X_1:B,M) $ and $I(X_i:X_1,Y_i,M) \leq I(X_i:X_1,B,M)$, ii) the fact that $-\sum^{N-1}_{i=1} H(X_i,X_1,B,M) \leq - H(X_1,\dots,X_{n},B,M)-(n-2)H(X_1,B,M) $ (as can be easily be proved inductively using the strong subadditivity property of entropies), iii) the monotonicity $H(M\vert X_1,\dots,X_{n},B) \geq 0$, iv) the independence relation $I(X_1,\dots, X_{n}:B)=0$ and v) the positivity of the mutual information $I(B:M) \geq 0$
\end{proof}
Note that this proof can be easily adapted to the case where the message $M$ sent from Alice to Bob is a quantum state. In this case there two differences. First, because the message is disturbed in order to create the guess $Y_i$, we cannot assign a entropy to $M$ and $Y_i$ simultaneously. That is, in the LHS side of the inequality \eqref{ICtighter2} we replace $I(X_i:Y_i,M) \rightarrow I(X_i:Y_i) $ and $I(X_1:X_i \vert Y_i,M) \rightarrow I(X_1:X_i \vert Y_i) $. The second difference is in step iii), because we have used the monotonicity $H(M\vert X_1,\dots,X_{n},B) \geq 0$ that is not valid for a quantum message. Instead of that, we can use a weak monotonicity inequality, namely $H(M\vert X_1,\dots,X_{n},B) + H(M) \geq 0$. Therefore, in the final inequality \eqref{ICtighter2}, $I(X_i:Y_i,M) \rightarrow I(X_i:Y_i) $ and $I(X_1:X_i \vert Y_i,M) \rightarrow I(X_1:X_i \vert Y_i)$ and $H(M)$ is replaced by $2H(M)$ (here, $H$ standing for the von Neumann entropy), leading exactly to what should be expected of a super dense coding \cite{Bennett1992}.
\subsection{Proving that in the triangle scenario the classical and quantum marginal cones coincide}
Since 1998 it is known that, for a number of variables $n \geq 4$, there are inequalities valid for Shannon entropies that cannot be derived from the elemental set of polymatroidal axioms (submodularity and monotonicity) \cite{Zhang1998}. These are the so called non-Shannon type inequalities \cite{Yeung2008}. More precisely, the existence of these inequalities imply that the true entropic cone (denoted by $\overline{\Gamma}^{*}_n$) is a strict subset of the Shannon cone, that is, the inclusion $ \overline{\Gamma}^{*}_n \subseteq \Gamma_n$ is strict for $n \geq 4$.
Remember that a convex cone has a dual description, either in terms of its facets or its extremal rays. In terms of its half-space description the strict inclusion $ \overline{\Gamma}^{*}_n \subset \Gamma_n$ implies that while all Shannon type inequalities are valid for any true entropy vector, they may fail to be tight. In terms of the extremal rays, this implies that some of the extremal rays of the Shannon cone are not populated, that is, there is no well defined probability distribution with an entropy vector corresponding to it.
Sometimes, the projection of the outer approximation $\Gamma_n$ onto a subspace, described by the marginal cone $\Gamma_{\mathcal{M}}$, may lead to the true cone in the marginal space, that is $\Gamma_{\mathcal{M}}=\overline{\Gamma}^{*}_{\mathcal{M}}$ \cite{FritzChaves2013}. A sufficient condition for that to happen is that all the extremal rays of $\Gamma_{\mathcal{M}}$ are populated. Using this idea, in the following we will prove that all the extremal rays describing the Shannon marginal cone of classical triangle scenario are populated, proving that in this case the Shannon and true marginal cones coincide. We will then use the linear program framework delineated previously in order to prove that all the corresponding inequalities are also valid for underlying quantum states, therefore proving that entropically the set of classical and quantum correlations coincide.
Proceeding with the three steps program delineated in the main text, one can see that the marginal scenario $\left\{ A, B, C \right\}$ of the triangle scenario is completely characterized by the following non-trivial Shannon type inequalities (and permutations thereof) \cite{Chaves2014}
\begin{widetext}
\begin{eqnarray}
\label{triangle_nontrivial_1}
&& I(A:B)+I(A:C) -H(A) \leq 0, \\
\label{triangle_nontrivial_2}
&& I(A:B:C)+I(A:B)+I(A:C)+I(B:C) -H(A,B) \leq 0, \\
\label{triangle_nontrivial_3}
&& I(A:B:C)+I(A:B)+I(A:C)+I(B:C)- \frac{1}{2}( H(A)+H(B)+H(C) ) \leq 0,
\end{eqnarray}
\end{widetext}
plus the polymatroidal axioms for the three variables $A,B,C$. Given the inequality description of the marginal cone we have used the software PORTA \cite{porta} in order to recover the extremal rays of it. There are only $10$ extremal rays, that can be organized in the $4$ different types listed in Table \ref{extremal_rays} .
\begin{table*}
\begin{tabular}{|c| c| c c c c c c c|} \hline
\multicolumn{9}{|c|}{Extremal rays of the marginal triangle scenario}\\
\hline
type
&\textbf{\#} of permutations
&$H_{C}$&$H_{B}$&$H_{BC}$&$H_{A}$&$H_{AC}$&$H_{AB}$&$H_{ABC}$
\\
\hline
1
&3
&0&0&0&1&1&1&1
\\
\hline
2
&3
&0&1&1&1&1&1&1
\\
\hline
3
&1
&1&1&2&1&2&2&2
\\
\hline
4
&3
&3&3&5&2&4&4&6
\\
\hline
\end{tabular}
\caption{The four kinds of extremal rays defining the marginal entropic cone of the triangle scenario.} \label{extremal_rays}
\end{table*}
Below we list the probability distributions reproducing the $4$ different types of entropy vectors:
\begin{widetext}
\begin{equation}
\label{type1}
p_1\left( a,b,c \right) =\left\{
\begin{array}{ll}
1/2 & \text{if } a=\left\{1,2\right\} \text{ and } b=c=\left\{1\right\}\\
0 & \text{, otherwise}%
\end{array}
\right. ,
\end{equation}
\begin{equation}
\label{type2}
p_2\left( a,b,c \right) =\left\{
\begin{array}{ll}
1/2 & \text{if } a=b=\left\{1,2\right\} \text{ and } c=\left\{1\right\}\\
0 & \text{, otherwise}%
\end{array}
\right. ,
\end{equation}
\begin{equation}
\label{type3}
p_3\left( a,b,c \right) =\left\{
\begin{array}{ll}
1/4 & \text{if } a\oplus b \oplus c=0 \text{ with } a,b,c= \left\{ 1,2 \right\}\\
0 & \text{, otherwise}%
\end{array}
\right. ,
\end{equation}
and
\begin{equation}
\label{type4}
p_4\left( a,b,c \right) =\left\{
\begin{array}{ll}
1/64 & \text{if } a\oplus b + a\oplus c + b\oplus c=0 \text{ with } a=\left\{1,\dots,4\right\} \text{ and } b,c=\left\{1,\dots,8\right\}\\
0 & \text{, otherwise}%
\end{array}
\right. ,
\end{equation}
\end{widetext}
Since all the extremal rays are populated, this proves that $\Gamma_{\mathcal{M}}=\overline{\Gamma}^{*}_{\mathcal{M}}$ for the marginal scenario $\mathcal{M}= \left\{A,B,C \right\}$ of the triangle scenario.
To prove that the same entropic cone holds for the associated quantum causal structure (Fig. \ref{fig:triangle_and_IC} a) we just need to prove that all the inequalities defining $\Gamma_{\mathcal{M}}$ hold true in the quantum case. Clearly, the polymatroidal axioms for $\left\{A,B,C \right\}$ also hold true in the quantum case, since these variables are classical. Using the linear programming framework detailed above, one can also prove that the inequalities \eqref{triangle_nontrivial_1}, \eqref{triangle_nontrivial_2} and \eqref{triangle_nontrivial_3} hold if the underlying hidden variables stand for quantum states.
The sets of jointly existing variables in the quantum case are given by $S_0=\left\{A_1,A_2, B_1,B_2 , C_1,C_2 \right\}$, $S_1=\left\{A, B_1,B_2 , C_1,C_2 \right\}$, $S_2=\left\{A_1,A_2, B, C_1,C_2 \right\}$, $S_3=\left\{A_1,A_2, B_1,B_2 , C \right\}$, $S_4=\left\{A, B , C_1,C_2 \right\}$, $S_5=\left\{A, B_1,B_2 , C \right\}$, $S_6=\left\{A_1,A_2, B, C \right\}$ and $S_7=\left\{A, B, C \right\}$. The fact that the quantum states are assumed to be independent is translated in the CI $H(A_1,A_2, B_1,B_2 , C_1,C_2 )= H(\rho_{A_1,B_1})+ H(\rho_{A_2,C_1})+ H(\rho_{C_2,C_2})$. The causal constraint that the observable variables have no direct influence on each other (all the correlation are mediated by the underlying quantum states) is encoded in the CI given by $I(A:B \vert A_1)=I(A:B \vert B_1)=0$ (similarly for permutation of the variables). Below we also list all used data processing inequalities:
\begin{eqnarray*}
I(A:B) & & \leq I(A:B_1 ,B_2) \\ \nonumber
I(A:B) & & \leq I(A_1 ,A_2:B) \\ \nonumber
I(A:B) & & \leq I(A_1 ,A_2:B_1 ,B_2) \\ \nonumber
I(A:B,C) & & \leq I(A:B, C_1 ,C_2) \\ \nonumber
I(A:B,C) & & \leq I(A:B_1 ,B_2,C) \\ \nonumber
I(A:B,C) & & \leq I(A:B_1 ,B_2, C_1 ,C_2) \\ \nonumber
I(A:B,C) & & \leq I(A_1 ,A_2:B, C_1 ,C_2) \\ \nonumber
I(A:B,C) & & \leq I(A_1 ,A_2:B_1 ,B_2,C) \\ \nonumber
I(A:B,C) & & \leq I(A_1 ,A_2:B_1 ,B_2, C_1 ,C_2) \\ \nonumber
I(A,B_1:B_2) & & \leq I(A_1 ,A_2,B_1:B_2) \\ \nonumber
I(A,C_1:C_2) & & \leq I(A_1 ,A_2,C_1:C_2)
\end{eqnarray*}
and similarly for permutations of all variables. Again, note that some of these DP inequalities may be redundant, that is, they may be implied by other DP inequalities together with the basic inequalities.
\subsection{Proving the monogamy relations of quantum networks}
In the following we provide an analytical proof of the monogamy inequality \eqref{ineq_mn} in the main text.
We start with the case $n=3$. For a Hilbert space $\mathcal{H}$ we denote the set of quantum states, i.e. the set of positive semidefinite operators with trace one, on it by $\mathbb{S}(\mathcal{H})$.
\begin{thm}\label{qtri}
Let $\rho_{A_1A_2B_1B_2C_1C_2}=\rho_{A_1 B_2}\otimes\rho_{B_1 C_2}\otimes\rho_{C_1 A_2}$ be a sixpartite quantum state on $\mathcal{H}=\mathcal{H}_{A_1}\otimes\mathcal{H}_{A_2}\otimes\mathcal{H}_{B_1}\otimes\mathcal{H}_{B_2}\otimes\mathcal{H}_{C_1}\otimes\mathcal{H}_{C_2}$. Let further $\Phi_N: \mathbb{S}\left(\mathcal{H}_{N_1}\otimes\mathcal{H}_{N_2}\right)\to \mathbb{S}\left(\mathcal{H}_N\right)$ be an arbitrary measurement for $N=A,B,C$. Then
\begin{equation}
I(A:B)+I(A:C)\leq H(A).
\end{equation}
\end{thm}
\begin{proof}
Data processing yields
\begin{eqnarray}
I(A:B)+I(A:C)&\leq& I(A:B_1B_2)+I(A:C_1C_2).
\end{eqnarray}
Then we exploit the chain rule twice and afterwards data processing again,
\begin{eqnarray}
I(A:B_1B_2)&=&I(A:B_2)+I(A:B_1|B_2)\nonumber\\
&=&I(A:B_2)+I(AB_2:B_1)-I(B_2:B_1)\nonumber\\
&\leq &I(A:B_2)+I(A_1A_2B_2:B_1)\nonumber\\
&=&I(A:B_2).
\end{eqnarray}
We have therefore
\begin{equation}
I(A:B)+I(A:C)\leq I(A:B_2)+I(A:C_1),
\end{equation}
from which it follows that
\begin{eqnarray}\label{eq:proof-qtri}
&I(A:B_2)+I(A:C_1)\nonumber\\
& = 2H(A)+H(B_2)+H(C_1)-H(AB_2)-H(AC_1)\nonumber\\
& \leq H(A)+H(B_2)+H(C_1)-H(AB_2C_1)\nonumber\\
& = H(A)-H(A|B_2C_1)\nonumber\\
&\leq H(A),
\end{eqnarray}
where in the second line we used strong subadditivity and in the last line we used that the entropy of a classical state conditioned on a quantum state is positive.
\end{proof}
This proof can easily be generalized to the case of an arbitrary number of random variables resulting from a classical-quantum Bayesian network in with each parent connects has at most two children.
\begin{cor}
Let
\begin{equation*}
\rho=\bigotimes_{\substack{i,j=1\\ i<j}}^n\rho_{(ij),(ji)}
\end{equation*}
be an $n(n-1)$-partite quantum state on
\begin{equation*}
\mathcal{H}=\bigotimes_{\substack{i,j=1\\i\neq j}}^n \mathcal{H}_{(ij)},
\end{equation*}
and let
\begin{equation*}
\Phi_{i}: \mathbb{S}\left(\bigotimes_{j\neq i} \mathcal{H}_{(ij)}\right)\to \mathbb{S}\left(\mathcal{H}_i\right)
\end{equation*}
be an arbitrary measurement for $i=1,...,n$. Then
\begin{equation}
\sum_{i=2}^n I(1:i)\le H(1)
\end{equation}
\end{cor}
\begin{proof}
First, utilize the independences in the same way as in the proof of Theorem \ref{qtri} to conclude
\begin{equation}
\sum_{i=2}^n I(1:i)\le\sum_{i=2}^n I(1:(i1)).
\end{equation}
Now continue by induction. For $n=3$ we have, according to the proof of Theorem \ref{qtri},
\begin{equation}
I(1:(21))+I(1:(31))\le H(1).
\end{equation}
Now assume
\begin{equation}
\sum_{i=2}^{n-1} I(1:(i1))\le H(1).
\end{equation}
Using the proof of Theorem \ref{qtri} again and stopping before the last inequality in \eqref{eq:proof-qtri} we get
\begin{equation}
I(1:(n-1 1))+I(1:(n1))\le I(1:(n-1 1)(n1)),
\end{equation}
i.e. we get
\begin{eqnarray}
\sum_{i=2}^{n} I(1:(i1))\le \sum_{i=2}^{n-1}I(1:(i1)')\le H(1),
\end{eqnarray}
where we defined the primed systems by $\mathcal{H}_{(n-11)'}=\mathcal{H}_{(n-11)}\otimes \mathcal{H}_{(n1)}$, observing that this yields a classical-quantum bayesian network with $n-1$ nodes and and connectivity two and used the induction hypothesis.
\end{proof}
\subsection{Proving the monogamy relation for GPTs}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\prep{-3}{0}{$\sigma_{AB}$}
\prep{0}{0}{$\sigma_{AC}$}
\prep{3}{0}{$\sigma_{BC}$}
\meas{-3}{2}{$X$}
\meas{0}{2}{$Y$}
\meas{3}{2}{$Z$}
\draw (-3.33,0) -- (-3.33,2);
\draw (-2.67,0) -- (-0.33,2);
\draw (-0.33,0) -- (-2.67,2);
\draw (3.33,0) -- (3.33,2);
\draw (2.67,0) -- (0.33,2);
\draw (0.33,0) -- (2.67,2);
\end{tikzpicture}
\end{center}
\caption{The GBN for the triangle}\label{gtri}
\end{figure}
We want to prove the inequality
\begin{equation}\label{theineq}
\sum_{\substack{j\in\{1,...,n\}\\ j\neq i}} I(V_i:V_j)\le H(V_i)
\end{equation}
For random variables that constitute a \emph{generalized Bayesian network} \cite{Henson2014} with respect to a DAG where each parent correlates at most two of them, i.e. the random variables are results of measurements on a set of arbitrary non-signalling resources shared between two parties. The case of three random variables has been proven in \cite{Henson2014}, the purpose of this appendix is to prove the generalization to an arbitrary number of random variables. Also we want to proof that for any fixed connectivity number for the parent nodes there are entropic corollaries. To this end we have to introduce a framework to handle generalized probabilistic theories that are non-signaling and have a property called local discriminability that was developed in \cite{Chiribella2010}.
An \emph{operational probabilistic theory} has two basic notions, \emph{systems} and \emph{tests}. Tests are the objects that represent any physical operation that is performed, e.g. the preparation of a state, or a measurement. A test has input and output systems and can have a classical random variable as measurement outcome as well. An outcome together with the corresponding output system state is called \emph{event}. The components are graphically represented by a directed acyclic graph (DAG) where the nodes represent tests, and the edges represent systems. We use the convention that the diagram is read from bottom to top, i.e. a tests input systems are represented by edges coming from below and its output systems are edges emerging from the top of the node:
\begin{center}
\begin{tikzpicture}
\draw [black] (-.6,2.6) rectangle (.6,1.4);
\draw (0,3.6) -- (0,2.6);
\draw (0,1.4) -- (0,0.4);
\node at (0,2) {$X$};
\node at (.2,.9) {$A$};
\node at (.2,3.1) {$B$};
\end{tikzpicture}
\end{center}
If a system has trivial input or trivial output we omit the edge and represent the node by a half moon shape. Tests with trivial input are called \emph{preparations}, tests with trivial output are called \emph{measurements}.
\begin{center}
\begin{tikzpicture}
\draw [thick, black] (-2.5,1.2) arc [radius=1, start angle=180, end angle= 360];
\draw [thick] (-2.5,1.2) -- (-.5,1.2);
\draw [thick] (2.5,1.2) -- (.5,1.2);
\draw [thick] (.5,1.2) arc [radius=1, start angle=180, end angle=0];
\draw (1.5,0) -- (1.5,1.2);
\draw (-1.5, 1.2) -- (-1.5,2.4);
\node at (-1.5,.8) {$X$};
\node at (1.5,1.6) {$Y$};
\node at (1.7,.5) {$B$};
\node at (-1.3,1.9) {$A$};
\end{tikzpicture}
\end{center}
If we do not need to talk about the systems we omit the labels of the edges, and preparation tests are given greek letter labels, as they have, without loss of generality, only a single event.
\begin{center}
\begin{tikzpicture}
\draw [thick, black] (-2.5,1.2) arc [radius=1, start angle=180, end angle= 360];
\draw [thick] (-2.5,1.2) -- (-.5,1.2);
\draw [thick] (2.5,1.2) -- (.5,1.2);
\draw [thick] (.5,1.2) arc [radius=1, start angle=180, end angle=0];
\draw (1.5,0) -- (1.5,1.2);
\draw (-1.5, 1.2) -- (-1.5,2.4);
\node at (-1.5,.8) {$\sigma$};
\node at (1.5,1.6) {$Y$};
\end{tikzpicture}
\end{center}
Finally we assume, just as it is done in \cite{Henson2014}, that there exists a unique way of discarding a system, which we denote by
\begin{center}
\begin{tikzpicture}
\earth{0}{1.2}
\draw (0,0) -- (0,1.2);
\node at (1,0) {.};
\end{tikzpicture}
\end{center}
We call this the \emph{discarding test}, and it is shown in \cite{Chiribella2010} that its existence and uniqueness is equivalent to the non-signaling condition.
These elements can now be connected by using the output system of one test as input system for another. An arrangment of tests is called a \emph{generalized Bayesian network} (GBN). We also say that the arrangement forms a GBN \emph{with respect to} a DAG $\mathcal{G}$, or that a GBN has \emph{shape} $\mathcal{G}$, if the tests are arranged according to it, analogous to the classical case introduced in the main text.
The main ingredient for the proof of \eqref{theineq} for $n=3$ in \cite{Henson2014} is the following
\begin{lem}[\cite{Henson2014}, Thm. 23.]\label{cutfree32}
For any probability distribution $p(x,y,z)$ of random variables that are the classical output of a GBN with respect to the DAG in Figure \ref{gtri} there is a probability distribution $p'$ such that
\begin{eqnarray}
p'(x,z)&=&p'(x)p'(z)\\
p'(x,y)&=&p(x,y)\\
p'(y,z)&=&p(y,z)
\end{eqnarray}
\end{lem}
For our purposes we need a generalization of this result. The GBN for the scenario of any parent connecting at most $m$ children can be described as follows. The $n$ random variables $V_1,...,V_n$ arise from $n$ measurement tests. For any $m$ of these measurement tests there is a preparation test whose output systems are input for exactly these measurements. We denote the preparation test corresponding to a subset $I\subset \{1,...,n\},\ |I|=m$ by $\sigma_{I}$. In total there are therefore $n \choose m$ preparations. This GBN can be found in Figure \ref{gnm}, and we denote the corresponding DAG by $\mathcal{G}_{n,m}$.
\begin{widetext}
\begin{center}
\begin{figure}[!h]
\begin{tikzpicture}
\prep{-5}{0}{\tiny$\sigma_{12...m}$}
\prep{-3}{0}{\tiny$\sigma_{1...m-1m+1}$}
\node at (1,-.3) {...};
\prep{5}{0}{\tiny$\sigma_{n-m...n}$}
\meas{-5}{2}{$V_1$}
\meas{-3}{2}{$V_2$}
\node at (-1,2.3) {...};
\meas{1}{2}{$V_m$}
\node at (3,2.3) {...};
\meas{5}{2}{$V_n$}
\draw (-5.33,0) -- (-5.33,2);
\draw (-5,0) -- (-3.33,2);
\node at (-4.2,.4) {...};
\draw (-4.67,0) -- (0.67,2);
\draw (-3.33,0) -- (-4.67,2);
\draw (-3.0,0) -- (-2.67,2);
\node at (-2.5,.3) {...};
\node at (4.6,.3) {...};
\draw (5.33,0) -- (5.33,2);
\end{tikzpicture}
\caption{A generalized Bayesian network for $\mathcal{G}_{n,m}$}\label{gnm}
\end{figure}
\end{center}
\end{widetext}
\begin{figure}
\begin{center}
\pgfmathsetmacro{\t}{1.}
\pgfmathsetmacro{\op{S}}{2.5}
\begin{tikzpicture}
\prep{\t *-3}{0}{\small$\sigma_{12}$}
\prep{\t *-1}{0}{\small$\sigma_{13}$}
\prep{\t *1}{0}{\small$\sigma_{13}'$}
\prep{\t *3}{0}{\small$\sigma'_{23}$}
\meas{\t *-2}{\op{S}}{$V_2$}
\meas{\t*0}{\op{S}}{$V_3$}
\meas{\t *2}{\op{S}}{$V_4$}
\draw (\t *-3.3,0) -- (\t *-2.3,\op{S});
\draw (\t *-2.7,0) -- (\t *-0.3,\op{S});
\draw (\t *-1.3,0) -- (\t *-1.7,\op{S});
\draw (\t *-0.7,0) -- (\t *-0.6,.3*\op{S});
\draw (\t *0.7,0) -- (\t *0.6,.3*\op{S});
\draw (\t *1.3,0) -- (\t *1.7,\op{S});
\draw (\t *2.7,0) -- (\t *.3,\op{S});
\draw (\t *3.3,0) -- (\t *2.3,\op{S});
\earth{\t *-.6}{.3*\op{S}}
\earth{\t *.6}{.3*\op{S}}
\end{tikzpicture}
\end{center}
\caption{An example of the modified GBN from Lemma \ref{cutfree} for $n=3$ and $i=2$}\label{g34}
\end{figure}
\begin{lem}\label{cutfree}
For any probability distribution arising from a GBN of shape $\mathcal{G}_{n,m}$ and any index $i\in \{1,...,n\}$ there is a probability distribution $p'$ such that any bivariate marginal involving $V_i$ is equal to the corresponding marginal of $p$ and $p'(v_1,...v_{i-1},v_{i+1},...,v_n)$ is compatible with $\mathcal{G}_{n-1,m-1}$.
\end{lem}
\begin{proof}
Analoguous to the proof of Lemma \ref{cutfree32} in \cite{Henson2014} we define a new GBN from the old one as follows:
\begin{itemize}
\item Any preparation test $\sigma_I$ with $i\in I$ is left as it is.
\item Any preparation test $\sigma_I$ with $i\notin I$ is copied. In one copy the first outgoing edge is discarded and in the second copy all edges except the first are discarded.
\end{itemize}
The modified GBN is depicted in Figure \ref{g34} for $n=3$, $m=2$ and $i=2$.
It can be seen in an analogous way as in the proof of Theorem 23 in \cite{Henson2014} that using the probability distribution that arises from this GBN as $p'$ has the desired properties.
\end{proof}
We are now ready to prove the inequality \eqref{theineq}.
\begin{thm}
Let $V_1,...,V_n$ be random variables defined by a GBN of shape $\mathcal{G}_{n,2}$. Then for any $i\in\{1,...,n\}$
\begin{equation}
\sum_{\substack{j\in\{1,...,n\}\\ j\neq i}} I(V_i:V_j)\le H(V_i)
\end{equation}
\end{thm}
\begin{proof}
Without loss of generality we take $i=1$.
We proceed by induction over $n$. For $n=2$ the inequality is trivially true. Assume now that the statement is true for $n-1$. We construct the probability distribution $p'$ according to Lemma \ref{cutfree} and observe that $p'(v_2,...,v_n)$ arises from $\mathcal{G}_{1,n-1}$, i.e. it is a product distribution. Denote the modified random variables by $V_1',...,V_n'$ and calculate
\begin{widetext}
\begin{eqnarray}
\sum_{j=2}^n I(V_1:V_j)&=& \sum_{j=2}^n I(V'_1:V'_j)\nonumber\\
&=&I(V'_2:V'_3)-I(V'_2:V'_3|V'_1)+I(V'_1:V'_2V'_3)+\sum_{j=4}^n I(V'_1:V'_j)\nonumber\\
&\le&I(V'_1:V'_2V'_3)+\sum_{j=4}^n I(V'_1:V'_j),
\end{eqnarray}
\end{widetext}
where the inequality follows from the independence of $V_2'$ and $V_3'$ and srong subadditivity.
Now observe that with $X=(V'_2,V'_3)$ the distribution of $V'_1, X, V'_4,...,V'_n$ is compatible with $\mathcal{G}_{n-1,2}$ and therefore we have, using the induction hypothesis,
\begin{equation}
I(V'_1:V'_2V'_3)+\sum_{j=4}^n I(V'_1:V'_j)\le H(V_1').
\end{equation}
But $p'(v_1)=p(v_1)$ and therefore \eqref{theineq} is proven.
\end{proof}
For general $m$ the situation is somewhat less simple, but for the special case of $n=m+1$ we can still prove a nontrivial inequality.
\begin{thm}
Let $V_1,...,V_{m+1}$ be random variables corresponding to a GBN of shape $\mathcal{G}_{m+1,m}$. Then
\begin{equation}\label{eq:nequalsmp1}
\sum_{k=2}^{m+1}I(V_1:V_k)\le \sum_{k=0}^{m-3}\frac{(m-k-1)(m-k-1)!}{(m-1)!}H(V_{k+1}),
\end{equation}
and there is a set of random variables $X_1,...,X_{m+1}$ incompatible with $\mathcal{G}_{m+1,m}$ that violates this inequality.
\end{thm}
Note that this inequality is, in particular, also true for quantum-classical bayesian networks and, to our knowledge, provides the only known entropic corollaries in this case, too.
\begin{proof}(by induction)
For $m=1$ the statement is trivially true, as then the two random variables are independent and therefore $I(V_1:V_2)=0\le 0$. Assume now the inequality was proven for $m-1$. Construct random variables $V_1',...,V_{m+1}'$ according to Lemma \ref{cutfree}. Then calculate
\begin{widetext}
\begin{eqnarray}
\sum_{k=2}^{m+1}I(V_1:V_k)&=&\sum_{k=2}^{m+1}I(V'_1:V'_k)\nonumber\\
&=&\frac{1}{m-1}\sum_{k=3}^{m+1}\left[I(V'_2:V'_k)-I(V'_2:V'_k|V'_1)+I(V'_1:V'_2V'_k)+(m-2)I(V'_1:V'_k)\right]\nonumber\\
&\le&\frac{1}{m-1}\sum_{k=3}^{m+1}\left[I(V'_2:V'_k)+I(V'_1:V'_2V'_k)+(m-2)I(V'_1:V'_k)\right]\nonumber\\
&\le&(m-1)H(V'_1)+\frac{1}{m-1}\sum_{k=0}^{m-4}\frac{(m-k-2)(m-k-2)!}{(m-2)!}H(V_{k+2})\nonumber\\
&=&(m-1)H(V'_1)+\sum_{k=1}^{m-3}\frac{(m-k-1)(m-k-1)!}{(m-1)!}H(V_{k+1})\nonumber\\
&=&\sum_{k=0}^{m-3}\frac{(m-k-1)(m-k-1)!}{(m-1)!}H(V_{k+1}),
\end{eqnarray}
\end{widetext}
where the first inequality follows from strong subadditivity and the second inequality follows from the induction hypothesis and the trivial bound $I(X:Y)\le H(X)$. This completes the proof of the first assertion, i.e. that the inequality is fulfilled by random variables from a GBN of shape $\mathcal{G}_{m+1,m}$. To see that more general random variables violate this inequality, let $X_i=X, i=1,...,m+1$, where $X$ is an unbiased coin. In other words, the $X_i$ are maximally corellated. Then $H(X_i)=1$ and $I(X_1:X_i)=1$. Therefore we have
\begin{equation}
\sum_{k=2}^{m+1}I(X_1:X_k)=m,
\end{equation}
but
\begin{widetext}
\begin{eqnarray}
\sum_{k=0}^{m-3}\frac{(m-k-1)(m-k-1)!}{(m-1)!}H(X_{k+1})&=&\sum_{k=0}^{m-3}\frac{(m-k-1)(m-k-1)!}{(m-1)!}\nonumber\\
&=&m-\frac{2}{(m-1)!}<m,
\end{eqnarray}
\end{widetext}
hence the inequality \eqref{eq:nequalsmp1} is violated
\end{proof}
Note that this inequality yields nontrivial constraints for the entropies of random variables resulting from a GBN of any shape $\mathcal{G}_{n,m},\ n>m$, as it can be applied to any $m+1$ of the $n$ variables.
|
1407.4006
|
\section{How to obtain the $\mathcal H$}
The scope of possible functions {$\mathcal H$} who
satisfy~(\ref{matsyuk:H=1}) is rather large. But, since every
pa\-ra\-me\-ter-independent variational problem, posed on $T^rM$, generates
a corresponding formulation on $C^r(1,M)$, and vice versa, one may
effectively try a pull-back of the Hamiltonian formulation of
the problem on $C^r(1,M)$ to $T^rM$.
Let a variational problem on $\mathbb R\times T^rM$ be given in terms of the
semi-basic (relative to $\mathbb R$) differential $1$-form $\mathcal L \,
d\tau$, where $\mathcal L$ is defined on $T^rM$ solely and satisfies the
Zermelo conditions. And let $L\, dx^{\so}$ be that representative of the
corresponding sheaf of equivalent semi-basic (relative to $M$) differential
$1$-forms on the fibred manifold $C^r(1,M)$, who in the above described coordinates
is given by the following relation,
\begin{equation*}
\mathcal L\, d\tau - (L\cc pr)\, dx^{\so} = {} - (L\circ pr)\,\vartheta \,,
\end{equation*}
where
\begin{equation}\label{matsyuk:theta}
\vartheta = dx^{\so} - u^{\so}d\tau
\end{equation}
is one of the contact forms on $J^1(\mathbb R,M)\approx \mathbb R\times TM$.
Hence\nopagebreak
\begin{equation}\label{matsyuk:cal L}
\mathcal L=u^{\so}\,L\circ pr.
\end{equation}
The canonical momenta are being introduced here as usual:
\begin{equation}\label{matsyuk:bp}
\begin{gathered}
\bp'=\dfrac{\partial L}{\partial \bv'} \\
\bp=\dfrac{\partial L}{\partial \bv} - D_t \bp'\,,
\end{gathered}
\end{equation}
where
\begin{equation}\label{matsyuk:Dt}
D_t = \sv^{i} \frac{\partial }{\mathstrut\partial x^i} +
\sv'^i \frac{\partial }{\mathstrut\partial \sv^i} +
\sv''^i \frac{\partial }{\mathstrut\partial \sv'^i}
\end{equation}
denotes the operator of total derivative with respect to $x^{\so}$.
The correspondence between
the operators (\ref{matsyuk:Dtau}) and (\ref{matsyuk:Dt}) of total
derivatives on relevant jet spaces, $J^2(\mathbb R, M)$ and (locally)
$J^2(\mathbb R, \mathbb R^{\,\mathrm {dim}M-1})$
seems evident, as in fact it is: whenever $\mathsf f$ is a local function on
$C^2(1,M)$, then
\begin{equation}\label{matsyuk:Dtau=Dt}
\mathcal D_\tau(\mathsf f\circ pr)=u^{\so}\,D_t \mathsf f\circ pr
\end{equation}
It would, however, be
of some instructive good to obtain (\ref{matsyuk:Dtau=Dt}) by direct
differentiation of the projection (\ref{matsyuk:pr}), which, in the
3\textsuperscript d order, reads in our coordinates:
\begin{equation}\label{matsyuk:u=v}
\begin{gathered}
\bv\circ pr=\dfrac\bu {u_{\so}} \\
\bv'\circ pr=\dfrac{\buu}{u_{\so}^2} - \dfrac{\dot
u_{\so}}{u_{\so}^3}\,\bu\\
\bv''\circ pr=\dfrac{\buuu}{u_{\so}^3} -
3\,\dfrac{\dot u_{\so}}{u_{\so}^4}\,\buu + 3\left(\dfrac{\dot
u_{\so}^2}{u_{\so}^5} - \dfrac{\ddot u_{\so}}{u_{\so}^4}\right)\bu\,.
\end{gathered}
\end{equation}
With relation~(\ref{matsyuk:Dtau=Dt}) in hand, we are ready now
to establish the correspondence between the pair of
momenta $\wp=(\wp_{\so}, \bwp)$ and $\wp'=(\wp'_{\so}, \bwp')$
in~(\ref{matsyuk:cal p}), calculated for the Lagrange function
$\cal L$ given by (\ref{matsyuk:cal L}), and the pull-back of the
momenta in~(\ref{matsyuk:bp}): {\allowdisplaybreaks
\begin{subequations}\label{matsyuk:all p=p}
\renewcommand{\theequation}{\theparentequation .\arabic{equation}}
\begin{align}
\label{matsyuk:wp'0=p'0}
\wp'_{\so}&=\uo\dfrac{\partial (L\cc pr)}{\partial \dot u_{\so}}=
-\dfrac1{\uo^2}\,\bu\left(\dfrac{\partial L}{\partial \bv'}\cc pr\right)=
-\dfrac1{\uo^2}\,\bu\,(\bp'\cc pr)\,;\\
\label{matsyuk:bwp'=bp'}
\bwp'&=\uo\dfrac{\partial (L\cc pr)}{\partial \buu}=
\dfrac1{\uo}\left(\dfrac{\partial L}{\partial \bv'}\cc pr\right)=
\dfrac1{\uo} (\bp'\cc pr)\,;
\end{align}
\begin{align}
\label{matsyuk:wp0=p0}
\begin{split}
\wp_{\so}&=L\cc pr+\uo\,\dfrac{\partial (L\cc pr)}{\partial u_{\so}}
-\mathcal D_\tau \wp'_\so \quad \text{by the reason of (\ref{matsyuk:u=v}), (\ref{matsyuk:Dtau=Dt}) and (\ref{matsyuk:wp'0=p'0})}
\\
&=L\cc pr
-\uo\left[
\dfrac1{\uo^2}\,\bu\left(\frac{\partial L}{\partial \bv}\cc pr\right)
+\dfrac{2}{\uo^3}\,\buu\left(\frac{\partial L}{\partial \bv'}\cc pr \right)
-\dfrac{3\dot u_{\so}}{\uo^4} \,\bu\left(\dfrac{\partial L}{\partial\bv' }\cc pr\right)\right] \\
&\phantom{=L\cc pr\;}-2\dfrac{\dot u_\so}{\uo^3}\,\bu(\bp'\cc pr)+\dfrac{1}{\uo^2}\,\buu (\bp'\cc pr)
+\dfrac1{\uo}\,\bu(D_t\bp'\circ pr)\\
&=L\cc pr
-\dfrac1{\uo}\,\bu\left(\frac{\partial L}{\partial \bv}\cc pr\right)
+\dfrac{\dot u_\so}{\uo^3}\,\bu(\bp'\cc pr)-\dfrac{1}{\uo^2}\,\buu (\bp'\cc pr)
+\dfrac1{\uo}\,\bu(D_t\bp'\circ pr)\\
&=L\cc pr -\bv\bp\circ pr -\bv'\bp'\circ pr\,;
\end{split}
\\[3\jot]
\label{matsyuk:bwp=bp}
\begin{split}
\bwp&=\uo\,\dfrac{\partial (L\cc pr)}{\partial \bu}
-\mathcal D_\tau \bwp' \quad \text{by the reason of (\ref{matsyuk:u=v}), (\ref{matsyuk:Dtau=Dt}) and (\ref{matsyuk:bwp'=bp'})}
\\
&=\uo
\left[
\dfrac1{\uo}\,\left(\dfrac{\partial L}{\partial \bv}\cc pr\right)
-\dfrac{\dot u_{\so}}{\uo^3} \left(\dfrac{\partial L}{\partial\bv' }\cc pr\right)
\right]
+\dfrac{\dot u_\so}{\uo^2}\,(\bp'\cc pr)-D_t\bp'\circ pr)\\
&=\dfrac{\partial L}{\partial \bv}\circ pr - D_t \bp'\circ pr = \bp\circ pr\,.
\end{split}
\end{align}
\end{subequations}
}
From (\ref{matsyuk:wp0=p0}) and (\ref{matsyuk:bwp=bp}) it follows that
{\allowdisplaybreaks
\begin{subequations}\label{matsyuk:wp=p final}
\renewcommand{\theequation}{\theparentequation .\arabic{equation}}
\begin{align}
\wp\, u&=\uo\, L\cc pr - \uo\,\bv'\bp'\circ pr\,,\\
\intertext{whereas from (\ref{matsyuk:wp'0=p'0}) and
(\ref{matsyuk:bwp'=bp'}) in view of (\ref{matsyuk:u=v}) it follows that}
\wp'\dot u&= \uo\,\bv'\bp'\circ pr\,,
\end{align}
\end{subequations} } and (\ref{matsyuk:Z2}) keeps true immediately.
In our further considerations we choose the approach
of the generalized Hamiltonian theory as
exposed in \cite{matsyuk:Krupkova}. Then, in our coordinates, it is best to
describe the system evolution by the kernel of the differential two
form
\begin{equation}\label{matsyuk:H form}
\omega=-dH\wedge dx^{\so} + d\bp\wedge d\bx + d\bp'\wedge d\bv\,,
\end{equation}
where the exterior product sign $\wedge$ comprises the contraction
of vector differential forms, if necessary. Now, it is tentative
that on the manifold $\mathbb R\times T^3M$ the evolution of this
same system be described by a differential two form of the same
shape,
\begin{equation}\label{matsyuk:cal H form}
\varOmega=-d\,\mathcal H\wedge d\tau + d\wp\wedge dx + d\wp'\wedge du\,,
\end{equation}
where the momenta $\wp$ and $\wp'$ due to the Lagrange
function~(\ref{matsyuk:cal L}).
Conventionally one puts
$H=\bp\bv+\bp'\bv'-L$.
Under this assumption it is straightforward to calculate the difference
between (\ref{matsyuk:cal H form}) and (\ref{matsyuk:H form}), taking into
account the relations
(\ref{matsyuk:bwp'=bp'}, \ref{matsyuk:bwp=bp})
and the Zermelo condition
(\ref{matsyuk:Z1}):
\begin{equation}\label{matsyuk:W-w}
\varOmega-pr^*\omega=
d(pr^*H+\wp_\so )\wedge dx^{\so}
-d\,\mathcal H\wedge d\tau\,.
\end{equation}
We wish that this difference be proportional to the contact form
(\ref{matsyuk:theta}), namely,
\begin{equation}\label{matsyuk:20mod}
\varOmega-pr^*\omega=\alpha\wedge\vartheta\,.
\end{equation}
The simplest reasonable way to comply in (\ref{matsyuk:W-w}) with
(\ref{matsyuk:20mod}) is to put
\begin{equation}\label{matsyuk:dcalH}
d\,\mathcal H=u^{\so}d(pr^*H+\wp_{\so})
\end{equation}
and
\begin{equation}\label{matsyuk:psi}
\mathcal H=\uo pr^*H+\varPsi\,.
\end{equation}
Now proceed to determine this deviating function $\varPsi$.
From (\ref{matsyuk:psi}) we have:
\begin{equation}\label{matsyuk:pr*dH}
pr^*dH=\dfrac{d\,\mathcal H}{\uo}
+ \left(\varPsi-\mathcal H \right)\dfrac{d\uo}{\uo^2}-\dfrac{d\,\varPsi}{\uo}\,.
\end{equation}
It suffices now to substitute (\ref{matsyuk:pr*dH}) into (\ref{matsyuk:dcalH})
to obtain the relation
\[
\dfrac{\mathcal H-\varPsi}{\uo}\,d\uo-\uo\,d\wp_\so ={}-d\,\varPsi\,,
\]
from where it becomes clear that
\[\begin{cases}
\varPsi=\uo\wp_\so+c\\\mathcal H=c
\end{cases}\]
and also, by the reason of (\ref{matsyuk:H=1}), $c=1$.
Hence
\begin{equation}\label{matsyuk:cal H}
\mathcal H=\uo pr^*H+\uo\wp_\so +1
\end{equation}
\section{{\itshape Zitterbewegung} of quasiclassical relativistic\\ particle}
As far back as 1946 Fritz Bopp developed a second-order Lagrange
function for the description of classical particle motion from the
second step approximation with respect to the parameter of retard
interaction~\cite{matsyuk:Bopp}. It seems prominent that the Bopp
Lagrangian may be cast into a simple shape in terms of the first
curvature of the particle's world line,
\begin{equation*}\label{matsyuk:k in u}
k=\dfrac{\|\dot u\wedge u\|}{\|u\|^3}\,,
\end{equation*}
as follows:
\begin{equation}\label{matsyuk:Bopp}
\mathcal L\overset{\mathrm{def}}= a \mathcal L_r + A \mathcal L_e
= \frac a 2 \|u\| k^2+\frac A 2 \|u\|\,,
\end{equation}
where we assume $a\not=0$ to confine with~(\ref{matsyuk:rank}).
This Lagrange function satisfies the Zermelo conditions~(\ref{matsyuk:Zermelo}).
The first addend in~(\ref{matsyuk:Bopp}), $\mathcal L_r$, turns out to be of the type considered
by H.~Rund in~\cite{matsyuk:Rund} (see also~\cite{matsyuk:Grasser}).
The second addend, $\mathcal L_e$, is the free particle Lagrange function.
According to (\ref{matsyuk:cal L}), the corresponding local Lagrange density
on $C^2(1,M)$ may be expressed in coordinates $x^{\so}$, $\bv$ and
$\bv'$:
\begin{multline}\label{matsyuk:L}
L\,dx^{\so}\overset{\mathrm{def}}= a L_r dx^{\so}+A L_e dx^{\so}\\
= \frac{a}{2}\sqrt{(1+\bv\2)}
\left(\frac{\bv'\2}{(1+\bv\2)^2}-\frac{(\bv\bcd\bv')^2}{(1+\bv\2)^3}\right)dx^{\so}
+\frac{A}{2}\sqrt{(1+\bv\2)}dx^{\so}\,.
\end{multline}
The momenta (\ref{matsyuk:bp}) for this Lagrangian read:
\[
\bp'_r=\dfrac{\bv'}{(1+\bv\2)^{3/2}}-\dfrac{\bv\bcd\bv'}{(1+\bv\2)^{5/2}}\,\bv \\
\]
\begin{multline*}
\bp_r=-\dfrac{\bv''}{(1+\bv\2)^{3/2}}+3\,\dfrac{\bv\bcd\bv'}{(1+\bv\2)^{5/2}}\,\bv' \\
+\dfrac{\bv\bcd\bv''}{(1+\bv\2)^{5/2}}\,\bv
-\dfrac{1}{2}\dfrac{\bv'\2}{(1+\bv\2)^{5/2}}\,\bv
-\dfrac{5}{2}\dfrac{(\bv\bcd\bv')^2}{(1+\bv\2)^{7/2}}\,\bv\,.
\end{multline*}
We introduce the standard
Hamilton function
\begin{multline}\label{matsyuk:serif H}
H=\bp\bv+\bp'\bv'-L
\\ \overset{\mathrm {def}}=
aH_r+AH_e=a\bp_r\bv+a\bp'_r\bv'-aL_r+A\bp_e\bv-AL_e\,,
\end{multline}
because $\bp'_e=0$. It is necessary to exclude the variable $\bv'$
in~(\ref{matsyuk:serif H}). We calculate:
\[
\begin{cases}
\bp'_r\bv'=2L_r \\
\bp'_r\2+(\bp'_r\bv)^2=2\,\dfrac{L_r}{(1+\bv\2)^{3/2}}\,,
\end{cases}
\]
and finally the Hamilton function reads
\begin{equation}\label{matsyuk:H final}
H=\bp\bv+\frac 1{2a}\left(1+\bv\2\right)^{3/2}\left(\bp'\2+(\bp'\bv)^2\right)
- \frac A{2}\sqrt{1+\bv\2}\,.
\end{equation}
In his paper~\cite[page~199]{matsyuk:Bopp}, Fritz Bopp asserted: ``Der
klassischen Bewegung \"uberlagert sich eine Zitterbewegung, die
durch die neuen Variabeln $\bv$ und $\bp'$ beschrieben wird. Sie
f\"uhrt zu spinartigen Effekten\dots''\nopagebreak
{\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\footnote[1]{Upon the classical motion some vibrational one
superimposes itself that is described by the new variables $\bv$
and $\bp'$. It leads to the effects of spin type\dots} }
Now the Hamilton function on $T^3M$ may be obtained from(\ref{matsyuk:cal H}):
\begin{equation}\label{matsyuk:cal H final}
\mathcal H=\wp u+\frac {1}{2a}\|u\|^3\wp'\2-\frac A 2 \|u\|+1\,.
\end{equation}
Alternatively, one could get the same expression directly from
the assertion
\begin{equation}\label{matsyuk:cal H directly}
\mathcal H=\wp u + \wp'\dot u - \mathcal L + 1\,,
\end{equation}
assuming $\mathcal L$ be taken from~(\ref{matsyuk:Bopp}). In view of~(\ref{matsyuk:rank}),
one cannot resolve the Legendre transformation~(\ref{matsyuk:cal p}) in full,
but nevertheless it is possible to eliminate the variable $\dot u$ from~(\ref{matsyuk:cal H directly}).
First we calculate the momenta for~(\ref{matsyuk:Bopp})
\begin{gather*}
\wp'=\dfrac{a}{\|u\|^5}\left[u\2\dot u-(u\cdot\dot u)u\right] \\
\wp= \dfrac{Au}{2\|u\|} {}-a\left[\dfrac{\ddot u}{\|u\|^3}
-3\,\dfrac{u\cdot\dot u}{\|u\|^5}\,\dot u
-\dfrac{u\cdot\ddot u}{\|u\|^5}\,u+\dfrac{\dot u\2}{2\|u\|^5}\,u
+\dfrac{5}{2}\dfrac{(u\dot u)^2}{\|u\|^7}\,u
\right].
\end{gather*}
In the next step we express all those quantities in~(\ref{matsyuk:cal H directly})
wherein the $\dot u$ enters, in terms of $\wp'$ and $u$ alone
\begin{equation}\label{matsyuk:}
\left\{
\begin{aligned}
\wp'\dot u&=\dfrac{\|u\|^3}{a}\,\wp'\2 \\
\mathcal L_r&=\dfrac{\|u\|^3}{2a^2}\,\wp'\2\,,
\end{aligned}
\right.
\end{equation}
and substitute into~(\ref{matsyuk:cal H directly}) to finally
achieve the Hamilton function~(\ref{matsyuk:cal H final}).
The approach to building up the Hamilton function in present paper
differs from
that of H.~S.~P.~Gr\"asser. I was inspired by his treatment of
general Lagrange function, quadratic in velocities, in the
framework of Finsler space, of which ours is a very
special case. But the physical model herein considered demands to include
also the free particle term $\mathcal L_e$.
It is not of much labour now
to calculate the fourth-order Euler-Poisson equation of the variational
problem with the Lagrange function~(\ref{matsyuk:Bopp}) from the starting
point of the Hamilton system~(\ref{matsyuk:H})
with the expression~(\ref{matsyuk:cal H final}) in hand.
For~(\ref{matsyuk:H}) we have:
\begin{equation*}
\left\{
\begin{aligned}
\dfrac{dx}{d\tau}&=\lambda u\,,
\\
\dfrac{du}{d\tau}&=\lambda \dfrac{\|u\|^3}{a}\,\wp'+\mu u\,,
\\
\dfrac{d\wp}{d\tau}&=0\,, \\
\dfrac{d\wp'}{d\tau}&=\dfrac{A}{2}\lambda \dfrac{u}{\|u\|}
-\lambda \wp-\dfrac{3}{2}\lambda \dfrac{\|u\|}{a}\,\wp'\2u-\mu\wp'\,.
\end{aligned}
\right.
\end{equation*}
The multiplier~$\mu$ may be obtained from the second equation by contracting it with~$u$ and recalling the Zermelo condition~(\ref{matsyuk:Z1}):
\begin{equation*}
\mu=\frac{u\cdot u}{\|u\|^2}\,.
\end{equation*}
Only at this stage one has the right to put some constraints on the choice of
the parameter $\tau$. We put $u\2=1$ to obtain
\begin{subequations}\label{matsyuk:H Zitterbewegung u=1}
\renewcommand{\theequation}{\theparentequation.\arabic{equation}}
\begin{align}
\dfrac{du}{d\tau}&=\dfrac{\wp'}{a} \label{matsyuk:dot u u=1}\\
\dfrac{d\wp'}{d\tau}&=\dfrac{A}{2}\,u
-\wp-\dfrac{3}{2a}\wp'\2 u\,, \label{matsyuk:.wp' u=1}
\end{align}
\end{subequations}
and it is clear that $\lambda=1$ and $\mu=0$, so that the
evolution equation~(\ref{matsyuk:Poisson}) regains the traditional shape now.
Next we differentiate the equation~(\ref{matsyuk:dot u u=1}) and substitute
the equation~(\ref{matsyuk:.wp' u=1}) therein, to obtain
\begin{gather}
\ddot u = \dfrac{A}{2a}\,u
-\dfrac{\wp}{a}-\dfrac{3}{2a^2}\wp'\2 u\,,
\label{matsyuk:..u}
\\
\dfrac{\wp\dot u}{a}={}- \ddot u \cdot \dot u \,,\label{matsyuk:..u.u}
\end{gather}
and, on the other hand, the contraction of (\ref{matsyuk:dot u u=1}) with
(\ref{matsyuk:.wp' u=1}) gives
\begin{equation}\label{matsyuk:.wp'wp' u=1}
\wp'\cdot \dot\wp'={}-a\,\wp\dot u\,.
\end{equation}
Differentiating (\ref{matsyuk:..u}) once again produces
\begin{equation}\label{matsyuk:...u}
\dddot u = \dfrac{A}{2a}\,\dot u{}-\dfrac{3}{a^2}\,(\wp'\cdot \dot\wp')\,u
- \dfrac{3}{2a^2}\,\wp'\2\dot u\,,
\end{equation}
in where we substitute (\ref{matsyuk:.wp'wp' u=1}), (\ref{matsyuk:dot u u=1}),
and, sequentially,
(\ref{matsyuk:..u.u}), to finish at the resulting fourth-order
equation of motion
\begin{equation}\label{matsyuk:...u final}
\dddot u + \left(\dfrac{3}{2}\,\dot u\2 - \dfrac{A}{2a}\right)\,\dot u
+3\,(\dot u \cdot \ddot u)\,u=0\,.
\end{equation}
As soon as in the actual parametrization $k^2=\dot u\2$, on the constrained
manifold of constant relativistic acceleration $k_\so$
the equation~(\ref{matsyuk:...u final}) {\em reduces to the equation of helical
motion of relativistic spinning particle considered first by F.~Riewe~\cite{matsyuk:Riewe} and then by
G.~C.~Constantelos~\cite{matsyuk:Constantelos}:}
\begin{equation}\label{matsyuk:Riewe}
\dfrac{d^2}{ds^2}\,\ddot x^\alpha +\varpi^2\ddot x^\alpha =0\,,
\end{equation}
where we have put
$\varpi=\frac{3}{2}\,k_\so^2-\frac{A}{2a}$.
\bigskip
In the previous paper~\cite{matsyuk:Matsyuk}, I proved that {\em this fourth order
equation of motion~(\ref{matsyuk:Riewe}) may be rigorously developed from
the third order general equation of motion of classical dipole particle
proposed by Mathisson in 1937 in~\cite{matsyuk:Mathisson}}.
|
1311.0752
|
\section{Introduction and Main Results}\label{sec:intro}
Pulsars are believed to be rapidly rotating neutron stars with extremely strong magnetic fields, whose pulses are caused by misalignment of the field and rotation axes. Such a configuration is inconsistent with a vacuum exterior \cite{goldreich-julian1969}, so that pulsars must have a plasma magnetosphere. The strong magnetic field ensures that the energy (including rest mass) and momentum of the charged particles is negligible compared to that of the fields. Conservation then dictates that the Lorentz force density must vanish everywhere in the plasma, $F_{ab}J^b=0$. Eliminating the current via Maxwell's equation $\nabla_b F^{ab} = 4 \pi J^a$, we may write the complete set of equations as
\begin{equation}
\nabla_{[a} F_{bc]}=0, \quad F_{ab} \nabla_c F^{bc}=0. \label{FFE}
\end{equation}
These are the equations of \textit{force-free electrodynamics}, a non-linear, deterministic set of equations for the electromagnetic field of a magnetically dominated plasma \cite{uchida1997,komissarov2002}.
Force-free electrodynamics is very different from vacuum electrodynamics. One dramatic example is the opening of magnetic field lines \cite{goldreich-julian1969, michel1974,ingraham1973,gralla-jacobson2014}. If a rotating, conducting star is endowed with a magnetic dipole and immersed in vacuum, the field lines form closed loops, as usual. If, on the other hand, the star is surrounded by a force-free plasma, lines leaving the star near its poles actually ``open up'', proceeding all the way to infinity, never to return to the star (Fig.~\ref{fig:pic}). Closed field lines are confined to a region near the star, so that the outer magnetosphere contains only open lines, which run in opposite directions on opposite sides of a current sheet. For aligned magnetic and rotation axes (``aligned rotor''), this sheet is on the equatorial plane, whereas for inclined axes it traces an oscillatory pattern at the rotational frequency.
A second, key difference from vacuum electrodynamics is that stationary, axisymmetric force-free fields can transport energy and angular momentum away from an isolated source. For example, even an aligned rotor loses energy to a force-free magnetosphere, at a rate comparable to the inclined case \cite{spitkovsky2006}. For spinning black holes, a stationary, axisymmetric magnetosphere can extract the hole's rotational energy, as first shown by Blandford and Znajek \cite{blandford-znajek1977}. While vacuum electrodynamics relies on acceleration to produce radiation that transports energy, force-free fields can carry away energy in steady state.
It is nevertheless natural to ask how force-free energy transport proceeds when acceleration is added to the mix. If a magnetized neutron star is accelerated, how does the magnetosphere respond, and how is the energy output modified? These questions have direct astrophysical application in modeling emission from compact object binaries, as potential electromagnetic counterparts to gravitational-wave observations \cite{schutz1986,holz-hughes2005} or precursor emission to gamma-ray bursts \cite{mcwilliams-levin2011}. While numerical simulations have now successfully treated some important configurations \cite{palenzuela-etal2013,palenzuela-etal2013b,paschalidis-etienne-shapiro2013}, the high computational cost of three-dimensional runs precludes a systematic exploration using numerical techniques alone. It is therefore of interest to develop analytical tools to address the question of the accelerated pulsar magnetosphere.
\begin{figure}[t]
\includegraphics[scale=.45]{pic.pdf}
\caption{Sketch of the pulsar magnetosphere. Outside of a zone of closed field lines near the star, magnetic field lines (blue) run in opposite directions on opposite sides of a current sheet (brown). (The field lines also wind around azimuthally, not shown in this projection.) Despite the complicated geometry, the current density (black) is approximately null and radial in the open zone.}\label{fig:pic}
\vspace{-3mm}
\end{figure}
The present approach is motivated by the observation that in numerical simulations of the magnetosphere of non-moving pulsars \cite{contopoulos-kazanas-fendt1999,spitkovsky2006,mckinney2006,timokhin2006,kalapotharakos-contopoulos-kazanas2012}, the four-current vector becomes very nearly null and radial at a few light cylinder radii from the pulsar \cite{kalapotharakos-contopoulos-kazanas2012,constantinoscomment}. It is therefore natural to suppose that, for a pulsar in motion, the outer magnetosphere continues to host a null current pointing towards the star. For relativistic motion, the current should point towards the pulsar location at the retarded time. Thus we expect that the outer magnetosphere of a moving pulsar has null four-current along the light cones of the star's worldline.
In this paper we find \textit{all} solutions with null current along the light cones of a timelike worldline in flat spacetime. We use techniques developed recently in \cite{brennan-gralla-jacobson2013}, combined with technology developed in \cite{newman-penrose1966,held-newman-posadas1970,newman1974,posadas-yanga1985}. For fields that are smooth everywhere off the worldline, the result takes a simple form. Let $u^a$ be the four-velocity of the worldline, extended to all spacetime by parallel transport along the (future) null cones, and let $\ell^a$ be the tangent to the null generators of the cones that satisfies $u^a \ell_a=1$. The general solution to Eqs.~\eqref{FFE} that is smooth away from the worldline and has four-current $J^a \propto \ell^a$ is given by
\begin{equation}\label{soln}
F_{ab} = F^\textrm{q}_{ab} - 2 \ell_{[a} \nabla_{b]}\psi,
\end{equation}
where $F^\textrm{q}_{ab}$ is the field of a magnetic monopole of charge $q$ moving on the worldline (the magnetic dual of the Lienard-Wiechert field) and $\psi$ is an \textit{arbitrary scalar field} satisfying $\ell^a \nabla_a \psi=0$. In light of the non-linearity of Eqs.~\eqref{FFE}, it is remarkable that such a broad class of solutions can be written down analytically. For a stationary worldline the solutions reduce to those of \cite{brennan-gralla-jacobson2013} restricted to flat spacetime, which in turn contain those of Michel \cite{michel1973b} and Lyutikov \cite{lyutikov2011} as special choices of $\psi$. The solutions \eqref{soln} are magnetically dominated ($F^{ab}F_{ab}>0$) when $q\neq0$ and null ($F^{ab}F_{ab}=0$) when $q=0$.
The power radiated on each cone by Eq.~\eqref{soln} is
\begin{equation}\label{power}
\mathcal{P}(u) = \frac{2}{3} q^2 a^2 + \frac{1}{4\pi} \int \nabla_a \psi \nabla^a \psi \ \! dS ,
\end{equation}
where $a$ is the magnitude of the four-acceleration at the vertex of the cone (proper time $u$), and the surface of integration (area element $dS$) is a ``retarded time rest frame sphere'', the intersection of the light cone and a spacelike plane orthogonal to the four-velocity at time $u$. (One may think of this as a sphere at future null infinity, but the integral is independent of the sphere on account of $\ell^a \nabla_a \psi=0$.) The first term of Eq.~\eqref{power} arises from the monopole field (this is simply the Larmor formula), while the second term is due to the second term in Eq.~\eqref{soln}. The cross term turns out to be a total derivative, and has vanished by Stokes' theorem.
Real pulsars do not contain monopoles, and the outer magnetosphere instead has a \textit{split} monopolar structure, where two regions of opposite polarity are separated by a current sheet. The split case introduces two subtleties. First, we may no longer assume globally smooth fields. For fields that are only locally smooth, we find that the charge $q$ may depend on time $u$, and an additional term proportional to the time derivative $\dot{q}$ appears, Eq.~\eqref{phi2weirdo} below. For the present work we set $\dot{q}=0$, in which case Eq.~\eqref{soln} gives the general locally smooth solution. As explained below, $q$ corresponds to intrinsic pulsar parameters (magnetic field strength and rotation rate), so $\dot{q}=0$ restricts to pulsars whose intrinsic properties do not change significantly in time.
The second subtlety of the split case is that Stokes' theorem fails, in general, to eliminate the cross-term in the power radiated. In the simplest models of current sheets \cite{bogovalov1999,gralla-jacobson2014} the field strength undergoes a sign change at the sheet, so that the cross-term is continuous, and no extra terms arise. However, in the most general case allowed by the electromagnetic junction conditions, we must supplement Eq.~\eqref{power} by boundary integrals taken on the intersection of the current sheet and the sphere, Eq.~\eqref{Psheet} below. In \cite{kalapotharakos-contopoulos-kazanas2012} it was shown that the shape of the dipole pulsar's current sheet precisely matches the simple model of \cite{bogovalov1999}, where the field strength undergoes a sign flip. For this reason we expect current sheets formed in the exterior of rotating stars to generically have this simple behavior
\begin{figure}[t]
\includegraphics[scale=.45]{worldline.pdf}
\caption{For an accelerated pulsar we expect the far-zone four-current to be along the light cones of the worldline. We find that exact solutions with such current are classified by a number $q$ and a function $\psi$ on the sphere cross the worldline. These parameters therefore encode the relevant details of the near-zone pulsar physics.}\label{fig:worldline}
\vspace{-3mm}
\end{figure}
We therefore expect that the outer magnetosphere of an accelerated pulsar will be described by Eq.~\eqref{soln}, with sign reversed on either side of a current sheet,\footnote{If the pulsar is given a quadrupole or higher moment magnetic field, one would expect additional current sheets, which can also be described by our solution.} with the power radiated given by Eq.~\eqref{power}. The solution has three free parameters/functions: the worldline, the monopole charge $q$, and the function $\psi$. We imagine fixing these as follows. First, perform numerical simulations of non-moving pulsars with a variety of physical parameter choices (spin, magnetic dipole, etc.) and, in each case, determine the associated $q$ and $\psi(t-r,\theta,\phi)$ by fitting the exterior magnetosphere to Eq.~\eqref{soln} with a stationary worldline. One thus has a map between pulsar parameters at time $t$ and a function $\psi(\theta,\phi)$ on the sphere, which describes the field on the associated light cone. Now suppose the pulsar is accelerated. Provided the acceleration does not significantly affect the near-zone physics,\footnote{This should at least be true for small acceleration, $a \ll \Omega$, where $\Omega$ is the angular frequency of the pulsar.} one should be able to simply use the ``same'' $q$ and $\psi$ for an accelerated worldline. That is, the same $q$ is used, while $\psi$ is promoted by demanding that it agree with the non-moving case on each light cone of the accelerated worldline (Fig.~\ref{fig:worldline}). In this way the outer magnetosphere and radiated power can be obtained without the need to simulate the accelerated pulsar.
We may determine the effect of acceleration on the power radiated without performing this procedure explicitly. The second term in Eq.~\eqref{power} agrees with the energy flux for an unaccelerated pulsar whose parameters agree instantaneously with the accelerated one. The first term may therefore be regraded as the correction due to acceleration. For a dipole pulsar with magnetic moment $\mu$ and angular velocity $\Omega$, dimensional analysis and linearity of the field in $\mu$ imply that $q \propto \mu \Omega$ and $\psi \propto \mu \Omega^2$. The second term in Eq.~\eqref{power} has the usual pulsar energy loss scaling $\mu^2 \Omega^4$, while the first term gives the acceleration correction as
\begin{equation}\label{Paccel-intro}
\mathcal{P}_{\textrm{accel.}} =\frac{2}{3} q^2 a^2 \propto \mu^2 \Omega^2 a^2.
\end{equation}
For comparison, note that the power radiated by an accelerated constant dipole in vacuum scales as $\mu^2 \dot{a}^2$, where dot is a time derivative \cite{ioka-taniguchi2000}.
In Sec.~\ref{sec:astro} we estimate the size of this effect for astrophysical binaries, concluding that it is too small to be observable with present methods. However, it is an significant fraction of the ordinary pulsar power for binaries near merger, and it would be interesting to compare with numerical simulations of binary systems. During the inspiral it should be possible to regard each member as approximately following an accelerated trajectory in flat spacetime, and the scaling $\mu^2 \Omega^2 a^2$ should appear as part of the energy flux.
Thus far, numerical simulations of magnetized binaries have been performed only in the irrotational case, $\Omega \approx 0$. It would be interesting to perform simulations with non-zero values of spin in order to see if the characteristic $\mu^2 \Omega^2 a^2$ energy flux appears. In principle this could be distinguished from other effects like unipolar induction \cite{goldreich-lynden-bell1969} by its dependence on spin and acceleration. Alternatively one could perform a simulation of a pulsar with an unmagnetized, non-conducting companion, where there should be no unipolar induction.
From a purely theoretical standpoint, this work provides a nice coda to the story of the pulsar magnetosphere. Perhaps the most dramatic aspect of this story is the opening of field lines, wherein the force-free plasma converts dipoles to (split) monopoles. In a sense, our results indicate that this conversion extends to radiation, too: An accelerated pulsar radiates not as a dipole $\mu$, but rather as a monopole $q\propto\mu \Omega$.
In Sec.~\ref{sec:technology} we review some computational technology, which we use to solve the force-free equations in Sec.~\ref{sec:solution}. We compute the energy flux in Sec.~\ref{sec:power} and discuss astrophysical applicability in Sec.~\ref{sec:astro}. Latin indices are abstract spacetime indices (holding independent of coordinates), while Greek indices label components in a coordinate system. The signature of our (flat) metric is $(+,-,-,-)$.
\section{Technology}\label{sec:technology}
We begin by reviewing some technology for the light cones congruence \cite{newman-penrose1966,held-newman-posadas1970,newman1974,posadas-yanga1985}. Consider flat spacetime in Cartesian Minkowski coordinates $x^\mu$, and let $(\zeta,\bar{\zeta})$ be complex stereographic coordinates for two-spheres in this fixed frame. (Complex stereographic coordinates are related to spherical coordinates by $\zeta=e^{i\phi}\cot \tfrac{\theta}{2}$.) Consider a timelike worldline parameterized by proper time $u$ as $x^\mu=z^\mu(u)$. The four-velocity is $u^\mu=\dot{z}^\mu$, where dot denotes a $u$-derivative. Define a new set of coordinates $(u,r,\zeta,\bar{\zeta})$ by
\begin{equation}\label{newcoords}
x^\mu=z ^\mu (u)+r \ell^\mu(u,\zeta,\bar{\zeta}),
\end{equation}
where $\ell^\mu(u,\zeta,\bar{\zeta})$ are the Minkowski coordinate components of the null vector pointing in the spatial direction $(\zeta,\bar{\zeta})$, and normalized so that $\ell_a u^a=1$. This latter condition gives $r$ the interpretation of the spatial distance between the point $(u,r,\zeta,\bar{\zeta})$ and the worldline point $z^\mu(u)$, as measured in the rest frame of the worldline at time $u$. Since these points are null-related, we refer to $u$ as the retarded time. The new coordinates are defined everywhere except for the worldline $r=0$, where Eq.~\eqref{newcoords} is not differentiable.
Letting an arbitrary factor $v(u,\zeta,\bar{\zeta})$ absorb the normalization, we may write $v \ell^\mu = \{1,\hat{n}(\zeta,\bar{\zeta})\}$, where $\hat{n}=\vec{x}/|\vec{x}|$ is the radial unit vector in the fixed frame. In terms of $(\zeta,\bar{\zeta})$ we then have
\begin{align}\label{lformula}
\ell^\mu=\frac{1}{vP}\left(P,\zeta+\bar{\zeta},\frac{\zeta-\bar{\zeta}}{i},\zeta \bar{\zeta}-1 \right),
\end{align}
where $P=1+\zeta \bar{\zeta}$. If we write $u^\mu=\gamma\{1,\vec{\beta}\}$ then $\vec{\beta}$ is the three-velocity of the worldline relative to the fixed frame. From $u^a \ell_a=1$ and $v \ell^\mu = \{1,\hat{n}\}$ we then obtain the explicit formula $v=\gamma(1-\vec{\beta}\cdot \hat{n})$. This form helps for checking a convenient identity satisfied by $v$,
\begin{equation}\label{videntity}
v^{-2} = 1 + \Delta \log v,
\end{equation}
where $\Delta$ is the Laplacian on the unit two-sphere.
In Eq.~\eqref{newcoords}, $r \ell^\mu$ is naturally regarded as a ``displacement vector'' between $z^\mu(u)$ and the field point $x^\mu$. For later purposes it is convenient to let $\ell^a$ be the vector field whose Minkowski coordinates are given by Eq.~\eqref{lformula} at each point $(u,r,\zeta,\bar{\zeta})$ on the manifold. This vector field is tangent to the congruence of future-directed null geodesics emanating from the worldline, and ill-defined on the worldline itself. We will also regard $v(u,\zeta,\bar{\zeta})$ as a scalar field on the manifold (minus the worldline), whose particular space-time dependence encodes the three-velocity $\vec{\beta}(u)$. Finally we extend the four-velocity off the worldline by parallel transport along the light cones, i.e., $u^\mu(u,r,\zeta,\bar{\zeta})=\dot{z}^\mu(u)$.
To compute the metric components in the new coordinates it is useful to note that $\partial_\zeta \ell^\mu$ is complex-null and orthogonal to $\ell^\mu$ and $u^\mu=\dot{z}^\mu$. From Eq.~\eqref{newcoords} we then find
\begin{align}\label{metric}
ds^2=(1-2r\frac{\dot{v}}{v})du^2+2 du dr-\frac{4r^2}{v^2 P^2}d\zeta d\bar{\zeta}.
\end{align}
The metric of a (unit) two-sphere in complex stereographic coordinates is given by $4P^{-2} d\zeta d\bar{\zeta}$. The extra factor of $v^2$ in Eq.~\eqref{metric} reflects the fact that the two-surface $u=r=\textrm{const}$ is a constant-distance sphere in the rest frame associated with retarded time $u$, whereas $(\zeta,\bar{\zeta})$ were defined relative to the fixed frame. We refer to $u=r=\textrm{const}$ spheres as rest frame spheres.
By construction, we have $\ell = \partial_r$ in the new coordinates. From Eq.~\eqref{metric} we may select three other null vectors satisfying the Newman-Penrose (NP) \cite{newman-penrose1962} requirements $\ell^a n_a=1$ and $m^a \bar{m}_a=-1$ (other inner products vanishing),
\begin{subequations}\label{tetrad}
\begin{align}
\ell^\mu &=(0,1,0,0)\\
n^\mu&=\left(1,-\frac{1}{2}\left(1-2r\frac{\dot{v}}{v}\right),0,0\right)\\
m^\mu&=\left(0,0,\frac{v P}{\sqrt{2}r},0\right)\\
\bar{m}^\mu&=\left(0,0,0,\frac{v P}{\sqrt{2}r}\right).
\end{align}
\end{subequations}
The vectors $\ell$ and $n$ are null normals to the rest frame spheres, while $m$ and $\bar{m}$ are complex-null tangents. From Eq.~\eqref{newcoords}, the $(u,r,\zeta,\bar{\zeta})$ coordinate components of the extended four-velocity are
\begin{align}\label{xidot}
u^\mu = (1,r\frac{\dot{v}}{v},0,0) = n^\mu + \tfrac{1}{2} \ell^\mu.
\end{align}
Finally, a unit vector orthogonal to $u^a$ and to rest frame spheres is given in these coordinates by
\begin{align}\label{R}
R^\mu = (1,-1-r\frac{\dot{v}}{v},0,0) = n^\mu - \tfrac{1}{2} \ell^\mu.
\end{align}
The spin coefficients for the tetrad \eqref{tetrad} are
\begin{align}
\rho=2\mu=-\frac{1}{r}, \quad \alpha = -\bar{\beta} = \frac{\partial_{\bar{\zeta}}(P v)}{2\sqrt{2}r},\nonumber \\ \quad \gamma = -\frac{\dot{v}}{2v}, \quad \nu=-\frac{Pv}{\sqrt{2}} \partial_{\bar{\zeta}}\left( \frac{\dot{v}}{{v}} \right), \label{spin-coefficients}
\end{align}
with all other coefficients vanishing. From $\kappa=\sigma=\textrm{Im}[\rho]=0$ we see that the null congruence along $\ell$ is geodesic, shear-free, and twist-free \cite{newman-penrose1962}. Using $u^a=n^a+\tfrac{1}{2} \ell^a$ and the NP equations \cite{newman-penrose1962} we may compute the frame components of the four-acceleration $a^a=u^b \nabla_b u^a$ (extended off the worldline by parallel transport along null cones), finding $a^a=\nu m^a + \bar{\nu} \bar{m}^a + \gamma(\ell^a - 2 n^a)$. Two particularly useful quantities are the projection on to $\ell^a$,
\begin{equation}\label{aell}
a^a \ell_a = \frac{\dot{v}}{{v}},
\end{equation}
and the magnitude,
\begin{equation}\label{a2}
a^a a_a = - P^2 v^2 \partial_\zeta \left( \frac{\dot{v}}{v} \right) \partial_{\bar{\zeta}} \left( \frac{\dot{v}}{v} \right) - \left( \frac{\dot{v}}{v} \right)^2.
\end{equation}
A final bit of technology we will find useful are the $\eth$ and $\bar{\eth}$ operators. These operators are defined on functions with a definite spin weight as
\begin{align}
\eth \eta & = P^{1-s} \partial_\zeta (P^s \eta) \label{eth} \\
\bar{\eth} \eta & = P^{1+s} \partial_{\bar{\zeta}} (P^{-s} \eta). \label{ethbar}
\end{align}
The spin-weight $s$ of a function $\eta$ is refers to its behavior $\eta\rightarrow \exp[i \theta s] \eta$ under rotations of the sphere tetrad vectors $m \rightarrow \exp[i \theta]m$. The application of $\eth$ raises the spin-weight of a quantity by one, while $\bar{\eth}$ lowers by one. Any smooth function of spin-weight $-1$ can be written as $\eth$ of a spin-weight zero function (and similarly for $\bar{\eth}$ and spin-weight +1). Acting on spin-weight zero functions, we have $\eth \bar{\eth}=\bar{\eth} \eth=\Delta$, where $\Delta=P^2 \partial_\zeta \partial_{\bar{\zeta}}$ is the sphere Laplacian. Both $\eth$ and $\bar{\eth}$ obey the Leibniz rule and have the property that the sphere-integral of $\eth f$ (or $\bar{\eth}f$) is vanishing for any spin-weighted function $f$. Thus total derivatives may be freely thrown away under integrals. In anticipation of this use we rewrite Eq.~\eqref{a2} as
\begin{equation}\label{larmorsavior}
\eth \left( \frac{\dot{v}}{v} \right) \bar{\eth} \left( \frac{\dot{v}}{v} \right) = - \frac{2}{3} \frac{a^a a_a}{v^2} + \frac{1}{6} \eth \bar{\eth} \left(\frac{\dot{v}}{v}\right)^2,
\end{equation}
where $v$ and $\dot{v}$ have spin-weight zero, and Eq.~\eqref{videntity} has been used.
\section{Solution}\label{sec:solution}
We follow the general approach of \cite{brennan-gralla-jacobson2013}, using some techniques from \cite{newman1974,posadas-yanga1985}. The electromagnetic NP scalars are defined as \cite{newman-penrose1962}
\begin{subequations}\label{NPdef}
\begin{align}
\phi_0=&F_{ab} \ell^a m^b \\
\phi_1=&\tfrac{1}{2} F_{ab}(\ell^a n^b +\bar{m}^a m^b)\\
\phi_2=&F_{ab} \bar{m}^a n^b.
\end{align}
\end{subequations}
We assume that the current is along the null congruence, $J^a = \mathcal{J} \ell^a$ with $\mathcal{J}\neq 0$. The force-free condition $F_{ab}J^b=0$ then becomes $F_{ab} \ell^b=0$, or equivalently
\begin{equation}\label{FFphi}
\phi_0=0, \qquad \textrm{Re}[\phi_1]=0.
\end{equation}
Using Eqs.~\eqref{FFphi}, \eqref{spin-coefficients} and \eqref{tetrad} in the spin-coefficient version of Maxwell's equations \cite{teukolsky1973}, we find
\begin{gather}
\left( \partial_r + \frac{2}{r} \right) \phi_1 = 0, \quad \chi \partial_\zeta \phi_1 = 0, \label{espresso}\\
\left( \partial_r + \frac{1}{r} \right) \phi_2 - \chi \partial_{\bar{\zeta}} \phi_1 = 0, \label{americano}\\
\chi^2 \partial_\zeta\left(\frac{\phi_2}{\chi}\right) - \left( \partial_u - \frac{1}{2} \left(1-2r\frac{\dot{v}}{v} \right) \partial_r - \frac{1}{r} \right) \phi_1 = 2 \pi \mathcal{J},\label{tea}
\end{gather}
where $\chi=(v P)/(\sqrt{2} r)$.
Eqs.~\eqref{espresso}, together with the fact that $\phi_1$ is pure imaginary, imply that $\phi_1=i q(u)/(2r^2)$ for a real function $q(u)$, where the factor of $1/2$ is convenient. This function must be independent of angles $(\zeta,\bar{\zeta})$ to satisfy the equations locally, but our application to split monopole magnetospheres requires us to allow different local solutions to be patched together, so that $q(u)$ becomes a piecewise-constant function on the sphere. We will write $q(u;\mathcal{D})$ to remind the reader that this function may take different constant values on different domains $\mathcal{D}$ of the sphere,
\begin{equation}\label{phi1}
\phi_1 = \frac{i q(u;\mathcal{D})}{2r^2}.
\end{equation}
The magnetic monopole charge $Q$ of a field configuration may be defined as $1/(4\pi)$ times the magnetic flux through a closed surface. (For regular fields satisfying Maxwell's equations such an integral is always zero, but our fields are singular on the worldline.) From Eqs.~\eqref{NPdef} together with the fact that $m$ and $\bar{m}$ span rest frame spheres $\mathcal{S}$, we see that
\begin{equation}\label{qtot}
Q = \frac{1}{4\pi} \int_{\mathcal{S}} F = \int q(u;\mathcal{D}) \ d\Omega,
\end{equation}
where the first statement views $F$ as a two-form. A solution with non-zero $Q$ cannot be realized physically, since it would require the presence of a magnetic monopole charge on the worldline. (In the picture of matching the solution to a pulsar interior, the matching could only succeed if the pulsar contained magnetic monopoles.) Thus for physical solutions we require that
\begin{equation}\label{nomonopole}
\int q(u;\mathcal{D}) \ d\Omega = 0.
\end{equation}
If we regard $q(u;\mathcal{D})$ as the effective local monopole charge, this statement means that the total effective monopole charge must vanish.
Note that we work locally on each domain $\mathcal{D}$, so that $\eth q=0$. Requiring Eqs.~\eqref{espresso}-\eqref{tea} to also be satisfied at the domain boundaries (i.e., in a distributional sense on spacetime) would enforce a strictly null, radial, current even on any current sheets. This is too restrictive for the application to pulsar magnetospheres. Instead, one must allow (non-force-free) charge-current to flow in the sheets. We defer the specific selection of appropriate domains to the task of constructing a detailed model of an outer magnetosphere.
Plugging Eq.~\eqref{phi1} into Eq.~\eqref{americano}, we find that
\begin{equation}\label{phi2f}
\phi_2=\frac{f(u,\zeta,\bar{\zeta})}{r}
\end{equation}
for a complex function $f(u,\zeta,\bar{\zeta})$. Eq.~\eqref{tea} then yields
\begin{equation}\label{cake}
\frac{1}{\sqrt{2}}\eth\left(\frac{f}{v}\right) - \frac{iq}{2}\partial_u\left(\frac{1}{v^2}\right) = 2 \pi \frac{r^2}{v^{2}} \mathcal{J} + \frac{i \dot{q}}{2 v^2},
\end{equation}
where we use the $\eth$ operator introduced in Eq.~\eqref{eth}, and $f/v$ has spin-weight $-1$. Using the identity \eqref{videntity} we may express Eq.~\eqref{cake} as
\begin{equation}\label{torte}
\eth\left(\frac{f}{\sqrt{2}v} - \frac{iq}{2} \bar{\eth} \frac{\dot{v}}{v} \right) = 2 \pi \frac{r^2}{v^{2}} \mathcal{J} + \frac{i \dot{q}}{2 v^2}.
\end{equation}
Since the term in parentheses is a spin $-1$ function of $(u,\zeta,\bar{\zeta})$, it may be expressed as $\tfrac{1}{2} \bar{\eth} \psi$ for a spin-zero function $\psi(u,\zeta,\bar{\zeta})$ (with a convenient factor of $\tfrac{1}{2}$),
\begin{equation}\label{cupcake}
\bar{\eth} \psi = \sqrt{2}\frac{f}{v} - iq \ \bar{\eth} \left(\frac{\dot{v}}{v}\right).
\end{equation}
Then Eq.~\eqref{torte} becomes
\begin{equation}\label{pie}
\Delta \psi = 4 \pi \frac{r^2}{v^{2}} \mathcal{J} + \frac{i \dot{q}}{2 v^2},
\end{equation}
where $\eth \bar{\eth}=\Delta$ is the sphere Laplacian.
To solve Eq.~\eqref{pie} we use the identity \eqref{videntity} and split $\psi$ into real and imaginary parts $\psi=\psi^R+i\psi^I$, finding
\begin{equation}\label{psiR}
\Delta \psi^R = 4\pi \frac{r^2}{v^2} \mathcal{J}
\end{equation}
and
\begin{equation}\label{psiI}
\Delta (\psi^I - \dot{q} \log v) = \dot{q}.
\end{equation}
Eq.~\eqref{psiR} provides no constraint on $\psi^R$, since we allow the current $\mathcal{J}$ to take on whatever value is set by the solution of Eqs.~\eqref{FFE}. On the other hand, Eq.~\eqref{psiI} should be solved for $\psi^I$. This equation asks for a function on the sphere whose Laplacian is constant. There is no such globally regular solution, but using $\Delta=P^2 \partial_\zeta \partial_{\bar{\zeta}}$ it is easy to see that the general solution is
\begin{equation}\label{weirdo}
\psi^I - \dot{q} \log v = \dot{q} \log P + \alpha(\zeta) + \beta(\bar{\zeta})
\end{equation}
where $\alpha$ and $\beta$ are free functions representing the freedom of adding homogeneous solutions. However, since only $\bar{\eth} \psi$ appears in the field strength (see Eqs.~\eqref{cupcake} and \eqref{phi2f}), we may set $\alpha=0$ without loss of generality. But the constraint that $\psi^I$ be real then implies that $\beta$ is constant (recall $\zeta=e^{i \phi} \cot\tfrac{\theta}{2})$, in which case we may also set $\beta=0$. In this case combining Eqs.~\eqref{phi2f}, \eqref{cupcake}, and \eqref{weirdo} yields
\begin{equation}\label{phi2weirdo}
\phi_2 = \frac{1}{\sqrt{2}} \frac{v}{r} \bar{\eth}\left( \psi^R + iq \frac{\dot{v}}{v} + i \dot{q} \log(P v) \right).
\end{equation}
While the last term in Eq.~\eqref{phi2weirdo} diverges at the ``south pole'' $\zeta,\bar{\zeta} \rightarrow \infty$, the choice of this pole is arbitrary (picked out here by working in a particular set of coordinates), and a regular solution could be constructed by taking regular portions of this term on each domain $\mathcal{D}$. We will discuss this type of construction in a future paper. For this paper we set $\dot{q}=0$.
Setting $\dot{q}=0$ and collecting everything together, our solution for the NP scalars is
\begin{align}
\phi_0 & = 0 \label{phi0soln} \\
\phi_1 & = \frac{i q}{2r^2} \label{phi1soln} \\
\phi_2 & = \frac{1}{\sqrt{2}} \frac{v}{r} \bar{\eth}\left( \psi + iq \frac{\dot{v}}{v} \right), \label{phi2soln}
\end{align}
where $\psi(u,\zeta,\bar{\zeta})$ is a free real function on the sphere cross time, and $q(\mathcal{D})$ is a piecewise constant function on the sphere. The domains $\mathcal{D}$ may in general change with time $u$, but the value of the $q$ within each domain must remain constant. While every choice of $\psi$ and $q(\mathcal{D})$ gives rise to a solution away from the domain boundaries, ensuring that the field satisfies appropriate junction conditions at the domains (i.e., that the boundaries do not host magnetic monopole sources) will restrict the choice.
When $\psi$ vanishes (or is constant), Eqs.~\eqref{phi0soln}-\eqref{phi2soln} are the spin coefficient form of the point monopole field \cite{newman1974,posadas-yanga1985}, locally in each domain $\mathcal{D}$. To see that the $\psi$ term gives the correction listed in Eq.~\eqref{soln}, contract $\ell_{[a} \nabla_{b]} \psi$ with tetrad vectors to compute the associated NP scalars, using the formulae \eqref{tetrad} for our tetrad.
For a field with $\phi_0=0$, the quadratic invariants are given by $*F^{ab}F_{ab}=-4\textrm{Im}[\phi_1^2]$ and $F^{ab}F_{ab}=-8\textrm{Re}[\phi_1^2]$ \cite{brennan-gralla-jacobson2013}. Thus our solutions have $*F^{ab}F_{ab}=0$ (which is true of any non-vacuum solution of Eqs.~\eqref{FFE}) and $F^{ab}F_{ab}=2q^2/r^4$. In particular the solutions are magnetically dominated ($F^{ab}F_{ab}>0$) when the monopole charge is non-vanishing, and otherwise null ($F^{ab}F_{ab}=0$).
The charge-current for the solution is given by Eq.~\eqref{pie} with $\dot{q}=0$,
\begin{equation}\label{Jsoln}
\mathcal{J} = \frac{1}{4\pi} \frac{v^2}{r^2} \Delta \psi.
\end{equation}
Integrating Eq.~\eqref{Jsoln} with respect to the sphere element $d\Omega$ shows that $\int \mathcal{J} v^{-2}d\Omega=0$, or equivalently $\int J^a dS_a=0$, where $dS_{\mu}=r^2/v^2 R_\mu d\Omega$ is the oriented area element on rest frame spheres. Thus no three-current flows through any such sphere, and, since $J^a$ is null, the net charge on the sphere vanishes. In particular, the worldline does not act as a source or sink of current, and the magnetosphere is charge-neutral overall.
\section{Power}\label{sec:power}
We now compute the power radiated by Eqs.~\eqref{phi0soln}-\eqref{phi2soln}, making the additional assumption that the magnitude of $q(\mathcal{D})$ is the same in each domain. Since power is Lorentz-invariant, we may use the frame defined by the four-velocity at each retarded time. The flux through a large rest frame sphere (sphere at future null infinity) is
\begin{equation}
\mathcal{P}(u) = \lim_{r \rightarrow \infty} \int T_{ab} u^a R^b \frac{r^2}{v^2} d\Omega = \frac{1}{2\pi} \lim_{r \rightarrow \infty} \int |\phi_2|^2 \frac{r^2}{v^2} d\Omega.\nonumber
\end{equation}
For the second equality we have used the NP form of the electromagnetic stress-tensor $T_{ab}$ \cite{teukolsky1973}, the formulas $u^a = n^a + \tfrac{1}{2}\ell^a$ and $R^a=n^a-\tfrac{1}{2} \ell^a$, and the facts that $\phi_0=0$ and $\phi_1=O(1/r^2)$ for our solution. Using the explicit formula \eqref{phi2soln} yields
\begin{align}
\mathcal{P}(u) & = \frac{1}{4\pi} \int \eth \left( \psi - i q \frac{\dot{v}}{v} \right) \bar{\eth} \left( \psi + i q \frac{\dot{v}}{v} \right) d\Omega \nonumber \\
& = \frac{1}{4\pi} \int \Bigg\{ \eth \psi \ \bar{\eth} \psi + q^2 \eth \left( \frac{\dot{v}}{v}\right) \bar{\eth} \left( \frac{\dot{v}}{v}\right) \nonumber \\
& \qquad \quad + i q \left[ \bar{\eth} \left( \frac{\dot{v}}{v} \eth \psi \right) - \eth \left( \frac{\dot{v}}{v} \bar{\eth} \psi \right) \right] \Bigg\} d\Omega \label{oliphaunt}
\end{align}
where we have used the Leibniz rule and the fact that $\eth$ and $\bar{\eth}$ commute on spin-zero functions.
We call the first term in Eq.~\eqref{oliphaunt} the pulsar power. To rewrite this term covariantly, note that the non-zero components of the inverse metric are $g^{ur}=1$, $g^{rr}=-1+2r(\dot{v}/v)$, and $g^{\zeta \bar{\zeta}}=-(P^2 v^2)/(2 r^2)$. Furthermore, the area element on the surface of integration is $dS=-r^2/v^2 d\Omega$. We then have $\eth \psi \bar{\eth} \psi d\Omega=P^2 \partial_\zeta \psi \partial_{\bar{\zeta}} \psi (-v^2/r^2) dS = g^{\mu \nu} \nabla_\mu \psi \nabla_\nu \psi dS$. This latter expression is covariant, and we have
\begin{equation}\label{Ppulsar}
\mathcal{P}_{\textrm{pulsar}}(u) = \frac{1}{4\pi} \int \nabla_a \psi \nabla^a \psi \ \! dS,
\end{equation}
deriving the second term in Eq.~\eqref{power}. While the integral arose on a large sphere, it is in fact independent of the sphere radius.
We call the second term in Eq.~\eqref{oliphaunt} the acceleration power. To evaluate this term we use Eq.~\eqref{larmorsavior}. Since $q^2$ is assumed constant over the sphere, $q^2 \tfrac{1}{6} \eth \bar{\eth} (\dot{v}/v)$ is a total derivative and does not contribute. We are then left with
\begin{equation}\label{Paccel}
\mathcal{P}_{\textrm{accel.}}(u) = \frac{1}{4\pi} \int \left(- \frac{2}{3} q^2 \frac{a^a a_a}{v^2} \right) \ d\Omega = - \frac{2}{3} q^2 a^a a_a,
\end{equation}
where we pull the constants $q^2$ and $a_a a^a$ out of the integral and then use Eq.~\eqref{videntity}, throwing away the total derivative. This derives the first term in Eq.~\eqref{power}.
We call the remaining contribution to Eq.~\eqref{oliphaunt} the sheet power. Since $\bar{\eth}(A \eth B)-\eth(A \bar{\eth}B)$ is equal to $P^2 [ \partial_{\bar{\zeta}}(A \partial_\zeta B) - \partial_\zeta(A \partial_{\bar{\zeta}} B)]$ for spin-zero functions $A$ and $B$, we identify this contribution (last line of Eq.~\eqref{oliphaunt}) as the integral of the two-form $-2 q d(\tfrac{\dot{v}}{v}d\psi)$, where $d$ is the exterior derivative. Then by Stokes theorem on each domain and $\dot{v}/v=a^a \ell_a$ (Eq.~\eqref{aell}), we have
\begin{equation}\label{Psheet}
\mathcal{P}_{\textrm{sheet}}(u) = \frac{-1}{2\pi} \int_S d\left( q \frac{\dot{v}}{v} d\psi \right) = \frac{-1}{2\pi}\int_{C_{\pm}} q \ \! a^a \ell_a d \psi.
\end{equation}
Here $C$ represents the oriented curve(s) on the sphere present at the boundary between domains $\mathcal{D}$. The notation $C_{\pm}$ indicates that an integral is to be performed using the limiting value of the integrand from either side of the curve, with opposite orientations on opposite sides, as required by Stokes' theorem.
The integral arises for a large sphere, but since neither the domains $\mathcal{D}$ nor the integrand $q a^a \ell_a d\psi$ depend on $r$, the integral does not depend on the radius of the sphere. The curve $C$ may be characterized in an invariant manner as the intersection between the current sheet, the light cone at time $u$, and a spacelike plane orthogonal to the four-velocity. The choice of spacelike plane corresponds to the radius $r$, and the above properties ensure that the integral is independent of this choice. As discussed in the introduction, we expect the contributions from $C_+$ and $C_-$ to cancel for the current sheets that arise in pulsar magnetospheres, so that this term makes no contribution to the energy flux.
\section{Astrophysical Applicability}\label{sec:astro}
Our exact solutions involve the idealization that the force-free plasma fills all of space. In reality, the force-free magnetosphere of a compact object will only extend a finite distance. (This distance is hard to estimate, since the force-free description does not include information on the particle density.) To apply our solutions and their predicted scaling \eqref{Paccel-intro}, the force-free magnetosphere should extend for at least several characteristic lengths of the trajectory. In the case of a comparable mass binary, this characteristic length is the orbital radius.
For a pulsar member of a comparable mass Newtonian binary in circular orbit, Eq.~\eqref{Paccel-intro} becomes
\begin{equation}
\mathcal{P}_{\textrm{accel.}} \approx 10^{36} \frac{B_{12}{}^2 M_{1.4}{}^2 R_{10}{}^6}{P^2D_{10}{}^4} \textrm{erg/s},
\end{equation}
where $B_{12} \times 10^{12} \textrm{ Gauss}$ is the surface magnetic field strength, $R_{10} \times 10 \textrm{ km}$ is the stellar radius, $M_{1.4} \times 1.4 \ M_{\odot}$ is the stellar mass, $P \times 1 \textrm{ s}$ is the rotational period, and $D_{10} \times 10 \textrm{ km}$ is the orbital separation. The relative strength of the acceleration power $\mu^2 \Omega^2 a^2$ to the pulsar power $\mu^2 \Omega^4$ is
\begin{equation}\label{Pratio}
\frac{\mathcal{P}_{\textrm{accel.}}}{\mathcal{P}_{\textrm{pulsar}}} \sim \frac{a^2}{\Omega^2} \approx 10^6 \frac{M_{1.4}{}^2P^2}{D_{10}{}^4} \approx 10^{-5} \left( \frac{M_{1.4} P^3}{P_{\textrm{orb.}}^4}\right)^{2/3},
\end{equation}
where $P_{\textrm{orb.}}$ is the orbital period of the binary in seconds.
The energy lost due to the acceleration will come at the expense of some combination of the rotational (spin) and translational (orbital) kinetic energy of the body. Any orbital energy decrease will be undetectably small, since $\mathcal{P}_{\textrm{accel.}}$ is vastly subdominant (by a factor of $\sim10^{-30} D_{10}$) to the power in gravitational-wave emission. The effect on spin-down is also small, but may become relevant for binaries near merger: The ratio $\mathcal{P}_{\textrm{accel.}}/\mathcal{P}_{\textrm{pulsar}}$ can range from $\sim 10^{-15}$ for known binary pulsars ($P_{\textrm{orb.}}\sim \textrm{hours}$) all the way to order unity for binaries near merger ($P_{\textrm{orb.}}\sim .01 \textrm{ s}$). Unfortunately, there is little prospect for receiving electromagnetic signals from this pre-merger inspiral period.
\acknowledgements{We thank Ted Jacobson, Constantinos Kalapotharakos and Maura McLaughlin for helpful conversations. S.G. acknowledges support from NASA through the Einstein Fellowship Program, Grant PF1-120082. D.B. was supported in part by the NSF under grant No. PHY-0903572.}
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1311.1122
|
\section{Introduction}
\label{sec:intro}
Modern portfolio theory has shown that investing in certain asset
classes promising higher returns has always been linked with a higher
variability (also called volatility) of those returns, hence resulting
in increased risks for the investor. Hence, it is one of the
main tasks of financial engineering to accurately estimate the
variability of the return of a given asset (or portfolio of
assets) and take this figure into account in various tasks, including
risk management, finding optimal portfolio strategies for a given
risk aversion level, or derivatives pricing.
For instance, properly estimating volatilities is essential
for applying the~\cite{BlaSch73} option pricing method;
another prominent example is the estimation of return distribution
quantiles for computing value-at-risk (VaR) figures, a method which is
typically recommended by international banking regulation authorities,
such as the Basel Committee on Banking Supervision. This approach is
also widely applied in practice in internal risk management systems of
many financial institutions worldwide.
In practice, sufficient statistical information about the past
behavior of an asset's return is often not available, hence one
is commonly forced to use the so-called \emph{square-root-of-time}
rule. Essentially, this rule transforms high-frequency risk estimates
(for instance, gathered with a 1-day period over several years) into a
lower frequency $T$ (like a 1-year period) by multiplying the
volatility by a factor of $\sqrt{T}$. As an illustration, the Basel
regulations recommend to compute the VaR for the ten day regulatory
requirement by estimating a 1-day VaR and by multiplying this value by
$\sqrt{10}$, where the VaR is the value that solves the equation
\begin{equation*}
\varepsilon = \int_{-\infty}^{-\text{VaR}} \hat{f}(r)dr
\end{equation*}
given the density $\hat{f}(r)$ of the
bank's return estimated probability distribution and a confidence
level $\varepsilon$, fixed for instance to $1\%$.
It is well-known that scaling volatilities with the
square-root-of-time is only accurate under a certain number of
assumptions that are typically not observed in practice: according to
~\cite{DanZie06},
returns need to be homoscedastic and conditionally serially
uncorrelated at all leads, an assumption slightly weaker than the one
of independently and identically distributed (iid)
returns. Dan{\'\i}elsson and Zigrand show furthermore that
for the square-root-of-time rule to be correct for all quantiles and
horizons implies the iid property of the zero-means returns, but also
that the returns are normally distributed.
In this paper, we are interested in studying the effects of applying
the square-root-of-time rule on the \emph{semivariance} of a
continuous jump-diffusion process. As a first step, the semivariance
being a downside risk measure, we quickly recall its history and
properties in the following section.
\subsection{Downside Risk and Semivariance}
\label{subsec:downsiderisk}
Downside risk measures have appeared in the context of portfolio
theory in the 1950's, with the development by~\cite{Mar52}
and~\cite{Roy52} of decision-making tools helping to manage
risky investment portfolios. \cite{Mar52} showed how to
exploit the averages, variances and covariances of the return
distributions of assets contained in a portfolio in order to compute
an efficient frontier on which every portfolio either maximizes the
expected return for a given variance (i.e., risk level), or minimizes
the variance for a given expected return. In the scenario of
Markowitz, a utility function, defining the investor's sensitivity to
changing wealth and risk, is used to pick the proper portfolio on the
optimal border.
On his side, \cite{Roy52} was willing to derive a
practical method allowing to determine the best risk-return trade-off;
as he was not convinced that it is feasible to model in practice
the sensitivity to risk of a human being with a utility function, he
chose to assume that an investor would prefer the investment with
the smallest probability of going below a disaster level, or a target
return. Recognizing the wisdom of this claim, \cite{Mar59}
figured out two very important points, namely that only
the downside risk is relevant for an investor, and that return
distributions might be skewed, i.e., not symmetrically distributed, in
practice. In that spirit, Markowitz suggested to use the following
variability measure, that he called a \emph{semivariance}, as it only
takes into account a subset of the return distribution:
\begin{equation}
\label{eq:TSV}
\int_{-\infty}^{\tau} (\tau-r)^2f(r)dr
\end{equation}
where $f(r)$ denotes the density of the
returns probability distribution, $R$ denotes a random variable
distributed according to $f(r)$ and $\tau$ is a return target level. If
$\tau$ is equal to $\mu_R = \int rf(r)dr$, then~\eqref{eq:TSV}
is called the \emph{below-mean} semivariance
of $R$, while if $\tau$ is arbitrary, \eqref{eq:TSV} is
called the \emph{below-target} semivariance of $R$, where $\tau$ is
defined to be the target return. In other words, only the deviations
to the left of the returns distribution average, or a fixed return
target are accounted for in the computations of the
variability. Similarly, the square root of a semivariance is called a
\emph{semideviation}, with analogy to the standard deviation. Note
that for a symmetrical, i.e., non-skewed return distribution, the
variance of a random variable $R$ is equal to twice its below-mean
semivariance.
The~\cite{Sha66} ratio is a measure of the risk-adjusted return
of an asset, a portfolio or an investment strategy, that quantifies
the excess return per unit of deviation; it is defined as
\begin{equation}
\frac{\mathrm{E}[R_\mathsf{A} -
R_\mathsf{B}]}{\sqrt{\mathrm{Var}[R_\mathsf{A} -R_\mathsf{B}]}},
\end{equation}
where $R_\mathsf{A}$ and $R_\mathsf{B}$ are random variables
modeling the returns of assets $\mathsf{A}$ and $\mathsf{B}$,
respectively. A prominent variant of the Sharpe ratio,
called the \emph{Sortino ratio} (see~\cite{SorVDM91}), is relying on
the semideviation instead of the standard deviation of the returns
distribution. It is well-known and easily understood that the Sharpe
and Sortino ratios tend to give very different results for
highly-skewed return distributions.
Finally, we would like to note that the concept of semivariance has
been generalized, resulting in the development of \emph{lower partial
moments} by~\cite{Baw75} and~\cite{Fish77}. Essentially, the square
is replaced by an arbitrary power $a$ that can freely vary:
\begin{equation}
\int_{-\infty}^{\tau} (\tau-r)^af(r)dr
\end{equation}
Varying $a$ might help in modeling the fact that an investor is more
(through larger values of $a$) or less (through smaller values of $a$)
sensitive to risk. In this paper, we have chosen to stick to $a=2$ for
simplicity reasons. In the following, we recall the concepts of
jump-diffusion models.
\subsection{Jump-Diffusion Models}
\label{subsec:jumpmodels}
Jump-diffusion models are continuous-time stochastic processes
introduced in quantitative finance by~\cite{Mer76}, extending
the celebrated work of~\cite{BlaSch73} on option
pricing. These models are a mixture of a standard diffusion process
and a jump process. They are typically used to reproduce stylized
facts observed in asset price dynamics, such as mean-reversion and
jumps. Indeed, modeling an asset price as a standard
Brownian process implies that it is very unlikely that large jumps
over a short period might occur, as it is sometimes the case in real
life, unless for unrealistically large volatility values. Hence,
introducing the concept of jumps allows to take into account those
brutal price variations, which is especially useful when considering
risk management, for instance.
Various specifications have been proposed in the literature and we
refer the reader to~\cite{ConTan04} for an extensive review. In what
follows, we consider first (in~\S\ref{sec:generalization}) the
standard jump-diffusion model with time invariant coefficients,
constant volatility and Gaussian distributed jumps. Later,
in~\S\ref{sec:jump_vol}, we will also consider more elaborated
stochastic processes, involving random jumps in returns and in
volatility.
A basic jump-diffusion stochastic process is a
mixture of a standard Brownian process with constant drift $\mu$ and
volatility $\sigma$ and of a (statistically independent) compound
Poisson process with parameter $\lambda$ and whose jump size is
distributed according to an independent normal law $\mathcal{N}(\mu_Q,
\sigma^2_Q)$. More precisely, this model can be expressed as the
following stochastic differential equation:
\begin{equation}
dX(t) = X(t) (\mu dt + \sigma dW(t) + J(t)dP(t)),
\label{eqn:process}
\end{equation}
where $X(t)$ denotes the process that describes the price of a financial
asset, with $\Pr[X(0) > 0] = 1$, where $\mu \in \mathbb{R}$ is the
process drift coefficient, $\sigma^2> 0$ is the process
variance, $W(t)$ is a standard Wiener process, $P(t)$ is a Poisson
process with constant intensity $\lambda > 0$ and $J (t)$ is the
process generating the jump size, that together with $P(t)$ forms a
compound Poisson process. The solution of the stochastic differential
equation~\eqref{eqn:process} is given by
\begin{equation}
X(t) = X(0) e^{ \left(\mu-\frac{\sigma^2}{2}\right)t + \sigma W(t) +
\sum_{k=1}^{P(t)}Q_k},
\label{eqn:process2}
\end{equation}
where $Q_k$ is implicitly defined according to $J (T_k) =
e^{Q_k}-1$, and $T_k$ is the time at which the $k$-th jump of the
Poisson process occurs.
If $P(t) = 0$, the sum is zero by convention. We assume that the $Q_k$
form an independent and identically normally distributed sequence
with mean $\mu_Q$ and variance $\sigma^2_Q$.
The log-return of $X(t)$ over a $t$-period is defined as $Y_t =
\log X(t) - \log X(0)$ and, from~\eqref{eqn:process2}, its dynamic is
given by
\begin{equation}
Y_t = \left(\mu-\frac{\sigma^2}{2}\right) t + \sigma W(t) +
\sum_{k=1}^{P(t)}Q_k,
\label{eqn:logret}
\end{equation}
The distribution of $Y_t$ is an infinite mixture of Gaussian
distributions
\begin{equation*}
\mathcal{N}\left( \left(\mu-0.5\sigma^2\right) t + k\mu_Q, \sigma^2 t
+ k \sigma_Q^2) \right)
\end{equation*}
and has a density function given by
\begin{equation}
f_{Y_t} (y) = \sum_{k=0}^{+\infty} \left( \frac{e^{-\lambda t}
(\lambda t)^k}{k!} \frac{1}{\sqrt{2 \pi (\sigma^2 t + k
\sigma_Q^2)}} e^{-\frac{1}{2}\frac{(y-((\mu-0.5\sigma^2) t +k
\mu_Q )}{\sigma^2 t + k \sigma_Q^2}} \right) .
\label{eqn:dens0}
\end{equation}
\subsection{Contributions and Outline of this Paper}
\label{subsec:outline}
Our contributions in this paper can be summarized as follows: first of
all, we derive in~\S\ref{sec:explicitform} an explicit formula for
computing the semivariance of a standard jump-diffusion process when
the volatility is constant. To the best of our knowledge, it is the
first time that such a formula is
provided. Second, we propose in~\S\ref{sec:generalization} a
generalization of the~\cite{BalTor83,BalTor85} approximation of a
jump-diffusion process. Indeed, the simplification brought by Ball and
Torous is based on the fact that, during a sufficiently short time
period, and assuming a small jump intensity parameter, only a single
jump can occur.
By doing so, the authors want to capture large and infrequent events
as opposed to frequent but small jumps. However, our analysis
in~\S\ref{sec:app} shows that limiting the jump intensity parameter
may result in an underestimation of the risk; hence, from a risk
management perspective, this approach does not seem to be
appropriate. It is the reason why we have preferred not to impose any
arbitrary condition on the Poisson
process intensity parameter and to estimate it by the maximum
likelihood method. Our extension of the work of Ball and Torous also
implies that more that one jump may occur during a single day. This is
a consequence of the fact that, when the intensity
parameter is sufficiently large, the probability of obtaining
more than one jump is then not negligible anymore. The only remaining
constraint that we keep in our approach is the fact that $\lambda$
should be smaller than a (large) upper bound. However, we show
in~\S\ref{subsec:charac} that this constraint is actually not a strong
limitation. Third, we apply our results in~\S\ref{sec:app} to compute
an estimation of the semivariance based on the Barclays US High Yield
Index returns, showing that the standard square-root-of-time rule
indeed may underestimate risk in certain periods and overestimate it
in other ones. For this, we make use of a customized optimization
algorithm based on differential evolution to maximize a likelihood
function. Last but not least, we discuss in~\S\ref{sec:jump_vol} the
extension of our work to a jump diffusion model with jumps in returns
and in volatility. Therein, we first recall the importance of
considering random jumps in returns and in volatility. Then, we
describe the stochastic volatility model that we use, which is an
extension of the model proposed by~\cite{EraJohPol03}. The statistical
estimation of its parameters is
addressed in the next section. We use in particular Markov Chain
Monte Carlo (MCMC) methods to
derive their values. We propose in~\S\ref{sub:semi} a method to
compute an annualized semideviation once the model parameters have
been determined. Finally, we present some experimental results.
\section{An Explicit Form for the Semivariance of a Jump-Diffusion
Process}
\label{sec:explicitform}
We derive in this section an explicit formula for the semivariance
of a standard jump-diffusion model with time invariant coefficients,
constant volatility and Gaussian distributed jumps. To the best of our
knowledge, this is the first time that an explicit formula is provided
for computing the semivariance.
In the following, let us denote respectively by $\phi(x)$ and
$\Phi(x)$ the probability density function and the cumulative
distribution function of a standard normal distribution
$\mathcal{N}(0, 1)$ with mean $0$ and variance $1$, i.e.,
\begin{equation*}
\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \text{ and }
\Phi(x) = \int_{-\infty}^{x} \phi(t)\,dt \text{ for } x \in
\mathbb{R}.
\end{equation*}
\begin{theorem}
\label{the:semivariance}
The semivariance of the density~\eqref{eqn:dens0} is given by
\begin{equation}
\sum_{k=0}^{+\infty} \frac{e^{-\lambda t} (\lambda t)^k}{k!} \left(
(D-\mu_k)^2 \Phi(D_k) + \sigma_k (D-\mu_k) f(D_k) +\sigma_k^2
\Phi(D_k) \right) ,
\label{eqn:semivariance}
\end{equation}
where $\mu_k = \left(\mu - \frac{\sigma^2}{2}\right)t + k \mu_Q$,
$\sigma_k^2 = \sigma^2 t + k \mu_Q^2$, $D_k =
\frac{D-\mu_k}{\sigma_k}$ and $k = 0,...,+\infty$.
\end{theorem}
The proof is given in Appendix~\ref{app:proof}. For a pure diffusion process without jump, the previous
formula~\eqref{eqn:semivariance} simplifies to the following one.
\begin{corollary}
The semivariance of a pure diffusion process with drift $\mu$ and
volatility $\sigma$ is equal to
\begin{equation}
(D-\mu_0)^2 \Phi(D_0) + \sigma_0 (D-\mu_0) f(D_0) +\sigma_0^2 \Phi(D_0),
\label{eqn:pure}
\end{equation}
where $\mu_0 = \left(\mu - \frac{\sigma^2}{2}\right)t$, $\sigma_0^2 =
\sigma^2 t$ and $D_0 = \frac{D-\mu_0}{\sigma_0}$.
\end{corollary}
The proof is a direct consequence of Proposition~\ref{prop:prop0}
given in Appendix~\ref{app:proof} and of the fact
that~\eqref{eqn:logret} can be rewritten as
\begin{equation}
Y_t = \left(\mu-\frac{\sigma^2}{2}\right) t + \sigma W(t)
\end{equation}
when we consider a pure diffusion process.
\section{Generalization of the Ball-Torous Approach}
\label{sec:generalization}
The task of fitting a jump-diffusion model to real-world data is not
as easy at it appears, and this fact has been early recognized, see
for instance~\cite{Bec81} or~\cite{Hon98}. Essentially, the reason
lies in the fact that the likelihood function of an infinite mixture
of distribution can be unbounded, hence resulting in
inconsistencies. However, by making some assumptions about the
parameters of this model, it is possible to accurately estimate
them. In the following, we present the approach
of~\cite{BalTor83,BalTor85}.
Therein, the authors present a simplified version of a jump-diffusion
process by assuming that, if the jumps occurrence rate is small, then
during a sufficiently short time period only a single jump can
occur. Accordingly, for small values of $\lambda \Delta t$, $\Delta
P(t)$ can be approximated by a Bernoulli distribution of parameter
$\lambda\Delta t$, and the density of $\Delta Y(t)$ can then be
written as
\begin{equation}
f_{\Delta Y} (y) = (1 -\lambda \Delta t) f_{\Delta D}(y) + \lambda
\Delta t ( f_{\Delta D} \star f_Q) (y),
\label{eqn:dens}
\end{equation}
where $f_{\Delta D}$ denotes the probability density function of the
diffusion part (including the drift), $f_Q$ the probability density function of the jump
intensity, and $\star$ denotes the convolution operator. As mentioned
in~\S\ref{subsec:jumpmodels}, $f_{\Delta D}$ follows
a normal law with mean $(\mu - \sigma^2/2) \Delta t $ and variance
$\sigma^2 \Delta t$. If $f_Q$ is distributed according to a normal
law statistically independent of the diffusion part, then the
convolution $f_{\Delta D} \star
f_Q$ of $f_{\Delta D}$ and $f_Q$ is normal with mean $(\mu -
\sigma^2/2) \Delta t + \mu_Q$ and variance $\sigma^2 \Delta t +
\sigma^2_Q$.
For a sequence of observed log-returns $\Delta y_1,\dots, \Delta y_T$,
the log-likelihood $\log\mathcal{L}$ of the model parameters
$\bm{\theta} = (\lambda, \mu, \sigma, \mu_Q, \sigma_Q)^\mathsf{T}$
is obtained in a straightforward manner from~\eqref{eqn:dens} as
\begin{equation}
\log\mathcal{L}(\bm{\theta}\,|\,\Delta y_1, ..., \Delta y_T) (y) =
\sum_{t=1}^{T} \log f_{\Delta Y} (\Delta y_t\,|\,\bm{\theta}),
\label{eqn:loglikelihood}
\end{equation}
and the maximum likelihood estimator $\bm{\hat{\theta}}$ is obtained
by maximizing~\eqref{eqn:loglikelihood}. \cite{Kie78} has
shown that there may exist several local minima in such a mixture
setting, a fact that we have also observed in the experimental setup
that is the subject of the next section.
While fitting the Ball-Torous model to real-world data
(see~\S\ref{sec:app} for more details and explanations about our
experimental setup) using~\eqref{eqn:loglikelihood}, we have
figured out that the assumption $\lambda\Delta t \ll 1$ might easily be
violated in practice. Indeed, we have observed on our data that, for
$\Delta t = \frac{1}{252}$ (i.e., $\Delta t$ representing one day in a
252-day trading year), the best obtained estimation for $\lambda$
ranged into the interval $[1, 252]$. This means that the value
$\lambda\Delta t$ was often nearer to $1$ than to $0$, and this
obviously questions the validity in practice of the assumption made
by Ball and Torous, at least in our experimental setup. In the
following, we propose a new methodology revolving around relaxing this
assumption to a milder one, namely that $\lambda\Delta t< 1$, and we
justify its use.
\subsection{Our Methodology}
\label{subsec:our_methodology}
The methodology we describe in this section can be interpreted as an
extension of the work of~\cite{BalTor83}. In this
approach, the authors make the assumption that $\lambda \Delta t$ is
small, or in other words, that the expected number of jumps per
$\Delta t$ period is very small. However, in practice, this assumption
might not always be satisfied, as we observed it on our data. We
propose to relax this assumption and to replace it by the milder one
$\lambda \Delta t < 1$. For $\Delta t = 1/252$, assuming $252$ trading
days in a year, this translates to $\lambda < 252$: concretely, it
means that on average, there is no more than a single jump per day or,
equivalently, no more than 252 jumps per year. We easily agree that
this milder assumption may seem arbitrary at first sight. However, we
show in~\S\ref{subsec:charac} that it is not constraining at all,
since one can easily prove that a jump-diffusion process converges
in distribution to a pure diffusion process for increasing values of
$\lambda$. To summarize, with our methodology, we are able to fit any
jump diffusion without having any strong restriction on $\lambda$,
which is a major improvement in comparison to the work
of~\cite{BalTor83}.
Obviously, the reason why Ball and Torous decided to limit the jump
rate to a small value was to capture the apparition of brutal and rare
events. As a matter of fact, it is clearly more desirable on a
practical point of view to be able to model rare and large downside
market movements than frequent and small ones. This assumption implies
small $\lambda\Delta t$ values, and consequently, it means that the
occurrence of more than a single jump per $\Delta t$ period is
sufficiently unlikely that it can be neglected.
However, we have experimentally figured out that the semivariance
computed on our relaxed model, i.e., allowing also frequent and small
jumps, may result in significantly higher values than on the model
assuming that $\lambda \Delta t$ is small (see
Figure~\ref{graph:sds-jump}). In other words, we observed that the Ball
and Torous model seems to underestimate the downside risk, compared to
our relaxed model, at least on our data set. A direct consequence of
not limiting the jumps occurrence rate $\lambda \Delta$ to a small
values is that the probability of having more than one jump during a
single day may become not negligible when $\lambda$ is large enough.
By allowing more than a single jump per $\Delta t$ period, one can
also take into account the fact that several bad news might influence
the market during a trading day, i.e., during a $\Delta t$
period. Obviously, if $\Delta t$ becomes sufficiently small, maybe as
small as a single second, it would be more difficult to justify in
practice several jumps during a time interval. However, when
discretizing the process in periods as large as a single day, we are
convinced that allowing more than a single jump better reflects the
reality.
Consequently, in the following, we will assume that up to $m \geq 1$
jumps are possible during a time interval of $\Delta t$, where
$m\in\mathbb{N}$ is a \emph{finite} value. The probability
distribution of the possible number $k$ of jumps that
may occur during a time interval $\Delta t$ is then the following:
\begin{equation}
\label{eq:pk_values}
p_k = \frac{e^{-\lambda \Delta t} (\lambda \Delta t)^k}{k!} \text{
for } k = 0,...,m-1 \text{ and }
p_m = 1-\sum_{k=0}^{m-1} p_k.
\end{equation}
The resulting probability distribution, that we denote by
$\tilde{f}_{\Delta Y} (y)$ and which can be compared
to $f_{\Delta Y} (y)$ in~\eqref{eqn:dens}, can be expressed as
\begin{equation*}
\tilde{f}_{\Delta Y} (y) = p_0f_{\Delta D}(y)+\sum_{k=1}^{m} p_k
\left( f_{\Delta D} \star f_{Q^{(k)}}\right) (y),
\end{equation*}
where $f_{Q^{(k)}}$ denotes the convolution of $k$ density functions
$f_Q$.
In what follows, we propose ourselves to look at the approximation
error of replacing a Poisson law by a truncated one. For a random
variable following a Poisson law of parameter $\lambda$, the
probability of obtaining a value strictly larger than $m$ is given by
the following function $f(\lambda)$:
\begin{equation}
\label{eqn:poisson}
f(\lambda) = 1 - \sum_{k=0}^{m} p_k\text{ and } p_k =
\frac{e^{-\lambda} \lambda^k}{k!} \text{ with } k = 0,\dots,m.
\end{equation}
It is easy to see that $f(\lambda)$ is an increasing function in
$\lambda$. Indeed, the derivative of $f(\lambda)$ is given by
\begin{equation*}
f'(\lambda) = - \sum_{k=0}^{m} \frac{d\,p_k}{d\lambda}.
\end{equation*}
Then, we have $\frac{dp_k}{d\lambda} = - p_k +
p_{k-1}$ for $k = 0,\dots,m$, since
\begin{equation*}
\frac{dp_k}{d\lambda} = \frac{d}{d\lambda} \frac{e^{-\lambda}
\lambda^k}{k!} = - \frac{e^{-\lambda}\lambda^{k}}{k!} +
\frac{ke^{-\lambda}\lambda^{k-1}}{k!} = -p_k + p_{k-1},
\end{equation*}
and where $p_{-1}$ is set to $0$ as
\begin{equation*}
\frac{dp_0}{d\lambda} = \frac{d}{d\lambda} e^{-\lambda} =
-e^{-\lambda} = - p_0.
\end{equation*}
Following this observation and telescoping the sum, the derivative
$f'(\lambda)$ can be rewritten as $f'(\lambda) = - \sum_{k=0}^{m}
\frac{dp_k}{d\lambda} = p_m > 0$. Now, if we substitute $\lambda$
by $\lambda \Delta t$ and assuming $\lambda \Delta t < 1$, then we can
observe that the supremum in~\eqref{eqn:poisson} is obtained for $\lambda \Delta t= 1$. We
conclude that, assuming that $\Delta t < 1$, the probability of having
more than $m$ jumps is upper-bounded by the following expression:
\begin{equation}
\sum_{k=m+1}^{+\infty} \frac{e^{-1}}{k!} = 1 - \sum_{k=0}^{m}
\frac{e^{-1}}{k!}
\label{eqn:error}
\end{equation}
Table~\ref{table:bounds} gives a numerical upper bound for the
probability of obtaining values strictly larger than $m$ for different
values of $m$ when $\lambda \Delta t < 1$.
\begin{table}[t]
\begin{center}
\begin{tabular}{|l|c|r|}
\hline
$m$ & Upper Bound\\
\hline
1& 0.264 \\
2& 0.080\\
3& 0.019\\
4& 0.003\\
5& 0.001\\
\hline
\end{tabular}
\end{center}
\caption{Upper bounds for the probabilities of obtaining more than $m$
jumps for a Poisson law with $\lambda\Delta t < 1$.}
\label{table:bounds}
\end{table}
Based on the previous results, we conclude that ``truncating'' the
Poisson random variable combined with our assumption that $\lambda
\Delta t < 1$ gives a very good approximation of the return
distribution. Assuming that $t = n \Delta t$ and that no more than $m$
jumps can happen in a $\Delta t$ interval, we can derive a very accurate
approximation for the semivariance given
in~\eqref{eqn:semivarianceformula} by simply considering the first
$(mn+1)$ terms of~\eqref{eqn:semivariance} given in
Theorem~\ref{the:semivariance}:
\begin{equation}
\sum_{k=0}^{mn} \frac{e^{-\lambda t} (\lambda t)^k}{k!} \left(
(D-\mu_k)^2 \Phi(D_k) + \sigma_k (D-\mu_k) f(D_k) +\sigma_k^2
\Phi(D_k) \right).
\label{eqn:semivariance_trunc}
\end{equation}
For a Poisson process, the expected value
and the variance are equal to $\lambda$ for $t$ = 1 year. In other
words, if $\lambda < 252$ and assuming that $m = 5$, it means that we
consider the first $5 \times 252 + 1 = 1261$ terms of the
expression. Concretely, it means that we are ignoring events occurring
at more than $\frac{1260-252}{\sqrt{252}} \approx 63$ standard
deviations from the average of the Poisson process; a new time, this
implies that the approximation given in~\eqref{eqn:semivariance_trunc}
is an almost exact formula.
\subsection{Discussion on the Assumption about $\lambda$}
\label{subsec:charac}
So far, we assumed that $\lambda \Delta t <
1$, a choice that might appear arbitrary at first sight. The purpose
of this section is to show that this assumption is actually not a
strong limitation. If we assume that $\Delta t = 1/252$ (1 day) and
that $\lambda \Delta t \geq 1$, or equivalently, that $\lambda \geq
252$, then its means that, on average, we have more than 1 jump a
day. In that case, the jump returns are small and often their order of
magnitude is less than the order of magnitude of a typical daily
return. In this particular situation, the effect of the jump returns
is hard to distinguish from the effect of the diffusion part of the
process. In other words, it is difficult to make the distinction between an
abnormal return coming from a jump or from the diffusion process. This
issue becomes even more evident when we have a look at the annualized
return distribution. When $P(t)$ is sufficiently large, then we can
invoke the central limit theorem to approximate $Q =
\sum_{k=1}^{P(t)}Q_k$, as the $Q_k$'s are iid random variables with
finite mean and variance. The mean and the variance of a compound
Poisson distribution derive in a simple way from the laws of total
expectation and of total variance. Formally, let us denote by
$\mathrm{E}_X[X]$ and $\mathrm{Var}[X]$ the expected value and the
variance of a random variable $X$, respectively. Furthermore, let
$\mathrm{E}_{X|Y}[X|Y]$ denote the conditional expectation of the
random variable $X$ conditioned by $Y$. Note that
$\mathrm{E}_{X|Y}[X|Y]$ is a random variable, and therefore, one
can compute its expected value. The law of total expectation tells us
that $E_Y\left[E_{X|Y}\left[X|Y\right]\right] = E_X[X]$, while the law
of total variance is formulated as $\mathrm{Var}\left[Y\right] =
\mathrm{E}_X\left[\mathrm{Var}[Y|X]\right] +
\mathrm{Var}_X\left[\mathrm{E}_{Y|X}[Y|X]\right]$.
Let us remind that in our jump-diffusion process, the amplitude $Q$ of a
single jump is assumed to follow a normal distribution with mean
$\mu_Q$ and standard deviation $\sigma_Q$, respectively, and that the
number of jumps $P$ in a given interval is modeled by a Poisson law
of parameter $\lambda \Delta t$. Finally, let us remind that $Q$ and
$P$ are independent random variables. We have
\begin{equation*}
\mathrm{E}[Q] = \mathrm{E}_P \left[ \mathrm{E}_{Q|P} [Q|P] \right] =
\mathrm{E}_P \left[P\cdot\mathrm{E}_Q [Q] \right] =
\mathrm{E}_P[P]\cdot \mathrm{E}_Q[Q] = \lambda t \mu_Q
\end{equation*}
as well as
\begin{eqnarray*}
\mathrm{Var}[Q] &=& \mathrm{E} \left[ \mathrm{Var}_{Q|P} [Q] \right] +
\mathrm{Var} \left[ \mathrm{E}_{Q|P} [Q] \right]
= \mathrm{E} \left[ P\cdot \mathrm{Var}(Q) \right] + \mathrm{Var}
\left[P\cdot \mathrm{E} [Q] \right] \\
&=& \mathrm{E}[P] \mathrm{Var}[Q] + (\mathrm{E}[Q])^2
\mathrm{Var} [P]
= \lambda t \sigma_Q^2 + \mu_Q^2 \lambda t = \lambda t ( \sigma_Q^2 +
\mu_Q^2)
\end{eqnarray*}
Based on the previous development, we can conclude that the jump
diffusion process converges in distribution to a normal distribution
when $P(t)$ becomes large:
\begin{equation}
Y_t \sim \mathcal{N}\left( (\mu-\frac{\sigma^2}{2}+ \lambda \mu_Q) t, (\sigma^2 +
\lambda(\sigma_Q^2+\mu_Q^2))t \right) .
\label{eqn:logret2}
\end{equation}
In short, the annualized return distribution converges to the return
distribution that we would obtain for a pure diffusion process but
with different drift and volatility parameters. Concretely, when
$\lambda \Delta t$ is larger than 1, then the interest of such a model
is limited and the use of a pure diffusion process is preferable at
least in our context.
\section{Applications}
\label{sec:app}
In order to practically illustrate our results, we have chosen to
focus on the \emph{Barclays US High Yield Bond Index}
between January 3rd, 2007 and July 31st, 2012. We explain in Appendix~\ref{app:NAV}
why we can use a jump-diffusion process to model a bond benchmark index.
The daily log-returns (see Figure~\ref{fig:barclays}) of this index form a good example of a
highly-skewed, long-tailed distribution with a negative sample
skewness of $-1.63$ and a sample kurtosis of $24.08$. An histogram
approximating the probability density function of the log-returns, as
well as a normal law whose parameters are the log-returns sample mean
$\hat{\mu} \approx 0.033\%$ and standard deviation $\hat{\sigma}
\approx 0.42\%$ are depicted in Figure~\ref{fig:histogram}.
The semivariance of the log-returns can easily be computed with help
of~\eqref{eq:TSV}. However, it is not clear at all how we should
proceed to annualize this semivariance, except in one case: if we
assume that the log-returns follow a pure diffusion process, with no
drift, and if the threshold $\tau$ is set to zero, then the variance
of the process increases linearly with time, and the annualized
semivariance is just the daily semivariance multiplied by the square
root of time. However, as soon as we introduce jumps in the stochastic
process, there is no reason to apply this rule anymore. Indeed, we
have derived in~\eqref{eqn:semivariance} an exact formula that
enables us to accurately annualize a daily semivariance into an
annualized semivariance.
The purpose of this section consists in performing a rolling analysis
and to compare the semivariances that we obtain from three
distinct methodologies:
\begin{enumerate}
\item an approach relying on fitting our data to a jump-diffusion
process, and computing an annualized semivariance according to
our generalized Ball-Torous model
and~\eqref{eqn:semivariance_trunc}, which we have
shown to be an almost exact approximation
of~\eqref{eqn:semivariance};
\item an approach relying on fitting our data to a pure diffusion
process (i.e., without jumps), and computing an annualized
semivariance according to~\eqref{eqn:pure};
\item and the square-root-of-time rule.
\end{enumerate}
The objective of this analysis is to experimentally determine which
one of these three methodologies seems to provide the best results. In
our computations, all the semivariances are based on 1 year of
historical data, that is, 252 points, and we perform a rolling
analysis over the whole period. We also have decided to set $\tau=0$
in all of our tests, since the use of the square-root-of-time is
only justified for this particular value.\footnote{Note that the
formulas derived in \S\ref{sec:explicitform} and
\S\ref{sec:generalization} can be applied to any value of $\tau$.}
The semivariance in $t$ is computed as follows: first, we compute the
daily semivariance based on historical data from $t-251$ days to $t$;
then, we transform the daily semivariance to obtain an annualized
semivariance by replacing $t$ by $\Delta t$
in~\eqref{eqn:semivariance_trunc} and~\eqref{eqn:pure},
respectively, or by applying the square-root-of-time rule.
This section is organized as follows: in~\S\ref{subsec:DEA}, we
discuss our parameter fitting procedure, that rely on the use of a
differential evolution algorithm. We first recall the salient features
of this type of optimization procedures, and we describe the
difficulties encountered, mainly in terms of stability of the obtained
results. Then, in~\S\ref{subsec:discussion_results}, we describe and
discuss the experimental results obtained while computing annualized
semivariances according to the three different approaches quickly
described above.
\begin{center}
\begin{figure}
\includegraphics[width=0.9\linewidth]{barclays}
\caption{\label{fig:barclays}Barclays US High Yield Bond Index --
Daily log-returns (in \%) and index value from January 3rd, 2007 and July
31st, 2012.}
\end{figure}
\end{center}
\begin{center}
\begin{figure}
\includegraphics[width=0.9\linewidth]{histogram}
\caption{\label{fig:histogram}Histogram of the daily returns. The
returns show negative sample skewness equal to $-1.63$ and a
sample kurtosis equal to $24.08$. The sample mean $\hat{\mu}$
and standard deviation $\hat{\sigma}^2$ are in \%.}
\end{figure}
\end{center}
\subsection{Optimization with Differential Evolution}
\label{subsec:DEA}
The main challenge that we faced from an optimization point of view is
the fact the objective function~\eqref{eqn:loglikelihood} possesses
several local optima. To overcome this issue, we have decided to use
the optimizer described in~\cite{ArdMul13}. The main motivations
for us to choose it is its easiness to use, its capacity to handle
local optima and its ability to manage constraints about the
parameters. Moreover, it was successfully tested in~\cite{ArdOspGir11b}
on diffusion problems. Finally, this kind of algorithm has proved to
be flexible enough for us to customize it for the rolling analyses
that we are dealing with in this paper. We call the variant we will
describe later \emph{differential evolution algorithm with memory}.
Optimization methods based on Differential Evolution (DE) is a
class of search algorithms introduced by~\cite{StoPri97} that belongs
to the class of evolutionary
algorithms. These algorithms assume no special mathematical property
about the target function, like differentiability for gradient
descents or quasi-Newton methods. It means that DE can handle
non-continuous or very ill-conditioned optimization problems by
searching a very large number of candidate solutions; we note that
this number of candidate solutions can be fitted to the computational
power one has at disposal. The price to pay on this lack of assumption
is that optimization algorithms relying on DE are never guaranteed to
converge towards an optimal solution. However, such meta-heuristics
still seem to deliver good performances on continuous problems, see
e.g.~\cite{PriStoLam05}.
Roughly speaking, DE-based optimization algorithms exploit
bio-inspired operations such as crossover, mutation and selection of
candidate populations in order to optimize an objective function over
several successive generations. Let $f: \mathbb{R}^d\longrightarrow
\mathbb{R}$ denote the $d$-dimensional \emph{objective function} that
must be optimized. For the sake of simplicity, we assume that $f$ has
to be minimized; this is not restrictive, as $f$ can easily be
maximized by minimizing $-f$. The DE algorithm begins with a
population of $v$ initial vectors $\bm{x}_i \in \mathbb{R}^d$, with $1
\leq i \leq v$, that can either be randomly chosen or provided by the
user. For a certain number of rounds, fixed in advance, one iterates
then the following operations: for each vector $\bm{x}_i \in
\mathbb{R}^d$, one selects at random three other vectors $\bm{x}_a$,
$\bm{x}_b$ and $\bm{x}_c$, with $1 \leq a, b, c, i \leq t$ being all
distinct, as well as a random index $1 \leq \rho \leq d$. Then,
the successor $\bm{x}_i^\prime$ of $\bm{x}_i$ is computed as follows:
given a \emph{crossover probability} of $\pi \in [0, 1]$, a
\emph{differential weight} $\omega \in [0, 1]$ and $\bm{x_i} =
(x_{i,1}, \dots, x_{i,d})^\mathsf{T}$, for each $1 \leq j \leq d$, one draws a
number $r_j$ uniformly at random. If $r_j < \pi$, or if $j = \rho$,
then one sets $x_{i,j}^\prime = x_{a, j} + \omega\cdot(x_{b, j} -
x_{c,j})$; otherwise, $x_{i, j}^\prime = x_{i,j}$ keeps
untouched. Once this mutation done, the resulting vector
$\bm{x}^\prime$ is evaluated with respect to the objective function,
and replaces $\bm{x}_i$ in the candidate population if it is
better. Otherwise, $\bm{x}^\prime_i$ is rejected. For more details, we
refer the reader to~\cite{PriStoLam05} and~\cite{StoPri97}.
For practical purposes, several implementations of DE-based
optimization algorithms are currently available, as illustrated by the
web-based list of DE-based software for general purpose optimization
maintained by~\cite{storn-website}. In what follows, we rely on
the package \texttt{DEoptim},
see~\cite{ArdMul13,MulArdGilWinCli11,ArdBouCarMulPet11}, which
implements DE-based optimization in the R language and environment for
statistical computing, see \cite{r-lang}.
For the optimizer, we have used a maximum number of iterations of
$250$, default values for the crossover probability of $\pi = 0.5$ and
for the differential weight $\omega = 0.8$,
as well as an initial population of $v=200$ vectors. We also have
constrained the value of $\lambda$ to the interval $[0,252]$ to be
compliant with the assumption $\lambda \Delta t < 1$. The main
challenge that we have faced for our rolling analysis is the lack of
stability over time of some parameters of the objective function. This
is illustrated in the upper graph of Figure~\ref{graph:stab1}, that
depicts the values computed by the DE-based optimizer for
$\lambda$.\
\begin{center}
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{DEmemory}
\end{center}
\caption{Evolution of $\lambda$ without and with the stabilization
procedure. The stabilization procedure relies on feeding the initial
population with the 50 last solutions, if available.}
\label{graph:stab1}
\end{figure}
\end{center}
We can observe that this lack of stability can be extreme for certain
periods of time. For instance, in the first half of 2010, we can see
that $\lambda$ jumps from a high value near to its upper bound $252$
to a very low value the next date. This is due to the fact that
objective function possesses several optima and that for
each of these solutions, $\lambda$ can be very different.
In order to overcome this issue, we propose to add ``memory''
to the algorithm. Indeed, for each date, we compute a solution that
maximizes the log-likelihood~\eqref{eqn:loglikelihood}. Our idea
consists in storing these solutions and to feed the initial population
with the last 50 solutions of the optimization problem. Concretely, to
determine the solution at a specific date, the initial population --
whose size is $v=200$ -- is fed with the solutions of the
optimization problem at times $t-1,t-2,\ldots,t-50$. There is no reason
why the solution at time $t$ is the same as time $t-1$, except if the
objective function remain the same; however, it is very likely that
the solution at $t-1$ will provide a good starting point for the
current optimization problem if the objective function has changed,
and directly an optimal solution if the objective function has not
changed. With this procedure, we also have the guarantee that the
solution for the date $t$ will not switch from one point to another
with the same log-likelihood value at the next optimization problem of
date $t+1$ if the objective function has not changed. The effect of
this stabilization procedure are illustrated in the lower graph of
Figure~\ref{graph:stab1}. The stabilization procedure is very important
for the interpretation of our results. Without this stabilization
procedure, it would be far more difficult and even impossible to
interpret the results for some periods of time.
\subsection{Discussion of our Results}
\label{subsec:discussion_results}
\begin{center}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{mainresult}
\end{center}
\caption{Comparison of the semideviation (in \%) obtained via the
square-root-of-time rule with the semideviation (in \%) based on a
jump-diffusion process. We can observe that the semideviation
based on the square-root-of-time rule significantly underestimates
the risk in periods of high volatility and overestimates the
risk in periods of lower stress.}
\label{graph:res}
\end{figure}
\end{center}
We start by comparing in Figure~\ref{graph:res} the evolution
of the annualized semideviation, i.e., the square root of the
semivariance, computed thanks to the square-root-of-time rule to the
semideviation computed through fitting a jump-diffusion process to the
data under the constraint $\lambda \leq 252$ on which we applied our
formula given in~\eqref{eqn:semivariance_trunc}. It is quite
striking to see that the risk estimated on the jump-diffusion process
is up to 4 times larger than the one computed thanks to the
square-root-of-time rule. At the beginning of 2009, the first one is
larger than $40\%$ while the other is just below $10\%$. The risk is
not only underestimated in period of crisis, but also seems to be
overestimated when the market rallies or in period of low or medium
stress. Indeed, we can observe that the jump-diffusion semideviation
is almost zero from the end of 2009 to the end of 2011, while the
square-root-of-time semideviation remains at a level around $2\%$
during that period.
\begin{center}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{params-interpretation}
\caption{Time evolution of the jump-diffusion process parameters}
\label{graph:params-interpretation}
\end{center}
\end{figure}
\end{center}
\subsubsection{Relationship between Important Process Parameters}
To better understand how the jump-diffusion semideviation works, it is
very important to understand the relationship between the different
process parameters. Figure~\ref{graph:params-interpretation}
illustrates the evolution through time of some important parameters
of the jump-diffusion process, namely $\lambda$, $\mu$ and $\mu_Q$.
We can observe a direct relationship between $\lambda$ and
$\mu_Q$, namely that when $\lambda$ gets small, then $\mu_Q$ gets
large (in absolute value). This behavior is quite easy to interpret:
in normal market conditions, $\lambda$ is rather high, while $\mu_Q$
is rather low. The combination of these two parameters under normal
market conditions captures a well-known effect in credit portfolios, a
negative asymmetry in their return distribution.
The parameter $\mu$ has an offsetting effect counterbalancing the
compound Poisson process. For example, let us assume that $\lambda =
252$ and that $\mu_Q$ has an average size of 1 basis point. In that
case, the cumulative annual impact of the accidents driven by the
Poisson process is on average a negative return of $252 \times 1 =
252$ basis points. Let us imagine that the annual return that we can
expect from our investment is $7\%$, then it is very likely that the
statistical estimation of $\mu$ would be
around $10\%$ (note that from~\eqref{eqn:logret}, we also have to
take into account the role of $\sigma$ to determine the annual
expected return). In crisis situation, $\lambda$ really captures the
frequency of extreme events and its value typically drops from 252 to
smaller values, sometimes close to 1. By contrast, $\mu$ is by
construction less sensitive to these extreme accidents, even though we
can see that it is impacted, by looking at
Figure~\ref{graph:params-interpretation}. Indeed, we can observe that
$\mu$ has dropped to negative values during the 2008 financial crisis.
It is also quite interesting to have a look at what happened during
the rally phase that started in March 2009. We can observe that $\mu$
has peaked to values larger than $60\%$. It is worth
mentioning that even though the value $\mu$ has been very large
at the end of 2009 due to a strong rally in the market, $\mu_Q$ stayed
negative during that phase. It is only in year 2010 that $\mu_Q$ had
positive values. This situation is quite exceptional, as we are used
to negative skewness in the returns for credit portfolios. However,
under some very particular circumstances, they can exhibit
positive skewness and the values of $\mu_Q$ cannot be interpreted any
more as losses, but as gains.
Note also that the jump-diffusion semideviation can be very close to
zero in very special cases, and in particular when $\mu$ is much
larger than $\lambda \cdot \mu_Q$, when $\mu_Q$ is negative or in
the situation where both $\mu$ and $\mu_Q$ are positive.
This just means that in this kind of situation, the probability of
obtaining a annualized return below zero is very low, resulting in a
semideviation close to zero.
The two worst crisis periods that the financial markets have
experienced during the last five years are, by order of importance,
the 2008 financial crisis and the US Debt Ceiling crisis in
2011. After weeks of negotiation, President Obama signed the Budget
Control Act of 2011 into law on August 2, the date estimated by the
department of the Treasury where the borrowing authority of the US
would be exhausted. Four days later, on August 5, the credit-rating
agency Standard \& Poor's downgraded the credit rating of US
government bond for the first time in the country's history. Markets
around the world experienced their most volatile week since the 2008
financial crisis with the Dow Jones Industrial Average plunging for
635 points (or $5.6\%$) in a single day. However, the yields from the
US Treasuries dropped as investors were more concerned by the European
sovereign-debt crisis and by the economic prospect for the world
economy and fled into the safety of US government bonds. This is a
reason why we could see the jump-diffusion semideviation increasing
substantially during that period.
\subsubsection{Relationship between $\lambda$ and the Semideviation}
It is worth studying the relationship between $\lambda$ and the
semideviation of the jump-diffusion process. In Figure~\ref{graph:sds-jump}, we give the
evolution of the semideviation based on different values of $\lambda$.
\begin{center}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{sds-jump.pdf}
\caption{\label{graph:sds-jump}Evolution of the semideviations
(in \%) for different values of
$\lambda$. The curve in red corresponds to the constraint $\lambda =
252$, in green to $\lambda = 100$, in blue to $\lambda = 50$ and
in violet to $\lambda = 10$. Note that the curves in red and in
green are almost identical.}
\end{center}
\end{figure}
\end{center}
This analysis clearly shows that the assumption made about $\lambda$
has a huge impact on the computation of the semideviation.
In particular, it has been a standard practice since the publication
of the work of~\cite{BalTor83} to use a Bernoulli
jump diffusion process as an approximation of the true diffusion
process by making the assumption that $\lambda \Delta t$ is
small. However, our analysis in Figure~\ref{graph:res} indicates that a
model with a larger $\lambda$ fits best the data. Obviously, the
motivation of Ball and Torous was to capture large and infrequent
events by opposition to small and frequent jumps. However, imposing to
$\lambda \Delta t$ to be small has the undesired consequence of
underestimating the risk in periods of high volatility. Indeed, the
graph in Figure~\ref{graph:sds-jump} shows that the semideviation
obtained with $\lambda = 252$ can be as much as 50 \% larger in a
period of stress than the semideviation computed with the constraint
that $\lambda = 10$. This observation was the main motivation for us to
extend the work of Ball and Torous by allowing models where the
frequency of the jumps is not imposed anymore, but driven by the
data.
In \S\ref{subsec:charac}, we have shown that when $\lambda$ is large,
then the jump-diffusion return distribution should be close to the
distribution of a pure diffusion process but with different
parameters. Rather than superposing the semideviation computed out of
a jump-diffusion process with $\lambda = 252$ with a pure diffusion
process, we have put in Figure~\ref{graph:sds-jump-pure} the
jump-diffusion semideviation and its \emph{difference} with the
semideviation obtained from a pure diffusion process.
\begin{center}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{sds-jump-pure.pdf}
\end{center}
\caption{\label{graph:sds-jump-pure}Comparison between the
semideviation (in \%) obtained from a jump
diffusion process with $\lambda = 252$ with a pure diffusion
process with $\lambda=0$. The blue and green lines represent the
jump-diffusion and pure diffusion semideviations, respectively,
while the red one is the difference between the two.
Even though the magnitude of the semideviations are
similar, we can observe that the difference is substantial
during the 2008 financial crisis. }
\end{figure}
\end{center}
The difference is small, except during the 2008 financial crisis. During
that period, the difference can be larger by several \%. The central
limit theorem tells us that the annualized return distribution should
converge to a normal distribution whose parameters are given
in~\eqref{eqn:logret2}.
But the parameters of the pure diffusion process have been determined
by maximization of the log-likelihood and may give results quite
different from the ones in~\eqref{eqn:logret2}. Moreover, we can see
that the two semideviations are diverging in period of crisis, i.e., when it is
difficult to calibrate the models due to the appearance of extreme
events that make the determination of the parameters of the models
quite challenging.
To illustrate this fact, we have depicted the
evolution of the log-likelihood value for both processes on
Figure~\ref{graph:likelihood}. We have also compared the
jump-diffusion and the Bernoulli jump-diffusion semideviations which
should be the same based on the results
from~\S\ref{subsec:charac}. Our results are displayed on
Figure~\ref{graph:sds-jump-bern}. The difference is negligible over
all the period, meaning that our methodology provides the same results
as the one based on the model of Ball and Torous provided $\lambda$ is
constrained to be small.
\begin{center}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{likelihood}
\caption{\label{graph:likelihood}Comparison of the log-likelihood
for the jump-diffusion
process (in blue) with $\lambda = 252$ and with the pure diffusion
process (in red). In both cases, we can see that the log-likelihood
is smaller in periods of high market stress, which means that the
calibration of the model is also more challenging.}
\end{center}
\end{figure}
\end{center}
\begin{center}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{sds-jump-bern.pdf}
\caption{\label{graph:sds-jump-bern}Comparison between the
semideviation (in \%) obtained from a jump
diffusion process (in blue) and the difference with a Ball and Torous
jump-diffusion (in red). In both cases, we have set $\lambda =
10$. The difference is very small compared to the level of the
semideviation.}
\end{center}
\end{figure}
\end{center}
\subsubsection{Evolution of $\sigma$}
Finally, we compare in Figure~\ref{graph:sigmas} the time
evolution of the volatility $\sigma$ when fitting a jump process and a
pure diffusion process. One can observe that the $\sigma$ obtained for a
pure diffusion process is larger in magnitude than the one obtained
for a jump diffusion process; this comes without any surprise, as in
the second case, the volatility is also captured by the jumps. A
second fact that we would like to outline is that $\sigma$ appears to
be quite unstable for the jump process. This can be interpreted as a
possible misspecification of the model, as the jumps
appear to influence the value of $\sigma$. In other words, assuming a
constant $\sigma$ might be a wrong hypothesis for our data, and this
can be translated as a further motivation to study processes with stochastic volatility including
jumps in returns and in volatility, as we will do it in the next section.
\begin{center}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{sigmas.pdf}
\caption{\label{graph:sigmas}Comparison between the time
evolution of the parameter $\sigma$ obtained when fitting a
jump process (in blue) and a pure
diffusion process (in green).}
\end{center}
\end{figure}
\end{center}
\section{Extension to Stochastic Volatility Models with Jumps in
Returns and Volatility}
\label{sec:jump_vol}
We explain now how to extend the methodology we developed before to
processes involving stochastic volatility with jumps in returns and
volatility. This section is structured as
follows. In~\S\ref{sub:import}, we recall
the importance of considering jumps in returns and in
volatility. Then, in~\S\ref{subsec:eraker}, we describe the stochastic
volatility model with jumps in returns and in volatility that we
consider, which is an extension of the model analyzed by
~\cite{EraJohPol03}. The only difference is
that we have replaced the Bernoulli jump process by the more general
model that we have developed in~\S\ref{sec:generalization}. We
continue by addressing in~\S\ref{sub:prac} the statistical estimation
of the model parameters and of their posterior distributions. We have
followed the approach developed in \cite{NumRen10} and our
implementation is based on a script developed in the R language
provided by the same authors that we have modified
in order to be able to cope with our approach. Once the
parameters have been estimated, we show in~\S\ref{sub:semi} how to
compute an annualized semivariance. In a nutshell, the first step
consists in generating the returns over a 1-year horizon and then to
compute the semivariance thanks to numerical integration. Finally, we
exhibit some experimental results in~\S\ref{stochvol:res}.
\subsection{Importance of Jumps in Returns and Volatility}
\label{sub:import}
Modeling accurately the price of stocks is an important topic in
financial engineering. A major breakthrough was the~\cite{BlaSch73}
model, that however suffers from at least two major
drawbacks. The first one is that the stock price follows a lognormal
distribution and the second one is that its volatility is
constant. Several studies (see~\cite{CheGalGhyTau99}, for instance)
have emphasized that the asset returns' unconditional distributions
show a greater degree of kurtosis than implied by the normality
assumption, and that volatility clustering is present in the data,
suggesting random changes in returns' volatility. An extension of this
model is to integrate large movements in the stock prices making
possible to model financial crashes, like the Black Monday (October
19th, 1987). They were introduced in the form of jump-diffusion
models, see \cite{Mer76,CoxRos76}. An important extension of the
Black-Scholes model was to use stochastic volatility rather than
considering it as constant, see for instance
\cite{HulWhi87,Sco87,Hes93}. \cite{Bat96} and~\cite{Sco97} combined
these two approaches
and introduced the stochastic volatility model with jumps in
returns. Event though their new approach has helped better characterize
the behavior of stock prices, several studies have shown that models
with both diffusive stochastic volatility and jumps in return were
incapable of fully capturing the empirical features of equity index
returns, see for instance~\cite{BakCaoChe97,Bat00,Pan02}. Their main
weakness is that they do not capture well the empirical fact that the
conditional volatility of returns rapidly increases. By adding jumps
in the volatility, then the volatility process is better
specified. \cite{DufPanSin00} were the first
to introduce a model with jumps in both returns and
volatility. \cite{EraJohPol03} have shown
that the new model with jumps in volatility performed better than
previous ones, and resulted in no major misspecification in the
volatility process.
\subsection{Extension of the work of Eraker, Johannes, and Polson}
\label{subsec:eraker}
As mentionned above, \cite{EraJohPol03} consider a jump
diffusion process with jumps in returns and in volatility. These jumps
arrive simultaneously, with the jump sizes being correlated. According
to the model, the logarithm of an asset's price $Y_t = \log(S_t )$
solves
\begin{equation}
\left( \begin{array}{c} dY_t \\ dV_t \end{array} \right) =
\left( \begin{array}{c} \mu \\ \kappa (\theta-V_t^-) \end{array}
\right) dt
+ \sqrt{V_{t^-}} \left( \begin{array}{cc} 1 & 0 \\ \rho \sigma_\nu &
\sqrt{(1-\rho^2)} \sigma_\nu \end{array} \right) d\bm{W}_t
+ \left( \begin{array}{c} \xi^y\\ \xi^\nu \end{array}
\right) dN_t
\label{eqn:stochVolDef}
\end{equation}
where $V_t^− = \lim_{s \uparrow t} V_s$, $\bm{W}_t = ( W_t^{(1)} \
W_t^{(2)})^\mathsf{T}$ is a standard Brownian motion in $\mathbb{R}^2$ and
$(\cdot)^\mathsf{T}$ denotes the transpose of a matrix or a vector. The
jump arrival $N_t$ is a Poisson process with
constant intensity $\lambda$, and this
model assumes that the jump arrivals for the returns and the
volatility are contemporaneous. The variables $\xi^y$ and $\xi^\nu$
denote the jump sizes in returns and volatility, respectively. The
jump size in volatility follows an exponential distribution $\xi^\nu
\sim \exp(\mu_v)$ while the jump sizes in returns and volatility are
correlated with $\xi^y|\xi^\nu \sim \mathcal{N}(\mu_y+\rho_j
\xi^\nu,\sigma_y^2)$.
Their methodology relies on Markov Chain Monte Carlo (MCMC)
methods. The basis for their MCMC estimation is the
time-discretization of~\eqref{eqn:stochVolDef}:
\begin{eqnarray}
Y_{t + \Delta t} - Y_t & = & \mu \Delta t + \sqrt{V_{t } \Delta t}
\varepsilon_{t + \Delta t}^y + \xi_{t + \Delta t}^y J_{t + \Delta t},
\nonumber \\
V_{t + \Delta t} - V_t & = & \kappa(\theta-V_{t}) \Delta t +
\sigma_\nu \sqrt{V_{t} \Delta t} \varepsilon_{t + \Delta t}^\nu +
\xi_{t + \Delta t}^\nu J_{t + \Delta t}, \label{eqn:disc}
\end{eqnarray}
where $J_{t + \Delta t}^k = 1$ indicates a jump
arrival. $\varepsilon_{t + \Delta t}^y$, $\varepsilon_{t + \Delta
t}^\nu$ are standard normal random variables with correlation $\rho$
and $\Delta t$ is the time-discretization interval (i.e., one
day). The jump sizes retain their distributional structure and the
jump times are Bernoulli random variables with constant intensity
$\lambda \Delta t$. The authors apply
then Bayesian techniques to compute the model parameters
$\bm{\Theta}$. The posterior distribution summarizes the sample
information regarding the parameters $\bm{\Theta}$ as well as the
latent volatility, jump times, and jump sizes:
\begin{equation}
\Pr(\bm{\Theta},J, \xi^y,\xi^\nu,V|\bm{Y}) \propto
\Pr(\bm{Y}|\bm{\Theta},J,\xi^y,\xi^\nu,V)
\Pr(\bm{\Theta},J,\xi^y,\xi^\nu,V)
\end{equation}
where $J,\xi^y,\xi^\nu,V$ are vectors containing the time series of
the relevant variables. The posterior combines the likelihood
$\Pr(\bm{Y}|\bm{\Theta},J,\xi^y,\xi^\nu,V)$ and
the prior
$\Pr(\bm{\Theta},J,\xi^y,\xi^\nu,V|\bm{Y})$. As
the posterior distribution is not known in closed form, the MCMC-based
algorithm generates samples by iteratively drawing from the following
conditional posteriors:
\begin{eqnarray*}
\text{Parameters:} & \Pr(\bm{\Theta}_i | \bm{\Theta}_{-i}, J,
\xi^y, \xi^\nu, V, \bm{Y}) &
i = 1,\dots, k \\
\text{Jump times:} & \Pr(J_t = 1 | \bm{\Theta}, \xi^y, \xi^\nu, V, \bm{Y}) & i
= 1,\dots, T \\
\text{Jump sizes:}& \Pr(\xi_t^y | \bm{\Theta}, J_t = 1, \xi^\nu, V, \bm{Y}) & i
= 1,\dots, T \\
& \Pr(\xi_t^\nu | \bm{\Theta}, J_t = 1, V, \bm{Y}) & i = 1,\dots, T \\
\text{Volatility:}& \Pr(V_t| V_{t+\Delta t}, V_{t-\Delta t},\bm{\Theta}, J,
\xi^y, \xi^\nu, \bm{Y}) & i = 1,\dots, T
\end{eqnarray*}
where $\bm{\Theta}_{-i}$ denotes the elements of the parameter vector,
\emph{except} $\bm{\Theta}_i$. Drawing from these distributions is
straightforward, with the exception of the volatility, as its distribution
is not of standard form. To sample from it, the authors use a
random-walk Metropolis algorithm. For $\rho$, they use an independence
Metropolis algorithm with a proposal density centered
at the sample correlation between the Brownian increments. For a
review of these MCMC techniques, we recommend \cite{JohPol09}
where the authors provide a review of the theory behind MCMC
algorithms in the context of continuous-time financial econometrics.
The algorithm of~\cite{EraJohPol03} produces
a set of draws $\left\{\bm{\Theta}^{(g)}, J^{(g)}, \xi^{y(g)},
\xi^{\nu(g)}, V^{(g)}\right\}_{g=1}^G$ which are samples from
$\Pr(\bm{\Theta},J,\xi^y,\xi^\nu,V|\bm{Y})$.
Our approach relies on the methodology developed
in~\cite{EraJohPol03}. The only difference is that we have replaced
the Bernoulli jump process by the jump process described
in~\S\ref{sec:generalization}.
This change is quite straightforward. In~\eqref{eqn:disc}, $J_{t +
\Delta t}$ can take only two values and its posterior is Bernoulli
with parameter $\lambda\Delta t$. To compute the Bernoulli
probability, the authors use the conditional independence of
increments to volatility and returns to get
\begin{eqnarray*}
&\Pr(J_{t+\Delta t} = 1 | V_{t + \Delta t}, V_t, Y_{t + \Delta t},
\xi_{t+\Delta t}^y, \xi_{t+\Delta t}^\nu, \bm{\Theta}) \propto & \\
& \lambda \Delta t \times \Pr(Y_{t + \Delta t}, V_{t + \Delta t} | V_t,
J_{t+\Delta t} = 1, \xi_{t+\Delta t}^y , \xi_{t+\Delta t}^\nu,
\bm{\Theta}).&
\end{eqnarray*}
Our generalization is straightforward. Following the approach
explained in~\S\ref{sec:generalization}, $J_{t + \Delta t}$ can take
any value between 0 and $m$. Then
\begin{eqnarray*}
&\Pr(J_{t+\Delta t} = m | V_{t + \Delta t}, V_t, Y_{t + \Delta t},
\xi_{t+\Delta t}^y, \xi_{t+\Delta t}^\nu, \bm{\Theta}) \propto & \\
& p_k \times \Pr(Y_{t + \Delta t}, V_{t + \Delta t} | V_t, J_{t+\Delta
t} = m, \xi_{t+\Delta t}^y , \xi_{t+\Delta t}^\nu, \bm{\Theta}),&
\end{eqnarray*}
with $p_k = \frac{e^{-\lambda \Delta t} (\lambda \Delta t)^k}{k!}$ for $k =
0,\dots, m-1$ and with $p_m = \sum_{k=0}^{m-1}p_k$.
\subsection{Parameters Estimation}
\label{sub:prac}
We have followed the approach developed by~\cite{NumRen10} for the
estimation of the model parameters. These
authors describe an implementation in R of the
methodology exposed by~\cite{EraJohPol03} and apply it on FTSE~100
daily returns. We assume that we have daily data at times $t_i$ for $i
= 1,\dots,T$ and
with $t_{i+1}-t_i = \Delta t = 1\text{ day}$. The bivariate density
function under consideration is given by
\begin{equation}
f(\bm{B}_{t_i}) = \frac{1}{2 \pi | \bm{\Sigma} | ^{ 0.5}} \exp \left(
-\frac{1}{2}(\bm{B}_{t_i}-\mathrm{E}[\bm{B}_{t_i}])^\mathsf{T} \bm{\Sigma}_{t_i}^{-1}
(\bm{B}_{t_i}-\mathrm{E}[\bm{B}_{t_i}]) \right),
\label{eqn:bi}
\end{equation}
where $|\cdot|$ denotes the determinant of matrix. The likelihood function is
simply given by $\prod_{i=1}^n f(\bm{B}_{t_i})$ with
\begin{equation}
\bm{B}_{t_i} = \left( \begin{array}{c} \Delta y_{t_i} \\ \Delta
\nu_{t_i} \end{array} \right)
\end{equation}
\begin{equation}
\mathrm{E}[\bm{B}_{t_i}] = \left( \begin{array}{c} \mu + \xi_{t_i}^y \xi
J_{t_i} \\ \kappa(\theta - V_{t_{i-1}}) + \xi_{t_i}^\nu
J_{t_i} \end{array} \right) \\
\end{equation}
\begin{equation}
\bm{\Sigma}_{t_i} = \mathrm{Cov}[\Delta y_{t_i}, \Delta \nu_{t_i}] =
\left( \begin{array}{cc} V_{t_{i-1}} & \rho \sigma_\nu V_{t_{i-1}}
\\
\rho \sigma_\nu V_{t_{i-1}} & \sigma_\nu^2 V_{t_{i-1}}
\end{array} \right) \\
\end{equation}
where $\Delta y_{t_i} = Y_{t_i} - Y_{t_{i-1}}$ and $\Delta
\nu_{t_i}=V_{t_i} - V_{t_{i-1}}$. The joint distribution is given by
the product of the likelihood times the distributions of the state
variables times the priors of the parameters, more specifically:
\begin{eqnarray*}
&&\left[ \prod_{i=1}^T f(\bm{B}_{t_i}) \right] \times \left[
\prod_{i=1}^T
f(\xi_{t_i}^y) \times f(\xi_{t_i}^\nu)\times f(J_{t_i}) \right] \\
&& \times \left[ f(\mu) \times f(\kappa) \times f(\theta) \times
f(\rho) \times f(\sigma_\nu^2) \times f(\mu_y) \times f(\rho_J)
\times f(\sigma_y^2) \right] \\
&& \times \left[ f(\mu_{\nu}) \times f(\lambda) \right]
\end{eqnarray*}
The distributions of the state variables are respectively given by
$\xi_{t_i}^\nu\sim \exp(\mu_\nu)$, $\xi_{t_i}^y \sim \mathcal{N}(\mu_y
+ \rho_J \xi_{t_i}^\nu$, $\sigma_y^2)$, and $J_{t_i} \sim
\mathcal{B}(\lambda)$. In~\cite{NumRen10}, the authors impose
little information through priors, that follow the same distributions
than in~\cite{EraJohPol03}: $\mu \sim \mathcal{N}(0,1)$, $\kappa \sim
\mathcal{N}(0,1)$, $\theta \sim \mathcal{N}(0,1)$, $\rho \sim
\mathcal{U}(-1,1)$, $\sigma_\nu^2 \sim\mathcal{IG}(2.5, 0.1)$, $\mu_y
\sim \mathcal{N}(0,100)$, $\rho_J \sim \mathcal{N}(0,1)$, $\sigma_y^2 \sim
\mathcal{IG}(5,20)$, $\mu_\nu\sim \mathcal{G}(20,10)$, and $\lambda \sim
\beta(2,40)$.\footnote{Here,
$\mathcal{B}(\cdot)$, $\mathcal{G}(\cdot, \cdot)$,
$\mathcal{IG}(\cdot, \cdot)$,
$\mathcal{U}(\cdot, \cdot)$, $\beta(\cdot, \cdot)$ denote the
Bernoulli, gamma, inverse gamma, uniform and beta distributions,
respectively.} Following our
approach, we only made the following change: we have relaxed the
assumption about the Bernoulli process to consider the more general
process proposed in~\S\ref{sec:generalization}. It may be
also tempting to change the assumptions about the distribution of
$\lambda$, but the $\beta$ distribution is
so flexible that it can cope with a wide variety of assumptions about
these parameters. For example, the $\beta(1,1)$ gives a uniform
variable which can be used if we want to have a uninformative
distribution.
It is important to store the sampled parameters and the vector of
state variables at each iteration. The mean of each parameter over the
number of iterations gives us the parameter estimate. The convergence
is checked using trace plots showing the history of the chain for each
parameter. ACF plots is used to analyze the correlation structure of
draws. Finally, the quality of the fit may be assessed through the
mean squared errors and the normal QQ plot. The standardized error is
defined as follows:
$$ \frac{Y_{t+\Delta t}-Y_t - \mu \Delta t - \xi_{t+\Delta t}^y
J_{t+\Delta t}^y}{\sqrt{V_{t+\Delta t} \Delta t}}=
\varepsilon_{t+\Delta t}^y.$$
\subsection{Computing the Semideviation}
\label{sub:semi}
Once we have managed to estimate the parameters of the model
on daily data, we still have to compute an annualized
semideviation. The first step consists in simulating the returns over
the $\tau$ horizon, where $\tau$ is typically 1-year if we are
interested in computing an annualized semideviation. We start by
simulating a Poisson process $N_{\tau}$ with jump intensity
$\lambda$. The output of this simulation consists of the number $N$ of
jumps occurring between times 0 and $\tau$, in addition to the times $0
\leq j_1 \leq \dots \leq j_N \leq \tau$ at which these jumps occur. For
each interval $[j_{i-1}, j_i]$, we simulate the diffusion parts of the
return and volatility processes according to an Euler discretization
schema for the Heston model, this procedure being explained later in
the section. This gives us preliminary values $Y_{j_i}^-$ and
$V_{j_i}^-$ for the return and variance processes at time $j_i$. Next,
we simulate jump sizes for the jumps in the two processes. These jumps
are given by $\xi_{j_i}^{y}$ and $\xi_{j_i}^{\nu}$, respectively. The
final values of the two processes at time $j_i$ can be calculated as:
\begin{eqnarray}
Y_{j_i} & = & Y_{j_i}^- + \xi_{j_i}^{y} \\
V_{j_i} & = & V_{j_i}^- + \xi_{j_i}^{\nu}
\end{eqnarray}
If $j_N \neq \tau$, then no jump occurs in the interval $[j_N, \tau]$,
and we can apply the Euler discretization schema for the Heston model to
get the values of $Y_{\tau}$ and $V_{\tau}$.
In the following, we explain how to implement the Euler discretization
schema for the Heston model. Discretizing the two
equations~\eqref{eqn:stochVolDef} and ignoring the jump part, the
difference between $Y_{t+\Delta t}$ and $Y_t$ and the difference
between $V_{t+\Delta t}$ and $V_t$ is simply given by
\begin{eqnarray}
Y_{t+\Delta t} - Y_t & = & \mu \Delta t + \sqrt{V_t} \left( \Delta
W_t^{(1)}\right), \nonumber \\
V_{t+\Delta t} - V_t & = & \kappa(\theta-V_{t}) \Delta t +
\sigma_\nu \sqrt{V_{t}} \left(\rho \Delta W_{t}^{(1)} + \sqrt{1-\rho^2}
\Delta W_{t}^{(2)}\right). \label{eqn:Heston}
\end{eqnarray}
As already mentioned before, we typically use $\Delta t=1$ day to
discretize both processes. To simulate the Brownian increments $\Delta
W_{t}^{(1)}$ and $\Delta W_{t}^{(2)}$, we use the fact that each
increment is independent of the others. Each such increment is normally
distributed with mean 0 and variance $\Delta t$. However,
using~\eqref{eqn:Heston} for the simulations may generate negative
volatilities. This is a well-known effect that has been addressed
several times in the literature. Following~\cite{LorKoeDij06}, we
slightly modify~\eqref{eqn:Heston} to prevent the simulation
from generating negative values:
\begin{equation}
V_{t+\Delta t} - V_t = \kappa(\theta-V_{t}^+) \Delta t + \sigma_\nu
\sqrt{V_{t}^+} \left( \rho \Delta W_{t}^{(1)} + \sqrt{1-\rho^2} \Delta
W_{t}^{(2)}\right),\label{eqn:Heston2}
\end{equation}
where $V_t^+$ is the maximum between $0$ and $V_t$. If the time
between
different jumps is smaller that the discretization step $\Delta t$,
then it simply means that we have several jumps occurring during that
time interval. In that case, we just have to simulate the
appropriate number of jumps, both for the return and for the
volatility, during that period. In the end, the return over the period
$[0, \tau]$ is just $Y_{\tau}$. By repeating this procedure $M$ times,
we get $M$ realizations of the accumulated returns over the period
$[0,\tau]$, which make it possible to estimate its density. In our
tests, we have used a Gaussian kernel to estimate it. This can easily
be done in R by using the \texttt{density()} function. Then, one can
obtain the semivariance by numerically integrating the expression
given in~\eqref{eq:TSV}. For this, we have used the QUADPACK
library via the use of the \texttt{integrate()} function in R. QUADPACK is a
well known numerical analysis library implemented in FORTRAN~77 and
containing very efficient methods for numerical integration of
one-dimensional functions.
\subsection{Results}
\label{stochvol:res}
We ran the algorithm using 100'000 iterations with 10'000 burn-in
iterations. Table~\ref{tbl:param} provides parameter posterior means and their computed standard
errors.
\begin{table}
\begin{center}
\begin{tabular}{lcc}
\hline
Parameter & Value (100k iter.) \\
\hline
$\mu$& 0.0333 (0.0132)\\
$\mu_y$& -0.0222 (0.2055)\\
$\mu_\nu$& 0.0028 (0.0007) \\
$\theta$& 0.2087 (0.0147)\\
$\kappa$& 0.9059 (0.0433)\\
$\rho$& -0.0058 (0.2595)\\
$\rho_J$& 0.0050 (2.0007)\\
$\lambda$& 0.0236 (0.0293)\\
$\sigma_y$& 1.0829 (0.1379)\\
$\sigma_\nu$& 0.2060 (0.0204)\\
\hline
\end{tabular}
\end{center}
\caption{Estimated parameters based on 100'000 iterations. The time
unit is the day rather than a year, meaning that parameters should
be interpreted on a daily basis. $\mu$, $\mu_y$, $\mu_\nu$,
$\sigma_y$, and $\sigma_\nu$ are in \%.}
\label{tbl:param}
\end{table}
The return mean $\mu$ is close to the daily return mean from the
data ($0.0325 \%$). The long-term mean of the $V_t$, given by
$\mathrm{E}[V_t] = \theta+(\mu_\nu \lambda) / \kappa$, is equal to
$0.2088 \%$. This value is not far away from the variance of returns,
which is equal to $0.1751$. The jump returns are normally distributed
with mean $-0.0222 \%$
and standard deviation $0.2055 \%$. Concretely, it means that there is a
$68 \%$ likelihood of having a jump return between $-23\text{ bps}$
and $+18\text{ bps}$. The parameters $\sigma_y$ and $\sigma_\nu$ are
$1.0829 \%$ and $0.2060 \%$, respectively. Finally, the intensity of the
jumps is $0.0236$. This means that, on average, there are
approximately 6 jumps per year. This is a bit low in comparison with
the number of jumps that we have observed in our data, as we would rather
expect an intensity around $0.05$. However, this empirical intensity
was computed considering that differences in returns above $2.57$
deviations from the mean could be considered as jumps. Some caution
must be taken, since this empirical intensity is very sensitive to the
number of standard deviations used as threshold for defining the
jumps. More details about the implementation can be found in
Appendix~\ref{sec:input}.
In a second step, we have computed an annualized semideviation based
on that process. We have followed the methodology explained
in~\S\ref{sub:semi} to first compute the simulated returns and
then the semideviation.
\begin{center}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{SimulatedReturns}
\caption{\label{graph:simulatedReturns}This graph shows the
histogram of the returns based on 1'000 simulations.}
\end{center}
\end{figure}
\end{center}
Our tests have shown that 1'000 simulations were sufficient to get an
accurate estimation of the density from which we have derived a
semideviation. Note that this semideviation is based on the whole
historical data. It was not possible for us to do the same rolling
analysis that we have performed before, assuming a constant
volatility, for the two following reasons. The first one is that the
estimation of the parameters is very time-consuming. Around 7 hours
were necessary to generate the parameters for the whole period with an
\textit{Intel\/$^{\mbox{\scriptsize{\textregistered}}}$
Core$^{\mbox{\scriptsize{\texttrademark}}}$~i5-3470 CPU @ 3.20
GHz}. Assuming that we would have to perform a rolling analysis
based on 1-year of history over the same period, then we would have to
run more than 1000 times this analysis. This is not impossible to
realize but would typically require parallel computing techniques. The
second reason for not having performed this analysis is that we have
to check the convergence of the process for each period we consider in
the rolling analysis, a task that is difficult to fully automate. As a
reminder, the convergence can be checked with trace plots showing the
history of the chain for each parameter, with ACF plots to analyze the
correlation structure of draws, and the quality of the fit may be
assessed through the mean squared errors and the normal QQ plot. So
these two reasons have prevented us from doing this rolling analysis
when we consider the model with stochastic volatility and jumps in
returns and in volatility. However, we have performed some analyses
based on 1 year of data for the two worst periods: during the credit
crunch crisis in 2008 and in 2011. Table~\ref{tbl:stochvolres}
summarizes the results and compares the semideviations obtained from
the different models that we have explored in this paper.
\begin{table}
\begin{center}
\begin{tabular}{ccccccccccccccccc}
\hline
Year & SD1 & SD2 & SD3 & SD4 \\
\hline
2008-2012 & 4.70 & 1.21 & 1.39 & 1.74 \\
2008 & 8.71 & 31.59 & 31.72 & 32.16 \\
2011 & 3.69 & 1.31 & 1.34 & 2.90 \\
\hline
\end{tabular}
\caption{\label{tbl:stochvolres}Annualized semideviations (in \%) for different periods. SD1
is based on the square-root-of-time rule, SD2 on a jump-diffusion
model with constant volatility, SD3 on a pure diffusion model with a constant volatility, and
SD4 is based on a jump diffusion model with stochastic volatility
and jumps in returns and in volatility.}
\end{center}
\end{table}
We can observe that the semideviation
SD4 obtained with a stochastic volatility model seems to be more in line
with the semideviations SD2 (based on a jump-diffusion model) and SD3
(based on a pure diffusion model) rather than SD1 (square-root-of-time
rule). This is not very surprising since SD4 may be interpreted as a
refinement of SD2, explaining why the two analytics should not be so
different.
\section{Conclusion and Future Venues of Research}
In this paper, we propose a generalization of the Ball-Torous
approximation of a jump-diffusion process and we show how to compute
the semideviation based on
several jump-diffusion models. We have in particular considered
a very exhaustive example based on the Barclays US High Yield Index,
whose returns show negative skewness and fat tails. It is a common
practice to impose a bound on the Poisson process intensity parameter
to make sure that only the large accidents are captured by the jumps
process. However, our analysis clearly shows that the risk may be
underestimated in such a situation. Without constraining the intensity
parameter, the semideviation may be more than $50$ \% larger than the
one obtained by arbitrary limiting this parameter in order to capture
only large jumps.
We see at least two reasons why our approach
should be preferred. The first one is that we do not impose any
arbitrary constraint on the intensity parameter. With our method, this
parameter is determined in order to obtain the best fit to the
data. Jumps may occur very often in certain periods and can be very
uncommon (as rare as a few ones per year) in others. The second reason
is that imposing an arbitrary limit on the intensity parameter may
result in an underestimation of the risk. Capturing the risk as
accurately as possible is one of the most important mission in risk
management. It is the reason why the ``traditional" approach based on
the work of Ball and Torous does not seem appropriate in our
context. Moreover, one of our conclusions is that the use of the
square-root-of-time rule may either underestimate the risk in periods
of high volatility or underestimate it in periods of lower stress.
We also provide in this paper a generalization of the work of
Eraker, Johannes and Polson, who consider a jump diffusion model with
stochastic volatility and jumps in returns and in volatility. As in
the case where the volatility is constant, we have extended their
approach by replacing the jump Bernoulli process by a more general
process being able to model several accidents per day when the
intensity parameter is high. The parameter estimation methods are
based on MCMC methods. In particular, we have used a Gibbs sampler
when the conditional distributions were known and variants of the
Metropolis algorithm when it was not the case. We also provide a
procedure to compute an annualized semideviation once the parameters
of the model have been estimated.
It was not possible for us to go as far as we would like to when we
have considered the model with stochastic volatility with jumps in
returns and in volatility. It would have been very useful to be
able to do the same rolling analyses that we have performed when the
volatility was kept constant. However, such analyses would have been
too much time consuming from a computational point of view and it
would have been complicated to automate the check of the convergence
of the process that needs to be done for each period in the rolling
analysis. This was clearly a limitation when we have tried to
determine empirically the relationship between the daily semideviation
and its annualized version.
We really think that the use of the semideviation (or semivariance)
could benefit the finance industry by
providing a useful and powerful risk measure. This risk metric has
been proposed a very long time ago but its difficulty to be computed
and its lack of nice properties for its scaling made it hard to
implement. However, we have shown in this article that it is still
possible to calculate it even when we consider very complex stochastic
processes and that some useful formula for its time scaling can be
derived under some mild assumptions. Finally, we hope that this paper will
help democratize its use in the asset management industry.\\
~\\
\noindent\textbf{Disclaimer:} The views expressed in this article are
the sole responsibility of the authors and do not necessarily reflect
those of Pictet Asset Management SA. Any remaining errors or
shortcomings are the authors’ responsibility.
|
2008.12708
|
\section{Introduction}
The problem of bio-inspired swarm control has attracted new perspectives from a variety of research areas, such as control theory, biology, and artificial intelligence~\cite{martinez2007motion}. These bio-inspired approaches provide valuable insights in designing multi-agent systems and/or robotic swarms. In these systems, studies aim to address the question of how to control a swarm of individual agents based on natural interactions among them and between them and their operating environment~\cite{carelli2006centralized,oh2017bio}. Challenges in answering this question arise from how to define flexible geometric constraints for the agents, and then how to maintain them in a required or connected formation in order to successfully achieve a specific mission~\cite{scharf2004survey,serrani2003robust}.
The shepherding problem is inspired from sheep-herding in agriculture wherein a single or multiple shepherds or sheepdogs are used to guide a large group of sheep. Shepherding is considered as a flocking behaviour when one or multiple external agents acting as shepherds, drive a swarm of individual agents, called \emph{flocking} or \emph{sheep agents} towards a given target. The herding idea has been applied to the field of multi-agent systems and swarm robotics~\cite{strombom2018robot}. There are a variety of possible applications~\cite{strombom2018robot,nalepka2019practical} of the shepherding problem within these fields such as herding living animals such as driving a large group of bird or sheep in a field area, assisting in controlling human crowd activities, cleaning environmental hazards such as oil-spills, or guiding cells to fix tissue in internal medicine~\cite{cohen2014galvanotactic}.
In recent years, Str\"{o}mbom et al.~\cite{strombom2014solving, strombom2018robot} introduced a heuristic approach to address the shepherding problem. In this approach, Str\"{o}mbom et al. use two main behaviours: \emph{collecting} and \emph{driving} in order to explain the interaction between a shepherd considered as one intelligent agent and a swarm of sheep being treated as autonomous agents. The collecting behaviour aims to maintain the entire sheep grouped within a connected network, while the driving behaviour enables guidance of the group/swarm of sheep towards a goal. The Str\"{o}mbom approach shows good shepherding performance in successfully collecting and driving a large number of sheep towards the given target. However, Str\"{o}mbom et al. evaluate their approach in an ideal environment possessing an unrealistically low noise level for both the shepherd and the sheep. In practice, the shepherd might face various sources of significant noise associated with the response of the sheep to the influence of the sheepdog, called the \emph{actuation noise}, and noise coming from the sensing ability of the shepherd, called the \emph{perception noise}. Under extreme weather conditions, or obstacles, the sheep or autonomous agents might move very imprecisely in response to shepherding commands, and the shepherd can insufficiently sense the position of the sheep. These errors may lead to poor performance of the shepherd. In the literature, there is insufficient attempts to understand the impact of these errors on the overall performance of the shepherding in the successful completion of the task.
In this paper, we investigate the level of evaluations sufficient to achieve stability in performance in the Str\"{o}mbom approach~\cite{strombom2014solving}, before studying performance impact under increasing actuation and perception noises. Furthermore, we identify appropriate thresholds of switching between the collecting and driving behaviours, called the collecting frequency, in order to improve the performance of the shepherd. To identify these thresholds, we slightly decrease and increase the threshold value used in the Str\"{o}mbom method. Our experiments are conducted in the same simulation environment as that introduced in the Str\"{o}mbom approach. The results from the experiments show that the performance of the shepherd is more sensitive to perception noise than the actuation noise. Moreover, a guiding set of appropriate thresholds are identified which should help improve the shepherding efficiency by adapting the switch between collecting and driving behaviours to suit the amount of perception and actuation noises present.
The remainder of the paper is organized as follows. In Section~\ref{sect:relatedwork}, we provide a brief review of the research focused on the shepherding problem in order to identify a gap in the evaluation of shepherding performance under noise conditions. Following this section, we formally define shepherding using an appropriate notional system and a corresponding mathematical objective in Section~\ref{sect:shepherding}. The proposed evaluation framework is introduced in Section~\ref{sect:framework}. The framework is conducted in a simulated shepherding task in Section~\ref{sect:experiment}. Section~\ref{sect:results} presents the results of the framework. Conclusions are drawn in Section~\ref{sect:conclusions}, followed by a discussion on future work.
\section{Related work}\label{sect:relatedwork}
In nature, the behaviours of flocking of birds, herding of land animals, or schooling of fish can be widely seen~\cite{reynolds1987flocks}. In these behaviours, a large number of individual agents will be influenced by one controlling agent in order to successfully achieve different goals such as finding food or foraging. Studying the various kinds of swarm behaviours in nature can greatly assist in the design of distributed and coordinated control methods for robotic swarms or multi-agent systems~\cite{kennedy1995particle,eberhart1995new}.
Early research on the shepherding problem was carried out by Schultz et al. ~\cite{schultz1996roboshepherd}. In this research, the authors use genetic algorithms to learn rules for a shepherd or sheep-dog agent so that it might drive a swarm of sheep towards a desired target. In an another approach, Lien et al.~\cite{lien2004shepherding} conducted experiments in order to simulate four main behaviours: herding, covering, patrolling, and collecting. The combination of these four behaviours for the shepherd shows effective shepherding strategies. However, both pieces of research are more suitable for driving a small number of sheep (less than 40)~\cite{bennett2012comparative}.
Towards guiding a large number of sheep agents (more than 40), Str\"{o}mbom et al.~\cite{strombom2014solving,strombom2018robot} introduced a heuristic approach to the shepherding problem. The authors use two main behaviours: collecting and driving, in order to guide the entire sheep. The approach is promising, enabling the shepherd to guide up to 300 sheep effectively. Adopting the idea behind the Str\"{o}mbom approach, some other research has attempted to use learning methods, such as reinforcement learning~\cite{nguyen2019deep}, apprenticeship learning~\cite{nguyen2019apprenticeship}, and machine education~\cite{gee2019transparent,clayton2019machine}. However, in both the Str\"{o}mbom method and the research adopting the approach of Str\"{o}mbom, the shepherd works in an ideal environment with just a small amount of noise added to the shepherd and the sheep to avoid deadlocks. In practice, the operating environment of these agents might include various significant noise sources impacting on the performance of the shepherd. These noises come from unexpected responses and behaviours of the sheep, called the actuation noise, and the sensing ability of the shepherd, called the perception noise. To date, there has not been any published work on contrasting these noises on the performance of a shepherd for swarm guidance.
\section{Shepherding Problem}\label{sect:shepherding}
In the shepherding problem, Str\"{o}mbom et al.~\cite{strombom2014solving} introduce a heuristic approach in which the movement of sheep is computed, and from there an effective control strategy is created for the shepherd. In this paper, the Str\"{o}mbom et al.~\cite{strombom2014solving} approach is described by providing the notations as well as the mechanism that we will use later in the experimental design.
The operating environment of the shepherding problem is a 2-D square paddock having length of $L$. In this environment, two kinds of agents, which are a set of sheep (called influenced agents) $\Pi = \{ \pi_1, \dots, \pi_i, \dots, \pi_N \}$, and a set of shepherds (called influencing agents) $B = \{ \beta_1, \dots, \beta_j, \dots, \beta_M \}$, are initialized. There are three main behaviours for each shepherd, and four basic behaviours for each sheep at a time step $t$. These behaviours are shown as below.
\begin{enumerate}
\item {For shepherd $\beta_j$:
\begin{itemize}
\item \emph{Driving behaviour} $\sigma_1$: When all sheep are collected in a cluster, i.e. all the distances from the observed sheep to the center of sheep\textquoteright s mass are lower than a threshold $f(N)$ calculated in Equation~\ref{eq:ShepherdBehaviorSelectionThresholdEquation}, a normalized force vector, $F^t_{\beta_jcd}$, is applied for the shepherd as a velocity vector in order to reach a driving point. This point is located behind the sheep\textquoteright s mass on the line drawn from the center of the sheep\textquoteright s mass and the target position.
\begin{equation}\label{eq:ShepherdBehaviorSelectionThresholdEquation}
f(N) = R_{\pi\pi} N^\frac{2}{3}
\end{equation}
\item \emph{Collecting behaviour} $\sigma_2$: When a sheep is deemed to have gone astray from the others i.e. the distance from the sheep to the center of the sheep\textquoteright s mass is greater than the threshold $f(N)$, a normalized force vector, $F^t_{\beta_jcd}$, is applied for the shepherd as a velocity vector in order to reach a collecting point. This point is positioned behind the outer or furthest sheep on the line drawn from the center of the sheep\textquoteright s mass to the furthest sheep.
\item \emph{Jittering behaviour} $\sigma_3$: To avoid an impasse during moving, a small random noise $F^t_{\beta_j\epsilon}$ with weight $W_{e\beta_j}$, is added added to the total force.
\end{itemize}
The total force $F^t_{\beta_j}$ of the shepherd $\beta_j$ (total force behaviour $\sigma_8$) is a weighted combination of the forces produced by the driving/collecting behaviour and the jittering behaviour. This total force is shown in Equation~\ref{eq:ShepherdTotalForceEquation}
\begin{equation}\label{eq:ShepherdTotalForceEquation} F^t_{\beta_j} = F^t_{\beta_jcd} + W_{e\beta_j} F^t_{\beta_j\epsilon} \end{equation}
}
\item {For sheep $\pi_i$}:
\begin{itemize}
\item \emph{Escaping behaviour} $\sigma_4$: This behaviour happens when the distance between the sheep $\pi_i$ at position $P^t_{\pi_i}$ and the shepherd $\beta_j$ at position $P^t_{\beta_j}$ is less than the sensing range, $R_{\pi\beta}$, a repulsive force $F^t_{\pi_i\beta_j}$ is provided the sheep $\pi_i$. The condition to trigger the behaviour is shown in Equation~\ref{eq:SheepDistanceToShepherdEquation}.
\begin{equation}\label{eq:SheepDistanceToShepherdEquation} \|P^t_{\pi_i}-P^t_{\beta_j}\| \le R_{\pi\beta}
\end{equation}
\item \emph{Collision avoidance behaviour} $\sigma_5$: This behaviour happens when there is a repulsion between the sheep $\pi_i$ and the other sheep $\pi_{k\ne i}$. The condition of activating the repulsion force between the two sheep is that the distance between them is less than the sensing range among sheep, $R_{\pi\pi}$. This condition is shown in Equation~\ref{eq:SheepDistanceToOtherSheepEquation}.
\begin{equation}\label{eq:SheepDistanceToOtherSheepEquation} \exists {k}, \ such \ that \ \|P^t_{\pi_i}-P^t_{\pi_{k}}\| \le R_{\pi\pi}
\end{equation}
Then, we have the summed force vectors, $F^t_{\pi_i\pi_{-i}}$, from all the other sheep within the threshold range, $R_{\pi\pi}$, applied onto sheep $\pi_i$.
\item \emph{Grouping behaviour} $\sigma_6$: This behaviour appears when the sheep $\pi_i$ under a force $F^t_{\pi_i\Lambda^t_{\pi_i}}$ will be attracted to move towards the center of the mass of its sheep neighbors, $\Lambda^t_{\pi_i}$.
\item \emph{Jittering behaviour} $\sigma_7$: Similar to the jittering behaviour of each shepherd, to avoid impasse, a small random noise is added to the total force $F^t_{\pi_i\epsilon}$ with weight $W_{e\pi_i}$.
\end{itemize}
The total force, $F^t_{\pi_i}$, of the sheep $\pi_i$ is represented by a weighted sum of individual force vectors $F^t_{\pi_i\beta_j}$, $F^t_{\pi_i\pi_{-i}}$,
$F^t_{\pi_i{\Lambda^t_{\pi_i}}}$, and $F^t_{\pi_i\epsilon}$; that is,
\begin{equation}\label{eq:SheepTotalForceEquation}
F^t_{\pi_i} = W_{\pi_\upsilon} F^{t-1}_{\pi_i} +
W_{\pi \Lambda} F^t_{\pi_i{\Lambda^t_{\pi_i}}} + W_{\pi\beta} F^t_{\pi_i\beta_j} + W_{\pi\pi} F^t_{\pi_i\pi_{-i}} + W_{e\pi_i} F^t_{\pi_i\epsilon}
\end{equation}
\end{enumerate}
The shepherds\textquoteright and sheep\textquoteright s positions are calculated according to Equations~\ref{eq:UpdatedShepherdPositionEquation}, and~\ref{eq:UpdatedSheepPositionEquation}. Meanwhile, the given $S^t_{\beta_j}$ and $S^t_{\pi_i}$ are the speed of the shepherd $\beta_j$ and the speed of the sheep $\pi_i$ at time step $t$. In the original Str\"{o}mbom approach, the speeds of both the shepherds and sheep are constant.\\
\begin{equation}\label{eq:UpdatedShepherdPositionEquation}
P^{t+1}_{\beta_j} = P^{t}_{\beta_j} + S^t_{\beta_j} F^t_{\beta_j}
\end{equation}
\begin{equation}\label{eq:UpdatedSheepPositionEquation}
P^{t+1}_{\pi_i} = P^t_{\pi_i} + S^t_{\pi_i} F^t_{\pi_i}
\end{equation}
\section{A Proposed Evaluation Framework.}\label{sect:framework}
In this paper, we evaluate the performance of the shepherd produced by the Str\"{o}mbom approach~\cite{strombom2014solving} under the two noises: the actuation ($\lambda$) and perception ($\alpha$). The actuation noise appears when the sheep move randomly around the location they are supposed to move to. The range of the random movement decides the degree of this noise. Meanwhile, the perception noise happens when there is deviation between the sheep\textquoteright s actual position and the position that the shepherd observes. The deviation range between these two positions defines the degree of the perception noise.
The standard normal distribution, with a mean of zero and standard deviation of 1, is used to create the actuation and perception noises. The procedure of updating the position of sheep $\pi_i$ under the actuation noise, called $ActP^{t+1}_{\pi_i}$, is shown in Equation~\ref{eq:addActuationNoise}.
\begin{equation}\label{eq:addActuationNoise}
ActP^{t+1}_{\pi_i} = P^t_{\pi_i} + S^t_{\pi_i}\times( F^t_{\pi_i} + \lambda\times StandardNormal())
\end{equation}
For the perception noise, the perceived position of sheep $\pi_i$ at timestep $t+1$ is denoted $PerP^{t+1}_{\pi_i}$. This position is sensed by the shepherd by Equation~\ref{eq:addPerceptionNoise}.
\begin{equation}\label{eq:addPerceptionNoise}
PerP^{t+1}_{\pi_i} = ActP^{t+1}_{\pi_i} + \alpha\times StandardNormal()
\end{equation}
According to the Str\"{o}mbom approach~\cite{strombom2014solving}, there are two main behaviours: collecting and driving. To switch between these behaviours, the shepherd needs to check whether any sheep is further from the center of mass than the threshold ($f(N)$) as calculated in Equation~\ref{eq:ShepherdBehaviorSelectionThresholdEquation}. We aim to identify appropriate thresholds ($f(N)$) of triggering between the collecting and driving behaviours, informed by the estimates of noise levels, in order to improve the performance of the shepherd.
To find these appropriate thresholds, we try values above and below the threshold of the Str\"{o}mbom approach shown in Equation~\ref{eq:ShepherdBehaviorSelectionThresholdEquation} by decreasing or increasing a predefined value, called $\Delta_{f}$. In our evaluation, we conduct three decreased levels of threshold ($-1,-2,-3$) and similarly three increased levels of threshold ($1,2,3$). These six levels will be multiplied with the $\Delta_f$. Thus, we have seven threshold values from $f(N) - 3\times\Delta_f$ to $f(N) + 3\times\Delta_f$ as shown in Table~\ref{tab:thresholdvalues}. It can be understood that when the threshold value increases, the collecting frequency will decrease, and the shepherd focuses on the driving behaviour. We set $\Delta_f=5(meter)$ in this paper.\\
\begin{table}[ht]
\centering
\caption{The Threshold to Switch Between Collecting and Driving.} \label{tab:thresholdvalues}
\begin{tabular}{c|c|c}
Collection Frequency & Parameter & Value (meter) \\ \hline
Extreme & $f_{-3}$ & $f(N) - 3\times\Delta_f$\\
Very High & $f_{-2}$ & $f(N) - 2\times\Delta_f$\\
High & $f_{-1}$ & $f(N) - 1\times\Delta_f$\\
Normal & $f_{0}$ & $f(N)$\\
Infrequent & $f_{+1}$ & $f(N) + 1\times\Delta_f$\\
Very Infrequent & $f_{+2}$ & $f(N) + 2\times\Delta_f$\\
Rare & $f_{+3}$ & $f(N) + 3\times\Delta_f$\\
\hline
\end{tabular}
\end{table}
For both actuation and perception noises, we set six noise levels increasing $0.01$ from $0.01$ to $0.06$ and $0.1$ from $0.1$ to $0.6$, respectively. To add these noises to the operation, we multiply these noise levels with a fixed change value, $\Delta_{n}$, which is set to the investigated maximum threshold value $f_{+3}$. Besides of the six noise level for the perception noise, we also investigate the performance of the shepherd in the same perception condition of the Str\"{o}mbom model~\cite{strombom2014solving} without noise. Hence, we have seven noise levels for both as given in Table~\ref{tab:noisealphalevels} and \ref{tab:noiseactuationlevels}.
\begin{table}[ht]
\centering
\caption{Levels of Perception Noise ($\alpha$)} \label{tab:noisealphalevels}
\begin{tabular}{c|c|c}
Level & Perception Noise & Value (meter) \\ \hline Noise Free & $\alpha_0$ & $0$\\
Very little & $\alpha_1$ & $0.1\times\Delta_n$\\
Little & $\alpha_2$ & $0.2\times\Delta_n$\\
Small & $\alpha_3$ & $0.3\times\Delta_n$\\
Medium & $\alpha_4$ & $0.4\times\Delta_n$\\
High & $\alpha_5$ & $0.5\times\Delta_n$\\
Very High & $\alpha_6$ & $0.6\times\Delta_n$\\
\end{tabular}
\end{table}
\begin{table}[ht]
\centering
\caption{Levels of Actuation Noise ($\lambda$).} \label{tab:noiseactuationlevels}
\begin{tabular}{c|c|c|c}
Level & Actuation Noise & Value (meter) \\ \hline Noise Free & $\lambda_0$ & $0$\\
Very little & $\lambda_1$ & $0.01\times\Delta_n$\\
Little & $\lambda_2$ & $0.02\times\Delta_n$\\
Small &$\lambda_3$ & $0.03\times\Delta_n$\\
Medium & $\lambda_4$ & $0.04\times\Delta_n$\\
High & $\lambda_5$ & $0.05\times\Delta_n$\\
Very High & $\lambda_6$ & $0.06\times\Delta_n$\\
\end{tabular}
\end{table}
In this work, the perception noise values are considerably higher than that of the actuation noise. The reason is because under the actuation noise, the sheep also have repulsion and attraction forces among them; thus, the movement of the sheep under the actuation is more spread. Meanwhile under the perception noise, the shepherd has wide range of its view (65 meters in the Str\"{o}mbom approach), and then it is sill able to control the sheep acceptably without reaching the true collecting and driving points. Therefore, the perception noise values need to be larger in order to measure the change of the performance.
The performance ($PF$) of the shepherd is validated based on combining the three above factors: the changing radius of the mass ($f$), and the two noise conditions ($\lambda$ and $\alpha$). This relation is illustrated in Equation~\ref{eq:relationthreefactors} in which the function - $g$ includes three variables ($f,\lambda, \alpha$).
\begin{equation}\label{eq:relationthreefactors}
PF_{\beta_i} = g(f,\lambda,\alpha)
\end{equation}
with $\beta_i$ is the shepherd $i$-th.
In the next section, we provide the design of the experiments in this paper.
\section{Experiments}\label{sect:experiment}
In this paper, we simulate the environment given by the Str\"{o}mbom model~\cite{strombom2014solving} as introduced in Section~\ref{sect:shepherding}. The same parameters regarding the environment initialization and the interaction between agents for the simulation are listed in Table~\ref{tab:EnvironmentalParameters}.
In each simulation, a given number of sheep are randomly initialized at the centre of the paddock with their coordinates in the range of between 1/4 and 3/4 of the length/width of the environment. The shepherd are randomly initialized at the lower left corner, with their coordinates not exceeding 1/10 of the length/width of the environment, near the target position (at $(0,0)$). The shepherd\textquoteright s mission is to collect outer sheep into a group and herd the entire sheep towards the target. The mission is achieved when all of the sheep have reached the target within a limit of 1000 steps, and the simulation ends. The illustration of the environment is shown in Figure~\ref{fig:envexperimnet}.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.3\textwidth]{figures/Experiment.png}
\caption{The experiment environment.}
\label{fig:envexperimnet}
\end{figure}
\begin{table}
\caption{Environmental parameters in the simulation.}
\begin{tabular}{c|p{1.8in}|c}
Parameter & \multicolumn{1}{c|}{Meaning} & Value \\ \hline
$L$ & Length and Width of Environment & 150\\
$N$& Number of Sheep & 100 \\
$M$& Number of Shepherds & 1 \\
$R_{\pi\beta}$& Sensing range of a sheep for the shepherd & 65\\
$R_{\pi\pi}$ & Sensing range of a sheep for another sheep & 2 \\
$W_{\pi\pi}$ & Sheep repulsion strength from other sheep & 2\\
$W_{\pi\beta}$ & Sheep repulsion strength from the shepherd & 1 \\
$W_{\pi \Lambda}$ & Sheep attraction strength to sheep centre of mass & 1.05 \\
$W_{\pi_\upsilon}$ & Inertial strength of sheep previous direction & 0.5 \\
$W_{e\pi_i}$ & Strength of sheep movement noise & 0.3 \\
$W_{e\beta_j}$ & Strength of the shepherd movement noise & 0.3 \\
$\vert\Omega_{\pi_i\pi}\vert$ & Number of sheep (neighborhood) a sheep can sense & 25 \\
$S_\pi$ & Maximum speed of sheep & 1 \\
$S_{\beta}$ & Maximum speed of the shepherd & 2\\
$\mathbb{D}$ & Minimum distance between the sheep\textquoteright s global centre of mass and the target for successful mission & 5
\end{tabular}
\label{tab:EnvironmentalParameters}
\end{table}
\subsection{Experimental Setups}
In this paper, we evaluate the performance of the shepherd of the Str\"{o}mbom model~\cite{strombom2014solving} under the two noises: actuation - $\lambda$ and perception -$\alpha$. Furthermore, we aim to identify the appropriate thresholds ($f$) of triggering between the collecting and driving behaviours that might lead to higher performance for the shepherd under these two noises. Thus, in total we conduct $7\times7\times7 = 343$ setups in which there are 7 changing levels of the threshold ($f$), 7 perception noise levels ($\alpha$), and 7 actuation noise levels ($\lambda$).
\subsection{Evaluation Metrics}\label{sect:EvaluationMetrics}
In order to evaluate the performance of the shepherd under these three factors ($f,\lambda, \alpha$), we use two assessment metrics as below:
\begin{itemize}
\item \textbf{Number of steps (NS)}: the number of time steps for the sheep to be herded to the target location.
\item \textbf{Success rate ($\%$) (SR)} is the percentage of mission completion computed on a number of testing cases. The mission success is achieved when all the sheep are collected and driven to the goal position.
\item \textbf{Standard Error of Mean (SEM)} of NS indicates stability of the evaluation. This metric is calculated in Equation~\ref{eq:SEM}.
\begin{equation}\label{eq:SEM}
SEM = Std/\sqrt{n}
\end{equation}
where $Std$ is the standard deviation of the number of steps, and $n$ is the number of episodes.
\item \textbf{Standard Error Percentage (SEM-P)} of Mean indicates which episode the evaluation should be stopped when the SEM-P is less a small threshold (in this work, we choose 3$\%$) of the mean.
\end{itemize}
\section{Results and Discussion}\label{sect:results}
In this paper, we conduct 300 random testing episodes for each of 343 setups. This number of testing episodes enables our evaluations to be able to reach a level of stability to ensure appropriateness of the analyses and precision in assessing the performance of the shepherd. The evaluation obtains the stable status when the SEM-P value of the performance of the shepherd in setups is less than the threshold of 3$\%$ of the mean.
Firstly, we investigate the effects of the actuation ($\lambda$) and perception ($\alpha$) noises on the performance of the shepherd. We show the evidences of the stability of the evaluation when the values of the SEM and SEM-P of the setups decrease gradually and maintain stable by the end of the 300 episodes. According to the SEM values of $f$, Figure~\ref{fig:alphax-lamda0} shows that when there is no actuation noise ($\lambda_0$), the performance of the shepherd is stably evaluated and maintained after 150 episodes at pairs of $\lambda_0$ and the perception noise levels $\alpha$ lower than the high level. At the high ($\alpha_5$) and very high ($\alpha_6$) perception noise levels, the stability as measured by the performance of the shepherd is not obtained, and then, the reliability of our ability to estimate the performance might be imprecise. The performance instability at the high and very high levels of the perception noise can be seen in Figure~\ref{fig:alphax-lamda0-semp}. In this Figure, we can see that just only two setups of the very infrequent $f_{+2}$ and the rare of collecting frequency $f_{+3}$ at the high level ($\alpha_5$), and the setup of the rare of collecting frequency $f_{+3}$ at the very high level ($\alpha_6$) have SEM-P values below the stop point of 3$\%$.
\begin{figure*}[!ht]
\centering
\subfigure[SEM of $f$ at $\lambda_0$ and $\alpha_0$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha0-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM of $f$ at $\lambda_0$ and $\alpha_1$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha1-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM of $f$ at $\lambda_0$ and $\alpha_2$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha2-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM of $f$ at $\lambda_0$ and $\alpha_3$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha3-sem.png}
\label{fig:driving4x4_sub}
}%
\\
\subfigure[SEM of $f$ at $\lambda_0$ and $\alpha_4$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha4-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM of $f$ at $\lambda_0$ and $\alpha_5$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha5-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM of $f$ at $\lambda_0$ and $\alpha_6$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha6-sem.png}
\label{fig:driving4x4_sub}
}%
\caption{Standard Error of Mean to Evaluate Stability of Alpha ($\alpha$) with Different Thresholds or Collecting Frequency ($f$) at Lambda ($\lambda_0$) in 300 Episodes.}
\label{fig:alphax-lamda0}
\end{figure*}
\begin{figure}[!ht]
\centering
\subfigure[SEM-P of $f$ at $\lambda_0$ and $\alpha_5$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha5-sem_p.png}
}%
\subfigure[SEM-P of $f$ at $\lambda_0$ and $\alpha_6$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha6-sem_p.png}
}%
\caption{Standard Error of Mean to Evaluate Stability of Alpha ($\alpha$) with Different Thresholds or Collecting Frequency ($f$) at Lambda ($\lambda_0$) in 300 Episodes.}
\label{fig:alphax-lamda0-semp}
\end{figure}
Similarly, according to the SEM values of $f$, Figure~\ref{fig:alpha0-lamdax} shows when there is no perception noise, the shepherd exhibits stable behaviour over 150 episodes at pairs of $\lambda$ from the noise free level to the medium level and $\alpha_0$. When high actuation noise ($\lambda_5$) is applied, the behaviour of the shepherd is more fluctuating in the first 150 episodes, and takes more additional 150 episodes to reach to the stable point wherein all the SEM-P values of the threshold $f$ reduces to below 3$\%$ can be seen in Figure~\ref{fig:alpha0-lamdax-semp}. Furthermore, with the very high actuation noise ($\lambda_6$), the shepherd is not able to achieve the mission.
\begin{figure*}[!ht]
\centering
\subfigure[SEM of $f$ at $\lambda_0$ and $\alpha_0$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alpha0-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM of $f$ at $\lambda_1$ and $\alpha_0$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda1-alpha0-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM of $f$ at $\lambda_2$ and $\alpha_0$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda2-alpha0-sem.png}
\label{fig:driving4x4_sub}
}%
\\
\subfigure[SEM of $f$ at $\lambda_3$ and $\alpha_0$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda3-alpha0-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM of $f$ at $\lambda_4$ and $\alpha_0$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda4-alpha0-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM of $f$ at $\lambda_5$ and $\alpha_0$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda5-alpha0-sem.png}
\label{fig:driving4x4_sub}
}%
\caption{Standard Error of Mean to Evaluate Stability of Lambda ($\lambda$) with Different Thresholds or Collecting Frequency ($f$) at Alpha ($\alpha_0$) in 300 Episodes.}
\label{fig:alpha0-lamdax}
\end{figure*}
\begin{figure}[!ht]
\centering
\subfigure[SEM-P of Mean of $f$ at $\lambda_4$ and $\alpha_0$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda4-alpha0-sem_p.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM-P of Mean of $f$ at $\lambda_5$ and $\alpha_0$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda5-alpha0-sem_p.png}
\label{fig:driving4x4_sub}
}%
\caption{Standard Error of Mean to Evaluate Stability of Lambda ($\lambda$) with Different Thresholds or Collecting Frequency ($f$) at Alpha ($\alpha_0$) in 300 Episodes.}
\label{fig:alpha0-lamdax-semp}
\end{figure}
The similar results coming from the other pairs of $\lambda$ and $\alpha$ show that the stability measured from performance of the shepherd is not reached at noise levels higher than medium in both the actuation and perception. Especially, the shepherd collapses at the very high level of actuation noise $\lambda_6$. Figure~\ref{fig:alpha-lamdax-notworking-evaluation} illustrates this instability when all the SEM and SEM-P values show instability by the end of our evaluation and can not reduce to the stable point wherein the SEM-P values need to be below $3\%$. We notes that in the unstable condition of the evaluation, the reliability to estimate the performance of the shepherd might be imprecise so in the next parts, we just focus on analysing the performance on the setups reaching stability.
\begin{figure}[!ht]
\centering
\subfigure[SEM of $f$ at $\lambda_2$ and $\alpha_5$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda2-alpha5-sem.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SEM-P of $f$ at $\lambda_2$ and $\alpha_5$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda2-alpha5-sem_p.png}
\label{fig:driving4x4_sub}
}%
\caption{Standard Error of Mean to Evaluate Stability of Alpha ($\alpha$) with Different Thresholds or Collecting Frequency ($f$) at Lambda ($\lambda_2$) in 300 Episodes.}
\label{fig:alpha-lamdax-notworking-evaluation}
\end{figure}
After validating the stability of the setups, we conduct the investigation on how the actuation and perception noises impact the performance of the shepherd. Figure~\ref{fig:lamda0-alpha0} shows actuation noise ($\lambda$) impacting more dramatically the performance of the shepherd than the perception noise ($\alpha$). We can see that when under the noise-free condition of perception $\alpha_0$, the shepherding task collapses at $\lambda_5$; meanwhile, without the actuation noise ($\lambda_0$), the shepherd is still able to successfully achieve the shepherding task until the very high level-$\alpha_7$ with approximately 600 steps. It notes that the value of the actuation noise is considerably smaller than that of the perception noise (10 times). Furthermore, it is interesting that the change of the threshold $f$ does not impact drastically on the performance of the shepherd under the actuation noise; meanwhile, for the perception noise, this change has obvious effects on the shepherd\textquoteright s performance. The decreasing collecting frequency allows to maintain the success rate with nearly $100\%$ as well as the smaller number of steps are maintained even though the perception noise increases.
\begin{figure}[!ht]
\centering
\subfigure[NS of f at $\alpha_0$ in 300 Episodes]
{
\includegraphics[width=0.25\textwidth]{figures/lambdax-alpha0-ns.png}
\label{fig:driving4x4_sub}
}%
\subfigure[NS of f at $\lambda_0$ in 300 Episodes]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alphax-ns.png}
\label{fig:collecting4x4_sub}
}%
\\
\subfigure[SR of f at $\alpha_0$ in 300 Episodes]
{
\includegraphics[width=0.25\textwidth]{figures/lambdax-alpha0-sr.png}
\label{fig:driving4x4_sub}
}%
\subfigure[SR of f at $\lambda_0$ in 300 Episodes]
{
\includegraphics[width=0.25\textwidth]{figures/lambda0-alphax-sr.png}
\label{fig:collecting4x4_sub}
}%
\caption{The Relationship between Lamda ($\lambda$), Alpha ($\alpha$), and the Threshold or the Collecting Frequency ($f$) in 300 Episodes when the Standard Error Percentage of Mean is below 3 percent (\%)}
\label{fig:lamda0-alpha0}
\end{figure}
Besides of this evaluation of the two noises on the performance of the shepherd, we conduct an additional evaluation between the changes of the collecting frequency and the performance. From this evaluation, a set of the appropriate thresholds $f$ leading to the shepherd's higher performance is provided in this paper. We focus on the setups reaching the stability measured from the performance. These setups are in the noise areas from the very little noise ($\lambda_1$ and $\alpha_1$) to the medium noise ($\lambda_4$ and $\alpha_4$). Figure~\ref{fig:alpha-lamdax} shows the evidences of obtaining these appropriate thresholds. It can be seen that for the very little noise level of the actuation as shown in Figure~\ref{fig:alpha-lambda1-fx-sr}, the decreasing collecting frequency leads to the considerably higher performance as well as the smaller number of steps when the perception noise increases. However, when the level of the actuation noise increases, under the little noise level of the perception as illustrated in Figure~\ref{fig:alpha-lambda2-fx-sr}, the shepherd should prefer the extreme and very high collecting frequency in order to have the higher success rate of nearly $100\%$ compared to approximately $95\%$ of the Str\"{o}mbom approach even though it takes more steps. Under the higher noise level of the perception, the decreasing collecting frequency will improve the total performance in both the success rate and the number of steps.
Similarly, Figure~\ref{fig:alpha-lambda3-fx-sr} shows the case of having the small noise level ($\lambda_3$) of the actuation, the trend of choosing the small threshold $f$ under the very little or little perception noises allows to improve the performance drastically when comparing with the Str\"{o}mbom approach. Furthermore, at the small noise level of the perception, it seems that the Str\"{o}mbom approach produces the best performance. Additionally, when the actuation noise is at the medium level, there is not considerable difference about the performance under the increasing perception noise and the thresholds $f$. It can be understood that the controlling ability of the shepherd is not enough to guide the sheep under the large noises, and nearly reaches the instability.
From this evaluation, it is interesting to demonstrate the robustness of the shepherding model against both sources of noise when the testing scenarios at $\alpha_2$, $\alpha_3$, and $\lambda_4$ could still have a 20\% minimum success rate. Regarding the changes of the threshold ($f$), when the noises increase, it is logical that this threshold should be increased to prefer the driving behaviour and reduce the collecting behaviour. Under the large noises, there appears more sheep going out of the mass, and then the shepherd might perform the collecting behaviour continuously, and then has no chance to drive the sheep towards the target. This causes the reason why the task collapses at the large noises and the small threshold radius.
\begin{figure*}[!ht]
\centering
\subfigure[SR of $\alpha$ at $\lambda_1$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda1-fx-sr.png}
\label{fig:alpha-lambda1-fx-sr}
}%
\subfigure[SR of $\alpha$ at $\lambda_2$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda2-fx-sr.png}
\label{fig:alpha-lambda2-fx-sr}
}%
\subfigure[SR of $\alpha$ at $\lambda_3$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda3-fx-sr.png}
\label{fig:alpha-lambda3-fx-sr}
}%
\subfigure[SR of $\alpha$ at $\lambda_4$ ]
{
\includegraphics[width=0.25\textwidth]{figures/lambda4-fx-sr.png}
\label{fig:alpha-lambda4-fx-sr}
}%
\\
\subfigure[NS of $\alpha$ at $\lambda_1$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda1-fx-ns.png}
\label{fig:alpha-lambda1-fx-ns}
}%
\subfigure[NS of $\alpha$ at $\lambda_2$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda2-fx-ns.png}
\label{ffig:alpha-lambda2-fx-ns}
}%
\subfigure[NS of $\alpha$ at $\lambda_3$]
{
\includegraphics[width=0.25\textwidth]{figures/lambda3-fx-ns.png}
\label{fig:alpha-lambda3-fx-ns}
}%
\subfigure[NS of $\alpha$ at $\lambda_4$ ]
{
\includegraphics[width=0.25\textwidth]{figures/lambda4-fx-ns.png}
\label{fig:alpha-lambda4-fx-ns}
}%
\caption{The Effects of Lamda ($\lambda$) and Alpha ($\alpha$) on Different Thresholds or Collecting Frequency ($f$) in 300 Episodes when the Standard Error Percentage of Mean is below 3 percent (\%).}
\label{fig:alpha-lamdax}
\end{figure*}
\section{Conclusion and Future Work}\label{sect:conclusions}
In this paper, we evaluate the performance of the shepherd introduced by the Str\"{o}mbom approach~\cite{strombom2014solving} under the actuation and perception noises. With 300 random episodes for 343 setups, the obtained results show that stability in performance is reached and maintained after the 150 first episodes at the noise levels not exceeding the high level identified for both actuation and perception. When the noises are at high levels, the stability breaks down, and then the reliability of our ability to estimate the performance is very likely to become imprecise. After validating stability, a valuable point is drawn that the actuation noise is more sensitive than perception noise for the performance of the shepherd. The performance of the shepherd deteriorates earlier at the high level of actuation noise though this noise\textquoteright s value is less than ten times that of perception noise at the same level.
Additionally, when the perception noise increases and the actuation noise is low, the lower collecting frequency leads to higher success rate. In contrast, when the actuation noise is higher and the perception noise is low, the higher collecting frequency contributes to higher success rate. These interesting results show promising evidences in order to design an adaptive behaviour controller, which allows to adjust the threshold $f$ to switch between the two collecting and driving behaviours, improving the performance of the shepherd under these noises. Our future work attempts to design this controller.
|
2008.12721
|
\subsubsection{Requirements}
\label{sec_horn_requirements}
The back-to-back horn array has two objectives: the front (sky) horns define the field of view (FoV) of the instrument, while the back horns illuminate the beam combiner with the desired edge taper. In table~\ref{tab_horn_requirements} we list the main requirements of the back-to-back horn array with notes detailing their relevance. Notice that we do not have a requirement for cross-polarization, because the horn array is placed behind the polarization modulation/separation stage, so that the cross-polarization does not introduce systematic effects.
\begin{table}[h!]
\renewcommand{\arraystretch}{1.5}
\caption{\label{tab_horn_requirements}Main requirements for the QUBIC back-to-back horns array}
\begin{center}
\begin{tabular}{m{4cm} m{2cm} m{8cm}}
\hline
Requirement& Value & Notes \\
\hline
\hline
Inter-axis distance\dotfill & 14\,mm & Driven by sampling of the angular power spectrum\\
Aperture\dotfill & 12\,mm & Driven by the FoV\\
Return loss\dotfill & $<-25$\,dB & Over the 130--240\,GHz bandwidth\\
Insertion loss\dotfill & $<0.1$\,dB & To ensure overal transmission of $\sim 95\%$\\
Mass\dotfill & 30\,g/horn & Must be suspended on the top of the optical combiner\\
\hline
\end{tabular}
\end{center}
\end{table}
For simplicity we built identical front and back horns. The design was based on a previous geometry \cite{2004SPIE.5498..812M} modified to accept the propagation of the 220\,GHz frequency band. In Figure~\ref{fig_corrugation_profile} we show the corrugations profile: the depth of the corrugations at the aperture is 0.5\,mm and at the throat is 0.7\,mm. This choice allowed us to obtain antennas which are sensitive to both the 150\,GHz an 220\,GHz bands.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=15cm]{figures/qubic_horns_profile.pdf}
\end{center}
\caption{\label{fig_corrugation_profile}The corrugations profile of the QUBIC horns.}
\end{figure}
\subsubsection{Electromagnetic simulations technique}
\label{sec_simulation_technique}
We have simulated the field produced at the mouth of the QUBIC corrugated horn using an electromagnetic mode-matching technique \cite{clarricoats}, depicted schematically in Figure~\ref{fig_mode_matching}. This technique regards the corrugated structure as a sequence of smooth walled cylindrical waveguide sections, each of which can support a set of propagating TE and TM modes. At each corrugation the sudden change in the radius results in a scattering of power into backward propagating reflected modes in the left-hand side guide segment and forward propagating transmitted modes in the right-hand segment.
The power coupling between modes is given by the overlap integral $\int e_{n,l}\, h_{m,r}\, dA$, where $e_{n,l}$ is the transverse electric field of mode $n$ on the left-hand side of the junction, $h_{m,r}$ is the magnetic field of mode $m$ on the right-hand side of the junction and $dA$ is a surface element on the transverse plane. The modes are then propagated through the length of waveguide section to the next scattering junction where the overlap integral between the modal components is computed again.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=10cm]{figures/mode-matching.pdf}
\end{center}
\caption{\label{fig_mode_matching}Schematic of the mode-matching model implemented in the electromagnetic simulations.}
\end{figure}
If $\vec A$ and $\vec C$ are column vectors of the mode coefficients of the fields incident from the left and the right, and $\vec B$ and $\vec D$ are the mode coefficients of the resulting reflected fields, then their relationship is described using a scattering matrix, $\mathbf{S}$:
\begin{equation}
\begin{bmatrix}
\vec B \\
\vec D
\end{bmatrix}
= \mathbf{S}\cdot
\begin{bmatrix}
\vec A \\
\vec C
\end{bmatrix}
=
\begin{bmatrix}
\mathbf{S_{1,1}} & \mathbf{S_{1,2}} \\
\mathbf{S_{2,1}} & \mathbf{S_{2,2}}
\end{bmatrix}
\cdot
\begin{bmatrix}
\vec A \\
\vec C
\end{bmatrix}
\end{equation}
whose elements are calculated using overlap integrals as described in \cite{olver}. The columns of the scattering matrix describe the amplitude of each output mode generated by a unit-amplitude input mode. The scattering matrix for the horn as a whole, is computed by cascading the matrices for each uniform section and junction. We assume no scattering at the horn aperture so $\vec C=0$.
The field at the mouth of the corrugated horn is then determined from $\vec D = \mathbf{S_{2,1}}\cdot \vec A$, where $\mathbf{S_{2,1}}$ is the sub-matrix that deals with the forward-propagating modes, and the reflected field is determined from $\vec B = \mathbf{S_{1,1}}\cdot \vec A$
. The transmitted and reflected power are found by multiplying the complex elements of the relevant column vector by their complex conjugate and summing them.
In our analysis we used 60 waveguide modes (30 TE and 30 TM), but most of these modes carry no power to the mouth of the horn at 150 GHz. The TE and TM modes with power have a coherent phase relationship and in this case correspond to the single hybrid $\mathrm{HE}_{1,1}$ mode.
In the 220 GHz band more than one column of the scattering matrices is non-zero and these represent possible independent modes of power transmission. We excite all modes equally at the input, $\vec A = [1, 1, 1, \ldots]^T$, and add the individual output fields incoherently. The reflected power is calculated as a percentage of the power that could be transmitted by the number of propagating modes and this can result in spikes at frequencies where a mode is just switching on but not carrying much power.
\subsubsection{Simulations results}
\label{sec_simulation_results}
In Figure~\ref{fig_horn_simulated_return_loss_xpol} we show the return loss (left panel) and maximum cross-polarization (right panel) in the two QUBIC bands. We can see at a glance that the performance in the 150\,GHz band is superior compared to the 220\,GHz band. In fact the design was initially tailored in the D-band and subsequently modified to accept also the higher band that could not be optimized in terms of performance like the lower frequency range.
The return loss at 150\,GHz is, on average, around $-$25\,dB, while in the higher frequency band it is compatible with $-$20\,dB up to 230\,GHz, and degrades to $\sim -10$\,dB on the right hand side of the frequency interval. We assessed the potential impact of the poor return loss in the highest part of the 220\,GHz band: a degradation of the return loss will induce a reduction of the horn transmission and therefore an overall decrease of the sensitivity. With a pessimistic $-$10\,dB return loss over the whole 220\,GHz, we estimate a degradation in the sensitivity of less than 2\%, which makes this out-of-spec a negligible issue.
The cross-polarization is very good ($\sim -35$\,dB) at 150\,GHz, while it is around $-$5\,dB at 220\,GHz. This is coherent with the design: at 150\,GHz we have a single-mode corrugated horn, for which we expect excellent polarization purity, while at 220\,GHz we have propagation of higher modes that do not preserve the polarization state. But this is not a problem for QUBIC, as already mentioned, because the polarization is selected before the radiation enters the horns. For this reason we show here the expected cross-polarization performance but we will not discuss it further in the rest of the paper.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{c c c}
\includegraphics[width=7cm]{figures/rl_sim.pdf} &
\,\,\,&
\includegraphics[width=7cm]{figures/xp_sim.pdf}
\end{tabular}
\end{center}
\caption{\label{fig_horn_simulated_return_loss_xpol}Simulated return loss (left) and maximum cross-polarization (right) in the two QUBIC frequency bands.}
\end{figure}
In Figure~\ref{fig_horn_simulated_pattern} we show the simulated beam patterns at 150 and 220\,GHz for the three main co-polar planes (E-plane, H-plane and 45$^\circ$ plane). At 150\,GHz we can appreciate the typical Gaussian profile of single-mode corrugated horns, while at 220\,GHz the main beam shape is a flat-top resulting from multi-mode propagation. The sidelobes are low, less than $\sim-$30\,dB at angles larger than $\sim 30^\circ$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm]{figures/bp_sim.pdf}
\end{center}
\caption{\label{fig_horn_simulated_pattern}Co-polar simulated beam patterns (E, H, and 45$^\circ$ planes) at 150\,GHz (top row) and 220\,GHz (bottom row).}
\end{figure}
\subsubsection{Experimental setup and procedures}
\label{sec_setup_procedures}
The experimental setup consisted of a Vector Network Analyser (VNA) equipped with millimeter extensions for full 2-ports characterization in the 110--170\,GHz and 170--260\,GHz bands. To sample the beam patterns, the TD feedhorn array was mounted on a goniometer fixed on an optical bench. Since the feedhorn output waveguide is circular, we used a set of adapters to connect the rectangular waveguide of the millimeter extension to the feed.
The following list summarizes the components in our experimental setup:
\begin{enumerate}
\item VNA Agilent Technologies PNA-X\textsuperscript{\textregistered} model N5246A
\item Agilent Technologies Millimeter Head Controller model N5261A
\item Two OML millimeter extensions model V06VNA2-T/R-A for full 2-port S-matrix characterization in the 110--170\,GHz band
\item Radiometer Physics 25\,dB gain corrugated circular feedhorn for 110--170\,GHz band measurements used to illuminate the array
\item D-band OML TRL calibration kit
\item Two VDI millimeter extensions model WR4.3VNATxRx-M for full 2-port S-matrix characterization in the 170--260\,GHz band
\item Millitech G band 23\,dB rectangular standard horn for 144--220\,GHz measurements used to illuminate the array (this horn is single-moded up to 240\,GHz and the beam patterns in the upper band are limited to 220\,GHz)
\item VDI TRL calibration kit for WR4.3 band
\item Home-made custom adapter (circular waveguide) to fit the horn non-standard flange to the UG-387U standard flange
\item Millitech rectangular-to-circular waveguide taper for the 110--170\,GHz band
\item Custom Microwaves taper for 110--170\,GHz to 170--260\,GHz
\item Edmund Optics manual X-Y-$\theta$ stages
\item Newport optical bench
\end{enumerate}
\begin{figure}
\begin{center}
\includegraphics[width=7.0cm]{figures/WR6setup.jpeg}
\includegraphics[width=7.0cm]{figures/WR4_3setup.jpeg}
\end{center}
\caption{\label{fig_em_measurements_setup}\textit{Left}: the setup used for beam pattern measurements in the lower band (110--170 GHz). \textit{Right}: the setup used for beam pattern measurements in the upper band (170--260 GHz). In this case the length of cables allowed us to span the range ($-45^\circ,+45^\circ$).}
\end{figure}
To measure the return loss we connected the array to the VNA by means of a cascade of adapters. To clean the data from the effects of the mismatch in the adapters chain we performed a time domain gating, retaining the back-scattered signals coming only from the horn. To this aim, before measuring the DUT scattering parameters, we carried out a TRL calibration of the system to identify the reference plane from which we calculated the gating window to mask the undesired signals.
In the beam pattern measurements the experimental setup was almost coincident with the one used for the return loss. The only extra components were a pair of corrugated circular standard gain horns (Radiometer Physics) one used to illuminate the feedhorn array and the other for reference. In these measurements we moved the DUT in azimuth with an angular step of 1$^\circ$ and selected the proper reference plane ($E$-plane or $H$-plane) by properly rotating the launcher and the DUT.
It is important to underline that these measurements were conducted considering only the principal propagation mode in both the 150 and 220\,GHz bands. Consequently also the results of the simulations displayed in Figures~\ref{fig_horns_measured_return_loss}, \ref{fig_measured_patterns_150GHz} and \ref{fig_measured_patterns_220GHz} regard single-mode propagation.
\subsubsection{Results}
\label{sec_results}
In this section we summarize the measured return loss and beam patters with simulations and show that we obtain an overall match within the uncertainties given by the mechanical differences among the horns.
\paragraph{Return loss.} In Figure~\ref{fig_horns_measured_return_loss} we show the results of the return loss measurement in both bands compared with the simulations. The orange area is the envelope of the return loss simulated for all the 128 feedhorns in the array, each with its own measured profile, while the blue area is the envelope of the measured return loss for all the tested horns (refer to Figure~\ref{fig_horns_measured_return_loss}).
We see that the measured reflection matches the simulation, within the scatter given by the mechanical differences among the horns. We also see that the average achieved return loss at 150\,GHz lies around $-$20\,dB, while in the higher band it is around $-$25\,dB up to 230\,GHz and then degrades to about $-$10\,dB as expected. The large scatter among simulations is likely to be caused by the out-of-spec in the mechanical tolerance discussed in Section~\ref{sec_second_prototype_metrology}. Given the improvements adopted in the manufacturing procedure we believe that this scatter is significantly reduced in the FI horn array.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm]{figures/td_measured_rl.pdf}
\end{center}
\caption{\label{fig_horns_measured_return_loss}Measured return loss compared with simulation.}
\end{figure}
The reader may notice that the measured return loss between 190 and 230\,GHz is about $-$25\,dB, therefore 5\,dB lower than the value resulting from the simulation of the nominal feedhorn (see the left panel of Figure~\ref{fig_horn_simulated_return_loss_xpol}). This is because the measurements and the simulations displayed in Figure~\ref{fig_horns_measured_return_loss} are relative to single-mode propagation, while the simulation in Figure~\ref{fig_horn_simulated_return_loss_xpol} considers all the possible modes that can propagate in the 220\,GHz band.
\paragraph{Beam patterns.} We show our beam pattern measurements compared with simulations in Figures~\ref{fig_measured_patterns_150GHz} and \ref{fig_measured_patterns_220GHz}. Also in these figures the orange area is the envelope of the simulated patterns for all the feedhorns in the arrays and the blue area is the envelope of the measured patterns.
In the 150\,GHz band (Figure~\ref{fig_measured_patterns_150GHz}) measured $E$- and $H$-plane diagrams for three frequencies: 145, 150 and 155\,GHz. Measurements match simulations very well (with a few dB discrepancy) down to about $-$30\,dB. The scatter increases at larger angles, where the detected power is smaller and the measurement becomes sensitive to signal reflections.
We have obtained similar results in the 220\,GHz band (Figure~\ref{fig_measured_patterns_220GHz}). In this case we measured only the $H$-plane diagram at five frequencies, equally spaced between 190 and 230\,GHz. Also in this band there is a very good match between measurements and simulations down to $-$30\,dB.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm]{figures/bp_meas_150.pdf}
\caption{\label{fig_measured_patterns_150GHz}Measured co-polar and cross-polar beam patterns at 150\,GHz compared with simulations. \textit{Left:} $E$-plane. \textit{Middle:} $H$-plane. \textit{Right:} 45$^\circ$ cross-polar plane.}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=16cm]{figures/bp_meas_220_1.pdf}
\mbox{}\\
\includegraphics[width=10.67cm]{figures/bp_meas_220_2.pdf}
\caption{\label{fig_measured_patterns_220GHz}Measured co-polar ($H$-plane) beam patterns in the 220\,GHz frequency band.}
\end{center}
\end{figure}
\section{Introduction}
\label{sec_introduction}
\input{introduction}
\section{The horn-switch system}
\label{sec_horns_switches_system}
\input{horns_switches_system}
\section{Back-to-back feed horns}
\label{sec_back2back_prototype}
\subsection{Feed horns requirements and design}
\label{sec_feed_horns_requirements_design}
\input{feed_horn_requirements_design}
\subsection{The technological demonstrator feed horn array}
\label{sec_td_horns}
\input{manufacturing_platelet}
\subsection{Mechanical measurements and achieved tolerance}
\label{sec_second_prototype_metrology}
\input{second_prototype_metrology}
\subsection{Electromagnetic measurements}
\label{sec_electromagnetic_measurements}
\input{horn_electromagnetic_measurements}
\subsection{The full instrument feed horn array}
\label{sec_fi_horns}
\input{full-instrument-horn-array}
\section{Switch system}
\label{sec_switches}
\subsection{Switch requirements and design}
\label{sec_switches_requirements_design}
\input{switches_requirements_design}
\subsection{Single channel prototype}
\label{sec_single_channel_prototype}
\input{single_channel_prototype}
\subsubsection{Switch electromagnetic measurements}
\label{sec_switch_electromagnetic_measurements}
\input{switch_electromagnetic_measurements}
\subsection{The technological temonstrator switch array}
\label{sec_td_switches}
\subsubsection{Switch manufacturing}
\label{sec_TD_switch_manufacturing}
\input{TD_switch_manufacturing}
\subsubsection{Switch cryogenic tests}
\label{sec_TD_switch_cryogenic_test}
\input{TD_switch_cryogenic_test}
\subsection{The full instrument switch array}
\label{sec_fi_switches}
\input{FI_switch_array}
\section{Conclusions}
\label{sec_conclusions}
\input{conclusions}
\section*{Acknowledgements}
\label{sec_acknowledgements}
\input{acnowledgements}
\bibliographystyle{JHEP}
\subsubsection{Experimental procedures}
\label{sec_metrology}
We tested the mechanical tolerance according to two different procedures. First we visually inspected the inner profile of a sacrificial brass sample that was cut to allow us to magnify the shape of the antenna teeth and grooves. Then we used a metrological machine (Werth ScopeCheck 200) to measure the position, diameter and deviation from circularity of each hole in the platelets of the final array.
\paragraph{Visual inspection.} The left panel in Figure~\ref{fig_details_of_apices} shows a section of the brass prototype. The enlargement in the right panel highlights the presence of cusps ($\lesssim 0.06$\,mm high) on the profile of all the corrugations that are the effect of a non uniform erosion of the metal during the etching process.
This non uniformity is a limitation which is inherent in the chemical etching process, so that we can expect that the antennas produced with this method present imperfections in their corrugated profile. In Section \ref{sec_impact_mechanical_imperfections} we discuss the impact of these defects on the feedhorn performance for QUBIC.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=7cm]{figures/feedhorn_section}
\includegraphics[width=7cm]{figures/feedhorn_apexes}
\end{center}
\caption{\label{fig_details_of_apices}\textit{Left.} A section of the first feedhorn prototype. \textit{Right.} Detail of the cusps on teeth and grooves resulting from non uniform chemical erosion.}
\end{figure}
\paragraph{Metrology.} Figure~\ref{fig_metrology_machine} shows the Werth ScopeCheck 200 that performs precision measurements using either an optical or a tactile device. In our setup we used the optical sensor, which can be moved in three dimensions over a glass work plane where we laid our platelets.
In the picture we see the two monitors (one to observe the hole profiles and the second to control the machine), and the control console in front of the computer keyboard. On the right part of the picture, in the background, we see the glass plane with the optical sensor. In Section~\ref{sec_metrology_results} we discuss the results of the metrological measurements.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm]{figures/metrological_machine}
\end{center}
\caption{\label{fig_metrology_machine}Werth ScopeCheck 200 metrology machine.}
\end{figure}
\subsubsection{Impact of mechanical imperfections on electromagnetic performance}
\label{sec_impact_mechanical_imperfections}
We assessed the impact of the imperfections in the feedhorn profile caused by the etching process by computing the return loss and the co-polar radiation patterns on the $E$ and $H$ planes considering two cases: (i) the nominal profile, and (ii) a profile modified inserting a step-like defect on teeth and grooves of all the corrugations (see Figure~\ref{fig_cusp_model}).
\begin{figure}[h!]
\begin{center}
\includegraphics[width=10cm]{figures/cusp_model.pdf}
\end{center}
\caption{\label{fig_cusp_model}Sketch of the model used to simulate the imperfections in the feedhorn profile.}
\end{figure}
\paragraph{Return loss.}
\label{sec_return_loss_cusps}
Figure~\ref{fig_rl_cusps} shows the effects of the defects on the return loss. We see that they do not change significantly the overall level, but shifts some of the resonances in frequency. In general, however, we can consider the impact on the return loss negligible.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm]{figures/rl_sim_cusp.pdf}
\end{center}
\caption{\label{fig_rl_cusps}Impact of defects on the feedhorn return loss.}
\end{figure}
\paragraph{Radiation pattern at the center frequency.}
\label{sec_pattern_simulation}
Figure~\ref{fig_beam_cusps} shows the simulated radiation patterns (E-plane, H-plane, and 45$^\circ$ plane) at 150\,GHz (top two rows) and 220\,GHz (bottom two rows) for the two cases studied. The bottom plot in each figure shows the difference in dB of the beam patterns for the two cases. In the main beam region ($-15^\circ < \theta < 15^\circ$) the difference is less than 0.05\,dB, and over all the $-90^\circ \leq \theta \leq 90^\circ$ range the difference is within $\pm$2\,dB.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=14cm]{figures/bp_sim_cusp.pdf}
\end{center}
\caption{\label{fig_beam_cusps}Impact of defects on the co-polar radiation patterns. The black line in the bottom panel shows the difference between the two beam patterns. The top plots refer to 150\,GHz simulations, the bottom plots refer to 220\,GHz simulations.}
\end{figure}
\subsubsection{Results of metrological measurements}
\label{sec_metrology_results}
We have carried out metrological measurements of both feedhorn arrays and compared the manufacturing precision with the maximum achievable tolerance of the chemical etching process, which is $\pm$0.05\,mm. In this section we will refer to the two arrays as array-1 and array-2.
We measured the holes of each antenna and alignment pin for all the aluminum plates, compared the measured positions and diameters with their nominal values and calculated the form tolerance (FT) of each hole (see the sketch in Figure~\ref{fig_form_tolerance} for a definition of this parameter). This rich set of measurements allowed us to obtain the actual mechanical profiles of all the feeds in the array that we used to simulate their actual electromagnetic behavior. We then compared this family of simulations with the electromagnetic parameters measured in the laboratory, as explained at the beginning of Section~\ref{sec_electromagnetic_measurements}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=10cm]{figures/form-tolerance.pdf}
\end{center}
\caption{\label{fig_form_tolerance}Definition of the form tolerance parameter.}
\end{figure}
The boxplots of Figure~\ref{fig_metrology_results} show the deviation of the measured Cartesian center coordinates of the antenna holes from their nominal value ($\Delta x$, top-left, and $\Delta y$, top-right), the deviation between the measured and nominal hole diameters (bottom-left), and the corresponding distribution of FT values (bottom-right). The red line corresponds to the expected deviation, while the green area highlights the expected manufacturing tolerance. The measurements are related to the array-1.
As one can see, all the antenna positions comply with the manufacturing tolerance, while more than $90\%$ of the antenna diameters are out of specification, generally larger than expected and distributed around two peaks: $\Delta d_1$ = 0.07\,mm and $\Delta d_2$ = 0.15\,mm, with a maximum deviation of 0.25\,mm. The measured shape tolerances show no significant deviation from circularity.
We obtained similar results for the alignment pins of array-1 and for both antenna holes and pins of array-2, but they are not reported here for simplicity. We measured also the top and bottom plates of the arrays and they are in compliance with the milling precision tolerance of 0.03\,mm.
The out-of-spec was due to a loose control of the chemical etching time. Indeed, one can see that the measurements are grouped in blocks of plates and the average deviations from the nominal diameters follow a bi-modal distribution. Discussing with the company that performed the etching we understood that the plates were treated in batches, and the time was dependent also on other items (independent of QUBIC) in their production line. These problems were solved in the production of the horns for the final instrument (see Section~\ref{sec_fi_horns}) by strictly controlling the etching time.
In Section~\ref{sec_electromagnetic_measurements} we discuss the effect of this out-of-spec on the TD horns electromagnetic performance, where we compare the measured return loss and beam patterns with simulations run with the nominal and measured antenna profiles.
The boxplots in Figure~\ref{fig_metrology_results} also highlight an oscillatory, almost sinusoidal pattern in the measurements of the holes centers coordinates as a function of the antenna number. This is likely a systematic effect in our measurement. In fact, this behavior correlates with the row-by-row scanning of the antenna holes in the square antenna array. Unfortunately, however, we could not clearly identify this effect, neither in the measurement strategy nor in the measurement machine, so that this remains a reasonable hypothesis that is not demonstrated yet. For this reason we preferred not to decorrelate this effect from the data and left it as an additional source of uncertainty.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=7cm]{figures/M1_BP_scarti_X_64Antenne.pdf}
\includegraphics[width=7cm]{figures/M1_BP_scarti_Y_64Antenne.pdf}
\includegraphics[width=7cm]{figures/M1_BP_scarti.pdf}
\includegraphics[width=7cm]{figures/M1_HISTO_FT.pdf}
\end{center}
\caption{\label{fig_metrology_results}Results of the metrological measurements of the antenna holes of array 1. \textit{Top-left} and \textit{top-right}: deviation of measured $x$ and $y$ center coordinates from the nominal value. \textit{Bottom-left:} deviation of measured diameters from the nominal value. \textit{Bottom-right}: measured form tolerances. The red line corresponds to the expected deviation, while the green area highlights the expected manufacturing tolerance.}
\end{figure}
|
1204.1578
|
\section{Introduction}
The role that spiral arms play in affecting the process and efficiency of star formation in galaxies is not yet clear. They may simply organise the gas of the interstellar medium (ISM), along with its molecular clouds,
into regions of higher density, thus increasing the local star-formation rate density.
Increases in star-formation efficiency (SFE) could result from such crowding, via a rise in the incidence or strength of local feedback, e.g. from earlier massive star formation, inducing additional star formation.
On the other hand, spiral arms may be more direct triggers of star formation. They may raise the probability of cloud-cloud collisions or increase the efficiency with which molecular clouds form from the neutral gas, via the shocks expected as the ISM gas enters the arm, or by altering the internal state, i.e.\ the average mass, velocity dispersion or lifetime of molecular clouds so as to affect their internal SFE.
\citet{HT98} found evidence, in H{\sc i} and CO data in W3/4/5, that the molecular fraction of the gas content in the outer Perseus spiral arm was around ten times higher than in the inter-arm regions on the line of sight. This result implies that spiral arms dramatically raise the efficiency with which molecular clouds are produced out of the neutral gas. In turn, this suggests that spiral density waves trigger additional star formation, via the creation of new molecular clouds, as modelled by, e.g., \citet{Dobbs06}. In contrast, recent observations of two external spiral galaxies by \citet{Foyle} indicate that the H$_2$/H{\sc i} fraction and the infrared- and UV-traced SFE are not significantly enhanced in spiral arms relative to the inter-arm gas. Also, \citet{leroy} conclude that the fraction of GMCs formed from H{\sc i} is governed by ISM physics acting on relatively small scales, i.e.\ the H$_2$ formation/destruction rate balance and stellar feedback. \citet{krumholz09} predict that, except in starburst conditions, molecular-cloud properties are dominated by internal radiative feedback and not the environment.
\citet{Dobbs11} suggest that spiral arms are mainly organising features,
whose main effect on the interstellar medium is to delay and crowd the gas, which is deflected from circular orbits while within the arm. The SFR is increased indirectly by enabling longer-lived and more massive giant molecular clouds. \citet{Roman10} conclude from observations that molecular clouds within spiral arms are more long-lived than in the inter-arm gas, which would lengthen the star-formation timescale for a typical cloud, increasing the SFE as a result. If larger or denser clouds are formed, it may be significant. \citet{krumholz10} predict that the column density of clouds affects the mass function of clusters that form within them via radiative heating which suppresses fragmentation in the higher-column clouds but does not significantly affect the overall SFR/E.
It is also likely that spiral arms are different from each other \citep[e.g.][]{Benjamin05}.
Also, the inner and outer portions of spiral arms may influence star formation differently. The entry
shock experienced by the ISM gas entering a spiral arm should only exist inside the co-rotation radius, where there is a differential velocity between the spiral pattern speed and the orbital rotation speed of the
galactic ISM. Outside this radius (thought to be just beyond the Solar circle in the Milky Way \citep{lepine}, supernovae may be the dominant mechanism determining the state of the ISM and, hence, star formation (e.g.\ \citealp{kobayashi}, \citealp{dib}). The presence of the Galactic bar may also affect star formation, especially near the bar ends where the
pattern rotates at the same speed as the adjacent orbiting gas and where crowding between orbits in the bar potential and external circular orbits may create a higher cloud collision rate.
Despite the abundance of clues, or perhaps because of it, our knowledge of the effect of spiral arms on the star-formation rate or efficiency is still poorly developed and the main questions about the relationship between large-scale Galactic structure and star formation remain unanswered.
This paper reports the results of an investigation into the dependence of the efficiency of star formation on Galactocentric radius and proximity to Galactic spiral arms. The SFE is estimated by the ratio of the luminosity produced by infrared-selected, massive young stellar objects (YSOs) and H{\sc ii} regions to the CO-traced mass in molecular clouds. In Sections 2 and 3 we describe the data and present the results. In Section 4 we discuss the findings and the implications for understanding the effect of Galactic structure on star formation.
\section[]{Data}
The data used for this study consist of the sample of molecular clouds extracted from the $^{13}$CO J=1--0 BU/FCRAO Galactic Ring Survey (GRS: \citealp{Jackson}) by \citet{Rathborne09} which have been
positionally matched to mid-infrared-selected, massive young stellar objects
and H{\sc ii} regions from the Red MSX Source (RMS) survey by \citet{Urquhart11}.
The result is a sample of molecular clouds with IR-detected high-mass star
formation, as well as a complementary sample of clouds without RMS detections, all with
kinematic distances, estimated cloud masses and total source luminosities. Distance
ambiguities were resolved mainly by \citet{Roman09} with additional determinations by
\citet{Urquhart11}, in which details of the construction of the matched sample can also be found.
The GRS cloud-mass estimates were revised upwards by a factor of several in a re-analysis by \citet{Roman10}. The spatial coverage of the data is $17^\circ\!.9 < l < 55^\circ\!.7$ and $|\,b\,| \le 1^\circ$, and the velocity range is $-5$ to 135\,km\,s$^{-1}$ for $l < 40^{\circ}$ and $-5$ to 85\,km\,s$^{-1}$ at $l > 40^{\circ}$, both set by the coverage of the GRS. The RMS sample is complete to $L_{\rm bol} > 10^4$\,L$_\odot$ out to a heliocentric distance of $\sim$14\,kpc and covers a Galactocentric radius ($R_{\rm GC}$) range of 2.5 to 8.5\,kpc. The combined sample consists of 176 RMS sources with 123 GRS cloud associations and 423 GRS clouds with no matching RMS detection above $10^4$\,L$_\odot$.
Only GRS clouds with masses above $5\times10^4$\,M$_{\odot}$\ are included in the sample. This is a more conservative limit than adopted by \citet{Roman10} ($1.1\times10^4$\,M$_{\odot}$). The sample was also limited to sources with heliocentric distances greater than 2\,kpc, in order to remove local sources that might affect the results at $R_{\rm GC} \simeq 8$\,kpc.
Source luminosities were obtained by constructing SEDs from various public data sources
and fitting them with the YSO model fitter of \citet{robitaille}. Luminosities are effectively bolometric
although necessarily dominated by infrared data (see \citealp{Mottram11b} for details).
\section{Analysis and Results}
\begin{figure}
\vspace{6.8cm}
\special{psfile=sfe_fig1.eps hscale=52 vscale=52 hoffset=-2 voffset=5}
\caption{
Galactic surface density of molecular mass in the GRS region as traced by clouds in the catalogue of \citet{Rathborne09}, using the revised masses of \citet{Roman10} and a lower limit of $5 \times 10^4$\,M$_{\odot}$ per cloud. The $x$ scale is Galactocentric radius and the bin size is 0.5\,kpc. Errors are derived from the mass uncertainties listed by \citet{Roman10}.
}
\label{COmass}
\end{figure}
\begin{figure}
\vspace{6.8cm}
\special{psfile=sfe_fig2.eps hscale=52 vscale=52 hoffset=-2 voffset=5}
\caption{
The Galactic surface density of RMS source luminosity above the completeness limit of $10^4$\,L$_{\odot}$, as a function of Galactocentric radius. Sources within 2\,kpc of the Sun and those associated with GRS clouds of $M<5 \times 10^4$ M$_{\odot}$\ have been removed. The error bars are based on the average uncertainty on the total IR luminosity of 34\% estimated by \citet{Mottram11}
}
\label{Lperarea}
\end{figure}
\begin{figure}
\vspace{6.8cm}
\special{psfile=sfe_fig3.eps hscale=53 vscale=53 hoffset=-7 voffset=5}
\caption{
The ratio of integrated RMS source luminosity to mass in GRS clouds ($L_{\rm bol}/M_{\rm CO}$) as a function of Galactocentric radius. The $y$ scale is plotted logarithmically in order to show detail at low levels.
}
\label{SFEbasic}
\end{figure}
Figure \ref{COmass} shows the mean mass surface density of molecular gas in $^{13}$CO-traced GRS clouds ($\Sigma_{M_{\rm CO}}$), selected as described in Section 2 above, as a function of $R_{\rm GC}$. This result is the same as that in Fig.\ 8 of \citet{Roman10} and has been seen in a number of previous works (e.g.\ \citealp{lbx84}; \citealp{liszt93}). There is a large peak in $\Sigma_{M_{\rm CO}}$ at $R_{\rm GC} = 4-5$\,kpc. This region corresponds to the Scutum spiral-arm origin and inner tangent and the end of the Galactic bar at Galactic longitude $l \sim 30^{\rm o}$, where the massive star-forming regions W43 and G29.96 are located (see, e.g., Bally et al., 2010, \citealp{Luong11}). The Scutum arm material occupies a relatively large range of velocities and kinematic distances (\citealp{Luong11}, \citealp{eden}) but a much narrower range in $R_{\rm GC}$ and so is a well defined feature using this scale.
At radii less than $\sim$4\,kpc, $\Sigma_{M_{\rm CO}}$ falls rapidly to comparatively very low values. At larger radii, $\Sigma_{M_{\rm CO}}$ declines slightly less steeply, with two additional peaks at $R_{\rm GC}$ = 6--6.5 kpc and at 7.5--8 kpc. These two zones correspond to the average radii of the inner segments of the Sagittarius and Perseus spiral arms, respectively, as they pass through the GRS region.
Figure \ref{Lperarea} shows the integrated luminosity of RMS MYSOs and H{\sc ii} regions per unit Galactic surface area ($\Sigma_{L_{\rm bol}}$) as a function of $R_{\rm GC}$.
Three peaks can again be identified in the distribution at 4--4.5 kpc, 6--6.5 kpc and 7.5--8 kpc, corresponding to those seen in the $\Sigma_{M_{\rm CO}}$
distribution of Figure \ref{COmass}. This time, however, the contrast between the three features is less marked and the contrast with the background levels is greater. The background level of $\Sigma_{L_{\rm bol}}$ also falls steadily with radius beyond $R_{\rm GC} \simeq 5$\,kpc, but less steeply than the mass surface density. These features were also seen, in the same data, by \citet{Urquhart11}.
Figure \ref{SFEbasic} presents the ratio of the total luminosity $L_{\rm bol}$ associated with RMS sources to total molecular mass $M_{\rm CO}$ in GRS clouds in each radial bin. From $R_{\rm GC} = 2.5$ to 5.5\,kpc, $L_{\rm bol}/M_{\rm CO}$ has a low value ($\sim 0.18$\,L$_{\odot}$/M$_{\odot}$) and is nearly flat and featureless. At $R_{\rm GC} > 5.5$\,kpc, $L_{\rm bol}/M_{\rm CO}$ begins to rise and goes through two discrete, significant peaks, the first at $R_{\rm GC}$ = 6--6.5\,kpc, where the value rises to $\sim 0.63$\,L$_{\odot}$/M$_{\odot}$, 70\% higher than the adjacent bins, and a much stronger one of $\sim 5.9$ L$_{\odot}$/M$_{\odot}$\ at $R_{\rm GC}$ = 7.5--8\,kpc, around six times larger than in the bins either side. The latter two features again correspond to the radii of the Sgr and Per arms, respectively (see Figure \ref{galplan}).
\begin{figure}
\vspace{14cm}
\special{psfile=milky_way_distribution_grs_north.eps hscale=63 vscale=63 hoffset=0 voffset=-18}
\caption{
The distribution of RMS YSOs that have assigned distances within the GRS region, superimposed on part of the sketch plan of the Galaxy by Robert Hurt of the {\em Spitzer} Science Center, in consultation with Robert Benjamin. The key demonstrates the approximate luminosity of each RMS source. The figure is adapted from \citet{Urquhart11}
}
\label{galplan}
\end{figure}
The GRS survey does not cover the two spiral arms beyond Perseus which are the Norma arm and the distant, outer Scutum-Centaurus arm recently detected by Dame \& Thaddeus (2011). Although at least partly within its spatial range, these are both outside the Solar circle and so at negative relative velocities, and thus not detected by the GRS. The outer Scutum-Centaurus arm also lies partly above the latitude range of the GRS, following the upward warp of the Plane.
\section{Discussion}
\subsection{Star-formation efficiency vs the massive YSO luminosity function}
The quantities plotted in Figures \ref{COmass} and \ref{Lperarea}, $\Sigma_{M_{\rm CO}}$ and $\Sigma_{L_{\rm bol}}$, depend on the number density of molecular clouds as well as on any physical differences in the sources. Their ratio, $L_{\rm bol}/M_{\rm CO}$ (Figure \ref{SFEbasic}), is independent of source crowding and therefore reveals differences in at least the outcome of the star-formation process as a function of Galactic radius. $L_{\rm bol}/M_{\rm CO}$ is determined by a combination of the mean star-formation efficiency (SFE), which is the star-formation rate (SFR) per unit gas mass, integrated over the relevant timescale, and the luminosity function (LF) of the massive young stars. If the SFE is high enough to cause significant depletion of the molecular gas mass reservoir in a star-formation time, its dependence on the SFR may become non-linear.
An increase in the value of $L_{\rm bol}/M_{\rm CO}$ can therefore be produced by one or more of the following: a rise in the SFR per unit gas mass, a shallower LF (i.e.\ weighted towards high-luminosity sources), or a long time period. The timescale sampled by the data is limited to the lifetimes of those evolutionary stages traced by the RMS survey. These are the massive YSO (MYSO) and compact H{\sc ii}-region stages, the durations of which have both been determined to be $<5\times 10^{5}$\,yr by \citet{Mottram11}. This timescale is short enough, compared to the lifetime of a molecular cloud, that the data provide a snapshot of the current star formation. We therefore consider that it is not necessary to account for differing lengths of time over which star formation might have continued.
Where the LF of forming stars is invariant, the ratio $L_{\rm bol}/M_{\rm CO}$ is simply a measure of the current SFE. In this case, Figure \ref{SFEbasic} indicates a significantly higher SFE in the two outer spiral arms (Sgr and Per) compared to that in the region with $R_{\rm GC} < 6$\,kpc and, importantly, relative to that in the neighbouring inter-arm clouds. This suggests that physical conditions in the molecular clouds within these arms are altered in some way. Additionally, Fig.\ \ref{SFEbasic} shows that the average SFE is significantly higher in the inner Perseus arm compared to the inner Sgr arm, and so that conditions for star formation may be different from arm to arm.
Despite very strong peaks in both
$\Sigma_{M_{\rm CO}}$ and $\Sigma_{L_{\rm bol}}$ at $R_{\rm GC}$ = 4--5\,kpc (Figures \ref{COmass} and \ref{Lperarea}), $L_{\rm bol}/M_{\rm CO}$ in this inner region is low and nearly constant.
These inner radii contain the Scutum spiral-arm tangent and the near end of the Galactic bar, including the massive star-forming regions W43 and G29.96. The former has been described as starburst-like (e.g.\ \citealt{bally}, \citealt{Luong11}), but Figure \ref{SFEbasic} implies that the intense concentration of star formation found there is largely the result of the huge amount of molecular gas along that line of sight and not to a significantly elevated SFE. This result is consistent with those of \citet{eden}, who find no significant difference between the mass fraction of dense clumps in the molecular clouds associated with W43 and that in the foreground and background clouds on the same line of sight.
\begin{figure}
\vspace{6.8cm}
\special{psfile=sfe_fig4.eps hscale=52 vscale=52 hoffset=-2 voffset=5}
\caption{
The number of RMS MYSO sources above $10^4$\,L$_{\odot}$\ per unit molecular gas mass traced by GRS clouds ($N_{\rm RMS}/M_{\rm CO}$). The error bars represent Poisson errors on the RMS source counts combined with uncertainties on mass estimates from \citet{Roman10}, with the latter adjusted for a distance uncertainty of 1\,kpc.
}
\label{NperCOmass}
\end{figure}
While there is little strong observational evidence to support the hypothesis that the IMF of massive stars is sensitive to the initial conditions for star formation \citep{Bastian10}, it is still possible that $L_{\rm bol}/M_{\rm CO}$ may also depend on the LF of the massive young stars that are forming. In such a case, the same SFE but with a flatter IMF would yield a higher value of $L_{\rm bol}/M_{\rm CO}$. Since the latter is made up of the number of YSOs formed per unit cloud mass and the LF of those YSOs, we can partly separate the two effects by examining independently the numbers of RMS sources per unit cloud mass, $N_{\rm RMS}/M_{\rm CO}$, and the RMS source LF.
Figure \ref{NperCOmass} shows $N_{\rm RMS}/M_{\rm CO}$ as a function of $R_{\rm GC}$.
Against a rising background value beyond $R_{\rm GC} \simeq 5$\,kpc, there is a small peak in $N_{\rm RMS}/M_{\rm CO}$ at 6.0--6.5 kpc, the radius of the Sgr arm. The apparent significance of this peak is barely 3 sigma because of the large Poisson error bars, but it represents an increase of around 70\% over the neighbouring points and is enough to account for the peak in $L_{\rm bol}/M_{\rm CO}$ at the same radius in Figure \ref{SFEbasic}. There is therefore no need to suspect any change in LF associated with the Sgr arm and an increase in the number of YSOs produced per unit gas mass, i.e., a simple increase in the SFE, appears to be a sufficient explanation.
In contrast, Figure \ref{NperCOmass} shows no evidence of any rise in $N_{\rm RMS}/M_{\rm CO}$ in the $R_{\rm GC} = 7.5-8.0$-kpc bin, and so a simple increase in SFE does not explain the large peak in $L_{\rm bol}/M_{\rm CO}$ in the Per arm seen in Fig.\ \ref{SFEbasic} and we need to look for changes in the LF as a function of $R_{\rm GC}$.
\begin{figure}
\vspace{6.8cm}
\special{psfile=sfe_fig5.eps hscale=53 vscale=53 hoffset=-7 voffset=5}
\caption{
The mean luminosity of RMS sources ($\avg{L_{\rm bol}}$) with $L_{\rm bol} \ge 10^4$\,L$_{\odot}$, as a function of Galactic radius, i.e.\ the total luminosity divided by the number of sources per bin. The blue squares show the effect of correcting for a $N^{1/3}$ dependence in $\avg{L_{\rm bol}}$ (see text).
}
\label{meanluminosityfig}
\end{figure}
The LF of MYSOs has been determined for the whole Galaxy by \citet{Mottram11}, but the present sample is not large enough to generate an explicit LF for each 0.5-kpc $R_{\rm GC}$ bin. However, we can crudely examine the LF as a function of $R_{\rm GC}$ via the mean luminosity $\avg{L_{\rm bol}}$, both explicitly and and by statistical tests for differences in the luminosity distributions.
Figure \ref{meanluminosityfig} shows the value of $\avg{L_{\rm bol}}$, i.e.\ the total $L_{\rm bol}$ divided by the number of RMS sources in each $R_{\rm GC}$ bin. This time we see a significant increase by a factor of $\sim$7 at the 8-kpc radius of the Per arm but no change at the Sgr arm (6.5\,kpc) and a flat or slowly rising value ($\sim (4-7)\times10^4$\,L$_{\odot}$) between 3 and 5 kpc. Some caution is required here, since the distribution of luminosities $N(L_{\rm bol})$ is a power law with a lower cutoff imposed by incompleteness and an upper cutoff at $N=1$, due to integer values of $N$. Hence $L_{\rm bol}/N$ is an increasing function of $N$, i.e.\ the sample size. If the power-law exponent is $-1.5$, then $\avg{L_{\rm bol}} \propto N^{1/3}$ exactly. The normalised correction for this bias is shown in Figure \ref{meanluminosityfig} but is obviously smaller than the $\sqrt{N}$ uncertainties on $N$. The true LF of RMS-traced massive star-forming regions may be shallower this (\citealp{Mottram11}) and the severity of the bias rises sharply for exponents more positive than $-1$. However, since there are fewer RMS sources in the 7.5-8.0 kpc Perseus-arm bin (19) than in the Sagittarius- and Scutum-arm bins (36 and 33, respectively), such a bias cannot explain the much higher $\avg{L_{\rm bol}}$ in the former. This result therefore implies a significant flattening of the MYSO luminosity function in the Per arm.
Two statistical tests of the distribution of $L_{\rm bol}$ were performed, looking for differences between the RMS sources in the Perseus-arm subsample ($R_{\rm GC} = 7.5 - 8.0$\,kpc) and the rest of the sample. The Mann-Whitney U test, which is mainly sensitive to displacements (i.e.\ differences in the means) between two samples which are not normally distributed, produced a probability that the two samples come from the same distribution of $p=0.0004$. The Kolmogorov-Smirnov test for general differences in the two distributions resulted in $p = 0.028$. The null hypothesis (that the samples come from the same distribution) can therefore be rejected at least at the 2-$\sigma$ level.
As well as the discrete features associated with the spiral arms, Figures \ref{SFEbasic} and \ref{NperCOmass} show a gradual increase in the baseline value of both $L_{\rm bol}/M_{\rm CO}$ and $N_{\rm RMS}/M_{\rm CO}$ by factors of about 5 between $R_{\rm GC} = 4$ and 8\,kpc. This gradient can be largely explained by Figure \ref{avemassfig}, which shows a steady decrease, beyond $R_{\rm GC}$ = 4\,kpc, in the mean molecular-cloud mass $\avg{M_{\rm CO}}$ by a similar factor. The latter result was also seen by \citet{Roman10} and will be discussed in more detail in a forthcoming paper.
Figure \ref{avemassfig} also shows a discrete peak in $\avg{M_{\rm CO}}$ at the radius of the Per arm, but none at the Sgr arm or the Scutum tangent region.
\subsection{The effect of individual sources within the arms}
The peaks in $L_{\rm bol}/M_{\rm CO}$ at $R_{\rm GC} \sim 6$ and 8\,kpc are both due to the presence of individual star-forming complexes with very high local values of $L_{\rm bol}/M_{\rm CO}$ within the corresponding spiral arms. At $\sim$6 kpc, W51A and W51B are associated with the GRS cloud G049.49--00.41 which has a Galactocentric distance of 6.5\,kpc (\citealp{Rathborne09}). The mass of this cloud is determined to be $(1.8 \pm 0.5)\times10^5$\,M$_{\odot}$\ (\citealp{Roman10}) and its integrated luminosity in RMS sources with $L>10^4$\,L$_{\odot}$\ is estimated to be $1.37\times10^6\,$L$_{\odot}$, giving $L_{\rm bol}/M_{\rm CO} = 7.6 \pm 2.3$\,L$_{\odot}$\,M$_{\odot}^{-1}$. Nine RMS sources are associated with this cloud \citep{Urquhart11}, seven of which are above the $10^4$-L$_{\odot}$\ completeness limit. Just three of these sources have luminosities above $10^5$\,L$_{\odot}$\ (RMS\,49.4903--00.3694, 49.5373--00.3929 and 49.4564--00.3559).
\citet{kang} estimate the total molecular gas mass in W\,51 as $2.3\times 10^5$\,M$_{\odot}$\ in at least 8 clouds while \citet{carpenter} obtain $1.2\times 10^6$\,M$_{\odot}$. \citet{harvey} set the total luminosity at $3\times10^6$\,L$_{\odot}$, using an assumed distance of 7\,kpc. The distance has been estimated by trigonomentric parallax at $5.4\pm 0.3$ kpc \citep{sato}, which implies $R_{\rm GC} = 6.3$\,kpc. At this distance, the Harvey et al.\ luminosity estimate becomes $1.8\times 10^6$\,L$_{\odot}$. The molecular gas content of W51 has been studied in detail by \citet{parsons}.
At $R_{\rm GC} \simeq 8$\,kpc, the well-known star-forming region W49A is associated with GRS cloud G43.19--00.01, whose mass is $(2.2\pm 0.3)\times10^5$\,M$_{\odot}$\ and total RMS source luminosity is $6.87\times10^6$\,L$_{\odot}$. This gives $L_{\rm bol}/M_{\rm CO} = 32\pm 6$\, L$_{\odot}$\,M$_{\odot}^{-1}$. Nine RMS MYSOs are associated with this cloud, all with luminosity above $10^4$\,L$_{\odot}$. Five of these have $L>10^5$\,L$_{\odot}$\ and the integrated luminosity is dominated by two sources (RMS\,43.1679--00.0095 and 43.1650--00.0285) with $L_{\rm bol}>10^6$\,L$_{\odot}$.
\citet{roberts} suggest that W49A is a good Galactic analogue for an extragalactic starburst system, having
gas temperatures of 50--100\,K and densities $\sim$$10^6$\,cm$^{-3}$. Its distance was determined from maser proper motions to be $11.4\pm1.2$\,kpc \citep{gwinn}. Its total luminosity was estimated at $6\times 10^6$\,L$_{\odot}$\ by \citet{harvey77} and at $>10^7$\,L$_{\odot}$\ by \citet{sievers} and \citet{dwt} and it contains more than a dozen compact H{\sc ii} regions. It has been suggested that the
high star-formation rate in W49 is the result of a cloud-cloud collision resulting from orbit crowding in the spiral arm \citep{serabyn}. The total associated gas mass in several clouds in the region has been variously assayed at $1.7\times 10^6$\,M$_{\odot}$\ \citep{miyawaki} and $1.1\times 10^6$\,M$_{\odot}$\ \citep{williams}.
If we remove the GRS cloud containing the W49A RMS sources, the $R_{\rm GC}$ = 7.5--8.0-kpc bin then has integrated $L_{\rm bol}/M_{\rm CO} = 1.03\pm 0.15$\,L$_{\odot}$\,M$_{\odot}$$^{-1}$, $N_{\rm RMS}/M_{\rm CO} = (0.9 \pm 0.3) \times 10^{-5}$\,M$_{\odot}$$^{-1}$ and $\avg{L_{\rm bol}} = (1.1 \pm 0.4) \times 10^5$\,L$_{\odot}$, similar to the levels in the adjacent bins in each case (Figs \ref{SFEbasic}, \ref{NperCOmass} and \ref{meanluminosityfig}). Removing the cloud associated with W51A \& B, the 6.0--6.5-kpc bin has $L_{\rm bol}/M_{\rm CO} = 0.38\pm 0.03$\,L$_{\odot}$\,M$_{\odot}$$^{-1}$, $N_{\rm RMS}/M_{\rm CO} = (0.5 \pm 0.1) \times 10^{-5}$\,M$_{\odot}$$^{-1}$ and $\avg{L_{\rm bol}} = (6.5\pm1.3) \times 10^4$\,L$_{\odot}$, which results in a slightly reduced peak in $L_{\rm bol}/M_{\rm CO}$, relative to the neighbouring bins.
This suggests that, without W51, the Sgr arm still has somewhat increased SFE while the Per arm has a flatter than normal LF solely due to the presence of the W49A massive YSOs.
\begin{figure}
\vspace{6.8cm}
\special{psfile=sfe_fig6.eps hscale=52 vscale=52 hoffset=-2 voffset=5}
\caption{
The average GRS cloud mass as a function of
Galactocentric distance.
}
\label{avemassfig}
\end{figure}
\subsection{Implications for the effect of spiral-arm structure on star formation}
What does the foregoing analysis tell us about spiral arms as potential large-scale triggers of star formation? Figure \ref{Lperarea} shows increases in $\Sigma_{L_{\rm bol}}$ by factors of about 2.2, 2.9 and 30 in the Scu, Sgr and Per arm segments, respectively, relative to the adjacent inter-arm regions. Around 70\%, 60\% and 80\%, again respectively, of these rises are due to simple source crowding. The remaining 20--40\% of the increase in $\Sigma_{L_{\rm bol}}$ is due to changes in the luminosity per unit cloud mass and may be associated with a physical effect on the clouds caused by the presence of the arms.
There is some indication in Figure \ref{avemassfig} that cloud masses may be larger in the Per spiral arm, while no similar increase is seen in the Sgr arm. This may be an indication that clouds within the Per arm, or at least in W49, may be in an altered state. If clouds are more massive, they may be less well supported on large scales. Interestingly, the simulations of \citet{krumholz11} predict that global collapse of massive molecular clouds with radiative feedback should produce a top-heavy stellar IMF because of the suppression of fragmentation of the cloud, while accretion onto the forming protostars continues. The overall SFE is not significantly affected.
Models by \citet{Dobbs08} predict that smaller clouds aggregate into larger ones within spiral arms and, as they leave the arms, clouds are subject to shear and lose molecular gas. Dobbs, Burkert and Pringle (2011) suggest that, while more massive GMCs accumulate in the arms, there is no direct effect on the SFR, only an indirect influence via longer-lived and more strongly bound clouds. Pathological SF regions like W49A may be due to cloud-cloud collisions made more likely by longer cloud lifetimes and orbit crowding within spiral arms.
There is no clear evidence in the current results that can distinguish between changes in the clouds, caused by being within an arm, and increased triggered star formation caused by higher feedback between crowded star-forming regions. However, the lack of significant increases in $L_{\rm bol}/M_{\rm CO}$ in the crowded Scutum tangent region suggests that feedback between clouds is not the dominant factor.
\citet{dib12} found no significant connection between the shear resulting from the Galactic rotation and the star-formation efficiency in the GRS molecular clouds.
The Scutum tangent lies at the bar end where there should be co-rotation between the pattern speed of the bar and the ISM. We might expect an altered star-forming environment in this region, at least from either the permanent presence of the bar potential and/or collisions between clouds in the circular orbits just outside the bar and gas following the $x_1$ orbits within it, as they reach the bar end. On the other hand, since the CO mass surface density is seen to drop rapidly inside 3\,kpc, there may not be enough molecular gas in the bar for this to be a significant effect.
\section{conclusions}
Around 70\% of the increase in the SFR density in spiral arms is due to simple source crowding within
the arms. The remaining $\sim 30$\% is the result of an increase in the luminosity coming from embedded massive YSOs, relative to the mass of molecular gas present. In the segment of the Sagittarius spiral arm included in the GRS data, this increase in $L_{\rm bol}/M_{\rm CO}$ is accounted for by a rise in the number of MYSOs per unit molecular gas mass, with no detected change in their mean luminosity compared to the nearby inter-arm areas. This implies an increase in the basic SFE in molecular clouds in the Sgr arm, and this is probably caused by a higher SFR per unit gas mass, given the short timescales sampled by the RMS data, without any change in luminosity function. In the Perseus arm segment, no increase in the number of RMS sources per unit cloud mass is detected, relative to the nearby inter-arm regions. Instead, the rise in $L_{\rm bol}/M_{\rm CO}$ is wholly accounted for by an increase in the average luminosity of the MYSOs which implies a significant change in the average luminosity function, i.e.\ in the IMF. Further, the changes in the Per arm are attributed wholly to the W49A star-forming complex. If this were removed from the data, star formation in the Per arm would be similar to that in the inter-arm gas. W49A appears to contain unusual star formation and may be genuinely starburst-like, while the major star-forming regions in Sgr (W51) and Scu (W43) may be part of a normal distribution of star-formation properties.
Compared to \citet{Foyle},who measured increases in SFR/$M$ of less than 10\% in the arms of external spiral galaxies, compared to inter-arm regions, our results show larger increases, of $\sim$30\% in the Sgr and Per arms. In the Scutum tangent region, however, the enhancement in star-formation rate density is almost entirely due to source crowding. This indicates variations within and between arms and that is is important to consider the scale to which such results correspond.
\section*{Acknowledgments}
This publication makes use of molecular line data from the Boston University-FCRAO Galactic Ring Survey (GRS). The GRS is a joint project of Boston University and Five College Radio Astronomy Observatory, funded by the National Science Foundation under grants AST-9800334, AST-0098562, \& AST-0100793.
This paper made use of information from the Red MSX Source survey database at $\mbox{www.ast.leeds.ac.uk/RMS}$ which was constructed with support from the Science and Technology Facilities Council (STFC) of the UK. TJTM and LKM were supported, in part and whole, respectively, by STFC grant ST/001847/1. The authors thank the referee, James Binney, for suggestions that have significantly improved the paper.
|
0912.4356
|
\section{Introduction}
Strongly correlated many-body systems are of interest in very diverse areas of physics. In particular,
nuclei have been explored in depth by means of electron scattering reactions for very different
kinematical situations. More than 50 years of experimentation have proved that electron
scattering provides one of the best tools for investigating the structure of nuclear systems
and their constituents~\cite{Fru84,Bof96,Kel96,Ras89,Walecka01}.
The electromagnetic interaction with which electrons probe
nuclei is well under control and is weak enough so that the process can be treated in first order
photon exchange. Under this assumption (Born Approximation), it is possible to isolate the different
components of the nuclear response by changing appropriately the electron kinematical variables.
Indeed, assuming the Plane Wave Born Approximation (PWBA), i.e., only one virtual photon exchanged and
electrons described as free particles, the inclusive $(e,e')$ quasielastic (QE) differential
cross section is written as~\cite{Fru84,Bof96,Kel96},
\begin{equation}
\frac{d\sigma}{d\varepsilon'd\Omega'}
=\sigma_M\left[v_LR^L(q,\omega)+v_TR^T(q,\omega)
\right]\, ,
\label{eq1}
\end{equation}
where $(\varepsilon',\Omega')$ are the energy and solid angle of the
scattered electron, $\sigma_M$ is the Mott cross section, and $v_L(v_T)$ the
longitudinal (transverse) leptonic kinematic factors that in the extreme relativistic limit
(ERL) for the electrons are simply given as $v_L=(Q^2/q^2)^2$ and $v_T=\tan^2(\theta_e/2)-Q^2/2q^2$
with $\theta_e$ the electron scattering angle and $(\omega,q)$ the energy and momentum
transferred in the process ($Q^2=\omega^2-q^2$).
The hadronic $R^K(q,\omega)$ response functions are constructed from the
nuclear electromagnetic tensor $W^{\mu\nu}$ given in terms of
the initial and final many-body nuclear state, and the
nuclear electromagnetic many-body current operator~\cite{Bof96,Kel96,Walecka01,Ras89}.
In the case of QE kinematics and the momentum of the exchanged photon large enough
(its wavelength being of the order of or smaller than the nucleon size),
knockout of a single nucleon is the dominant contribution to the nuclear response.
Under these conditions, the impulse approximation (IA) holds, and the
inclusive $(e,e')$ cross section can be given as the integrated semi-inclusive
single-nucleon knockout cross sections. This approximation, which is implicit in scaling
analyses, has been shown to work properly in the kinematic region dominated by
the QE process~\cite{Chinn89,Jourdan,Udias,highp,Udias3}.
Lowest nucleon resonances are mainly excited with purely transverse photons, hence they
do not affect the longitudinal response which essentially captures the purely
nucleonic contribution to the nuclear response.
Assuming that the nucleons are the only relevant degrees of freedom, sum rules have been
derived in both relativistic~\cite{Chinn89} and nonrelativistic
schemes~\cite{McVoy65}. These
sum rules can be stated separately for the longitudinal and the
transverse contributions to the inclusive cross section. In particular,
the Coulomb sum rule (CSR) states that by integrating the longitudinal strength over the full range of
energy loss $\omega$ at large enough momentum transfer $q$, one should get the total charge
(number of protons) of the nucleus. While the experimental realization of the transverse sum rule
gets contributions from resonance excitations and thus, will likely be above theoretical
estimates based only upon nucleonic degrees of freedom, the experimental CSR
is suitable to comparison with theoretical predictions. Not only the asymptotic value of the
CSR for large $q$, but also the evolution of the CSR with increasing $q$, is of interest
in order to test nuclear models and/or descriptions of the reaction mechanism.
Indeed, enormous experimental efforts have been made at different laboratories,
Saclay~\cite{Saclay1,Saclay2,Morg01},
Bates~\cite{Bates}, JLAB~\cite{CSR-JLAB}, to get separated longitudinal and transverse contributions from
QE electron scattering data.
The analysis of data and its impact on the CSR for different nuclei have been discussed in the literature,
leading to different conclusions concerning the role played by several ingredients:
nucleon correlations, final state interactions, modification of the nucleon form factors
by the nuclear medium, etc. Jourdan concluded that the integrated longitudinal
($L$) response function saturates
for $q$ high enough at the 100\% of the CSR limit~\cite{Jourdan},
and thus it is not suppressed, showing no $A$-dependent quenching. On the contrary,
from the analysis of data taken at Saclay~\cite{Saclay1}, Morgenstern and Meziani
have concluded the existence of a significant quenching of the CSR,
and have interpreted such suppression as due to the change of the nucleon
properties inside the nuclear medium~\cite{Morg01}.
Being aware of the present controversy,
the most comprenhensive effort to measure separated longitudinal inclusive
responses from several nuclei and different values of momentum transfer $q$,
in a large enough range of energy and with unprecedent high statistics and small systematic errors,
has been recently completed at JLAB~\cite{CSR-JLAB}. This experiment is
currently under analysis and preliminary results will be released in the next months.
An alternative procedure to get some insight into the CSR relies on the
information provided by the scaling properties of the longitudinal separated data. As already
shown in previous works~\cite{Day90,DS199,MDS02}, world $(e,e')$ data have clearly demonstrated
the validity of scaling and superscaling (independence of the scaled response on the kinematics
and on the nuclear target) behavior. In particular, the analysis of the
separated $L$ contribution has led to introduce a {\sl ``universal''}
superscaling function, which contains the relevant information about
the initial and final state nuclear dynamics explored by the probe~\cite{MDS02}.
Superscaling was originally introduced within
the simple Relativistic Fermi Gas (RFG) model that, albeit showing
perfect scaling and superscaling properties, yields
a superscaled function shape not in accordance with data~\cite{MDS02,scaling88}.
The experimental superscaling function has an asymmetric shape, with a
long tail exhibiting strength for energy transfers well beyond the RFG domain.
Further, most nonrelativistic models also lack the significative
strength at high-$\omega$~\cite{neutrino2}.
The presence of this tail is of relevance for the CSR analysis,
as the sum rule requires integration of the strength in the whole energy transfer range
(up to infinity), which is of course not feasible from the experimental point of view.
The integration of the experimental strength ends at some finite value
of the transferred energy,
located where the asymmetric tail of the superscaling function resides.
Thus, what is left out of the integration region from theoretical estimates of the CSR
would highly depend on whether the model does or does not reproduce this asymmetric tail.
The main aim of this work is trying to shed some light to the CSR problem,
making use of the experience acquired during the analysis of the scaling and
superscaling phenomenon~\cite{DS199,MDS02}.
\section{Scaling and Superscaling}
The usual procedure to get the scaling function consists in dividing the
inclusive differential cross section (\ref{eq1}) by the
appropriate single-nucleon $eN$ elastic cross section, weighted by the
corresponding proton and neutron numbers~\cite{DS199,MDS02,Barbaro:1998gu}
involved in the process,
\begin{equation}
f(\psi',q)\equiv k_F\frac{\left[\displaystyle\frac{d\sigma}{d\varepsilon'd\Omega'}\right]}
{\sigma_M\left[V_LG_L(q,\omega)+V_TG_T(q,\omega)\right]} \, .
\label{fscaling}
\end{equation}
$\psi'(q,\omega)$ is the dimensionless scaling variable extracted
from the RFG analysis that incorporates the typical momentum scale for the selected
nucleus~\cite{MDS02}.
The fully relativistic expressions for $G_{L(T)}$ involve proton and neutron
form factors $G_{E(M)pn}$, weighted by proton and neutron numbers, and
an additional dependence on the nuclear scale given through the Fermi momentum $k_F$
(explicit expressions are given by Eqs.~(16-19) in ~\cite{MDS02}).
Analogously, the analysis of the separated longitudinal ($L$) and transverse ($T$)
contributions leads to scaling functions,
\begin{equation}
f_{L(T)}(\psi',q)\equiv k_F\frac{R^{L(T)}(q,\omega)}{G_{L(T)}(q,\omega)}
\label{fltscaling} \, .
\end{equation}
At transferred energies above the QE peak, scaling is violated
in the transverse channel by effects beyond the impulse approximation~\cite{Day90,DS199}.
However, the available data for the $L$ response are compatible with scaling in all
the QE region. This has made it possible to extract
an experimental scaling function $f_L^{exp}$, that effectively represents the
nucleon contribution to the nuclear response under QE kinematics~\cite{DS199,MDS02}.
In this work we use the Relativistic Impulse Approximation (RIA) that
leads to a hadronic tensor evaluated from the transition
single-nucleon current matrix elements. These are constructed making use of the relativistic bound-state,
the scattering wave function and the relativistic single-nucleon electromagnetic current operator.
We guide our analysis with calculations where the bound nucleon states correspond to self-consistent
Dirac-Hartree solutions, derived within a relativistic mean field (RMF)
approach. The outgoing nucleon wave function is given as a solution of the
Dirac equation in presence of a relativistic potential which takes care of
the final state interactions (FSI) between the ejected nucleon and the residual
nucleus. In previous works~\cite{PRL05,jac06,Chiara03} it has been investigated the
role played by different descriptions of FSI: i) the use of the same relativistic
mean field employed to describe the initial bound states and ii) considering the
phenomenological relativistic optical potential derived by Clark {\it et al.}~\cite{Clark}
but with their imaginary part set to zero in order to consider all final
channels and not only the elastic one,
that we denoted as rROP. Finally, ignoring all distortion from FSI leads to the relativistic plane
wave impulse approximation (RPWIA) where the knocked out nucleon is treated as a plane wave.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.8]{exp_fsi_cc2_new2.eps}
\caption{(Color online) Superscaling function for $^{12}C(e,e')$ evaluated with the RPWIA,
rROP and RMF approaches compared to the experimental function. In the RMF case,
separate $L$ and $T$ contributions are shown.
}
\label{fLT_scaling}
\end{center}
\end{figure}
In recent works~\cite{PRL05,jac06,Amaro07} it has been shown that the RMF model,
where the same relativistic potentials
are applied to the initial and the final state, reproduces satisfactorily the magnitude and detailed
shape of $f_L^{exp}$, while other models fail to reproduce the long tail appearing at high energy
transfer $\omega$ (large positive values of the scaling variable $\psi'$).
This is clearly illustrated in Fig.~\ref{fLT_scaling} where we present
the superscaling function evaluated within
the RIA and different descriptions of FSI. Results correspond to $^{12}$C and $q=700$ MeV/c.
The scaling function obtained using the real part of
the relativistic optical potential (rROP) is compared to the plane wave limit (RPWIA) and
to results obtained in presence of the scalar and vector terms in the relativistic mean
field potential (RMF), where the separate $L$ and $T$ contributions to the scaling function are
plotted. Scaling of zeroth kind, {\it i.e.,} $f_L=f_T=f$,
has been shown to be fulfilled within RPWIA and rROP approaches~\cite{jac06,jac_plb}.
In all the cases, the CC2 prescription for the current operator has been
selected~\cite{jac06}. As observed, $f_T$ obtained within RMF is increased
with regard to $f_L$. This is due to the off-shellness of the nucleons,
modest for RPWIA and rROP and much more important for RMF because of the
stronger potentials involved in the final nucleon states. While the function $f_L$
hardly changes (a consequence of current conservation), $f_T$
exhibits a significant dependence with off-shell nucleon effects~\cite{jac06,jac_plb}.
It is worth mentioning that correlations also shift strength towards larger energy values,
as they allow for multi-nucleon knockout. Correlations have been a common ingredient
of theoretical predictions of CSR~\cite{omar1,omar2,omar3} and can also explain the asymmetrical
shape of the superscaling function~\cite{BCS}. In this work our focus is not the
explanation of the observed asymmetry, that has been discussed in previous
work~\cite{PRL05,jac06,Amaro07,jac_plb}, but rather explore its effect on the
predicted CSR values. The comparison with the experimental $L$ superscaling function,
also provided in Fig.~\ref{fLT_scaling}, shows that the RMF approach
follows closely the behavior of data describing also the asymmetrical shape
of $f_L^{exp}$.
Moreover, the RMF model, as studied in previous work~\cite{jac06, jac_plb},
fullfills the continuity equation and dispersion relations, hence being adequate to inclusive
scattering where all nucleon propagation channels, not only the elastic one described by
the optical potentials, must be incorporated. The different behavior presented by
RMF and rROP (Fig.~\ref{fLT_scaling}) is linked directly to the strong potentials present
in the RMF for large values of the energy transfer. On the contrary, rROP potentials tend to
weaken significantly with increasing energy values. This is also consistent with
the similar behavior shown by rROP and RPWIA results. The potentials modify the effective
values of the momenta at the nucleon vertex, giving rise to a shift of strength to
(asymptotical) larger values of $\omega$ (see Refs.~\cite{PRL05,jac06,Amaro07,jac_plb}).
Finally, use of relativistic optical potentials (with imaginary term) in inclusive
reactions has been applied within the relativistic Green's function (GF) approach, leading
to similar results to RMF, {\it i.e.,} with presence of the asymmetry in the scaling
function for intermediate $q$-values~\cite{pavia}.
\section{The Coulomb sum rule}
Including relativistic corrections~\cite{Forest} and the structure
of the nucleons, the explicit expression for the CSR, widely used by
experimentalists in the analysis of the separate $L$-data~\cite{Saclay1,Bates},
is written as
\begin{equation}
CSR(q)=\frac{1}{Z}\int_{\omega^+}^\infty \frac{R^L(q,\omega)}{\widetilde{G}_E^2(Q^2)}d\omega
\, \label{CSR_1}
\end{equation}
with the effective electric form factor given by
\begin{equation}
\widetilde{G}_E^2(Q^2)=\left[G^2_{Ep}(Q^2)+\frac{N}{Z}G_{En}^2(Q^2)\right]\frac{(1+\tau)}{(1+2\tau)}
\, , \label{effective}
\end{equation}
where $N$ and $Z$ are the neutron and proton numbers of the target, respectively, and
$G_{Ep}$ and $G_{En}$ the Sachs electric form factors for proton and neutron. The
term $\tau$ is the usual dimensionless quantity, $\tau\equiv |Q^2|/4M_N^2$
with $M_N$ the nucleon mass. The lower
limit in the integration $\omega^+$ includes all inelastic contributions but excludes the
elastic peak.
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.63]{Figure1.eps}
\vspace{0.25cm}
\caption{(Color online) Coulomb sum rule as a function of the energy transfer for $^{12}$C (top panel),
$^{40}$Ca (middle) and $^{56}$Fe (bottom). In each case, results obtained using the expression of
the CSR given by Eq.~(\ref{CSR_1}) are compared with predictions based on the scaling
analysis (\ref{CSR_2}) for three different values of the momentum transfer.
}
\label{CSR1_vs_CSR2}
\end{center}
\end{figure}
An analog of the CSR can be also introduced in terms of the superscaled function
and the scaling variable by taking
into account the explicit expression of the longitudinal superscaling function,
as well as its physics significance,
\begin{equation}
CSR_{scal}(q)=\int_{-\infty}^{\infty}d\psi' f_L(\psi') \, . \label{CSR_2}
\end{equation}
Here the integration limits, denoted by $(-\infty,+\infty)$, extend in reality to the range
allowed by kinematics and the experimental setup. Note that the scaling variable depends on the
transferred momentum and energy, $q,\omega$. Expression (\ref{CSR_1}) of the CSR used by
experimentalists does not exactly correspond
to Eq.~(\ref{CSR_2}) due to the fully relativistic expressions involved in
the longitudinal scaling function~\cite{MDS02} and to the different integration variable.
Thus, in order to set down the impact of the particular CSR expression on the analysis of data,
in what follows we compare results corresponding to Eqs.~(\ref{CSR_1}) and (\ref{CSR_2}). The analysis is
presented in Fig.~\ref{CSR1_vs_CSR2} where we have considered three nuclei: $^{12}$C (top panel),
$^{40}$Ca (middle) and $^{56}$Fe (bottom). In each case we show how the CSR behaves
as a function of the energy transfer $\omega$ for three different values of the momentum transfer
$q$: $0.3$, $0.5$ and $0.7$ GeV/c. We compare the results corresponding to Eq.~(\ref{CSR_1}),
denoted as CSR (dashed line), with the ones evaluated through the scaling function (\ref{CSR_2}),
denoted as Scaling (solid line). We conclude that, apart from some minor discrepancies ascribed
to the different single-nucleon expressions considered and the influence of the nuclear scale
introduced in the longitudinal scaling function, both expressions for the CSR lead to
similar results, hence drawing analogous conclusions.
Notice that in all the cases the result given by Eq.~(\ref{CSR_2}) lies slightly
below the one of (\ref{CSR_1}) for intermediate values of the energy transfer.
All results in Fig.~\ref{CSR1_vs_CSR2} have been obtained with the RIA-RMF model.
Comparing the results obtained for the three nuclei, the CSR dependence with the target is
seen to be very tiny. The CSR saturates to almost the same value
for the three nuclei: $\sim 0.9$ for $q=0.5$ and $0.7$ GeV/c and $\sim 0.7$ for $q=0.3$ GeV/c.
Moreover, the behavior of the CSR is similar for the
three targets, getting saturation, at each $q$-value, for very close transferred energies.
Results in Fig.~\ref{CSR1_vs_CSR2} allow us to focus on the
CSR predictions given by Eq. (\ref{CSR_2}) and to compare them to data arranged according
to~(\ref{CSR_1}).
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.54,angle=270]{CSR_model.eps}
\vspace{0.65cm}
\caption{(Color online) Coulomb sum rule according to Eq.~(\ref{CSR_2}) for $^{12}$C (top panels)
and $^{40}$Ca (bottom) vs the energy transfer $\omega$. Results
are shown for three values of the momentum transfer $q$, comparing RPWIA (dashed) and RMF
(solid) approaches.
}
\label{CSR_model}
\end{center}
\end{figure}
As shown in Fig.~\ref{fLT_scaling}, the function $f_L^{exp}(\psi')$ presents
a long tail extended to large $\omega$-values, which is not reproduced by
RPWIA and rROP calculations (neither by the majority of
nonrelativistic models employed in the literature~\cite{neutrino2}).
The presence of this important strength in $f_L^{exp}(\psi')$ may affect significantly
the results for the CSR. Hence, in what follows we study how the CSR depends
on the specific approach considered. To make easier the analysis we only consider two extreme
cases: RPWIA, namely no FSI, and RMF, {\it i.e.,} the presence of strong scalar and vector
potentials in the final state. The shape of $f_L(\psi')$ in both cases is quite different,
being the tail at large $\omega$-values largely absent in RPWIA.
Results are presented in Fig.~\ref{CSR_model} for $^{12}$C (top panels) and $^{40}$Ca (bottom) and
three $q$ values: $q=300$ MeV/c (left panels), $500$ MeV/c (middle) and 700 MeV/c (right).
One observes that the CSR saturates to $\sim 1$ for all $q$-values and
the two nuclei in the case of RPWIA.
On the contrary, the RMF description leads to a saturation value smaller than 1, which
grows with $q$ up to being of the order of $0.9$ for $q\geq 0.5$ GeV/c, {\it i.e.,}
where Pauli blocking is
not in effect and thus also scaling holds. It is also important to point out that saturation is reached
faster in RPWIA. This is consistent with the general symmetry shown by $f_L^{RPWIA}(\psi')$
in contrast to the long tail
presented by data and the RMF model (see Fig.~\ref{fLT_scaling}).
Part of the strength that has been shifted to
high $\omega$-values in the RMF case (because of FSI) cannot be reached within the kinematical constrains,
hence leading to RMF-CSR results being smaller than the RPWIA ones.
Comparing the results obtained for $^{12}$C and
$^{40}$Ca as well as $^{56}$Fe (not shown in the figure but following a very similar trend),
the CSR dependence on the target is very tiny (see discussion in previous figure).
This outcome is in accordance with second kind scaling property.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.5,angle=270]{SCALING_FIG4.eps}
\caption{(Color online) Integrated scaling function versus the momentum transfer $q$. Results are shown
for $^{12}$C comparing three different models, integrated up to the maximum
energy transfer allowed by kinematics (see text for details).
}
\label{CSR_q}
\end{center}
\end{figure}
The behavior of the CSR with the momentum transfer $q$ is illustrated in Fig.~\ref{CSR_q}, where
we show the results for $^{12}$C evaluated through Eq.~(\ref{CSR_2}) with different
descriptions of the FSI: RPWIA, rROP and RMF, integrated up to the maximum energy transfer
allowed by kinematics. The CC2 current operator has
been considered, as results with this prescription
agree fairly well with the superscaling function. As observed, RPWIA leads
to unity for all $q$-values, even in the region where the CSR is not
saturated in the RFG due to Pauli blocking,
{\it i.e.,} $q\leq 400-450$ MeV/c.
This is in contrast with the other two models with FSI turned on,
which show a CSR that increases with $q$ up to becoming stable.
Concerning specific CSR-saturated values, the rROP gets its maximum ($\sim 0.95$)
for $q=0.4$ GeV/c, starting to diminish slightly in the region $0.4\leq q\leq 0.9$ GeV/c up to being of the
order of $\sim 0.9$. The CSR result obtained within RMF increases with $q$
up to reaching $\sim 0.9$ for $q\approx 400-500$ MeV/c. This CSR value remains stabilized in the
whole $q$ region explored in the figure, that is, up to $q=1$ GeV/c.
For higher $q$, not presented in Fig.~\ref{CSR_q}, it can be shown that the CSR-RMF
(likewise rROP) result starts to increase approaching 1 for $q\sim 1.6$ GeV/c. However,
for so large $q$-values, caution should be drawn on the assumptions implied by our
theoretical description as well as by the extraction of the CSR from data. Finally,
note that the strong potentials involved in the RMF, both for initial and final nucleon states,
make the strength to be shifted to higher values of the (asymptotic) nucleon
momentum~\cite{Amaro07}.
Comparison between CSR theoretical results and experimental data requires
to extract the Coulomb sum rule from the longitudinal response data
by performing the integrals in Eqs.~(\ref{CSR_1}), (\ref{CSR_2}) using as upper
integration limits the specific $\omega$-cutoff values employed by the experimentalists.
In particular, in the case of $^{40}$Ca, different
experiments, Bates~\cite{Bates} and Saclay~\cite{Saclay1},
have considered different $\omega_{max}$-values
as integration limits, as shown in Tables~\ref{table1} and~\ref{table2}.
One has to keep in mind that in the case of Bates data and for $q\geq 425$ MeV/c,
the value of the maximum energy transfer included in the experimental CSR
diminishes as the momentum transfer $q$ goes up due to the
uncertainties associated with the L/T separation. This explains why in Table~\ref{table1},
while the total CSR estimated under the RMF reaches a rather constant value ($\sim 0.88$)
for transferred momenta larger than 425 MeV/c, the
predicted CSR under RMF employing the experimental energy transfer cutoff
gets smaller for $q$ increasing, after $\sim 400$ MeV/c.
\begin{table}[h]
\vspace{1cm}
\begin{center}
\begin{tabular}{cccccc} \hline \hline
\ \ $q$ [MeV/c] \ \ & \ \ $\omega_{max}$ [MeV] \ \ &\ \ CSR (total)\ \ & \ \ CSR($\omega_{max}$)\ \ & \ \ $\%$(diff.)\ \ &\ \ CSR(RFG)
\\ \hline
300 & 140 & 0.7493 & 0.6917 & 7.7 & 0.8197 \\
325 & 160 & 0.7889 & 0.7378 & 6.5 & 0.8640 \\
350 & 190 & 0.8207 & 0.7874 & 3.9 & 0.8975 \\
375 & 220 & 0.8458 & 0.8234 & 2.6 & 0.9289 \\
400 & 250 & 0.8638 & 0.8485 & 1.8 & 0.9483 \\
425 & 260 & 0.8759 & 0.8548 & 2.4 & 0.9649 \\
450 & 240 & 0.8822 & 0.8159 & 7.5 & 0.9683 \\
475 & 230 & 0.8842 & 0.7552 & 14.6 & 0.9707 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Integrated CSR evaluated within the RMF as a function of the momentum transfer $q$. Second
column presents the maximum energy loss as indicated in Bates experiment~\cite{Bates}. We
present the CSR results evaluated by extending the integration up to the maximum energy transfer allowed by
kinematics (column 3) and up to the cutoff value used in Bates (column 4). Finally, column 5
reflects the difference between both results (percentage) and column 6 presents for
reference the RFG predictions.}
\label{table1}
\end{table}
\begin{table}[h]
\vspace{1cm}
\begin{center}
\begin{tabular}{ccccc} \hline \hline
\ \ $q$ [MeV/c]\ \ &\ \ $\omega_{max}$ [MeV]\ \ & \ \ CSR (total) \ \ & \ \ CSR($\omega_{max}$)\ \ & \ \ $\%$(diff.)
\\ \hline
330 & 175 & 0.7889 & 0.7586 & 3.8 \\
370 & 195 & 0.8458 & 0.7953 & 6.0 \\
410 & 235 & 0.8638 & 0.8387 & 2.9 \\
450 & 265 & 0.8822 & 0.8490 & 3.8 \\
500 & 290 & 0.8825 & 0.8335 & 5.6 \\
550 & 310 & 0.8788 & 0.8003 & 8.9 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Same as Table~\ref{table1}, but for the kinematics considered at Saclay~\cite{Saclay1}.}
\label{table2}
\end{table}
In Fig.~\ref{CSR_exp} we present results for $^{40}$Ca corresponding to RMF
(top panel) and RPWIA (bottom) approaches. In each case, three $q$-values have
been considered, $q=300$, 400 and 500 MeV/c. The CSR is shown as a function
of the scaling variable $\psi'(q,\omega)$. We also plot, for each $q$,
the value of the scaling variable $\psi'$ corresponding to the specific
$\omega$-cutoffs given in the experimental papers.
These span the regions: $140\lesssim\omega\lesssim 150$ MeV/c for $q=300$ MeV/c,
$230\lesssim\omega\lesssim 250$ MeV/c for $q=400$ MeV/c, and
$220\lesssim\omega\lesssim 290$ MeV/c for $q=500$ MeV/c. In the latter ($q=500$ MeV/c) the
lower $\omega$-value represents the limit employed at Bates~\cite{Bates}
and the larger one the cutoff included in Saclay~\cite{Saclay1}.
These regions are presented as shadowed areas, where the color indicates the specific $q$-value
which is directly connected with the corresponding (same color) theoretical CSR result.
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.5]{CSR_Ca40_exp_v3.eps}
\vspace{0.2cm}
\caption{(Color online) Coulomb sum rule as a function of the scaling variable $\psi'$ for $^{40}$Ca.
Top panel refers to results obtained within the RMF approach and bottom to RPWIA.
The vertical shadowed bands refer to the extreme values
of $\psi'$ corresponding to the energy transfer cutoffs considered in the analysis of the experiments.
Each color refers to a different $q$-value, namely red ($q=300$ MeV/c), green ($q=400$ MeV/c) and
blue ($q=500$ MeV/c). Lower limits in each band correspond to Bates values and higher ones to Saclay
integration cutoff.}
\label{CSR_exp}
\end{center}
\end{figure}
Results in Fig.~\ref{CSR_exp} illustrate clearly the amount of saturation reached by the CSR
at the maximum $\omega$-loss taken from the experiment. Let us consider the case $q=300$ MeV/c
(solid red line and red shadowed band). Here the CSR saturates to $\sim 0.75$ for RMF and
$\sim 0.97$ for RPWIA if the integration is extended over the whole allowed range
(see Table~\ref{table1}).
On the contrary, CSR results integrated up to the shadowed area are approximately $\sim 0.70$ (RMF)
and $\sim 0.93$ (RWPIA).
This means that saturation of CSR is reached at the order of $\sim 93\%$
for RMF and $\sim 96\%$ for RPWIA at the experimental energy cutoff. In other words,
the $\omega$-values beyond the
experimental accessible region correspond to a $\sim 7\%$ contribution
to the fully integrated CSR in the RMF, and only $\sim 4\%$ in
the RPWIA case. These results reflect the increased tail of the longitudinal response in the RMF case.
It is worth recalling, however, the different CSR values emerging from the
the two approaches, $0.75$ in RMF and almost 1 ($0.97$) in RPWIA.
Similar comments apply also to higher $q$-values, $400$ MeV/c (green color) and $500$ MeV/c (blue),
although here the discrepancy between RMF and RPWIA results gets reduced because of the significant
enhancement of the CSR value in the RMF approach. For $q=400$ MeV/c, the RMF-CSR experimental
cutoff result is on average $\sim 97\%$ of the RMF-CSR for the whole range, whereas in RPWIA saturation
is already reached at the experimental energy loss. Finally, in the case of $q=500$ MeV/c some comments apply
because of the wide blue shadowed area linked to the very different $\omega$-cutoffs
considered at Bates and Saclay, $\omega_{max}=220$ MeV and 290 MeV, respectively.
For the Saclay experiment~\cite{Saclay1}, {\it i.e.,} upper limit in the shadowed band,
the CSR model evaluated up to the experimental cutoff includes
$\sim 95\%$ ($\sim 100\%$) contribution of the total CSR strength in the RMF (RPWIA)
approach. On the contrary, the contribution (integrated up to $\omega_{max}$) reduces to
$\sim 75\%$ ($\sim 95\%$) for RMF (RPWIA) in the case of the maximum energy loss used at
Bates~\cite{Bates} (lower limit of the band, $\omega_{max}=220$ MeV).
As it will be shown later on, this makes a significant difference when
comparing theoretical calculations with the CSR extracted from both experiments.
It is important to point out again that the CSR obtained in the whole allowed $\omega$ range
within the RMF approach saturates to
$\sim 0.88$ for $q\geq 400, 500$ MeV/c, that is, $\sim 12\%$
below the RPWIA result. Further the RMF-CSR result accumulated
up to the experimental energy cuttof employed in Bates,
is on average $\sim 15-18\%$ below the corresponding
RPWIA result for $q=400,\ 500$ MeV/c, and $\sim 25\%$ below for $q$ around 300 MeV/c.
This is consistent with the behavior shown in Fig.~\ref{CSR_q}.
It is worth noticing that the contribution to the CSR of the
strength outside the experimentally integrated region is different if considering RPWIA and/or RMF.
In a model like RMF which agrees with the experimentally deduced
longitudinal scaling response of Bates, the contribution
of the unobserved tail beyond the cutoff employed at Bates is around 7\% for $q\sim 300$ MeV/c,
2-3\% around $q\sim 400$ MeV/c and increases up to 15\% for the largest $q$-value
(475 MeV/c) measured at Bates.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.6,angle=270]{CSR_data_final_v6.eps}
\caption{(Color online) Coulomb sum rule compared to data. Top-left panel shows results obtained
with the RMF for $^{12}$C (green triangles) and $^{40}$Ca (red circles) and RFG with Pauli blocking
(black diamonds). In all cases, integration in Eq.~(\ref{CSR_2}) has been extended to the whole
region allowed by kinematics. Theoretical results are compared with
data from Bates corresponding to $^{40}$Ca. Top-right panel: as in previous case but RMF calculations
evaluated using the $\omega$-cutoff values given in \cite{Bates}.
Bottom panels present the ratio between RMF results
and RFG ones compared to data from Bates and Saclay (see text for details).
}
\label{theory_data}
\end{center}
\end{figure}
To conclude, a comparison between theory and experimental data is provided in
Fig.~\ref{theory_data}. First, in the left-top panel theoretical results for the CSR
evaluated with the RMF approach applied to $^{40}$Ca (red circles) and $^{12}$C
(green triangles) are presented. In both cases, CSR has been
obtained making use of (\ref{CSR_2}) and extending the upper
integration limit to the maximum value permitted by the kinematics, i.e.,
once CSR has already reached saturation. Results in Fig.~\ref{theory_data}
show the independence of the CSR on the nuclear target, within the present approaches.
For reference, we also include
the CSR evaluated with the RFG model (black diamonds). Here,
the CSR result approaches almost 1 for $q\sim 500$ MeV/c, {\it i.e.,} $q\geq 2k_F$. For lower
$q$-values Pauli-blocking effects are important giving rise to a significant reduction in
the CSR value. Notice that, although the integral of $f_L^{RFG}$ (likewise the CSR)
should be exactly one in the QE domain, the value in Fig.~\ref{theory_data},
slightly lower than 1, reflects the shift energy included in the definition
of the scaling variable $\psi'$ (see \cite{MDS02,jac06} for details).
Theoretical results are compared with the Coulomb sum rule for $^{40}$Ca
extracted from data measured at Bates~\cite{Bates} for $q$-values in the
domain, $300\leq q\leq 475$ MeV/c. On general grounds, we observe that
RMF results agree fairly well with data, lying slightly below for the
smaller $q$-values, $[300,350]$ MeV/c, and above data for $q>400$ MeV/c
where the experimental uncertainty is significantly larger.
Notice however, that the behavior shown by data, with
a depletion occurring for $q\geq 400$ MeV/c is not reproduced by theoretical
CSR calculations, which increase smoothly with $q$ approaching saturation.
This discrepancy is mainly linked to the upper integration $\omega$-limits used in the
analysis of data. Whereas theoretical CSR results were obtained through
Eq.~(\ref{CSR_2}) extending the integral up to the maximum $\omega$,
likewise $\psi'$, value permitted by kinematics, Bates CSR data on the contrary,
have been extracted
making use of Eq.~(\ref{CSR_1}) with the upper $\omega_{max}$ limit fixed, for
each $q$, according to the values given in Table II of
Ref.~\cite{Bates} (see also Table~\ref{table1}). In particular, notice the relatively
low $\omega_{max}$ values used by experimentalists for $q \sim 450,500$ MeV/c.
As we have already mentioned, significant strength in the CSR may be left out when using
relatively low energy transfer cutoffs.
This is clearly illustrated in the right-top panel of Fig.~\ref{theory_data}, where
we compare again Bates CSR data with RMF theoretical results, but these now evaluated using
as upper integration limits the same $\omega_{max}$ values considered in the experiment
(see Table~\ref{table1} in this work and Table II in~\cite{Bates}). Compared with previous results,
a decrease in the RMF-CSR is observed,
depending its magnitude on the specific momentum transfer considered: from $\sim 2-4\%$ for the
central $q$ ($[350-425]$ MeV/c) and $\sim 6-8\%$ for $q=300, 325, 450$ MeV/c,
up to $\sim 15\%$ for $q=475$ MeV/c.
This explains the depletion presented by the CSR
(theory and data) for larger $q$. Concerning the comparison between theory and data,
we observe that Bates CSR data are reproduced within the RMF approach. Only for
$q=300$ and $325$ MeV/c, RMF results underestimate data by $\sim 10-12\%$. This
discrepancy can be partly ascribed to the different expressions used to evaluate
the CSR, Eq.~(\ref{CSR_1}) for experimental data and (\ref{CSR_2}) for RMF.
As shown in Fig.~\ref{CSR1_vs_CSR2}, using (\ref{CSR_1}) and/or (\ref{CSR_2})
lead to slightly different CSR results, being the former larger
for the $\omega_{max}$-values considered in the experiment. Hence,
the discrepancy between theory and experiment in Fig.~\ref{theory_data} reduces by
$\sim 3\%-5\%$ when Eq.~(\ref{CSR_1}) is also considered within the RMF approach. Further,
for the lowest values of momentum transfer, discrete inelastic excitations of the nuclei may be present in the
data, while they are not considered in the purely QE nucleon knockout estimations of the models.
\hbox{}From this analysis, we conclude that our theoretical model describes quite consistently
Bates data, with a minor underestimation (within the experimental error bars), indicating that no
additional quenching of the relativistic models is needed, other
than the $\sim 10-15\%$ strength that is shifted outside the experimentally available region
for the $L$ channel. This is consistent with the good agreement found between the RMF model
and the experimental longitudinal scaling function.
The previous argument is also reinforced by results shown in the bottom panels
of Fig.~\ref{theory_data}. Here we present the ratio of integrated response functions to
the longitudinal strength calculated from the RFG model.
We compare data from Bates
experiment (blue squares) with those given by Meziani et al.~\cite{Saclay1} (green triangles)
and the theoretical results evaluated within the RMF approach (red circles). As in the previous
discussion, the left-bottom panel refers to RMF results evaluated by extending the integral
(\ref{CSR_2}) up to the whole region allowed by kinematics, and dividing by the RFG results.
This explains why the RMF approach leads to very similar values ($\sim 0.9$) for all $q$, as the comparison
of RMF to RFG results is rather constant with $q$ if integration includes the whole tail region.
On the contrary, in the
right-bottom panel, theoretical RMF-CSR results have been evaluated by using the specific
$\omega_{max}$-limits considered at Bates~\cite{Bates} for each momentum transfer
(red circles). We also show the RMF results obtained by using the momentum transfers $q$
and energy losses $\omega_{max}$ given by Meziani et al.,~\cite{Saclay1} corresponding to
Saclay experiment (black diamonds). Apart from the slightly different $q$ values used
in Bates and Saclay, the difference between
the RMF results corresponding to both
experimental setups comes from the energy transfer cutoffs considered
(see Tables~\ref{table1} and~\ref{table2}).
The effect of the cutoff
is particularly visible for $q\sim 475-500$ MeV/c where the larger $\omega_{max}$-values
considered in Saclay lead to higher RMF-CSR results, making the theoretical prediction
to depart even further from data.
Therefore, from the general analysis shown in Fig.~\ref{theory_data}, we
observe that RMF calculations are compatible with Bates data in the whole $q$-region,
apart from some deviation (underpredicting data) for the lowest $q=300,325$ MeV/c. On the contrary,
data from Saclay experiment
show an important
depletion ($\gtrsim 40\%$) with regards to the theoretical RMF predictions, even when these data
should include in principle more contribution from the high energy tail (compared with Bates).
This depletion is not present in Bates data
and is neither supported by our theoretical estimates. According to the analysis carried out in this work,
this difference in behaviour of Saclay and Bates data cannot be due to
strength outside the experimental bounds for the energy transfer.
\section{Conclusions}
The study of the CSR and its extraction from the analysis of the separated
$L$ contribution to QE electron scattering data has been extensively
reviewed by different authors leading to rather controversial results. This controversy
is directly linked to the interpretation of experiments as well as to the theoretical
descriptions and the role played by different ingredients. Whereas in some works it is
concluded that a significative quenching occurs in the observed CSR, others show that
only a very mild reduction (or no reduction at all) is observed from the analysis of data.
Being aware that new, high precision, data expected from Jlab at high energy transfer would help
in disentangling between different approaches, in this work we try to shed some
light on this problem analyzing also its connection with the general scaling properties observed by
inclusive QE electron scattering.
Scaling arguments applied to $(e,e')$ data have clearly proved
to high accuracy how well scaling is respected by QE data. Moreover,
a ``universal'' superscaling function has been extracted from the analysis of separated
longitudinal data, showing a representative shape with a long tail that extends to high
values of the energy transfer. As we have shown, this extended tail, with regards to
usual nonrelativistic models or plane wave approaches, must be kept in mind if
making estimates of the contribution to the CSR coming from outside the experimentally
explored region.
A careful analysis has been performed using different theoretical approaches: RPWIA, RMF, rROP.
Results have shown that the CSR is basically independent on the nuclear system considered. Obviously,
for heavy nuclei Coulomb distortion of the electron wave functions would need to be taken into account
in order to extract reliable longitudinal response from the data, but this will not likely affect the
theoretical estimations made in this work. Concerning how the Coulomb
sum rule reaches its saturation value, we have observed that RPWIA gets saturation faster than RMF.
This is in accordance with the general shape shown by the superscaling function in both cases,
being the tail for large $\omega$-values absent in RPWIA. Furthermore, whereas RPWIA leads to a saturated
CSR very close to 1 for all $q$-values, even when the integration is limited
to the range experimentally considered, the RMF-CSR integrated in the whole allowed range
gets about $\sim 0.87$ for $q\geq 0.4-0.5$ GeV/c, and this value keeps
stabilized for $q$ up to the maximum $q\sim 1$ GeV/c explored in this work.
In order to compare our theoretical predictions with experiment, we have analyzed the role
played by the cutoff $\omega$-value considered as upper integration limit in the expression
of the CSR.
\hbox{}From our results, we conclude that the Coulomb sum rule from RMF reaches
$\sim 85-95\%$ of its saturated value if truncation at the experimental
$\omega$-cutoff is taken. The largest strength lost in CSR occurs for $q=475$ MeV/c,
and is of the order of $\sim 15\%$.
The comparison with CSR results obtained from data measured at Bates for $^{40}$Ca has shown
its accordance with the RMF approach. Similar comments apply to the ratio of integrated
response functions to the $L$ strength evaluated with the RFG. These results show
that no further quenching than the one predicted in the relativistic mean field impulse approximation
is needed to describe the longitudinal response measured at BATES, that shows a depletion of the free value
of the order of $\sim 10-20\%$. This is in contrast with data measured at Saclay
showing a reduction of the $L$ channel of the order of $\sim 30-40\%$ \cite{Saclay1,Saclay2,Morg01}, in spite
of the fact that in these experiments the cuttof in the energy transfer is larger than for Bates experiments.
The reasons for this difference would hopefully be clarified by the recent experiment
at JLAB~\cite{CSR-JLAB}.
\section*{Acknowledgements}
This work was partially supported by DGI (MICINN-Spain) contracts
FIS2008-04189, FPA2007-62216, by the UCM and Comunidad de Madrid
(Grupo de F\'{\i}sica Nuclear, 910059),
the Spanish Consolider-Ingenio programme CPAN (CSD2007-00042), by the
Junta de Andaluc\'{\i}a, and by the INFN-CICYT collaboration
agreements INFN08-20 \& FPA2008-03770-E/INFN.
This work has benefited from discussions with
M.B. Barbaro and T.W. Donnelly.
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1403.6276
|
\section{Introduction}\label{sec:intro}
Beryllium has one single stable isotope, $^{9}$Be. Spectral lines of Be have been identified in the Sun, in the near UV region that is observable from the ground, quite some time ago by \citet{1895ApJ.....1...14R}. The lines tentatively identified included the Be I lines at $\lambda$ 3321.011, 3321.079 and 3321.340 \AA, and the Be II resonance lines at $\lambda$ 3130.422 and 3131.067 \AA\ \citep[wavelengths from][]{1997JPCRD..26.1185K,2005PhyS...72..309K}.
However, it has been shown that the identification of the Be I lines was erroneous \citep{1968SoPh....5..159G,1968ApL.....2..235H}. Only the Be II $^{2}$S--$^{2}$P$_{0}$ resonance lines at $\sim$ 3130 \AA\ are now used to determine Be abundances in late-type stars. Other Be lines can be found in the UV below 3000 \AA, but are only observable from space and seem to be of limited use \citep{1996A&A...313..909G}. Isotopic shifted lines of unstable $^{7}$Be and $^{10}$Be have been searched for, but were never detected \citep{1968PASP...80..622W,1973ApJ...185L..27B,1975A&A....42...37C}.
High signal-to-noise (S/N) spectra in the near UV is hard to obtain because of atmospheric extinction. In addition, the spectral region of the Be lines is crowded with other atomic and molecular lines (see the solar spectrum in Fig. \ref{fig:sun}). Some of these blending lines are still unidentified, as for example the one contaminating the blue wing of the 3131.067 \AA\ line \citep[see e.g.][]{1974SoPh...36...11R,1995A&A...302..184G,1997ApJ...480..784P,2011A&A...535A..75S}.
Early analyses of Be abundances were reviewed by \citet{1969ARA&A...7...99W} and \citet{1976PASP...88..353B}. The solar Be abundance seems to have been first determined by \citet{1929ApJ....70...11R}. The approximation of local thermodynamic equilibrium (LTE) has been shown to result in correct abundances for the Sun \citep{1975A&A....42...37C,1979AJ.....84.1756S,2005ARA&A..43..481A}. The solar Be abundance also seems to be largely insensitive to 3D hydrodynamical effects \citep{2004A&A...417..769A,2009ARA&A..47..481A}. The current value of the solar meteoritic abundance of Be is A(Be)\footnote{The abundance of an element X in this notation is A(X) = $\log \epsilon$(X) = $\log$ [N(X)/N(H)] + 12, i.e. an abundance by number in a scale where the number of hydrogen atoms is 10$^{12}$.} = 1.32 \citep{MakishimaNakamura06,2010ppc..conf..379L} while the photospheric abundance is A(Be) = 1.38 \citep{2009ARA&A..47..481A}. Nevertheless, most 1D LTE model atmosphere analyses of Be in the Sun tend to find values between A(Be) = 1.10 and 1.15. The difference is ascribed to near-UV continuum opacity missing in the computations \citep{1998Natur.392..791B,2001ApJ...546L..65B}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm]{sun}
\end{center}
\caption{The Be lines in the solar spectrum}
\label{fig:sun}
\end{figure}
Beryllium is not produced in significant amounts by the primordial nucleosynthesis, because there are no stable elements with mass number 5 or 8 to act as an intermediate step in synthesising $^{9}$Be \citep{1967ApJ...148....3W,1993ApJ...406..569T,2012ApJ...744..158C}. In addition, Be is rapidly destroyed by proton capture reactions when in regions inside a star with a temperature above $\sim$ 3.5 $\times 10^{6}$ K \citep{1954ApJ...119..113G,1955PhRv...97.1237S}. Therefore, stars do not produce Be.
In stars like the Sun, for example, Be is only present in the external regions of lower temperature \citep[see Fig.\ 1 of][]{1999ApJ...511..466B}. In evolved stars, where the convective envelope has increased in size and mixed the surface material with the interior, Be, unlike Li, is usually not detected \citep{1977ApJ...214..124B,2006A&A...447..299G,2010A&A...510A..50S}. Beryllium has never been detected in Li-rich giants \citep{1997A&A...321L..37D,1999A&A...345..249C,2005A&A...439..227M}.
Long ago, it was understood that Be can only be produced by the spallation of heavier nuclei, mostly from carbon, nitrogen, and oxygen \citep{1955ApJS....2..167F,1967AnPhy..44..426B}. The only known way to produce significant amounts of Be is by cosmic-ray induced spallation in the interstellar medium (ISM), as first shown by \citet{1970Natur.226..727R} and \citet{1971A&A....15..337M}.
Two channels of cosmic-ray spallation might work to produce Be. In the so-called \emph{direct process}, Be is produced by accelerated protons and $\alpha$-particles that collide with CNO nuclei of the ISM \citep{1975A&A....40...99M,1990ApJ...364..568V,1993ApJ...403..630P}. In the \emph{inverse process}, accelerated CNO nuclei collide with protons and $\alpha$-particles of the ISM \citep{1997ApJ...488..730R,1998ApJ...499..735L,1998A&A...337..714V,2002ApJ...566..252V,2012A&A...542A..67P}.
If the first channel dominates in the early Galaxy, Be should behave as a secondary element, as its production rate would be proportional to the metallicity of the ISM. If the second channel dominates, Be should instead behave as a primary element. The two behaviors can be distinguished by the analysis of Be abundances in metal-poor stars. For primary elements, one should observe a linear correlation between $\log$(Be/H) and the metallicity [Fe/H]\footnote{[A/B] = log [N(A)/N(B)]$_{\rm \star}$ $-$ log
[N(A)/N(B)]$_{\rm\odot}$} with a slope close to one. For secondary elements, the slope should be around two.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm]{full_box.jpg}
\end{center}
\caption{The Be abundances as a function of metallicty in the metal-poor stars analyzed by \citet{2009A&A...499..103S}. Also shown are the slopes expected for primary and secondary elements. Clearly, the inverse process dominates the Be production and it behaves as a primary element}
\label{fig:slopes}
\end{figure}
Linear relations with the two slopes are compared in Fig. \ref{fig:slopes}. Also shown are the Be abundances of metal-poor stars determined by \citet{2009A&A...499..103S}. It is clear then that Be behaves as a primary element, meaning that the inverse process dominates \citep[as first suggested by][]{1992ApJ...401..584D}. The determination of abundances of Be in metal-poor stars were first attempted by \citet{1984A&A...139..394M} and \citet{1988A&A...193..193R}. But it was with the works of \citet{1992ApJ...388..184R} and \citet{1992Natur.357..379G} that the linear correlation with slope of one became well established. Beryllium abundances in metal-poor stars have been further investigated in several works since then \citep[e.g.][]{1993AJ....106.2309B,1997A&A...319..593M,
1999AJ....117.1549B,2009A&A...499..103S,2009MNRAS.392..205T,2009ApJ...701.1519R,2011ApJ...738L..33T,2011ApJ...743..140B}.
With the inverse process dominating, and considering that cosmic-rays are globally transported across the Galaxy, then the Be production in the early Galaxy should be a widespread process. It follows that, at a given time, the abundance of Be across the Galaxy should have a smaller scatter than the products of stellar nucleosynthesis (such as Fe and O), as suggested by \citet{2000IAUS..198..425B} and \citet{2001ApJ...549..303S}. In this case, Be abundances could be used as a time scale for the early Galaxy \citep{2005A&A...436L..57P,2009A&A...499..103S}. This is an interesting application that still needs to be further constrained.
\section{CUBES}
CUBES (Cassegrain U-Band Brazil-ESO Spectrograph) is a new medium-resolution ground based near UV spectrograph for use at the VLT (Very Large Telescope). CUBES is being constructed by Brazilian institutions together with ESO (see Barbuy et al. this volume, for more details). The properties of CUBES are still not finalized, but it will have a resolution of at least R = 20\,000 and should provide access to the wavelength range between 3000--4000 \AA.
CUBES will be more efficient than UVES \citep{2000SPIE.4008..534D} in the near UV. UVES (Ultraviolet and Visual Echelle Spectrograph) is the current spectrograph at the VLT that is able to obtain high-resolution spectra in this region. There is an expected gain of about three magnitudes at 3200 \AA. This gain in sensitivity will expand the number of targets that can be observed. Among the main science cases for CUBES are the study of abundances of Be, C, N, O, and of heavy neutron-capture elements in metal-poor stars (see also Siqueira-Mello, this volume and Bonifacio, this volume).
Most of the future new instruments and facilities (such as the E-ELT, European Extremely Large Telescope) are now being optimized for the red and near-infrared spectral regions. Thus, CUBES has the potential of being an unique instrument in terms of sensitivity, spectral range, and resolution. With respect to current ESO instruments, CUBES will outperform both UVES and X-Shooter \citep{2011A&A...536A.105V}. CUBES should also be more efficient than other ground-based near-UV capable spectrographs, such as HIRES (High Resolution Echelle Spectrometer) at the Keck Observatory and HDS (High Dispersion Spectrograph) at the Subaru telescope.
Regarding space-based telescopes, the HST (Hubble Space Telescope) has two UV spectrographs, the Space Telescope Imaging Spectrograph (STIS) and the Cosmic Origins Spectrograph (COS). Nevertheless, the HST will likely not be operational when CUBES comes online ($\sim$ 2017/2018). The WSO-UV (World Space Observatory - Ultraviolet), a 170 cm space based telescope, that should be launched in 2016 will provide access to the UV region (see Shustov, this volume). The WSO-UV is however optimized for the region between 1150-3200 \AA. It is thus not a competitor but complementary to CUBES.
\begin{figure}[t]
\includegraphics[width=7cm]{be_poor_nuva}
\caption{Beryllium abundances as a function of metallicty. The stars from \citet{2009A&A...499..103S} are shown as filled circles, together with stars BD+03 740 ([Fe/H] = $-$2.85), BD$-$13 3442 ([Fe/H] = $-$2.80), G64-12 ([Fe/H] = $-$3.20), G64-37 ([Fe/H] = $-$3.15), and LP 815-43 ([Fe/H] = $-$2.75) as open circles. Also shown is the upper limit for star BD+44 493 ([Fe/H] = $-$3.80), as an open triangle}
\label{fig:poor}
\end{figure}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=12cm]{cubes_mp1_r20k}
\end{center}
\caption{Comparison of three spectra with R = 20\,000 of a metal-poor star with [Fe/H] = $-$3.00. The black spectrum was calculated without Be, the blue with log(Be/H) = $-$13.10 (simulating an abundance plateau), and the red one with an abundance decreased by 0.60 dex. The lines at the red spectra can never be detected}
\label{fig:mp1}
\end{figure*}
\section{Science with beryllium abundances and CUBES}
\subsection{Extremely metal-poor stars}
The linear relation with slope close to one between log(Be/H) and [Fe/H] is well established down to [Fe/H] = $-$2.5/$-$3.0 \citep[but see also][for a slightly different view]{2009ApJ...701.1519R}. Going to the extremely metal-poor regime ([Fe/H] $\leq$ $-$3.00), however, the situation is less clear.
The detection of Be in two stars (LP 815-43 and G64-12) with [Fe/H] $\sim$ $-$3.00 by \citet{2000A&A...362..666P,2000A&A...364L..42P} seems to suggest some deviation from the linear trend,
with a possible flattening of the relation between log(Be/H) and [Fe/H]. The Be abundance of a third star (G64-37), analyzed by
\citet{2006ApJ...641.1122B}, seems to be consistent with this flattening, although these authors argue that there is only evidence
for a dispersion of the Be abundances. The Be abundances of these three stars, together with the stars from \citet{2009A&A...499..103S}, and of stars BD+03 740 and BD$-$13 3442 ([Fe/H] = $-$2.85 and $-$2.80, respectively) are shown in Fig. \ref{fig:poor}. The Be abundances and metallicities of these extremely metal-poor stars were redetermined by \citet{2012ASPC..458...79S}.
A Be upper limit of log(Be/H) $<$ $-$13.80 was determined for the carbon-enhanced metal-poor star BD+44$^{\circ}$ 493 with [Fe/H] = $-$3.80 by \citet{2009ApJ...698L..37I,2013ApJ...773...33I}. This limit is in principle consistent with an extension of the linear trend between Be and metallicity to lower values of Fe (see Fig. \ref{fig:poor}). Care in the interpretation is needed because the origin of the carbon enhancement in this type of stars is also not well established. If any kind of transfer of material from evolved stars (which are depleted in Be) is involved, the surface abundance of Be would be diluted.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=12cm]{cubes_mp2_r20k}
\end{center}
\caption{Comparison of three spectra with R = 20\,000 of a metal-poor star with [Fe/H] = $-$3.50. The black spectrum was calculated without Be, the blue with log(Be/H) = $-$13.10 (simulating an abundance plateau), and the red one with an abundance decreased by 1.00 dex. The lines at the red spectra can never be detected}
\label{fig:mp2}
\end{figure*}
Whether the flattening exists or not is thus still not clear, because Be abundances have been determined in only a few stars with metallicity around or below [Fe/H] = $-$3.00 \citep[see][]{2011ApJ...743..140B,2012ASPC..458...79S}. A number of different scenarios could cause such flattening, each one with its own astrophysical implication. \citet{1997ApJ...488..515O}, for example, suggest that a Be plateau, although an order of magnitude below of the current detections, could be the result of an inhomogeneous primordial nucleosynthesis. The identification of such a primordial plateau of Be, similar to the one observed for Li \citep{1982A&A...115..357S}, might be an important test of the standard big bang model. Other possibilities include: 1) the accretion of metal-enriched interstellar gas onto metal-poor halo stars, while crossing the Galactic plane (see \citeauthor{1995ApJ...447..184Y} \citeyear{1995ApJ...447..184Y} for the case of beryllium, but see also \citeauthor{2014ApJ...784..153H} \citeyear{2014ApJ...784..153H} for a more recent assessment of this effect on stellar metallicities); 2) pre-Galactic production by cosmic-rays in the intergalactic medium \citep{2008ApJ...673..676R,2008ApJ...681...18K}; and 3) although a plateau is not predicted, Be enriched stars with [Fe/H] $\geq$ $-$3.2 might be associated to population III quark-novae \citep{2013MNRAS.428..236O}.
To address the existence of the flattening, it is important to expand the number of stars with [Fe/H] $<$ $-$3.00 where Be abundances have been determined. All additional known stars with this or lower metallicity are too faint to be observed with current instrumentation. CUBES will then offer an unique opportunity to obtain the spectra needed to test this possibility.
For this science case, it is important to establish whether a sample of stars has Be abundances at the level of a given plateau or whether the Be abundances follow an extension of the linear relation between log(Be/H) and [Fe/H]. Therefore, the key requirement is to detect the Be lines at low-metallicities and derive proper abundances, not only upper-limits.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=12cm]{ngc6752_r20k}
\end{center}
\caption{Comparison between turn-off stars of NGC6752 with R = 20\,000. A difference in abundances of $\Delta$Be = 0.60 dex can be detected in all cases except the lowest S/N case, where a detection is marginal at best}
\label{fig:n6752}
\end{figure*}
To test if this will be possible with CUBES, spectra of two different stars were simulated, one with [Fe/H] = $-$3.00 and the other with Fe/H] = $-$3.50. Synthetic spectra were computed with the same codes and line lists used in \citet{2009A&A...499..103S}. The spectra of each star was computed with two different Be abundances. The higher abundance for the two stars is the same, chosen to simulate the presence of a plateau. This abundance is log(Be/H) = $-$13.10, i.e. the Be abundance found for star G64-12 by \citet{2000A&A...364L..42P}. The smaller Be abundance, was decreased by 0.60 dex for the star with [Fe/H] = $-$3.00 and by 1.00 dex for the star with [Fe/H] = $-$3.50:
\begin{enumerate}
\item Metal-poor star 1: $T_{\rm eff}$ = 6300 K; log $g$ = 4.30; [Fe/H] = $-$3.00; $\xi$ = 1.20 km s$^{-1}$
\item Metal-poor star 2: $T_{\rm eff}$ = 6300 K; log $g$ = 4.30; [Fe/H] = $-$3.50; $\xi$ = 1.20 km s$^{-1}$
\end{enumerate}
The Be abundances reflect a scale where A(Be)$_{\odot}$ = 1.10, as derived by \citet{2009A&A...499..103S}. In addition, all the spectra were computed with the abundance of the alpha-elements increased, [$\alpha$/Fe] = +0.40.
The spectra were computed with three different spectral resolutions, R = 15\,000, 20\,000, and 25\,000, although only the case of R = 20\,000 is shown here, as this currently seems to be the favored CUBES value. To simulate the observations, noise was added to the spectra with four different levels in 10 different realizations (kindly computed by H. Kuntschner). The signal-to-noise levels were S/N = 20, 50, 100, and 150 for the case with R = 25\,000. For the other cases the S/N was scaled to try to simulate the gain in S/N given by the decrease in resolution. Thus, the values used were S/N = 22, 56, 112, 168 for the R = 20\,000 case and S/N = 26, 65, 129, 194 for the R = 15\,000 case.
Example are shown in Figs. \ref{fig:mp1} and \ref{fig:mp2}. It is possible to see that detecting low Be abundances (log(Be/H) $\leq$ $-$13.70) is not possible, in the simulated spectra. This is also valid for the cases with different resolutions not shown here. Nevertheless, it is also shown that if there is a plateau around log(Be/H) = $-$13.10, the Be lines can be detected in both metal-poor stars, if S/N $\gtrsim$ 100. Therefore, with CUBES data it would be possible to detect the flattening, if it has a level of Be abundances similar to the one currently found for stars with [Fe/H] $\sim$ $-$3.00. The exact limit of Be detection for CUBES remains to be determined. To calculate that, more information is needed also to understand which level of S/N can be reached.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=12cm]{OmeCen_to1_r20k}
\end{center}
\caption{Comparison between stars at the metal-poor turn-off of $\Omega$ Cen with R = 20\,000. A difference in abundances of $\Delta$Be = 0.60 dex can not be easily detected in the lowest S/N case. It seems possible in the three other cases}
\label{fig:cen1}
\end{figure*}
\subsection{Stars in globular clusters}
Globular clusters host multiple generations of stars \citep[see e.g.][and references therein]{2012A&ARv..20...50G}. Second generation stars are formed out of material contaminated by proton-capture nucleosynthetic products. The polluters are usually thought to be either first-generation massive asymptotic-giant-branch stars (AGBs) or first-generation fast-rotating massive stars \citep{2009A&A...499..835V,2007A&A...464.1029D}.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=12cm]{OmeCen_to2_r20k}
\end{center}
\caption{Comparison between stars at the metal-rich turn-off of $\Omega$ Cen with R = 20\,000. A difference in abundances of $\Delta$Be = 0.60 dex can be detected in all cases}
\label{fig:cen2}
\end{figure*}
That these peculiarities have a pristine origin, and are not the result of deep mixing inside the stars themselves, became clear when the same chemical pattern was found in turn-off stars \citep{2001A&A...369...87G}. The most obvious observational result of mixing pristine and processed material is the appearance of the correlations and anti-correlations between light-element abundances. Proton-capture reactions can decrease the abundances of Li, C, O, and Mg and enhance the abundances of N, Na, and Al. All the globular clusters analyzed so far with sufficient depth show signs of these chemical inhomogeneities \citep{2009A&A...505..139C}.
Beryllium, another light element, should also be affected by the polluting material. In particular, and as discussed before, Be is not produced inside stars but only destroyed. Thus the polluting material from the first generation of evolved stars, regardless of the exact nature of the polluting star, is completely devoid of Be. A star formed with only pristine material should have the original Be abundance of that material. \emph{Stars with different amounts of polluted material should have diluted the surface Be abundance to different levels.} Stars with bigger fraction of polluted material should have stronger correlations and anti-correlations between the light elements and should be strongly depleted in Be.
Lithium abundances should in principle behave in a similar way. However, AGB stars can produce Li via the Cameron-Fowler mechanism \citep{1971ApJ...164..111C}. Even though some extreme fine tuning might be needed to explain the full range
of oxygen abundances in stars with seemly normal Li, this option can not be completely excluded as of yet \citep[see e.g.][]{2010A&A...524L...2S}. In this situation, Be offers an important complementary test. As Be is never produced in stars, it is an unique tracer of the amount of pollution suffered by the globular cluster stars.
In this science case for CUBES, the objective is to detect differences in the Be abundance among different turn-off stars of the same cluster. These different stars should have been polluted to different levels. If different Be abundances are detected, then one can use them to quantify the fraction of polluting material. On the other hand, if stars with different levels of pollution are found to have the same Be abundance, this would definitely show that all the simple dilution-pollution scenarios currently proposed to explain the chemical properties of globular clusters are too simplistic and not valid. It would show that the true formation scenario was more complex than that.
Measurements of Be abundances have been attempted in two globular clusters so far: i) In NGC6397, Be was detected in two turn-off stars by \citet{2004A&A...426..651P}, and an upper limit was derived for the extremely Li-rich main-sequence star NGC6397 1657 \citep{2014A&A...563A...3P} and ii) two turn-off stars in NGC6752 were analyzed by \citet{2007A&A...464..601P}, resulting in one detection and one upper limit. The results seem to indicate that the stars have the same Be abundance, even though they have different abundances of oxygen (and thus were polluted to different levels). Nevertheless, the results are uncertain due to the low S/N of the spectra (S/N $\sim$ 10--15), even though they were obtained with long total exposure time ($\sim$ 900 min). CUBES will allow better spectra to be acquired with shorter exposure times, increasing also the number of globular clusters were Be abundances can be investigated.
To test this science case, synthetic spectra of three different stars were computed. One star represents turn-off stars in the cluster NGC6752, the other stars represent the metal-poor and metal-rich turn-off members of $\Omega$ Cen. For each star, two different Be abundances were adopted. The higher abundance in NGC6752 is the one derived by \citet{2007A&A...464..601P}, log (Be/H) = $-$12.05. The smaller abundance was decreased by 0.60 dex. For the stars in $\Omega$ Cen, the Be abundances were scaled according to the metallicity -- log(Be/H) = $-$12.6 and $-$13.2 for turn-off star 1 and log(Be/H) = $-$11.7 and $-$12.3 for turn-off star 2:
\begin{enumerate}
\item Turn-off star in NGC 6752: $T_{\rm eff}$ = 6400 K; log $g$ = 4.30; [Fe/H] = $-$1.50; $\xi$ = 1.20 km s$^{-1}$
\item Turn-off star 1 in $\Omega$ Cen: $T_{\rm eff}$ = 6700 K; log $g$ = 4.20; [Fe/H] = $-$2.00; $\xi$ = 1.20 km s$^{-1}$
\item Turn-off star 2 in $\Omega$ Cen: $T_{\rm eff}$ = 6550 K; log $g$ = 4.20; [Fe/H] = $-$1.10; $\xi$ = 1.20 km s$^{-1}$
\end{enumerate}
The same combinations of resolving power and S/N used before were adopted. The results are shown in Figs. \ref{fig:n6752}, \ref{fig:cen1}, and \ref{fig:cen2}. It can be seen that basically in all cases a difference in Be abundances can be detected, the only exception being the metal-poor turn-off of $\Omega$ Cen with low-S/N. Therefore, with CUBES it will be possible to use Be as a tracer of the dilution-pollution processes in globular clusters. More sound quantitative predictions have to wait until the CUBES properties are better defined. Then, it will be possible to define the list of clusters were the measurements will be possible and better determine the limit of detection of the Be abundances in these stars.
\section{Summary}
Abundances of the fragile element Be can be used in different areas of astrophysics, including studies of the Galactic chemical evolution, of stellar evolution, and of the formation of globular clusters. Only the Be II resonance lines at $\lambda$ 3130.422 and 3131.067 \AA\ are used for abundance studies. The lines are in the near UV spectral region, a region strongly affected by atmospheric absorption. It is very time demanding to obtain the high-resolution, high-S/N spectra needed to study Be.
Because of that, Be abundances have not been used to its full potential yet. To increase the number of stars where Be abundances can be determined, and to decrease the time cost of obtaining high-quality spectra, new UV sensitive instruments are needed. CUBES is one such instrument. It has an expected sensitivity gain of about three magnitudes at 3200 \AA\ over UVES.
In the near future, CUBES will likely be the only instrument offering the opportunity to measure Be abundances in samples of extremely metal-poor stars and in turn-off stars of globular clusters. Here, preliminary simulations of CUBES-like spectra for these types of stars were presented.
For the first case, the simulations indicate that with CUBES spectra it will be possible to investigate the suggested flattening of the relation between Be and Fe at low metallicities ([Fe/H] $<$ $-$3.00). This flattening might have implications, for example, to primordial nucleosynthesis models.
With CUBES, it will also be possible to investigate Be abundances in turn-off stars of globular clusters. Beryllium can be used as a tracer of the fraction of polluting material accreted by the second generation stars. If stars with different levels of pollution are found to have the same Be abundances, this would argue against the currently favored dilution-pollution scenarios invoked to explain the chemical properties of globular clusters.
\acknowledgments
I thank the organizers of the ``Challenges in UV Astronomy'' workshop for the invitation to give the review talk on beryllium abundances. I acknowledge financial support by the National Science Center of Poland through grant 2012/07/B/ST9/04428. I am pleased to thank L. Pasquini and H. Kuntschner for sharing their enthusiasm for the CUBES project.
\bibliographystyle{plain}
\input{smiljanic_beryllium_nuva_astroph.bbl}
\end{document}
|
1406.0210
|
\section{Introduction}
The discovery of brown dwarfs with significant near-infrared variability \citep[e.g.][]{artigau09, radigan12} has indicated that clouds are not homogeneously distributed in some brown dwarf atmospheres. While L dwarf photospheres are thought to be covered by thick silicate and iron clouds, these clouds have disappeared by mid-T spectral types. One explanation to simultaneously explain the color evolution through the L/T transition, near-infrared variability and the re-emergence of the 0.99 $\mu$m FeH band in early to mid T dwarfs is the growth of holes in the clouds that allow flux from deeper, hotter regions to emerge \citep{ackerman01, burgasser02, marley10}. Early T dwarfs indeed seem to be the most frequent strong variables \citep{radigan14}, but significant variability has also been found in cloudy L and (mostly) clear mid-T dwarfs \citep{buenzli12, heinze13, buenzli14, radigan14, wilson14}. Furthermore, spectral variability measured with HST \citep{buenzli12, apai13} is inconsistent with deep cloud holes.
\section{Are L/T transition objects partly cloudy?}
The first two confident detections of variable brown dwarfs interpreted as patchy cloud coverage were the T2.5 dwarf SIMP0136 \citep{artigau09} and the T1.5 dwarf 2M2139 \citep{radigan12}, both lying squarely in the L/T transition. For SIMP0136, the amplitude of the variations in J and K band (peak-to-peak $\approx$6\% and $3\%$) could be explained by several model combinations \citep{radigan12} with combinations of cloudy and clear models, as well as with different sedimentation efficiency parameters \citep[$f_{sed}$,][]{ackerman01}, corresponding to different cloud thickness. For 2M2139, much larger amplitude variations in J, H and K band (between $\approx 15-30\%$ depending on wavelength and observing date) combined with a spectrum suggested that atmospheres with fully clear sections could not well reproduce the observations \citep{radigan12}. Many combinations of cloud thicknesses and temperatures remained possible.
Spectral variability observations obtained with HST/WFC3 of both 2M2139 and SIMP0136 \citep{apai13} from about 1.08 to 1.66~$\mu$m revealed that the variability amplitude is significantly lower in the deep water absorption band at 1.4~$\mu$m than in J or H band, but otherwise remarkably constant outside of the water absorption feature. For both objects the characteristics are very similar except for the amplitude, and we only focus on 2M2139 (Fig. 1). The water band centered at 1.15 $\mu$m varies only very slightly less than the J band peak emission, and no difference at all is seen in the K I feature at 1.25 $\mu$m compared to the continuum. The ratio is only marginally smaller in the H band than the J band peak.
\citet{apai13} also showed that any combination of cloudy and clear models could be excluded. Combinations of thin (E-type, $T_{\mathrm{eff}}=1100$~K) and thick (B-type, $T_{\mathrm{eff}}=800$~K) clouds \citep{burrows06} could well reproduce the color variations. However, these models could not fit the variability in absorption bands, and the very thick and cold B-type cloud is rather atypical for brown dwarfs. Here we model for the first time the full spectral variability. We find that a model combination with approximately equal covering fraction of a cool thick cloud ($f_{\mathrm{sed}} = 1,\,T_{\mathrm{eff}} = 1100$) and a thin warmer cloud ($f_{\mathrm{sed}} = 4,\, T_{\mathrm{eff}}=1400$ K) provides the best match to both the spectrum and the variability (Fig. \ref{fig1}). A sole increase in the fractional coverage of the thin cloud cannot explain the further evolution through the L/T transition. At a slightly later stage, formation of deeper holes or additional thinning of the cloud is still needed to explain the bluer color and a re-emergence of FeH.
Our model manages to explain most of the characteristics of the spectral variability, while other combinations of $f_{\mathrm{sed}}$ fail to match the variability ratio across the J band and/or the relative amplitude between the 1.4~$\mu$m band and the J band. No model adequately reproduces the variability on the red side of the 1.4 $\mu$m water feature, nor the spectral shape at $1.3-1.5$~$\mu$m and $1.13-1.2$~$\mu$m, perhaps due to incomplete model opacities. The predicted K-band amplitude is consistent with observations \citep{radigan12}, but the spectral mis-match in K band suggests that models including vertical mixing \citep{stephens09} might improve the fit. Furthermore, these model combinations are not self-consistent in the sense that the two models have different, independent temperature-pressure profiles. An attempt at more self-consistent patchy cloud models was made in \citet{marley10}, but only for partly cloudy atmospheres with fully clear holes. These models cannot adequately reproduce our observed spectral variability. Calculations of self-consistent patchy cloud models with different cloud thicknesses should be the next step in the modeling of brown dwarf spectral variability.
\begin{figure*}[t!]
\resizebox{\hsize}{!}{\includegraphics[clip=true]{fig1_color.eps}}
\caption{\footnotesize
Top left: Maximum and minimum HST/WFC3 spectrum (black) for 2MASSJ21392676+0220226 and an average spectrum from the SPEX prism library \citep[blue,][]{burgasser06}. Overplotted is our best patchy cloud model (red). Bottom left: Ratio of maximum to minimum spectrum for the observations (black) and the model (red) with parameters indicated. Right: Measured light curve over 6 HST orbits integrated over three wavelength regions.}
\label{fig1}
\end{figure*}
\subsection{The curious case of Luhman 16B}
The recent detection of the very nearby binary L/T transition dwarf WISEJ1049 \citep{luhman13}, aka Luhman 16AB, has provided an extraordinary benchmark object for the detailed study of cloud structure at the L/T transition. The T0.5 type B component \citep{burgasser13} was found variable in i+z band \citep{gillon13} with very fast light curve evolution. Simultaneous multi-wavelength photometry with the GROND instrument \citep{biller13} revealed a behavior not in line with the spectroscopic observations of the previously discussed two variable brown dwarfs with similar spectral type. Particularly curious is a non-detection of variability in the J band on one night (but strong detection one week earlier), while significant variability was simultaneously found in z' and H together with anti-correlated variability in r' and i' and out-of-phase variability in K band. No current patchy cloud model can produce significant variability ($\approx$10\%) in z' and H band but no or significantly lower variability in J band. The out-of-phase variability in K band is also different than for 2M2139, where quasi-simultaneous observations by \citet{radigan12} suggest that JHK light curves are all in phase. No comparable observations at r' and i' exist for the other two brown dwarfs because they are too faint, and thus anti-correlation at these wavelengths may not be unusual. \citet{biller13} propose that the phase shift correlates with the probed pressure, similar to the T6.5 dwarf 2M2228 (cf Sect. \ref{t6}). However, preliminary analysis of new HST spectroscopic variability observations of Luhman 16B (Buenzli et al. in prep) show that the variability in the water band at 1.4~$\mu$m, which probes pressures even lower than the r' and i' bands, is completely in phase with the J and H band variability. In fact, the HST spectral variability appears to be remarkably similar for Luhman 16B to 2M2139 (Fig. \ref{fig1}) and SIMP0136, contrary to the GROND observations. Furthermore, \citep{burgasser14} obtained a 45 min ground-based spectroscopic variability sequence that also suggested the strongest variability at Y and J band with some decrease towards H and K band. If the GROND observations are correct, Luhman 16B undergoes drastic changes not only in the light curve shape, but also in its spectral variability characteristics that currently cannot be explained by patchy cloud models.
\section{Variability beyond the L/T transition}
\label{t6}
Silicate clouds are thought to have sunk below the visible photosphere beyond spectral type of $\approx$T4, but substantial variability has been observed in several such brown dwarfs. The most notable is the T6.5 dwarf 2M2228, discovered as variable in J band by \citet{clarke08} and characterized in detail with simultaneous HST spectral and Spitzer photometric time series by \citet{buenzli12}. Some of its variability may be explained by patchy sulfide clouds that can potentially condense at these temperatures \citep{morley12}. However, the largest variability amplitude is found in the water band at 1.4 $\mu$m, anti-correlated to the variability in the J and H band peak. Furthermore, the light curves in the IRAC 4.5 $\mu$m channel and in the methane absorption band at 1.65~$\mu$m have intermediate phases between J, H and the water band, indicating a correlation of the shift with the probed atmospheric pressure.
Simple patchy cloud models are unlikely to be able to match these observations, and more complex circulation patterns that invoke temperature perturbations may also play a role. \citet{robinson14} investigated if a periodic temperature perturbation as predicted by dynamical models \citep{showman13} could propagate upward and introduce phase dependent variability. While their model currently neglects clouds and rotation, it shows that variability with the approximate amplitude and phase shift can, in principle, arise in this manner. Additional mid T dwarfs have been found variable in J band \citep{radigan14} or in H$_2$O or CH$_4$ bands \citep{buenzli14}, suggesting that 2M2228 is not unique.
\section{Conclusions}
Brown dwarf variability is ubiquitous and points to complex cloud structure and evolution. Most current cloud and circulation models are one-dimensional and cannot yet sufficiently model these effects, but significant progress is already underway. The discovery of the unusually bright and variable Luhman 16B has provided a target that allows a very detailed view into the cloud structure at the L/T transition. It is one of very few L/T transition dwarfs accessible to Gaia, which will provide important points for obtaining the binary orbit. This will eventually lead to an independent mass measurement, crucial for calibrating models.
\begin{acknowledgements}
EB acknowledges support from the Swiss National Science Foundation (SNSF). Support for HST program \#12314 and 13280 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. This research has benefitted from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at http://pono.ucsd.edu/\~{}adam/browndwarfs/spexprism.
\end{acknowledgements}
\bibliographystyle{aa}
|
1406.0610
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\section{Introduction}
One of the central problems in the theory of integrable systems is their finite-dimensional reductions. We start with the dispersionless Kadomtsev--Petviashvili (dKP) hierarchy
as an illistration. Let $\lambda(z,\text{\bf t})$ be a meromorphic function in variable $z$ and depending on an infinite family of generalized times $\text{\bf t}=(t_0=x, t_1,\dots,t_n\dots)$ with
the expansion
\[
\lambda(z,\text{\bf t})=z+\sum\limits_{n=0}^{\infty}\frac{A^n(\text{\bf t})}{z^{n+1}},
\]
about infinity (here $A^n$ means index instead of exponent). The Poisson structure $\{\cdot\, , \cdot\}$ is defined by
\[
\{F,G\}=\frac{\partial F}{\partial z}\frac{\partial G}{\partial x}-\frac{\partial F}{\partial x}\frac{\partial G}{\partial z}.
\]
Then the dKP hierarchy (see \cite{KodamaG89}) is an infinite number of commuting flows
\begin{equation}\label{dKPE}
\frac{\partial \lambda}{\partial t_{n}}=\{\mathcal L_{n+1},\lambda\},\quad n\geq 0,
\end{equation}
where $\mathcal L_n=\frac{1}{n}(\lambda^n)_{\geq 0}$, $n=1,2,\dots$, denotes the polynomial part of $\lambda^n$. In particular, $(\lambda)_{\geq 0}=z$, $(\lambda^2)_{\geq 0}=z^2+2A^0$, $(\lambda^3)_{\geq 0}=z^3+3zA^0+3A^1$, etc. For $n=2$ and $s=t_1$, the second equation in the hierarchy is equivalent to
\begin{equation}\label{BE}
A_s^n+A_x^{n+1}+nA^{n-1}A_x^0=0,
\end{equation}
known as the Benney moment equation.
Benney~\cite{Benney} investigated long non-linear waves
propagating on a free surface showing that the governing equations
have an infinite number of conservation laws.
The compatibility condition for \eqref{dKPE} is
\begin{equation}\label{compa}
\frac{\partial \mathcal L_m}{\partial t_n}-\frac{\partial \mathcal L_n}{\partial t_m}+\{\mathcal L_m,\mathcal L_n\}=0,
\end{equation}
which means that the flows \eqref{dKPE} commute. In particular, for $m=3$, $n=2$, $y=t_2$ and $u:=A^0$ we arrive at the equation
\begin{equation}\label{KZ}
u_{ss}+\frac{\partial}{\partial x}(u_{y}+uu_x)=0,
\end{equation}
known as the dKP equation or the Zabolotskaya--Khokhlov equation~\cite{ZabKhokh}.
A finite-dimensional reduction suggests that the function $\lambda$ depends on the generalized times $\text{\bf t}$ via a finite number of functions $u_k=u_k(\text{\bf t})$, i.e.,
$$\lambda(z, \text{\bf t})=\lambda(z, \text{\bf u}(\text{\bf t}))=\lambda(z, u_1(\text{\bf t}),\dots, u_N(\text{\bf t})).$$ A well-known polynomial reduction was proposed by Zakharov~\cite{Zakharov}.
Kodama and Gibbons~\cite{KodamaG89} realized that
dKP equation possesses infinitely many multi
component two dimensional reductions. They
presented several examples and the dependence
$\lambda$ was found in these particular cases.
They considered the vector-function $\text{\bf u}$ satisfying a system of hydrodynamic type
\[
\frac{\partial \text{\bf u}}{\partial t_n}=\xi_n(\text{\bf u})\frac{\partial \text{\bf u}}{\partial x}, \quad n>1.
\]
Gibbons and
Tsar\"ev~\cite{GT} were first who noticed that the chordal
L\"owner equation plays an essential role in the classification of
reductions of the Benney equations. If we denote
\[
u=u_0=A^0,\dots, u_N=A^N,\quad A^n=A^n(\text{\bf u}),\quad n>N,
\]
then we obtain $N(N-1)/2$ compatibility conditions for $A^{N+1}$ and the function $z=z(\lambda,\text{\bf t})$ inverse to $\lambda$ satisfies the equations
\begin{equation}\label{p1}
\partial_k z=-\frac{\partial_ku}{z-\mu_k},
\end{equation}
where $\partial_k=\frac{\partial}{\partial r^k}$ and $\mu_k$ are the zeros of the function $\lambda_z$ and the values $r^k=\lambda(\mu_k,\text{\bf t})$ are the Riemann invariants. Formally, the equation above is similar to the chordal
L\"owner equation which we will discuss in what follows.
The consistency conditions for \eqref{p1} are
\begin{equation}\label{p2}
\partial_i \mu_k=\frac{\partial_k u}{\mu_i-\mu_k},\quad \partial_i\partial_k u=2\frac{\partial_i u \partial_k u}{(\mu_i-\mu_k)^2},
\end{equation}
which is the Gibbons--Tsar\"ev system.
A generalization was suggested by Ma{\~n}as, Mart{\'\i}nez Alonso and Medina \cite{MMM}, where
the authors were looking for the function $\lambda$ and its inverse as a solution to the system of equation
\[
\frac{\partial z}{\partial u_k}=\sum\limits_{k=1}^N\frac{\eta_{ik}}{z-\mu_k},
\]
satisfying some compatibility conditions. Formally, again the above equations are of the form of multi-slit chordal L\"owner equation.
Later Takebe, Teo, and
Zabrodin~\cite{Takebe06} showed that the chordal and radial
L\"owner PDE served as consistency conditions for one-variable
reductions of dispersionless KP and Toda hierarchies,
respectively. In the chordal case, the function $\lambda$ satisfying \eqref{dKPE} depends on $\text{\bf t}$ via one function $s(\text{\bf t})$ and
\[
\frac{\partial \lambda}{\partial s}=-\frac{k}{z-\xi}\frac{\partial \lambda}{\partial z},
\]
with the compatibility condition of hydrodynamic type
\[
\frac{\partial s}{\partial t_n}=\chi_n \frac{\partial s}{\partial x},
\]
where $k$ is the $s$-derivative of the coefficient at $1/z$ in the Laurent expansion of $\lambda$, the functions $\chi_n(s)$ are constructed in a canonical way from the Lax function, and again we see the chordal L\"owner PDE. These approaches are somewhat in want of analytic background of the L\"owner theory.
On the other hand, another evolution process described by
Laplacian growth~\cite{GV} possesses an infinite number of
conservation laws, harmonic moments. Being a typical field problem
the moments of the Laplacian growth bring us to the dispersionless
Toda hierarchy~\cite{Mineev}. Unlike the Laplacian growth the
L\"owner evolution represents another group of models, in which
the evolution is governed by the infinite number of parameters,
namely the controllable dynamical system, where the infinite number
of degrees of freedom follows from the infinite number of driving
terms. Surprisingly, the same structural background, the Virasoro
algebra, appears again for this group~\cite{MarkinaVasiliev10}.
The idea of this paper is to revisit Gibbons and Tsar\"ev
observation and show that the chordal L\"owner evolution also
possesses an infinite number of conservation laws, moments. We
show that the L\"owner PDE is exactly the Vlasov equation under an
appropriate change of variables and the L\"owner ODE implies the
hydrodynamic type conservation equation. We start with the L\"owner evolution and
splitting time we arrive at integrable chains. This approach shows universality
of the L\"owner equation as an attraction point for several integrable chains, this was noticed in \cite{Pavlov2006}.
\section{Vlasov and L\"owner equations, conservation laws}
Let us consider a L\"owner chain of receding domains $\mathbb
H_t=\mathbb H\setminus\gamma_t$ in the upper half-plane $\mathbb
H=\{:\text{\rm Im}\, z>0\}$ and let $f\colon \mathbb H\to\mathbb H_t$ is
normalized near infinity as
\begin{equation}\label{norm}
f(z,t)=z+\frac{A^0}{z}+O\left(\frac{1}{z^2}\right),
\end{equation}
where $(-A^0(t))$ is the half-plane capacity of $\gamma_t$. Let
$\gamma_t$ be a Jordan curve in $\mathbb H$ except for an end
point on the real axis $\mathbb R$, $\gamma_t$ is parameterized by
$t$. Then $f$ satisfies the L\"owner PDE
\begin{equation}\label{LPDE}
(z-\xi_t)\frac{\partial f(z,t)}{\partial t}-\frac{d A^0}{dt}\frac{\partial f(z,t)}{\partial z}=0,
\end{equation}
with a real-valued continuous driving function $\xi_t$ and an
initial condition $f(z,0)=f_0(z)$. For every $t\geq0$, the
function $f(z,t)$ has a continuous extension on the closure of
$\mathbb H$, and the extended function denoted also by $f(z,t)$
satisfies equation \eqref{LPDE} at least almost everywhere. The
driving function $\xi_t$ generates the growing slit $\gamma_t$.
\begin{figure}[ht] \scalebox{0.7}{
\begin{pspicture}(2,1)(17,8)
\psline[linecolor=blue,linewidth=0.8mm](1,4)(3.6,4)
\psline[linecolor=blue,linewidth=0.8mm](6.6,4)(8,4)
\rput(13.7,5){$\mathbb H_t$}\rput(15.2,4.2){0}
\psline[linewidth=0.15mm]{->}(15,1)(15,7)
\psline[linewidth=0.15mm]{->}(11,4)(18,4)
\pscurve[linewidth=0.3mm](15,4)(14.9,4.2)(15.1,4.5)(14.9,5)(14.9,5.3)(14.7,6)
\psline[linecolor=blue,linewidth=0.8mm](12,4)(18,4)
\rput(3,5){$\mathbb H$}\rput(5.2,4.2){0}
\psline[linewidth=0.15mm]{->}(5,1)(5,7)
\psline[linewidth=0.15mm]{->}(2,4)(8,4)
\rput(14.5,6.5){$\gamma_t$}
\pscurve[linewidth=0.8mm,
linecolor=red]{->}(8,5)(9.5,5.5)(11,5)
\rput(9.5,5.9){$f(z,t)$}
\pscurve[linewidth=0.3mm,
linecolor=red]{->}(14.8,3.8)(9.5,2)(7,2)(3.7,3.6)
\psline[linecolor=red,linewidth=0.3mm]{->}(7.5,1.9)(6.5,3.6)
\rput(3.5,4.3){$g^-(0,t)$}
\rput(6.5,4.3){$g^+(0,t)$}
\rput(4.57,3.7){$\xi_t$}
\pscircle[fillstyle=solid,
fillcolor=black](3.5,4){.1}
\pscircle[fillstyle=solid,
fillcolor=red](14.7,6){.1}
\pscircle[fillstyle=solid,
fillcolor=red](4.6,4){.1}
\pscircle[fillstyle=solid,
fillcolor=black](6.5,4){.1}
\pscircle[fillstyle=solid,
fillcolor=black](15,4){.1}
\end{pspicture}}
\end{figure}
We will also use the two-parametric family of conformal maps
$$g(w,t,\tau):=f^{-1}(w(z,t),\tau)=f^{-1}(f(z,t),\tau),$$ where $0\leq \tau\leq
t<\infty.$ We also denote $g(w,t,0)=:g(w,t)$. The function $g$
maps the half-plane $\mathbb H$ onto a subset of $\mathbb H$. It
satisfies the L\"owner ODE for the half-plane
\begin{equation}\label{LODE}
\frac{\partial g(w,t,\tau)}{\partial t}=-\frac{d A^0/d
t}{g(w,t,\tau)-\xi_t}, \quad 0\leq \tau\leq t<\infty,\quad
g(w,\tau,\tau)=w.
\end{equation}
Moreover,
$\lim\limits_{t\to\infty}g(w,t,\tau)=\lim\limits_{t\to\infty}f^{-1}(f(z,t),\tau)=f(z,\tau)$.
Define the time splitting as the real-valued functions $t=t(x,s)$, a solution to the
quasi-linear differential equation
\begin{equation}\label{hydro}
\xi_t\frac{\partial t}{\partial x}+\frac{\partial t}{\partial s}=0,
\end{equation}
satisfying the asymptotic behaviour
$\lim\limits_{x\to\infty}t(x,s)=\lim\limits_{x\to-\infty}t(x,s)<\infty$.
Assume that $\xi_t$ is a function which admits a cone of solutions
to \eqref{hydro} with the needed asymptotic behaviour.
Now, let us consider the superposition $f(z,t(x,s))$ and multiply both sides of \eqref{LPDE} by
$\frac{\partial t}{\partial x}$. By abuse of notation, we continue to write $f$ for the function
$f(z,t(x,s))=f(z,x,s)$. Then
\[
z\frac{\partial f}{\partial x}-\xi_t\frac{\partial f(z,t)}{\partial t}\frac{\partial t}{\partial x}-\frac{\partial A^0}
{\partial x}\frac{\partial f}{\partial z}=0.
\]
If we use equation \eqref{hydro}, then
\begin{equation}\label{Vlasov}
z\frac{\partial f}{\partial x}+\frac{\partial f}{\partial s}-\frac{\partial A^0}{\partial x}\frac{\partial f}
{\partial z}=0,
\end{equation}
which is the Vlasov equation, see~\cite{GT, Vlasov61}, describing time evolution of the distribution function of plasma
consisting of charged particles with long-range interaction.
In fluid descriptions of plasmas one does not consider the velocity distribution but rather the plasma moments
$A^n(t(x,s))\equiv A^n(x,s)$.
Among solutions to the Vlasov equation \eqref{Vlasov} let us
choose those which provide finite integrals for the moments $A^n$.
Namely, for a given solution $f(z,x,s)$ with the normalization
\eqref{norm}, choose a solution $\phi(z,x,s)=\varphi(f(z,x,s))$
where $\varphi$ is an appropriate rapidly decreasing at infinities
$z\to\pm\infty$ function, see, e.g., \cite{PavlovTsarev2013}.
For example, $\varphi(f)=\exp(-f^2)$ is appropriate. Then the moments
$A^n(x,s)$ are defined by
\[
A^n(x,s)=\int_{-\infty}^{\infty}w^n\,\phi(w,x,s)\,dw,\quad n\geq 1.
\]
The direct computations implies
\[
A^n_s=\int_{-\infty}^{\infty}w^n\,\frac{\partial \phi}{\partial
s}\,dw,\quad A^{n+1}_x=\int_{-\infty}^{\infty}w^{n+1}\,
\frac{\partial \phi}{\partial x}\,dw.
\]
Integrating by parts yields
\[
A^{n-1}=-\int_{-\infty}^{\infty}\frac{w^n}{n}\,\frac{\partial \phi
}{\partial w}\,dw.
\]
Now we can use the Vlasov equation \eqref{Vlasov} and arrive at the equation for the moments
\begin{equation}\label{mom}
A^n_s+A^{n+1}_x+nA^{n-1}A^0_x=0,
\end{equation}
which is an infinite autonomous system, known as Benney's moment equations, see~\cite{Benney}, which appear
in long wavelength hydrodynamics of an ideal incompressible fluid of a finite depth in a gravitational field.
Following \cite{GT, KM} let us define a function $\lambda(z,x,s)$ by the Cauchy principal value of a singular integral
\[
\lambda(z,x,s)=z+\int_{-\infty}^{\infty}\frac{\phi(w,x,s)}{z-w}dw=z+\sum_{n=0}^{\infty}\frac{A^n}{z^{n+1}},\quad
\text{ $z\to \infty$ in $\mathbb H$},
\]
where $z=g(w,t(x,s))$ and the coefficient $A^0$ is the same as for $f$. Then,
\[
\lambda_s=\frac{\partial \lambda}{\partial s}=z_s+\sum_{n=0}^{\infty}\left(\frac{A^n_s}{z^{n+1}}-\frac{(n+1)A^nz_s}
{z^{n+2}}\right),
\]
\[
\lambda_x=\frac{\partial \lambda}{\partial x}=z_x+\sum_{n=0}^{\infty}\left(\frac{A^n_x}{z^{n+1}}-\frac{(n+1)A^nz_x}
{z^{n+2}}\right),
\]
and
\[
\lambda_s+z\lambda_x=z_s+z\cdot z_x+A^0_x+\sum_{n=0}^{\infty}\frac{A^n_s+A^{n+1}_x-nA^{n-1}z_s-(n+1)A^nz_x}{z^{n+1}}.
\]
Making use of the moment equations we come to
\[
\lambda_s+z\lambda_x=z_s+z\cdot z_x+A^0_x+\sum_{n=0}^{\infty}\frac{-nA^{n-1}A^0_x}{z^{n+1}}-\sum_{n=1}^{\infty}
\frac{nA^{n-1}(z_s+z\cdot z_x)}{z^{n+1}}=
\]
\[
=A^0_x\lambda_z+(z_s+z\cdot z_x)\left(1-\sum_{n=1}^{\infty}\frac{nA^{n-1}}{z^{n+1}}\right)=
\lambda_z\left(A^0_x+z_s+z\cdot z_x\right).
\]
The L\"owner ODE \eqref{LODE} implies that $A^0_x=z_x(\xi_t-z)$ and the definition of the function $t(x,s)$ yields that
\begin{equation}\label{GibbonsEq}
A^0_x+z_s+z\cdot z_x=0,
\end{equation}
and therefore,
the equality $\lambda_s+z\lambda_x=0$ holds along the trajectories of the L\"owner ODE \eqref{LODE}. Equation
\eqref{GibbonsEq} received the name the Gibbons equation in \cite{Pavlov2007} following the original Gibbons' paper
\cite{Gibbons1982}.
Let us consider the map $z(\lambda,x,s)$ which is the inverse to $\lambda(z,x,s)$ with respect to
$\lambda \leftrightarrow z$,
\begin{eqnarray}
\label{ser1} \lambda(z,x,s)&=&z+\sum_{n=0}^{\infty}\frac{A^n}{z^{n+1}},\quad \text{ $z\to \infty$ in $\mathbb H$},\\
\label{ser2} z(\lambda, x,s)&=&\lambda-\sum_{n=0}^{\infty}\frac{H^n}{\lambda^{n+1}}.
\end{eqnarray}
Then,
\[
\sum_{n=0}^{\infty}\frac{A^n}{z^{n+1}}=\sum_{n=0}^{\infty}\frac{H^n}{\lambda^{n+1}},
\]
and
\[
\frac{\lambda}{z}\left(A^0+\frac{A^1}{z}+\dots\right)=H^0+\frac{H^1}{\lambda}+\dots
\]
So $H^0=A^0$. We continue by
\[
\lambda\left(\frac{\lambda}{z}-1\right)A^0+\frac{\lambda^2}{z^2}\left(A^1+\frac{A^2}{z}+\dots\right)=
H^1+\frac{H^2}{\lambda}+\dots,
\]
and conclude $H^1=A^1$. In the same fashion we come to
\[
\lambda^2\left(\frac{\lambda}{z}-1\right)A^0+\lambda\left(\frac{\lambda^2}{z^2}-1\right)A^1+
\frac{\lambda^3}{z^3}\left(A^2+\frac{A^3}{z}+\dots\right)=H^2+\frac{H^3}{\lambda}+\dots,
\]
and $H^2=A^2+(A^0)^2$.
Finally, we have
\[
\sum_{k=0}^{n}\lambda^{n-k}\left(\frac{\lambda^{k+1}}{z^{k+1}}A^k-H^k\right)+\frac{\lambda^{n+1}}{z^{n+2}}
\left(A^{n+1}+\frac{A^{n+2}}{z}+\dots\right)=\frac{1}{\lambda}\left(H^{n+1}+\frac{H^{n+2}}{\lambda}+\dots\right),
\]
and the coefficient $H^n$ is calculated as $H^n=A^n+P(A^0,\dots, A^{n-1})$, where $P(A^0,\dots, A^{n-1})$ is a
polynomial of $A^0,\dots, A^{n-1}$, $n\geq 2$.
The first coefficients are
\[
H^0=A^0,\quad H^1=A^1,\quad H^2=A^2+(A^0)^2, \quad H^3=A^3+3A^0A^1,
\]
\[
H^4=A^4+4A^0A^2+2(A^1)^2+2(A^0)^3.
\]
Analogous coefficients were calculated in, e.g., \cite{KM, Tammi}.
This way the L\"owner ODE \eqref{LODE} becomes the conservation
equation in the following sense. According to \eqref{GibbonsEq}
\[
\frac{d}{d s}\int_{-\infty}^{\infty}z(\lambda, x,
s)\,dx=-\int_{-\infty}^{\infty} (A^0_x+z\cdot z_x)\,dx,
\]
where we integrate with respect to $x\in\mathbb R$ in the Cauchy
principal value sense. The requirements on the asymptotic
behaviour of $t(x,s)$ as $x\to\pm\infty$ imply that
\[
\int_{-\infty}^{\infty} (A^0_x+z\cdot z_x)\,dx=0,
\]
Therefore
\[
\frac{d}{d s}\int_{-\infty}^{\infty}z(\lambda, x, s)\,dx=0,
\]
which corresponds to the momentum conservation law. So the
conserved quantities of the evolution are the moments
\[
I^n=\int_{-\infty}^{\infty}H^n(x, s)\,dx,\quad n\geq 0.
\]
Analogous integrals of motion were studied in the original work by
Benney~\cite{Benney} as well as in \cite{KM, PT, Zakharov}.
The Poisson structure allows us to reformulate the Benney moment
equation \eqref{mom} as an evolution equation with a Hamiltoinian
function. The Kupershmidt-Manin Poisson structure \cite{KM, KM2}
starts with the operators of differentiation and multiplication to
the right for the moments $A^n\frac{\partial}{\partial x}$ as
skew-symmetric operators with respect to the $L^2(\mathbb
R)$-paring, acting to the right by
\[
\{A^m,A^n\}(\cdot)=-mA^{n+m-1}\frac{\partial}{\partial x}(\cdot)-n\frac{\partial}{\partial x}
\left(A^{n+m-1}(\cdot)\right).
\]
Then for any two observables $F(A)$ and $G(A)$, the Poisson bracket can be written as
\[
\{F,G\}(A)=\sum\limits_{m,n=0}^{\infty}\int\limits_{-\infty}^{\infty}\frac{\delta F}{\delta A^m}\{A^m,A^n\}
\frac{\delta G}{\delta A^n}dx.
\]
Writing $\bar{H}^k=\frac{1}{k}\int\limits_{-\infty}^{\infty}
H^kdx=\frac{1}{k}I^k$, we have the hierarchy of commuting flows
with the Hamiltonians $\bar{H}^k\colon \{\bar{H}^k,\bar{H}^j\}=0$,
in the form of evolution equations
\[
\frac{\partial A^{m}}{\partial t_k}=\sum\limits_{n=0}^{\infty}\{A^m,
A^n\}\frac{\delta \bar{H}^k}{\delta A^n},
\]
so that equation \eqref{mom} becomes the second equation in this hierarchy.
\section{Finite-dimensional time}
The L\"owner PDE \eqref{LPDE} can be generalized to the form
\begin{equation}\label{mLPDE}
\frac{\partial f(z,t)}{\partial
t}=\sum_{k=1}^m\frac{\mu_k(t)}{z-\xi_k(t)}\frac{d
A^0}{dt}\frac{\partial f(z,t)}{\partial z},\;\;\;f(z,0)=f_0(z),
\end{equation}
with piecewise continuous coefficients $\mu_k(t)\geq0$,
$k=1,\dots,m$, $\sum_{k=1}^m\mu_k(t)=1$, and real-valued
continuous driving functions $\xi_1(t),\dots,\xi_m(t)$. A solution
$f(z,t)$ to \eqref{mLPDE} maps $\mathbb H$ onto $\mathbb
H\setminus\cup_{k=1}^m\gamma_k(t)$ where $\gamma_k(t)$ are growing Jordan
curves (slits) in $\mathbb H$ except for their endpoints on $\mathbb R$.
The driving functions $\xi_k(t)$ generate slits $\gamma_k(t)$, and the
coefficients $\mu_k(t)$ govern the relative dynamics of the slits
$\gamma_k(t)$ with respect to each other.
Instead of \eqref{mLPDE}, it is possible to consider the
generalized L\"owner PDE with a generalized time-vector ${\bf t}=(t_1,\dots,t_m)$
\begin{equation}\label{vecLPDE}
(z-\xi_k({\bf t}))\frac{\partial f(z,{\bf t})}{\partial
t_k}=\frac{\partial A^0}{\partial t_k}\frac{\partial f(z,{\bf
t})}{\partial z},\;\;\;f(z,{\bf 0})=f_{\bf
0}(z),\;\;\;k=1,\dots,m.
\end{equation}
In this model, $A^0({\bf t})=A^0(t_1,\dots,t_m)$ is not an
arbitrary function of $\mathbf{t}$. At every point ${\bf t}=(t_1,\dots,t_m)$,
\[
\frac{\partial A^0}{\partial t_1}=\frac{\partial A^0}{\partial
t_2}=\dots=\frac{\partial A^0}{\partial t_m}.
\]
For every ${\bf t}=(t_1,\dots,t_m)$, the solution $f(z,{\bf t})$ to
system \eqref{vecLPDE} maps $\mathbb H$ onto $\mathbb
H\setminus\cup_{k=1}^m\gamma_k(t_k)$, where $\gamma_k(t_k)$ is an endpoint of the slit $\gamma_k$ generated by the driving function
$\xi_k({\bf t})$.
Similarly to the scalar $t$, let us denote by
\[
g(w,\tau,{\bf t}):=f^{-1}(w(z,\tau),{\bf t})=f^{-1}(f(z,\tau),{\bf
t}),\;\;\; \tau=(\tau_1,\dots,\tau_m),
\]
where $0\leq\tau_k\leq t_k<\infty$ for any $k=1,\dots,m.$ We also
write $g(w,{\bf 0},{\bf t})=:g(w,{\bf t})$, where $\mathbf{0}$ states for the null-vector. The function $g$ maps
the half-plane $\mathbb H$ onto a subset of $\mathbb H$. It
satisfies the system of L\"owner's ODE in the half-plane
\begin{equation}\label{vecLODE}
\frac{\partial g(w,\tau,{\bf t})}{\partial t_k}=-\frac{\partial
A^0/\partial t_k}{g(w,\tau,{\bf t})-\xi_k({\bf t})}, \quad 0\leq
\tau_j\leq t_j<\infty, \quad g(w,\tau,\tau)=w,
\end{equation}
\[
j=1,\dots,m,\;\;\;k=1,\dots,m.
\]
Moreover,
$\lim\limits_{t_k\to\infty}g(w,\tau,(\tau_1,\dots,\tau_{k-1},t_k,\tau_{k+1},\dots,\tau_m))=
f(z,\tau)$.
Again we can define the vector-function ${\bf t}={\bf t}(x,s)$, as
a solution to the system of quasi-linear differential equations
\begin{equation}\label{vechydro}
\xi_k({\bf t})\frac{\partial t_k}{\partial x}+\frac{\partial
t_k}{\partial s}=0,\;\;\;k=1,\dots,m,
\end{equation}
satisfying the asymptotic behaviour
$\lim\limits_{x\to\infty}t_k(x,s)=\lim\limits_{x\to-\infty}t_k(x,s)<\infty$,
$k=1,\dots,m.$ Assume that functions $\xi_k(t_k)$ admit their
cones of solutions to \eqref{vechydro} under the necessary asymptotic
behaviour.
Equation \eqref{vecLPDE} implies that $f(z,{\bf
t}(x,s))=:f(z,x,s)$ satisfies
\[
z\frac{\partial f}{\partial t_k}\frac{\partial t_k}{\partial
x}-\xi_k({\bf t})\frac{\partial f}{\partial t_k}\frac{\partial
t_k}{\partial x}-\frac{\partial A^0}{\partial t_k}\frac{\partial
t_k}{\partial x}\frac{\partial f}{\partial z}=0
\]
which together with \eqref{vechydro} gives
\[
z\frac{\partial f}{\partial t_k}\frac{\partial t_k}{\partial
x}+\frac{\partial f}{\partial t_k}\frac{\partial t_k}{\partial
s}-\frac{\partial A^0}{\partial t_k}\frac{\partial t_k}{\partial
x}\frac{\partial f}{\partial z}=0,\;\;\;k=1,\dots,m.
\]
Summing up the latter equations for $k=1,\dots, m$, we obtain the
Vlasov equation for $f(z,{\bf t}(x,s))$
\[
z\frac{\partial f(z,{\bf t}(x,s))}{\partial x}+\frac{\partial
f(z,{\bf t}(x,s))}{\partial s}-\frac{\partial A^0}{\partial
x}\frac{\partial f(z,{\bf t}(x,s))}{\partial z}=0.
\]
Similarly to the scalar case, there appear the moments $A^n({\bf
t}(x,s))=:A^n(x,s)$ satisfying equation \eqref{mom}, the functions
$\lambda(z,x,s)$ and $z(\lambda,x,s)$ and the hierarchy of
commuting flows with the Hamiltonians $\bar H^k$.
Equations \eqref{vecLPDE} can be reduced to \eqref{mLPDE}. Indeed,
assume that $\mu_1(t)>0$ and construct the following reduction
\[
\frac{dt_k}{dt_1}=\frac{\mu_k}{\mu_1},\;\;\;t_k(0)=0,\;\;\;k=2,\dots,m.
\]
Then, after multiplying by $\frac{\partial t_k}{\partial t_1}$ and
summing up, equations \eqref{vecLPDE} for $t_1=t$ and $f(z,{\bf
t}(t))=:f(z,t)$ become
\[
\frac{\partial f(z,{\bf t}(t))}{\partial t}=\frac{\partial
f}{\partial t_1}+\frac{\partial f}{\partial t_2}\frac{\partial
t_2}{\partial t}+\dots+\frac{\partial f}{\partial
t_m}\frac{\partial t_m}{\partial t}=
\frac{1}{\mu_1}\left[\frac{\mu_1}{z-\xi_1}\frac{\partial
A^0}{\partial t_1}+\dots+\frac{\mu_m}{z-\xi_m}\frac{\partial
A^0}{\partial t_m}\right]\frac{\partial f}{\partial z}
\]
which is equivalent to \eqref{mLPDE} with $\tilde A^0=A^0/\mu_1$.
\section{Infinite-dimensional time}
The limiting case of equation \eqref{mLPDE} as $m\to\infty$ leads to
the L\"owner-Kufarev type equation
\begin{equation}\label{count}
\frac{\partial f}{\partial t}=\int_{\mathbb
R}\frac{d\nu_t(\xi)}{z-\xi(t)}\;\frac{dA^0}{dt}\;\frac{\partial
f}{\partial z},
\end{equation}
where, for every $t\geq0$, $d\nu_t(\xi)$ is a probability measure
with a compact support $I_t\subset\mathbb R$. A solution $f(z,t)$
to \eqref{count} maps $\mathbb H$ onto $\mathbb H\setminus K_t$
where, in general, the omitted set $K_t\cap\mathbb H$ is not
reduced to a countable set of slits. The set $K_t$ is generated by
the measure $d\nu_t(\xi)$.
However, the domain $\mathbb H\setminus K_t$ is the Carath\'eodory
kernel for the sequence of domains $\mathbb
H\setminus\cup_{k=1}^m\gamma_k(t)$ as $m\to\infty$. Here the slits
$\gamma_k$ are dense in $K_t$. In this interpretation the measure
$d\nu_t(\xi)$ is represented as a limit of point mass measures
with a dense set of mass points in the support of $d\nu_t$.
In this case it is impossible to generalize directly system
\eqref{vecLPDE} passing to an infinite set of equations
corresponding to the countable set of coordinates $(t_1,t_2,\dots)$.
Let us build a model with a successive dynamics of every slit
$\gamma_1,\gamma_2,\dots$. For $k=1,2,\dots$, denote by
\[
P_k(z,t_k)=\frac{\partial A^0}{\partial t_k}\frac{\partial
f(z,t_k)}{\partial z}\;\;\text{for}\;\;T_{k-1}<t_k<T_k,\;\;T_0=0,
\]
and
\[
P_k(z,t_k)=0\;\;\text{for}\;\;t_k\in\mathbb
R\setminus(T_{k-k},T_k),\;\;\;k=1,2,\dots\;.
\]
Now, instead of \eqref{count}, we are able to introduce a system
of PDE with an infinite set of coordinates ${\bf
t}:=(t_1,t_2,\dots)$,
\begin{equation}\label{infLPDE}
(z-\xi_k(t_k))\frac{\partial f}{\partial
t_k}=P_k(z,t_k),\;\;\;k=1,2,\dots\;.
\end{equation}
Suppose a function $f_0(z)$ is expanded near infinity as
\begin{equation}\label{norm2}
f_0(z)=z+\sum_{n=0}^{\infty}\frac{A^n}{z^{n+1}},
\end{equation}
and let $f_0$ serve as an initial data for the L\"owner chain $f(z,t)$
governed by \eqref{count} and as an initial data for the first equation of system \eqref{infLPDE}. Successively, the function $f(z,t_k)$ serves as an initial data for the $(k+1)$-th equation in \eqref{infLPDE}. It is clear that the resulting chain
$f(z,t)=f(z,{\bf t})$, ${\bf t}=(t_1,t_2,\dots)$, in contrast to the chain obtained from~\eqref{count}, is piecewise differentiable. The functions $f(z,t)$ are normalized
as in \eqref{norm2} with $A^n=A^n(t)$. So there exist driving
functions $\xi_1(t_1),\xi_2(t_2),\dots$ such that $f(z,{\bf t})$,
${\bf t}=(t_1,t_2,\dots)$, is a solution to the infinite system of
PDE \eqref{infLPDE}.
Let us apply the results by Takebe, Teo and Zabrodin
\cite{Takebe06} to construct a one-variable reduction of
dispersionless KP hierarchy for the system of PDE \eqref{infLPDE}.
Let $g(w,{\bf t}):=f^{-1}(w,{\bf t})$ be the inverse to $f(z,{\bf
t})$. Then $g$ is normalized at infinity as
\[
g(w,{\bf t})=w+\sum_{n=1}^{\infty}\frac{b_n({\bf t})}{w^n}.
\]
Denote by $\Phi_k(w,{\bf t})=[g^k(w,{\bf t})]_{\geq 0}$, $k\geq1$, the Faber polynomials
for $g(w,{\bf t})$. Let us forget for the moment the dependence on ${\bf t}$ and let us write simply $g(w)$ and $\Phi_k(w)$. The first Faber polynomials are
\[
\Phi_0=1,\quad \Phi_1=w,\quad \Phi_2=w^2-2b_1,\quad \Phi_3=w^3-3b_1w-3b_2,
\]
and the recurrence formula
\[
\Phi_{n+1}=w\Phi_n-\sum\limits_{k=1}^{n-1}b_{n-k}\Phi_k-(n+1)b_n
\]
holds for all $n\geq 1$. The Faber polynomials are related to the Grunsky coefficients which implies that
\[
\log\frac{g(w)-\xi}{w}=-\sum\limits_{n=1}^{\infty}\frac{1}{nw^n}\Phi_n(\xi).
\]
Changing variables $\xi=e^u$ and differentiating both sides with respect to $u$ yields
\[
\frac{1}{g(w)-\xi}=\sum\limits_{n=1}^{\infty}\frac{1}{nw^n}\Phi'_n(\xi).
\]
Returning back to equation~(\ref{infLPDE}) we conclude that the
function $g=f^{-1}$, w.r.t. the first variable, satisfies the system of equations
\[
\frac{\partial g}{\partial t_k}=-\frac{\partial A^0}{\partial
t_k}\sum_{n=1}^{\infty}\frac{\Phi_n'(\xi_k,t_k)}{nw^n},\;\;\;T_{k-1}<t_k<T_k,
\]
and
\[
\frac{\partial g}{\partial t_k}=0\;\;\text{for}\;\;t_k\in\mathbb
R\setminus(T_{k-1},T_k),\;\;\;k=1,2,\dots\;.
\]
This implies that, for all $k\geq1$,
\[
n\frac{\partial b_k}{\partial t_k}=-\frac{\partial A^0}{\partial
t_k}\Phi_k'(\xi_k,t_k),\;\;\;T_{k-1}<t_k<T_k,
\]
and
\[
\frac{\partial b_k}{\partial t_k}=0,\;\;\;t_k\notin(T_{k-1},T_k).
\]
There is an evident way to write dependence on ${\bf t}$ through a
single variable $t=t_1$. Set
\[
\tau({\bf t})=t_1,\;\;\text{if}\;\;t_1\in(0,T_1),
\]
\[
\tau({\bf
t})=kt_k(t_1)\;\;\text{if}\;\;t_1\in(T_{k-1},T_k),\;\;k\geq2,
\]
where
\[
\frac{dt_k}{dt_1}=\Phi_k'(\xi_k,t_k)\;\;\text{for}\;\;t_1\in(T_{k-1},T_k)
\]
and
\[
\frac{dt_k}{dt_1}=0\;\;\text{for}\;\;t_1\notin(T_{k-1},T_k),\;\;\;k\geq2.
\]
In the spirit of the results of Takebe, Teo and Zabrodin \cite{Takebe06} we conclude that the non-intersecting intervals $(T_{k-1},T_k)$ imply
that given $f(z,\tau({\bf t}))$ as the solution to system \eqref{infLPDE} with the initial condition $f_0$, one
has the Lax function $\mathcal{L}=f(z,\tau({\bf t}))$ which solves the dKP hierarchy by
\[
\frac{\partial \mathcal L}{\partial t_k}=\{\mathcal L_k,\mathcal L\},\quad T_{k-1}<t_k<T_k,
\]
where $\mathcal L_k=\frac{1}{k}[ \mathcal L^k]_{\geq 0}$, and the Poisson bracket is given by
\[
\{F,G\}=\frac{\partial F}{\partial w}\frac{\partial G}{\partial x}-\frac{\partial F}{\partial x}\frac{\partial G}{\partial w}, \quad T_{k-1}<x:=t_1<T_k.
\]
The Benney equations again can be recovered as the second equation of dKP in the following way. Set $s=t_2$. Then,
\[
\frac{\partial \mathcal L}{\partial s}=\{(z^2+2A^0),\mathcal L\},\quad T_{1}<s,x<T_2,
\]
where
\[
\mathcal{L}=z+\sum_{n=0}^{\infty}\frac{A^n}{z^{n+1}}.
\]
Equating the coefficients in front of powers of $z$ leads to the equations~\eqref{mom}. The higher equations in the hierarchy lead
to interesting PDEs due to the conditions on commuting flows (compatibility conditions) \eqref{compa}
For example, if $n=2$ and $m=3$ imply the dKP equation \eqref{KZ} (Zabolotskaya-Khokhlov equation~\cite{ZabKhokh}) for $A^{0}$.
\section{Vlasov kinetic equation}
Let us return back to the consistency conditions \eqref{p2} (the Gibbons-Tsar\"ev system), where $u(\mathbf{r})$ is a conservation law density and $\mu_{k}(\mathbf{r}
) $ are the characteristic velocities of $N$ component hydrodynamic type system
\begin{equation}
r_{s}^{i}+\mu_{i}(\mathbf{r})r_{x}^{i}=0,\quad i=1,\dots, N, \label{raz}
\end{equation}
written in the Riemann invariants. System \eqref{raz} is integrable by the generalized hodograph method, see \cite{T85, T91},
and has infinitely many conservation laws and commuting flows
\begin{equation}
r_{y}^{i}+\lambda_{i}(\mathbf{r})r_{x}^{i}=0, \label{dva}
\end{equation}
where $\lambda_{i}$ are functions of two entries $\mu_{i}$ and $u$ only: $
\lambda_{i}=F(\mu_{i},u)$. We will make use of the Tsar\"ev system
\begin{equation}
\frac{\partial _{i}\lambda_{k}}{\lambda_{i}-\lambda_{k}}=\frac{\partial
_{i}\mu_{k}}{\mu_{i}-\mu_{k}},\text{ \ }i\neq k, \label{tsar}
\end{equation}%
which is a direct consequence of the commutation $(r_{t}^{i})_{y}=(r_{y}^{i})_{t}$. Substitution of the ansatz $\lambda_{i}=F(\mu_{i},u)$ into (\ref{tsar}) yields
\begin{equation}
(\mu_{k}-\mu_{i})\frac{\partial F(\mu_{i},u)}{\partial u}=\frac{F(\mu_{k},u)-F(\mu_{i},u)}{\mu_{k}-\mu_{i}}-\frac{\partial F(\mu_{i},u)}{
\partial \mu_{i}},\text{ \ }i\neq k. \label{a}
\end{equation}
Interchanging indices and summing up both formulas we obtain
\begin{equation}
(\mu_{k}-\mu_{i})\left( \frac{\partial F(\mu_{i},u)}{\partial u}+\frac{
\partial F(\mu_{k},u)}{\partial u}\right) =\frac{\partial F(\mu_{k},u)}{
\partial \mu_{k}}-\frac{\partial F(\mu_{i},u)}{\partial \mu_{i}}.
\label{b}
\end{equation}
In the limit $\mu_{k}\rightarrow \mu_{i}$ this formula becomes
\begin{equation}
2\frac{\partial F(\mu_{i},u)}{\partial u}=\frac{\partial ^{2}F(\mu_{i},u)}{
(\partial \mu_{i})^{2}}. \label{j}
\end{equation}
Then (\ref{b}) reads
\begin{equation*}
(\mu_{k}-\mu_{i})\left( \frac{\partial ^{2}F(\mu_{i},u)}{(\partial \mu
^{i})^{2}}+\frac{\partial ^{2}F(\mu_{k},u)}{(\partial \mu_{k})^{2}}\right)
=2\left( \frac{\partial F(\mu_{k},u)}{\partial \mu_{k}}-\frac{\partial
F(\mu_{i},u)}{\partial \mu_{i}}\right) .
\end{equation*}
Taking derivative of this relationship with respect to $\mu_{k}$, we obtain
\begin{equation*}
\frac{\partial ^{3}F(\mu_{k},u)}{(\partial \mu_{k})^{3}}=\frac{\frac{%
\partial ^{2}F(\mu_{k},u)}{(\partial \mu_{k})^{2}}-\frac{\partial
^{2}F(\mu_{i},u)}{(\partial \mu_{i})^{2}}}{\mu_{k}-\mu_{i}},\text{ \ }%
i\neq k.
\end{equation*}
Interchanging indices we conclude that
\begin{equation*}
\frac{\partial ^{3}F(\mu_{k},u)}{(\partial \mu_{k})^{3}}=a^{\prime }(u)
\end{equation*}
for any index $k$. Thus
\begin{equation}
F(\mu_{k},u)=\frac{1}{6}a(u)(\mu_{k})^{3}+b(u)(\mu_{k})^{2}+c(u)\mu_{k}+d(u), \label{c}
\end{equation}
where functions $a(u),b(u),c(u),d(u)$ still have not been determined yet.
However the
substitution (\ref{c}) into (\ref{j}) leads to $a(u)=$const$,b(u)=$const$
,2c^{\prime }(u)=a,d^{\prime }(u)=b$. So, (\ref{c}) admits the form
\begin{equation}
F(\mu_{k},u)=\frac{a}{6}(\mu_{k})^{3}+\frac{a}{2}u\mu_{k}+b[(\mu_{k})^{2}+u]. \label{k}
\end{equation}
Finally, the substitution (\ref{k}) into (\ref{a}) implies $a=0$. Thus we
have found the so called `dispersive relation'
\begin{equation}
\lambda_{i}=(\mu_{i})^{2}+u. \label{tri}
\end{equation}
Following \cite{Fer} we write the L\"{o}wner system
\begin{equation}
\partial _{i}z=\frac{\partial _{i}u}{\mu_{i}-z}. \label{zero}
\end{equation}
Then, see (\ref{raz}), (\ref{dva}), (\ref{tri}), (\ref{zero})),
\begin{equation*}
z_{x}=\sum \partial _{i}z\cdot r_{x}^{i}=\sum \frac{\partial _{i}u}{\mu
_{i}-z}r_{x}^{i},
\end{equation*}%
\begin{equation*}
-z_{s}=-\sum \partial _{i}z\cdot r_{s}^{i}=\sum \frac{\partial _{i}u}{\mu
_{i}-z}\mu _{i}r_{x}^{i}=\sum \partial _{i}u\cdot r_{x}^{i}+z\sum \frac{%
\partial _{i}u}{\mu _{i}-z}r_{x}^{i}=u_{x}+zz_{x},
\end{equation*}%
\begin{equation*}
-z_{y}=-\sum \partial _{i}z\cdot r_{y}^{i}=\sum \frac{\partial _{i}u}{\mu
_{i}-z}\lambda_{i}r_{x}^{i}=\sum \frac{\partial _{i}u}{\mu_{i}-z}[(\mu
_{i})^{2}+u]r_{x}^{i}
\end{equation*}%
\begin{equation*}
=\sum \partial _{i}u\cdot (\mu_{i}-z)r_{x}^{i}+2z\sum \partial _{i}u\cdot
r_{x}^{i}+(z^{2}+u)\sum \frac{\partial _{i}u}{\mu_{i}-z}r_{x}^{i}
\end{equation*}%
\begin{equation*}
=(v_{x}-zu_{x})+2zu_{x}+(z^{2}+u)z_{x}=v_{x}+zu_{x}+uz_{x}+z^{2}z_{x},
\end{equation*}%
where we have introduced a new function $v(\mathbf{r})$ such that $\partial
_{i}v=\mu_{i}\partial _{i}u$. Indeed the potential function $v$ exists because the compatibility
condition $\partial _{i}(\partial _{k}v)=\partial _{k}(\partial _{i}v)$
leads to the identity according to the Gibbons--Tsar\"ev system (\ref{p2}).
Thus
we reconstructed two equations
\begin{equation}
z_{s}+\left( \frac{z^{2}}{2}+u\right)_x=0,\text{ \ }z_{y}+\left( \frac{z^{3}%
}{3}+uz+v\right)_x=0. \label{gen}
\end{equation}%
Their compatibility condition $(z_{s})_{y}=(z_{y})_{s}$ leads to the equations
\begin{equation}
v_{x}+u_{s}=0,\text{ \ }v_{s}+u_{y}+uu_{x}=0, \label{dKP}
\end{equation}
which are equivalent to \eqref{KZ}.
Now we introduce the so called vertex equation%
\begin{equation}
\partial _{\tau (\zeta )}z(\lambda )=\partial _{x}\ln (z(\lambda )-z(\zeta
)), \label{ierar}
\end{equation}%
where $z(\lambda )$ is just a short notation for $z(\lambda
;t_0,t_1,t_2,...)$. Here we use infinitely many
`time' variables $t_k$, which will be discussed below.
Let us consider the formal expansion \eqref{ser2} as $\zeta \rightarrow \infty $ and %
\begin{equation}
\partial _{\tau (\zeta )}=-\frac{1}{\zeta }%
\partial _{t_0}-\frac{1}{\zeta ^{2}}\partial _{t_1}-\frac{1}{\zeta ^{3}}%
\partial _{t_2}-... \label{f}
\end{equation}%
Then one can obtain infinitely many equations
\[
\partial_{t_n}z(\lambda)=\partial_x \frac{\Phi_{n+1}(z(\lambda))}{n+1},\quad n\geq 0,
\]
where $\Phi_n$ stands for the Faber polynomials (see Section 4), or for the first polynomials,
\[
\partial _{t_0}z(\lambda )=z_x(\lambda ),\text{ \ }\partial
_{t_1}z(\lambda )=\left( \frac{z^{2}(\lambda )}{2}+H^{0}\right)_x,
\]
\[
\partial _{t_2}z(\lambda )=\left( \frac{z^{3}(\lambda )}{3}%
+H^{0}z(\lambda )+H^{1}\right)_x,...
\]
identifying $x=t_0,s=-t_1,y=-t_2$ as well as $u=H^{0},v=H^{1}$ (see (%
\ref{gen})). Substituting a similar formal expansion ($\lambda
\rightarrow \infty $)%
\begin{equation}
z(\lambda )=\lambda -\frac{H^{0}}{\lambda }-\frac{H^{1}}{\lambda ^{2}}-\frac{%
H^{2}}{\lambda ^{3}}-... \label{d}
\end{equation}%
in these generating functions of conservation laws leads to infinitely
many local conservation laws. For instance%
\begin{equation}
(H^{k})_{s}+\left( H^{k+1}-\frac{1}{2}\underset{m=0}{\overset{k-1}{\sum }}%
H^{m}H^{k-1-m}\right) _{x}=0,\text{ }k=0,1,2,... \label{one}
\end{equation}%
This is nothing but the Benney hydrodynamic chain \eqref{BE},
written in the conservative form where all conservation law densities $H^{k}$
are polynomials with respect to moments $A^{m}$ as in Section~2.
Alternative expansions ($\zeta \rightarrow 0$)%
\begin{equation}
z(\zeta )=H^{-1}+\zeta H^{-2}+\zeta ^{2}H^{-3}+...,\text{ \ }\partial _{\tau
(\zeta )}=\partial _{t_{-1}}+\zeta \partial _{t_{-2}}+\zeta ^{2}\partial
_{t_{-3}}+... \label{z}
\end{equation}%
lead to another generating functions of conservation laws, for instance%
\begin{equation*}
\partial _{t_{-1}}z(\lambda )=\partial _{x}\ln (z(\lambda )-H^{-1}),\text{ \
}\partial _{t_{-2}}z(\lambda )=-\partial _{x}\frac{H^{-2}}{z(\lambda )-H^{-1}%
},...
\end{equation*}%
Substituting expansion (\ref{d}) and the expansion ($\lambda \rightarrow 0$)%
\begin{equation*}
z(\lambda )=H^{-1}+\lambda H^{-2}+\lambda ^{2}H^{-3}+...
\end{equation*}%
implies extra infinitely many conservation laws (cf. (\ref{one})). If for
instance, we substitute the above expansion in (\ref{gen}),
two additional conservation laws%
\[
\partial_sH^{-1}=-\left( \frac{1}{2}(H^{-1})^{2}+H^{0}\right)_x=-\partial_x\frac{\Phi_2(H^{-1})}{2},
\]
\[
\partial_yH^{-1}=-\left( \frac{1}{3}(H^{-1})^{3}+H^{0}H^{-1}+H^{1}\right)_x=-\partial_x\frac{\Phi_3(H^{-1})}{3}
\]
follow. Infinitely many conservative laws (cf. (\ref{one}))%
\begin{equation*}
(H^{-1})_{s}+\left( H^{0}+\frac{1}{2}(H^{-1})^{2}\right) _{x}=0,
\end{equation*}%
\begin{equation*}
(H^{k})_{s}+\left( H^{k+1}-\frac{1}{2}\overset{k-1}{\underset{m=0}{\sum }}%
H^{m}H^{k-1-m}\right) _{x}=0,\text{ }k=0,1,...
\end{equation*}%
also can be written as the modified Benney hydrodynamic chain (see details in
\cite{MaksBenney})%
\begin{equation}
B_{s}^{k}+B_{x}^{k+1}+\frac{1}{2}B^{0}B_{x}^{k}+\frac{k+1}{2}%
B^{k}B_{x}^{0}+kB^{k-1}\left( \frac{1}{2}B^{1}-\frac{1}{8}(B^{0})^{2}\right)
_{x}=0, \label{1}
\end{equation}%
where $H^{-1}=B^{0},H^{0}=B^{1},H^{1}=B^{2}+B^{0}B^{1}-\frac{1}{12}%
(B^{0})^{3}$,... This Modified Benney hydrodynamic chain is related to
the modified dKP equation (cf. (\ref{dKP}))%
\begin{equation}
H_{s}^{-1}+\left( H^{0}+\frac{1}{2}(H^{-1})^{2}\right) _{x}=0,\text{ \ }%
H^{0}_s=H^{-1}_y+\left( H^{0}H^{-1}+\frac{1}{3}(H^{-1})^{3}\right) _{x},
\label{MdKP}
\end{equation}%
which can be obtained from the compatibility condition $(\tilde{z}_{s})_{y}=(%
\tilde{z}_{y})_{s}$, where%
\begin{equation}
\tilde{z}_{s}+\left( \frac{\tilde{z}^{2}}{2}+H^{-1}\tilde{z}\right) _{x}=0,%
\text{ \ }\tilde{z}_{y}+\left( \frac{\tilde{z}^{3}}{3}+H^{-1}\tilde{z}%
^{2}+(H^{0}+(H^{-1})^{2})\tilde{z}\right) _{x}=0. \label{dist}
\end{equation}%
One can derive the modified L\"{o}wner equations%
\begin{equation*}
\partial _{i}\tilde{z}=\tilde{z}\frac{\partial _{i}H^{-1}}{\mu_{i}-\tilde{z}%
-H^{-1}},
\end{equation*}%
which are equivalent to the original L\"{o}wner equations (\ref{zero}) by
substituting $z=\tilde{z}+H^{-1}$ and $(\mu_{i}-H^{-1})\partial
_{i}H^{-1}=\partial _{i}H^{0}$.
Both hydrodynamic chains have the same local Hamiltonian structure%
\begin{equation*}
A_{s}^{k}=-\frac{1}{2}[(k+m)A^{k+m-1}\partial _{x}+mA_{x}^{k+m-1}]\frac{\partial H^{2}}{%
\partial A^{m}},\text{ \ }
\end{equation*}
\begin{equation*}
B_{s}^{k}=-\frac{1}{2}[(k+m)B^{k+m-1}\partial
_{x}+mB_{x}^{k+m-1}]\frac{\partial H^{1}}{\partial B^{m}},
\end{equation*}%
where their Hamiltonian densities are%
\begin{equation*}
H^{2}=A^{2}+(A^{0})^{2},\text{ \ }H^{1}=A^1=B^{2}+B^{0}B^{1}-\frac{1}{12}(B^{0})^{3}.
\end{equation*}%
Since all moments $A^{k}$ can be expressed via moments $%
B^{0},B^{1},...,B^{k},B^{k+1}$, the Kupershmidt--Manin Poisson brackets%
\begin{equation*}
\{B^{k},B^{m}\}=[(k+m)B^{k+m-1}\partial _{x}+mB_{x}^{k+m-1}]\delta
(x-x^{\prime })
\end{equation*}%
can be recalculated via moments $A^{s}$. This means that the Benney
hydrodynamic chain has at least two local Hamiltonian structures (see details
in \cite{MaksBenney}).
As in the previous particular case, we are looking for $N$ components commuting the
hydrodynamic reductions%
\begin{equation*}
r_{\tau (\zeta )}^{i}=w^{i}(\mathbf{r},\zeta )r_{x}^{i}.
\end{equation*}%
Then (\ref{ierar}) reduces to the form%
\begin{equation*}
\partial _{i}z(\lambda )=\frac{\partial _{i}z(\zeta )}{1-[z(\lambda
)-z(\zeta )]w^{i}(\mathbf{r},\zeta )}.
\end{equation*}%
Taking into account (\ref{zero}), one can obtain%
\begin{equation*}
w^{i}(\mathbf{r},\zeta )=\frac{1}{\mu_{i}-z(\zeta )}.
\end{equation*}%
Using expansions (\ref{f}), (\ref{z}), one can expand the generating
function (with respect to parameter $\zeta $ at infinity and about zero,
respectively) of infinitely many higher commuting flows%
\begin{equation*}
r_{\tau (\zeta )}^{i}=\frac{1}{\mu_{i}-z(\zeta )}r_{x}^{i}.
\end{equation*}
|
1811.08620
|
\section{Introduction}
\label{sec:intro}
The solution of many partial differential equations frequently must satisfy a maximum principle, or more generally, certain variables must obey a lower and/or upper bound. In this paper we will denote all these cases with positivity preserving. In particular, if the partial differential equations model physical processes then these bounds are also crucial to obtain a meaningful physical solution. For example, a density, concentration or pressure in fluid flow must be nonnegative, and a probability distribution should be in the range $[0,1]$. A numerical solution should therefore strictly obey the bounds on the exact solution, otherwise the problem can become ill-posed and the solution would be meaningless. Also, the numerical algorithm can easily become unstable and lack robustness if the numerical solution violates these essential bounds.
In recent years, the development of positivity preserving discontinuous Galerkin (DG) finite element methods therefore has been a very active area of research. The standard approach to ensure that the numerical solution satisfies the bounds imposed by the partial differential equations is to use limiters, but this can easily result in loss of accuracy, especially for higher order accurate discretizations.
In a seminal paper Zhang and Shu \cite{zhang2010maximum} showed how to design maximum principle and positivity preserving higher order accurate DG methods for first order scalar conservation laws. Their algorithm consists of a several important steps: i.) starting from a bounds preserving solution at time $t_n$ ensure that the element average of the solution satisfies the bounds at the next time level $t_{n+1}$ by selecting a suitable time step in combination with a monotone first order scheme; ii.) limit the higher order accurate polynomial solution at the quadrature points in each element without destroying the higher order accuracy; iii.) higher order accuracy in time can then be easily obtained using explicit SSP Runge-Kutta methods \cite{shu1988efficient}. This algorithm has been subsequently extended in many directions, e.g. various element shapes, convection-diffusion equation, Euler and Navier-Stokes equations and relativistic hydrodynamics \cite{zhang2012maximum,zhang2013maximum,zhang2010positivity,zhang2011positivity,zhang2017positivity,qin2016bound}. Other approaches to obtain higher order positivity preserving DG discretizations can be found in e.g. \cite{chen2016third,guo2015positivity,guo2017bound}.
All these DG discretizations use, however, an explicit time integration method. For many partial differential equations this results in an efficient numerical discretization, where to ensure stability the time step is restricted by the Courant-Friedrichs-Lewy (CFL) condition. On locally dense meshes and for higher order partial differential equations, which often have a time step constraint $\triangle t\leq C h^p$, with $p> 1$ and $h$ the mesh size, these time-explicit algorithms can become computationally very costly. The alternative is to resort to implicit time integration methods, but positivity preserving time-implicit DG discretizations are still very much in their infancy. Meister and Ortleb developed in \cite{meister2014unconditionally} a positivity preserving DG discretization for the shallow water equations using the Patankar technique \cite{patankar1980numerical}. Qin and Shu \cite{qin2018implicit} extended the framework in \cite{zhang2010maximum,zhang2010positivity} to implicit positivity preserving DG discretizations of conservation laws in combination with an implicit Euler time integration method. An interesting result of the analysis in \cite{qin2018implicit} is that to ensure positivity in the algorithm of Qin and Shu a lower bound on the time step is required. The approaches in \cite{meister2014unconditionally,qin2018implicit} require, however, a detailed analysis of the time-implicit DG discretization to ensure that the bounds are satisfied and are not so easy to extend to other classes of problems.
In this paper, we will present a very different approach to develop positivity preserving higher order accurate DG discretizations that are combined with a Diagonally Implicit Runge-Kutta (DIRK) time integration method. In analogy with obstacle problems we consider the bounds imposed by a maximum principle or positivity constraint as a restriction on the DG solution space. The constraints are then imposed using a limiter and directly coupled to the time-implicit higher order accurate DG discretization using Lagrange multipliers. { The resulting equations are the well-known Karush-Kuhn-Tucker (KKT) equations, which are frequently encountered in constrained optimization and solved with a semi-smooth Newton method \cite{facchinei2007finite,ito2008lagrange}, and also used in constrained optimization based discretizations of partial differential equation in e.g. \cite{Bochev2016CMA,Bochev2016bc,evans2009,BochevSJSC2017}.} The key benefit of the approach discussed in this paper, which we denote KKT Limiter and so far has not been applied to positivity preserving time-implicit DG discretizations, is that no detailed analysis is required to ensure that the DG discretization preserves the bounds for a particular partial differential equation. They are imposed explicitly and not part of the DG discretization. Also, since the limiter is only active in areas where positivity must be enforced, it does not affect the higher order DG discretization elsewhere since the Lagrange multipliers will be zero there. The approach discussed in this paper presents a general framework how to couple DG discretizations with limiters and, very importantly, how to efficiently solve the resulting nonlinear algebraic equations.
The algebraic equations resulting from the KKT formulation of the positivity preserving time-implicit DG discretization are only semi-smooth. This excludes the use of standard Newton methods since they require $C^1$ continuity \cite{deuflhard2011newton}. The obvious choice would be to use one of the many semi-smooth Newton methods available for nonlinear constrained optimization problems \cite{facchinei2007finite,ito2008lagrange}, but the algebraic equations for the positivity preserving time-implicit DG discretization have a different structure than for most constrained optimization problems.
For instance, the conditions to ensure a non-singular Jacobian \cite{facchinei2007finite} for methods based on the Fischer-Burmeister or related complementarity functions \cite{munson2001semismooth,chen2010smoothing} are not met by the KKT-limiter in combination with a time-implicit DG discretization. This frequently results in nearly singular Jacobian matrices, poor convergence and lack of robustness. We therefore developed an efficient active set semi-smooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semi-smooth Newton method can be proven using a specially designed quasi-directional derivative as outlined in \cite{han1992globally}, see also \cite{ito2008lagrange,ito2009semi}.
The organization of this paper is as follows. In Section \ref{sec:KKT_Limiter} we formulate the KKT equations, followed in Section \ref{sec:semismooth_Newton} by a discussion of an active set semi-smooth Newton method that is suitable to solve the nonlinear algebraic equations resulting from the positivity preserving time-implicit DG discretization. Special attention will be given to the quasi-directional derivative, which is an essential part to ensure convergence of the semi-smooth Newton method. In Section \ref{KKTDGdiscretization} we discuss the DG discretization in combination with an DIRK time integration method and positivity constraints. In Section \ref{Numerical Experiments} numerical experiments for the advection, Burgers, Allen-Cahn, Barenblatt, and Buckley-Leverett equations are provided. Conclusions are drawn in Section \ref{sec:conclusions}. In the Appendix more details on the quasi-directional derivative are given.
\section{Karush-Kuhn-Tucker limiting approach}
\label{sec:KKT_Limiter}
In this section we will directly couple the bounds preserving limiter to the time-implicit discontinuous Galerkin discretization using Lagrange multipliers. We will denote this approach as the KKT-Limiter.
Define the set
\begin{equation*}
K:=\{x\in\mathbb{R}^n\;\vert\; h(x)=0,\; g(x)\leq 0 \},
\end{equation*}
where $h:\mathbb{R}^n\rightarrow\mathbb{R}^l$ and $g:\mathbb{R}^n\rightarrow\mathbb{R}^m$ are twice continuously differentiable functions denoting, respectively, the $l$ equality and $m$ inequality constraints to be imposed on the DG discretization. The variable $x$ denotes the degrees of freedom and $n$ the number of degrees of freedom in the unlimited DG discretization. For the continuously differentiable function $L:\mathbb{R}^n\rightarrow\mathbb{R}^n$, representing the unlimited discontinuous Galerkin discretization, the KKT-equations are
\begin{subequations}
\begin{align}
\mathcal{L}(x,\mu,\lambda):=L(x)+\nabla h(x)^T\mu+\nabla g(x)^T\lambda&=0,\\
-h(x)&=0,\label{equality_constraint}\\
0\geq g(x)\perp\lambda&\geq 0,\label{compatibility_eq}
\end{align}\label{KKT_equations}
\end{subequations}
with $\mu\in\mathbb{R}^l$, $\lambda\in\mathbb{R}^m$ the Lagrange multipliers.
The compatibility condition \cref{compatibility_eq} is component-wise equal to:
\begin{equation*}
0\geq g_j(x),\qquad\lambda_j\geq 0\quad\text{and}\quad g_j(x)\lambda_j=0,\quad j=1,\cdots,m,
\end{equation*}
which is equivalent with
\begin{equation*}
\min(-g(x),\lambda)=0,
\end{equation*}
where the $\min$-function is applied component-wise.
The KKT-equations, with $F(z)\in\mathbb{R}^{n+l+m}$, can now be formulated as
\begin{equation}
0=F(z):=\left(\begin{matrix}
\mathcal{L}(x,\mu,\lambda)\\[5pt]
-h(x)\\[5pt]
\min(-g(x),\lambda)
\end{matrix}\right),\label{KKTnonlineareq}
\end{equation}
where $z:=(x,\mu,\lambda)$. In the next section we will discuss a global active set semi-smooth Newton suitable for the efficient solution of \cref{KKTnonlineareq} in combination with a DIRK-DG discretization. In Section \ref{KKTDGdiscretization} the DG discretization and KKT-Limiter will be presented for a number of scalar conservation laws.
\section{Semi-Smooth Newton Method}\label{sec:semismooth_Newton}
Standard Newton methods assume that $F(z)$ is continuously differentiable \cite{deuflhard2011newton}, but $F(z)$ given by \cref{KKTnonlineareq} is only semi-smooth \cite{facchinei2007finite}. In this section we will present a robust active set semi-smooth Newton method for \cref{KKTnonlineareq} that is suitable for the efficient solution of the KKT-equations resulting from a higher order DG discretization combined with positivity preserving limiters and a Diagonally Implicit Runge-Kutta time integration method \cite{hairer2010solving}.
\subsection{Differentiability concepts}
For the definition of the semi-smooth Newton method we need several more general definitions of derivatives, which will be discussed in this section. For more details, we refer to e.g. \cite{clarke1990optimization,facchinei2007finite,ito2008lagrange,shapiro1990concepts}. Since we use the semi-smooth Newton method directly on the algebraic equations of the limited DIRK-DG discretization we only consider finite dimensional spaces here.
Let $D\subseteq \mathbb{R}^m$ be an open subset in $\mathbb{R}^m$.
Given $d\in \mathbb{R}^m$, the {directional derivative} of $F:D\rightarrow\mathbb{R}^n$ at $x\in D$ in the direction $d$ is defined as
\begin{equation}
F^\prime(x;d):=\lim_{t\downarrow 0^+}\frac{F(x+t d)-F(x)}{t}.\label{eq:directional_derivative}
\end{equation}
A function $F:D \rightarrow\mathbb{R}^n$ is {locally Lipschitz continuous} if for every $x\in D$ there exists a neighborhood $N_x\subseteq D$ and a constant $C_x$, such that
\begin{equation*}
\vert F(y)-F(z)\vert\leq C_x\vert y-z\vert,\qquad\forall y,z\in N_x.
\end{equation*}
If $F$ is locally Lipschitz on $D$ then according to Rademacher's
theorem $F$ is differentiable almost everywhere with derivative $F^\prime(x)$.
The B-subdifferential $\partial_BF(x)$ of $F(x)$ is then defined as
\begin{equation*}
\partial_BF(x):=\lim_{\bar{x}\rightarrow x,\bar{x}\in D_F}F^\prime(\bar{x}),
\end{equation*}
with $D_F$ the points where $F$ is differentiable, and the generalized derivative in the sense of Clarke is defined as
\begin{equation*}
\partial F(x):=\text{ convex hull of $\partial_BF(x)$}.
\end{equation*}
For example, $F(x)=\vert x\vert$ at $x=0$ has $\partial_BF(0)=\{-1,1\}$ and $\partial F(0)=[-1,1]$.
A function $F: D\rightarrow\mathbb{R}^n$ is called semi-smooth if \cite{qi1993nonsmooth}
\begin{equation*}
\lim_{V\in\partial F(x+td^\prime), d^\prime\rightarrow d, t\downarrow 0^+} Vd^\prime\quad\text{exists for all $d\in\mathbb{R}^m$}.
\end{equation*}
A function $F:D\rightarrow \mathbb{R}^n$ is {Bouligand-differentiable} (B-differentiable) at $x\in D$ if it is directionally differentiable at $x$ and
\begin{equation*}
\lim_{d\rightarrow 0}\frac{F(x+d)-F(x)-F^\prime(x;d)}{\vert d\vert}=0.
\end{equation*}
A locally Lipschitz continuous function $F$ is {B-differentiable} at $x$ if and only if it is directionally differentiable at $x$ \cite{shapiro1990concepts}.
Given $d\in\mathbb{R}^m$, the {Clarke generalized directional derivative} of $F: D\rightarrow
\mathbb{R}^n$ at $x\in D$ in the direction of $d$ is defined by \cite{clarke1990optimization}
\begin{equation*}
F^0(x;d):=\lim_{y\rightarrow x}\sup_{t\downarrow 0^+}\frac{F(y+t d)-F(y)}{t}.
\end{equation*}
\subsection{Global active set semi-smooth Newton method}
For the construction of a global semi-smooth Newton method for \cref{KKTnonlineareq} we will use the merit function $\theta(z)=\frac{1}{2}\vert F(z)\vert^2$, with $z=(x,\mu,\lambda)$. The Clarke directional derivative of $\theta$ and $F$ have the following relation.
Let $F:D\subseteq\mathbb{R}^p\rightarrow\mathbb{R}^p$, with $D$ an open set and $p=n+l+m$, be a locally Lipschitz continuous function then the Clarke generalized directional derivative of $\theta(z)$ can be expressed as \cite{ito2008lagrange}
\begin{equation}
\theta^0(z;d)=\limsup_{y\rightarrow z, t\downarrow 0^+}\frac{(F(z),(F(y+td)-F(y))}{t},\label{thetaclark}
\end{equation}
and there exists an $F^0:D\times\mathbb{R}^p\rightarrow\mathbb{R}^p$ such that
\begin{equation}
\theta^0(z;d)=(F(z),F^0(z;d))\qquad\text{for $(z,d)\in D\times\mathbb{R}^p$}.\label{thetader1}
\end{equation}
Here $(\cdot,\cdot)$ denotes the Euclidian inner product.
The crucial point in designing a Newton method is to obtain proper descent directions for the Newton iterations. A possible choice is to use the Clarke derivative $\partial F$ as generalized Jacobian \cite{facchinei2007finite,ito2008lagrange}, but this derivative is in general difficult to compute. In \cite{pang1990newton,pang1991b} it was proposed to use $d$ as the solution of
\begin{equation}
F(z)+F^\prime(z;d)=0,\label{direcderivsearch}
\end{equation}
which for the KKT-equations results in a mixed linear complementarity problem \cite{pang1991b}. Unfortunately, \cref{direcderivsearch} does not always have a solution, unless additional conditions are imposed. A better alternative is to use the quasi-directional derivative $G$ of $F$
\cite{han1992globally,ito2008lagrange,ito2009semi}.
Let $F:D\subseteq\mathbb{R}^p\rightarrow\mathbb{R}^p$ be directionally differentiable and locally Lipschitz continuous. Assume that $S=\{z\in D\;\vert\; \vert F(z)\vert\leq \vert F(z^0)\vert\}$ is bounded. Then $G:S\times\mathbb{R}^p\rightarrow\mathbb{R}^p$ is called the quasi-directional derivative of $F$ on $S\subset\mathbb{R}^p$ if for all $z,\bar{z}\in S$ the following conditions hold \cite{han1992globally,ito2008lagrange,ito2009semi}
\begin{subequations}
\begin{align}
&(F(z),F^\prime(z;d))\leq (F(z),G(z;d)),\label{quasi_der_cond1}\\[5pt]
&G(z;td)=tG(z;d)\quad\text{for all $d\in\mathbb{R}^p,z\in S$ and $t\geq 0$},\label{quasi_der_cond2}\\[5pt]
&(F(\bar{z}),F^0(\bar{z};\bar{d}))\leq\limsup_{z\rightarrow \bar{z},d\rightarrow \bar{d}}(F(z),G(z;d))\quad\text{for all $z\rightarrow \bar{z},d\rightarrow \bar{d}$}.\label{Gupperbound}
\end{align}\label{quasidirectional_derivative}
\end{subequations}
The search direction $d$ in the semi-smooth Newton method is now the solution of
\begin{equation}
F(z)+G(z;d)=0,\quad\text{with}\; z\in S,d\in\mathbb{R}^p,\label{searcheq}
\end{equation}
which results for the KKT-equations \cref{KKTnonlineareq} in a mixed linear complementarity problem.
Using \cref{thetader1}, \cref{Gupperbound} and \cref{searcheq} this immediately results in the bound
\begin{equation*}
\theta^0(\bar{z};\bar{d})\leq \limsup_{z\rightarrow \bar{z},d\rightarrow \bar{d}}(F(z),G(z;d))=-\lim_{z\rightarrow \bar{z}}\vert F(z)\vert^2=-2\theta(\bar{z}).
\end{equation*}
Hence the search direction $d$ obtained from \cref{searcheq} always provides a descent direction for the merit function $\theta(z)$. The merit function $\theta(z)$ and the quasi-directional derivative $G(z,d)$ can therefore be used to define a global line search semi-smooth Newton algorithm, which is stated in Algorithm \ref{Newton_algorithm}. The key benefit of using the quasi-directional derivative $G$ in this Newton algorithm is that, under the additional assumption $\Vert G(z;d)\Vert\geq L\Vert d\Vert$, with $L>0$ constant, we immediately obtain a proof of the convergence of this algorithm, given by \cite{han1992globally}, Theorem 1.
In the next section we will present the quasi-directional derivative $G$ for the KKT-equations \cref{KKTnonlineareq} and define the active sets used to solve \cref{searcheq} with the semi-smooth Newton algorithm presented in Section \ref{active_set_algorithm}. In Section \ref{KKTDGdiscretization} Algorithm \ref{Newton_algorithm} will then be used to solve the nonlinear equations resulting from the DG discretization using a KKT-limiter in combination with a Diagonally Implicit Runge-Kutta (DIRK) method.
\subsection{Quasi-directional derivative}
In order to compute the quasi-directional derivative $G$, satisfying the conditions stated in \cref{quasidirectional_derivative}, we first need to compute the directional and Clarke generalized directional derivatives of the function $F(z)$ defined in \cref{KKTnonlineareq}.
Define $z\in\mathbb{R}^p$, with $p=n+l+m$ as $z=(x,\mu,\lambda)$ with $x\in\mathbb{R}^n$, $\mu\in\mathbb{R}^l$, $\lambda\in\mathbb{R}^m$.
Define $d\in\mathbb{R}^p$ as $d=(u,v,w)$ with $u\in\mathbb{R}^n$, $v\in\mathbb{R}^l$, $w\in\mathbb{R}^m$.
The {directional derivative} $F^\prime(z;d)\in\mathbb{R}^p\times \mathbb{R}^p$ of $F(z)$ defined in \cref{KKTnonlineareq} in the direction $d$ is equal to
\begin{subequations}
\begin{alignat}{2}
F_i^\prime(z;d)&=D_x\mathcal{L}_i(z)\cdot u+D_\mu \mathcal{L}_i(z)\cdot v+D_\lambda \mathcal{L}_i(z)\cdot w,&\qquad &i\in N_n,\\
F_{i+n}^\prime(z;d)&=-D_xh_i(x)\cdot u,&& i\in N_l,\\
F_{i+n+l}^\prime(z;d)&=-D_xg_i(x)\cdot u,&& i\in \alpha(z),\\
&=\min(-D_xg_i(x)\cdot u,w_i),&& i\in\beta(z),\label{directionaldermin}\\
&=w_i,&& i\in\gamma(z),
\end{alignat}\label{directionalderivative}
\end{subequations}
where the following sets are used
\begin{align*}
N_q&=\left\{j\in\mathbb{N}\;\vert\;1\leq j\leq q\right\},\\
\alpha(z)&=\left\{j\in\mathbb{N}_m\;\vert\;\lambda_j>-g_j(x)\right\},\\
\beta(z)&=\left\{j\in\mathbb{N}_m\;\vert\;\lambda_j=-g_j(x)\right\},\\
\gamma(z)&=\left\{j\in\mathbb{N}_m\;\vert\;\lambda_j<-g_j(x)\right\},
\end{align*}
with $q=n$ or $q=l$. The calculation of most of the terms in \cref{directionalderivative} is straightforward, except \cref{directionaldermin}, which can be computed using a Taylor series expansion of the arguments of $\min(-g_i(x),\lambda_i)$ in the limit of the directional derivative \cref{eq:directional_derivative}, combined with the relation $\min(a+b,a+d)-\min(a,a)=\min(b,d)$ and the fact that $i\in\beta(z)$.
The Clarke Generalized derivative of $F(z)$ can be computed using the relations \cref{thetaclark}-\cref{thetader1} and is equal to
\begin{subequations}
\begin{alignat}{2}
F_i^0(z;d)&=D_x\mathcal{L}_i(z)\cdot u+D_\mu \mathcal{L}_i(z)\cdot v+D_\lambda \mathcal{L}_i(z)\cdot w,&\quad &i\in N_n,\\
F_{i+n}^0(z;d)&=-D_xh_i(x)\cdot u,&&\hspace*{-30pt} i\in N_l,\\
F_{i+n+l}^0(z;d)&=-D_xg_i(x)\cdot u,&&\hspace*{-30pt} i\in \alpha(z),\label{Clarke_direc_deriva}\\
&=\max(-D_xg_i(x)\cdot u,w_i),&&\hspace*{-30pt} i\in\beta(z),F_{i+n+l}(z)>0,\label{Clarke_direc_derivb}\\
&=\min(-D_xg_i(x)\cdot u,w_i),&&\hspace*{-30pt} i\in\beta(z),F_{i+n+l}(z)\leq 0,\label{Clarke_direc_derivc}\\
&=w_i,&&\hspace*{-30pt} i\in\gamma(z).
\end{alignat}\label{Clarke_direc_deriv}
\end{subequations}
The calculation of \cref{Clarke_direc_derivb,Clarke_direc_derivc} in $F^0(z;d)$ is non-trivial and is detailed in Appendix \ref{Appendix1}.
Using the results for the directional derivative and the Clarke generalized directional derivative we can now state a quasi-directional derivative $G:D\times\mathbb{R}^p\rightarrow\mathbb{R}^p$, satisfying the conditions \cref{quasidirectional_derivative}, which for any $\delta>0$ is equal to
\begin{subequations}
\begin{alignat}{2}
G_i(z;d)&=D_x\mathcal{L}_i(z)\cdot u+D_\mu \mathcal{L}_i(z)\cdot v+D_\lambda \mathcal{L}_i(z)\cdot w,&\quad &i\in N_n,\\
G_{i+n}(z;d)&=-D_xh_i(x)\cdot u,&&\hspace*{-40pt} i\in N_l,\\
G_{i+n+l}(z;d)&=-D_xg_i(x)\cdot u,&&\hspace*{-40pt} i\in \alpha_\delta(z),\\
&=\max(-D_xg_i(x)\cdot u,w_i),&&\hspace*{-40pt} i\in\beta_\delta(z),F_{i+n+l}(z)>0,\\
&=\min(-D_xg_i(x)\cdot u,w_i),&&\hspace*{-40pt} i\in\beta_\delta(z),F_{i+n+l}(z)\leq 0,\\
&=w_i,&&\hspace*{-40pt} i\in\gamma_\delta(z),
\end{alignat}\label{quasidirectionDeriv1}
\end{subequations}
with the sets
\begin{align*}
\alpha_\delta(z)&=\left\{j\in\mathbb{N}_m\;\vert\;\lambda_j>-g_j(x)+\delta\right\},\\
\beta_\delta(z)&=\left\{j\in\mathbb{N}_m\;\vert\;-g_j(x)-\delta\leq\lambda_j\leq -g_j(x)+\delta\right\},\\
\gamma_\delta(z)&=\left\{j\in\mathbb{N}_m\;\vert\;\lambda_j<-g_j(x)-\delta\right\}.
\end{align*}
The main benefit of introducing the $\delta$-dependent sets is that in practice it is hard to test for the set $\beta(z)$, which would generally be ignored in real computations due to rounding errors. One would then miss a number of important components in the quasi-directional derivative, which can significantly affect the performance of the Newton algorithm. The set $\beta_\delta$ gives, however, a computational well defined quasi-directional derivative $G(z;d)$. In Appendix \ref{Appendix2} a proof is given that $G(z;d)$ satisfies the conditions stated in \cref{quasidirectional_derivative}, which is the condition required in \cite{han1992globally}, Theorem 1, to ensure convergence of the semi-smooth Newton method.
The formulation of the quasi-directional derivative $G$ \cref{quasidirectionDeriv1} is, however, not directly useful as a Jacobian in the semi-smooth Newton method due to the $\max$ and $\min$ functions. In order to eliminate these functions we introduce the following sets
\begin{align*}
I_{\beta_\delta}^{11}(z,d)&:=\{i\in\beta_\delta(z)\;\vert\; F_{i+n+l}(z)>0,-D_xg_i(x)\cdot u>w_i\},\\
I_{\beta_\delta}^{12}(z,d)&:=\{i\in\beta_\delta(z)\;\vert\; F_{i+n+l}(z)>0,-D_xg_i(x)\cdot u\leq w_i\},\\
I_{\beta_\delta}^{21}(z,d)&:=\{i\in\beta_\delta(z)\;\vert\; F_{i+n+l}(z)\leq 0,-D_xg_i(x)\cdot u>w_i\},\\
I_{\beta_\delta}^{22}(z,d)&:=\{i\in\beta_\delta(z)\;\vert\; F_{i+n+l}(z)\leq 0,-D_xg_i(x)\cdot u\leq w_i\},
\end{align*}
and define
\begin{subequations}
\begin{align}
I_\delta^1(z,d)&:=\alpha_\delta(z)\cup I_{\beta_\delta}^{11}(z,d)\cup I_{\beta_\delta}^{22}(z,d),\\
I_\delta^2(z,d)&:=\gamma_\delta(z)\cup I_{\beta_\delta}^{12}(z,d)\cup I_{\beta_\delta}^{21}(z,d).
\end{align}\label{active_sets}
\end{subequations}
The quasi-directional derivative $G(z;d)$ can now be written in a form suitable to serve as a Jacobian in the active set semi-smooth Newton method defined in Algorithm \ref{Newton_algorithm} to solve \cref{KKTnonlineareq}
\begin{equation*}
G(z;d)=\widehat{G}(z)d,
\end{equation*}
with
\begin{equation}
\widehat{G}(z)=\left(\begin{matrix}
D_x\mathcal{L}_i(z)\vert_{i\in N_n} & D_\mu \mathcal{L}_i(z)\vert_{i\in N_n} & D_\lambda \mathcal{L}_i(z)\vert_{i\in N_n} \\
-D_x h_i(x)\vert_{i\in N_l}&0&0\\
-D_x g_i(x)\vert_{i\in I^1_\delta(z,d)} & 0 & \delta_{ij}\vert_{i,j\in I_\delta^2(z,d)}
\end{matrix}\right)\in\mathbb{R}^{p\times p},\label{eq:quasi-direc-deriv}
\end{equation}
with $\delta_{ij}$ the Kronecker symbol. By updating the sets $I_\delta^1(z;d)$ and $I_\delta^2(z;d)$ as part of the Newton method the complementary problem \cref{searcheq} is simultaneously solved with the solution of \cref{KKTnonlineareq}. In general, after a few iterations the proper sets $I^{1,2}_\delta(z;d)$ will be found and the semi-smooth Newton method then converges like a regular Newton method. Also, one should note that {\it only} the contribution $D_x\mathcal{L}_i(z)$ in \cref{eq:quasi-direc-deriv} depends on the DG discretization in $\mathcal{L}_i(z)$. Hence, the KKT-Limiter provides a general framework to impose limiters on time-implicit numerical discretizations and could for instance also be applied to time-implicit finite volume discretizations.
\subsection{Active set semi-smooth Newton algorithm}\label{active_set_algorithm}
\algsetup{indent=2em}
\begin{algorithm}[tbh]
\small
\caption{Active set semi-smooth Newton method}
\label{Newton_algorithm}
\begin{algorithmic}[1]
\STATE (A.0) ({\it Initialization}) Let $\bar{\alpha}\geq 0$, $\beta,\gamma\in(0,1)$, $\sigma\in(0,\bar{\sigma})$, $\delta>0$ and $b>C\in\mathbb{R}^+$ arbitrarily large, but bounded. Choose $z^0, d^0\in\mathbb{R}^p$ and tolerance $\epsilon$.
\STATE (A.1) Scale $z^0$.
\STATE (A.2) ({\it Newton method})
\FOR{$k=0,1,\cdots$ until $\Vert F(z^k)\Vert\leq\epsilon$ \AND $\Vert d^k\Vert\leq\epsilon$}
\STATE Compute the quasi-directional derivative matrix $\widehat{G}_k:=\widehat{G}(z^k)$ given by \cref{eq:quasi-direc-deriv} and the active sets $I^1_\delta(z;d)$, $I^2_\delta(z;d)$ of $\widehat{G}_k$ given by \cref{active_sets}.
\STATE Apply row-column scaling to $ (\widehat{G}^T_k\widehat{G}_k+\bar{\alpha}\Vert F(z^k)/F(z^0)\Vert I)$, with $I$ the identity matrix, such that the matrix has a norm $\Vert\cdot\Vert_{L^\infty}\cong 1$.
\IF {there exists a solution $h^k$ to
\begin{equation}
(\widehat{G}^T_k\widehat{G}_k+\bar{\alpha}\Vert F(z^k)/F(z^0)\Vert I)h^k=-\widehat{G}_k^TF(z^k),\label{leastsq}
\end{equation}
with $\vert h^k\vert\leq b\vert F(z^k)\vert$ \AND
$$\vert F(z^k+h^k)\vert<\gamma\vert F(z^k)\vert,$$}
\STATE Set $d^k=h^k$, $z^{k+1}=z^k+d^k$, $\alpha_k=1$ and $m_k=0$.
\ELSE
\STATE Choose $d^k=h^k$.
\STATE Compute $\alpha_k=\beta^{m_k}$, where $m_k$ is the first positive integer $m$ for which,
$$\theta(z^k+\beta^{m_k}d^k)-\theta(z^k)\leq-\sigma\beta^m\theta(z^k).$$
\vspace*{-10pt}
\STATE Set $z^{k+1}=z^k+\alpha_kd^k$.
\ENDIF
\ENDFOR
\end{algorithmic}\label{algorithm_1}
\end{algorithm}
As default values we use in Algorithm \ref{algorithm_1} $\bar{\alpha}=10^{-12}$, $\beta=\gamma=\frac{1}{2}$, $\sigma=10^{-9}$, $\delta=10^{-12}$ and $\epsilon=10^{-8}$.
An important aspect of Algorithm \ref{Newton_algorithm} is that we simultaneously solve the mixed linear complementarity equations \cref{searcheq} for the search direction $d$ as part of the global Newton method using an active set technique. This was motivated by \cite{harker1990damped} and will reduce the mixed linear complementarity problem \cref{searcheq} into a set of linear equations. The use of the active set technique is also based on the observation in \cite{ito2009semi} of the close relation between an active set Newton method and a semi-smooth Newton method. After the proper sets $I^1_\delta(z;d)$, $I^2_\delta(z;d)$ are obtained for the quasi-directional derivative $G(z;d)$ the difference with a Newton method for smooth problems \cite{deuflhard2011newton} will be rather small. The mixed linear complementarity problem can, however, have one, multiple or no solutions and, in order to deal also with cases where the matrix $G$ is poorly conditioned, we will use a minimum norm least squares or Gauss-Newton method to solve the algebraic equations \cref{leastsq}.
For the performance of a Newton algorithm proper scaling of the variables is crucial. Here, we use the approach outlined in \cite{deuflhard2011newton} and the Newton method is applied directly to the scaled variables. Also, the matrix $\widehat{G}^T_k\widehat{G}_k+\bar{\alpha}\Vert F(z^k)/F(z^0)\Vert I$ in the Newton method will have a much larger condition number than the matrix $\widehat{G}_k$. In order to improve the conditioning of this matrix we use simultaneous iterative row and column scaling in the $L^\infty$-matrix norm, as described in \cite{amestoy2008parallel}. This algorithm very efficiently scales the rows and columns such that an $L^\infty$-matrix norm approximately equal to one is obtained. This gives a many orders of magnitude reduction in the matrix condition number and generally reduces the condition number of the matrix \cref{leastsq} to the same order as the condition number of the original matrix $\widehat{G}_k$.
\section{KKT-Limiter DG discretization}\label{KKTDGdiscretization}
Given a domain $\Omega\subseteq\mathbb{R}^d$, $d={\rm dim}(\Omega)$, $d=1,2$, with Lipschitz continuous boundary $\partial\Omega$.
As general model problem we consider the following second order nonlinear scalar equation
\begin{equation}
\frac{\partial u}{\partial t}+\nabla\cdot F(u)+G(u)-\nabla\cdot(\nu(u)\nabla u)=0,\label{eq:2ndconservation_law}
\end{equation}
with $u(x,t):\mathbb{R}^d\times\mathbb{R}^+\rightarrow\mathbb{R}$ a scalar quantity, $F(u):\mathbb{R}\rightarrow\mathbb{R}^d$ the flux, $G(u):\mathbb{R}\rightarrow\mathbb{R}$ a reaction term and $\nu(u):\mathbb{R}\rightarrow\mathbb{R}^+$ a nonlinear diffusion term.
By selecting different functions $F, G$ and $\nu$ in \cref{eq:2ndconservation_law} we will demonstrate in Section \ref{Numerical Experiments} the KKT-Limiter on various model problems that impose different positivity constraints on the solution.
For the DG discretization we introduce the auxiliary variable $Q\in\mathbb{R}^d$ and rewrite
\cref{eq:2ndconservation_law} as a first order system of conservation laws
\begin{subequations}
\begin{align}
\frac{\partial u}{\partial t}+\nabla\cdot F(u)+G(u)-\nabla\cdot(\nu(u)Q)&=0,\\
Q-\nabla u&=0.
\end{align}\label{eq:1stconservation_law}
\end{subequations}
\subsection{DG discretization}\label{DGdiscretization}
Let $\mathcal{T}_h$ be a tessellation of the domain $\Omega$ with shape regular line or quadrilateral elements $K$ with maximum diameter $h>0$. The total number of elements in $\mathcal{T}_h$ is $N_K$. We denote the union of the set of all boundary faces $\partial K$, $K\in\mathcal{T}_h$, as $\mathcal{F}_h$, all internal faces ${\mathcal F}_h^i$ and the boundary faces as ${\mathcal F}^b_h$, hence $\mathcal{F}_h=\mathcal{F}_h^i\cup\mathcal{F}_h^b$. The elements connected to each side of a face $S\in\mathcal{F}_h$ are denoted by the indices $L$ and $R$, respectively. { For the KKT-Limiter it is important to use orthogonal basis functions, see Section \ref{Limiter_Constraints}. In this paper $\mathcal{P}_p(K)$ represent tensor product Legendre polynomials of degree $p$ on $d$-dimensional rectangular elements $K\in\mathcal{T}_h$, when $K$ is mapped to the reference element $(-1,1)^d$. For general elements one can use Jacobi polynomials with proper weights to obtain an orthogonal basis, see \cite{karniadakis2013spectral}, Section 3.2.}
Next, we define the finite element spaces
\begin{align*}
V_h^p&:=\Big\{v\in L^2(\Omega)\;\vert\; v\vert_K\in \mathcal{P}_p(K),\forall K\in\mathcal{T}_h\Big\},\\
W_h^p&:=\Big\{v\in (L^2(\Omega))^d\;\vert\; v\vert_K\in (\mathcal{P}_p(K))^d,\forall K\in\mathcal{T}_h\Big\},
\end{align*}
with $L^2(\Omega)$ the Sobolev space of square integrable functions.
Equation (\ref{eq:1stconservation_law}) is discretized using the Local Discontinuous Galerkin discretization from \cite{cockburn1998local}.
Define $L^1_h: V_h^p\times W_h^p\times V_h^p\rightarrow \mathbb{R}$ and $L^2_h: V_h^p\times W_h^p\rightarrow \mathbb{R}$ as
\begin{align}
L^1_h(u_h,Q_h;v):=&-\big(F(u_h)-\nu(u_h)Q_h,\nabla_h v\big)_\Omega+\big(G(u_h),v\big)_\Omega\nonumber\\[4pt]
+&\sum_{S\in\mathcal{F}_h^i}\big(H(u_h^L,u_h^R;n^L)-\widehat{\nu(u_h)}n^L\cdot\widehat{{Q}_h},v^L-v^R\big)_S\nonumber\\
+&\sum_{S\in\mathcal{F}_h^b}\big(H(u_h^L,u_h^b;n^L)-\widehat{\nu(u_h)}n^L\cdot Q_{h}^b,v^L\big)_S,\label{L1_discretization}\\
L^2_h(u_h;w):=&\big(u_h,\nabla_h\cdot w\big)_\Omega
-\sum_{S\in\mathcal{F}_h^i}\big(\widehat{u_h}n^L,w^L-w^R\big)_S\nonumber\\
-&\sum_{S\in\mathcal{F}_h^b}\big(u_h^bn^L,w^L\big)_S,\nonumber
\end{align}
where $(\cdot,\cdot)_D$ is the $L^2(D)$ inner product, $\nabla_h$ the element-wise nabla operator and the superscript $b$ refers to boundary data.
Here $n^L\in\mathbb{R}^d$ is the exterior unit normal vector at the boundary of the element $L\in\mathcal{T}_h$ that is connected to face $S$. The numerical flux $H$ is the Lax-Friedrichs flux
\begin{equation*}
H(u_h^L,u_h^R;n)=\frac{1}{2}\big(n\cdot(F(u_h^L)+F(u_h^R))-C_{LF}(u_h^R-u_h^L)\big),
\end{equation*}
with Lax-Friedrichs coefficient $C_{LF}=\sup_{u_h\in[u_h^L,u_h^R]}\vert\frac{\partial}{\partial u_h}(n\cdot F(u_h))\vert$. For $\widehat{{Q}_h}$ and $\widehat{u_h}$ we use the alternating fluxes
\begin{subequations}
\begin{align}
\widehat{Q_h}&=(1-\alpha)Q_{h}^L+\alpha Q_{h}^R,\\
\widehat{u_h}&=\alpha u_h^L+(1-\alpha)u_h^R,
\end{align}\label{upwind_flux}
\end{subequations}
with $0\leq\alpha\leq 1$. The numerical flux for the nonlinear diffusion is defined as
\begin{equation*}
\widehat{\nu(u_h)}=\frac{1}{2}(\nu(u_h^L)+\nu(u_h^R)).\end{equation*}
For $t\in(0,T]$ the semi-discrete DG formulation for \cref{eq:1stconservation_law} now can be expressed as: Find $u_h(t)\in V_h^p$, $Q_h(t)\in W_h^p$, such that for all $v\in V_h^p$, $w\in W_h^p$,
\begin{subequations}
\begin{align}
\Big(\frac{\partial u_h}{\partial t},v\Big)_\Omega+L^1_h(u_h,Q_h;v)&=0,\label{eq:uhequation}\\
(Q_h,w)_\Omega+L^2_h(u_h;w)&=0.\label{eq:Qhequation}
\end{align}
\end{subequations}
These equations are discretized in time with a Diagonally Implicit Runge-Kutta (DIRK) method \cite{hairer2010solving}. The main benefit of the DIRK method is that the Runge-Kutta stages can be computed successively, which significantly reduces the computational cost and memory overhead.
{
We represent $u_h$ and $Q_h$ in each element $K\in\mathcal{T}_h$, respectively, as $u_h\vert_K=\sum_{j=1}^{N_u}\widehat{U}_j^K\phi_j^K$ and $Q_h\vert_K=\sum_{j=1}^{N_Q}\widehat{Q}_j^K\psi_j^K$, with basis functions $\phi_j^K\in\mathcal{P}_p(K)$, $\psi_j^K\in\big(\mathcal{P}_p(K)\big)^d$ and DG coefficients $\widehat{U}_j^K\in\mathbb{R}$, $\widehat{Q}_j^K\in\mathbb{R}^{d}$. After replacing the test functions $v\in V_h^p$ in \cref{eq:uhequation} and $w\in W_h^p$ \cref{eq:Qhequation} with, respectively, the independent basis functions $\phi_i^K\in\mathcal{P}_p(K)$, $i=1,\cdots,N_u$, and $\psi_i^K\in\big(\mathcal{P}_p(K)\big)^d$, $i=1,\cdots,N_Q$, we obtain the algebraic equations for the DG discretization.
In order to simplify notation we introduce
$\widehat{L}_h^1(\widehat{U},\widehat{Q})=L_h^1(u_h,Q_h;\phi)\in\mathbb{R}^{N_uN_K}$ and $\widehat{L}_h^2(\widehat{U})=L_h^2(u_h;\psi)\in\mathbb{R}^{dN_QN_K}$, with $N_K$ the number of elements in $\mathcal{T}_h$ and $\phi=\phi_i^K$, $\psi=\psi_i^K$ the basis functions in element $K$. The algebraic equations for the DIRK stage vector
$\widehat{K}^{(i)}\in\mathbb{R}^{N_uN_K}$, $i=1,\cdots,s$ with the DG coefficients, then can be expressed as
\begin{align}
\widehat{L}_h(\widehat{K}^{(i)}):=&M_1\big(\widehat{K}^{(i)}-\widehat{U}^n\big)+\triangle t\sum_{j=1}^i a_{ij}\widehat{L}^1_h\big(\widehat{K}^{(j)},-M_2^{-1}\widehat{L}^2_h(\widehat{K}^{(j)})\big)=0.\label{DIRK_stage}
\end{align}
Here we eliminated the DG coefficients for the auxiliary variable $Q_h$ using \cref{eq:Qhequation}.} The matrices $M_1\in\mathbb{R}^{N_uN_K\times N_uN_K}$, $M_2\in\mathbb{R}^{dN_QN_K\times dN_QN_K}$ are block-diagonal mass matrices since we use orthogonal basis functions and $n$ denotes the index of time level $t=t_n$.
The coefficients $a_{ij}$ are the coefficients in the Butcher tableau, which determine the properties of the Runge-Kutta method \cite{hairer2010solving}. For DIRK methods $a_{ij}=0$ if $j>i$. The following DIRK methods are used: for basis functions with polynomial order $p=1$ \cite{Alexander1977diagonally}, Page 1012, Theorem 5, first method with $\alpha=1-\frac{1}{2}$; $p=2$ \cite{skvortsov2006diagonally}, Page 2117 (top); $p=3$ \cite{Alexander1977diagonally} Page 1012, Theorem 5, second method, see also \cite{skvortsov2006diagonally}, Page 2117 (top). { The order of accuracy of these DIRK methods is $p+1$ and their coefficients in the Butcher tableau satisfy $a_{sj}=b_j$, $j=1,\cdots,s$, which implies that these methods are stiffly accurate, see \cite{hairer2010solving}, Section IV.6, and the solution of the last DIRK stage is equal to the solution at the new time-step
\begin{equation*}
\widehat{U}^{n+1}=\widehat{K}^{(s)}.
\end{equation*}
Since each DIRK stage vector must satisfy the positivity constraints this then also immediately applies to the solution at time $t_{n+1}$.
The Jacobian $D_x\mathcal{L}(\widehat{K}^{(i)})\in\mathbb{R}^{N_uN_K\times N_uN_K}$, with $x=\widehat{K}^{(i)}$, in the quasi-directional derivative $G$ \cref{eq:quasi-direc-deriv} of DIRK stage $i$ of the unlimited DIRK-DG discretization \cref{DIRK_stage} is now equal to
\begin{equation*}
D_x\mathcal{L}(\widehat{K}^{(i)})=M_1+\triangle t a_{ii}\Big(\frac{\partial L_h^1}{\partial\widehat{K}^{(i)}}-\frac{\partial L_h^1}{\partial\widehat{Q}^{(i)}}M_2^{-1}\frac{\partial L_h^2}{\partial\widehat{K}^{(i)}}\Big).
\end{equation*}
\subsection{Limiter constraints}\label{Limiter_Constraints}
The limiter constraints for the DG discretization can be imposed directly by defining the inequality constraints in the KKT-equations. In each element $K\in\mathcal{T}_h$ we apply for each DIRK-stage $i=1,\cdots,s$, the following inequality constraints
\begin{itemize}
\item[{\it i.}] {\it Positivity constraint}
\begin{equation}
g^K_{1,k}(\widehat{K}^{K,(i)})=u_{\min}-\sum_{q=1}^{N_u}\widehat{K}^{K,{(i)}}_q\phi_q^K(x_k),\quad k=1,\cdots,N_p,\label{positivity_limiter}
\end{equation}
\item[{\it ii.}] {\it Maximum constraint}
\begin{equation}
g^K_{2,k}(\widehat{K}^{K,(i)})=\sum_{q=1}^{N_u}\widehat{K}^{K,{(i)}}_q\phi_q^K(x_k)-u_{\max},\quad k=1,\cdots,N_p.\label{maximum_limiter}
\end{equation}
Here the superscript $K$ refers to element $K\in\mathcal{T}_h$, and $(i)$ is the $i$-th DIRK-stage. The points $x_k$, $k=1,\cdots,N_p$, are the points in element $K$ where the inequality constraints are imposed and $u_{\min}$ and $u_{\max}$ denote, respectively, the allowed minimum and maximum value of $u$. The inequality constraints are imposed using the Lagrange multiplier $\lambda$, see \cref{compatibility_eq}.
\smallskip
\item[{\it iii.}] {\it Conservation constraint}
\smallskip
Since the basis functions $\phi_j^K,j=1,\cdots,N_u$ are orthogonal in each element $K$, we have $(1,\phi_j^K)_K=0$, for $j=2,\cdots, N_u$. Hence, at each Runge-Kutta stage $i$, limiting the DG coefficients $\widehat{K}^{K,(i)}_j$ with $j=2,\cdots,N_u$ has no effect on the element average $\bar{u}^{K,(i)}_h=\frac{1}{\vert K\vert}(u_h^{(i)},1)_K=\widehat{K}^{K,(i)}_1$, with $u_h^{(i)}$ the solution at stage $i$, and therefore does not influence the conservation properties of the DG discretization.
\smallskip
Limiting the DG coefficients $\widehat{K}^{K,(i)}_1$ can, however, effect the conservation properties of the DG discretization since $\bar{u}^{K,(i)}_h=\widehat{K}^{K,(i)}_1$. In order to ensure local conservation we therefore need to impose in each element the local conservation constraint
\begin{align}
h^K\big(\widehat{K}^{K,(i)}\big)&=\widehat{L}_{h,1}^K(\widehat{K}^{(i)})\nonumber\\
&=\vert K\vert \big(\widehat{K}^{K,(i)}_1-\widehat{U}^{n}_1\big)+(G(u_h^{(i)},\phi_1^K)_K\nonumber\\[3pt]
&+\sum_{S\in\mathcal{F}_h^i\cap\partial K}\big(H(u_h^{L,(i)},u_h^{R,(i)};n^L)-\nonumber\\&\hspace*{58pt}\widehat{\nu(u_h)}n^L\cdot
((1-\alpha)Q_h^{L,(i)}+\alpha Q_h^{R,(i)}),\phi^L_1-\phi^R_1\big)_S\nonumber\\[5pt]
&+\sum_{S\in\mathcal{F}_h^b\cap\partial K}\big(H(u_h^{L,(i)},u_h^b;n^L)-\widehat{\nu(u_h)}n^L\cdot Q_{h}^b,\phi^L_1\big)_S,\label{cons_constr}
\end{align}
with $\widehat{L}_{h,1}^K$ the equation for the element mean in element $K$ in \cref{DIRK_stage}. The conservation constraint \cref{cons_constr} is imposed using the Lagrange multiplier $\mu$, see \cref{equality_constraint}.
The conservation constraint explicitly ensures that at each Runge-Kutta stage the equation for the element mean $\bar{u}^{K,(i)}_h$ is exactly preserved in each element, hence the KKT limiter does not affect the conservation properties of the DG discretization.
\end{itemize}
\smallskip
The remaining Jacobians $D_xh_i(x) \in\mathbb{R}^{N_K\times N_uN_K}$, $D_xg_i(x) \in\mathbb{R}^{N_pN_K\times N_uN_K}$, and $D_\mu\mathcal{L}_i(z)\in\mathbb{R}^{N_uN_K\times N_K}$,
$D_\lambda\mathcal{L}_i(z)\in\mathbb{R}^{N_uN_K\times N_pN_K}$, with $x=\widehat{K}^{(i)}$, in the quasi-directional derivative matrix $\widehat{G}$ \cref{eq:quasi-direc-deriv} are now straightforward to calculate. }
It is important to ensure that the initial solution also satisfies the positivity constraints. An $L^2$-projection of the solution will in general not satisfy these constraints for a non-smooth solution. To ensure that the initial solution also satisfies the positivity constraints we apply a constrained projection using the active set semi-smooth Newton method given by Algorithm \ref{algorithm_1}. The only difference is now that instead of \cref{DIRK_stage} we use $L^2$-projection
\begin{equation*}
\widehat{L}_{h i}(\widehat{U}^0)=M^1\widehat{U}^0-(u_0,\phi_i)_\Omega,
\end{equation*}
and combine this with the positivity constraints \cref{positivity_limiter}-\cref{maximum_limiter}. Here, $u_0$ denotes the initial solution. As initial solution for the constrained projection we use in Algorithm \ref{algorithm_1} the standard $L^2$-projection without constraints.
The positivity constraints are imposed at all element quadrature points, since only the solution at these quadrature points is used in the DG discretization. In 1D we use Gauss-Lobatto quadrature rules and in 2D product Gauss-Legendre quadrature rules. Since the number of quadrature points in an element is generally larger than the number of degrees of freedom in an element this will result in an over-determined set of algebraic equations and a rank deficit Jacobian matrix if the number of active constraints in an element is larger than the degrees of freedom $N_u$ in element. In order to obtain in Algorithm \ref{algorithm_1} accurate search directions $h^k$ we use the Gauss-Newton method given by \cref{leastsq}. This approaches can efficiently deal with the possible rank deficiency of the Jacobian matrix.
In practice it will not be necessary to apply the inequality constraints in all elements and one can significantly reduce the computational cost and memory overhead by excluding those elements for which it is obvious that they will meet the constraints anyway.
\section{Numerical experiments}\label{Numerical Experiments}
In this section we will discuss a number of numerical experiments to demonstrate the performance of the DIRK-DG scheme with the positivity preserving KKT Limiter. All computations were performed using the default values for the coefficients listed for Algorithm \ref{algorithm_1}, except that for the accuracy tests discussed in Section \ref{accuracy_tests} we use $\epsilon=10^{-10}$. The upwind coefficient $\alpha$ in \cref{upwind_flux} is set to $\alpha=1$. In all 1D computations the local conservation constraint is imposed and satisfied with an error less than $10^{-12}$.
\subsection{Accuracy tests}\label{accuracy_tests}
It is important to investigate if the KKT-limiter negatively affects the accuracy of the DG discretization in case the exact solution is smooth, but where also a positivity preserving limiter is required to ensure that the numerical solution stays within the bounds. To investigate this we conduct the same accuracy tests as conducted in Qin and Shu \cite{qin2018implicit}, Section 5.1. Both the linear advection and inviscid Burgers equation are considered, which are obtained by setting $F(u)=u$ and $F(u)=\frac{1}{2}u^2$, respectively, and $G(u)=\nu(u)=0$ in \cref{eq:2ndconservation_law}.
\indent{\it Example 5.1} (steady state solution to linear advection equation). We consider
\begin{equation}
u_t+u_x=\sin^4x,\qquad u(x,0)=\sin^2x,\quad u(0,t)=0,\label{linadvec_eq}
\end{equation}
with outflow boundary condition at $x=2\pi$.
The exact solution $u(x,t)$ is positive for all $t>0$, see \cite{qin2018implicit}. As steady state solution we use the solution at $t=500$, when all residuals are approximately $10^{-16}$. During the computations the CFL number is dynamically adjusted between 10 and 89. For the time integration an implicit Euler method is used. In Tables \ref{linadvec_steady_nolimiter} and \ref{linadvec_steady_limiter} the results of the accuracy tests, without and with the KKT-limiter, are shown. The results in Table \ref{linadvec_steady_limiter} show that the KKT-limiter does not negatively affect the accuracy. For all test cases the optimal accuracy in the $L^2$- and $L^\infty$-norms is obtained. Also, the limiter is necessary, as can be seen from Table \ref{linadvec_steady_nolimiter}, and preserves the imposed positivity bound $u_{h\min}=10^{-14}$ for the numerical solution.
\begin{table}[htb]
\centering
\caption{Error table for steady state linear advection equation \cref{linadvec_eq} without limiter.}
\begin{tabular}{c|c|c|c|c|c|c}
\hline\hline
$p$ & $N$ &$L^2 $ error & Order &$ L^\infty$ error & Order &$\min u_h$ \\\hline
&20 &1.461068e-02&-&2.044253e-02&&-5.169578e-03 \\
&40 &3.702581e-03&1.98&5.287628e-03&1.95&-2.883487e-04 \\
1&80 &9.288342e-04&2.00&1.331962e-03&1.99&-1.208793e-05 \\
&160 &2.324090e-04&2.00&3.336614e-04&2.00&-4.036603e-07 \\
&320 &5.811478e-05&2.00&8.345620e-05&2.00&-1.282064e-08 \\\hline
&20 &9.287703e-04&-&1.776878e-03&-&-4.952018e-05 \\
&40 &1.177042e-04&2.98&2.489488e-04&2.84&-1.627459e-06 \\
2&80 &1.476405e-05&3.00&3.200035e-05&2.96&-5.149990e-08 \\
&160 &1.847107e-06&3.00&4.027944e-06&2.99&-1.614420e-09 \\
&320 &2.309385e-07&3.00&5.043677e-07&3.00&-5.049013e-11 \\\hline
&20 &5.653820e-05&-&1.230308e-04&-&-3.877467e-05 \\
&40 &3.583918e-06&3.98&7.803741e-06&3.98&-1.326415e-06 \\
3&80 &2.247890e-07&3.99&4.950122e-07&3.98&-4.237972e-08 \\
&160 &1.406175e-08&4.00&3.090593e-08&4.00&-1.331692e-09 \\
&320 &8.790539e-10&4.00&1.935324e-09&4.00&-4.167274e-11 \\\hline\hline
\end{tabular}
\label{linadvec_steady_nolimiter}
\end{table}
\begin{table}[htb]
\centering
\caption{Error table for steady state linear advection equation \cref{linadvec_eq} with limiter.}
\begin{tabular}{c|c|c|c|c|c|c}
\hline\hline
$p$ & $N$ &$L^2 $ error & Order &$ L^\infty$ error & Order &$\min u_h$ \\\hline
&20 &1.464990e-02&-&2.044253e-02&-&9.998946e-15\\
&40 &3.702367e-03&1.98&5.287628e-03&1.95&9.999813e-15\\
1&80 &9.288338e-04&2.00&1.331962e-03&1.99&1.000000e-14\\
&160 &2.324090e-04&2.00&3.336614e-04&2.00&1.000000e-14 \\
&320 &5.811478e-05&2.00&8.345620e-05&2.00&1.000000e-14\\\hline
&20 &9.290268e-04&-&1.776878e-03&-&1.000000e-14\\
&40 &1.177053e-04&2.98&2.489488e-04&2.84&1.000000e-14\\
2&80 &1.476406e-05&3.00&3.200035e-05&2.96&1.000000e-14\\
&160 &1.847107e-06&3.00&4.027944e-06&2.99&1.000000e-14 \\
&320 &2.309385e-07&3.00&5.043677e-07&3.00&1.000000e-14\\\hline
&20 &5.742649e-05&-&1.230309e-04&-&9.999990e-15\\
&40 &3.592170e-06&4.00&7.803745e-06&3.98&1.000000e-14\\
3&80 &2.248562e-07&4.00&4.950122e-07&3.98&1.000000e-14\\
&160 &1.406228e-08&4.00&3.090593e-08&4.00&1.000000e-14 \\
&320 &8.790580e-10&4.00&1.935323e-09&4.00&1.000000e-14\\\hline\hline
\end{tabular}
\label{linadvec_steady_limiter}
\end{table}
\indent{\it Example 5.2} (steady state solution to inviscid Burger's equation). We consider the inviscid Burgers equation
\begin{equation}
u_t+(\frac{1}{2}u^2)_x=\sin^3\Big(\frac{x}{4}\Big),\qquad u(x,0)=\sin^2\Big(\frac{x}{4}\Big),\quad u(0,t)=0,\label{Burgers_eq}
\end{equation}
with outflow boundary condition at $x=2\pi$. The exact solution $u(x,t)$ is positive for all $t>0$, see \cite{qin2018implicit}. As steady state solution we use the solution at $t=20.000$, when all residuals are approximately $10^{-16}$. During the computations the CFL number is dynamically adjusted between 10 and 954. For the time integration an implicit Euler method is used. In Tables \ref{Burgers_steady_nolimiter} and \ref{Burgers_steady_limiter} the results of the accuracy tests, without and with the KKT-limiter, show that the KKT-limiter does not negatively affect the accuracy. For all test cases optimal accuracy in the $L^2$- and $L^\infty$-norms is obtained. Also, the limiter is necessary and preserves the imposed positivity bound $u_{h\min}=10^{-14}$ for the numerical solution.
\begin{table}[htb]
\centering
\caption{Error table for steady state inviscid Burgers equation \cref{Burgers_eq} without limiter.}
\begin{tabular}{c|c|c|c|c|c|c}
\hline\hline
$p$ & $N$ &$L^2 $ error & Order &$ L^\infty$ error & Order &$\min u_h$ \\\hline
&20 &2.110016e-03&-&3.387013e-03&-&-2.347303e-03 \\
&40 &5.230241e-04&2.01&8.577912e-04&1.98&-5.865522e-04 \\
1&80 &1.297377e-04&2.01&2.151386e-04&2.00&-1.466204e-04 \\\hline
&20 &2.122765e-05&-&3.024868e-05&-&-1.048636e-05 \\
&40 &2.623666e-06&3.02&3.731754e-06&3.02&-6.681764e-07 \\
2&80 &3.266401e-07&3.01&4.634046e-07&3.01&-4.196975e-08\\ \hline
&20 &2.985321e-07&-&1.895437e-06&-&1.895437e-06 \\
&40 &1.452601e-08&4.36&1.196963e-07&3.99&1.196963e-07\\
3&80 &7.368455e-10&4.30&7.500564e-09&4.00&7.500564e-09\\
&160 &3.948207e-11&4.22&4.346084e-10&4.11& 4.346084e-10\\\hline\hline
\end{tabular}
\label{Burgers_steady_nolimiter}
\end{table}
\begin{table}[htb]
\centering
\caption{Error table for steady state inviscid Burgers equation \cref{Burgers_eq} with limiter.}
\begin{tabular}{c|c|c|c|c|c|c}
\hline\hline
$p$ & $N$ &$L^2 $ error & Order &$ L^\infty$ error & Order &$\min u_h$ \\\hline
&20 &2.208009e-03&-&3.637762e-03&-&9.999813e-15 \\
&40 &5.358952e-04&2.04&9.282398e-04&1.97&1.000003e-14\\
1&80 &1.313948e-04&2.03&2.339566e-04&1.99&1.000003e-14\\\hline
&20 &2.116746e-05&-&3.024864e-05&-&1.000003e-14\\
&40 &2.622584e-06&3.01&3.731752e-06&3.02&1.000139e-14\\
2&80 &3.266221e-07&3.01&4.634046e-07&3.01&1.000040e-14\\\hline
&20 &2.985321e-07&-&1.895437e-06&-&1.895437e-06\\
&40 &1.452601e-08&4.36&1.196963e-07&3.99&1.196963e-07\\
3&80 &5.610147e-10&4.70&1.574760e-09&6.25&1.000105e-14\\
&160 &3.232240e-11&4.11& 9.038604e-11&4.12&1.000017e-14\\\hline\hline
\end{tabular}
\label{Burgers_steady_limiter}
\end{table}
\subsection{Time dependent tests}
In this section we will present results of simulations of the linear advection, Allen-Cahn, Barenblatt and Buckley-Leverett equations. The order of accuracy of the DIRK time integration method is always $p+1$, with $p$ the polynomial order of the spatial discretization. The minimum value of the residual $F(z)$ and Newton update $d$ in Algorithm \ref{algorithm_1} to stop the Newton iterations is $\epsilon=10^{-8}$ for each DIRK stage. This is a quite strong stopping criteria and in practice the values are often smaller at the end of each DIRK-stage. It is also important to make sure that the Newton stopping criterion is in balance with the accuracy required for the constraints. If the algebraic equations are not solved sufficiently accurate then it is not likely that the KKT-constraints will be satisfied.
The time step for the DIRK method is dynamically computed, based on the CFL or diffusion number. If the Newton method does not converge within a predefined number of iterations, then the computation for the time step will be restarted with $\triangle t/2$. This is generally more efficient than conducting many Newton iterations. In the next time step the time step will then be increased to $1.2\triangle t$, until the maximum CFL-number is obtained. In practice, depending on the severity of the nonlinearity, the time step will be constantly adjusted during the computations.
\indent{\it Example 5.3} (1D linear advection equation). We consider \cref{linadvec_eq} with a zero right hand side in the domain $\Omega=[0,10]$ and periodic boundary conditions. The exact solution is
\begin{equation*}
u(x,t)=\max(\cos(2\pi(x-t)/10),0), \quad \text{for}\; x\in\Omega, t\in[0,T].
\end{equation*}
A constrained projection of $u(x,0)$ onto the finite element space $V_h^p$ is used as initial solution $u_h(x,0)$. The computational mesh contains 100 elements and the maximum CFL number is 1. In Figures \ref{Linear_Advection1}, \ref{Linear_Advection3}, and \ref{Linear_Advection4} the exact and numerical solution at time $t=20$ are plotted for, respectively, polynomial orders 1, 2 and 3. At this time the wave has travelled twice through the domain and the numerical solution matches very well with the exact solution. Also, plotted is the value of the Lagrange multipliers used to impose the positivity constraint $u_{h\min}=10^{-10}$. These plots clearly show that the limiter is only active at locations where the constraint must be imposed and not in the smooth part of the solution. In Figure \ref{Linear_Advection2}, the solution for polynomial order $p=1$ without the KKT-Limiter is plotted, which clearly shows that without the limiter the solution is significantly below the $u=0$ minimum of the exact solution $u(x,t)$.
\begin{figure}[tbhp]
\centering
\hspace*{-5pt}\subfloat[$p=1$]{\label{Linear_Advection1}\includegraphics[width=0.48\textwidth,height=0.25\textheight]{Figures/Linear_Advection/Variables1D_N100_p1_UL200000}}\hspace*{4pt}
\subfloat[$p=1$]{\label{Linear_Advection2}\includegraphics[width=0.45\textwidth,height=0.25\textheight]{Figures/Linear_Advection/Variables1D_N100_p1_nolimiter_T200000}}\hspace*{8pt}\\
\subfloat[$p=2$]{\label{Linear_Advection3} \includegraphics[width=0.48\textwidth,height=0.25\textheight]{Figures/Linear_Advection/Variables1D_N100_p2_UL200000}}\hspace*{5pt}
\subfloat[$p=3$]{\label{Linear_Advection4}\includegraphics[width=0.48\textwidth,height=0.25\textheight]{Figures/Linear_Advection/Variables1D_N100_p3_UL200000}}
\caption{Example 5.3, advection equation 1D, (a), (c), (d) numerical solution $u_h$ with positivity preserving limiter, polynomial order, respectively, $p=1$, 2, and $3$, (b) numerical solution $u_h$ without positivity preserving limiter, polynomial order $p=1$. Computational mesh $100$ elements. Values of the Lagrange multiplier used in the positivity preserving limiter larger than $10^{-10}$ are indicated
in (a), (c) and (d) with a red circle.}
\label{1DAdvection_Equation_Overview}
\end{figure}
\begin{figure}[htb]
\centering
\subfloat[]{\label{Advection_2D_Solution}\includegraphics[width=0.49\textwidth]{Figures/Linear_Advection/Variables2D_N30x30_p3_T06343}}
\subfloat[]{\label{Advection_2D_Lagrange_Multiplier}\includegraphics[width=0.49\textwidth]{Figures/Linear_Advection/Variables2D_N30x30_p3_Lambda06343}}
\caption{Example 5.4, advection equation 2D, (a) solution $u_h$, (b) Lagrange multiplier. Computational mesh $30\times 30$ elements, polynomial order $p=3$. Values of the Lagrange multiplier used in the positivity preserving limiter larger than $10^{-10}$ are indicated in (b) with a red asterisk.}
\label{2DAdvection_Equation_Overview}
\end{figure}
\indent{\it Example 5.4} (2D linear advection equation). The KKT-Limiter is also tested on a 2D linear advection equation, which is obtained by setting $F(u)=cu$, with $c=(-1,-2)$, and $G(u)=\nu(u)=0$ in \cref{eq:2ndconservation_law}. The domain $\Omega=[0,3]^2$ with periodic boundary conditions is used in the computations. The computational mesh contains $30\times 30$ elements. The exact solution is
\begin{equation*}
u(x,t)=\max(\cos(2\pi(x+t)/3)\cos(2\pi(y+2t)/3),0)\quad \text{for}\; x\in\Omega, t\in[0,T].
\end{equation*}
A constrained projection of $u(x,0)$ onto the finite element space $V_h^p$ is used as initial solution $u_h(x,0)$. The maximum CFL number is 1. In Figure \ref{Advection_2D_Solution} the numerical solution is shown at $t=6.3428$ and in Figure \ref{Advection_2D_Lagrange_Multiplier} the values of the Lagrange multipliers used to enforce the positivity constraint $u_{h\min}=10^{-10}$. Comparing Figures \ref{Advection_2D_Solution} and \ref{Advection_2D_Lagrange_Multiplier} clearly shows that the KKT-Limiter is only active in those parts of the domain where the solution needs to satisfy the positivity constraint and not in the smooth part.
{ \indent{\it Example 5.5} (1D Burgers equation).
\begin{figure}[htb]
\centering
\subfloat[]{\label{Burgers_1D_Solution_A}\includegraphics[width=0.49\textwidth]{Figures/Burgers/Burgers_maxCosine_t0316}}\hspace*{3pt}
\subfloat[]{\label{Burgers_1D_Solution_B}\includegraphics[width=0.49\textwidth]{Figures/Burgers/Burgers_maxCosine_t065}}\\
\subfloat[]{\label{Burgers_1D_Solution_C}\includegraphics[width=0.49\textwidth]{Figures/Burgers/Burgers_maxCosine_Nonconservative_t065}}
\caption{Example 5.5, Burgers equation 1D, (a)-(c) solution $u_h$ and Lagrange multiplier. The solution in (a) and (b) is computed with local conservation imposed as an explicit constraint, whereas (c) shows the solution without explicitly imposing local conservation. Computational mesh $80$ elements, polynomial order $p=3$. Values of the Lagrange multiplier used in the positivity preserving limiter larger than $10^{-10}$ are indicated in with a red circle.}
\label{1DBurgers_Equation_Overview}
\end{figure}
In order to test the KKT-Limiter on problems with time-dependent shocks we consider the 1D Burgers equation on a domain $\Omega=[-1,1]$ with initial condition $u_0=\max(\cos(\pi x),0)$ and periodic boundary conditions. The polynomial order is $p=3$. As lower and upper bounds in the positivity preserving limiter we use, respectively $u_{h\min}=10^{-10}$ and $u_{h\max}=1$, and no monotonicity constraint is imposed. The initially smooth part of the solution develops into a shock. The onset of the shock is shown in Figure \ref{Burgers_1D_Solution_A} and the later stages of the shock at $t=0.65$ in Figure \ref{Burgers_1D_Solution_B}. Figure \ref{Burgers_1D_Solution_C} shows the solution when the conservation constraint \cref{cons_constr} is not explicitly enforced. The difference in the shock solution for the discretizations with and without the explicitly imposed conservation constraint is very small. The main reason for this is that the KKT-Limiter is only active in regions where the constraints must be imposed and does not affect the discretization at other places in the domain. This can be seen from the values of the Lagrange multipliers that are used to impose the positivity constraints, which are indicated with red circles, and are only non-zero in the vicinity of the shock and at locations where the solution has a discontinuous derivative. The KKT-Limiter to ensure the positivity constraints therefore has a very small effect on the conservation properties of the DG discretization as can be seen by comparing Figures \ref{Burgers_1D_Solution_B} and \ref{Burgers_1D_Solution_C}.}
\indent{\it Example 5.6} (Allen-Cahn equation). The Allen-Cahn equation is a reaction-diffusion equation that describes phase transition. The Allen-Cahn equation is obtained by setting $G(u)=u^3-u$, $\nu(u)=\bar{\nu}$, and $F(u)=0$ in \cref{eq:2ndconservation_law}. The solution of the Allen-Cahn equation should stay within the range $[0,1]$. Hence, we apply both the positivity and maximum preserving limiters, respectively, \cref{positivity_limiter}-\cref{maximum_limiter} with bounds $u_{h{\rm min}}=10^{-14}$ and $u_{h{\rm max}}=1-10^{-10}$. A constrained projection of $u(x,0)$ onto the finite element space $V_h^p$ is used as initial solution $u_h(x,0)$.
\indent{\it Example 5.6a} (Allen-Cahn equation 1D). As test case we use the traveling wave solution
\begin{equation*}
u(x,t)= \frac{1}{2}\left(1-\tanh\left(\frac{x-st}{2\sqrt{2\bar{\nu}}}\right)\right),
\end{equation*}
with wave velocity $s=3\sqrt{\bar{\nu}/2}$. The computational domain is $\Omega = [-\frac{1}{2},2]$. If the mesh resolution is sufficiently dense such that the jump in the traveling wave solution is well resolved, then no limiter is required. For small values of the viscosity the solution will, however, violate the positivity constraints, except on very fine meshes. In Figures \ref{Allen_Cahn1} and \ref{Allen_Cahn2}, respectively, the numerical solution $u_h$ and its derivative $Q_h$ and the exact solutions are shown for the viscosity $\bar{\nu}=10^{-5}$ on a mesh with 100 elements and polynomial order 3 for the basis functions. The values of the Lagrange multiplier used to impose the positivity constraints are also shown in Figure \ref{Allen_Cahn1}. The solution has a very thin and steep transition region, but the wave speed is still correctly computed by the LDG scheme and the KKT limiter ensures that both the positivity and maximum constraint are satisfied.
\indent{\it Example 5.6b} (Allen-Cahn equation 2D). For the 2D test case the computational domain is $\Omega = [-\frac{1}{2},2]^2$ and the computational mesh contains $30\times 30$ elements. The viscosity coefficient is selected as $\bar{\nu}=10^{-4}$. As test case we use the initial solution
\begin{equation*}
u(x,0)= \frac{1}{4}\left(1-\tanh\left(\frac{x}{2\sqrt{2\bar{\nu}}}\right)\right)\left(1-\tanh\left(\frac{y}{2\sqrt{2\bar{\nu}}}\right)\right),
\end{equation*}
which values are also used as boundary condition for $t>0$.
At this mesh resolution a positivity preserving limiter is necessary. The numerical solution shown in Figure \ref{Allen_Cahn3} has steep gradients and the positivity preserving limiter ensures that the bounds are satisfied. The locations where the limiter is active can be seen in Figure \ref{Allen_Cahn4}, which shows the values and locations of the Lagrange multipliers used to impose the bounds in the DG discretization.
\begin{figure}[tbhp]
\centering
\subfloat[$u_h$ - with limiter]{\label{Allen_Cahn1}\includegraphics[width=0.49\textwidth,height=0.22\textheight]{Figures/Allen_Cahn/Allen_Cahn_N100_p3_visc_1e-5_Variables_TU100000}}\hspace*{5pt}
\subfloat[$Q_h$ - with limiter]{\label{Allen_Cahn2}\includegraphics[width=0.49\textwidth,height=0.22\textheight]{Figures/Allen_Cahn/Allen_Cahn_N100_p3_visc_1e-5_Variables_TD100000}}
\caption{Allen-Cahn equation 1D, Example 5.6a, (a) numerical solution $u_h$ and exact solution $u$, (b) derivative of numerical solution $Q_h$ and exact derivative $Du$. Computational mesh $100$ elements, polynomial order $p=3$. Values of the Lagrange multiplier used in the positivity and maximum preserving limiters larger than $10^{-10}$ are indicated in (a) with a red circle.}
\label{Allen_Cahn_Overview1}
\end{figure}
\begin{figure}[tbhp]
\centering
\subfloat[]{\label{Allen_Cahn3}\includegraphics[width=0.49\textwidth,height=0.25\textheight]{Figures/Allen_Cahn/2D_Results/Allen_Cahn2D_30x30_p3_visc_1e-4_Variables_T50000}}
\subfloat[]{\label{Allen_Cahn4}\includegraphics[width=0.49\textwidth,height=0.25\textheight]{Figures/Allen_Cahn/2D_Results/Allen_Cahn2D_30x30_p3_visc_1e-4_Variables_Lambda50000}}
\caption{Allen-Cahn equation 2D, Example 5.6b, (a) numerical solution $u_h$ (b) Lagrange multiplier. Computational mesh $30\times 30$ elements, polynomial order $p=3$. Values of the Lagrange multiplier used in the positivity and maximum preserving limiters larger than $10^{-10}$ are indicated in (b) with a red asterisk.}
\label{Allen_Cahn_Overview2}
\end{figure}
\indent{\it Example 5.7} (Barenblatt equation). The Barenblatt equation, which models a porous medium, is obtained by setting $\nu(u)=mu^{m-1}$, $m>1$, and $F(u)=0$, $G(u)=0$ in \cref{eq:2ndconservation_law}.
The exact solution is
\begin{equation*}
u(t,x)=t^\alpha\left(\left(C-\frac{\beta(m-1)}{2m}\frac{\vert x\vert^2}{t^{2\beta}}\right)_+\right)^{\frac{1}{m-1}},
\end{equation*}
with $\alpha=\frac{n}{n(m-1)+2}$, $\beta=\frac{\alpha}{n}$, $n={\rm dim}(\Omega)$, $(x)_+=\max(x,0)$ and $C>0$. We selected $C=1$ and $m=8$. The solution should be positive or zero for $t>0$. The initial solution for the computations is the constrained projection of $u(x,1)$ onto the finite element space $V_h^p$. In the computations Dirichlet boundary conditions are imposed, where the solution for $t>0$ is fixed at the same level as the initial solution.
\indent{\it Example 5.7a} (1D Barenblatt equation).
We first consider the 1D Barenblatt equation on the domain $\Omega=[-7,7]$ using a computational mesh of 100 elements. In Figure \ref{Barenblatt1D_nolimiter} the numerical solution without the use of a limiter is shown. It is clear that near the boundary of $u(t,x)>0$, where the derivative of $u$ becomes unbounded, significant negative values of $u_h$ are obtained. These cause severe numerical problems and do not allow the continuation of the computations.
\indent{\it Example 5.7b} (2D Barenblatt equation).
In Figures \ref{Barenblatt1} and \ref{Barenblatt2}, respectively, the numerical solution $u_h$ of the 2D Barenblatt equation and the values of the Lagrange multiplier are shown at time $t=2$ on a mesh of $50\times 50$ elements. In these computations the KKT Limiter was used, which successfully prevents the numerical solution $u_h$ from becoming negative, which is shown in Figure \ref{Barenblatt3}. The imposed constraint is $u_{h{\rm min}}=10^{-10}$. Figure \ref{Barenblatt3} also shows an excellent agreement between the exact solution $u$ and the numerical solution $u_h$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Barenblatt/Barenblatt1D_N100_p3_nolimiter_Variables_T11352}
\caption{Barenblatt equation 1D, Example 5.7a, numerical solution $u_h$ without limiter and exact solution $u$. Computational mesh $100$ elements, polynomial order $p=3$. }
\label{Barenblatt1D_nolimiter}
\end{figure}
\begin{figure}[tbhp]
\centering
\subfloat[]{\label{Barenblatt1}\includegraphics[width=0.49\textwidth]{Figures/Barenblatt/Barenblatt2D_N50x50_p3_Variables_T20000}}
\subfloat[]{\label{Barenblatt2}\includegraphics[width=0.45\textwidth]{Figures/Barenblatt/Barenblatt2D_N50x50_p3_Variables_Lambda20000}}\\[5pt]
\subfloat[]{\label{Barenblatt3}\includegraphics[width=0.45\textwidth]{Figures/Barenblatt/Barenblatt2D_N50x50_p3_Variables_c20000}}
\caption{Barenblatt equation 2D, Example 5.7b, (a) solution $u_h$, (b) Lagrange multiplier, (c) numerical solution $u_h$ and exact solution $u$ in cross-section at $y=0$. Computational mesh $50\times 50$ elements, polynomial order $p=3$. Values of the Lagrange multiplier used in the positivity preserving limiter larger than $10^{-10}$ are indicated in (b) with a red asterisk.}
\label{Barenblatt_Overview}
\end{figure}
\indent{\it Example 5.8} (1D Buckley-Leverett equation). The Buckley-Leverett equation models two phase flow in a porous medium. We consider two cases, respectively, with and without gravity. Since the solution has to be strictly inside the range $[0,1]$ we use both the positivity and maximum preserving limiter, with bounds $u_{h{\rm min}}=10^{-10}$ and $u_{h{\rm max}}=1-10^{-10}$, respectively. The computational domain is $\Omega =[0,1]$. A Dirichlet boundary condition at $x=0$, based on the initial solution, and an outflow boundary condition at $x=1$ are imposed. The viscosity coefficient is $\bar{\nu}=0.01$. Since we do not have an exact solution to compare with we compute the numerical solution on two meshes, viz. with 100 and 200 elements.
The two test cases given by Examples 5.8a and 5.8b are also considered in \cite{kurganov2000new}.
\indent{\it Example 5.8a} (1D Buckley-Leverett equation without gravity). The 1D Buckley-Leverett equation without gravity is obtained by setting $G(u)=0$, and $\nu(u)$ and $F(u)=f(u)$, respectively, as
\begin{equation*}
\nu(u)=
\begin{cases}
4\bar{\nu} u(1-u),\hspace*{10pt}&\text{if}\; 0\leq u\leq 1,\\
0,&\text{otherwise}.
\end{cases}
\end{equation*}
\begin{equation}
f(u)=
\begin{cases}
0,&\text{if}\; u<0,\\
\frac{u^2}{u^2+(1-u)^2},\hspace*{6pt}&\text{if}\; 0\leq u\leq 1,\\
1,&\text{if}\; u>1.
\end{cases}\label{BLflux_nogravity}
\end{equation}
The initial condition is
\begin{equation*}
u(x,0)=\begin{cases}
0.99-3x\quad& 0\leq x\leq 0.33,\\
0&\frac{1}{3} <x\leq 1.
\end{cases}
\end{equation*}
The numerical solution $u_h$ and its derivative $Q_h$ are shown in, respectively, Figures \ref{Buckley_Leverett1} and \ref{Buckley_Leverett2}. Also, the values of the Lagrange multiplier used to enforce the constraints is shown in Figure \ref{Buckley_Leverett1}. The limiter is only active in the thin layer between the phases and is crucial to obtain sensible physical solutions. The results of 100 and 200 elements match well.
\indent{\it Example 5.8b} (1D Buckley-Leverett equation with gravity). A much more difficult test case is provided by the Buckley-Leverett equation with gravity, which is obtained by modifying the flux $F(u)$ as
\begin{equation*}
F(u)=\begin{cases}
f(u)(1-5(1-u)^2),\quad& u\leq 1,\\
1& u>1,
\end{cases}
\end{equation*}
with $f(u)$ given by \cref{BLflux_nogravity}. The initial solution is
\begin{equation*}
u(x,0)=\begin{cases}
0\quad& 0\leq x\leq a,\\
\frac{1}{mh}(x-a)& a< x \leq 1-\frac{1}{\sqrt{2}},\\
1&1-\frac{1}{\sqrt{2}} <x\leq 1,
\end{cases}
\end{equation*}
with $a=1-\frac{1}{\sqrt{2}}-mh$, $h$ the mesh size and $m=3$. The linear transition for $x$ in the range $[a,1-\frac{1}{\sqrt{2}}]$ is used to remove the infinite value in the derivative, which would otherwise result in unbounded values of $Q_h$ at $t=0$.
The Buckley-Leverett equations with gravity result a strongly nonlinear problem where the equations change type and is a severe test for the KKT-Limiter and semi-smooth Newton algorithm. The solution $u_h$ and values of the Lagrange multiplier are shown in Figure \ref{Buckley_Leverett3} and the derivative $Q_h$ in Figure \ref{Buckley_Leverett4}. The results on the two meshes compare well and the limiter ensures that the positivity and maximum bounds are satisfied.
\begin{figure}[tbhp]
\centering
\subfloat[$u_h$ - no gravity]{\label{Buckley_Leverett1}\includegraphics[width=0.49\textwidth,height=0.25\textheight]{Figures/Buckley_Leverett_NoGravity/Buckley_Leverett_NoGravity_N100_N200_p3_Variables_TU02000_u0-099}}\hspace*{5pt}
\subfloat[$Q_h$ - no gravity]{\label{Buckley_Leverett2}\includegraphics[width=0.49\textwidth,height=0.25\textheight]{Figures/Buckley_Leverett_NoGravity/Buckley_Leverett_NoGravity_N100_N200_p3_Variables_TD02000_u0-099}}\\[5pt]
\subfloat[$u_h$ - gravity]{\label{Buckley_Leverett3}\includegraphics[width=0.49\textwidth,height=0.25\textheight]{Figures/Buckley_Leverett_Gravity/Buckley_Leverett_Grav_N100_N200_p3_Variables_TU02000}}\hspace*{5pt}
\subfloat[$Q_h$ - gravity]{\label{Buckley_Leverett4}\includegraphics[width=0.49\textwidth,height=0.25\textheight]{Figures/Buckley_Leverett_Gravity/Buckley_Leverett_Grav_N100_N200_p3_Variables_TD02000}}
\caption{Example 5.8a, Buckley-Leverett equation without gravity 1D, (a) numerical solution $u_h$, (b) numerical solution derivative $Q_h$; Example 5.8b, Buckley-Leverett equation with gravity 1D, (c) numerical solution $u_h$, (d) numerical solution derivative $Q_h$. Computational meshes $100$ and $200$ elements, polynomial order $p=3$. Values of the Lagrange multiplier used in the positivity and maximum preserving limiters larger than $10^{-10}$ are indicated with a red circle in (a) and (c).}
\label{Buckley_Leverett_Overview}
\end{figure}
The number of Newton iterations necessary to obtain a minimum value $10^{-8}$ for the residual $F(z)$ and Newton update $d$ in Algorithm \ref{algorithm_1} to stop the Newton iterations for each DIRK stage strongly varies. It depends on the type of equation, time-step and nonlinearity. In general, the time step is chosen such that the number of Newton iterations for each DIRK stage is between 5 and 20. For most time dependent problems the CFL number is then close to one, which is necessary to ensure time-accuracy. Only for the Buckley-Leverett equation with gravity the time step frequently had to be less than one in order to deal with the strong nonlinearity of the problem. In the computations we did not observe a minimum time step to ensure positivity as noticed in \cite{qin2018implicit}.
\section{Conclusions}
\label{sec:conclusions}
In this paper we present a novel framework to combine positivity preserving limiters for discontinuous Galerkin discretizations with implicit time integration methods. This approach does not depend on the specific type of discontinuous Galerkin discretization and is also applicable to e.g. finite volume discretizations. The key features of the numerical method is the formulation of the positivity constraints as a Karush-Kuhn-Tucker problem and the development of an active set semi-smooth Newton method that accounts for the non-smoothness of the algebraic equations. The algorithm was successfully tested on a number of increasingly difficult test cases, which required that the positivity constraints are satisfied in order to obtain meaningful results. The KKT Limiter does not negatively affect the accuracy for smooth problems and accurately preserves the positivity constraints. Future work will focus on the extension of the KKT Limiter to ensure also monotonicity of the solution.
|
1808.07827
|
\subsection{Basic notations and concepts}
\paragraph{String notation.} We denote by $\Sigma$ a finite alphabet of symbols, its Kleene-closure by $\Sigma^*$ and a string element by $\sigma \in \Sigma^*$. If $\sigma = \sigma_0\sigma_1\cdots\sigma_n$,
the length of $\sigma$ is $|\sigma|=n+1$ and the element in the $i$-th position is $\sigma_i$. Given two strings $\sigma, \sigma' \in \Sigma^*$, $\sigma\sigma'$ is their concatenation. A language is a set of strings, i.e., $\mbox{\tt L} \in \wp(\Sigma^*)$. We
use the following notations: $\Sigma^i\mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}}\sset{\sigma\in\Sigma^*}{|\sigma|=i}$ and $\Sigma^{< i}\mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}}\bigcup_{j< i}\Sigma^j$. Given $\sigma\in\Sigma^*$, $i,j\in\mathbb{N}$ ($i\leq j\leq |\sigma|$) the substring between $i$ and $j$ of $\sigma$ is the string $\sigma_i\cdots\sigma_{j-1}$, and we denote it by $\substringf{\sigma}{i}{j}$. Let $\mathbb{Z}$ be the set of integers. We denote by $\Sigma^*_{\scriptsize{\mathbb{Z}}} \mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}} \{+,-, \epsilon\}\cdot\{0,1, \dots, 9\}^+$ the set of \textit{numeric strings}, i.e., strings corresponding to integers. $\mathcal{I}: \Sigma^*_{\scriptsize{\mathbb{Z}}} \rightarrow \mathbb{Z}$ maps numeric strings to the corresponding integers. Dually, we define the function $\mathcal{S}: \mathbb{Z} \rightarrow \Sigma^*_{\scriptsize{\mathbb{Z}}}$ that maps each integer to its numeric string representation (e.g., 1 is mapped to the string \code{"1"}, not \code{"+1"}, -5 is mapped to \code{"-5"}).
\noindent
\textbf{Regular languages and finite state automata.} We follow \cite{hopcroft1979} for automata notation. A finite state automaton (FA) is a tuple $\mbox{\tt A} = (Q, q_0, \Sigma, \delta, F)$ where $Q$ is a finite set of states, $q_0 \in Q$ is the initial state, $\Sigma$ is a finite alphabet, $\delta \subseteq Q \times \Sigma \times Q$ is the transition relation and $F \subseteq Q$ is the set of final states. In particular, if $\delta:Q\times\Sigma\rightarrow Q$ is a function then $\mbox{\tt A}$ is called deterministic FA (DFA).\footnote{We consider DFA also those FAs which are not complete, namely such that a transition for each pair $(q,a)$ ($q\in Q$, $a\in\Sigma$) does not exists. They can be easily transformed in a DFA by adding a sink state receiving all the missing transitions.}
The class of languages recognized by FAs is the class of regular languages. We denote the set of all DFAs as \mbox{\sc Dfa}.
Given an automaton $\mbox{\tt A}$, we denote the language accepted by $\mbox{\tt A}$ as $\mathscr{L}(\mbox{\tt A})$. A language $\mbox{\tt L}$ is regular iff there exists a FA $\mbox{\tt A}$ such that $\mbox{\tt L} = \mathscr{L}(\mbox{\tt A})$. From the Myhill-Nerode theorem\cite{davis1994}, for each regular language there uniquely exists a minimum automaton, i.e., with the minimum number of states, recognizing the language.
Given a regular language $\mbox{\tt L}$, we denote by $\mathsf{Min}(\mbox{\tt L})$ the minimum DFA $\mbox{\tt A}$ s.t. $\mbox{\tt L}=\mathscr{L}(\mbox{\tt A})$.
\noindent
\textbf{The programming language.} We consider an $\mathsf{IMP}$ language (Fig.~\ref{fig:mujs-syntax}) that
contains representative string operations taken from the set of methods offered by the JavaScript built-in class \code{String}\cite{w3school-string}. Other JavaScript string operations can be modeled by composition of the given string operations or as particular cases of them. Primitive values are $\mathbb{V}=\mathbb{S}\cup\mathbb{Z}\cup\mathbb{B}\cup\{\code{NaN}\}$ with $\mathbb{S}\mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}}\Sigma^*$ (strings on the alphabet $\Sigma$),
$\mathbb{B}\mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}}\{\mbox{\tt true},\mbox{\tt false}\}$ and $\code{NaN}$ a special value denoting not-a-number.
\noindent
\textbf{Implicit type conversion.} In order to capture the semantics of the language $\mathsf{IMP}$, inspired by the JavaScript semantics, we need to deal with \textit{implicit type conversion}\cite{arceri2017}. For each primitive value, we define an auxiliary function converting primitive values to other primitive values (Fig.~\ref{fig:type-juggling}). Note that all the functions behave like the identity when applied to values not needing conversion, e.g., $\code{toInt}$ on integers. Then, $\code{toStr}: \mathbb{V} \rightarrow \mathbb{S}$ maps any input value to its string representation; $\code{toInt}: \mathbb{V} \rightarrow \mathbb{Z} \cup \{\code{NaN}\}$ returns the integer corresponding to a value, when it is possible: For $\mbox{\tt true}$ and $\mbox{\tt false}$ it returns respectively $1$ and $0$, for strings in $\Sigma^*_{\footnotesize{\mathbb{Z}}}$ it returns the corresponding integer, while all the other values are converted to $\code{NaN}$.
For instance, $\code{toInt}(``42") = 42$, $\code{toInt}(``42hello") = \code{NaN}$. Finally,
$\code{toBool}: \mathbb{V} \rightarrow \mathbb{B}$ returns $\mbox{\tt false}$ when the input is $0$,
and $\mbox{\tt true}$ for all the other non boolean primitive values. For example, implicit type conversion is applied when the guards of \code{while} and \code{if} statements do not evaluate to booleans (e.g., \code{while (1)} \{x=x+1;\}, the guard is implicitly converted to \code{true}).
\begin{figure}[t]
\begin{framed}
\vbox{%
\setlength{\grammarparsep}{3pt plus 1pt minus 1pt}
\setlength{\grammarindent}{4em}
\renewcommand{\syntleft}{} \renewcommand{\syntright}{}
{\footnotesize
\begin{grammar} <\mbox{\bf Exp}> ::= \mbox{\bf Id} ~|~ v $\in\mathbb{V}$
~|~ \mbox{\bf Exp} \ \mbox{\tt +} \mbox{\bf Exp}
~|~ \mbox{\bf Exp} \ \mbox{\tt -} \mbox{\bf Exp}
~|~ \mbox{\bf Exp} \ \mbox{\tt *} \mbox{\bf Exp}
~|~ \mbox{\bf Exp} \ \mbox{\tt /} \mbox{\bf Exp}
~|~ \mbox{\bf Exp} \ \mbox{\tt \&\&} \mbox{\bf Exp}
\alt \mbox{\bf Exp} \ \mbox{\tt ||} \mbox{\bf Exp}
~|~ \mbox{\tt !} \mbox{\bf Exp}
~|~ \mbox{\bf Exp} \ \mbox{\tt >} \mbox{\bf Exp}
~|~ \mbox{\bf Exp} \ \mbox{\tt \textless} \mbox{\bf Exp}
~|~ \mbox{\bf Exp} \ \mbox{\tt ==} \mbox{\bf Exp}
~|~ \mbox{\bf Exp}\lstinline{.}\substring{\mbox{\bf Exp}}{\mbox{\bf Exp}}
\alt \mbox{\bf Exp}\lstinline{.}\charat{\mbox{\bf Exp}}
~|~ \mbox{\bf Exp}\lstinline{.}\indexof{\mbox{\bf Exp}}
~|~ \mbox{\bf Exp}\lstinline{.}\mbox{\tt length}
<\mbox{\bf Block}> ::= \lstinline@{@ \lstinline@}@
~|~ \lstinline@{@ \mbox{\bf Stmt} \ \lstinline@}@
<\mbox{\bf Stmt}> ::= \mbox{\bf Id} \ \mbox{\tt =} \mbox{\bf Exp} \mbox{\tt ;}
~|~ \mbox{\tt if} (\mbox{\bf Exp}) \mbox{\bf Block} \ \mbox{\tt else} \mbox{\bf Block}
~|~ \mbox{\tt while} (\mbox{\bf Exp}) \mbox{\bf Block}
~|~ \mbox{\bf Block}
~|~ \mbox{\bf Stmt} \ \mbox{\bf Stmt}
~|~ \ulitright{;}
\vspace{-0.5cm}
\end{grammar}}
}%
\end{framed}
\caption{$\mathsf{IMP}$ syntax}
\label{fig:mujs-syntax}
\end{figure}
\noindent
\textbf{Semantics.} Program states are partial maps from identifiers to primitive values, i.e., $\mktype{States}: \mbox{\bf Id} \rightarrow \mathbb{V}$. The concrete big-step semantics $\sem{\cdot} : \mbox{\bf Stmt} \times \mktype{States} \rightarrow \mktype{States}$ follows~\cite{arceri2017}, and it includes dynamic typing and implicit type conversion. Also the expression semantics, $\sem{\cdot}:\mbox{\bf Exp}\times\mktype{States}\rightarrow\mathbb{V}$, is standard; we only provide the formal and precise semantics of the $\mathsf{IMP}$ string operations. Let $\sigma, \sigma'\in\mathbb{S}$ and $i,j\in\mathbb{Z}$ (values which are not strings or numbers respectively, are converted by the implicit type conversion primitives. Negative values are treated as zero).
\begin{description}
\item[{\tt substring}:] It extracts substrings from strings, i.e., all the characters between two indexes. The semantics is the function {\sc Ss}$: \mathbb{S} \times \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{S}$ defined as:
%
\[
\substringl{\sigma}{i}{j} \mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}}
\begin{cases}
\substringl{\sigma}{j}{i} & j < i \\
\substringf{\sigma}{i}{\mathsf{max}(j,|\sigma|)} & \mbox{otherwise}
\end{cases}
\]
\item[{\tt charAt}:] It returns the character at a specified index. The semantics is the function {\sc Ca}$: \mathbb{S} \times \mathbb{Z} \rightarrow \mathbb{S}$ defined as follows:
%
\[
\charats{\sigma}{i} \mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}} \begin{cases}
\sigma_i & 0 \leq i < |\sigma| \\
\epsilon & \mbox{otherwise}
\end{cases}
\]
\item[{\tt indexOf}:] It returns the position of the first occurrence of a given substring. The semantics is the function {\sc Io}$:\mathbb{S} \times \mathbb{S} \rightarrow \mathbb{Z}$ defined as follows:
%
\[
\indexofs{\sigma}{\sigma'}\mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}} \begin{cases}
\mathsf{min}\sset{i}{\sigma_i\dots\sigma_j = \sigma'} & \exists i,j.\:\sigma_i\dots\sigma_j = \sigma' \\
-1 & \mbox{otherwise}
\end{cases}
\]
\item[{\tt length}:] It returns the length of a string $\sigma \in \mathbb{S}$. Its semantics is the function {\sc Le}$:\mathbb{S}\rightarrow\mathbb{Z}$ defined as $\lengths{\sigma} \mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}} |\sigma| $.
\item[{\tt concat}:] The string concatenation is handled by $\mathsf{IMP}$ plus operator (\code{+}). The concrete semantics relies on the concatenation operator reported in Sect.~\ref{sect:background}, i.e., $\concs{\sigma}{\sigma'} = \sigma\sigma'$.
\end{description}
\begin{figure}[t]
\begin{minipage}{0.27\textwidth}
\begin{adjustbox}{scale=0.76}
$
\ctostr{v} =
\begin{cases}
v & v \in \mathbb{S} \\
``\code{NaN}" & v = \code{NaN} \\
``\mbox{\tt true}" & v = \mbox{\tt true} \\
``\mbox{\tt false}" & v = \mbox{\tt false} \\
\mathcal{S}(v) & v \in \mathbb{Z}
\end{cases}
$
\end{adjustbox}
\end{minipage}
~
\begin{minipage}{0.27\textwidth}
\begin{adjustbox}{scale=0.76}
$
\ctoint{v} =
\begin{cases}
v & v \in \mathbb{Z} \\
1 & v = \mbox{\tt true} \\
0 & v = \mbox{\tt false} \vee v = \code{NaN} \\
\mathcal{I}(v) & v \in \mathbb{S} \wedge v \in \Sigma^*_{\scriptsize{\mathbb{Z}}} \\
\code{NaN} & v \in \mathbb{S} \wedge v \not\in \Sigma^*_{\scriptsize{\mathbb{Z}}}
\end{cases}
$
\end{adjustbox}
\end{minipage}
~\hspace*{0.55cm}
\begin{minipage}{0.25\textwidth}
\begin{adjustbox}{scale=0.76}
$
\ctobool{v} =
\begin{cases}
v & v \in \mathbb{B} \\
\mbox{\tt true} & v \in \mathbb{Z} \smallsetminus \{0\} \vee v \in \mathbb{S} \smallsetminus \{\epsilon\} \\
\mbox{\tt false} & v = 0 \vee v = \epsilon \vee v = \code{NaN}
\end{cases}
$
\end{adjustbox}
\end{minipage}
\caption{$\mathsf{IMP}$ implicit type conversion functions.}
\label{fig:type-juggling}
\end{figure}
\subsection{The finite state automata domain for strings}\label{sect:fa-domain}
In this section, we describe the automata abstract domain for strings \cite{park2016, wid-approach, yu2008}, namely the domain of regular languages over $\Sigma^*$. In particular, our aim is that of characterize automata as a domain for abstracting the computation of program semantics in the abstract interpretation framework.
The exploited idea is that of approximating strings as regular languages represented by the minimum DFAs~\cite{davis1994} recognizing them. In general, we have more DFAs that recognize a regular language, hence the domain of automata is indeed the quotient $\mathsf{\mbox{\sc Dfa}_{/\equiv}}$ w.r.t. the equivalence relation induced by language equality: $\forall \mbox{\tt A}_1,\mbox{\tt A}_2\in\mbox{\sc Dfa}.\:\mbox{\tt A}_1 \equiv \mbox{\tt A}_2 \Leftrightarrow \mathscr{L}(\mbox{\tt A}_1) = \mathscr{L}(\mbox{\tt A}_2)$. Hence, any equivalence class $[\mbox{\tt A}]_{\equiv}$ is composed by the automata that recognize the same regular language.
We abuse notation by representing equivalence classes in the domain $\mathsf{\mbox{\sc Dfa}_{/\equiv}}$ w.r.t.\ $\equiv$ by one of its automata (usually the minimum), i.e., when we write $\mbox{\tt A}\in\mathsf{\mbox{\sc Dfa}_{/\equiv}}$ we mean $[\mbox{\tt A}]_{\equiv}$. The partial order $\sqsubseteq_{\mbox{\tiny \DFA}}$ induced by language inclusion is
$\forall \mbox{\tt A}_1, \mbox{\tt A}_2 \in \mathsf{\mbox{\sc Dfa}_{/\equiv}} \ . \ \mbox{\tt A}_1 \sqsubseteq_{\mbox{\tiny \DFA}} \mbox{\tt A}_2 \Leftrightarrow \mathscr{L}(\mbox{\tt A}_1) \subseteq \mathscr{L}(\mbox{\tt A}_2)$, which is well defined since automata in the same $\equiv$-equivalence class recognize the same language.
\noindent
The least upper bound (lub) $\sqcup_{\mbox{\tiny \DFA}}: \mathsf{\mbox{\sc Dfa}_{/\equiv}} \times \mathsf{\mbox{\sc Dfa}_{/\equiv}} \rightarrow \mathsf{\mbox{\sc Dfa}_{/\equiv}}$ on the domain $\mathsf{\mbox{\sc Dfa}_{/\equiv}}$, corresponds to the standard union between automata:
$\forall \mbox{\tt A}_1, \mbox{\tt A}_2 \in \mathsf{\mbox{\sc Dfa}_{/\equiv}}.\:\mbox{\tt A}_1 \sqcup_{\mbox{\tiny \DFA}} \mbox{\tt A}_2 \mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}} \mathsf{Min}(\mathscr{L}(\mbox{\tt A}_1) \cup \mathscr{L}(\mbox{\tt A}_2))$.
It is the minimum automaton recognizing the union of the languages $\mathscr{L}(\mbox{\tt A}_1)$ and $\mathscr{L}(\mbox{\tt A}_2)$. This is a well-defined notion since regular languages are closed under union.
The greatest lower bound $\sqcap_{\mbox{\tiny \DFA}} : \mathsf{\mbox{\sc Dfa}_{/\equiv}} \times \mathsf{\mbox{\sc Dfa}_{/\equiv}} \rightarrow \mathsf{\mbox{\sc Dfa}_{/\equiv}}$ corresponds to automata intersection, since regular languages are closed under finite intersection:
$\forall \mbox{\tt A}_1, \mbox{\tt A}_2 \in \mathsf{\mbox{\sc Dfa}_{/\equiv}}.\: \mbox{\tt A}_1 \sqcap_{\mbox{\tiny \DFA}} \mbox{\tt A}_2 \mbox{\raisebox{0ex}[1ex][1ex]{$\stackrel{\mbox{\tiny def}}{\; =\;}$}} \mathsf{Min}(\mathscr{L}(\mbox{\tt A}_1) \cap \mathscr{L}(\mbox{\tt A}_2)).$
\begin{theorem}\label{thm:fa-moore-family}$\langle \fa, \leqfa, \lubfa, \glbfa,\mathsf{Min}(\varnothing),\mathsf{Min}(\Sigma^*)\rangle$ is a sub-lattice but not a complete meet-sub-semilattice of $\wp(\Sigma^*)$.\end{theorem}
In other words, there exists no Galois connections between $\mathsf{\mbox{\sc Dfa}_{/\equiv}}$ and $\wp(\Sigma^*)$, i.e., there may exist no minimal automaton abstracting a language.\footnote{Note that, some works have studied automatic procedures to compute, given an input language $L$, the \textit{regular cover} of $L$ \cite{domaratzki2001} (i.e., an automaton containing the language $L$). In particular, \cite{campeanu2002, domaratzki2001} have studied regular covers guaranteeing that the automaton obtained is the best w.r.t. a \textit{minimal relation} (but not minimum).}
However, this is not a concern, since the relation between concrete semantics and abstract semantics can be weakened while still ensuring soundness \cite{cousot1992}. A well known example is the convex polyhedra domain \cite{cousot1978}.
\noindent
\textbf{Widening.} The domain $\mathsf{\mbox{\sc Dfa}_{/\equiv}}$ is an infinite domain, and it is not ACC, i.e., it contains infinite ascending chains. For instance, consider the set of languages $\{\sset{a^j b^j}{0\leq j\leq i}\}_{i\geq 0}\subseteq\wp(\Sigma^*)$ forming an infinite ascending chain, then also the set of the corresponding minimal automata forms an ascending chain on $\mathsf{\mbox{\sc Dfa}_{/\equiv}}$. This clearly implies that any computation on $\mathsf{\mbox{\sc Dfa}_{/\equiv}}$ may lose convergence \cite{cousot1992}.
Most of the proposed abstract domains for strings \cite{costantini2015,jsai2014, tajs2009, safe2012} trivially satisfy ACC by being finite, but they may lose precision during the abstract computation \cite{cousot1992-2}.
In these cases, domains must be equipped with a widening operator approximating the lub in order to force convergence (by necessarily losing precision) for any increasing chain \cite{cousot1992-2}.
As far as automata are concerned, existing widenings are defined in terms of a state equivalence relation merging states that recognize the same language, up to a fixed length $n$ (set as parameter for tuning the widening precision) \cite{silva2006,DBLP:conf/cav/BartzisB04}. We denote this parametric widening with $\nabla_n$, $n \in \mathbb{N}$\cite{silva2006}.
\begin{example} Consider the following $\mathsf{IMP}$ fragment
\begin{CenteredBox}
\begin{lstlisting}[numbers=none]
str = ""; while (x++ < 100) { str += "a"; }
\end{lstlisting}
\end{CenteredBox}
Since the value of the variable \code{x} is unknown, also the number of iterations of the \code{while}-loop is unknown. In these cases, in order to guarantee soundness and termination, we apply the widening operator. In Fig.~\ref{fig:1-wid} we report the abstract value of the variable \code{str} at the beginning of the second iteration of the loop, while in Fig.~\ref{fig:2-wid} the abstract value of the variable \code{str} at the end of the second iteration is reported. Before starting a new iteration, in the example, we apply $\nabla_1$ between two automata, namely we merge all the states having the same outgoing character. The minimization of the obtained automaton is reported in Fig.~\ref{fig:3-wid}. The next iteration will reach the fix-point, guaranteeing soundness and termination.
\begin{figure}[t]
\centering
(a)\begin{subfigure}[t]{0.25\textwidth}
\centering
\begin{adjustbox}{scale=0.85}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm, semithick]
\node[initial,state,scale=0.45, accepting, initial text =] (A) {};
\node[state,scale=0.45, accepting] (B) [right of=A] {};
\path[->] (A) edge node {$a$} (B);
\end{tikzpicture}
\end{adjustbox}
\caption{}
\label{fig:1-wid}
\end{subfigure}%
~
(b)\begin{subfigure}[t]{0.3\textwidth}
\centering
\begin{adjustbox}{scale=0.85}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm, semithick]
\node[initial,state,scale=0.45, initial text =] (A) {};
\node[state,scale=0.45, accepting] (B) [right of=A] {};
\node[state,scale=0.45, accepting] (C) [right of=B] {};
\path[->] (A) edge node {$a$} (B);
\path[->] (B) edge node {$a$} (C);
\end{tikzpicture}
\end{adjustbox}
\caption{}
\label{fig:2-wid}
\end{subfigure}
~
(c)\begin{subfigure}[t]{0.3\textwidth}
\centering
\begin{adjustbox}{scale=0.85}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm, semithick]
\node[initial,state,scale=0.45, initial text =, accepting] (A) {};
\path[->] (A) edge [loop above] node {$a$} (A);
\end{tikzpicture}
\end{adjustbox}
\caption{}
\label{fig:3-wid}
\end{subfigure}
\caption{(a) $\mbox{\tt A}_1$ s.t. $\mathscr{L}(\mbox{\tt A}_1) = \{\epsilon, a\}$ (b)$\mbox{\tt A}_2$ s.t. $\mathscr{L}(\mbox{\tt A}_2) = \{a, aa\}$ (c) $\mbox{\tt A}_1 \nabla_1 \mbox{\tt A}_2$}
\label{fig:wid}
\end{figure}
\end{example}
\section{Introduction}
\input{intro}
\section{Background}\label{sect:background}
\input{background}
\section{An abstract domain for string manipulation}\label{sect:abs-domain}
\input{absdomain}
\section{The $\mathsf{IMP}$ abstract semantics}\label{sect:str-op-abs-sem}
\input{abssem}
\subsection{Abstract semantics of {\tt substring}}
\input{substr}
\subsection{Abstract semantics of \code{charAt}}
\input{charat}
\subsection{Abstract semantics of \code{length}}
\input{length}
\subsection{Abstract semantics of \code{indexOf}}
\input{indexof}
\subsection{Abstract semantics of concatenation}
\input{concat}
\subsection{Concerning abstract implicit type conversion}
\input{tj2}
\input{static-analyzer}
\section{Discussion and related work}\label{sect:rel}
\input{related}
\bibliographystyle{eptcs}
|
1905.10929
|
\section{Introduction}
{\it Introduction}.---%
The spin Hall (SH) effect is the generation of spin current along the transverse direction by an applied electric field \cite{Dyakonov1971,Hirsch1999}.
Because it allows us to manipulate magnetic quanta, i.e., spins, without applying a magnetic field,
this would become a key component in creating efficient spintronic devices.
By combining the SH effect and its reciprocal effect, the inverse SH effect \cite{Saitoh2006},
a variety of phenomena have been demonstrated (for recent review, see Refs.~\cite{Murakami2011,Sinova2015}).
As in the anomalous Hall effect ~\cite{Nagaosa2010},
the relativistic spin-orbit coupling (SOC) plays the fundamental role for the SH effect,
and both intrinsic mechanisms \cite{Sinova2004,Murakami2004} and extrinsic mechanisms \cite{Smit1955,Berger1970,Crepieux2001,Tse2006}
have been proposed.
Whereas many theoretical studies considered static disorder or impurities at zero temperature,
the effect of nonzero temperature $T$ in the SH effect has been addressed using phenomenological electron-phonon coupling \cite{Gorini2015,Xiao2018}
or first-principle scattering approach \cite{Wang2016}.
At present, the intensity of the SH effect is too weak for practical applications \cite{Hoffmann2013}.
One of the pathways to enhance the spin-charge conversion efficiency or the SH angle $\Theta_{SH} = \sigma_{SH}/\sigma_c$,
where $\sigma_{SH(c)}$ corresponds to the SH (charge) conductivity, is to reduce the charge conductivity $\sigma_c$.
For example, Ref.~\cite{Fujiwara2013} proposed to use $5d$ transition-metal oxides, IrO$_2$, where the strong SOC comes from Ir, rather than metallic materials.
The SH effect in the surface state of topological insulators with spin-momentum locking has been also studied \cite{Ong2018}.
More recently, Jiao {\it et al.} reported the significant enhancement in SH effect in metallic glasses at finite temperatures~\cite{Jiao2018}.
Because such enhancement is not expected in crystalline systems \cite{Vila2007},
it was suggested that local structural fluctuations \cite{Gorini2015,Karnad2018} are responsible for this effect, similar to the phonon skew-scattering mechanism.
Thus, the fluctuations of lattice or some other degrees of freedom at finite temperatures
could provide a route to improve the efficiency of the SH effect.
For magnetic systems, the effect of finite temperatures has been studied for the anomalous Hall effect in terms of skew scattering \cite{Kondo1962} and
resonant skew scattering \cite{Fert1972,Coleman1985,Fert1987}.
Theories for the resonant skew scattering were further developed by considering strong quantum spin fluctuations
for systems with the time-reversal symmetry (TRS), therefore for the SH effect rather than the anomalous Hall effect \cite{Guo2009,Gu2010a,Gu2010b}.
Later, the relation between the anomalous Hall effect below the ferromagnetic transition temperature $T_C$ and the SH effect above $T_C$ was investigated
by including nonlocal magnetic correlations in Kondo's model \cite{Gu2012,Wei2012}.
A recent investigation on Fe$_x$Pt$_{1-x}$ alloys also reported the enhancement in the SH effect near $T_C$ \cite{Ou2018}.
So far, the magnetic fluctuation at finite temperatures has been theoretically treated on a single-site level \cite{Guo2009,Gu2010a,Gu2010b}
or using static approximations \cite{Kondo1962,Fert1972,Coleman1985,Fert1987,Gu2012}.
When localized moments have long-range dynamical correlations near a magnetic instability, it is required to go beyond such a treatment
(for example, see Refs.~\cite{Moriya1973,Hertz1976,Moriya1985,Millis1993}).
This could open new pathways for novel spintronics.
In this paper, we address the effect of such magnetic fluctuations onto the SH effect by calculating the SH conductivity of a model system
in which conduction electrons are interacting with dynamically fluctuating local magnetic moments.
We start from defining our model Hamiltonian and then identify two different mechanisms for the SH effect.
The similarity and dissimilarity with the SH effect arising from impurity potential scattering or phonon scattering are discussed.
The SH conductivity is computed using the Matsubara formalism by combining the self-consistent renormalization theory \cite{Moriya1985}.
We show that the SH conductivity is enhanced at low temperatures when the system is in close vicinity to the ferromagnetic critical point at $T=0$.
Possible realization of this effect in $4d$ or $5d$ metallic compounds is discussed.
{\it Model and formalism}.---%
To be specific, we consider the $s$-$d$ or $s$-$f$ Hamiltonian proposed by Kondo \cite{Kondo1962,supp},
$H = H_0 + H_K$ with $H_0=\sum_{\vect k, \nu} \varepsilon_{\vect k} a_{\vect k \nu}^\dag a_{\vect k \nu}$ and
\begin{eqnarray}
&&\hspace{-1em} H_K =
-\frac{1}{N} \sum_{n}^{N_m} \sum_{\vect k, \vect k'} \sum_{\nu, \nu'} e^{i(\vect k' -\vect k) \cdot \vect R_n}
a_{\vect k \nu}^\dag a_{\vect k' \nu'} \nonumber \\
&&\hspace{-1em} \times \biggl[ 2(\vect J_n \cdot \vect s_{\nu \nu'}
\bigl\{ {\cal F}_0 + 2 {\cal F}_1(\vect k \cdot \vect k') \bigr\}
+ i {\cal F}_2 \vect J_n \cdot (\vect k' \times \vect k) \nonumber \\
&&\hspace{-1em} + i {\cal F}_3 \Bigl\{( \vect J_n \cdot \vect s_{\nu \nu'}) \bigl( \vect J_n \cdot (\vect k' \times \vect k) \bigr)
+\bigl( \vect J_n \cdot (\vect k' \times \vect k) \bigr) (\vect J_n \cdot \vect s_{\nu \nu'}) \nonumber \\
&&\hspace{-1em} - \frac{2}{3} (\vect J_n \cdot \vect J_n) \bigl( \vect s_{\nu \nu'} \cdot (\vect k' \times \vect k) \bigr) \Bigr\} \biggr] .
\label{eq:Kondo}
\end{eqnarray}
Here, $a_{\vect k \nu}^{(\dag)}$ is the annihilation (creation) operator of a conduction electron with momentum $\vect k$ and spin $\nu$,
$\varepsilon_{\vect k} = \frac{\hbar^2 k^2}{2m}- \mu$ is the dispersion relation measured from the Fermi level $\mu$
with the carrier effective mass $m$,
$\vect s_{\nu \nu'}=\frac{1}{2} \vecs \sigma_{\nu \nu'}$is the conduction electron spin with $\vecs \sigma$ the Pauli matrices,
and $N (N_m)$ is the total number of lattice sites (local moments).
$\vect J_n$ is the local spin moment at position $\vect R_n$, when the SOC is weaker than the crystal field splitting and could be treated as a perturbation,
or the local total angular momentum, when the SOC is strong so that the total angular momentum is a constant of motion.
Parameters ${\cal F}_l$ are related to $F_l$ defined in Ref.~\cite{Kondo1962}
as discussed in the Supplemental Material \cite{supp}.
In this work, we focus on three-dimensional systems.
While the current analysis could be applied to other dimensions, lower-dimensional systems require more careful treatments.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth, clip]{scatter2.pdf}
\caption{Scattering processes involving
(a) ${\cal F}_{0,1}$ terms, (b) ${\cal F}_2$ terms, and (c) ${\cal F}_3$terms.
Yellow arrows indicate conduction electrons, and green arrows indicate local moments.
In the ${\cal F}_{2 (3)}$ scattering processes, the electron deflection depends on the direction of the local moment (the electron spin),
leading to the side-jump-type (skew-scattering-type) contribution to $\sigma_{SH}$.
}
\label{fig:scatter}
\end{center}
\end{figure}
In Eq.~(\ref{eq:Kondo}), ${\cal F}_{0,1}$ terms correspond to the standard $s$-$d$ or $s$-$f$ exchange interaction,
acting as the spin-dependent potential scattering as schematically shown in Fig.~\ref{fig:scatter}~(a).
${\cal F}_{2,3}$ terms represent the exchange of angular momentum between a conduction electron and a local moment.
These terms are odd (linear or cubic) order in $J_n$ and $s$ and induce the electron deflection depending on the direction of $\vect J_n$ or $\vect s$
as depicted in Figs.~\ref{fig:scatter}~(b) and ref{fig:scatter}~(c).
As discussed below, the ${\cal F}_2$ term and the ${\cal F}_3$ term, respectively, generate
the side-jump- and the skew-scattering-type contributions to the SH conductivity.
In order to see the different types of contributions,
we analyze the velocity operator, from which the charge current and the spin current operators are defined.
Importantly, a side-jump-type contribution to the SH effect arises from the anomalous velocity as in the conventional SH effect.
The velocity operator is defined by $\vect v = (i/\hbar) [H,\vect r]$.
Among various terms, lowest order contributions to the spin Hall conductivity come from
\begin{eqnarray}
\vect v \!\!\!&=&\!\!\! \sum_{\vect k}\frac{\hbar \vect k}{m} a_{\vect k \nu}^\dag a_{\vect k \nu}
- \frac{i }{\hbar N} \sum_{n} \sum_{\vect k, \vect k'} \sum_{\nu, \nu'} e^{i(\vect k'-\vect k) \cdot \vect R_n}
\nonumber \\
\!\!\!&\times&\!\!\! \bigl\{ {\cal F}_2 \vect J_n
+ 2 {\cal F}_3 (\vect J_n \cdot \vect s_{\nu \nu'}) \vect J_n \bigr\}
\times (\vect k'-\vect k) a_{\vect k \nu}^\dag a_{\vect k' \nu'}.
\label{eq:velocity}
\end{eqnarray}
Here, a term involving ${\cal F}_1$ is neglected because it is proportional to $(\vect k + \vect k')$ and
does not contribute to $\sigma_{SH}$ at the lowest order.
The second terms involving ${\cal F}_{2,3}$ are the anomalous velocity.
The charge current and the spin current are then given by using the velocity operator as
$\vect j^c = -e \vect v$ and $\vect j^s = -e \{\frac{1}{N} \sum_{\vect k} s_{\nu \nu'}^z a_{\vect k \nu}^\dag a_{\vect k \nu'}, \vect v\}$, respectively.
Note that $\vect j^c$ and $\vect j^s$ have the same dimension.
Now, we consider the side-jump-type mechanism arising from the anomalous velocity in Eq.~(\ref{eq:velocity})
combined with the spin-dependent potential scattering ${\cal F}_{0,1}$ in Eq.~(\ref{eq:Kondo}).
At this moment, one could notice some analogy between the current model and the previous ones utilizing the potential scattering $V_n$
\cite{Berger1970,Crepieux2001,Tse2006} as
${\cal F}_{0,1} J_n s \leftrightarrow V_n$ and ${\cal F}_2 J_n \leftrightarrow \lambda^2 V_n s $, i.e.,
the spin $s$ dependence is switched from the anomalous velocity to the scattering term.
Therefore, the second-order processes involving ${\cal F}_{0,1}$ and ${\cal F}_2$ terms could generate the side-jump-type contribution to the SH effect.
The diagramatic representation of this side-jump-type contribution to the SH conductivity is presented in Fig.~\ref{fig:sidejump}.
Note that this contribution is ${\cal O}({\cal F}_{0,1} {\cal F}_2 \langle J_n J_{n'} \rangle)$.
If the ${\cal F}_3$ term in the anomalous velocity is used, it would become
${\cal O}({\cal F}_{0,1} {\cal F}_3 \langle J_n J^2_{n'} \rangle)$, odd order in the local moment.
Such a contribution vanishes when the local moments have the TRS in a paramagnetic phase above magnetic transition temperature.
How about the skew-scattering-type contribution?
Unlike the side-jump-type contribution, the ${\cal F}_2$ does not contribute to $\sigma_{SH}$ arising from
the third-order perturbation processes combined with ${\cal F}_{0,1}$ terms.
This is because such processes are ${\cal O}({\cal F}^2_{0,1} {\cal F}_2 \langle J_n J_{n'} J_{n''} \rangle)$ and vanish by the TRS in the local moments.
In fact, the skew-scattering-type contribution arises from the third-order processes involving ${\cal F}_{0,1}$ and ${\cal F}_3$ terms
as ${\cal O}({\cal F}^2_{0,1} {\cal F}_3 \langle J_n J_{n'} J^2_{n''} \rangle)$.
Therefore, such skew-scattering-type contributions are possible without introducing unharmonic (third-order) magnetic correlations,
while it is second order in the spin fluctuation propagator ${\cal O}(D^2)$ as discussed below.
This contrasts with the phonon skew scattering, where unharmonic phonon interactions are essential \cite{Gorini2015}.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth, clip]{sidejump2.pdf}
\caption{Diagrammatic representation for the side-jump contribution.
Solid (wavy) lines are the electron Green's functions (the spin fluctuation propagators).
Squares (circles) are the spin (charge) current vertices, with filled symbols representing the velocity correction with ${\cal F}_2$,
i.e., side jump.
Filled triangles are the interaction vertices with ${\cal F}_{0,1}$. }
\label{fig:sidejump}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth, clip]{skew2.pdf}
\caption{Diagrammatic representation for the skew-scattering contribution.
Filled pentagons are the interaction vertices with ${\cal F}_3$.
The definitions of the other symbols or lines are the same as in Fig.~\ref{fig:sidejump}. }
\label{fig:skew}
\end{center}
\end{figure}
{\it Matsubara formalism and spin fluctuation}.---%
In what follows, we use the Matsubara formalism to compute the SH conductivity given by
\begin{eqnarray}
\sigma_{SH}(i\Omega_l) = \frac{i}{i\Omega_l V} \int_0^{1/T} \hspace{-1em} d\tau e^{i \Omega_l \tau}
\langle T_\tau j_x^s (\tau) j_y^c (0) \rangle,
\label{eq:sigmaSH}
\end{eqnarray}
where $\Omega_l$ is the bosonic Matsubara frequency, and $V$ is the volume of the system.
At the end of the analysis, $i \Omega_l$ is analytically continued to real frequency as $i\Omega_l \rightarrow \Omega + i 0^+$.
We will then consider the dc limit, $\Omega \rightarrow 0$, to obtain $\sigma_{SH}$.
This formalism allows one to treat conduction electrons coupled with dynamically fluctuating local moments $\vect J_n$.
To describe the latter, we consider a generic Gaussian action given by
$A_{Gauss} = \frac{1}{2}\sum_{\vect q, l}
D^{-1}_{\vect q}( i\omega_l) J_{\vect q}( i\omega_l) J_{-\vect q} (-i\omega_l)
$
with
$D^{-1}_{\vect q} (i\omega_l) = \delta + A q^2 + |\omega_l|/\Gamma_q$.
Here, $\omega_l = 2 l \pi T$ is the bosonic Matsubara frequency,
and $A$ is introduced as a constant so that $A q^2$ has the unit of energy.
$\delta$ is the distance from a ferromagnetically ordered state and is related to the magnetic correlation length as $\xi^2 \propto \delta^{-1}$.
$J_{\vect q}( i\omega_l)$ is a space and imaginary-time $\tau$ Fourier transform of $J_{n}(\tau)$, where we made the $\tau$ dependence explicit.
In principle, $\delta$ depends on temperature and is determined by solving self-consistent equations for a full model including non-Gaussian terms
\cite{Moriya1973,Hertz1976,Moriya1985,Millis1993,Nagaosa1999}.
$\Gamma_q$ represents the momentum-dependent damping.
In clean metals close to the ferromagnetic instability, $\Gamma_q = \Gamma q$.
When elastic scatting exists due to impurities or disorders, $q$ has a small cutoff $q_c \sim \ell^{-1} = 1/v_F \tau_c $ with
$\ell$ being the mean free path of conduction electrons, $v_F = \hbar k_F/m$ the Fermi velocity, and $\tau_c$ the carrier lifetime.
Therefore, the damping term at $q \alt q_c$ has to be replaced by $\Gamma q_c$ \cite{Lee1992}.
With this propagator $D$, the spatial and temporal correlation of $\vect J_n$ is given by
$\langle T_\tau J_{n} (\tau) J_{n'} (0) \rangle
= \frac{T}{N} \sum_{\vect q, l} e^{-i\omega_l \tau + i \vect q \cdot (\vect R_n - \vect R_{n'})}
D_{\vect q} (i\omega_l)$.
Theoretical analyses based on this model have been successful to explain many experimental results on itinerant magnets \cite{Moriya1985}.
Because of the phase factor $ e^{ i \vect q \cdot (\vect R_n - \vect R_{n'})}$,
the ferromagnetic fluctuation is essential for the SH effect.
When the spin fluctuation has characteristic momentum $\vect Q \ne \vect 0$,
$ e^{ i \vect Q \cdot (\vect R_n - \vect R_{n'})}$ has destructive effects.
{\it Spin-Hall conductivity}.--
With the above preparations, now we proceed to examine the SH conductivity.
Based on the diagrammatic representations in Figs.~\ref{fig:sidejump} and \ref{fig:skew},
$\sigma_{SH}$ is expressed in terms of electron Green's function $G$ and the propagator of local magnetic moments $D$.
The full expression is presented in Ref.~\cite{supp}.
We carry out the Matsubara summations, the energy integrals and the momentum summations as detailed in Ref.~\cite{supp} to find
\begin{equation}
\sigma^{side \, jump}_{SH} \approx \frac{2 e^2 n_m^2}{m} \, \tau_c \, I(T,\delta) \,
\biggl( \frac{1}{3} {\cal F}_0 k_F^2 - \frac{2}{5} {\cal F}_1 k_F^4 \biggr) {\cal F}_2 N_F
\label{eq:sigmaSJ2}
\end{equation}
for the side-jump contribution and
\begin{equation}
\sigma^{skew \, scat.}_{SH} \approx
\frac{4 e^2 \hbar n_m^3}{ m^2 }
\, \tau_c^2 \, I^2(T,\delta) \,
\Bigl( {\cal F}_0 + {\cal F}_1 k_F^2 \Bigr)^2 {\cal F}_3 \frac{2 k_F^4}{15} N_F
\label{eq:sigmaSS6}
\end{equation}
for the skew-scatting contribution.
Here,
$n_m=N_m/N$ is the concentration of local moments,
and
$N_F=m k_F/2 \pi^2 \hbar^2$ is the electron density of states per spin at the Fermi level.
The function $I(T,\delta)$ defined in Ref.~\cite{supp}
is the direct consequence of the coupling between conduction electrons and the dynamical spin fluctuation.
There are a number of limiting cases where the analytic form of $I(T,\delta)$ is available.
For clean systems ($\Gamma_q = \Gamma q$, i.e., no momentum cutoff) at low temperatures,
where $\delta + A (a T/\hbar v_F)^2 \ll \hbar v_F/\Gamma$ is satisfied,
$I(T,\delta) \approx \frac{1}{8 \pi \delta} (a T/\hbar v_F)^3$
with $a$ being the lattice constant.
When the system is on the quantum critical point for the ferromagnetic ordering,
$\delta$ is scaled as $\delta \propto T^{4/3}$~\cite{Moriya1985}.
Thus, $I(T,\delta) \propto T^{5/3}$ is expected.
For clean systems at high temperatures, where $\delta + A (a T/\hbar v_F)^2 \gg \hbar v_F/\Gamma$ is satisfied,
$I(T,\delta) \approx \frac{\hbar v_F}{4 \pi^2 \Gamma \delta^2} (a T/\hbar v_F)^3$.
At such high temperatures, $\delta$ is linearly dependent on $T$ \cite{Moriya1985,Ueda1975}.
Therefore, one expects $I(T,\delta) \propto T$.
Similar analyses are possible for dirty systems, where $\Gamma_q$ has a small momentum cutoff.
In this case, one expects $I(T,\delta) \propto T$ at both low temperatures and high temperatures
(see Ref.~\cite{supp} for details).
In addition to $I(T,\delta)$, the temperature dependence of $\sigma_{SH}$ is induced by the carrier lifetime $\tau_c$.
This quantity comes from several different contributions as
\begin{eqnarray}
\tau_c^{-1} = \tau_{sf}^{-1}+\tau^{-1}_{ee} + \tau^{-1}_{ep}+ \tau_{dis}^{-1} + \ldots
\end{eqnarray}
Here,
$\tau_{sf}^{-1}$ is from the scattering due to the spin fluctuation.
Using $H_K$ and the same level of approximation,
$\tau_{sf}^{-1}$ is given by $\tau_{sf}^{-1} \approx \frac{2 n_m^2}{\hbar} I(T,\delta) ({\cal F}_0+2 {\cal F}_1 k_F^2 )^2$ \cite{supp}.
$\tau_{sf}^{-1}$ and $I(T,\delta)$ have the same $T$ dependence as schematically shown in Fig.~\ref{fig:Tdep}~(a).
$\tau_{ee}^{-1}$ and $\tau_{ep}^{-1}$ are from the electron-electron interactions and the electron-phonon interactions, respectively.
Their leading $T$ dependence is given by $\tau_{ee}^{-1} \approx \tau_{ee,0}^{-1}(T/T_F)^2$ \cite{Baber1937}
and $\tau_{ep}^{-1} \approx \tau_{ep,0}^{-1}(T/T_D)^5$ \cite{Bloch1930,Ziman1960},
where $T_{F(D)}$ is the Fermi (Debye) temperature.
$\tau_{dis}^{-1}$ is from the disorder effects, and its $T$ dependence is expected to be small.
Figure~\ref{fig:Tdep}~(b) summarizes the $T$ dependence of $\tau^{-1}_{dis,ee,ep}$.
The overall $T$ dependence of $\sigma_{SH}$ is determined by the combination of $I(T,\delta)$ and $\tau_c$.
The strong enhancement is thus expected at the ferromagnetic critical point, where the magnetic correlation length $\xi \propto \delta^{-1/2}$ diverges as $T^{-2/3}$.
This results in $\tau_{sf}^{-1}$ and hence the electrical resistivity $\sigma_c^{-1}$ scaled as $T^{5/3}$ \cite{Ueda1975}.
Since $\tau_{sf}^{-1} \propto I(T,\delta)$, $\sigma^{side \, jump}_{SH}$ and $\sigma^{skew \, scat.}_{SH}$ are expected to be maximized
when the spin fluctuation dominates $\tau_c$ as
\begin{equation}
\sigma^{side \, jump}_{SH, max} \approx
\frac{e^2\hbar}{m} \frac{{\cal F}_2 k_F^2 }{3 {\cal F}_0} N_F
\label{eq:sigmaSJ3}
\end{equation}
and
\begin{equation}
\sigma^{skew\, scatt.}_{SH, max}
\approx
\frac{e^2 \hbar^3}{m^2 n_m} \frac{2 {\cal F}_3 k_F^4}{15 {\cal F}_0^2} N_F,
\label{eq:sigmaSS7}
\end{equation}
respectively,
at low but nonzero temperature $T_{max}$.
This $T_{max}$ is approximately given by $T_F (5 \tau_{ee,0}/\tau_{dis})^{1/2}$ when $T_F \ll T_D$ or
$T_D (\tau_{ep,0}/ 2 \tau_{dis})^{1/5}$ when $T_F \gg T_D$.
As the temperature is lowered to zero, $\sigma_{SH}$ goes to zero as $\sigma^{side \, jump}_{SH} \propto \tau_{dis} I (T,\delta) \propto T^{5/3}$ and
$\sigma^{skew \, scat.}_{SH} \propto \tau_{dis}^2 I^2(T,\delta) \propto T^{10/3}$
because of the nonzero $\tau_{dis}^{-1}$,
and the residual SH conductivity is due to disorders or impurities.
At higher temperatures,
the carrier lifetime is suppressed by the electron-electron or electron-phonon interaction, and therefore
$\sigma_{SH}$ is decreased.
The overall $T$ dependence of $\sigma_{SH}^{skew \, scat.}$ is schematically shown in Fig.~\ref{fig:Tdep}~(c).
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth, clip]{Tdep2.pdf}
\caption{Schematic temperature dependence of (a) $\tau_{sf}^{-1}$ and $I(T,\delta)$,
(b) $\tau_{dis}^{-1}$ (dashed line), $\tau_{ee}^{-1}$ (dotted line), and $\tau_{ep}^{-1}$ (dash-dotted line),
and (c) $\sigma_{SH}^{skew scat.}$.
Red lines and blue lines correspond to the clean system and the dirty system, respectively.
At a low (intermediate, high) temperature regime, $\tau_c^{-1}$ is dominated by $\tau_{dis}^{-1}$ ($\tau_{sf}^{-1}$, $\tau_{ee}^{-1}$ or $\tau_{ep}^{-1}$),
creating $\sigma_{SH, max}^{skew \, scat.}$ at $T_{max}$.}
\label{fig:Tdep}
\end{center}
\end{figure}
In dirty systems, $\Gamma_q$ involves a small cutoff momentum.
Because $\tau_{dis}$ is dominant, we expect $\sigma^{side \, jump}_{SH} \propto T$
and $\sigma^{skew \, scat.}_{SH} \propto T^2$ at low temperatures as discussed in Ref.~\cite{supp}.
When the temperature is increased above $T \sim \min\{ T_F,T_D\}$,
$\sigma_{SH}$ decreases with $T$ because $\tau_c$ is suppressed.
Thus, $\sigma_{SH}$ is expected to be maximized at around $T_{max}$ as discussed for clean systems,
yet the maximum value depends explicitly on $\tau_c$'s.
In fact, the enhancement in $\tau_{sf,ee,ep}^{-1}$ with increasing $T$
always induces a momentum cutoff in the damping term $\Gamma_q$ at high temperatures.
Therefore, we expect that clean systems and dirty systems behave similarly at high temperatures,
i.e., $\sigma_{SH}^{side \, jump} \propto \tau_c T$ and
$\sigma_{SH}^{skew \, scat.} \propto \tau_c^2 T^2$.
{\it Discussion}.--
How realistic is the current spin fluctuation mechanism?
Here, we provide rough estimations of $\sigma_{SH,max}^{side \, jump}$ and $\sigma^{skew \, scat.}_{SH, max}$.
According to a free electron model, ${\cal F}_0$ is expected to be $\sim 0.1$~eV for both transition metal and actinide compounds \cite{Kasuya1959}.
(In Ref.~\cite{Kasuya1959}, $J_0$, corresponding to ${\cal F}_0$ in this study, was estimated to be $0.7\times 10^{-12}$~erg for the $s$-$d$ interaction in Mn and
$2.5 \times 10^{-13}$~erg for the $s$-$f$ interaction in Gd.)
Since ${\cal F}_{2,3} k_F^2$ involve the integral of higher-order spherical Bessel functions, $j_{1,3}$, i.e., $p$-wave scattering,
than ${\cal F}_0$, $j_0$, i.e., $s$-wave scattering \cite{Kondo1962},
${\cal F}_{2(3)} k_F^2$ would be an order (two orders) of magnitude smaller than ${\cal F}_0$.
Therefore, taking a rough estimation ${\cal F}_2 k_F^2 \sim 0.01$~eV, ${\cal F}_3 k_F^2 \sim 0.001$~eV and
typical values of $k_F/\pi \sim 10^9~{\rm m}^{-1}$ and $\frac{\hbar^2 k_F^2}{2m}=\mu \sim 10$~eV \cite{Martin} for $s$ electrons in metallic compounds,
optimistic estimations are $\sigma^{side \, jump}_{SH, max} \sim 10^{3}~\Omega^{-1} {\rm m}^{-1}$ and
$\sigma^{skew \, scat.}_{SH, max} \sim 10^{5}~\Omega^{-1} {\rm m}^{-1}$.
The difference in magnitude between $\sigma^{side \, jump}_{SH, max}$ and $\sigma^{skew \, scat.}_{SH, max}$ comes from
the small factor ${\cal F}_2/{\cal F}_0$ in $\sigma^{side \, jump}_{SH, max}$ and the large factor $\mu/{\cal F}_0$ in $\sigma^{skew \, scat.}_{SH, max}$.
Thus, $\sigma^{skew \, scat.}_{SH, max}$ could be comparable to the largest $\sigma_{SH}$ reported so far \cite{Hoffmann2013}.
Could there be systems that show the SH effect by the proposed mechanisms?
The crucial ingredients are the coupling between conduction electrons and localized but not ordered magnetic moments.
Suitable candidate materials would be $4d$ or $5d$ metallic compounds with partially filled $d$ shells, such as Ir, Pt, W and Re.
Because of the large SOC than $3d$ compounds,
the intrinsic mechanism could contribute to the SH effect.
One route to enhance $\sigma_{SH}$ further is doping with magnetic $3d$ transition metal elements to enhance the ferromagnetic spin fluctuation.
It would be possible to distinguish between the intrinsic mechanism and the extrinsic mechanisms discussed in this work
by comparing crystalline samples and disordered samples such as metallic glasses.
In fact, metallic glasses might be a good choice in trying to enhance the SH angle $\Theta_{SH}$.
Since the carrier lifetime in metallic glasses is dominated by the structure factor,
the temperature dependence of $\tau_c \sim \tau_{dis}$ is small \cite{Ziman1961,Ziman1967}.
Using the same formalism, the longitudinal charge conductivity is given by $\sigma_c = 2e^2 \tau_c k_F^3/3m \pi^2$.
Therefore, $\Theta_{SH} = \sigma_{SH}/\sigma_c$ is more sensitive to the spin fluctuation contribution than $\sigma_{SH}$ itself.
Since $\sigma_{SH}^{skew \, scat.}$ is dominant, the spin fluctuation contribution $I(T,\delta)$ could be extracted from $\sigma_{SH}/\sigma_c^2$.
Recently, Ou {\it et al.} reported very large $\Theta_{SH}>0.34$ in Fe$_x$Pt$_{1-x}$ alloys near $T_C$ \cite{Ou2018}.
While the detailed analyses remain to be carried out,
with the typical conductivity in their sample $\sigma_c \sim 10^{6}~ \Omega^{-1} {\rm m}^{-1}$ and
our theoretical $\sigma^{skew\, scat.}_{SH, max} \sim 10^{5}~\Omega^{-1} {\rm m}^{-1}$,
$\Theta_{SH}$ is estimated to be $\sim 0.1$, that is comparable to this report.
To summarize, we investigated the effect of fluctuating magnetic moments on the spin Hall effect in metallic systems.
We employed the microscopic model developed by Kondo for the coupling between conduction electrons and localized moments \cite{Kondo1962} and
analyzed the fluctuation of local moments using the self-consistent renormalization theory by Moriya \cite{Moriya1985}.
As in the conventional spin Hall effect due to the impurity scattering, a side-jump-type mechanism and a skew-scattering-type mechanism appear.
Because of the dynamical spin fluctuation, the spin Hall conductivity has a nontrivial temperature dependence,
leading to the enhancement at nonzero temperatures near the ferromagnetic instability.
The skew scattering mechanism we proposed could generate a sizable spin Hall effect.
The research by S.O. and T.E. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.
N.N. was supported by JST CREST Grant No. JPMJCR1874 and JPMJCR16F1, Japan, and JSPS KAKENHI Grants No. 18H03676 and No. 26103006.
|
1704.04022
|
\section{Introduction}
There is inherent correlation between rotation and polarization in
materials as shown in the Barnett effect \cite{Barnett:1935} and
the Einstein-de Haas effect \cite{dehaas:1915}.
We expect that the same phenomena also exist in heavy ion
collisions. Huge global angular momenta are generated in non-central
heavy ion collisions at high energies \cite{Liang:2004ph,Liang:2004xn,Voloshin:2004ha,Betz2007,Becattini:2007sr,Gao2008}.
How such huge global angular momenta are transferred to the hot and
dense matter created in heavy ion collisions and how to measure them
are two core questions in this field. There are some models to address
the first question: the microscopic spin-orbital coupling model \cite{Liang:2004ph,Liang:2004xn,Gao2008,Chen:2008wh},
the statistical-hydro model \cite{weert:1982,zubarev:1979,Becattini:2009wh,Becattini:2012tc,Becattini:2013fla,Becattini:2015nva,Hayata:2015lga}
and the kinetic model with Wigner functions \cite{Gao:2012ix,Chen:2012ca,Fang:2016vpj,Fang:2016uds}.
For the second question, it was proposed that the global angular momentum
can lead to the local polarization of hadrons, which can be measured
by the polarization of $\Lambda$ hyperons and vector mesons \cite{Liang:2004ph,Liang:2004xn}.
The global polarization is the net polarization of local ones in an
event which is aligned in the direction of the event plane. Recently
the STAR collaboration has measured the global polarization of $\Lambda$
hyperons in the beam energy scan program \cite{STAR:2017ckg,Abelev:2007zk}.
At all energies below 62.4 GeV, positive polarizations have been found
for $\Lambda$ and $\bar{\Lambda}$. On average over all data, the
gobal polarization for $\Lambda$ and $\bar{\Lambda}$ are $\Pi_{\Lambda}=(1.08\pm0.15)\%$
and $\Pi_{\bar{\Lambda}}=(1.38\pm0.30)\%$.
As will be discussed at the end of Sec. \ref{exp}, this implies that the
matter created in ultra-relativistic heavy ion collisions is the most
vortical fluid ever produced in the laboratory.
In this note, we give a brief overview about recent developments in
theories and experiments on the global and local spin polarization
in heavy ion collisions.
\section{Theoretical models in particle polarization}
In this section we first give a brief introduction to the global orbital angular momentum and local vorticity,
which are basic concepts in this topic. Then we introduce three theoretical models which
have been widely used in this field. All these models address the same global polarization problem
in different angles and are consistent to each other. The spin-orbital coupling model is a microscopic model
and the Wigner function and statistical-hydro model are macroscopic models and of statistical type.
The thermal average of the local orbital angular momentum in the microscopic model
gives the vorticity of the fluid in macroscopic models. The same freeze-out formula for the polarization of
fermions are abtained from the Wigner function and statistical-hydro model, which has been used to
calulate observables in experiments.
The Wigner function model is a quantum kinetic approach where quantum effects like
the chiral magnetic and vortical effect and chiral anomaly can be naturally incorporated.
The statistical-hydro model is a generalization of the statistical model
for a thermal system without rotation to a hydrodynamical one with rotation.
With the statistical-hydro model one can easily derive the spin-vorticity coupling term
for a system of massive fermions and then the spin polarization density
which is proportional to vorticity.
\subsection{Global orbital angluar momentum and local vorticity}
Let us consider two colliding nuclei with the beam momentum per nucleon
$\mathbf{p}_{\mathrm{beam}}\equiv p_{\mathrm{beam}}\mathbf{e}_{z}$
(projectile) and $-\mathbf{p}_{\mathrm{beam}}$ (target). The impact
parameter $\mathbf{b}\equiv b\mathbf{e}_{x}$ whose modulus is the
transverse distance between the centers of the projectile and target
nucleus points from the target to the projectile. The normal direction
of the reaction plane or the direction of the global angular momentum
is along $\hat{\mathbf{b}}\times\hat{\mathbf{p}}_{\mathrm{beam}}=-\mathbf{e}_{y}$.
We should keep in mind that due to event-by-event fluctuations of the nucleon positions,
the global orbital angular momentum does not in general point to $-\mathbf{e}_{y}$.
The discussion in this subsection is the ideal case only for theoretical simplicity.
The magnitude of the total orbital angular momentum $L_{y}$ and the
resulting longitudinal fluid shear can be estimated within the wounded
nucleon model of particle production \cite{Liang:2004ph,Gao2008}.
The transverse distributions (integrated over y) of participant nucleons
in each nucleus can be written as
\begin{equation}
\frac{dN_{\mathrm{part}}^{\mathrm{P,T}}}{dx}=\int dydz\rho_{A}^{\mathrm{P,T}}(x,y,z,b),
\end{equation}
where $\rho_{A}^{\mathrm{P,T}}$ denotes the number of participant
nucleons in the projectile and target, respectively. One can use models
to estimate $\rho_{A}^{\mathrm{P,T}}$ such as the hard-sphere or
Woods-Saxon model. Then we obtain
\begin{equation}
L_{y}=-p_{\mathrm{in}}\int dxx\left(\frac{dN_{\mathrm{part}}^{\mathrm{P}}}{dx}-\frac{dN_{\mathrm{part}}^{\mathrm{P}}}{dx}\right).
\end{equation}
The average collective longitudinal momentum per parton can be estimated
as
\begin{equation}
p_{z}(x,b;\sqrt{s})=p_{0}\frac{dN_{\mathrm{part}}^{\mathrm{P}}/dx-dN_{\mathrm{part}}^{\mathrm{T}}/dx}{dN_{\mathrm{part}}^{\mathrm{P}}/dx+dN_{\mathrm{part}}^{\mathrm{T}}/dx},
\end{equation}
where $p_{0}=\sqrt{s}/[2c(s)]$ denotes the maximum average longtitudinal
momentum per parton. The average relative orbital angular momentum
for two colliding partons separated by $\Delta x$ in the transverse
direction is then $l_{y}\equiv-(\Delta x)^{2}dp_{z}/dx$.
Note that $l_y$ is expected to be proportional to the local vorticity.
As we all know the strongly coupled quark gluon plasma (sQGP)
can be well described by relativistic hydrodynamic models. So the
sQGP can be treated as a fluid which is characterized by local quantities
such as the momentum, energy and particle-number densities $\mathbf{p}(\mathbf{r})$,
$\epsilon(\mathbf{r})$ and $n(\mathbf{r})$, respectively. The total
angluar momentum of a fluid can be written as $\mathbf{L}=\int d^{3}r\,\mathbf{r}\times\mathbf{p}(\mathbf{r})$.
The fluid velocity is defined by $\mathbf{v}(\mathbf{r})=\mathbf{p}(\mathbf{r})/\epsilon(\mathbf{r})$.
In non-relativistic theory, the fluid vorticity is defined by $\boldsymbol{\omega}=\frac{1}{2}\nabla\times\mathbf{v}(\mathbf{r})$.
Note that a 1/2 prefactor is introduced in the definition of the vorticity,
which is different from normal convention,
this is to be consistent to the convention of
the vorticity four-vector in relativistic theory.
For a rigid-body rotation with a constant angular velocity $\bar{\boldsymbol{\omega}}$,
the velocity of a point on the rigid body is given by $v=\bar{\boldsymbol{\omega}}\times\mathbf{r}$.
We can verify that $\boldsymbol{\omega}=\frac{1}{2}\nabla\times(\bar{\boldsymbol{\omega}}\times\mathbf{r})=\bar{\boldsymbol{\omega}}$,
i.e. for a rigid body in rotation the vorticity is identical to the
angular momentum. With the local vorticity, the total angluar momentum
can be re-written as $\mathbf{L}=\int d^{3}r\,\epsilon(\mathbf{r})[r^{2}\boldsymbol{\omega}-(\boldsymbol{\omega}\cdot\mathbf{r})\mathbf{r}]$.
We see that $\mathbf{L}$ is an integral of the moment of inertia
density and the local vorticity. The time evolution of the local velocity
and vorticity field can be simulated through the hydrodynamic model
\cite{Csernai:2013bqa,Csernai:2014ywa,Pang:2016igs}, the AMPT model
\cite{Jiang:2016woz,Li:2017slc} or the HIJING model with a smearing
technique \cite{Deng:2016gyh}.
\subsection{Spin-orbital coupling model}
We first consider a simple model for a spin-1/2 quark scattered in
a static Yukawa potential $V(\mathbf{r})=e^{-m_{D}|\mathbf{r}|}/(4\pi|\mathbf{r}|)$
with $m_{D}$ being the screening mass. The scattering amplitude is
\begin{equation}
\mathcal{M}(p_{\mathrm{i}},\lambda^{\prime}\rightarrow p_{\mathrm{f}},\lambda)=Qu_{\lambda}^{\dagger}(p_{\mathrm{f}})V(\mathbf{q})u_{\lambda^{\prime}}(p_{\mathrm{i}}),
\end{equation}
where $V(\mathbf{q})=1/(\mathbf{q}^{2}+m_{D}^{2})$ is the Fourier
transform of $V(\mathbf{r})$ with $\mathbf{q}=\mathbf{p}_{\mathrm{f}}-\mathbf{p}_{\mathrm{i}}$,
$Q$ denotes the coupling constant, and $u_{\lambda^{\prime}}(p_{\mathrm{i}})$
and $u_{\lambda}(p_{\mathrm{f}})$ are Dirac spinors of the quark
before and after the scattering where $(p_{\mathrm{i}},\lambda^{\prime})$
and $(p_{\mathrm{f}},\lambda)$ are (4-momentum, spin) of the quark
in the initial and final state, respectively. The spin-dependent cross
section can be obtained
\begin{eqnarray}
\sigma_{\lambda} & = & \frac{1}{2E_{\mathrm{i}}v_{\mathrm{i}}}\frac{1}{2}\sum_{\lambda^{\prime}}\int\frac{d^{3}p_{\mathrm{f}}}{(2\pi)^{3}2E_{\mathrm{f}}}\bigg|\mathcal{M}(p_{\mathrm{i}},\lambda^{\prime}\rightarrow p_{\mathrm{f}},\lambda)\bigg|^{2}(2\pi)\delta(E_{\mathrm{f}}-E_{\mathrm{i}}),
\end{eqnarray}
where $v_{\mathrm{i}}=|\mathbf{p}_{\mathrm{i}}|/E_{\mathrm{i}}$ and
$E_{\mathrm{i}}=\sqrt{\mathbf{p}_{\mathrm{i}}^{2}+m^{2}}$. The polarized
and total cross sections can thus be obtained by $\Delta\sigma=\sigma_{+}-\sigma_{-}$
and $\sigma=\sigma_{+}+\sigma_{-}$. In small angle scatterings, the
corresponding differential cross sections are in the form $d^{2}\sigma/d^{2}\mathbf{x}_{T}\sim K_{0}(m_{D}|\mathbf{x}_{T}|)$
and $d^{2}\Delta\sigma/d^{2}\mathbf{x}_{T}\sim\mathbf{n}\cdot(\mathbf{x}_{T}\times\mathbf{p}_{\mathrm{i}})$,
where $\mathbf{x}_{T}$ is the impact parameter of the scattering
in a small local cell \cite{Liang:2004ph}. We see that the polarized
cross section is proportional to the spin-orbital coupling, $\mathbf{n}\cdot(\mathbf{x}_{T}\times\mathbf{p}_{\mathrm{i}})$,
where $\mathbf{n}$ is the spin quantization direction and $\mathbf{L}=\mathbf{x}_{T}\times\mathbf{p}_{\mathrm{i}}$
is the orbital angular momentum. In order to see the connection of the polarization
with the spin-ortibal coupling energy $\Delta E_{\mathrm{LS}}$
(as in the nuclear shell model), we rewrite the polarization of the particle for small angle scatterings in the static limit ($\mathbf{p}_{\mathrm{i}}\sim 0$),
\begin{equation}
\Pi\sim\frac{\Delta\sigma}{\sigma}\sim\frac{m_{D}|\mathbf{p}_{\mathrm{i}}|}{E_{\mathrm{i}}(E_{\mathrm{i}}+m)}\sim\frac{m_{D}|\mathbf{p}_{\mathrm{i}}|}{m^{2}}\sim\frac{\Delta E_{\mathrm{LS}}}{E_{0}}
\end{equation}
which $\Delta E_{\mathrm{LS}}$ is given by
given by
\begin{equation}
\Delta E_{\mathrm{LS}}\sim\mathbf{L}\cdot\mathbf{S}\frac{1}{m^{2}r}\cdot\frac{dV}{dr}\sim\frac{1}{m^{2}}(E_{0}m_{D}^{2})\frac{|\mathbf{p}_{\mathrm{i}}|}{m_{D}}
\end{equation}
where $E_{0}$ is an energy scale, $L\sim|\mathbf{p}_{\mathrm{i}}|/m_{D}$
is the angular momentum of the particle, $r^{-1}dV/dr\sim E_{0}m_{D}^{2}$
is the potential gradient divided by the typical range of the potential
$r\sim1/m_{D}$.
One can elaborate the spin-orbital coupling model by considering a
more realistic quark-quark scattering at a transverse distance of
$\mathbf{x}_{T}$, whose polarized differential cross section is proportional
to the spin-orbital coupling $\mathbf{n}\cdot(\mathbf{x}_{T}\times\mathbf{p}_{\mathrm{i}})$,
similar to the case of the static potential \cite{Gao2008}.
\subsection{Wigner function method}
As the spin-orbital coupling involves a particle's angular momentum,
we have to know a particle's position and momentum simultaneously.
In the classical theory, we use the phase space distribution function of particles,
while in quantum theory we have to use the Wigner function,
a quantum analogue of the distribution function.
In relativistic quantum theory, the spin four-vector of a massive particle
is defined as the Pauli-Lubanski pseudo-vector ,
$\hat{S}^{\mu}=-\frac{1}{2m}\hat{J}_{\nu\rho}^{\mathrm{S}}\hat{P}_{\sigma}$,
which satisfies $[\hat{S}^{\mu},\hat{P}^{\nu}]=0$, $\hat{S}^{\mu}\hat{P}_{\mu}=0$
and $\hat{S}^{\mu}\hat{S}_{\mu}=-S(S+1)$ with $S$ is spin quantum
number of the particle. For massive fermions with spin 1/2,
we can express its spin tensor density in terms of the Wigner function \cite{Fang:2016vpj},
\begin{eqnarray}
\left\langle M^{\alpha\beta}(x)\right\rangle & = & \frac{1}{2}\lim_{y\rightarrow0}\mathrm{Tr}\left[\gamma_{0}\sigma^{\alpha\beta}\psi(x-\frac{y}{2})\bar{\psi}(x+\frac{y}{2})\right]\nonumber \\
& = & \frac{1}{2}\int d^{4}p\mathrm{Tr}\left[\gamma_{0}\sigma^{\alpha\beta}W(x,p)\right].
\end{eqnarray}
Then we can define the spin tensor component of the Wigner function
as
\begin{eqnarray}
\mathscr{M}^{\alpha\beta}(x,p) & \equiv & \frac{1}{2}\mathrm{Tr}\left[\gamma_{0}\sigma^{\alpha\beta}W(x,p)\right]\nonumber \\
& = & \frac{1}{2}\left[-\epsilon^{0\alpha\beta\rho}\mathscr{A}_{\rho}+ig^{\alpha0}\mathscr{V}^{\beta}-ig^{\beta0}\mathrm{Tr}(\gamma^{\alpha}W)\right],
\end{eqnarray}
If we take $\alpha\beta=ij$ (spatial indices), we have a simple relation
\begin{eqnarray}
\mathscr{M}^{ij}(x,p) & = & \frac{1}{2}\epsilon^{ijk}\mathscr{A}^{k}(x,p),
\end{eqnarray}
where $\epsilon_{ijk}$ is 3-dimensional anti-symmetric tensor. We
see that one can treat the axial vector component as the spin pseudo-vector
phase space density. So the polarization (or spin) pseudo-vector density
(with a factor 1/2) is \cite{Fang:2016vpj}
\begin{eqnarray}
\Pi^{\mu}(x) & \approx & \frac{1}{2}\int d^{4}p\mathscr{A}^{\mu}(x,p)
\end{eqnarray}
at the non-relativistic limit. To match the spin four-vector (Pauli-Lubanski pseudo-vector)
in relativistic case, we should multiply a Lorentz factor $E_{p}/m$ as,
\begin{eqnarray}
\Pi^{\mu}(x) & \approx & \frac{1}{2m}\int d^{4}pE_{p}\mathscr{A}^{\mu}(x,p).
\end{eqnarray}
The axial component of the Wigner function can be solved perturbatively
in an expansion of powers of space-time derivative $(\partial _\mu)^n$ and field strength $(F_{\mu\nu})^n$,
whose zeroth and first order solution are
\begin{eqnarray}
A_{(0)}^{\alpha} & = & m\left[\theta(p_{0})n^{\alpha}(\mathbf{p},\mathbf{n})-\theta(-p_{0})n^{\alpha}(-\mathbf{p},-\mathbf{n})\right]\delta(p^{2}-m^{2})A,\nonumber \\
A_{(1)}^{\alpha}(x,p) & = & -\frac{1}{2}\hbar\tilde{\Omega}^{\alpha\sigma}p_{\sigma}\frac{dV}{d(\beta p_{0})}\delta(p^{2}-m^{2})-Q\hbar\tilde{F}^{\alpha\lambda}p_{\lambda}V\frac{\delta(p^{2}-m^{2})}{p^{2}-m^{2}},\label{eq:a0-a1}
\end{eqnarray}
where $V=f_{+}+f_{-}$ and $A=f_{+}-f_{-}$ with the phase space distribution
$f_{s}$ for the spin state $s=\pm$ being defined by
\begin{eqnarray}
f_{s}(x,p) & = & \frac{2}{(2\pi)^{3}}\left[\theta(p_{0})f_{{\rm FD}}(p_{0}-\mu_{s})+\theta(-p_{0})f_{{\rm FD}}(-p_{0}+\mu_{s})\right],\label{eq:dist}
\end{eqnarray}
where $p_{0}\equiv p_{\mu}u^{\mu}$ with $u^{\mu}$ being the fluid
velocity, $f_{{\rm FD}}$ is the Fermi-Dirac distribution function,
and $\mu_{s}$ is the chemical potential corresponding to the spin
state $s$. In Eq. (\ref{eq:a0-a1}), the 4-vector of the spin quantization
direction is given by
\begin{eqnarray}
n^{\mu}(\mathbf{p},\mathbf{n}) & = & \Lambda_{\;\nu}^{\mu}(-\mathbf{v}_{p})n^{\nu}(\mathbf{0},\mathbf{n})=\left(\frac{\mathbf{n}\cdot\mathbf{p}}{m},\mathbf{n}+\frac{(\mathbf{n}\cdot\mathbf{p})\mathbf{p}}{m(m+E_{p})}\right),\label{eq:polar-cmoving}
\end{eqnarray}
where $\Lambda_{\;\nu}^{\mu}(-\mathbf{v}_{p})$ is the Lorentz transformation
with $\mathbf{v}_{p}=\mathbf{p}/E_{p}$ and $n^{\nu}(\mathbf{0},\mathbf{n})=(0,\mathbf{n})$
is the spin quantization direction in the rest frame of the fermion.
We note that the polarization pseudo-vector density at the zeroth
order is vanishing if $\mu_{s}$ does not depend on the spin $s$.
The polarization density at the first order ($\sim \omega^{\alpha}, B^{\alpha}$) is obtained by integration
over 4-momentum for $A_{(1)}^{\alpha}(x,p)$,
\begin{eqnarray}
\Pi_{(1)}^{\alpha} & \approx & \frac{1}{2m}\hbar\beta\int\frac{d^{3}p}{(2\pi)^{3}}\left\{ \left[E_{p}\omega^{\alpha}+QB^{\alpha}\right]\frac{e^{\beta(E_{p}-\mu)}}{[e^{\beta(E_{p}-\mu)}+1]^{2}}\right.\nonumber \\
& & \left.+\left[E_{p}\omega^{\alpha}-QB^{\alpha}\right]\frac{e^{\beta(E_{p}+\mu)}}{[e^{\beta(E_{p}+\mu)}+1]^{2}}\right\} ,\label{eq:pol-1}
\end{eqnarray}
where $Q>0$ is the fermion's electric charge. The momentum spectra
of the polarization pseudo-vector at the freezout hypersurface can
be obtained
\begin{eqnarray}
E_{p}\frac{d\Pi^{\alpha}(p)}{d^{3}p} & \approx & \frac{\hbar}{2m}\beta\frac{1}{(2\pi)^{3}}\int d\Sigma_{\lambda}p^{\lambda}\nonumber \\
& & \times\left(\tilde{\Omega}^{\alpha\sigma}p_{\sigma}\pm Q\tilde{F}^{\alpha\sigma}u_{\sigma}\right)f_{\mathrm{FD}}^{\pm}(x,p)\left[1-f_{\mathrm{FD}}^{\pm}(x,p)\right],\label{eq:pol-freeze}
\end{eqnarray}
where $f_{\mathrm{FD}}^{\pm}$ are Dermi-Dirac distribution functions
for fermions ($+$) and anti-fermions ($-$), respectively, and $\Sigma_{\lambda}$
denotes the freezeout hypersurface. In Eqs. (\ref{eq:pol-1},\ref{eq:pol-freeze}),
we have used $\tilde{F}^{\rho\lambda}=\frac{1}{2}\epsilon^{\rho\lambda\mu\nu}F_{\mu\nu}$,
$\tilde{\Omega}^{\xi\eta}=\frac{1}{2}\epsilon^{\xi\eta\nu\sigma}\Omega_{\nu\sigma}$
with $\Omega_{\nu\sigma}=\frac{1}{2}(\partial_{\nu}u_{\sigma}-\partial_{\sigma}u_{\nu})$,
where $\epsilon^{\mu\nu\sigma\beta}$ and $\epsilon_{\mu\nu\sigma\beta}$
are anti-symmetric tensors with $\epsilon^{\mu\nu\sigma\beta}=1(-1)$
and $\epsilon_{\mu\nu\sigma\beta}=-1(1)$ for even (odd) permutations
of indices 0123, so we have $\epsilon^{0123}=-\epsilon_{0123}=1$.
Instead of $\Omega_{\nu\sigma}$, $\tilde{\Omega}^{\xi\eta}$, $F_{\mu\nu}$
and $\tilde{F}^{\rho\lambda}$, we will also use the vorticity vector
$\omega^{\rho}=\frac{1}{2}\epsilon^{\rho\sigma\alpha\beta}u_{\sigma}\partial_{\alpha}u_{\beta}$,
the electric field $E^{\mu}=F^{\mu\nu}u_{\nu}$, and the magnetic
field $B^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}u_{\nu}F_{\lambda\rho}$.
One can use Eq. (\ref{eq:pol-freeze}) to calculate the polarization
of spin-1/2 baryons at freezeout hypersurface in heavy ion collisions
and compare with experiments.
We note that the above formalism is to describe the polarization of massive fermions
with spin 1/2 such as massive quarks or octet baryons of $(1/2)^+$.
For massless fermions for which the spin vector is not well defined
but with helicity or chirality, the polarization can be caused by
the chiral magnetic and vortical effect \cite{Kharzeev:2007jp,Fukushima:2008xe,Kharzeev:2015znc}.
\subsection{Statistical-hydro model}
The polarization of a partical in a locally rotating fluid can be
described by the statistical-hydro model. The derivation of relativistic
hydrodynamics in quantum statistical theory was proposed in late 1970s
\cite{zubarev:1979} and early 1980s \cite{weert:1982} and further
developed by several authors \cite{Becattini:2009wh,Becattini:2012tc,Becattini:2013fla,Becattini:2015nva,Hayata:2015lga}.
With the density operator, one can calculate the energy-momentum tensor
and current as functions of space-time, $T^{\mu\nu}(x)=\mathrm{Tr}\left[\hat{\rho}\hat{T}^{\mu\nu}(x)\right]\equiv\left\langle \hat{T}^{\mu\nu}(x)\right\rangle $
and $j^{\mu}(x)=\mathrm{Tr}\left[\hat{\rho}\hat{j}^{\mu}(x)\right]\equiv\left\langle \hat{j}^{\mu}(x)\right\rangle $.
One can employ the principle of maximum entropy to derive the density
operator at local equilibrium. We then use Lagrange multiplier to
maximize the entropy under the condition of fixed $T^{\mu\nu}(x)$
and $j^{\mu}(x)$,
\begin{eqnarray}
S & = & \mathrm{Tr}\left(\hat{\rho}\ln\hat{\rho}\right)+\int_{\Sigma(\tau)}d\Sigma_{\mu}\left\{ \left[\left\langle \hat{T}^{\mu\nu}(x)\right\rangle -T^{\mu\nu}(x)\right]\beta_{\nu}(x)\right.\nonumber \\
& & \left.-\left[\left\langle \hat{j}^{\mu}(x)\right\rangle -j^{\mu}(x)\right]\zeta(x)\right\} ,
\end{eqnarray}
where $\Sigma_{\mu}=\Sigma n_{\mu}$ is the space like hypersurface
with $n_{\mu}$ being the time-like vector, $\beta_{\nu}=\beta u_{\nu}$
with $u_{\nu}$ being the fluid velocity. in which leads to $\hat{\rho}_{\mathrm{LE}}$
at local equilibrium (LE),
\begin{equation}
\hat{\rho}_{\mathrm{LE}}=\frac{1}{Z}\exp\left[\int_{\Sigma(\tau)}d\Sigma_{\mu}\left(-T^{\mu\nu}\beta_{\nu}+\zeta\hat{j}^{\mu}\right)\right].
\end{equation}
Given $n_{\mu}$, one can determine the local equilibrium value of
$\beta^{\alpha}$ and $\zeta$ by $n_{\mu}\left\langle \hat{T}^{\mu\nu}(x)\right\rangle _{\mathrm{LE}}=n_{\mu}T^{\mu\nu}(x)$
and $n_{\mu}\left\langle \hat{j}^{\mu}(x)\right\rangle _{\mathrm{LE}}=n_{\mu}j^{\mu}(x)$.
The global equilibrium of the fluid can be found by imposing the stationary
condition under which the density operator does not depend on a particular
choice of space-like hypersurface $\Sigma$, so we have $\int_{\Sigma_{1}}d\Sigma_{\mu}\hat{\Phi}^{\mu}=\int_{\Sigma_{2}}d\Sigma_{\mu}\hat{\Phi}^{\mu}$,
where $\hat{\Phi}^{\mu}\equiv-\hat{T}^{\mu\nu}\beta_{\nu}+\zeta\hat{j}^{\mu}$,
or in another form
\begin{equation}
\oint_{\Sigma_{1}+\Sigma_{2}+\Sigma_{T}}d\Sigma_{\mu}\hat{\Phi}^{\mu}=\int_{V}d^{4}x\partial_{\mu}\hat{\Phi}^{\mu}=0,
\end{equation}
where $\Sigma_{T}$ is the transverse surface to $\Sigma_{1}$ and
$\Sigma_{2}$. The above equation leads to
\begin{eqnarray*}
\partial_{\mu}\hat{\Phi}^{\mu} & = & -\frac{1}{2}\hat{T}^{\mu\nu}(\partial_{\mu}\beta_{\nu}+\partial_{\nu}\beta_{\mu})+(\partial_{\mu}\zeta)\hat{j}^{\mu}=0.
\end{eqnarray*}
So we obtain the stationary conditions
\begin{equation}
\partial_{\mu}\beta_{\nu}+\partial_{\nu}\beta_{\mu}=0,\;\;\partial_{\mu}\zeta=0,
\end{equation}
where the former condition is called the Killing condition whose solution
is in the form $\beta^{\mu}=\beta u^{\mu}+\varpi^{\mu\nu}x_{\nu}$,
where $\varpi^{\mu\nu}=-\frac{1}{2}(\partial^{\mu}\beta^{\nu}-\partial^{\nu}\beta^{\mu})$.
So we obtain the density operator at global equilibrium
\begin{equation}
\hat{\rho}_{\mathrm{GE}}=\frac{1}{Z}\exp\left[-\beta u_{\nu}\hat{P}^{\nu}+\frac{1}{2}\hat{J}^{\nu\rho}\varpi_{\nu\rho}+\zeta\hat{Q}\right],
\end{equation}
where $\hat{P}^{\nu}=\int_{\Sigma}d\Sigma_{\mu}\hat{T}^{\mu\nu}$,
$\hat{J}^{\nu\rho}=\int_{\Sigma}d\Sigma_{\mu}(x^{\nu}\hat{T}^{\mu\rho}-x^{\rho}\hat{T}^{\mu\nu})$
and $\hat{Q}=\int_{\Sigma}d\Sigma_{\mu}\hat{j}^{\mu}$. We can also
add the spin tensor to the angular momentum tensor density $\hat{S}^{\mu;\nu\rho}$:
\begin{eqnarray}
\hat{J}^{\nu\rho} & = & \int_{\Sigma}d\Sigma_{\mu}(x^{\nu}\hat{T}^{\mu\rho}-x^{\rho}\hat{T}^{\mu\nu}+\hat{S}^{\mu;\nu\rho})\nonumber \\
& = & \hat{J}_{\mathrm{OAM}}^{\nu\rho}+\hat{J}_{\mathrm{S}}^{\nu\rho}.
\end{eqnarray}
The spin tensor $\hat{J}_{\mathrm{S}}^{\nu\rho}$ gives the Pauli-Lubanski
pseudo-vector. The expectation value of spin vector is given
by $S^{\mu}=\mathrm{Tr}(\hat{\rho}_{\mathrm{GE}}\hat{S}^{\mu})$.
Then the polarization is obtained by $\Pi^{\mu}=S^{\mu}/S$.
Since the spin pseudo-vector $\hat{S}^{\mu}$ involves the momentum
operator, we need to know a particle's momentum to evaluate its polarization.
In general, this requires the knowledge of the Wigner function, which
allows to express the mean values of operators as integrals over space-time
and 4-momentum. The mean spin pseudo-vector of a spin-1/2 particle
with 4-momentum $p^{\mu}$, produced at $x^{\mu}$ on particlization
hypersurface, at the leading order in the thermal vorticity reads
\cite{Becattini:2013fla,Becattini:2016gvu}
\begin{equation}
\Pi^{\mu}(x,p)=-\frac{1}{8m}[1-f_{\mathrm{FD}}(x,p)]\epsilon^{\mu\nu\sigma\rho}p_{\nu}\varpi_{\sigma\rho},\label{eq:beca-pol}
\end{equation}
where $f_{\mathrm{FD}}(x,p)$ is the Fermi-Dirac distribution function.
The mean polarization of the particle with 4-momentum $p^{\mu}$ over
the particlization hypersurface is given by
\begin{equation}
\Pi^{\mu}(p)=\frac{\int d\Sigma_{\rho}p^{\rho}f_{\mathrm{FD}}(x,p)\Pi^{\mu}(x,p)}{\int d\Sigma_{\rho}p^{\rho}f_{\mathrm{FD}}(x,p)}.\label{eq:beca-pol-free}
\end{equation}
Note that at a constant temperature, Eqs. (\ref{eq:beca-pol},\ref{eq:beca-pol-free})
are consistent to Eq. (\ref{eq:pol-freeze}) \cite{Fang:2016vpj}.
The parameters in Eq. (\ref{eq:beca-pol-free}) are those in the hydrodynamical models
which give the temperatures, chemical potentials and fluid velocities
on the freezeout hypersurface.
\section{Experimental measurements of global polarization}
\label{exp}
The global polarization can be measured by the $\Lambda$ hyperon's
weak decay into a proton and a negatively charged pion. Due to its
nature of weak interaction, the proton is emitted preferentially along
the direction of the $\Lambda$'s spin in the $\Lambda$'s rest frame,
so the parity is broken in the decay process. In this sense, we say
that $\Lambda$ is \textit{self-analyzing} since we can determine
the $\Lambda$'s polarization by measuring the daughter proton's momentum
\cite{Overseth:1967zz}. The solid angle distribution for the daughter
proton in the $\Lambda$'s rest frame is given by
\begin{eqnarray*}
\frac{dN}{d\Omega^{*}} & = & \frac{1}{4\pi}\left(1+\alpha_{H}\hat{\mathbf{p}}_{\mathrm{p}}^{*}\cdot\boldsymbol{\Pi}_{\Lambda}\right)=\frac{1}{4\pi}\left(1+\alpha_{H}\Pi_{\Lambda}\cos\theta^{*}\right),
\end{eqnarray*}
where $\hat{\mathbf{p}}_{\mathrm{p}}^{*}$ is the direction of the
daughter proton's momentum in the $\Lambda$'s rest frame, $\boldsymbol{\Pi}_{\Lambda}$
is the $\Lambda$'s polarization vector with its modulus $\Pi_{\Lambda}<1$,
$\theta^{*}$ is the angle between the momentum of the daughter proton's
and that of $\Lambda$, and $\alpha_{H}=0.642\pm0.013$ is the $\Lambda$'s
decay constant measured in experiments. The $\Lambda$'s polarization
can be determined by an event average of the proton's momentum direction
in the $\Lambda$'s rest frame,
\begin{equation}
\Pi_{\Lambda}=\frac{3}{\alpha_{H}}\left\langle \cos\theta^{*}\right\rangle _{\mathrm{ev}}.
\end{equation}
We assume the beam direction is along $\mathbf{e}_{z}$, $\hat{\mathbf{p}}_{\mathrm{beam}}=(0,0,1)$,
and the direction of the impact parameter is $\hat{\mathbf{b}}=(\cos\psi_{\mathrm{RP}},\sin\psi_{\mathrm{RP}},0)$
where $\psi_{\mathrm{RP}}$ is the azimuthal angle of the reaction
plane. The global polarization $\mathbf{L}$ is along $\hat{\mathbf{b}}\times\hat{\mathbf{p}}_{\mathrm{beam}}=(\sin\psi_{\mathrm{RP}},-\cos\psi_{\mathrm{RP}},0)$.
The direction of the daughter proton's momentum in the $\Lambda$'s
rest frame is assumed to be $\hat{\mathbf{p}}_{\mathrm{p}}^{*}=(\sin\theta_{\mathrm{p}}^{*}\cos\phi_{\mathrm{p}}^{*},\sin\theta_{\mathrm{p}}^{*}\sin\phi_{\mathrm{p}}^{*},\cos\theta_{\mathrm{p}}^{*})$.
If $\boldsymbol{\Pi}_{\Lambda}$ is in the direction of the global
polarization $\mathbf{L}$, we have
\begin{equation}
\cos\theta^{*}=\hat{\mathbf{p}}_{\mathrm{p}}^{*}\cdot\hat{\boldsymbol{\Pi}}_{\Lambda}=\sin\theta_{\mathrm{p}}^{*}\sin(\psi_{\mathrm{RP}}-\phi_{\mathrm{p}}^{*}).
\end{equation}
We can obtain the proton's distribution in $\phi_{\mathrm{p}}^{*}$
after an integration over $\theta_{\mathrm{p}}^{*}$,
\begin{eqnarray}
\frac{dN}{d\phi_{\mathrm{p}}^{*}} & = & \int_{0}^{\pi}d\theta_{\mathrm{p}}^{*}\sin\theta_{\mathrm{p}}^{*}\frac{dN}{d\Omega^{*}}\nonumber \\
& = & \frac{1}{8}+\frac{1}{8}\alpha_{H}\Pi_{\Lambda}\sin(\psi_{\mathrm{RP}}-\phi_{\mathrm{p}}^{*}).
\end{eqnarray}
Then we obtain $\Pi_{\Lambda}$ by taking an event average of $\sin(\psi_{\mathrm{RP}}-\phi_{\mathrm{p}}^{*})$
\cite{Abelev:2007zk},
\begin{equation}
\Pi_{\Lambda}=-\frac{8}{\pi\alpha_{H}}\left\langle \sin(\phi_{\mathrm{p}}^{*}-\psi_{\mathrm{RP}})\right\rangle _{\mathrm{ev}}.\label{eq:pol-lambda}
\end{equation}
The above equation is similar to that used in directed flow measurements
\cite{Barrette:1996rs,Alt:2003ab,Adams:2005ca}, which allows us
to use the corresponding anisotropic flow measurement technique \cite{Voloshin:1994mz,Poskanzer:1998yz}.
The reaction plane angle in Eq. (\ref{eq:pol-lambda}) is estimated
by calculating the angle of the first order event plane, so we need
to correct the final results by the reaction plane resolution $R_{\mathrm{EP}}^{(1)}$.
Then we can rewrite Eq. (\ref{eq:pol-lambda}) in terms of the first-order
event plane angle $\Psi_{\mathrm{EP}}^{(1)}$ and its resolution $R_{\mathrm{EP}}^{(1)}$
\cite{Abelev:2007zk},
\begin{equation}
\Pi_{\Lambda}=-\frac{8}{\pi\alpha_{H}R_{\mathrm{EP}}^{(1)}}\left\langle \sin\left(\phi_{\mathrm{p}}^{*}-\Psi_{\mathrm{EP}}^{(1)}\right)\right\rangle _{\mathrm{ev}}.\label{eq:pol-lambda-1}
\end{equation}
The first-order event plane angle is estimated experimentally by measuring
the sidewards deflection of the forward- and backward-going fragments
and particles in the STAR's BBC detectors.
\begin{figure}
\caption{\label{fig:STAR}STAR results for the global $\Lambda$ polarization \cite{STAR:2017ckg}. }
\begin{center}\includegraphics[scale=0.4]{star-result.eps}\end{center}
\end{figure}
The STAR's recent measurements for the global $\Lambda$ polarization
at all collisional energies in the Beam Energy Scan (BES) program
are shown in Fig. \ref{fig:STAR} \cite{STAR:2017ckg}. At each energy, a positive polarization
at the level of $(1.1-3.6)\sigma$ is observed for $\Lambda$ and
$\bar{\Lambda}$. Taking all data at different energies into account,
the global polarization for $\Lambda$ and $\bar{\Lambda}$ are $\Pi_{\Lambda}=(1.08\pm0.15)\%$
and $\Pi_{\bar{\Lambda}}=(1.38\pm0.30)\%$ respectively.
Although the experimental uncertainties are too large to state so with confidence,
there may be some indication for anti-Lambdas to have larger polarization than Lambdas.
Such a difference could in principle be caused by magnetic coupling of their opposite magnetic moments to the magnetic field. However, a quick calculation shows
that even for the largest magnetic fields that could be expected in these collisions
the effect of this spin-magnetic coupling on the polarization signal would be
at most a small fraction of a percent and thus invisible in this experiment.
Another source of difference may possibly be due to more Pauli blocking effect for
fermions than anti-fermions in lower collisional energies where fermions
have non-vanishing chemical potentials \cite{Fang:2016vpj,Aristova:2016wxe}.
But still such a difference is too small to be observed given
the present experimental error bars \cite{STAR:2017ckg,Becattini:2016gvu}.
The global polarization decreases with increasing collision energy. This is consistent with the observation that longitudinal boost-invariance for the longitudinal expansion becomes a better approximation at higher energies \cite{Li:2017slc,Karpenko:2016jyx}, and that a boost-invariant longitudinal flow profile has a vanishing vorticity component orthogonal to the reaction plane.
The fluid vorticity can be estimated from the data by the hydro-statistical
model $\omega\approx T(\Pi_{\Lambda}+\Pi_{\bar{\Lambda}})$, where
$T$ is the temperature of the fluid at the moment of particle freezeout.
The polarization data averaged over collisional energies imply that
the vorticity is about $(9\pm1)\times10^{21}\,\mathrm{s}^{-1}$. This
is much larger than any other fluids that exist in the universe. Then
the sQGP created in heavy ion collisions is not only the hottest,
least viscous, but also the most vortical fluid that is ever produced
in the laboratory.
\textit{Acknowledgment}. QW thanks M. Lisa and F. Becattini for helpful
discussions. QW is supported in part by the Major State Basic Research
Development Program (MSBRD) in China under the Grant No. 2015CB856902
and 2014CB845402 and by the National Natural Science Foundation of
China (NSFC) under the Grant No. 11535012.
\bibliographystyle{elsarticle-num}
|
1101.5893
|
\section{Introduction}
\label{sec:intro}
Throughout, let $(W,S)$ be a finite Coxeter system with distinguished set of
generators $S$ and let $E$ be the real reflection representation of
$W$. Define $T=\{\, wsw\inverse\mid w\in W,\,s\in S\,\}$ to be the set of elements of
$W$ that act on $E$ as reflections. By a \emph{reflection subgroup of $W$}
we mean a subgroup of $W$ generated by a subset of $T$.
Reflection subgroups of $W$ play an important role in the theory of Coxeter
groups; for instance, by a fundamental theorem due to Steinberg,
\cite[Thm.~1.5]{Steinberg64}, the stabilizer of any subspace of $E$ is a
reflection subgroup of~$W$.
Our first aim in this note is to give a complete classification of all
reflection subgroups of $W$ up to conjugacy. In case $W$ is a Weyl group,
Carter \cite[p.~8]{Carter1972} has already outlined a procedure which leads
to the classification based on the algorithm of Borel--De
Siebenthal~\cite{BorelDeSiebenthal1949}. Here, we recast slightly Carter's
construction and give the classification for non-crystallographic Coxeter
groups as well.
Similar classifications have been described by Felikson and
Tumarkin~\cite{TumarkinFelikson2005}, and by Dyer and
Lehrer~\cite{DyerLehrer2009}.
Our methods differ from those used in the sources cited above in that
we use the notion of a parabolic closure of a reflection subgroup as
an inductive tool in our analysis.
Every reflection subgroup of $W$ is a maximal rank reflection subgroup of
some parabolic subgroup of $W$. Thus, classifying conjugacy classes of
reflection subgroups may be done recursively and reduces to first classifying
conjugacy classes of parabolic subgroups and then classifying maximal rank
subgroups of irreducible Coxeter groups. Conjugacy classes of parabolic
subgroups of an irreducible finite Coxeter group are described in Chapter~2
and Appendix~A of \cite{GeckPfeiffer2000}. Classifying maximal rank
reflection subgroups of $W$ amounts to listing, up to the action of $W$, all
subsets $Y$ of $T$ whose fixed point set in $E$ is trivial and which are
closed in the sense that $\Span{Y} \cap T=Y$. In case $W$ is a Weyl
group, the algorithm of Borel--De Siebenthal~\cite{BorelDeSiebenthal1949} is
computationally much more efficient than classifying subsets of $T$
with the two required properties.
We have implemented the classification algorithms in the computer
algebra system {\sf GAP}~\cite{GAP} with the aid of the package {\sf
CHEVIE}~\cite{chevie}. Thus, it is feasible to actually compute the
classification explicitly for $W$ of a fixed rank. Indeed, we present
the classification in cases $W$ is a Weyl group of exceptional type,
or a non-crystallographic Coxeter group of type $H_3$ and $H_4$, in
the form of explicit lists.
Our second aim in this note is to study the map which assigns to a
given conjugacy class of reflection subgroups the conjugacy class of
its Coxeter elements. It is well
known~\cite[Lem.~3.5]{OrlikSolomon1983} that if $R$ and $R'$ are
parabolic subgroups containing Coxeter elements $c$ and $c'$
respectively, then $R$ and $R'$ are conjugate subgroups if and only if
$c$ and $c'$ are conjugate in $W$. Thus, conjugacy classes of
parabolic subgroups are parametrized by a distinguished set of
conjugacy classes of elements in $W$. For general reflection
subgroups this need not be the case. However, the following somewhat
surprising result shows that when $T$ is a single conjugacy class,
conjugacy classes of reflection subgroups are still parametrized by
the conjugacy classes of their Coxeter elements in all but one case.
\begin{Theorem}
\label{thm:main}
Suppose that $T$ is a single conjugacy class. Let $R$ and $R'$ be
reflection subgroups containing Coxeter elements $c$ and $c'$,
respectively. Then $R$ and $R'$ are conjugate if and only if $c$ and
$c'$ are conjugate in $W$; unless $W$ is of type $E_8$ and $R$ and
$R'$ are of types $A_1 A_7$ and $A_3 D_5$, respectively. In this
case, $c$ and $c'$ are conjugate, while $R$ and $R'$ are not.
\end{Theorem}
This theorem is an immediate consequence of the classification of the
reflection subgroups of $W$ and our computation of the map $\gamma$,
which is defined as follows. Denote by $\mathcal{R}$ the set of conjugacy
classes of reflection subgroups of $W$ and by $\mathcal{C}$ the set of
conjugacy classes of elements of $W$. Then, denote by
\[
\gamma \colon \mathcal{R} \to \mathcal{C}
\]
the map defined by $\gamma([R]) = [c]$, which associates to the conjugacy
class $[R]$ of a reflection subgroup $R$ of $W$ the conjugacy class $[c]$ in
$W$ of a Coxeter element $c$ in $R$. This map $\gamma$ is well-known to be
a bijection for Coxeter groups $W$ of type $A_n$, with both the
conjugacy classes of reflection subgroups and the conjugacy classes of
elements of $W$ labeled by the partitions of~$n$. In \S\ref{sec:mick} we
state necessary and sufficient conditions for the map $\gamma$ to be an
injection
(note that if $\gamma$ is an injection, then the statement of Theorem \ref{thm:main} holds).
The image of $\gamma$ is computed explicitly for each type of
irreducible Coxeter group in \S\ref{sec:classical} - \S\ref{sec:brian} and
Tables \ref{tab:e6} - \ref{tab:h4}. The classes in the image of $\gamma$
are also known in the literature as \emph{semi-Coxeter classes}, e.g.,
see~\cite{CarterElkington1972}. Properties of the map $\gamma$ have not
been considered in the earlier literature on the subject.
The rest of this note is organized as follows. In \S\ref{sec:mick} we recall
some definitions, give some preliminary results, and state precisely when
the map $\gamma$ is injective or surjective.
\S\ref{sec:classical} contains explicit combinatorial rules
that describe the map $\gamma$ for classical types, and demonstrate
that $\gamma$ is surjective but not injective for Coxeter groups of
type $B_n$ ($n \geq 2$), and that $\gamma$ is injective but not
surjective for Coxeter groups of type $D_n$ ($n \geq 4$).
The classification of conjugacy classes of
reflection subgroups and the explicit computation of the map $\gamma$ is
given for classical Weyl groups in \S\ref{sec:classical}
(with the examples of $W(B_5)$ and $W(D_6)$ in Tables \ref{tab:b5} and \ref{tab:d6} respectively);
for exceptional
Weyl groups in \S\ref{sec:keith} and Tables \ref{tab:e6} - \ref{tab:g2}; and for
non-crystallographic Coxeter groups in \S\ref{sec:brian} and Tables
\ref{tab:h3} and \ref{tab:h4}.
\section{Preliminaries}
\label{sec:mick}
For general information on Coxeter groups, root systems, and groups
generated by reflections, we refer the reader to
Bourbaki~\cite{Bourbaki1968}.
For the rest of this note we fix a $W$-invariant, positive definite,
bilinear form on $E$.
Notice first that if $R= \langle Y \rangle$ is a reflection subgroup of
$W$, then $R$ is a Coxeter group in its own right.
Moreover, the orthogonal complement of the space of fixed points of
$R$ in $E$ is an $R$-stable subspace that affords the reflection
representation of $R$.
Recall that a \emph{parabolic subgroup} of $W$ is a subgroup of the form
\[
W_V=\{\, w\in W\mid w(v)=v \ \, \forall v\in V\,\},
\]
where $V$ is a subspace of $E$. By Steinberg's Theorem~\cite[Thm.~1.5]{Steinberg64}, parabolic subgroups
are generated by the reflections they contain and so are reflection
subgroups.
For a subset $X$ of $W$ let
\[
\mathrm{Fix}(X)= \{\, v\in E\mid x(v)=v\ \, \forall x\in X\,\}
\]
denote the set of fixed points of $X$ in $E$.
Following Solomon~\cite{Solomon1976} and Bergeron et al.~\cite{BergeronEtAl1992},
we define the \emph{parabolic closure}
of $X$ to be the parabolic subgroup $\cl (X)= W_{\mathrm{Fix}(X)}$ of~$W$. Obviously
$X\subseteq \cl (X)$ and it follows from Steinberg's Theorem that $\cl(\cl(
X)) = \cl (X)$. When $X=\{w\}$ we simply write $\mathrm{Fix}(w)$ and $\cl(w)$ instead
of $\mathrm{Fix}(\{w\})$ and $\cl(\{w\})$, respectively.
For a discussion of parabolic closures of finitely generated
subgroups of arbitrary Coxeter systems, see the recent paper by Dyer~\cite{Dyer2010}.
For $w, x\in W$ we denote the
$w$-conjugate $w^{-1}xw$ of $x$ by $x^w$ and for a subset $X$ of $W$
let $X^w = \{x^w \mid x \in X\}$ denote the $w$-conjugate of $X$.
The \emph{rank} of a Coxeter group is the cardinality of a Coxeter
generating set, or equivalently, the dimension of its reflection
representation. It follows from the next lemma that every reflection
subgroup is a maximal rank reflection subgroup of its parabolic closure.
\begin{Lemma}
\label{lemma:rank}
Let $R$ be a reflection subgroup of $W$. Then $R$ and its parabolic
closure $\cl(R)$ have the same rank as Coxeter groups.
\end{Lemma}
\begin{proof}
The rank of $R$ is the codimension of its fixed point space
$\mathrm{Fix}(R)$. The rank of $\cl(R)$, as stabilizer of $\mathrm{Fix}(R)$, is not
larger than the rank of $R$, and, since $R \subseteq \cl(R)$, not
smaller than the rank of $R$ either.
\end{proof}
As noted in the Introduction, the classification of conjugacy classes of
reflection subgroups of $W$ reduces to (1) classifying conjugacy classes of
parabolic subgroups of $W$ and (2) classifying maximal rank reflection
subgroups of irreducible Coxeter groups.
The conjugacy classes of parabolic subgroups of an irreducible finite
Coxeter group are described in Chapter~2 and Appendix~A of
\cite{GeckPfeiffer2000} (see also \cite[Prop.\ 6.3]{BalaCarter1976}). In
most cases, two parabolic subgroups are conjugate if and only if they have
the same type. However, in type $D_{2m}$ there are two conjugacy classes of
parabolic subgroups of type $A_{k_1} \times A_{k_2} \times \cdots \times
A_{k_r}$ with all $k_i$ odd so that $2m = \sum (k_i +1)$ and in type $E_7$
there are two classes of parabolic subgroups for each of the types
$A_1^3$, $A_1 A_3$, and $A_5$.
For a given $W$, classifying the maximal rank reflection subgroups of $W$ up
to conjugacy amounts to listing, up to conjugacy in $W$, all subsets $Y$
of $T$ such that
\[
\Span{Y} \cap T=Y \qquad \text{and} \qquad \mathrm{Fix}(Y)=\{0\}.
\]
For a Coxeter group of small rank (including the non-crystallographic
types $H_3$ and $H_4$) these sets can be systematically enumerated. As
described below, for crystallographic Coxeter groups, that is, Weyl groups,
using the algorithm of Borel--De Siebenthal~\cite{BorelDeSiebenthal1949} is
computationally more efficient than classifying subsets of $T$.
By a \emph{root system} in $E$ we mean a reduced root system in the
sense of Bourbaki~\cite[Ch.~VI]{Bourbaki1968}. Suppose $\Phi$ is a root
system in $E$. The Weyl group of $\Phi$, $W(\Phi)$, is the group of linear
transformations of $E$ generated by the reflections through the hyperplanes
orthogonal to the roots in $\Phi$. The \emph{dual} of $\Phi$ is the root
system $\tilde \Phi=\{\ \frac 1 {|\alpha|^2} \alpha\mid \alpha \in
\Phi\,\}$. Note that $W(\Phi) = W(\tilde \Phi)$. By a \emph{Weyl group} or a
\emph{crystallographic Coxeter group} we mean the Weyl group of a root
system in $E$.
Suppose that $W=W(\Phi)=W(\tilde\Phi)$ is a Weyl group. We may extract a
classification of the maximal rank reflection subgroups of $W$ from the
arguments in \cite{Carter1972}. Each maximal rank reflection subgroup of $W$
is again a Weyl group and thus is the Weyl group of a maximal rank subsystem
of $\Phi$ or $\tilde \Phi$. By work of Dynkin, two maximal rank subsystems are
isomorphic if and only if they are equivalent under the action of $W$; see
\cite[Prop.\ 32]{Carter1972} or \cite[Ch.~VI, \S 4, ex.~4]{Bourbaki1968}. By
the classification of root systems, two root systems are isomorphic if and
only if they have the same Dynkin diagram. We have already observed that a
root system and its dual have the same Weyl group. Thus, the conjugacy
classes of maximal rank reflection subgroups of $W$ are in one-one
correspondence with the set of Coxeter graphs arising from Dynkin diagrams
of maximal rank subsystems of $\Phi$ and $\tilde \Phi$.
The Borel--De Siebenthal algorithm produces all maximal rank subsystems of
$\Phi$ and $\tilde \Phi$ as follows (see~\cite[p.~8]{Carter1972}).
\begin{enumerate}
\item Add a node to the Dynkin diagram of $\Phi$ corresponding to the
negative of the highest root of $\Phi$. Take the extended Dynkin diagram
and remove one node in all possible ways.
\item Add a node to the Dynkin diagram of $\tilde \Phi$ corresponding to the
negative of the highest root of $\tilde \Phi$. Take the extended Dynkin
diagram and remove one node in all possible ways.
\item Repeat steps (1) and (2) with each of the resulting Dynkin diagrams
until no new diagrams appear.
\end{enumerate}
This algorithm does not apply to the non-crystallographic groups $W(H_3)$,
$W(H_4)$ and $W(I_2(m))$, but these groups are sufficiently small that all relevant information can be calculated directly
We now turn to the map $\gamma$ which assigns to a given conjugacy class of
reflection subgroups of $W$ the conjugacy class of its Coxeter elements.
Recall~\cite[Ch.~V, \S6, no.~1]{Bourbaki1968} that a \emph{Coxeter
element} in $W$ is the product of the elements of some Coxeter
generating set of $W$ taken in some order. All Coxeter elements of
$W$ are conjugate in $W$.
Suppose that $R$ is a reflection subgroup of $W$. Then $R$ is a Coxeter
group and so we may consider Coxeter elements in $R$. If $c$ is a
Coxeter element in $R$ and $w$ is in $W$, it is easy to see that $c^w$
is a Coxeter element in $R^w$. Thus, conjugate reflection subgroups of $W$
have conjugate Coxeter elements and the map $\gamma$ is well-defined.
The proof of Theorem \ref{thm:main} follows immediately from the
classification of reflection subgroups and the explicit computation of the
map $\gamma$ in Theorems \ref{thm:a} and \ref{thm:d} and Tables \ref{tab:e6}
- \ref{tab:e8}. It would be interesting to have a conceptual explanation of
why the single exception occurs in Theorem \ref{thm:main}. More generally,
from Theorems \ref{thm:a}, \ref{thm:b}, and \ref{thm:d} along with Tables
\ref{tab:e6} - \ref{tab:h4}, we derive necessary and sufficient conditions
for the map $\gamma$ to be injective.
\begin{Theorem}
Suppose that $W$ is irreducible and not of type $E_8$. Then the map
$\gamma \colon \mathcal{R}\to \mathcal{C}$ is injective if and only if $T$ is a single
conjugacy class in~$W$.
If $W$ is of type $E_8$, then $\gamma$ is not injective: the
conjugacy classes of reflection subgroups of types $A_1 A_7$ and $A_3
D_5$ both map to the same conjugacy class of elements of $W$.
\end{Theorem}
Hence, we conclude that the map $\gamma$
is injective if and only if $W$ has type $A_n$, $D_n$,
$E_6$, $E_7$, $H_3$, $H_4$ and $I_2(m)$ with $m$ odd. Moreover, it follows from the
computations in \S \ref{sec:classical} - \S \ref{sec:brian} that, if $W$ is irreducible, then
$\gamma$ is surjective when $W$ has type $A_n$, $B_n$, or $G_2$. Notice that when the map
$\gamma$ is surjective, every
conjugacy class of $W$ contains a representative that is a Coxeter element
in some reflection subgroup of $W$.
It is easy to see that $\cl(X^w)= \cl(X)^w$ when $X\subseteq W$ and $w\in
W$. In particular, conjugate reflection subgroups have conjugate parabolic
closures. Similarly, conjugate elements in $W$ have conjugate parabolic
closures. In particular, if $c\in R$ and $c'\in R'$ are Coxeter
elements in reflection subgroups $R$ and $R'$, and $c$ and $c'$ are
conjugate in $W$, then $\cl(c)$ and $\cl(c')$ are conjugate in~$W$.
\begin{Lemma}
\label{la:7}
Suppose $R$ is a reflection subgroup of $W$ and $x\in R$ is not contained
in any proper parabolic subgroup of $R$. Then $\cl(x)=
\cl(R)$. Consequently, if $V$ is a subspace of $E$ and $c$ is a
Coxeter element of $R$ that is conjugate to an element of $W_V$, then
$R$ is conjugate to a subgroup of $W_V$.
\end{Lemma}
\begin{proof}
It is shown in \cite[\S2]{Carter1972} that
$\mathrm{Fix}(x)=\mathrm{Fix}(R)$. It follows immediately that $\cl(x)= \cl(R)$.
For the second statement, let $w \in W$ be such that $c^w$ is in
$W_V$. Then $\cl(c^w) \subseteq W_V$ and so $R^w\subseteq \cl(R)^w
=\cl(c)^w= \cl(c^w) \subseteq W_V$.
\end{proof}
Now suppose that $R$ and $R'$ are reflection subgroups of $W$ containing
Coxeter elements $c$ and $c'$ respectively. Then, if $c$ and $c'$ are
conjugate in $W$, $\cl(R)$ and $\cl(R')$ are conjugate. In other words,
even if $\gamma$ is not injective, reflection subgroups with non-conjugate
parabolic closures must have non-conjugate Coxeter elements. This
observation shows that conjugacy classes containing Coxeter elements of
reflection subgroups are separated by the parabolic closures of reflection
subgroups that contain them.
Note that Lemma~\ref{la:7} generalizes \cite[Lem.~7]{Solomon1976}
which is the special case of Lemma \ref{la:7} when $R$ is a parabolic
subgroup of $W$. In the same way, Theorem~\ref{thm:main} generalizes
the forward implication of~\cite[Lem.~3.5]{OrlikSolomon1983}.
\section{The classical Weyl groups}
\label{sec:classical}
A \emph{partition} $\lambda = (\lambda_1, \dots, \lambda_k)$ is a
non-increasing finite sequence of positive integers $\lambda_1 \geq
\dots \geq \lambda_k > 0$. The integers $\lambda_i$ are called the
\emph{parts} of the partition $\lambda$. If $\sum_{i=1}^k \lambda_i =
n$, then $\lambda$ is a partition of $n$ and we write
$\lambda \vdash n$. The unique partition of $n = 0$ is the \emph{empty
partition}, denoted by $\emptyset$. We denote by $\ell(\lambda) = k$
the length of the partition $\lambda = (\lambda_1, \dots, \lambda_k)$,
e.g., $\ell(\emptyset) = 0$.
A partition of $n$ is \emph{even}, if all its parts are even, i.e., if
it has the form $\lambda = (2 \mu_1, \dots, 2 \mu_k)$ for some
partition $\mu$ of $n/2$. The \emph{join} $\lambda^1 \cup \lambda^2$
of two partitions $\lambda^1 \vdash n_1$ and
$\lambda^2 \vdash n_2$ is the partition of $n_1 + n_2$ consisting
of the parts of both $\lambda^1$ and $\lambda^2$, suitably arranged.
The \emph{sum} of a partition $\lambda = (\lambda_1, \dots,
\lambda_k)$ and an integer $m$ is the partition $\lambda + m =
(\lambda_1 + m, \dots, \lambda_k+m)$. We write $\lambda > m$ if
$\lambda_i > m$ for all parts $\lambda_i$ of $\lambda$.
Note that, vacuously, $\emptyset > m$ for all $m$.
The symmetric group $\mathfrak{S}_n$ on $n$ points is a Coxeter group of type
$A_{n-1}$ with Coxeter generators $s_i = (i, i+1)$, $i = 1, \dots, n-1$.
The \emph{cycle type} of $w \in \mathfrak{S}_n$ is the partition $\lambda$ of $n$
which has the lengths of the cycles of $w$ on $\{1, \dots, n\}$ as its parts
(here a fixed point contributes a cycle of length $1$). Of course, two
permutations in $\mathfrak{S}_n$ are conjugate if and only if they have the same
cycle type. The next theorem is well-known, and can easily be deduced from
Bourbaki~\cite[Ch.~VI, \S 4, ex.~4]{Bourbaki1968}.
\begin{Theorem}
\label{thm:a}
Let $W$ be a Coxeter group of type $A_n$. Then every reflection subgroup of
$W$ is a parabolic subgroup. Moreover, the map $\gamma$ from conjugacy
classes of reflection subgroups to conjugacy classes of $W$ is a
bijection. Both sets are in one-to-one correspondence with the set of all
partitions of $n$.
\end{Theorem}
An \emph{$r$-partition} is a sequence $\lambda = (\lambda^1, \dots,
\lambda^r)$ of $r$ partitions $\lambda^1, \dots, \lambda^r$. We say that
$\lambda$ is an $r$-partition of the integer $n$, and write $\lambda
\rpartitions{r} n$, if $\lambda^1 \cup \dots \cup \lambda^r \vdash n$.
We call $\lambda$ a \emph{double partition} if $r = 2$, and a \emph{triple
partition} if $r = 3$.
The Coxeter group $W(B_n)$ acts faithfully as a group of signed permutations
on the set of long roots $\{\pm e_i \mid i = 1, \dots, n\}$, permuting the
lines $\Span{e_i}$, $i = 1, \dots, n$. A cycle of $w$ in $W(B_n)$ is either
\emph{positive} or \emph{negative}, depending on whether the number of
positive roots $e_i$ with $\Span{e_i}$ in the cycle that are mapped to
negative roots is even or odd. The \emph{cycle type} of $w$ in $W(B_n)$ is a
double partition $\lambda = (\lambda^1, \lambda^2)$ of $n$, where
$\lambda^1$ records the lengths of the positive cycles of $w$ and
$\lambda^2$ records the lengths of the negative cycles. Again, two elements
of $W(B_n)$ are conjugate if and only if they have the same cycle type and,
in this way, the double partitions of $n$ naturally parametrize the conjugacy
classes of $W(B_n)$.
According to \cite[Prop.~2.3.10]{GeckPfeiffer2000}, the parabolic
subgroups of $W(B_n)$ are of the form $W(B_{n-m}) \times \prod_i
W(A_{\lambda_i-1})$, one conjugacy class for each partition
$\lambda \vdash m$, $0 \leq m \leq n$. By Borel--De Siebenthal,
the maximal rank reflection subgroups of $W(B_n)$ are of type $\prod_i
W(B_{\lambda^1_i}) \times \prod_i W(D_{\lambda^2_i})$, one class for
each double partition $\lambda \vdash^2 n$ with $\lambda^2 > 1$
(or $\lambda^2 = \emptyset$). It follows that the reflection
subgroups of $W(B_n)$ are direct products of Coxeter groups of types
$A$, $B$ and $D$, and their classes are naturally labeled by triple
partitions of~$n$.
\begin{Theorem}
\label{thm:b}
Let $W$ be a Coxeter group of type $B_n$, $n \ge 2$. Then the conjugacy classes of
reflection subgroups of $W$ are represented by
\begin{align*}
\{W_{\lambda} \mid \lambda \rpartitions{3} n,\, \lambda^3 > 1\},
\end{align*}
where $W_{\lambda} = \prod_i W(A_{\lambda^1_i-1}) \times \prod_i
W(B_{\lambda^2_i}) \times \prod_i W(D_{\lambda^3_i})$. The parabolic
closure of $W_{\lambda}$ has type $W(B_{n-m}) \times \prod_i
W(A_{\lambda^1_i-1})$, where $\lambda^1 \vdash m$. The Coxeter
elements of $W_{\lambda}$ have cycle type $(\lambda^1, \lambda^2 \cup
(\lambda^3-1) \cup 1^{\ell(\lambda^3)})$. In particular, the map $\gamma
\colon \mathcal{R} \to \mathcal{C}$ is surjective, but not injective.
\end{Theorem}
We illustrate the classification in type $B_n$ in Table \ref{tab:b5} below,
where
we list all conjugacy classes of reflection subgroups of $W(B_5)$ according
to Theorem \ref{thm:b}. Clearly, it follows from the data in Table
\ref{tab:b5} that $\gamma$ is not injective in this case.
The Coxeter group $W(D_n)$ is a normal subgroup of index $2$ in $W(B_n)$,
and as such it is a union of $W(B_n)$-conjugacy classes of elements. In
fact, the class of elements of cycle type $\lambda = (\lambda^1, \lambda^2)$
is contained in $W(D_n)$ if and only if $\ell(\lambda^2)$ is even, and it is
a single conjugacy class in $W(D_n)$, unless $\lambda^2 = \emptyset$ and
$\lambda^1$ is even. In the latter case, the $W(B_n)$-class splits into two
$W(D_n)$-classes, labelled $(\lambda^1, +)$ and $(\lambda^1,-)$. In this
way, the conjugacy classes of $W(D_n)$ are parametrized by certain
double partitions
of $n$.
According to \cite[Prop.~2.3.13]{GeckPfeiffer2000}, $W(D_n)$ has three
distinct kinds of parabolic subgroups: one class of subgroups of type
$W(D_{n-m}) \times \prod_i W(A_{\lambda_i-1})$ for each partition
$\lambda \vdash m$, $0 \leq m \leq n - 2$, two classes of subgroups of
type $\prod_i W(A_{\lambda_i-1})$ for each even partition
$\lambda \vdash n$, and one class of subgroups of type $\prod_i
W(A_{\lambda_i-1})$ for each non-even partition $\lambda \vdash n$.
By Borel--De Siebenthal, the maximal rank reflection
subgroups of $W(D_n)$ are of type $\prod_i W(D_{\lambda_i})$, one class for
each partition $\lambda \vdash n$ with $\lambda > 1$.
It follows that
reflection subgroups of $W(D_n)$ are direct products of Coxeter groups of
types $A$ and $D$, and their classes are naturally labeled by double
partitions of~$n$.
This yields the
following classification of the conjugacy classes of reflection subgroups of
$W(D_n)$, in terms of double partitions of $n$.
\begin{Theorem}
\label{thm:d}
Let $W$ be a Coxeter group of type $D_n$, $n \geq 4$. Then the conjugacy
classes of reflection subgroups of $W$ are represented by
\begin{align*}
\{W_{\lambda} \mid \lambda \rpartitions{2} n,\, \lambda^2 > 1\}
\end{align*}
if $n$ is odd, and by
\begin{align*}
\{W_{\lambda} \mid \lambda \rpartitions{2} n,\, \lambda^2 > 1 \text{ and }
\lambda^1 \text{ non-even in case } \lambda^2 = \emptyset\} \cup
\{W_{\lambda}^{\pm} \mid \lambda \vdash n \text{ and } \lambda \text{ even}\}
\end{align*}
if $n$ is even, where $W_{\lambda} = \prod_i W(A_{\lambda^1_i-1}) \times
\prod_i W(D_{\lambda^2_i})$ and $W_{\lambda}^{\epsilon} = \prod_i
W(A_{\lambda_i-1})$, where $\epsilon = \pm$.
The parabolic closure of $W_{\lambda}$ has type
$W(D_{n-m}) \times \prod_i W(A_{\lambda^1_i-1})$, where
$\lambda^1 \vdash m$; the parabolic closure of $W_{\lambda}^{\epsilon}$
is $W_{\lambda}^{\epsilon}$ itself. The Coxeter elements of $W_{\lambda}$
have cycle type $(\lambda^1, (\lambda^2-1) \cup 1^{\ell(\lambda^2)})$; the
Coxeter elements of $W_{\lambda}^{\epsilon}$ have cycle type $(\lambda,
\epsilon)$. In particular, the map $\gamma \colon \mathcal{R} \to \mathcal{C}$ is
injective, but not surjective.
\end{Theorem}
We illustrate the classification in type $D_n$ from Theorem \ref{thm:d} for
$n = 6$ in Table \ref{tab:d6} below.
\section{The exceptional Weyl groups}
\label{sec:keith}
For the exceptional Weyl groups all results are obtained by following the
recursive procedure outlined in \S\ref{sec:mick}, using the Borel--De
Siebenthal algorithm for the various factors of each standard parabolic
subgroup of $W$. The calculations were carried out with the use of {\sf
GAP}~\cite{GAP} and {\sf CHEVIE}~\cite{chevie}. Here the conjugacy classes
of the elements in $W$ are labeled as in Carter~\cite{Carter1972}.
In Tables \ref{tab:e6} - \ref{tab:g2} we list all reflection subgroups in
case $W$ is of exceptional type up to conjugacy. In the cases when $W$ has
only a single class of reflections, it is readily checked that $\gamma$ is
injective, as required for Theorem \ref{thm:main}.
Table \ref{tab:e8} contains the results for $W(E_8)$. Here the two maximal
rank reflection subgroups of types $A_1 A_7$ and $A_3 D_5$ have Coxeter
elements that are conjugate in $W$. Hence $\gamma$ is not injective.
In Tables \ref{tab:f4} and \ref{tab:g2} we list all conjugacy classes of
reflection subgroups of $W(F_4)$ and $W(G_2)$, respectively. In both
instances we see that $\gamma$ is not injective.
\section{The non-crystallographic cases}
\label{sec:brian}
As in the exceptional cases, the non-crystallographic instances were
computed using {\sf GAP}~\cite{GAP} and {\sf CHEVIE}~\cite{chevie}. The
Borel--De Siebenthal algorithm does not apply, but these groups are
sufficiently small that all the relevant information can be calculated
directly.
In Tables \ref{tab:h3} and \ref{tab:h4} we list all conjugacy classes of
reflection subgroups of $W(H_3)$ and $W(H_4)$, respectively. Here we see
that $\gamma$ is injective in both cases. The labeling of the conjugacy
classes is the one used by {\sf CHEVIE}.
The reflection subgroups of the dihedral group $W(I_2(m))$ can be described
as follows.
\begin{Theorem}\label{thm:i2}
Let $W$ be of type $I_2(m)$, $m = 5$ or $m >6$.
\begin{enumerate}
\item If $m$ is odd then the classes of reflection subgroups of $W$ are of
types $\emptyset$, $A_1$, and $I_2(d)$ where $d > 1$ is a divisor of
$m$. The map $\gamma$ is injective, but not surjective.
\item If $m$ is even then the classes of reflection subgroups of $W$ are
of types $\emptyset$, $A_1$, $\tilde{A}_1$, $I_2(d)$ where $d > 1$ is a
divisor of $m$ and $\tilde{I}_2(d)$ where $2d > 2$ is a divisor of
$m$. The map $\gamma$ is neither injective nor surjective.
\end{enumerate}
\end{Theorem}
The subgroups of a dihedral group are determined by a
straightforward computation. The theorem follows by filtering
out those subgroups that are generated by reflections.
\section{Tables}
In Tables \ref{tab:b5} - \ref{tab:h4} we present the classification of the
reflection subgroups of $W$ in various cases. The tables provide the
following information. In the first column of each table we list the types
of the reflection subgroups $R$ of $W$. In the second column in Tables
\ref{tab:b5} and \ref{tab:d6} we also give the partition representing $R$
according to Theorems \ref{thm:b} and \ref{thm:d}, respectively. The next
two columns give the cardinality of $R$ and the cardinality of the class
$[R]$ of $R$ (that is, $|W:N_W(R)|$). Finally, in the last column we list the
image of $\gamma$, i.e.\ the class $[c]$ of a Coxeter element $c$ of $R$ in
$W$. For the classical types, conjugacy classes are labeled by cycle
type. For the exceptional types, conjugacy classes are labeled as in
Carter's classification~\cite{Carter1972}.
Conjugacy classes of reflection subgroups with distinct parabolic closures
are separated by horizontal lines. For a given parabolic subgroup $P$ of
$W$, the row for $P$ is preceded by a horizontal line and followed by the
rows for reflection subgroups $R$ of $W$ with $\cl(R) = P$.
\begin{table}[p]
\bigskip
\extrarowheight2pt
\caption{Reflection subgroups of $W(B_5)$.}
\label{tab:b5}
\begin{tabular}[t]{ccrrc}\toprule
Type of $R$ & $\lambda$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[5pt] \toprule
$\emptyset$ & $1^5..$& 1 & 1 & $1^5.$\\
\noalign{\vglue 2pt} \hline
$B_1$ & $1^4.1.$ & 2 & 5 & $1^4.1$\\
\noalign{\vglue 2pt} \hline
$A_1$ & $21^3..$ & 2 & 20 & $21^3.$\\
\noalign{\vglue 2pt} \hline
$B_1 A_1$ & $21^2.1.$ & 4 & 60 & $21^2.1$\\
\noalign{\vglue 2pt} \hline
$A_1^{2}$ & $2^21..$ & 4 & 60 & $2^21.$\\
\noalign{\vglue 2pt} \hline
$A_2$ & $31^2..$ & 6 & 40 & $31^2.$\\
\noalign{\vglue 2pt} \hline
$B_2$ & $1^3.2.$ & 8 & 10 & $1^3.2$\\
$B_1^{2}$ & $1^3.1^2.$ & 4 & 10 & $1^3.1^2$\\
$D_2$ & $1^3..2$ & 4 & 10 & $1^3.1^2$\\
\noalign{\vglue 2pt} \hline
$B_1 A_1^{2}$ & $2^2.1.$ & 8 & 60 & $2^2.1$\\
\noalign{\vglue 2pt} \hline
$B_1 A_2$ & $31.1.$ & 12 & 80 & $31.1$\\
\noalign{\vglue 2pt} \hline
$A_1 A_2$ & $32..$ & 12 & 80 & $32.$\\
\noalign{\vglue 2pt} \hline
$B_2 A_1$ & $21.2.$ & 16 & 60 & $21.2$\\
$B_1^{2} A_1$ & $21.1^2.$ & 8 & 60 & $21.1^2$\\
$D_2 A_1$ & $21..2$ & 8 & 60 & $21.1^2$\\
\noalign{\vglue 2pt} \hline
$A_3$ & $41..$ & 24 & 40 & $41.$\\
\noalign{\vglue 2pt} \hline
$B_3$ & $1^2.3. $ & 48 & 10 & $1^2.3$\\
$B_1 B_2$ & $1^2.21.$ & 16 & 30 & $1^2.21$\\
$D_3$ & $1^2..3$ & 24 & 10 & $1^2.21$\\
$D_2 B_1$ & $1^2.1.2$ & 8 & 30 & $1^2.1^3$\\
$B_1^{3}$ & $1^2.1^3.$ & 8 & 10 & $1^2.1^3$\\
\noalign{\vglue 2pt} \hline
$B_2 A_2$ & $3.2.$ & 48 & 40 & $3.2$\\
$B_1^{2} A_2$ & $3.1^2.$ & 24 & 40 & $3.1^2$\\
$D_2 A_2$ & $3..2$ & 24 & 40 & $3.1^2$\\
\noalign{\vglue 2pt} \hline
$B_1 A_3$ & $4.1.$ & 48 & 40 & $4.1$\\
\noalign{\vglue 2pt} \hline
$B_3 A_1$ & $2.3.$ & 96 & 20 & $2.3$\\
$B_1 B_2 A_1$ & $2.21.$ & 32 & 60 & $2.21$\\
$D_3 A_1$ & $2..3$ & 48 & 20 & $2.21$\\
$D_2 B_1 A_1$ & $2.1.2$ & 16 & 60 & $2.1^3$\\
$B_1^{3} A_1$ & $2.1^3.$ & 16 & 20 & $2.1^3$\\
\bottomrule
\end{tabular}
\qquad
\begin{tabular}[t]{ccrrc}\toprule
Type of $R$ & $\lambda$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[5pt] \toprule
$A_4$ & $5..$ & 120 & 16 & $5.$\\
\noalign{\vglue 2pt} \hline
$B_4$ & $1.4.$ & 384 & 5 & $1.4$\\
$B_1 B_3$ & $1.31.$ & 96 & 20 & $1.31$\\
$B_2^{2}$ & $1.2^2.$ & 64 & 15 & $1.2^2$\\
$D_4$ & $1..4$ & 192 & 5 & $1.31$\\
$D_3 B_1$ & $1.1.3$ & 48 & 20 & $1.21^2$\\
$D_2 B_2$ & $1.2.2$ & 32 & 30 & $1.21^2$\\
$B_1^{2} B_2$ & $1.21^2.$ & 32 & 30 & $1.21^2$\\
$D_2 B_1^{2}$ & $1.1^2.2$ & 16 & 30 & $1.1^4$\\
$B_1^{4}$ & $1.1^4.$ & 16 & 5 & $1.1^4$\\
$D_2^{2}$ & $1..2^2$ & 16 & 15 & $1.1^4$\\
\noalign{\vglue 2pt} \hline
$B_5$ & $.5.$ & 3840 & 1 & $.5$\\
$B_1 B_4$ & $.41.$ & 768 & 5 & $.41$\\
$B_2 B_3$ & $.32.$ & 384 & 10 & $.32$\\
$D_5$ & $..5$ & 1920 & 1 & $.41$\\
$D_4 B_1$ & $.1.4$ & 384 & 5 & $.31^2$\\
$D_3 B_2$ & $.2.3$ & 192 & 10 & $.2^21$\\
$D_2 B_3$ & $.3.2$ & 192 & 10 & $.31^2$\\
$B_1^{2} B_3$ & $.31^2.$ & 192 & 10 & $.31^2$\\
$B_1 B_2^{2}$ & $.2^21.$ & 128 & 15 & $.2^21$\\
$D_3 B_1^{2}$ & $.1^2.3$ & 96 & 10 & $.21^3$\\
$D_2 B_1 B_2$ & $.21.2$ & 64 & 30 & $.21^3$\\
$B_1^{3} B_2$ & $.21^3.$ & 64 & 10 & $.21^3$\\
$D_2 B_1^{3}$ & $.1^3.2$ & 32 & 10 & $.1^5$\\
$D_2^{2} B_1$ & $.1.2^2$ & 32 & 15 & $.1^5$\\
$D_2 D_3$ & $..23$ & 96 & 10 & $.21^3$\\
$B_1^{5}$ & $.1^5.$ & 32 & 1 & $.1^5$\\
\bottomrule
\end{tabular}
\bigskip
\end{table}
\begin{table}[p]
\bigskip
\small
\extrarowheight1pt
\caption{Reflection subgroups of $W(D_6)$.}
\label{tab:d6}
\begin{tabular}[t]{ccrrc}\toprule
Type of $R$ & $\lambda$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[4pt] \toprule
$\emptyset$ & $1^6.$ & 1 & 1 & $1^6.$\\
\noalign{\vglue 1pt} \hline
$A_1$ & $21^4.$ & 2 & 30 & $21^4.$\\
\noalign{\vglue 1pt} \hline
$D_2$ & $1^4.2$ & 4 & 15 & $1^4.1^2$\\
\noalign{\vglue 1pt} \hline
$A_1^{2}$ & $2^21^2.$ & 4 & 180 & $2^21^2.$\\
\noalign{\vglue 1pt} \hline
$A_2$ & $31^3.$ & 6 & 80 & $31^3.$\\
\noalign{\vglue 1pt} \hline
$A_1^{3}$ & $2^3.+$ & 8 & 60 & $2^3.+$\\
\noalign{\vglue 1pt} \hline
$A_1^{3}$ & $2^3.-$ & 8 & 60 & $2^3.-$\\
\noalign{\vglue 1pt} \hline
$D_2 A_1$ & $21^2.2$ & 8 & 180 & $21^2.1^2$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2$ & $321.$ & 12 & 480 & $321.$\\
\noalign{\vglue 1pt} \hline
$D_3$ & $1^3.3$ & 24 & 20 & $1^3.21$\\
\noalign{\vglue 1pt} \hline
$A_3$ & $41^2.$ & 24 & 120 & $41^2.$\\
\noalign{\vglue 1pt} \hline
$D_2 A_1^{2}$ & $2^2.2$ & 16 & 180 & $2^2.1^2$\\
\noalign{\vglue 1pt} \hline
$D_2 A_2$ & $31.2$ & 24 & 240 & $31.1^2$\\
\noalign{\vglue 1pt} \hline
$A_2^{2}$ & $33.$ & 36 & 160 & $33.$\\
\noalign{\vglue 1pt} \hline
$D_3 A_1$ & $21.3$ & 48 & 120 & $21.21$\\
\noalign{\vglue 1pt} \hline
$A_1 A_3$ & $42.+$ & 48 & 120 & $42.+$\\
\noalign{\vglue 1pt} \hline
$A_1 A_3$ & $42.-$ & 48 & 120 & $42.-$\\
\bottomrule
\end{tabular}
\qquad
\begin{tabular}[t]{ccrrc}\toprule
Type of $R$ & $\lambda$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[4pt]\toprule
$A_4$ & $51.$ & 120 & 96 & $51.$\\
\noalign{\vglue 1pt} \hline
$D_4$ & $1^2.4$ & 192 & 15 & $1^2.31$\\
$D_2^{2}$ & $1^2.2^2$ & 16 & 45 & $1^2.1^4$\\
\noalign{\vglue 1pt} \hline
$D_2 A_3$ & $4.2$ & 96 & 120 & $4.1^2$\\
\noalign{\vglue 1pt} \hline
$D_3 A_2$ & $3.3$ & 144 & 80 & $3.21$\\
\noalign{\vglue 1pt} \hline
$D_4 A_1$ & $2.4$ & 384 & 30 & $2.31$\\
$D_2^2 A_1$ & $2.2^2$ & 32 & 90 & $2.1^4$\\
\noalign{\vglue 1pt} \hline
$A_5$ & $6.+$ & 720 & 16 & $6.+$\\
\noalign{\vglue 1pt} \hline
$A_5$ & $6.-$ & 720 & 16 & $6.-$\\
\noalign{\vglue 1pt} \hline
$D_5$ & $1.5$ & 1920 & 6 & $1.41$\\
$D_2 D_3$ & $1.32$ & 96 & 60 & $1.21^3$\\
\noalign{\vglue 1pt} \hline
$D_6$ & $.6$ & 23040 & 1 & $.51$\\
$D_2 D_4$ & $.42$ & 768 & 15 & $.31^3$\\
$D_3^{2}$ & $.3^2$ & 576 & 10 & $.2^21^2$\\
$D_2^{3}$ & $.2^3$ & 64 & 15 & $.1^6$\\
\bottomrule
\end{tabular}
\bigskip
\end{table}
\begin{table}[p]
\bigskip
\small
\extrarowheight1pt
\caption{Reflection subgroups of $W(E_6)$.}
\label{tab:e6}
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[4pt]
\toprule
$\emptyset$ & 1 & 1 & $1$\\
\noalign{\vglue 1pt} \hline
$A_1$ & 2 & 36 & $A_1$\\
\noalign{\vglue 1pt} \hline
$A_1^{2}$ & 4 & 270 & $2A_1$\\
\noalign{\vglue 1pt} \hline
$A_2$ & 6 & 120 & $A_2$\\
\noalign{\vglue 1pt} \hline
$A_1^{3}$ & 8 & 540 & $3A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2$ & 12 & 720 & $A_2+A_1$\\
\noalign{\vglue 1pt} \hline
$A_3$ & 24 & 270 & $A_3$\\
\noalign{\vglue 1pt} \hline
$A_1^{2} A_2$ & 24 & 1080 & $A_2+2A_1$\\
\noalign{\vglue 1pt} \hline
$A_2^{2}$ & 36 & 120 & $2A_2$\\
\noalign{\vglue 1pt} \hline
$A_1 A_3$ & 48 & 540 & $A_3+A_1$\\
\noalign{\vglue 1pt} \hline
$A_4$ & 120 & 216 & $A_4$\\
\bottomrule
\end{tabular}
\qquad
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[4pt] \toprule
$D_4$ & 192 & 45 & $D_4$\\
$A_1^{4}$ & 16 & 135 & $4A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2^{2}$ & 72 & 360 & $2A_2+A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 A_4$ & 240 & 216 & $A_4+A_1$\\
\noalign{\vglue 1pt} \hline
$A_5$ & 720 & 36 & $A_5$\\
\noalign{\vglue 1pt} \hline
$D_5$ & 1920 & 27 & $D_5$\\
$A_1^{2} A_3$ & 96 & 270 & $A_3+2A_1$\\
\noalign{\vglue 1pt} \hline
$E_6$ & 51840 & 1 & $E_6$\\
$A_1 A_5$ & 1440 & 36 & $A_5+A_1$\\
$A_2^{3}$ & 216 & 40 & $3A_2$\\
\bottomrule
\end{tabular}
\bigskip
\end{table}
\begin{table}[p]
\bigskip
\small
\extrarowheight1pt
\caption{Reflection subgroups of $W(E_7)$.}
\label{tab:e7}
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[4pt] \toprule
$\emptyset$ & 1 & 1 & $1$\\
\noalign{\vglue 1pt} \hline
$A_1$ & 2 & 63 & $A_1$\\
\noalign{\vglue 1pt} \hline
$A_1^{2}$ & 4 & 945 & $2A_1$\\
\noalign{\vglue 1pt} \hline
$A_2$ & 6 & 336 & $A_2$\\
\noalign{\vglue 1pt} \hline
$A_1^{3}$ & 8 & 315 & $3A_1'$\\
\noalign{\vglue 1pt} \hline
$A_1^{3}$ & 8 & 3780 & $3A_1''$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2$ & 12 & 5040 & $A_2+A_1$\\
\noalign{\vglue 1pt} \hline
$A_3$ & 24 & 1260 & $A_3$\\
\noalign{\vglue 1pt} \hline
$A_1^{4}$ & 16 & 3780 & $4A_1'$\\
\noalign{\vglue 1pt} \hline
$A_1^{2} A_2$ & 24 & 15120 & $A_2+2A_1$\\
\noalign{\vglue 1pt} \hline
$A_2^{2}$ & 36 & 3360 & $2A_2$\\
\noalign{\vglue 1pt} \hline
$A_1 A_3$ & 48 & 1260 & $A_3+A_1'$\\
\noalign{\vglue 1pt} \hline
$A_1 A_3$ & 48 & 7560 & $A_3+A_1''$\\
\noalign{\vglue 1pt} \hline
$A_4$ & 120 & 2016 & $A_4$\\
\noalign{\vglue 1pt} \hline
$D_4$ & 192 & 315 & $D_4$\\
$A_1^{4}$ & 16 & 945 & $4A_1''$\\
\noalign{\vglue 1pt} \hline
$A_1^{3} A_2$ & 48 & 5040 & $A_2+3A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2^{2}$ & 72 & 10080 & $2A_2+A_1$\\
\noalign{\vglue 1pt} \hline
$A_1^{2} A_3$ & 96 & 7560 & $A_3+2A_1'$\\
\noalign{\vglue 1pt} \hline
$A_2 A_3$ & 144 & 5040 & $A_3+A_2$\\
\noalign{\vglue 1pt} \hline
$A_1 A_4$ & 240 & 6048 & $A_4+A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 D_4$ & 384 & 945 & $D_4+A_1$\\
$A_1^{5}$ & 32 & 2835 & $5A_1$\\
\bottomrule
\end{tabular}
\qquad
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[4pt] \toprule
$A_5$ & 720 & 336 & $A_5'$\\
\noalign{\vglue 1pt} \hline
$A_5$ & 720 & 1008 & $A_5''$\\
\noalign{\vglue 1pt} \hline
$D_5$ & 1920 & 378 & $D_5$\\
$A_1^{2} A_3$ & 96 & 3780 & $A_3+2A_1''$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2 A_3$ & 288 & 5040 & $A_3+A_2+A_1$\\
\noalign{\vglue 1pt} \hline
$A_2 A_4$ & 720 & 2016 & $A_4+A_2$\\
\noalign{\vglue 1pt} \hline
$A_1 A_5$ & 1440 & 1008 & $A_5+A_1'$\\
\noalign{\vglue 1pt} \hline
$A_1 D_5$ & 3840 & 378 & $D_5+A_1$\\
$A_1^{3} A_3$ & 192 & 3780 & $A_3+3A_1$\\
\noalign{\vglue 1pt} \hline
$A_6$ & 5040 & 288 & $A_6$\\
\noalign{\vglue 1pt} \hline
$D_6$ & 23040 & 63 & $D_6$\\
$A_1^{2} D_4$ & 768 & 945 & $D_4+2A_1$\\
$A_3^{2}$ & 576 & 630 & $D_4(a_1)+2A_1$\\
$A_1^{6}$ & 64 & 945 & $6A_1$\\
\noalign{\vglue 1pt} \hline
$E_6$ & 51840 & 28 & $E_6$\\
$A_1 A_5$ & 1440 & 1008 & $A_5+A_1''$\\
$A_2^{3}$ & 216 & 1120 & $3A_2$\\
\noalign{\vglue 1pt} \hline
$E_7$ & 2903040 & 1 & $E_7$\\
$A_1 D_6$ & 46080 & 63 & $D_6+A_1$\\
$A_7$ & 40320 & 36 & $A_7$\\
$A_2 A_5$ & 4320 & 336 & $A_5+A_2$\\
$A_1 A_3^{2}$ & 1152 & 630 & $2A_3+A_1$\\
$A_1^{3} D_4$ & 1536 & 315 & $D_4+3A_1$\\
$A_1^{7}$ & 128 & 135 & $7A_1$\\
\bottomrule
\end{tabular}
\bigskip
\end{table}
\begin{table}[p]
\bigskip
\footnotesize
\extrarowheight1pt
\caption{Reflection subgroups of $W(E_8)$.}
\label{tab:e8}
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[4pt] \toprule
$\emptyset$ & 1 & 1 & $1$\\
\noalign{\vglue 1pt} \hline
$A_1$ & 2 & 120 & $A_1$\\
\noalign{\vglue 1pt} \hline
$A_1^{2}$ & 4 & 3780 & $2A_1$\\
\noalign{\vglue 1pt} \hline
$A_2$ & 6 & 1120 & $A_2$\\
\noalign{\vglue 1pt} \hline
$A_1^{3}$ & 8 & 37800 & $3A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2$ & 12 & 40320 & $A_2+A_1$\\
\noalign{\vglue 1pt} \hline
$A_3$ & 24 & 7560 & $A_3$\\
\noalign{\vglue 1pt} \hline
$A_1^{4}$ & 16 & 113400 & $4A_1''$\\
\noalign{\vglue 1pt} \hline
$A_1^{2} A_2$ & 24 & 302400 & $A_2+2A_1$\\
\noalign{\vglue 1pt} \hline
$A_2^{2}$ & 36 & 67200 & $2A_2$\\
\noalign{\vglue 1pt} \hline
$A_1 A_3$ & 48 & 151200 & $A_3+A_1$\\
\noalign{\vglue 1pt} \hline
$A_4$ & 120 & 24192 & $A_4$\\
\noalign{\vglue 1pt} \hline
$D_4$ & 192 & 3150 & $D_4$\\
$A_1^{4}$ & 16 & 9450 & $4A_1'$\\
\noalign{\vglue 1pt} \hline
$A_1^{3} A_2$ & 48 & 604800 & $A_2+3A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2^{2}$ & 72 & 403200 & $2A_2+A_1$\\
\noalign{\vglue 1pt} \hline
$A_1^{2} A_3$ & 96 & 453600 & $A_3+2A_1''$\\
\noalign{\vglue 1pt} \hline
$A_2 A_3$ & 144 & 302400 & $A_3+A_2$\\
\noalign{\vglue 1pt} \hline
$A_1 A_4$ & 240 & 241920 & $A_4+A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 D_4$ & 384 & 37800 & $D_4+A_1$\\
$A_1^{5}$ & 32 & 113400 & $5A_1$\\
\noalign{\vglue 1pt} \hline
$A_5$ & 720 & 40320 & $A_5$\\
\noalign{\vglue 1pt} \hline
$D_5$ & 1920 & 7560 & $D_5$\\
$A_1^{2} A_3$ & 96 & 75600 & $A_3+2A_1'$\\
\noalign{\vglue 1pt} \hline
$A_1^{2} A_2^{2}$ & 144 & 604800 & $2A_2+2A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2 A_3$ & 288 & 604800 & $A_3+A_2+A_1$\\
\noalign{\vglue 1pt} \hline
$A_1^{2} A_4$ & 480 & 362880 & $A_4+2A_1$\\
\noalign{\vglue 1pt} \hline
$A_3^{2}$ & 576 & 151200 & $2A_3''$\\
\noalign{\vglue 1pt} \hline
$A_2 A_4$ & 720 & 241920 & $A_4+A_2$\\
\noalign{\vglue 1pt} \hline
$A_2 D_4$ & 1152 & 50400 & $D_4+A_2$\\
$A_1^{4} A_2$ & 96 & 151200 & $A_2+4A_1$\\
\noalign{\vglue 1pt} \hline
$A_1 A_5$ & 1440 & 120960 & $A_5+A_1''$\\
\noalign{\vglue 1pt} \hline
$A_1 D_5$ & 3840 & 45360 & $D_5+A_1$\\
$A_1^{3} A_3$ & 192 & 453600 & $A_3+3A_1$\\
\noalign{\vglue 1pt} \hline
$A_6$ & 5040 & 34560 & $A_6$\\
\noalign{\vglue 1pt} \hline
$D_6$ & \llap{23040} & 3780 & $D_6$\\
$A_1^{2} D_4$ & 768 & 56700 & $D_4+2A_1$\\
$A_3^{2}$ & 576 & 37800 & $2A_3'$\\
$A_1^{6}$ & 64 & 56700 & $6A_1$\\
\bottomrule
\end{tabular}
\qquad
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[4pt] \toprule
$E_6$ & 51840 & 1120 & $E_6$\\
$A_1 A_5$ & 1440 & 40320 & $A_5+A_1'$\\
$A_2^{3}$ & 216 & 44800 & $3A_2$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2 A_4$ & 1440 & 241920 & $A_4+A_2+1$\\
\noalign{\vglue 1pt} \hline
$A_3 A_4$ & 2880 & 120960 & $A_4+A_3$\\
\noalign{\vglue 1pt} \hline
$A_1 A_6$ & 10080 & 34560 & $A_6+A_1$\\
\noalign{\vglue 1pt} \hline
$A_2 D_5$ & 11520 & 30240 & $D_5+A_2$\\
$A_1^{2} A_2 A_3$ & 576 & 302400 & $A_3+A_2+2A_1$\\
\noalign{\vglue 1pt} \hline
$A_7$ & 40320 & 8640 & $A_7''$\\
\noalign{\vglue 1pt} \hline
$A_1 E_6$ & 103680 & 3360 & $E_6+A_1$\\
$A_1^{2} A_5$ & 2880 & 120960 & $A_5+2A_1$\\
$A_1 A_2^{3}$ & 432 & 134400 & $3A_2+A_1$\\
\noalign{\vglue 1pt} \hline
$D_7$ & 322560 & 1080 & $D_7$\\
$A_1^{2} D_5$ & 7680 & 22680 & $D_5+2A_1$\\
$A_3 D_4$ & 4608 & 37800 & $D_4+A_3$\\
$A_1^{4} A_3$ & 384 & 113400 & $A_3+4A_1$\\
\noalign{\vglue 1pt} \hline
$E_7$ & 2903040 & 120 & $E_7$\\
$A_1 D_6$ & 46080 & 7560 & $D_6+A_1$\\
$A_7$ & 40320 & 4320 & $A_7'$\\
$A_2 A_5$ & 4320 & 40320 & $A_5+A_2$\\
$A_1 A_3^{2}$ & 1152 & 75600 & $2A_3+A_1$\\
$A_1^{3} D_4$ & 1536 & 37800 & $D_4+3A_1$\\
$A_1^{7}$ & 128 & 16200 & $7A_1$\\
\noalign{\vglue 1pt} \hline
$E_8$ & \llap{696729600} & 1 & $E_8$\\
$D_8$ & \llap{5160960} & 135 & $D_8$\\
$A_8$ & 362880 & 960 & $A_8$\\
$A_1 A_7$ & 80640 & 4320 & $A_7+A_1$\\
$A_1 A_2 A_5$ & 8640 & 40320 & $A_5+A_2+A_1$\\
$A_4^{2}$ & 14400 & 12096 & $2A_4$\\
$A_3 D_5$ & 46080 & 7560 & $A_7+A_1$\\
$A_2 E_6$ & 311040 & 1120 & $E_6+A_2$\\
$A_1 E_7$ & \llap{5806080} & 120 & $E_7+A_1$\\
$A_1^{2} D_6$ & 92160 & 3780 & $D_6+2A_1$\\
$D_4^{2}$ & 36864 & 1575 & $2D_4$\\
$A_1^{2} A_3^{2}$ & 2304 & 37800 & $2A_3+2A_1$\\
$A_2^{4}$ & 1296 & 11200 & $4A_2$\\
$A_1^{4} D_4$ & 3072 & 9450 & $D_4+4A_1$\\
$A_1^{8}$ & 256 & 2025 & $8A_1$\\
\bottomrule
\end{tabular}
\bigskip
\end{table}
\begin{table}[p]
\bigskip
\extrarowheight2pt
\caption{Reflection subgroups of $W(F_4)$.}
\label{tab:f4}
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[5pt] \toprule
$\emptyset$ & 1 & 1 & $1$\\
\noalign{\vglue 2pt} \hline
$A_1$ & 2 & 12 & $A_1$\\
\noalign{\vglue 2pt} \hline
$\tilde A_1$ & 2 & 12 & $\tilde A_1$\\
\noalign{\vglue 2pt} \hline
$A_1 \tilde A_1$ & 4 & 72 & $A_1+\tilde A_1$\\
\noalign{\vglue 2pt} \hline
$A_2$ & 6 & 16 & $A_2$\\
\noalign{\vglue 2pt} \hline
$\tilde A_2$ & 6 & 16 & $\tilde A_2$\\
\noalign{\vglue 2pt} \hline
$B_2$ & 8 & 18 & $B_2$\\
$\tilde A_1^{2}$ & 4 & 18 & $2A_1$\\
$A_1^{2}$ & 4 & 18 & $2A_1$\\
\noalign{\vglue 2pt} \hline
$A_2 \tilde A_1$ & 12 & 48 & $A_2+\tilde A_1$\\
\noalign{\vglue 2pt} \hline
$A_1 \tilde A_2$ & 12 & 48 & $\tilde A_2+A_1$\\
\noalign{\vglue 2pt} \hline
$B_3$ & 48 & 12 & $B_3$\\
$A_3$ & 24 & 12 & $A_3$\\
$\tilde A_1 B_2$ & 16 & 36 & $A_3$\\
$A_1^{2} \tilde A_1$ & 8 & 36 & $2A_1+\tilde A_1$\\
$\tilde A_1^{3}$ & 8 & 12 & $2A_1+\tilde A_1$\\
\noalign{\vglue 2pt} \hline
$C_3$ & 48 & 12 & $C_3$\\
$\tilde A_3$ & 24 & 12 & $B_2+A_1$\\
$A_1 B_2$ & 16 & 36 & $B_2+A_1$\\
$A_1 \tilde A_1^{2}$ & 8 & 36 & $3A_1$\\
$A_1^{3}$ & 8 & 12 & $3A_1$\\
\bottomrule
\end{tabular}
\qquad
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[5pt] \toprule
$F_4$ & 1152 & 1 & $F_4$\\
$B_4$ & 384 & 3 & $B_4$\\
$C_4$ & 384 & 3 & $B_4$\\
$\tilde D_4$ & 192 & 1 & $C_3+A_1$\\
$D_4$ & 192 & 1 & $D_4$\\
$\tilde A_1 B_3$ & 96 & 12 & $D_4$\\
$A_1 C_3$ & 96 & 12 & $C_3+A_1$\\
$B_2^{2}$ & 64 & 9 & $D_4(a_1)$\\
$A_1 \tilde A_3$ & 48 & 12 & $A_3+\tilde A_1$\\
$A_3 \tilde A_1$ & 48 & 12 & $A_3+\tilde A_1$\\
$A_2 \tilde A_2$ & 36 & 16 & $A_2+\tilde A_2$\\
$\tilde A_1^{2} B_2$ & 32 & 18 & $A_3+\tilde A_1$\\
$A_1^{2} B_2$ & 32 & 18 & $A_3+\tilde A_1$\\
$A_1^{2} \tilde A_1^{2}$ & 16 & 18 & $4A_1$\\
$\tilde A_1^{4}$ & 16 & 3 & $4A_1$\\
$A_1^{4}$ & 16 & 3 & $4A_1$\\
\bottomrule
\end{tabular}
\bigskip
\end{table}
\begin{table}[p]
\bigskip
\extrarowheight2pt
\caption{Reflection subgroups of $W(G_2)$.}
\label{tab:g2}
\begin{tabular}{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[5pt] \toprule
$\emptyset$ & 1 & 1 & $1$\\
\noalign{\vglue 1pt} \hline
$A_1$ & 2 & 3 & $A_1$\\
\noalign{\vglue 1pt} \hline
$\tilde A_1$ & 2 & 3 & $\tilde A_1$\\
\noalign{\vglue 1pt} \hline
$G_2$ & 12 & 1 & $G_2$\\
$\tilde A_2$ & 6 & 1 & $A_2$\\
$A_1 \tilde A_1$ & 4 & 3 & $A_1 + \tilde A_1$\\
$A_2$ & 6 & 1 & $A_2$\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[p]
\bigskip
\extrarowheight2pt
\caption{Reflection subgroups of $W(H_3)$.}
\label{tab:h3}
\begin{tabular}{crrr}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[5pt] \toprule
$\emptyset$ & 1 & 1 & $1$\\
\noalign{\vglue 1pt} \hline
$A_1$ & 2 & 15 & $2$\\
\noalign{\vglue 1pt} \hline
$A_1^{2}$ & 4 & 15 & $4$\\
\noalign{\vglue 1pt} \hline
$A_2$ & 6 & 10 & $5$\\
\noalign{\vglue 1pt} \hline
$I_2(5)$ & 10 & 6 & $3$\\
\noalign{\vglue 1pt} \hline
$H_3$ & 120 & 1 & $6$\\
$A_1^{3}$ & 8 & 5 & $10$\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[p]
\bigskip
\extrarowheight2pt
\caption{Reflection subgroups of $W(H_4)$.}
\label{tab:h4}
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[5pt] \toprule
$\emptyset$ & 1 & 1 & $1$\\
\noalign{\vglue 1pt} \hline
$A_1$ & 2 & 60 & $2$\\
\noalign{\vglue 1pt} \hline
$A_1^{2}$ & 4 & 450 & $4$\\
\noalign{\vglue 1pt} \hline
$A_2$ & 6 & 200 & $5$\\
\noalign{\vglue 1pt} \hline
$I_2(5)$ & 10 & 72 & $3$\\
\noalign{\vglue 1pt} \hline
$A_1 A_2$ & 12 & 600 & $8$\\
\noalign{\vglue 1pt} \hline
$I_2(5) A_1$ & 20 & 360 & $7$\\
\noalign{\vglue 1pt} \hline
$A_3$ & 24 & 300 & $9$\\
\bottomrule
\end{tabular}
\qquad
\begin{tabular}[t]{crrc}\toprule
Type of $R$ & $\Size{R}$ & $\Size{[R]}$ & Class
\\[5pt] \toprule
$H_3$ & 120 & 60 & $6$\\
$A_1^{3}$ & 8 & 300 & $20$\\
\noalign{\vglue 1pt} \hline
$H_4$ & 14400 & 1 & $11$\\
$H_3 A_1$ & 240 & 60 & $21$\\
$I_2(5)^{2}$ & 100 & 36 & $26$\\
$A_4$ & 120 & 60 & $27$\\
$A_2^{2}$ & 36 & 100 & $32$\\
$D_4$ & 192 & 25 & $25$\\
$A_1^{4}$ & 16 & 75 & $34$\\
\bottomrule
\end{tabular}
\end{table}
\clearpage
\bigskip
{\bf Acknowledgments}: The authors acknowledge the financial
support of the DFG-priority programme SPP1489 ``Algorithmic and Experimental
Methods in Algebra, Geometry, and Number Theory''. Part of the research for
this paper was carried out while the authors were staying at the
Mathematical Research Institute Oberwolfach supported by the ``Research in
Pairs'' programme.
The second author wishes to acknowledge support from Science Foundation
Ireland.
We are grateful to Robert Howlett for helpful discussions.
Finally, we thank the referee for helpful comments.
\bibliographystyle{plain}
|
1604.03361
|
\section{Introduction}
Singlet superconductivity and ferromagnetism are two antagonistic orders. The formation of singlet Cooper pairs requires electrons with anti-parallel spins, whereas the ferromagnetism tends to align electron spins parallel. Nevertheless, Fulde-Ferrell \cite{FF} and Larkin-Ovchinnikov \cite{LO} (FFLO) predicted superconductivity on a ferromagnetic background, however, in a very narrow range of parameters (see Fig. 22 of Fulde's review\cite{Fulde}). Therefore, only a few experimental realizations exist so far, in heavy fermion and organic superconductors (see the work of Zwicknagl and Wosnitza \cite{Zwicknagel10} for a review).\\
For the heavy fermion system, CeCoIn$_5$, specific heat data\cite{Bianchi03, Radovan03}, thermal conductivity\cite{Capan04}, and penetration depth measurements\cite{Martin05} show evidence for the existence of the FFLO state. In quasi-two-dimensional organic superconductors evidence has been obtained from specific heat data\cite{Lortz07} and magnetic torque studies\cite{Bergk10,Bergk11}. However, a spatial oscillation of the order parameter, which is the main feature of the FFLO state, has not yet been observed directly.\\
In layered organic superconductors, an unusual dependence of the transition temperature on the field direction has been predicted theoretically\cite{Croitoru12,Croitoru12_3,Croitoru12_2,Croitoru12_4,Croitoru13}. It is based on the interplay between the vector potential of a magnetic field (applied parallel to the layered structure), the interlayer coupling, and the nodal structure of the order parameter (and its spatial modulation). These calculations shed new light on the interpretation of the results of experimental investigations\cite{Yonezawa08,Yonezawa08_2} as fingerprints of the FFLO state.\\
In superconductor-ferromagnet (S/F) proximity effect systems, \textit{e.g.} in S/F bilayers, a quasi-one-dimensional FFLO-like state can be realized by Cooper pairs migrating from the superconductor into the ferromagnet \cite{Buzdin05, Eschrig11}. Due to the exchange splitting in the ferromagnet, the Cooper pair gains a non-zero momentum, resulting in an oscillating pairing wave function \cite{Buzdin05, Eschrig11, Tagirov98, LP,BVE}. Its reflection at the outer surface of the F-layer leads to interference effects, yielding a superconducting transition temperature, $T_c$, oscillating as a function of the F-layer thickness, $d_F$ \cite{Tagirov98, Zdravkov06, Sidorenko10}.\\
In the presence of two F-layers (\textit{i.e.} for F/S/F and S/F/F structures), the superconducting transition temperature depends on the relative orientation of their magnetizations \cite{Oh,Tag1}. Such systems represent superconducting spin valves, which can be switched between two states with different transition temperatures by magnetic fields, as demonstrated experimentally for the F/S/F \cite{Gu02,Potenza05,Nowak08} and S/F/F \cite{Nowak13, Leksin10, Leksin11} case.\\
For non-collinear orientations of the magnetizations, an unconventional odd-in-frequency triplet s-wave pairing \cite{BVE} is predicted, reducing the superconducting transition temperature \cite{Fominov10}. Thus, a triplet spin-valve effect \cite{Fominov10} can be established, which could be observed experimentally in S/F/F heterostructures \cite{Zdravkov13,Leksin12} and seems to play a crucial role in a recently realized F/S/F memory element \cite{Zdravkov13_2}. Moreover, in S/F/S Josephson junctions it is possible to realize $\pi$-junctions, in which the phase of the FFLO-like pairing wave function changes by $\pi$ across the device \cite{Ryaz1,Ryazanov01, Oboznov06}. This structure is already applied to fabricate $\pi$-shifters for superconducting digital quantum circuits \cite{Khabipov10,Feofanov10}.\\
Most of these devices are operated by applying a magnetic field to the system. If the S-layer is a type II superconductor (often Nb is used, which is a type II material) vortices appear above the lower critical field. However, also in the case of type I materials, the electron mean free path, $l$, in nanoscale thin film structures may be reduced so far, that the S-layer changes to type II behavior. For example, this is the case for In and Pb at $l=35$~nm and $460$~nm, respectively, calculated using equations and parameters from literature\cite{Tidecks80,Werner86,Werner87}.\\
For Nb it is $\mu_0H_{c1}$= 100~mT and $\mu_0H_{c2}\approx$ 400~mT (at 4.2~K for a polycrystalline rod with $T_{c0}$ = 9.1~K) \cite{Weber91}. Here, $H_{c1}$ and $H_{c2}$ are the lower and upper critical magnetic fields, respectively, $\mu_0$ is the vacuum permeability, and $T_{c0}$ the critical temperature. A detailed study of the temperature dependence of $H_{c1}$ and $H_{c2}$ for Nb is given by Finnemore et al. \cite{Finnemore66}.\\
In S/F structures with Nb as S-material, $H_{c1}$ is very small and, thus, the superconducting layer is soon driven into the Shubnikov phase if a magnetic field is applied. Intuitively, one would expect that the flux quanta penetrating the S-layer also have to be generated (and shielded) in the (superconducting) FFLO-like state in the F-layer.\\
While the vortex state and dynamics in low-$T_c$ and high-$T_c$ superconductors is widely investigated \cite{Blatter94,Huebener01,Huebener02}, there are only a few publications concerning the vortex matter in the FFLO state\cite{Bulaevski03,Ikeda07,Zwicknagel10,Dao13}. In this state a vortex lattice may get pinned at the oscillating FFLO order parameter\cite{Bulaevski03,Uji06,Ikeda07,Zwicknagel10}. While different lattices for the FFLO state have been theoretically proposed \cite{Zwicknagel10,Jiang07,Denisov09}, all of them seem to exhibit nodal planes of the order parameter, as present in the quasi-one-dimensional case, which should be favorable sites for vortex pinning. However, for the quasi-one-dimensional FFLO-like state the vortex state and dynamics is so far unexplored.
\section{Experimental Methods}
The samples of the present work, a thin film S/F bilayer (S=Nb, F=Cu$_{41}$Ni$_{59}$) and a thin Nb film, were deposited by magnetron sputtering on a Si substrate. The thickness of the layers and the composition of the ferromagnetic alloy of the S/F sample, S23\#5, were determined by Rutherford Backscattering Spectroscopy (RBS), yielding $d_S=14.1$~nm and $d_F=34.3$~nm and an alloy composition of $41$~at\% Cu and $59$~at\% Ni. To check the quality and the thickness of the single Nb film, Nb5/1, cross-sectional High Resolution Transmission Electron Microscopy (HRTEM) was applied, resulting in $d_S=7.3$~nm. For details concerning sample preparation and characterization see Appendix A.1.\\
The thin film S/F bilayer of the present work, as well as a Nb reference film, were investigated by measurements of the resistive transitions under applied field and a non-resonant microwave absorption study. To determine the upper critical fields of the samples for fields, applied perpendicular and parallel to the film plane, the superconducting resistive transitions as a function of temperature at fixed magnetic fields were measured in an Oxford Instruments Heliox Sorption Pumped $^3$He Insert (using a lock-in technique with a current of about $50$~$\mu$A at a frequency of $18.792$~Hz). The superconducting transition temperature corresponding to the fixed upper critical field is evaluated as the mid-point of the resistive transition.\\
The angular dependence of the upper critical field at temperatures close to the critical temperature, $T_{c0}$, was investigated by non-resonant microwave absorption. This technique has only been applied so far to study the properties of bulk superconductors \cite{Mamoon5, Mamoon4, Mamoon3, Owens, Shaltiel91, Shaltiel, Shaltiel08, Shaltiel08_2, Shaltiel09, Shaltiel10}. In most cases the Induced Microwave Dissipation by AC Magnetic Field (IMDACMF) technique has been used \cite{Shaltiel91, Shaltiel, Shaltiel08, Shaltiel08_2, Shaltiel09, Shaltiel10}, which we also apply for the measurements of the present work. Sketches of the experimental set up, including the relative orientation of the high frequency (HF) microwave field, the static (DC) magnetic field and the modulating (AC) magnetic field applied, are given by Shaltiel et al. \cite{Shaltiel, Shaltiel08, Shaltiel08_2, Shaltiel09, Shaltiel10}.\\
The basic mechanism of the microwave absorption in this technique is that the magnetic state is defined by the DC magnetic field, whereas the AC modulation tends to reduce the pinning energy of the vortices by 'shaking' them. The 'shaking' occurs, because the AC modulation yields a change of the flux through the sample and, thus, the need of additional or less vortices penetrating the sample and rearrangement of the whole vortex structure. This yields the possibility of vortex motion, resulting in absorption of the high-frequency microwave.\\
Thus, the AC magnetic field induces a modulation signal into the microwave power, $P$, reflected from the cavity\cite{Shaltiel09}. This microwave power is rectified by a diode and fed into a lock-in detector. The signal, $\text{d}P/\text{d}H$ (sometimes also called 'intensity' in literature), detected at the fundamental frequency (also denoted as first AC harmonic in literature) of $H_{AC}$, is obtained from the lock-in detector, \textit{i.e.} information about the microwave dissipation of the sample due to the AC modulation in the state determined by the DC magnetic field is obtained.\\
For details on the IMDACMF technique see Appendix A.2, where also the conversion of magnetic fields from the cgs emu unit system (used in Chap. IV-B) into the international SI system (applied in Chaps. III and IV-A) is given.
\section{Theoretical Framework}
Within the Ginzburg-Landau (GL) theory it can be shown for a thin superconducting film in a parallel magnetic field, $H_\parallel$, that, if the thickness $d$ is smaller than $\sqrt{5}\lambda(T)$ (condition for the transition to the normal state to be of second order), the parallel critical field, $H_{c\parallel}$, is given by\cite{TinkhamBook,Tinkham}
\begin{equation}
H_{c\parallel}(T)=2\sqrt{6}H_{cth}(T)\lambda(T)/d
\end{equation}
In a second order phase transition the superconducting order parameter, $\psi$, of the GL theory\cite{TinkhamBook} (with $|\psi|^2$, representing the density, $n_s$, of the superconducting charge carriers) approaches zero continuously, when $H_\parallel$ is increased to $H_{c\parallel}$. Here, $H_{cth}$ is the thermodynamical critical field of the bulk material\cite{TinkhamBook,Tinkham}. Moreover, $\lambda(T)$ is the penetration depth in weak fields\cite{Tinkham}, which is given by\cite{Tidecks90} $\lambda(T)=0.5^{1/2}\lambda_L(0)\left[T_{c0}/\left(\chi\left(T_{c0}-T\right)\right)\right]^{1/2}$, with \mbox{$\lambda_L^2(0)=3/(2e^2\mu_0N_0v_F^2)$}, where $e$ is the elementary charge, $N_0$ is the number of electronic states (in the free electron model) for one spin direction per volume and energy interval at the Fermi level, and $v_F$ is the Fermi velocity. Furthermore, $\chi=(1+0.752\xi_0/l)^{-1}$, with $l$ the electron mean free path and $\xi_0=\tilde{\gamma}\hbar v_F/(\pi^2kT_{c0})$ the Bardeen-Cooper-Schrieffer (BCS) coherence length, where $\hbar=h/(2\pi)$ with $h$ the Planck constant, $k$ the Boltzmann constant, and $\tilde{\gamma}=\text{exp}(\gamma)=1.781...$ with $\gamma=0.5772...$ the Euler-Mascheroni constant (both $\gamma$ and $\tilde{\gamma}$ sometimes found in literature as Euler constant).\\
Thus, for a thin film with $d\approx l \ll \xi_0$ one obtains just the expression for $\lambda_{eff}(T)$ considered by Tinkham to be appropriate to be used in Eq.(1) [See Tinkham's book\cite{TinkhamBook}, Chap. 4.6, together with Eqs.(3.136) and (3.123), which in the GL regime is given by Eq.(3.123b)].\\
Using the relation\cite{Tidecks90}
\begin{equation}
B_{cth}(T)\xi(T)\lambda(T) 2e = \hbar/\sqrt{2}
\end{equation}
it is possible to rewrite Eq.(1) as
\begin{equation}
H_{c\parallel}(T)=\sqrt{3}\Phi_0/(\pi\xi(T)d\mu_0)
\end{equation}
Here, $\Phi_0=h/(2e)=2.07\cdot10^{-15}$~Tm$^2$ is the elementary flux quantum. Furthermore, $\xi(T)=0.74\chi^{1/2}\xi_0\left[T_{c0}/\left(T_{c0}-T\right)\right]^{1/2}$ is the GL coherence length\cite{Tidecks90} and $B_{cth}=\mu_0 H_{cth}$.\\
For a thin superconducting film in a magnetic field perpendicular to the film plane, Tinkham developed an elementary theory\cite{Tinkham} for the critical field, $H_{c\perp}$. The theory is based on the GL theory and the London theory. It describes the superconducting transition within a model based on the concept of fluxoid quantization. Again, a second order phase transition is assumed. From a detailed discussion of the free enthalpy difference of the superconducting and normal conducting state, the maximum field with a non-vanishing order parameter can be determined, yielding
\begin{equation}
H_{c\perp}(T)=4\pi\mu_0\lambda^2(T)H_{cth}^2(T)/\Phi_0
\end{equation}
Using Eq.(2), this result can be rewritten as
\begin{equation}
H_{c\perp}(T)=\Phi_0/(2\pi\xi^2(T)\mu_0)
\end{equation}
This is equal to the expression for the upper critical field in bulk samples\cite{TinkhamBook}, \textit{i.e.} $H_{c\perp}(T)=H_{c2}(T)$. Combing Eqs.(5) and (2), we obtain $H_{c\perp}(T)=\sqrt{2}\kappa H_{cth}(T)$, where $\kappa = \lambda(T)/\xi(T)$ is the GL parameter, yielding $\kappa=0.956\lambda_L(0)/(\xi_0\chi)$ using the expressions given above. If the superconducting film is very thin, $l$ is limited by $d$\cite{TinkhamBook} and, thus, $\kappa$ varies with the film thickness.\\
In his elementary theory\cite{Tinkham}, Tinkham also considered the angular dependence of the critical field. From the calculated expression for the free enthalpy density difference, he concludes, that the perpendicular field component leads to an energy term scaling linearly with $H$, while the parallel field component results in a term quadratic in the field, which both have to be balanced against the condensation energy. The origin of this difference is, that the current loops of the perpendicular vortices can scale down, as $H$ increases, while vortices parallel to the thin film are fixed in one dimension by the film thickness.\\
From these arguments Tinkham concluded that for a given angle $\theta$ between the film plane and the magnetic field it is\cite{Tinkham}
\begin{equation}
\left|\frac{H_c(\theta)\text{sin}(\theta)}{H_{c\perp}}\right|+\left(\frac{H_c(\theta)\text{cos}(\theta)}{H_{c\parallel}}\right)^2=1
\end{equation}
Here, for $\theta=0^\circ$ and $\theta=90^\circ$ the field is parallel and perpendicular to the film plane, respectively.\\
In his book\cite{TinkhamBook}, Tinkham pointed out, that the limiting values [given by Eqs.(3) and (5)], as well as his formula for intermediated angles [given by Eq.(6)] are only valid if $d\ll\xi(T)$, so that $|\psi|$ can be regarded as constant over the thickness of the thin film.\\
It is possible to derive the 'Tinkham formula', given by Eq.(6), from the linearized GL equation as a thin film limit (introducing a suitable vector pontential). Details of this derivation are given in Appendix B.1.\\
The GL theory used to derive the results given above is only valid for temperatures just below the critical temperature $T_{c0}$. So far, as the phenomenological GL equations are applied, the results are valid independent of the strength of the electron-phonon interaction. This is the case for Eqs. (1)-(6) and those in Appendix B.1. The explicit expressions for $\lambda(T)$, $\xi(T)$, and $\kappa$, however, resulting from the microscopic derivation of the GL equations by Gorkov, are only valid in the weak coupling limit.\\
The GL theory and the theory of type II superconductors in a magnetic field has been extended to low temperatures. However, no 'Tinkham-like' formula for lower temperatures has been derived. For a detailed discussion, see Appendix B.2.\\
We will apply the theoretical results to our samples, although they are (or contain) films of Nb, which is not a weak coupling superconductor. There is a detailed discussion in Appendix B.3, why this may be allowed.\\
The critical temperature, $T_{c0}$, in the equations above is defined as the superconducting transition temperature, $T_c$, in the absence of currents and magnetic fields.
Strictly obeying the definition of $T_{c0}$, it can not be defined for S/F heterostructures (therefore it was denoted $T_c$ in our former works), because a magnetic material is present in the sample. Nevertheless, in the present work we identify the transition temperature of S/F heterostructures in zero magnetic field also with $T_{c0}$.\\
\section{Results and Discussion}
\subsection{Temperature Dependence of the Upper Critical Fields}
The temperature dependencies of the upper critical fields perpendicular and parallel to the film plane for the samples S23\#5 and Nb5/1 are shown in Fig. 1 (a) and (b), together with the linear regressions according to Eqs. (3) and (5). The upper critical fields follow the predicted temperature dependencies, \textit{i.e} $\mu_0H_{c\parallel}(T)\propto(1-T/T_{c0})^{1/2}$ and $\mu_0H_{c\perp}(T)\propto 1-T/T_{c0}$ over a wide range of temperatures. Deviations are observed for low temperatures (as expected, because the GL theory should not be valid here) and in the direct vicinity of the critical temperature (possible reasons will be discussed at the end of Chap.~IV-A).\\
Thus, the critical temperatures of the measurement, $T_{c0,MS}=6.42$~K and $6.07$~K for S23\#5 and Nb5/1, respectively, deviate from $T_{c0,GL}$ (simply called $T_{c0}$ in the following) determined by the extrapolation of the temperature behavior of the critical fields predicted by the GL theory.\\
The obtained critical temperatures $T_{c0}($S23\#5$)$ and $T_{c0}($Nb5/1$)$ are $6.34$~K and $5.95$~K, respectively. The slopes of the linear regressions \mbox{$\frac{\text{d}\left(\mu_0H_{c\perp}\right)}{\text{d}T}$} are $-0.316$~T/K and $-0.592$~T/K for S23\#5 and Nb5/1, respectively. For \mbox{$\frac{\text{d}(\mu_0H_{c\parallel})^2}{\text{d}T}$} we obtain $-8.93$~T$^2$/K and $-39.44$~T$^2$/K, respectively. The given values are obtained by a general fit to both the positive and negative field data, obtained for both increasing and decreasing temperature, with a single parameter $T_{c0}$ for both field directions.\\
\begin{figure}
\includegraphics[width=\columnwidth]{HcT.pdf}
\caption{(Color online) Temperature dependence of the upper critical fields, $H_{c\perp}(T)$ (a) and $H_{c\parallel}(T)$ (b), perpendicular and parallel to the film surface, respectively, for Nb5/1 (black dots) and the S/F bilayer S23\#5 (red squares). For the parallel field, $(\mu_0H_{c\parallel})^2(T)$ has been plotted.\\The inserts show an enlargement of the data near the critical temperature, $T_{c0}$. The solid lines in (a) and (b) show linear regressions according to Eqs. (5) and (3), respectively.}
\end{figure}
According to Eq.(5), it is
\begin{equation}
\xi(0)=0.74\chi^{1/2}\xi_0=\left[-\frac{2\pi T_{c0}}{\Phi_0}\cdot\frac{\text{d}\left(\mu_0H_{c\perp}\right)}{\text{d}T}\right]^{-1/2}
\end{equation}
Thus, we obtain $\xi(0)= 12.8$ nm and $9.7$ nm for S23\#5 and Nb5/1, respectively. We should emphasize, that while $\xi(0)$ for Nb5/1 is a direct property of the sample, it is in contrast only an effective coherence length, reflecting the whole (inhomogenous) superconducting state in S23\#5.\\
Another part of the same film Nb5 was investigated in our former work\cite{Sidorenko10} by the same measurements presented here. We obtained $T_{c0}=6.25$ K (measured also earlier as $T_{c0}=6.40$ K) and \mbox{$\frac{\text{d}\left(\mu_0H_{c\perp}\right)}{\text{d}T}=-0.558$ T/K}, resulting in $\xi(0)=9.7$ nm. While $T_{c0}$ is somewhat higher, $\xi(0)$ is in good agreement with the value obtained in the present investigation.\\
According to Eq.(3) it is
\begin{equation}
d=\left[-\frac{\pi^2\xi(0)^2T_{c0}}{3\Phi_{0}^2}\cdot\frac{\text{d}(\mu_0H_{c\parallel})^2}{\text{d}T}\right]^{-1/2}
\end{equation}
Inserting the respective quantities for S23\#5 and Nb5/1, we obtain $d=15.6$~nm and $7.7$~nm, respectively. For Nb5/1 the value is very close to $d_S=7.3$~nm, obtained from TEM investigations (see Appendix A.1). For S23\#5 the value is not directly related to the geometry of the sample. These thicknesses represent effective values entering the GL expression for the parallel critical field and, thus, will be referred to as $d_{GL}$ in the following.\\
Moreover, in our former work\cite{Sidorenko10}, we also investigated the temperature dependence of the critical fields of a Nb film of $14$~nm thickness (nearly equal to $d_S$ in S23\#5) by the same measurements presented here. We obtained $T_{c0}=8.00$ K (measured also earlier as $T_{c0}=8.05$ K) and \mbox{$\frac{\text{d}\left(\mu_0H_{c\perp}\right)}{\text{d}T}=-0.372$ T/K}, resulting in $\xi(0)=10.5$ nm. If we compare these values with the effective values obtained for S23\#5, we see that the S/F bilayer behaves similar to a thicker layer of a more weakly superconducting material. This is just as expected, because the superconducting layer is weakened by the proximity effect, but the superconductivity can extend into the F-layer.\\
Concerning the deviations of $\mu_0H_{c\perp}(T)$ in Nb5/1 from the GL behavior close to the critical temperature, we refer to Weber et al.\cite{Weber91}, who observed a similar bending up in their $\mu_0H_{c2}(T)$ measurements of Nb bulk samples. They could describe the deviations within the anisotropic Eliashberg theory, considering a mean square anisotropy of electron-phonon interaction and of the Fermi velocity. It is, however, unclear whether the deviations, observed for $\mu_0H_{c\perp}(T)$ of S23\#5 and $\mu_0H_{c\parallel}(T)$ of S23\#5 and Nb5/1, arise from similar effects.\\
\subsection{Angular Dependence of the Critical Field}
\subsubsection{Experimental Results}
As discussed above, the non-resonant microwave absorption signal $\text{d}P/\text{d}H$ is generated by the motion of the vortices in the superconducting phase. Consequently, the upper critical field $H_{c}$ can be evaluated as the point of vanishing absorption.\\
There are different temperatures mentioned in the captions of Figs. 2-4. First, the setpoint of the temperature controller of the Electron Paramagnetic Resonance (EPR) spectrometer, $T_{SP}$, which is slightly higher than the exact measurement temperature $T_{ME}$. Moreover, to be able to compare $(T_c,H_c)$-points obtained by resistive transitions at constant applied field, $H$, with ones obtained from the EPR measurements at constant temperature, we evaluate a midpoint temperature, $T_{MP}$, to which the EPR spectra correspond (assuming that the same $(T_c,H_c)$-point should be obtained from both methods). The determination of $T_{ME}$ and $T_{MP}$ is described in detail in Appendix A.3.\\
Figure 2 shows selected microwave absorption spectra for S23\#5 at $T_{MP}=4.64$ K, well below $T_{c0}$. The data is recorded from $\theta=0^\circ$ to $\theta=90^\circ$. The transition at $H_{c}$ is well defined and quite smooth and sharp. On the other hand, at this temperature, it is technically not possible to measure the upper critical field for fields applied parallel to the film plane, as it exceeds the limit of the magnet of 16 kOe.\\
However, close to $T_{c0}$ the upper critical field is strongly reduced. Thus, a proper choice of the measuring temperature will reduce the magnetic field below the limit of the magnet. To identify the lowest temperature, at which $H_{c\parallel}<16$ kOe, we evaluated the onset of $\text{d}P/\text{d}H$ with decreasing temperature at a constant field of 15 kOe, applied parallel to the film plane. However, for field sweep measurements at constant temperature on Nb5/1, it was still not possible to establish a measuring temperature corresponding to a $H_{c\parallel}$ below 16 kOe.\\
Moreover, if the measuring temperature approaches $T_{c0}$, the transition is increasingly broadened by the increasing influence of the temperature stability on $H_{c}$ arising from the steep slope of $H_{c\parallel}(T)$ close to $T_{c0}$ due to the square root temperature dependence (see Eq. (3)). The signal is also increasingly noisy, which we attribute to flux flow activation by temperature fluctuations.\\
In the following, the data is always recorded starting from $\theta=0^\circ$ to both, $\theta=-90^\circ$ and, subsequently, to $\theta=+90^\circ$.\\
Figure 3(a) shows a contour plot of the collected data cube of d$P/$d$H$ as a function of $H$ and $\theta$ at $T_{MP}=6.10$~K, close to $T_{c0}$. White color represents a signal below the cut-off value of -115, which represents zero effective signal. A clear cusp at 0$^\circ$ is observed. The white area at low fields around $0^\circ$ is an artifact from phase instability between the AC reference and the measured signal. However, near $H_{c}$ the coupling is reestablished. Unfortunately, it was not possible to obtain an evaluable signal from the sample for some angles, mainly for $\theta>45^\circ$.\\
Figure 3(b) shows the field dependence of $\text{d}P/\text{d}H$ for S23\#5 for selected angles at $T_{MP}=6.10$ K. Basically, $H_{c}$ is given by the mid-point of the observed transition. However, the width of transition (and thus its midpoint) is hard to evaluate as it is veiled by the noise. Nevertheless, the point of vanishing signal can be clearly evaluated (black circles in Fig. 3(b)). This corresponds to the upper end of the superconducting transition at the highest temperature within the temperature stability range.\\
The weak angle-independent signals at $1.8$~kOe and $3.3$~kOe can be assigned to paramagnetic resonances of the cavity background, while the signal at 12 kOe is the paramagnetic resonance of oxygen.\\
\begin{figure}
\includegraphics[width=\columnwidth]{SpectraLT2.pdf}
\caption{(Color online) Selection of microwave absorption spectra at $T_{MP}=4.64$~K for the S/F bilayer, S23\#5, well below the transition temperature, $T_{c0}$, as a function of the applied magnetic field, $H=H_{DC}$, for different angles, $\theta$, between the applied field and the film plane. The individual curves have been offset for better visibility.\\Here, $T_{SP}=5.00$~K and $T_{ME}=4.74$~K (for the definition of the different temperatures see the text, for further details see Appendix A.3). The insert shows a sketch of the sample and definition of the angle $\theta$.}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{Spectrum.pdf}
\caption{(Color online) (a) Contour plot of the raw microwave absorption signal of the S/F bilayer S23\#5 as a function of the applied field, $H=H_{DC}$, and the angle $\theta$, between the applied field and the film plane, at $T_{MP}=6.10$~K, close to the critical temperature, $T_{c0}$. Here, $T_{SP}=6.45$~K and $T_{ME}=6.20$~K. For details see the text and Appendix A.3.\\(b) Microwave absorption spectra, selected from (a) for different angles $\theta$. The individual curves have been offset for better visibility. The black circles show the points of vanishing signal, at which the upper critical field, $H_{c}$, is evaluated. For details see the text.}
\end{figure}
\subsubsection{Description by the Tinkham Formula}
We applied the Tinkham formula, Eq.(6), to the evaluated angular dependence of $H_{c}$ of both, the reference film Nb5/1 and the S/F bilayer sample S23\#5 close to $T_{c0}$. A detailed analysis of the validity conditions of the Tinkham formula is given in Appendix B.4, showing, that the they are fulfilled for Nb5/1 and, under certain assumptions, also for S23\#5.\\
\mbox{Figure 4} shows the results of the $H_{c}$ evaluation, as well as calculated predictions. The red solid lines in Fig. 4 (a) and (c) are obtained from Eq. (6) by using experimental values from Chap. IV-A (discussed in Appendix A.3). We used $H_{c\perp}(6.10$~K$)=0.882$~kOe and $H_{c\parallel}(6.10$~K$)=15.077$~kOe for S23\#5 and $H_{c\perp}(5.90$~K$)=0.900$~kOe and $H_{c\parallel}(5.90$~K$)=16.300$~kOe for Nb5/1, respectively.\\
In contrast to the case of Nb5/1, where the obtained data scatters around the Tinkham prediction, there is a systematic deviation in the case of S23\#5. Thus, the data obtained for S23\#5 has to be discussed in more detail.\\
In Fig. 4 (b) we show predictions obtained from the expressions due to the GL theory for these fields. From Eqs.(3) and (5) (using the results from Chap. IV-A for $T_{c0}$ and $\xi(0)$ and setting $d=d_{GL}$) we obtain $H_{c\parallel}(T)=57.3$ kOe$\cdot (1-T/T_{c0})^{1/2}$ and $H_{c\perp}(T)=20.1$ kOe$\cdot (1-T/T_{c0})$, respectively. Setting $d=d_S$ and $d=d_S+d_F$ yields $H_{c\parallel}(T)=63.4$ kOe$\cdot (1-T/T_{c0})^{1/2}$ and $H_{c\parallel}(T)=18.5$ kOe$\cdot (1-T/T_{c0})^{1/2}$, yielding (using $T=T_{MP}$ and Eq.(6)) the solid and dashed line in Fig. 4 (b), respectively. The case $d=d_{GL}$ is represented by the dotted line.\\
While the general shape of the data is roughly given by the solid line, in particular the data points for angles $|\theta|<40^\circ$ deviate from that prediction. For $|\theta|>10^\circ$ the measurements are better described by the dashed line. Reducing $d$ increases the value of $H_{c\parallel}$, defining the maximum of $H_c(\theta)$, and, thus, the value of $H_c$ around $\theta=0^\circ$. Consequently, the data can be described by changing $d$ (color coded) step by step from $d_S+d_F$ to $d_S$ with decreasing absolute value of the angle. This means, that the value of $H_{c\parallel}$ seems to be determined more and more by the S-layer when $\theta$ approaches zero, \textit{i.e.} the parallel orientation.\\
At a first view, this might indicate that the FFLO-state is weakened or destroyed more and more by the increasing value of the applied field. However, measurements and calculations of the transition temperature oscillations as a function of $d_F$ of F/S/F heterostructures\cite{Antropov13} have shown, that the FFLO-state is neither destroyed nor strongly suppressed for the magnetic fields applied here (at least for high thicknesses $d_{\text{CuNi}}$). To directly compare the results for F/S/F trilayers with those of S/F bilayers, the thickness of the S-layer has to be divided by two\cite{Sidorenko10, Zdravkov11} and only one of the two ferromagnetic layers has to be considered\cite{Zdravkov11} (\textit{i.e.} $d_F$ in the present work has to be compared with approximately $d_{\text{CuNi}}/2$ in our previous work\cite{Antropov13}). See our previous works for details\cite{Sidorenko10,Zdravkov11}.\\
Moreover, also stray fields can be excluded, because above a field of 2-3~kOe the F-layer is in the saturated state \cite{Ruotolo04,Sidorenko10,Kehrle12,Zdravkov13}. There is also no indication of a $T_c$ reduction by stray fields visible in the measurement of $B_{c\perp}(T)$ and $B_{c\parallel}(T)$ presented in Chap.~IV-A. Here, effects of stray fields should be observable at the coercive field of the F-layer (at negative values of $B_{c\perp}\approx-750$~Oe and $B_{c\parallel}\approx -250$~Oe\cite{Kehrle12,Zdravkov13}), where the stray field effects are expected to be most strongly expressed.\\
Thus, this effect seems to result from the special nature of the vortex in S/F bilayers.
\subsection{Vortex State in S/F bilayers}
\subsubsection{Conjecture about a New Vortex Structure}
A possible shape of a vortex in a S/F bilayer, based on the specific anisotropy induced by the quasi-one-dimensional FFLO-like state, optimizing the losses of condensation energy and the energy, needed by the current system to generate the flux quantum and shield the entire superconductor, is proposed in Fig. 5 and will be discussed in detail below.\\
\begin{figure}
\includegraphics[width=\columnwidth]{Tinkham.pdf}
\caption{(Color online) Upper critical field $H_{c}$ as a function of the angle, $\theta$, between the applied field and the film plane, for the S/F bilayer sample S23\#5 (a) and (b), and the reference film Nb5/1 (c).\\The solid lines in (a) and (c) are descriptions of the data according to Eq.(6), using experimental values for $H_{c\perp}$ and $H_{c\parallel}$, obtained in Chap.~IV-A, at the temperatures $T_{MP}=6.10$~K and $5.90$~K for S23\#5 and Nb5/1, respectively. The values for $T_{ME}$ are $6.20$~K and $6.18$~K, respectively (for details see Appendix A.3). In both measurements $T_{SP}=6.45$~K. The theoretical curve in (c) includes an angular offset, $\Delta\theta=-3.2^\circ$, as the exact angle of the maximum of $H_c$ is not precisely known.\\In (b) the experimental data for S23\#5 is plotted as open squares, while the solid, dashed, and dotted lines represent predictions based on Eq.(6) for different superconducting layer thicknesses, $d=d_S$, $d_S+d_F$, and $d_{GL}$, respectively. Moreover, $H_c(\theta)$ is presented for continuous $d$ (color coded), calculated from Eqs. (3),(5), and (6), to illustrate the decrease of $d$ from $d_{S}+d_{F}$ (dashed line) to $d_S$ (solid line). For details see the text.}
\end{figure}
\begin{figure*}
\includegraphics[width=\textwidth]{Vortex7.pdf}
\caption{(Color online) Left panel: Pairing wave function and Cooper pair density as a function of the space coordinate normal to the film plane in a S/F bilayer.\\Right panel: Sketch of a possible vortex structure taking into account the anisotropic Cooper pair density (color coded) in S/F bilayers.\\For details see the text.}
\end{figure*}
The oscillation of the pairing wave function inside the ferromagnetic layer leads to infinitely thin normal conducting layers parallel to the film, at the position of the nodes of the pairing wave function \cite{Buzdin05,Tagirov98,Sidorenko09} (see Fig. 5 left panel). This may lead to a vortex structure in our S/F bilayers, which has a certain similarity with that one in layered high-$T_c$ superconductors.\\
Within the Lawrence-Doniach theory\cite{TinkhamBook}, Blatter et al. showed in Chap. VIII-A of their review\cite{Blatter94}, that in layered high-$T_c$ superconductors for large angles, $\vartheta$, between the applied field and the ab-plane of such compounds, the vortex is realized by a stack of pancake vortices, which are perpendicular to the ab-plane and generated by current systems in the (strongly superconducting) ab-plane. When approaching smaller angles, \textit{i.e.} if tan$\vartheta<d_{int}/\xi$ (here $d_{int}$ is the spacing between the ab-planes)(see Chap. VIII-A3 of the review of Blatter et al.\cite{Blatter94}), a 'crossover' to a new vortex structure occurs, when the pancake vortices cannot overlap anymore and have to be connected by Josephson vortices, which are parallel to the ab-plane and realized by currents systems perpendicular to the ab-planes.\\
By introducing a vortex into a superconducting material, the superconductor gains the magnetic flux exclusion energy corresponding to one flux quantum, but has to expend the magnetic energy stored in the current system of the vortex, and looses the condensation energy due to the suppression of the order parameter. The energy gain from one vortex is always the same. However, the corresponding energy loss terms in S/F bilayers might (in analogy to the vortex structure in layered high-$T_c$ superconductors) be reduced by a transition from an inclined vortex, parallel to $\underline{H}$, to a series of 'short-link' vortices inclined by an angle between $\theta$ and 90$^\circ$ to the film plane, connected by vortices parallel to the film plane located in the areas of weak superconductivity generated by the FFLO-like state. While the kinetic energy of the shielding currents is increased in such a vortex structure due to the longer flux line, the energy loss due to the destruction of superconductivity inside the vortex is reduced by channeling the vortex through regions of weak superconductivity.\\
It is worthy to mention, that the current systems in Fig. 5 are only sketched schematically as circular. In more detail, we expect that perpendicular to the plane of Fig. 5 (\textit{i.e.} parallel to the film plane) the current system of the vortex segments parallel to the film plane is elongated as in the case of layered superconductors (see Chap. VIII-A1 of Blatter et al.\cite{Blatter94})\\
In Fig. 5, we approximated the pairing wave function to be constant inside the superconductor and to be an exponential decaying cosine function inside the ferromagnet with a step, due to imperfect transparency, at the interface. The local Cooper pair density is proportional to the absolute square of the pairing wave function. The color coding gives the local Cooper pair density from yellow to brown from low to high. Both quantities are plotted as a function of the space coordinate normal to the interface in Fig. 5 left panel.\\
Due to the exponential damping of the oscillation of the pairing wave function with increasing distance from the S/F interface, the reduction of condensation energy loss, which can be obtained by segmenting the vortex, decreases with the distance from the interface as well. For distances large compared to the decay length, the possible reduction of condensation energy loss would be essentially zero, thus, there will be no segmentation. To take into account this decrease in energy gain by segmentation, we propose the inclination angle of the 'short-link' vortices to decrease with increasing distance from the S/F interface.\\
In the case of S/F bilayers the segments parallel to the layers are only similar to Josephson vortices (see Chap. VIII-A1 of Blatter et al. \cite{Blatter94}). Nevertheless, we will calculate the angle, $\theta$, at which the 'crossover' to a segmented vortex mentioned above might occur, for the case that $d_{int}$ is the distance between the S/F interface and the first maximum of the Cooper pair density in the F-material. To calculate $d_{int}$ for S23\#5, we have to calculate the position of the first minimum of the pairing wave function $\Phi_F(x_F)$, \textit{i.e.} the first maximum of $|\Phi_F(x_F)|^2$. Here $x_F$ is the perpendicular distance from the S/F interface. This calculation, together with a general review of the oscillation properties of the pairing wave function in S/F bilayers is given in Appendix~C, yielding $d_{int}=23.9$~nm. Moreover, using $\xi(0)=12.8$~nm and $T_{c0}=6.34$~K, we get for $T=T_{MP}=6.10$~K that $\xi(T_{MP})=65.9$~nm.\\
Thus, we obtain $\text{tan}(\theta)=23.9$~nm~$/$~$65.9$~nm~$=0.363$, yielding $\theta=19.9^\circ$. For smaller $\theta$ a segmentation of the vortex into pancake-like vortices and vortices parallel to the film plane located in the minima of $|\Phi_F(x_F)|^2$ should be possible (within the analogy to the situation in high-$T_c$ superconductors).\\
According to Blatter et al.\cite{Blatter94} for this angular regime new phenomena are expected, whereas for large $\theta$ a continuous description applies. While it is not obvious, at which angles exactly the deviations from the Tinkham formula in our data occur, especially due to the asymmetry of the data, at least the rough angular regime of the deviations seems to fit.\\
Blatter et al. show in Figs. 32 and 33 of their review\cite{Blatter94} the spatial magnetic field distribution for a pancake vortex in a thin superconducting film and a layered superconductor with vanishing Josephson coupling, respectively. The screening currents in the neighboring layers squeeze the magnetic field into the planar direction. Moreover, in Fig. 35 they show a vortex line at small angles $\vartheta$ (corresponding to $\theta$ in the present work), where the core of the Josephson string is fully developed, guiding the magnetic flux between the superconducting layers. In our case, the shielding currents of the vortex guiding the magnetic flux through the minimum of $|\Phi_F(x_F)|^2$ will penetrate into the S-layer.\\
A possible consequence of the proposed vortex structure might be, that for decreasing absolute value of $\theta$, the segments parallel to the S-layer will increasingly dominate. Thus, the shielding properties of the S-layer become more and more important for the value of $H_{c\parallel}$, because the shielding currents will penetrate the S/F interface (yielding $d$ to be more and more governed by $d_S$ - see Fig. 4 (b)). Finally, in the S/F bilayer sample investigated in the present work, for $\theta=0^\circ$, we expect the vortex to have the core in the first node of the FFLO wave function.\\
A detailed justification of the proposed vortex is beyond the scope of the article. To derive a 'Tinkham formula' for S/F bilayers from the GL equation, it would be necessary to include the properties of the quasi-one-dimensional FFLO-like state into the GL equation for the order parameter and find a suitable vector potential, which generates both the applied field and the magnetization. Aside, that it is expected to be difficult to solve this equation, there is the general problem, that a pairing wave function exists in the F-material, but (at least strictly speaking) no superconducting order parameter (similar to the case of the superconductor/normalconductor proximity effect \cite{Deutscher69, Clarke69}).\\
To extend the approach of Tinkham's original work \cite{Tinkham}, when balancing the energy of the shielding current system against the loss of condensation energy, one has to consider both, the anisotropy of the magnetic and the superconducting state, in the free enthalpy terms, yielding much more complicated and space dependent equations. Moreover, the straight-forward superposition of the parallel and perpendicular enthalpy terms to obtain the Tinkham formula will most probably not be possible.\\
\subsubsection{Vortex Dynamics}
The vortex structure proposed has a pinning behavior, which is expected to be different from that one of a continuous vortex considered, to derive the angular dependence of $H_c$ by Tinkham\cite{Tinkham}. It is expected to behave more similar to the vortices in high-$T_c$ superconductors. Here, the Josephson vortices are much more strongly pinned than the pancake vortices. Thus, the ESR signal should decrease with the increase of the parallel field component. However, there is a mechanism leading to increased Josephson vortex mobility at very small angles\cite{Weidinger97}. While we see the general decrease of the signal amplitude with decreasing angle, a possible stabilization around $\theta=0^\circ$ can not be stated based on the obtained data. Moreover, since the direction of flux movement is not determined by an applied current in contrast to the work of Weidinger et al.\cite{Weidinger97}, the parallel flux through the sample can be changed by moving the vortices only along the nodal plane of the pairing wave function, so it is questionable if such a mechanism is applicable in our case.\\
Investigations of high-$T_c$ superconductors, using the IMDACMF method, with $H_{DC}$ perpendicular to the ab-plane of the layered structure show a non-vanishing signal $($d$P/$d$H)(T)$ at constant field only just below $T_c$, where thermal activated flux flow (TAFF) governs the motion of pancake vortices. For $H_{DC}$ parallel to the ab-planes, where only Josephson vortices are present, the signal intensity increase sharply at $T_c$ with a further increase down to low temperatures (see Fig. 4 of Shaltiel et al.\cite{Shaltiel09}). This leads to the conclusion, that in those materials at temperatures well below $T_c$ the induced microwave dissipation results from the interaction of the microwaves with the Josephson vortices\cite{Shaltiel09}.\\
This is different in the experiments in the present work. For the S/F bilayer we investigated $($d$P/$d$H)(T)$ at constant field for $H_{DC}$ parallel and perpendicular to the film plane. In both cases the signal has its largest value just below $T_c$ and decays to lower temperatures, but does not vanish (at least down to $4$~K).\\
\subsection{Comparison with Related Systems}
\subsubsection{Bulk FFLO Superconductors}
The angular dependence of the upper critical field for the FFLO state in bulk superconductors has been investigated by Dao et al.\cite{Dao13}, considering the role of crystal anisotropy on the vortex state. Contrary to conventional superconductivity, where only the crystal structure influences the type of the Abrikosov vortex lattice, the modulation of the order parameter in the FFLO phase has an influence on the vortex structure, too. In special situations, higher Landau level (LL) states lead to an angular dependence of $H_c$ with transitions between the higher LL states. If only one state is considered, a smooth $H_c(\theta)$ dependence is predicted (between $90^\circ$ and $30^\circ$). In the general case, transitions between different LL states lead to structures in the $H_c(\theta)$ dependence, for $|\theta|<20^\circ$. These structures, however, only appear for low temperatures. Above $T=0.5T_{c0}$ the structures vanish. The overall shape of the $H_c(\theta)$ curve is rounded at $\theta=0^\circ$, \textit{i.e.} it has a shape similar to the Lawrence-Doniach behavior mentioned above. The experiments of the present work are performed closer to $T_{c0}$, where no LL state transition generated structures are predicted. Moreover, we observe a sharp cusp-like behavior of $H_c(\theta)$ in our experiments.\\
\subsubsection{Thin Films and Layered Superconductors}
The angular dependence of the critical field was widely investigated for thin films\cite{Tinkham64,Harper68,Sidorenko81,Banerjee84}, multilayers \cite{Banerjee83,Banerjee84,Chun84,Sidorenko91,Sidorenko96,Klemm2012}, including fractal geometries\cite{Sidorenko96}, and high-$T_c$ superconductors\cite{Palstra88,Naughton88,Juang88,Iye90,Marcon92}. Although the experiments often follow the general behavior of Tinkham's prediction, deviations are found in detail, as observed in the present work. In multilayers and high-$T_c$ superconductors also a Lawrence-Doniach behavior of $H_c(\theta)$\cite{TinkhamBook}, which describes an anisotropic three-dimensional multilayer superconductor, is observed. We ascribe the observed deviations in the present work to a special vortex structure, generated by the quasi-one-dimensional FFLO-like state in the F-material of the S/F bilayer.\\
Since this vortex structure is related to that one of a layered superconductor (see Chap. VIII-A of the review of Blatter et al.\cite{Blatter94}), this conclusion is supported by the investigations of Prischepa et al. \cite{Prischepa05}, who found an angular dimensional cross-over of $H_c(\theta)$ at fixed temperature for Nb/Pd multilayers. Samples of odd and even numbers of normal/superconducting (N/S) bilayers of Pd/Nb (9 and 10, respectively, plus a capping Pd layer) were measured in the temperature range $T<T^*<T_{c0}$, where the square-root behavior of $H_{c\parallel}(T)$ indicates a two-dimensional behavior (for $T^*<T<T_{c0}$ a linear temperature law of $H_{c\parallel}(T)$ is observed, indicating three-dimensionality). Strong deviations from the Tinkham formula are obtained for $H_c(\theta)$ of the multilayer with an even number of bilayers. Only certain ranges of $H_c(\theta)$ follow Tinkham's prediction. For small angles, however, with $H_{c\perp}$ as a free parameter and for larger angles, with $H_{c\parallel}$ as the free parameter. The latter range is interpreted as an unusual three-dimensional mode, because the large angle tail can be described by the Lawrence-Doniach model.\\
The physical interpretation proposed is that for small angles the superconducting nucleus is localized in one period of the S/N structure, but for large angles, it is spread over more than one period by the perpendicular component of the external magnetic field, resulting in an object with a three dimensional feature (for a detailed discussion see the work of Prischepa et al.\cite{Prischepa05}). It is argued that, nevertheless, the Lawrence-Doniach description is not applicable, because it was deduced in the approximation of the homogeneous infinite medium. The applicability of the Tinkham formula is concluded to be the consequence of a relatively homogeneous order parameter in one S-layer.\\
This is very similar to our S/F bilayer, where the pairing wave function oscillates in the F-material due to the FFLO state, but is nearly constant in the S-layer. Allthough a S/F bilayer is not a multilayer, the oscillating pairing wave function generates strongly and weakly superconducting regions in the F-material. The perpendicular component of the external field shifts more and more 'weight' of the vortex into these strongly superconducting regions for increasing angle. Since our S/F bilayer is always in the two-dimensional regime, a Lawrence-Doniach model can not be applied to explain the measurements, but the Tinkham formula with a changing $H_{c\parallel}$, possibly arising from a varying effective superconducting thickness $d$ increasing from $d_S$ to $d_S+d_F$ for increasing angle, gives a reasonable description in the angular regime, where the segmentation of the vortex is expected.
\section{Conclusions}
In summary, in a S/F bilayer and a thin Nb film, we investigated the temperature dependence of $H_{c\perp}$ and $H_{c\parallel}$ by measurement of resistive transitions, and the angular dependence of $H_c$ by a non-resonant microwave absorption study.\\
Over a wide temperature range, the temperature dependence of $H_{c\perp}$ and $H_{c\parallel}$ follow the respective linear and square-root behavior, predicted by the Ginzburg-Landau theory. However, close to the critical temperature deviations are observed, and compared to those arising from anisotropies of the electron-phonon interaction and Fermi velocity in niobium.\\
We analyzed the results of the angular dependence of $H_c$ within the framework of Tinkham's theory of thin superconducting films. While the thin Nb film could be well described by this theory, the S/F bilayer data shows deviations at low inclination of the applied field to the film plane.\\
Based on the oscillations of the pairing wave function inside the ferromagnetic layer, induced by the quasi-one-dimensional FFLO-like state, and adopting the approach of a segmented vortex, as present in layered high-$T_c$ superconductors, we propose a new vortex structure, which reduces the energy in the system by alternating steps of vortex short-links through strongly superconducting regions and flux channeling through the weak superconducting minima of the Cooper pair density. Since the pairing wave function is damped as a function of the distance to the S/F interface, the Cooper pair density difference between strong and weak superconducting regions, and thus the possible energy gain, decreases. Therefore, we propose the inclination angle of the short-link part of the vortex to decay with increasing distance from the S/F interface.\\
Although the vortex structure proposed has some similarity to the segmented vortex in high-$T_c$ superconductors, the vortex dynamics seem to be different.\\
Moreover, we discuss our findings and interpretations in the context of investigations of an angular dimensional cross-over of the upper critical field in superconductor-normalconductor multilayers, where special vortex states are discussed to arise from the layered geometry.\\
While there are theoretical studies of the angular dependence of the upper critical field for the FFLO state in bulk superconductors, considering the influence of the modulation of the order parameter on the vortex state, there are so far no such calculations for the quasi-onedimensional FFLO-like state in S/F proximity effect systems. However, such theoretical considerations are strongly desirable, because an extension of the Tinkham formula to this situation seems not to be possible.
\section*{Acknowledgments}
The authors are grateful to S.~Heidemeyer, B.~Knoblich, and W.~Reiber for TEM sample preparation, to J.-M.~Kehrle for taking the TEM image, to W.~Reiber and S.~Gsell for the RBS measurements, and to G.~Obermeier and R.~Horny for technical support concerning the low temperature resistive measurements.\\
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under the Grant No. HO 955/9-1. R.M. and L.R.T. were supported in part by the Program of Competitive Growth of Kazan Federal University. V.I.Z. was partially supported by ERC advanced grant 'ASTONISH'. The IMDACMF investigations (A.L. and H.-A. K.v.N.) were partially supported by the Deutsche Forschungsgemeinschaft (DFG) within the Transregional Collaborative Research Center TRR 80 'From Electronics Correlations to Functionality' (Augsburg, Munich).\\
\section*{Appendix}
\subsection{Experimental Techniques}
\subsubsection{Sample Preparation and Characterization}
The S/F bilayer sample, investigated in the present work is part of a Nb/Cu$_{41}$Ni$_{59}$ thin-film sample series (S23) produced by magnetron sputtering at room temperature\cite{Sidorenko10}. All targets used in the preparation were first pre-sputtered for 10-15 minutes to remove possible contaminations. Afterwards, an $1$~mm thick commercial \{111\} silicon substrate (size $7$~mm x $80$~mm) was covered with an amorphous silicon buffer layer by RF sputtering to provide a clean surface for the subsequent layers. In the next step a thin niobium layer was produced by applying the 'spray technique' \cite{Zdravkov06,Sidorenko10,Sidorenko09}. In this technique the Nb target was continuously moved across the substrate during the DC sputtering process to ensure a layer of constant thickness and precise control of the growth rate, resulting in a flat niobium layer of thickness $d_S=14.1$~nm.\\
Subsequently, a wedge-shaped ferromagnetic layer was RF sputtered from a Cu$_{40}$Ni$_{60}$-alloy target by using the intrinsic spatial gradient of the deposition rate inside the chamber. Finally, to prevent degradation of the samples under atmospheric conditions, an amorphous silicon cap layer of about 5-10 nm thickness was deposited on top of the sample.\\
Individual samples were then cut perpendicular to the wedge gradient (36 slices, enumerated from the thick to the thin end of the wedge, usual width about $2.5$~mm) from the obtained layered structure. Due to the small thickness gradient of the wedge-like ferromagnetic layer and the small sample width, the thickness of the S and F-layer is regarded constant within each individual sample.\\
We have chosen the sample S23\texttt{\#}5 (size $4.4$~mm x $2.8$~mm), with $d_F=34.3$~nm. In this range of thicknesses, $T_{c0}$ becomes almost independent of $d_F$ - it is $T_{c0}~\approx$~6.4~K, for $d_F >23$~nm and, thus, for sample S23\texttt{\#}5. This does \emph{not} mean that interference effects of the pairing wave function in this range of thicknesses are absent, but only that the interference modulation of the flux of the pairing wave function through the S/F interface is too weak to influence the superconducting state in the whole S-layer. This statement can be justified by the behavior of $T_{c0}(d_F)$ for lower $d_S$ (\textit{e.g.} sample series S21 in our former work\cite{Sidorenko10}) where at comparable thicknesses $d_F$ interference effects are still observable. For details of the argumentation above, see Fig. 6 and Chap. IV of our former work\cite{Sidorenko10}. However, the interference effects are expected to decay as the amplitude of oscillation of the pairing wave function at the outer F-boundary decays with increasing $d_F$. Thus, the pairing wave function in the F-layer is more and more close to the one induced by the underlying FFLO proximity effect. Moreover, we expect the anisotropy of the pairing wave function inside the F-layer to be larger for constructive than for destructive interference (see Fig. 2 of our article on interference effects in S/F bilayers\cite{Sidorenko09}). Thus, we have chosen a relatively thick sample of the series, which should exhibit constructive interference.\\
To distinguish the effects, arising from the influence of the ferromagnetic layer, from those intrinsic to a thin niobium layer, a reference film (Nb5) was produced by the same deposition procedure, however, without the ferromagnetic layer on top, \textit{i.e.} a single niobium layer with constant thickness, which is sandwiched between the amorphous silicon buffer and cap layers. We cutted several parts from Nb5 for different measurements. The part Nb5/1 (size 7 mm x 4 mm) is used for low temperature measurements.\\
\begin{figure}
\includegraphics[width=\columnwidth]{TEM.pdf}
\caption{Cross-sectional HRTEM image of a part of the niobium film Nb5. The dark niobium layer shows a highly crystalline structure with Nb \{110\} planes. On the upper right side, lattice planes of the silicon substrate are visible and were used to confirm the scale.}
\end{figure}
One part of the reference film, Nb5, was subjected to cross-sectional High-Resolution Transmission Electron Microscopy (HRTEM) to check the thickness and quality of the layer. The cross-section specimen was prepared by conventional dimpling and ion thinning. The obtained HRTEM image is shown in Fig. 6. On the upper right side, the \{111\} planes of the silicon substrate can be clearly identified by their lattice constant of 3.134 \AA. A careful inspection also reveals the Si \{220\} planes with a spacing of 1.93 \AA, including an angle of $\approx$ 35$^\circ$ with the \{111\} planes which is in agreement with the theoretical value of 35.26$^\circ$.\\
The niobium layer is clearly visible due to the strong Z contrast to the silicon substrate. It shows a highly crystalline structure with lattice plane distances of 2.33 \AA. These distances can be attributed to the Nb \{110\} planes. The angle between these planes is $\approx$ 61.4$^\circ$, which corresponds to the theoretical value of 60$^\circ$. Therefore, the viewing direction in the niobium layer could be identified as [111].\\
Lattice types and constants are taken from literature\cite{Handbook}, angles and spacings between lattice planes are calculated according to the known crystal structure.\\
The thickness of the niobium layer is evaluated to be $d_S=$ 7.3 nm. We regard this value to be more accurate than the value of 6.8 nm, obtained by Rutherford Backscattering Spectroscopy (RBS) on Nb5/1 in \mbox{Chap. III-A} of our former work \cite{Sidorenko10}.\\
\begin{figure}
\includegraphics[width=\columnwidth]{RBS.pdf}
\caption{(Color online) RBS spectrum of the S/F bilayer S23\texttt{\#}4 (thicker next to S23\texttt{\#}5).}
\end{figure}
To determine the thicknesses and composition of the layers in the S/F bilayer sample we performed RBS with $\alpha$-particles at an energy of 3.5 MeV. However, we did not use S23\texttt{\#}5 to prevent altering of its properties by radiation damage. Instead, we investigated a subset of samples across the whole sample series and obtained the data of S23\texttt{\#}5 by linear interpolation of the results of S23\texttt{\#}4 and S23\texttt{\#}7. Figure 7 shows the RBS spectrum of S23\texttt{\#}4 together with the fit. The fit is in good agreement with the experimental data. The small unfitted feature at approximately 2.35 MeV is an artifact, arising from the sample holder. The different peaks are assigned to the corresponding layers. The shown spectrum is representative for all obtained spectra. For S23\texttt{\#}5 we obtain 14.1 nm and 34.3 nm for $d_S$ and $d_F$, respectively, and a composition of $41$~at\% copper and $59$~at\% nickel for the F-layer.
\subsubsection{Induced Microwave Dissipation by AC Magnetic Field (IMDACMF) Technique}
The non-resonant microwave absorption experiments, presented in Chap.~IV-B, have been performed in a Bruker ELEXYS 500 X-band EPR spectrometer. The microwave source feeds a 9.3~GHz rectangular H102 (also known as TE102) cavity. The sample is positioned at its center, where only the magnetic component of the microwave field is present. Moreover, the sample is exposed to collinear DC ($H_{DC}$) and AC ($H_{AC}$, amplitude 30~Oe, frequency~100 kHz) magnetic fields, applied perpendicular to the magnetic microwave field. The sample can be cooled to low temperatures using a continuous helium flow cryostat (ESR900, Oxford Instruments). The relative orientation of the magnetic DC and AC field with respect to the film plane of the sample (thin films on Si substrate) can be varied using a goniometer. The rotation axis is parallel to the microwave magnetic field (to keep the microwave magnetic field strength unchanged, see sketch in Fig. 1 in the work of Shaltiel \cite{Shaltiel09}) and the long side of the sample. This means for the S/F bilayer investigated in the present work, that it is rotated around the magnetically semi-easy axis\cite{Kehrle07} of the F-layer from its hard axis to its easy axis, which are parallel and perpendicular to the film surface, respectively\cite{Ruotolo04,Kehrle07,Zdravkov13}.\\
The samples were first zero-field cooled from room to liquid helium temperature. Initially, by comparing data obtained in both sweep directions of the magnetic field, we verified that the field sweep direction has nearly no influence on the signal. The data presented in \mbox{Chap.~IV-B} was recorded by sweeping the magnetic field from 0 to 16 kOe at a given angle. After reducing the field to zero, the angle was changed and the procedure was repeated again.\\
Since the Bruker EPR spectrometer used is calibrated in the cgs emu unit system, the applied magnetic fields are measured in Oe. In contrast, the theory in the present work and the results in Chap.~IV-A are given in the international SI system. To convert magnetic fields into the SI system the relation \mbox{$1$~Oe~$=10^3/(4\pi)$~A/m~$=79.58$~A/m} \cite{Goldfarb85} is used. Furthermore, with \mbox{$\mu_0=4\pi \cdot 10^{-7}\frac{\text{Vs}}{\text{Am}}$} one obtains the magnetic flux density related to $1$~Oe as $B=\mu_0H=10^{-4}$~Vs/m$^2$, \textit{i.e.} \mbox{$10$~kOe} yield \mbox{1 T}.
\subsubsection{Calibration of the IMDACMF Temperature Scale}
Usually, the sample temperature measured in the EPR spectrometer is found to be somewhat lower than the setpoint of the temperature controller, $T_{SP}$. Thus, the exact measurement temperature, $T_{ME}$, has to be calibrated. For this purpose, we compare the upper critical fields, $H_{c\perp}$ and $H_{c\parallel}$, obtained by IMDACMF (see Fig. 4), with the data for $H_{c\perp}(T)$ and $H_{c\parallel}(T)$, obtained in \mbox{Chap.~IV-A}, and their respective linear interpolations. The values, which lead to the best description (according to Eq.(6)) of the $H_c(\theta)$ data in Fig. 4 and the related temperatures are \mbox{$H_{c\perp}(5.90$~K~$)=0.900$~kOe} and \mbox{$H_{c\parallel}(5.90$~K~$)=16.300$~kOe} for Nb5/1 and \mbox{$H_{c\perp}(6.10$~K~$)=0.882$~kOe} and \mbox{$H_{c\parallel}(6.10$~K~$)=15.077$~kOe} for sample S23\#5. To obtain these values for Nb5/1, we allowed the parallel alignment to deviate from $\theta=0^\circ$ by $\Delta\theta=-3.2^\circ$, as the exact angle at which the maximum of $H_c$ occurs is not precisely known (for S23\#5 it is $\Delta\theta=0^\circ$).\\
However, the obtained temperatures are \textit{not} the actual measurement temperatures, $T_{ME}$, but the mid-point temperature, $T_{MP}$ of resistive transitions, which would lead to the same data points, if the results would have been obtained from resistive transitions at constant fields. As noted before, the data points are determined by evaluating the vanishing of the IMDACMF signal, when the sample enters the normal state. Consequently, $T_{ME}$ is higher than $T_{MP}$ by half of the transition width at constant field (about \mbox{$500$~mK~$/2 = 250$~mK for Nb5/1} and \mbox{$150$~mK~$/2 = 75$~mK for S23\#5)} and half (about \mbox{50 mK /2 = 25 mK)} of the temperature stability range of the EPR spectrometer. Thus, $T_{ME}=6.20$~K and $6.18$~K for the measurements of S23\#5 and Nb5/1 in Fig. 4 (a) and (c), respectively. In both cases, the setpoint temperature was $T_{SP}=6.45$~K, yielding an average difference of $0.26$~K between $T_{SP}$ and $T_{ME}$. Assuming the same temperature offset also for the measurements on S23\#5 at lower temperatures (see Fig.~2), for which $T_{SP}=5.00$~K, we estimate a corresponding $T_{ME}$ of $4.74$~K and (considering the transition width and temperature stability mentioned above) $T_{MP}$ to be $4.64$~K.\\
\subsection{Details on the Theoretical Framework}
\subsubsection{Derivation of the Tinkham Formula from the Linearized Ginzburg-Landau Equation}
The 'Tinkham formula' given in Eq.(6) can be derived from the linearized GL equation, obtained by neglecting the term proportional to $|\psi|^2\psi$ in the GL equation for the order parameter\cite{Tidecks90}, yielding
\begin{equation}
(1/2m')(-i\hbar\underline{\nabla}-e'\underline{A})^2\psi+\alpha\psi=0
\end{equation}
Here, $m'=2m$ is twice the electron mass, $e'=-2e$ twice the electron charge, $\underline{A}(\underline{r})$ the vector potential of the magnetic flux density with $\underline{B}(\underline{r})=\text{rot}(\underline{A}(\underline{r}))$, and $\alpha=-B_{cth}^2/(\mu_0n_{s0})$ with $n_{s0}=|\psi_0|^2$ the density of the particles described by $\psi$ in the absence of currents and magnetic fields.
Introducing the GL coherence length $\xi^2=-\hbar^2/(2m'\alpha)$ and $\Phi_0=h/(2e)$ one obtains
\begin{equation}
\left[\left(\frac{\underline{\nabla}}{i}-\frac{2\pi\underline{A}}{\Phi_0}\right)^2-\frac{1}{\xi^2}\right]\psi=0
\end{equation}
By choosing a coordinate system, in which $x$ is measured normal to the film from its midplane and a magnetic field lying in the xz-plane, the magnetic field is given by \mbox{$\underline{H}=H(\hat{\underline{x}}\text{sin}(\theta)+\hat{\underline{z}}\text{cos}(\theta))$} with $H=|\underline{H}|$. For a second order phase transition, in a first approximation, the magnetization $\underline{M}$ of a superconductor can be neglected in the direct vicinity of the critical magnetic field, so that $\underline{B}=\mu_0(\underline{H}+\underline{M})\approx\mu_0\underline{H}$. Thus, a vector potential with only a y-component corresponding to this field can be chosen as
\begin{equation}
\underline{A}(\underline{r})=\mu_0H(x\text{cos}(\theta)-z\text{sin}(\theta))\hat{\underline{y}}
\end{equation}
Inserting $\underline{A}$ from Eq.(11) into Eq.(10) yields a differential equation, which is hard to solve. However, with several simplifying assumptions, especially that $\psi$ is independent of $x$ (justified by $d\ll\xi$ in the thin film limit $d\rightarrow0$), it is possible to obtain\cite{TinkhamBook}
\begin{equation}
\begin{aligned}
-\frac{\text{d}^2\psi}{\text{d}z^2}+\bigg(\frac{2\pi\mu_0H\text{sin}(\theta)}{\Phi_0}\bigg)^2&z^2\psi=\\
\Bigg[\frac{1}{\xi^2}-\bigg(&\frac{\pi d\mu_0H\text{cos}(\theta)}{\sqrt{3}\Phi_0}\bigg)^2\Bigg]\psi
\end{aligned}
\end{equation}
The structure of this equation is completely equivalent to the one-dimensional Schr\"odinger equation of the harmonic oscillator, describing a particle of mass $m$ in a harmonic potential $Dx^2/2$ (with $D$ the spring constant), given by\cite{Gasiorowicz74}
\begin{equation}
\left(-\frac{\hbar^2}{2m}\frac{\text{d}^2}{\text{d}x^2}+\frac{m\omega^2}{2}x^2\right)u(x)=Eu(x)
\end{equation}
with the angular frequency $\omega=\sqrt{D/m}$ and the eigenvalues \mbox{$E=(n+1/2)\hbar\omega$} with \mbox{$n= 0,1,2\ldots$} the quantum number. In this case the eigenvalue $E$ is given by $(n+1/2)$ multiplied by twice the square root of the product of the prefactor of $-\text{d}^2/\text{d}x^2$ and the prefactor of $x^2$, that means $E=(n+1/2)\cdot2\left\{\left[\hbar^2/\left(2m\right)\right]\left[m\omega^2/2\right]\right\}^{1/2}$.\\
Applying this procedure to Eq.(12), identifying $x$ with $z$, yields
\begin{equation}
E=(n+1/2)\cdot2\left[1\cdot\left(\frac{2\pi\mu_0H\text{sin}(\theta)}{\Phi_0}\right)^2\right]^{1/2}
\end{equation}
On the other hand, it is
\begin{equation}
E=\frac{1}{\xi^2}-\bigg(\frac{\pi d\mu_0H\text{cos}(\theta)}{\sqrt{3}\Phi_0}\bigg)^2
\end{equation}
For $n=0$ the magnetic field for a given $\theta$ becomes maximal, so that $H=H_c(\theta)$, yielding
\begin{equation}
\left|\frac{2\pi\xi^2\mu_0}{\Phi_0}H_c(\theta)\text{sin}(\theta)\right|+\bigg(\frac{\pi\xi d\mu_0}{\sqrt{3}\Phi_0}H_c(\theta)\text{cos}(\theta)\bigg)^2=1
\end{equation}
Finally, equating the coefficients in Eq.(16) with Eqs.(3) and (5) results in Tinkham's formula, given in Eq.(6).\\
\subsubsection{Extensions of the Ginzburg-Landau Theory and the Theory of Type II Superconductors}
Extensions of the microscopic version of the GL theory and the theory of type II superconductors in a magnetic field towards lower temperatures were carried out by Maki\cite{Maki64,Maki64-2,Maki66}, Maki and Tsuzuki\cite{Maki65}, de Gennes\cite{deGennes64}, Caroli et al.\cite{Caroli66}, Tewordt\cite{Tewordt63,Tewordt65,Tewordt64,Tewordt65-2}, Neumann and Tewordt\cite{Neumann66,Neumann66-2}, Werthamer\cite{Werthamer63}, Helfland and Werthamer\cite{Helfland64,Helfland66}, Werthamer et al.\cite{Werthamer66}, and Werthamer and McMillan\cite{Werthamer67}. The topic is reviewed by Werthamer\cite{Werthamer69}, Cyrot\cite{Cyrot73}, and Fetter and Hohenberg\cite{Fetter69}. The ranges of validity of the extensions of the GL theory are summarized in Fig. 6 of Werthamer's review\cite{Werthamer69}. The ranges of applicability of the extensions to the description of type II superconductors are given in Fig. 13 of the Fetter and Hohenberg review\cite{Fetter69}. There is a range of applicability in a certain region of magnetic fields close to the $H_{c2}(T)$ line down to zero temperature. Nevertheless, there is no application of the results to get a 'Tinkham-like' formula for an extended temperature range (as far as known to the authors). The reason may be the complexity of the theoretical expressions, which often only allow a numerical solution.\\
\subsubsection{Niobium - An Intermediate Coupling Superconductor}
In the present work, Nb is used as S-material. According to Finnemore et al.\cite{Finnemore66} Nb is not a weak coupling, but an intermediate coupling superconductor. A quantity, characterizing the strength of the electron-phonon coupling, is the parameter $\lambda$, describing the effective mass enhancement, $m^*/m$, from the effective mass $m$ of the electron determined by the band structure, due to the electron-phonon interaction, given by\cite{McMillan68,Scalapino69} $m^*/m=1+\lambda$. The value of $\lambda$ for Nb is determined to be in the range of\cite{McMillan68,Weber91,Carbotte90} $0.8$ to $1.2$, which is between the values\cite{Carbotte90,Bergmann73} for In ($\lambda=0.8$) and Hg ($\lambda=1.6$). Indium can be regarded as almost weak coupling superconductor, while mercury is a strong coupling superconductor.\\ Another measure for the strength of the electron-phonon interaction is the ratio $2\mathbf{\Delta}(0)/(kT_{c0})$, where $\mathbf{\Delta}(0)$ is the energy gap at zero temperature. The prediction of the BCS theory, valid for weak coupling superconductors, of this ratio is\cite{TinkhamBook} $3.5$. For Nb values between $3.6$ and $3.8$ are obtained from experiments\cite{Buckel94,Gladstone69}, which are more close to the BCS value obtained for Sn and In, than to $4.3$ and $4.6$ obtained for the strong coupling superconductor Pb and Hg, respectively, obtained from tunneling experiments\cite{Buckel94}.\\
Werthamer and McMillan\cite{Werthamer67} calculated the strong coupling corrections to $H_{c2}(T)$ and carried out a numerical computation for Nb. They found, that the strong coupling effects constitute only a negligible portion of the discrepancy between the weak coupling theory and the experimental observation, which is mainly caused by Fermi surface anisotropy.\\
Thus, we will apply the weak coupling results of Chap.~III of the main text to the data for Nb films and the S/F bilayer of the present work.\\
\subsubsection{Validity Conditions for the Tinkham Formula}
With the parameters obtained in the main text, we now investigate, whether the conditions for the validity of Tinkham's formula are fulfilled, \textit{i.e.} if $d<\sqrt{5}\lambda(T)$ and $d<<\xi(T)$. Since studies of $\lambda(T)$ and $\xi(T)$ are not available for S/F bilayers, this can strictly only be done for Nb5/1. However, we will test the conditions also for S23\#5 under different assumptions.\\
The magnetic penetration depth of thin superconducting Nb films has been measured\cite{Siegel,Lemberger}. For film thicknesses between $7$~nm and $20$~nm, $\lambda(T=0~$K$)$ decreases from about $240$~nm to $140$~nm (see Fig. 6 of the work of Gubin et al.\cite{Siegel}). Since $\lambda(T)$ increases for increasing temperature, the values of $\lambda(T)$ are expected to be always much larger than the film thickness of $7.3$~nm in Nb5/1 and a Nb layer with a thickness of $14.1$~nm, as present in S23\#5. We expect, that $\lambda(0)=0.5^{1/2}\chi^{-1/2}\lambda_L(0)$ is increasing for a Nb film in the presence of an F-layer ($\lambda_L(0)$ should not change, $\chi$ should decrease, because $\xi_0$ increases, for details see below). From the phenomenological GL theory one gets\cite{Tidecks90} $\lambda(T)=(m'/(e'^2\mu_0\left|\psi_0\right|^2)^{1/2}$, where $\left|\psi_0\right|^2=n_{s0}$. Since the pairing wave function in the F-layer is smaller than in the S-layer, \textit{i.e.} the superconducting charge carrier density is reduced, one expects (identifying $|\psi_0|^2$ with $|\Phi_{F}|^2$) an even larger magnetic penetration depth there. Thus, in the investigated samples the condition $d<\sqrt{5}\lambda(T)$ is expected to be fulfilled for all temperatures.\\
Next, we calculate the GL coherence length for Nb5/1 and a freestanding Nb-film of thickness $14.0$~nm (similar to the one present in S23\#5). To calculate $\xi(T)$, we use that $\xi(0)$ is $9.7$~nm and $10.5$~nm for Nb5/1 and the Nb-film of thickness $14.0$~nm, respectively (see Chap.~IV-A of the main text). The critical temperatures of the films are $5.95$~K and $8.00$~K, respectively. Using $\xi(T)=\xi(0)(T_{c0}/(T_{c0}-T))^{1/2}$, we obtain $108$~nm and $22$~nm at $T=T_{MP}=5.90$~K and 6.10~K for Nb5/1 and S23\#5, respectively. Thus, $d=d_S<<\xi(T)$ is fulfilled for Nb5/1 and at least $d_S<\xi(T)$ is fulfilled for a freestanding Nb-film with similar $d_S$ as the one in S23\#5.\\
For the further discussion, we now calculate BCS coherence lengths according to the expression given in Chap.~III of the main text. Using $v_F=2.768\cdot10^5$~m/s for Nb, according to Weber et al.\cite{Weber91}, and inserting the respective critical temperatures yields $\xi_0=64.0$~nm and $47.7$~nm for Nb5/1 and the freestanding Nb film of $14.0$~nm, respectively. With $\xi(0)=0.74\chi^{1/2}\xi_0$ (see Eq.(7)), we then get $l=2.1$~nm and $3.5$~nm, respectively. Thus, both Nb films are in the dirty limit ($l<<\xi_0$).\\
To estimate $\xi(T)$ for the Nb film in S23\#5, we assume that its critical temperature is only suppressed due to the proximity effect by the F-layer. Thus, the $14.0$~nm freestanding Nb film, however, with a suppressed $T_{c0}$ of $6.34$~K, is a suitable reference system. For this (fictive) film, we obtain, according to Eq.(7), $\xi(0)=11.9$~nm, using $\xi_0=60.2$~nm and $l=3.5$~nm. Consequently, we obtain $\xi(T_{MP}=6.10$~K$)=61.3$~nm and, thus, again it is $d=d_S<<\xi(T)$.\\
To get an estimate of $\xi(T)$ for the whole sample S23\#5, we use $\xi(0)=12.8$~nm, as obtained in \mbox{Chap.~IV-A} of the main text, yielding $\xi(T_{MP}=6.10$~K$)=65.9$~nm, so that $d=d_S+d_F<\xi(T)$.\\
Here, we do not consider an enhancement factor to the slope $d\mu_0H_{c\perp}/dT$, calculated for Nb by Butler \cite{Butler80} (see the discussion by Weber et al. \cite{Weber91}, in our case the factor for the dirty limit \cite{Butler80} would be appropiate). This would lead to a slightly larger value of $\xi(0)$ and, thus, $l$. However, this would not change the presented conclusions.\\
The expressions for the magnetic penetration depth and the GL coherence entering the derivation of the Tinkham formula are those in a weak magnetic field\cite{Tinkham, TinkhamBook}, given in Chap. III of the main text. Thus, it is not necessary to consider a possible magnetic field dependence of these quantities. According to Douglass\cite{Douglass61}, for thin superconducting films in a parallel magnetic field, $\lambda(T,H)$ is larger than $\lambda(T,0)$ and approaches infinity for $H\rightarrow H_c$. Kogan proposed a magnetic field dependence of the coherence length $\xi(T,H)<\xi(T,0)$\cite{Kogan86}, which can be neglected in the dirty case, and a resulting influence on the superconducting transition temperature \cite{Kogan86,Kogan87}. However, this theory is controversially discussed in the literature\cite{Scotto91,Hara94,Seguchi92}. According to Kogan\cite{Kogan86} the proposed enhancement of the transition temperature should occur for $d_S$ below a critical thickness, $d_c$. For Nb5/1 it is, however, $d_S>d_c=5.7$~nm. For S23\#5, there is no uniform $l$, so $d_c$ cannot be calculated. In any case, we do not observe any evidence for this enhancement in both samples.\\
\subsection{Oscillation Properties of the Pairing Wave Function in the Quasi-One-Dimensional FFLO-Like State in S/F Bilayers}
The oscilatory behavior of $\Phi_F$ has been analyzed theoretically in detail\cite{Tagirov98}. The oscillation wavelengths and decay lengths for the case of a clean and dirty ferromagnet are summarized in Chap. IV of our previous work\cite{Sidorenko10}. Moreover, the topic is discussed in detail in the Appendix of the doctoral thesis of Kehrle\cite{Kehrle12}, where it is also shown, that the decay length in the clean case is given by twice the electron mean free path, $l_F$ in the F-material (the factor of 2 was omitted in our previous work \cite{Sidorenko10}).\\
For $\Phi_F(x_F)\propto \text{cos}(k_{FM}x_F)$ the pairing wave function has its first node at $k_{FM}x_{F1}=\pi/2$ and its first minimum at $k_{FM}x_{F2}=\pi$. Here, $k_{FM}=2\pi/\lambda_{FM}$ is the wave number and $\lambda_{FM}$ the oscillation wavelength. We thus get $x_{F1}=\lambda_{FM}/4$ and $x_{F2}=\lambda_{FM}/2$. Since the experimental results for oscillatory behavior of $T_c(d_F)$ are best described by the extension of the dirty case theory towards the clean case, as discussed in Chap. IV of our previous work\cite{Sidorenko10}, we apply the clean case expression for the oscillation wavelength, \textit{i.e.} $\lambda_{FM}=\lambda_{F0}=2\pi\xi_{F0}$. where $\xi_{F0}=\hbar v_F/E_{ex}$, with $E_{ex}$ the exchange splitting energy. According to our previous work\cite{Sidorenko10}, for sample series S23, it is $\xi_{F0}=10.8$~nm, yielding $\lambda_{FM}=67.9$~nm and, thus, $x_{F1}=17.0$~nm and $x_{F2}=33.9$~nm.\\
As discussed in our previous work\cite{Sidorenko09}, for $d_F=x_{F1}$, the reflection of $\Phi_F$ at the outer border of the F-layer leads to interference effects yielding the first minimum of $T_c(d_F)$. The experimental results for $T_c(d_F)$ of sample series S23 are shown in Fig. 6 of our previous work\cite{Sidorenko10}, yielding this minimum at $d_F=7.0$~nm. The experiments are well described by the theory. Thus, there is a phase shift of the pairing wave function at the S/F interface due to boundary conditions, so that $\Phi_F(x_F)\rightarrow \Phi_F(x_F+10$~nm~$)$ and, thus, $x_{F1}\rightarrow 7.0$~nm and $x_{F2}\rightarrow 23.9$~nm. Consequently, the distance of the first minimum of $\Phi_F(x_F)$ to the S/F interface is 23.9~nm.\\
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1604.03327
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\section{Introduction}
Cosmic inflation is a successful, well-studied paradigm which offers an elegant solution to many cosmological problems \cite{Guth:1980zm}. Besides, cosmological perturbations resulting from quantum fluctuations during inflation generate the seeds of the structures which we observe today. While many key predictions of inflation have been verified by CMB and LSS observations, still the primordial gravitational waves or B-mode polarization remains elusive \cite{Ade:2015lrj}. In 2014, the lensing B-mode signal has been directly detected by Polarbear \cite{Ade:2014afa} and shortly after, BICEP2 \cite{Ade:2014xna} pushed its constraints to a level that is competitive with temperature. The current upper limit on tensor fluctuations ($r_{0.05} < 0.07$ at $95 \%$ CL) comes from the latest joint analysis of Planck and BICEP2/Keck array measurements \cite{Array:2015xqh}. We are living in the golden age of observational cosmology and the quest for inflationary gravitational waves is the major goal of several observational projects. The road ahead seems promising for the detection of primordial gravitational waves and the discovery of new physics underlying inflation \cite{Kovetz:2015pia, Creminelli:2015oda, Schwarz:2015cma}. In case of single scalar field scenarios of inflation, by observing the primordial gravitational wave, we can determine both the energy scale of inflation, $V^{\frac14}\simeq10^{16}\textmd{Gev}\big(\frac{r}{0.01}\big)^{\frac14}$, and the inflaton field excursion, $\Delta\varphi\gtrsim\big(\frac{r}{0.01}\big)^{\frac12}M_{\rm pl}$ \cite{Lyth:1996im}. However, that relations can in principle be evaded in cases that the gravitational waves are coupled to some new fields during inflation which has a negligible contribution to the scalar sector.
Axion fields are abundant in string theory and therefore very well-motivated candidates for the inflaton field. Enjoying shift symmetry, their effective potential is protected from dangerous quantum corrections which guaranteed the flatness of the potential.
The axion field, $\varphi$, is classically coupled to gauge fields through a topological term $F\tilde F$, which is hence invariant under shift transformations of the form $\varphi\to \varphi +\varphi_0$ for an arbitrary $\varphi_0$ shift. On the other hand, quantum effects (\textit{i.e.} instanton contributions) induce a perturbatively exact cosine-type potential for the axion $V(\varphi)= \mu^4 (1+\cos(\varphi/f))$ which breaks the continuous shift symmetry to the discrete symmetry of $\varphi\to \varphi+2\pi f$ \cite{Weinberg-QFT-II}. Here, $\mu$ is the scale of the (approximate) shift symmetry breaking and $f$ is the axion decay constant. Since super-Planckian axion decay constant is hard to realize in string theory \cite{Svrcek:2006yi, Banks:2003sx}, the axion potential is under theoretical control if $H\!<\!f\!<\!M_{\rm pl}$. The lower limit on $f$ comes from the fact that the axion theory arises from integrating out modes heavier than $f$, hence, it can only work in inflation scales lower than that. For an exhaustive review of axion inflation see \cite{Pajer:2013fsa} and a comprehensive survey of axion inflation in string theory is presented in \cite{Long:2016jvd}.
The first model of axion inflation has been proposed more than 25 years ago in \cite{Freese:1990rb} and called \textit{natural inflation}. Although natural inflation could rectify the naturalness problem by means of the shift symmetry and radiative stability of the potential, does not fully resolve it. In fact, to have a successful inflationary background, this model needs a super-Planckian $f$ parameter which is not a natural scale within particle physics models. Natural inflation is now disfavoured by the joint BICEP2/Keck Array and Planck data. One of the most popular and well-motivated axion models of inflation is \textit{monodromy inflation} \cite{Silverstein:2008sg,Flauger:2009ab, McAllister:2014mpa, Easther:2013kla, Flauger:2014ana}. This inflationary mechanism is a string theoretic construction based on a single axion field and motivates a broad class of axion potentials of the form $V(\varphi)=\mu^{4-p}\varphi^p+\Lambda^4 e^{-c(\frac{\varphi}{\varphi_0})^{p_{\Lambda}}}\cos\bigg(\frac{\varphi_0}{f}(\frac{\varphi}{\varphi_0})^q+\theta_0\bigg)$. While the underlying periodicity of the theory continues to protect the inflaton potential from corrections, the periodic field space of the axion is now effectively unfolded due to the monodromy.
Besides their appealing theoretical stability, models of axion inflation are attractive phenomenologically due to their ability to generate observable primordial gravitational waves. These models can create detectable gravitational waves either as vacuum fluctuations of a large field model or sourced perturbations through their interaction with the gauge fields. Axions can naturally couple to gauge fields, Abelian or non-Abelian, and creates a richer phenomenology which leads to new observational and theoretical features. One possible construction is an axion driven inflation which interacts with a U(1) gauge field via $\varphi F\tilde F$. The \textit{Abelian} gauge field quanta is mixed to the gravitational waves at the \textit{nonlinear level} through the interaction $\delta A+\delta A\rightarrow\delta g$. That mechanism generates sourced chiral gravitational waves in addition to the standard (unpolarized) vacuum fluctuations \cite{Sorbo:2011rz}. However, the U(1) gauge field quanta is also coupled to the inflaton via $\delta A+\delta A\rightarrow\delta\varphi$ and generates large amounts of non-Gaussianity. In other words, the resulting sourced gravity wave signal is correlated to the large scale non-Gaussianity. Therefore, once the CMB constraints are imposed, the gravitational waves sourced by the U(1) gauge field are undetectable \cite{Barnaby:2010vf, Barnaby:2011qe, Barnaby:2011vw}. Authors of \cite{Namba:2015gja} evades that issue by considering an inflationary scenario in which the U(1) gauge field is coupled to a fast rolling axion field while both fields are only gravitationally coupled to the inflaton field.
Another natural possibility to study as the matter content of axion inflation is a (dark) SU(2) gauge field, $A^a_{~\mu}$. Thanks to the SU(2) algebra in such scenarios, there exists a homogeneous and isotropic field configuration for the gauge field \cite{Maleknejad:2011jw, Maleknejad:2011sq, Maleknejad:2012fw}. Therefore, the mixing between the \textit{non-Abelian} gauge field and perturbations in the scalar and tensor sectors are at the \textit{linear order} and coming from different fluctuations. Hence, the enhancement of gravitational wave and the modification in the scalar perturbations are uncorrelated. One of the possible realizations of axion inflationary models involving non-Abelian gauge fields is \textit{chromo-natural inflation} \cite{chromo-natural-short}.
In this model, the axion has a standard cosine potential and is coupled to the gauge field with $-\frac{\lambda}{4f}\textmd{tr}(F^a_{\mu\nu}F_a^{\mu\nu})$. The gauge field has an energy density $\rho_{_{\rm YM}}\sim \epsilon M_{\rm pl}^2H^2$ and $\frac{\lambda}{f}\sim\frac{\mathcal{O}(10^3)}{M_{\rm pl}}$ which leads to slow-roll inflationary background, without requiring super-Planckian $f$ \cite{Adshead:2012qe, Martinec:2012bv, Maleknejad:2012dt, Adshead:2013qp}.
Moreover, the tensor fluctuations of gauge field source a chiral spectrum of gravitational waves. Despite its technical naturality, chromo-natural inflation has been disfavored by Planck data \cite{Dimastrogiovanni:2012ew, Adshead:2013nka}. In particular, the scalar perturbations of the model are stable if the magnetic to electric ratio of the vev gauge field is more than $\sqrt{2}$, and it is otherwise unstable. The source of instability in the scalar sector is coming from the interaction term $\frac{\lambda}{f}\big(\frac{\rho_{_{\rm YM}}}{H^2}\big)^{\frac12}\frac{1}{k\tau}$ which gets relevant at the intermediate regime $-k\tau=\frac{\lambda}{f}\big(\frac{\rho_{_{\rm YM}}}{H^2}\big)^{\frac12}\sim\mathcal{O}(10^{2})$.
The tensor perturbations are however enhanced at large magnetic to electric ratio. Therefore, depending on the parameters, this model can either overgenerate gravitational waves or predicts a too red spectral tilt \cite{Adshead:2013nka, Obata:2016tmo}.
In this paper, we focus on a single field axion inflation in the presence of an SU(2) gauge field with a small vev ($\rho_{_{\rm YM}}\lesssim\epsilon^2M_{\rm pl}^2H^2$). For the sake of generality, here we consider an arbitrary potential for the axion that is able to support the slow-roll inflation. The gauge field is coupled to the axion through a Chern-Simons interaction $-\frac{\lambda}{4f}\textmd{tr}(F^a_{\mu\nu}F_a^{\mu\nu})$ with $\frac{\lambda}{f}\sim\mathcal{O}(10)$. This interaction with the gauge field is expected as it is compatible with all the symmetries of the axion. Moreover, due to the SU(2) algebra, the gauge field can have an isotropic and homogeneous field configuration. It has a negligible effect on the background evolution as $\rho_{_{\rm YM}}\lesssim\epsilon^2M_{\rm pl}^2H^2$ and the coupling between the gauge field and the axion is small. The quantum fluctuations of the gauge field, however, makes a significant contribution to the cosmic perturbation. In particular, the spin-2 fluctuations of the perturbed gauge field linearly coupled to the primordial gravitational waves and explicitly breaks the parity between the left- and right-handed polarization states. Therefore, our gravity waves has a circularly polarized power spectrum proportional to $\frac{\rho_{_{\rm YM}}}{M_{\rm pl}^2H^2}$ which can be comparable to the power spectrum of its vacuum fluctuations. That results in parity odd CMB correlations between E and B-modes and T and B-models. Moreover, the perturbed gauge field has some scalar degrees of freedom which are linearly coupled to the curvature perturbations via $\frac{\lambda}{f}\big(\frac{\rho_{_{\rm YM}}}{M_{\rm pl}^2H^2}\big)^{\frac12}\frac{1}{k\tau}$. In this scenario, the interaction terms are more relevant after horizon crossing, $-k\tau\sim\mathcal{O}(0.1)$. Therefore, the scalar sector is modified by the SU(2) gauge field at large scales. Our scalar perturbations are stable and almost adiabatic in case that the background magnetic to electric ratio of the gauge field is more than $\sqrt{2}$ while otherwise deviates from the adiabatic solution. There are parameter regimes in which the gauge field, at the same time, generates a detectable chiral gravitational wave signal and has a negligible contribution to the scalar fluctuations, in agreement with the current CMB observations. Hence, it satisfies in a modified version of the Lyth bound and the tensor power spectrum does not specify the scale of inflation.
This paper is organized as follows. Section \ref{basic-setup} presents the basic setup of the model. In section \ref{perturbation}, we classify its cosmic perturbation theory and work out the field equations. The scalar and tensor perturbations are studied in section \ref{Scalar perturbations} and \ref{tensor-section} respectively. Finally, we summarize in section \ref{conclusion}. Some technical details are presented in appendices A and B.
\section{Theoretical setup}\label{basic-setup}
We consider a generic axion-driven inflation model with a gauge field sector, both minimally coupled to Einstein gravity
\begin{eqnarray}\label{action}
\mathcal{L}_{\textmd{inf}} =\frac{R}{2}-\frac12\partial_\mu\varphi\partial^\mu\varphi-V(\varphi)+\mathcal{L}_{A}(A^a_{\mu},g_{\mu\nu},\varphi)\,,
\end{eqnarray}
where $\varphi$ is the axion field, $V(\varphi)$ is the axion potential and $\mathcal{L}_{A}$ is the gauge field sector. Here and throughout, the reduced Planck mass is set to unity, unless otherwise specified. For the purpose of this work and in order to be as model-independent as possible, $V(\varphi)$ is an arbitrary potential that is able to support the slow-roll inflation. In addition to the inflaton, we have a SU(2) gauge field which through the Chern-Simons interaction couples to the axion field
\begin{eqnarray}\label{action-A}
\mathcal{L}_{A}(A^a_{\mu},g_{\mu\nu},\varphi)=-\frac{1}{4}\bigg(F^a_{\mu\nu}F_a^{\mu\nu}+\frac{\lambda}{f}\varphi\ F^a_{\mu\nu}\tilde{F}_a^{\mu\nu}\bigg)\,,
\end{eqnarray}
where $\lambda$ is a dimensionless parameter, $f$ is the axion decay constant and $\tilde{F}^{a\mu\nu}=\frac12\epsilon^{\mu\nu\lambda\sigma}F^a_{\lambda\sigma}$.
The gauge field strength tensor is
\begin{equation}
F^a_{~\mu\nu}=\partial_\mu A^a_{~\nu}-\partial_\nu A^a_{~\mu}-g\epsilon^a_{~bc}A^a_{~\mu}A^b_{~\nu},
\end{equation}
where $g$ is the gauge coupling, $a, b, c...$ are the indices of the $su(2)$ algebra with generators $\{T_a\}$, defined by the commutation relation $[T_a,T_b]=i\epsilon_{ab~}^{~~c}T_c$.
\subsection{Geometry of the isotropic configuration}
In the flat FLRW metric
\begin{equation}
\label{FLRW}
ds^2=-dt^2+a(t)^2\delta_{ij}dx^{i}dx^{j},
\end{equation}
and after choosing the temporal gauge for the gauge field $(A^a_0=0)$, we have the following isotropic and homogeneous field configuration
\begin{equation}\label{ansatz}
\varphi=\varphi(t) \quad \textmd{and}\quad A^a_{~\mu}(t)= \psi(t)e^a_{~\mu},
\end{equation}%
where $\{e^{\alpha}_{~\mu}\}$ are tetrads of FRW metric (with $e^{a}_{~0}=0$) and the effective field value of the gauge field $\psi$ is a pseudo-scalar. The tetrad fields are the noncoordinate orthonormal basis satisfying
\begin{equation}
g_{\mu\nu}=e^{\alpha}_{~\mu}e^{\beta}_{~\nu}\eta_{\alpha\beta},
\end{equation}
where $\alpha,\beta=0,1,2,3$ and $\eta_{\alpha\beta}$ is the Minkowski metric. For the FRW metric, $\{e^{\alpha}_{~\mu}\}$ are specified as
\begin{equation}
e^{0}_{~\mu}=n_{\mu} \quad \textmd{and} \quad e^{a}_{~\mu}=a(t)\delta^a_{\mu} \quad a=1,2,3,
\end{equation}
where $n^{\mu}=(1,0,0,0)$ is the 4-velocity of the comoving observer.
The reason for the existence of such a homogeneous and isotropic solution is as follows \cite{Maleknejad:2011jw, Maleknejad:2011sq}.
Working in the temporal gauge $A^a_0=0$, under the action of an infinitesimal rotation $R(\vec{\theta})=e^{\vec\theta.\vec{M}}$, $A^a_i$ transforms as
\begin{equation}\label{trans-R}
A^a_i\xmapsto{R} (R(\vec{\theta})A^a)_i=(\delta^j_i-\theta_k\epsilon_i^{~jk})A^a_{j},
\end{equation}
where $M_i$s are generators of $SO(3)$ in 3-dimensional vector space, $(M_i)_{jk}=-\epsilon_{ijk}$.
On the other hand, setting $A^a_0=0$, only fixes $A^a_i$ up to global SU(2) gauge transformations\footnote{Under the action of a generic (local) gauge transformation $\Lambda(\lambda(t,\textbf{x}))=e^{i\lambda_aT^a}$, the gauge field transforms as $A_{\mu}\mapsto A_{\mu}-\frac{i}{g}\Lambda^{-1}D_{\mu}\Lambda$, where $D_{\mu}=\partial_{\mu}+ig A_{\mu}$ is the covariant derivative.} of the form $\Lambda(\lambda)=e^{i\lambda_aT^a}$. The residual (global) gauge transformation is in the form
\begin{equation}\label{trans-Gauge}
A^a_i\xmapsto{\Lambda} \big(\Lambda^{-1}(\vec{\lambda})A_i\Lambda(\vec{\lambda})\big)^{\!a}=(\delta^a_b-\lambda^c\epsilon^{a}_{~bc})A^b_{i}=R(\vec{\lambda})^a_{~b}A^b_{i}.
\end{equation}
From the combination \eqref{trans-R} and \eqref{trans-Gauge} we find that for all $\theta_k$s there exists a $\lambda_c=-\delta^k_c\theta_k$, so that $A^a_i\propto e^a_{~i}$ is invariant under the action of their combination. That then explains the existence of the isotropic and homogeneous configurations of the form \eqref{ansatz}. The isomorphism of $su(2)$ and $so(3)$ Lie algebras plays a key rule here and makes the identification of algebra and spatial indices of the local frame possible.
\subsection{Background evolution and slow-roll inflation}
The isotropic and homogeneous solution in \eqref{ansatz} gives the electric and magnetic field components as
\begin{equation}
E^a_i=-(H\psi+\dot{\psi})\delta^a_i \quad \textmd{and} \quad B^a_i=-g\psi^2\delta^a_i.
\end{equation}
The background energy densities of the axion and the gauge field are respectively
\begin{subequations}
\begin{align}
\rho_{\varphi}&=\frac12\dot{\varphi}^2+V(\varphi),\\
\rho_{_{\rm YM}}&=\frac12\bigg(\vec{E}^a.\vec{E}_a+\vec{B}^a.\vec{B}_a\bigg).
\end{align}
\end{subequations}
The field equations of $\varphi$ and $\psi$ are
\begin{subequations}\label{c-n.e.o.m}
\begin{align}
&\ddot\varphi+3H\dot\varphi+V_{\varphi}=-3\frac{\lambda g}{f} \psi^2(\dot\psi+H\psi)\,,\\\label{eq-psi}
&\ddot\psi+3H\dot\psi+(2H^2+\dot H)\psi+2g^2\psi^3=\frac{\lambda g}{f}\psi^2\dot\varphi\,,
\end{align}
\end{subequations}
which are coupled by the Chern-Simons interaction term.
Moreover, the continuity equations are
\begin{subequations}
\begin{align}
&\dot\rho_{\varphi}+3H(\rho_{\varphi}+P_{\varphi})=-\frac{\lambda }{f}\dot{\varphi}\vec{E}^a.\vec{B}_a,\\\label{rho-ym}
&\dot\rho_{_{\rm YM}}+4H\rho_{_{\rm YM}}=\frac{\lambda }{f}\dot{\varphi}\vec{E}^a.\vec{B}_a.
\end{align}
\end{subequations}
As we see explicitly in \eqref{rho-ym}, in the absence of the interaction term with the axion, $\rho_{_{\rm YM}}$ damps like $a^{-4}$. However, the Chern-Simons interaction breaks the conformal symmetry and prevents the damping of the gauge field (when $\dot\varphi\neq0$).
Considering the standard slow-roll inflation, we can quantify the slow-roll dynamics by
\begin{equation}\label{SL-H}
\epsilon\equiv-\frac{\dot H}{H^2}\quad \textmd{and} \quad \eta\equiv-\frac{\ddot H}{2H\dot H}=-\frac{(\epsilon H^2\dot{)}}{2\epsilon H^3}.
\end{equation}
We also demand the gauge field to have a slow varying evolution, therefore from \eqref{eq-psi} we realize that the dimensionless time derivatives of $\psi$
\begin{equation}\label{vartheta}
\epsilon_{\psi}\equiv \frac{\dot\psi}{H\psi} \quad \textmd{and} \quad \eta_{\psi}\equiv-\frac{\ddot{\psi}}{H\dot{\psi}},
\end{equation}
should also be very small during slow-roll inflation.
It is useful to define two new parameters
\begin{equation}
\xi\equiv\frac{\lambda\dot{\varphi}}{2fH} \quad \textmd{and} \quad \xi_{\psi}\equiv\frac{B}{E},
\end{equation}
where $E=(\vec{E}^a.\vec{E}_a)^{\frac{1}{2}}$ and $B=(\vec{B}^a.\vec{B}_a)^{\frac{1}{2}}$.
The ratio of the energy of dark radiation to total energy is
\begin{equation}
\frac{\rho_{_{\rm YM}}}{\rho}\simeq\frac{\psi^2}{2}(1+\xi_{\psi}^2),
\end{equation}
in which we neglect the sub-dominant term $\epsilon_{\psi}$. Hereafter, a $``\simeq$'' means up to the dominant order in slow-roll.
In our model, we are interested in the regime that
\begin{equation}
\frac{\rho_{_{\rm YM}}}{\rho}\lesssim\epsilon^2,
\end{equation}
thus, our slow-roll parameters are
\begin{equation}\label{SL-H-SL}
\epsilon\simeq\frac12\frac{\dot\varphi^2}{H^2}\quad \textmd{and} \quad \eta\simeq-\frac{\ddot\varphi}{H\dot\varphi}.
\end{equation}
Up to the dominate order in slow-roll, we have $\xi_{\psi}\simeq\frac{g\psi}{H}$ and $\xi\simeq\sqrt{\frac{\epsilon}{2}}\frac{\lambda}{f}$ which are related as
\begin{equation}\label{xi-eq}
\xi\simeq\frac{(1+\xi_{\psi}^2)}{\xi_{\psi}}.
\end{equation}
During the slow-roll inflation, the energy density of the gauge field is almost constant and $\rho_{_{\rm YM}}\simeq\frac{\xi}{2}\vec{E}^a.\vec{B}_a$. For a $\xi\sim1$, we have $\xi_{\psi}\sim1$, $\frac{\lambda}{f}\sim1/\sqrt{\epsilon}$ and $\psi\sim\epsilon$. Since the large coupling is hard to achieve in a controlled string compactification \cite{Baumann:2014nda}, we are interested in small $\lambda$, e.g. $f\sim0.01$ and $\lambda\sim0.1$. As the axion rolls down its potential, $\dot{\varphi}/H$ increases and part of the energy of the axion gradually injects to the gauge field, therefore $\rho_{_{\rm YM}}$ (as well as $\psi$ and $\xi_{\psi}$) slowly increases during inflation. After the end of inflation on the other hand, $\dot{\varphi}$ starts oscillating around the minimum of the potential and the gauge field acts like a dark radiation sector, \textit{i.e.} $A^a_{i}\propto a^{-1}$.
\section{Cosmic perturbation theory}\label{perturbation}
In this section, we work out the cosmic perturbation theory of the axion model \eqref{action} in the presence of an SU(2) gauge field.
We are interested in linear perturbations in this paper. At the perturbation level, fields are perturbed around the isotropic and homogeneous configuration \eqref{ansatz}. Due to the quantum fluctuations, all the non-Abelian gauge field modes are turned on and can contribute to the perturbation theory. Dealing with non-Abelian gauge fields bring new features and complications compared to the standard axion
scalar models. However, because of the isotropy of the background, one can still use the scalar, vector, and tensor decomposition for the perturbations \cite{Maleknejad:2012fw}.
\subsection{Classification of the fluctuations}
In this subsection, we turn to classify the field and metric fluctuations around the homogeneous and isotropic background solution.
The most general form of the perturbed FRW metric can be parametrized as
\begin{equation}\label{metric-pert}%
ds^2=-(1+2A)dt^2+2a(\partial_iB+V_i)dx^idt+a^2\left((1-2C)\delta_{ij}+2\partial_{ij}E+2\partial_{(i}W_{j)}+\gamma_{ij}\right)dx^idx^j\,,
\end{equation}
where $\partial_i$ denotes partial derivative respect to $x^i$ and $A,\ B,\ C$ and $E$ are scalar perturbations, $V_i,\ W_i$ parametrize vector perturbations (these are divergence-free three-vectors) and $\gamma_{ij}$, which is symmetric, traceless and divergence-free, is the tensor mode. The axion and the SU(2) gauge field are also perturbed around their homogeneous and isotropic background configurations (Eqn. \eqref{ansatz})
\begin{equation}\label{field-perturb}
\varphi(t,\textbf{x})=\varphi(t)+\delta\tilde{\varphi}(t,\textbf{x})\quad \textmd{and} \quad A^a_{~\mu}(t,\textbf{x})=\left\{
\begin{array}{ll} a\psi(t)\delta^a_i+\delta A^a_{~i}(t,\textbf{x})\, ,\qquad &\mu=i
\\ \delta A^a_{~0}(t,\textbf{x})\,, \qquad &\mu=0\,
\end{array}\right.
\end{equation}%
where (as explained in appendix \ref{gauge-invariant}) the 12 components of $\delta A^a_{~\mu}(t,\textbf{x})$ are%
\begin{subequations} \label{gauge-field-pert}
\begin{align}
\delta A^a_{~i}&=a\delta^a_i (\delta\psi-\psi C)+\delta^{aj}\big(\partial_{ij}(\tilde{Z}+a\psi E)+\partial_i (v_j+a\psi W_j)+a(\tilde{\gamma}_{ij}+\frac{\psi}{2}\gamma_{ij})\big)\\
&+\epsilon^{a~j}_{~i}\big(ga\psi\partial_{j}(Z-\tilde{Z})+w_j\big),\nonumber\\
\delta A^a_{~0}&=\delta^{k}_a\partial_k(Y+a\psi\dot{E})+\delta_a^j (u_j+\psi V_j).
\end{align}
\end{subequations}
Because of the gauge transformations generated by space-time diffeomorphisms as well as the gauge transformations of $A^a_{\mu}$, not all the above 23 metric and fields perturbations are physically meaningful. Eliminating all the gauge symmetries, 4 coordinate freedoms and 3 internal gauge transformations, we then can construct 16 gauge invariant degrees of freedom.
\begin{itemize}
\item{
On the \emph{scalar} sector, one can construct \emph{six} independent gauge-invariant combinations, two standard Bardeen potentials, the perturbed axion field and three gauge invariant combinations coming from the gauge field fluctuations
\begin{equation}\label{scalar-gin}
\begin{array}{ll} \Psi=C+a^2H(\dot{E}-\frac{B}{a})\\
\Phi=A-\frac{d}{dt}\left(a^2(\dot{E}-\frac{B}{a})\right)\\
\end{array}
\quad \textmd{and}\quad
\begin{array}{llll} \delta\varphi=\delta\tilde\varphi-\dot\varphi a^2(\dot{E}-\frac{B}{a}),\\
\delta\psi=\delta\psi,\\
M=g^2\psi^3aZ,\\
\tilde{M}=H\psi(\dot{\tilde{Z}}-Y).\,
\end{array}
\end{equation}%
}
\item{There are \emph{three} gauge invariant divergence-free \emph{vector} perturbations, one from the metric fluctuation and two from the gauge field perturbations
\begin{equation}\label{vector-gin}
\mathcal{Z}_i=a\dot{W}_i-V_i\,, \quad \textmd{and} \quad \begin{array}{ll} \mathcal{U}_i=\frac1g\dot{w}_i+u_i, \\
\mathcal{V}_i=\frac1g w_i+v_i.
\end{array}
\end{equation}
}
\item{On the \emph{tensor} sector, we have two tensor perturbations $\gamma_{ij}$ and $\tilde{\gamma}_{ij}$, which are both gauge invariant with two degrees of freedom. The tensor perturbations are, by definition, symmetric, traceless and divergence-free.}
\end{itemize}
\subsection{Independent field equations}
Working out the gauge-invariant combinations, we are now ready to field the linearized field equations that govern their
dynamics.
The linear order perturbed energy-momentum tensor around a background perfect fluid can be decomposed as %
\begin{align}
\delta T_{ij}=&\bar P\delta g_{ij}+a^2\left(\delta_{ij}(\delta P-\frac13\nabla^2\pi^S)+\partial_{ij}\pi^S
+\partial_i\pi^V_j+\partial_j\pi^V_i+\pi^T_{ij}\right)\,,\cr
\delta T_{i0}=&\bar P\delta g_{i0}-(\bar \rho+\bar P)(\partial_i\delta u+\delta u_i^V)\,,\cr
\delta T_{00}=&-\bar\rho\delta g_{00}+\delta \rho\,,\nonumber
\end{align}
where $\bar\rho$ and $\bar P$ are the background energy and pressure densities. Moreover, $\pi^S$, $\pi^V_i$, $\pi^T_{ij}$ represent the \textit{anisotropic inertia},
characterizing departures from the perfect fluid form of the
energy-momentum tensor, while $\delta u_i^V$ is the vorticity. They satisfy the following conditions
$$\partial_i\pi^V_i=\partial_i\pi^T_{ij}=\partial_i\delta u_i^V=0.$$
One can construct the following four gauge invariant combinations from $\delta\rho$, $\delta P$ and $\delta q$
\begin{subequations}\label{diff-inv-Tmunu}
\begin{align}
\delta\rho_g=&\delta\rho-\dot{\bar \rho}a^2(\dot{E}-\frac{B}{a})\,,\nonumber\\
\delta P_g=&\delta P-\dot{\bar P}a^2(\dot{E}-\frac{B}{a})\,,\nonumber\\
\delta q_g=&\delta q+(\bar\rho+\bar P)a^2(\dot{E}-\frac{B}{a})\,,\nonumber
\end{align}\end{subequations}
while $\pi^S$, $\pi^V_i$, $\pi^T_{ij}$ and $\delta u_i^V$ are gauge invariant quantities, where $\delta q=(\bar\rho+\bar P)\delta u$.
It is useful to decompose the energy-momentum tensor into the contribution of the axion and the gauge field as $$\delta T^{\mu\nu}=\delta T^{\mu\nu}_{\varphi}+\delta T^{\mu\nu}_{_{\rm YM}}.$$ The axion sector, $\delta T^{\mu\nu}_{\varphi}$, is specified by
\begin{subequations}
\begin{align}
\delta q_{\varphi}&=-\dot{\varphi}\delta\varphi,\\
\delta\rho_{\varphi}&=\dot\varphi\delta\dot\varphi-\dot\varphi^2\Phi+V_{\varphi}\delta\varphi,\\
\delta P_{\varphi}&=\dot\varphi\delta\dot\varphi-\dot\varphi^2\Phi-V_{\varphi}\delta\varphi,
\end{align}
\end{subequations}
while $\delta T^{\mu\nu}_{_{\rm YM}}$ has the following momentum, energy and pressure densities
\begin{subequations}
\begin{align}
\delta q_{_{\rm YM}}&=-2\dot M+2H\bigg(M+\xi^2_{\psi}\tilde M-\psi\delta\psi+\psi^2\Psi\bigg),\\
\delta\rho_{_{\rm YM}}&=3H^2\psi^2\bigg(\frac1H(\frac{\delta\psi}{\psi}\dot{)}-\Phi+(1+2\xi_{\psi}^2)\frac{\delta\psi}{\psi}\bigg)-\frac{k^2}{a^2}(\tilde M+2M),\\
\delta P_{_{\rm YM}}&=\frac13\delta\rho_{_{\rm YM}}.
\end{align}
\end{subequations}
Unlike the axion energy-momentum tensor, $\delta T^{\mu\nu}_{_{\rm YM}}$ deviates from the perfect fluid form. In other words, although the background energy-momentum tensor is in the form of a perfect fluid, at the perturbation level, $\delta T^{\mu\nu}_{_{\rm YM}}$ is an \textit{imperfect fluid} with non-vanishing anisotropic inertia and vorticity as
\begin{subequations}\label{devi}
\begin{align}
\label{piS}
a^2\pi^S&=2(M-\tilde M),\\
a\label{piv}\pi_i^V&=H\psi\bigg(H\xi^2_{\psi}\mathcal{V}_i+(\mathcal{U}_i-\dot{\mathcal{V}}_i-\psi\mathcal{Z}_i)\bigg)\,,\\
\label{piT}
\pi^T_{~ij}&=2H\psi\bigg((\xi^2_{\psi}-1)H\tilde{\gamma}_{ij}-\dot{\tilde{\gamma}}_{ij}-\frac{\psi}{2}\dot{\gamma}_{ij}
+\xi_{\psi}\partial_k\epsilon^{kl}_{~~(i}\big[\tilde{\gamma}_{j)l}+\frac{\psi}{2}\gamma_{j)l}\big]\bigg)\,,\\
\delta q^V_i&=
H\psi\bigg(\xi_{\psi}\nabla\times\big(\dot{\vec{\mathcal{V}}}-\vec{\mathcal{U}}+\psi\vec{\mathcal{Z}})-2\xi^2_{\psi}H\vec{\mathcal{U}}
-\xi_{\psi}H(\nabla\times\vec{\mathcal{V}})\bigg)_i.
\end{align}
\end{subequations}
As follows from \eqref{pi^s}-\eqref{dP}, \eqref{firstV}-\eqref{delta-q-V} and \eqref{T-gf}, there are ten independent Einstein equations, \emph{four} scalars, \emph{two} vectors and \emph{one} tensor. Since they are less than the number of (physical) gauge-invariant quantities, one needs more equations to have a complete set of equations. These extra equations are provided by the field equations which are given by the second order action. In fact, the scalar and vector parts of the gauge field equations
can be written as\footnote{These extra equations are the field equation of $A^a_{~0}$ component which are constraints
enforcing the gauge invariance of the action. Note that dealing with a gauge invariant action, $\dot A^a_{~0}$
does not appear in the Lagrangian density, $\mathcal{L}$, and the momentum conjugate to $A^a_{~0}$ is identically zero.}
\begin{subequations}
\label{constraint-2nd-order-action}
\begin{align}
\delta^{k}_a\partial_k\big(\frac{\partial\delta\!_{_{2}}(\sqrt{-g}\mathcal{L})}{\partial Y }\big)&=0,\\
\delta^a_i\big(\frac{\partial\delta\!_{_{2}}(\sqrt{-g}\mathcal{L})}{\partial u_i }\big)&=0\,,
\end{align}
\end{subequations}
where $\delta_2$ stands for second order in perturbations. The equation of motion for the perturbed axion field $\delta\varphi$ and the tensor mode $\tilde{\gamma}_{ij}$ will also be
obtained from the corresponding parts of the second order action. In the following table, we summarize the number of gauge-invariant perturbations and the independent equations governing the dynamics of each part of the system.
\begin{center}
\begin{tabular}{cp{1.5cm}p{3cm}p{3cm}p{3cm}p{2cm}}
\hline
& & \begin{footnotesize}
Gauge-invariants
\end{footnotesize} &\begin{footnotesize}
Einstein Eqn.s
\end{footnotesize} & \begin{small}
($\frac{\delta S}{\delta A})_{_{_{\!(\!1\!)}}}$
\end{small} & \begin{small}
$(\frac{\delta S}{\delta \varphi})_{_{_{\!(\!1\!)}}}$
\end{small}\\ [2.5ex]
\hline
&\begin{footnotesize}
Scalar
\end{footnotesize} & 6 & 4 & 1 & 1 \\ [1.1ex]
& \begin{footnotesize}
Vector
\end{footnotesize} & 3 & 2 & 1 & 0 \\[1.1ex]
& \begin{footnotesize}
Tensor
\end{footnotesize} & 2 & 1 & 1 & 0 \\ [1.1ex] \cline{1-6}
\end{tabular}
\vskip 0.1 cm
\textbf{Table I: Gauge-invariant perturbation modes and independent field equations}
\end{center}
In the table I, $(\frac{\delta S}{\delta A})_{_{_{\!(\!1\!)}}}$ and $(\frac{\delta S}{\delta \varphi})_{_{_{\!(\!1\!)}}}$ represent the linear order field equations of the gauge field and the axion field which are determined by the second order action. Here, we only present the final results, for more details we refer to \cite{Maleknejad:2012fw}.
For later convenience, here we introduce two Fourier space variables in terms of conformal time $\tau$ and comoving momentum $k$
\begin{equation}\label{T-def}
\tilde{\tau}\equiv-k\tau \quad \textmd{and} \quad \tilde\mathcal{H}\equiv\frac{\mathcal{H}}{k},
\end{equation}
where $\mathcal{H}=aH$. During the slow-roll inflation in which $\mathcal{H}\simeq-(1+\epsilon)/\tau$, we have
\begin{equation}
\tilde{\tau}\simeq\frac{k_{\rm phy}}{H}\quad \textmd{and} \quad \tilde\mathcal{H}\simeq\frac{(1+\epsilon)}{\tilde{\tau}}.
\end{equation}
in which $k_{\rm phy}$ is the physical momentum $k/a$.
\subsubsection{Scalar sector}
In the scalar sector of the perturbations, we have \emph{six} gauge-invariant combinations of \eqref{scalar-gin}, $\{\delta\varphi,\delta\psi,M,\tilde M,\Psi,\Phi\}$. These perturbations are governed by \emph{four} scalar Einstein equations, the field equation of $\delta A^a_{~0}$ ( Eqn. \eqref{constraint-2nd-order-action}) and $\delta\varphi$.
The scalar part of the perturbed Einstein equations take the form
\begin{subequations}%
\begin{align}\label{pi^s}
&a^2\partial_{ij}\pi^s=\partial_{ij}(\Psi-\Phi)\,,\\
\label{dq} &\partial_{i}(\delta q_g+2(\dot{\Psi}+H\Phi))=0\,,\\
\label{drho} &\delta\rho_g-3H\delta q_g+2\frac{k^2}{a^2}\Psi=0\,,\\
\label{dP}
&\delta P_g+\dot{\delta q}_g+3H\delta q_g+2\epsilon H^2\Phi-
\frac23\frac{k^2}{a^2}(\Psi-\Phi)=0\,. %
\end{align}
\end{subequations}%
Moreover, the scalar part of the field equation of $\delta A^a_{~0}$ (Eqn. \eqref{constraint-2nd-order-action}) is the constraint below
\begin{eqnarray}\label{A02}
H\delta q_g-H\psi^2\big(\frac{\delta\psi}{\psi}\dot{\big)}+(\dot{\varphi}+\frac{\lambda g\psi^3}{f})H\delta\varphi+H^2\psi^2(\frac{\delta\psi}{\psi}+\Phi)+\frac{k^2}{a^2}\tilde M=0.~~~~~~
\end{eqnarray}
The field equation of $\delta\varphi$ is
\begin{eqnarray}\label{deltachi-eq}
\delta\ddot\varphi+3H\delta\dot{\varphi}+\bigg(\frac{k^2}{a^2}+V_{\varphi\varphi}\bigg)\delta\varphi=2(\ddot\varphi+3H\dot\varphi)\Phi+\dot\varphi(\dot\Phi+3\dot\Psi)-\frac{\lambda}{f}\delta(\vec{E}^a.\vec{B}_a)
\end{eqnarray}
where $\delta(\vec{E}^a.\vec{B}_a)$ is the linear order perturbation of $\vec{E}^a.\vec{B}_a$ which is
\begin{equation}\label{wedge}
\delta(\vec{E}^a.\vec{B}_a)=3 g\psi^3H\bigg(\frac{1}{H}\bigl(\frac{\delta\psi}{\psi}\dot{\bigl)}+3\big(\frac{\delta\psi}{\psi}\big)-\Phi-\frac{k^2}{3a^2}(\frac{2M}{g^2\psi^4}+\frac{\tilde M}{H^2\psi^2})\bigg).
\end{equation}
Eqn.s \eqref{pi^s}-\eqref{dP}, \eqref{A02} and \eqref{deltachi-eq} provides enough number of equations for $\delta\varphi$, $\delta\psi$, $\Psi$, $\Phi$, $M$ and $\tilde M$. In sec. \ref{Scalar perturbations}, we solve these equations and study scalar fluctuations during the slow-roll inflation.
\subsubsection{Vector sector}
The vector perturbations of the metric and the gauge fields have three gauge invariant combinations of Eqn.\eqref{vector-gin}, $\{\mathcal{V}_i,\mathcal{U}_i,\mathcal{Z}_i\}$. The perturbed Einstein equations involves two vector equations, one constraint and one dynamical equation, given as
\begin{subequations}
\label{vector-eins}
\begin{align}
\label{firstV}
&\partial_i\left(2a^2\pi^V_j-\frac{1}{a}(a^2\mathcal{Z}_j\dot{)}\right)=0\,,\\\label{delta-q-V}
&2a\delta q_i^V+\nabla^2\mathcal{Z}_i=0\,.
\end{align}
\end{subequations}
Dealing with three unknowns, the last equation is provided by the vector part of the field equation of $\delta A^a_{~0}$.
Explicitly, using \eqref{delta-q-V} in the vector part of \eqref{constraint-2nd-order-action} yields to
\begin{equation}\label{vec-const}
g\psi^2\vec\nabla\times\vec{\mathcal{U}}_i-
\frac{\psi}{a}\nabla^2(\mathcal{U}_i-\dot{\mathcal{V}}_i-\psi\mathcal{Z}_i)+\frac{1}{2a}\nabla^2{\cal Z}_i=0 \,.
\end{equation}
This completes the set of equations we need for solving vector perturbations.
Then, the combination of \eqref{firstV}-\eqref{delta-q-V} and \eqref{vec-const} indicates that $\cal Z$ exponentially damps during inflation.
From the combination of \eqref{devi} and \eqref{vector-eins}, we then find that $\mathcal{Z}_i$ vanishes
after horizon crossing. Despite having gauge fields in our matter content, the power spectrum of the vector modes are unimportant in inflationary cosmology and CMB anisotropies.
\subsubsection{Tensor sector}
In the tensor sector, we have two gauge invariant tensors each with two degrees of freedom: the spin-2 fluctuations of the metric $\gamma_{ij}$ (\textit{gravitational waves}) and the gauge field $\tilde{\gamma}_{ij}$, which we call \textit{tensor waves}.
These tensor modes are governed by the tensor part of the Einstein equation and the field equation of $\tilde{\gamma}_{ij}$ given by the second order action. Tensor fluctuations of the SU(2) gauge field interact with the tensor perturbations of the metric and modify its linear order field equation. These new interactions in the quadratic action involve parity odd terms which generate chiral tensor modes. Here, we only focus on the tensor perturbations of the axion inflation in \eqref{action}. However, the above property is the generic feature of inflationary models in the presence of a non-Abelian gauge field \cite{Maleknejad:2014wsa}.
The perturbed Einstein equations involve one equation for $\gamma_{ij}$
\begin{equation} \label{T-gf}
\ddot \gamma_{ij}+3H \dot \gamma_{ij}-\frac{\nabla^2}{a^2}\gamma_{ij}=2\pi^T_{ij}\,,
\end{equation}
in which $\pi^T_{ij}$ is the tensor part of the anisotropic inertia\footnote{Comparing with the exact form of $\pi^T_{ij}$ in \eqref{devi}, here in \eqref{pi-T} we dropped two slow-roll suppressed derivatives of $\gamma_{ij}$ of the form $\frac{\rho_{\rm YM}}{H}\dot{\gamma}_{ij}$ and $\frac{\rho_{\rm YM}}{H}\epsilon^{kl}_{~~i}\partial_k\gamma_{jl}$. }
\begin{equation}\label{pi-T}
\pi^T_{~ij}=2H\psi\bigg((\xi^2_{\psi}-1)H\tilde{\gamma}_{ij}-\dot{\tilde{\gamma}}_{ij}
+\xi_{\psi}\partial_k\epsilon^{kl}_{~~(i}\tilde{\gamma}_{j)l}\bigg)\,,
\end{equation}
Note that $\pi^T_{ij}$ is proportional to $\psi$, the effective field value of the gauge field in the background level. Therefore, in order to have a linear order anisotropic inertia, the gauge fields should be turned on at the background level.
Moreover, the field equation of the tensor perturbation of the gauge field $\tilde{\gamma}_{ij}$
is provided by its second order action
\begin{eqnarray}
\label{2ndts}
&&\hspace*{-7mm}\delta\!_{_{2}}S_{\tilde h}\simeq\frac12\int d^3x dt a^3\biggl(
\big(\dot{\tilde{\gamma}}_{ij}\big)^2-\big(\frac{\partial_k\tilde{\gamma}_{ij}}{a}\big)^2-2\xi\xi_{\psi}H^2\tilde{\gamma}_{ij}^2+2(\xi
+\xi_{\psi})H\epsilon^{ijk}\tilde{\gamma}_{kl}\frac{\partial_i\tilde{\gamma}_{jl}}{a}\nonumber\\
&&~~~~~~+2H\psi\big(\dot{\gamma}_{ij}+\xi\epsilon^{ikl}\frac{\partial_k\gamma_{jk}}{a})\tilde{\gamma}_{ij}\biggr)\,.
\end{eqnarray}
Interestingly, both $\gamma_{ij}$ and $\tilde{\gamma}_{ij}$ have sound speeds equal to one. It is noteworthy to mention that the quadratic action above involves all the possible combinations of $\tilde\gamma\tilde\gamma$ with $n\leq2$ derivatives.
Among them, we have two parity violating terms, $\epsilon^{ijk}\tilde{\gamma}_{kl}\partial_i\tilde{\gamma}_{jl}$ and $\epsilon^{ijk}\tilde{\gamma}_{kl}\partial_i \gamma_{jl}$, which are originated from the Yang-Mills and Chern-Simons terms in the action.
Going to the Fourier space, we can diagonalize the system in terms of circular polarizations. In terms of the right- and left-handed polarizations, $\gamma_{ij}$ and $\tilde{\gamma}_{ij}$ are decomposed as
\begin{subequations}
\begin{align}
&\gamma_{ij}(\tau,\textbf{x})=\frac{1}{\sqrt{2}a}\sum_{\sigma=R,L}\int \frac{d^3k}{(2\pi)^\frac32} h_{\sigma}(\tau,\textbf{k})e^{\sigma}_{ij}(\textbf{k})e^{i\textbf{k}.\textbf{x}},\\ &\tilde{\gamma}_{ij}(\tau,\textbf{x})=\frac{1}{2\sqrt{2}a}\sum_{\sigma=R,L}\int \frac{d^3k}{(2\pi)^\frac32} \tilde{h}_{\sigma}(\tau,\textbf{k})e^{\sigma}_{ij}(\textbf{k})e^{i\textbf{k}.\textbf{x}},
\end{align}
\end{subequations}
where $\{h_{_{R,L}},\tilde{h}_{_{R,L}}\}$ are the canonically normalized fields and $e^{R,L}_{ij}$ are the circular polarization tensors which satisfy the conditions
\begin{subequations}
\begin{align}
&e^{\sigma}_{ij}e^{\sigma'*}_{ij}=2\delta^{\sigma\sigma'},\\
&\epsilon^{ijk}\hat{k}_ie^{\sigma}_{kl}=i\lambda_{\sigma}e^{\sigma j}_{l}, \quad \textmd{with} \quad \lambda_{_{R,L}}=\pm1.
\end{align}
\end{subequations}
For a wave vector $\textbf{k}=(0,0,k)$, the right- and left-handed modes are defined as $h_{_{R,L}}\equiv a(\gamma_{11}\pm i\gamma_{12})/2$.
From the second order action \eqref{2ndts}, we obtain the field equation of $\tilde{h}_{_{R,L}}(\tau,\textbf{k})$ as
\begin{eqnarray}\label{v-eq}
&&\tilde{h}''_{_{R,L}}+\bigg(k^2\mp2(\xi+\xi_{\psi})k\mathcal{H}+2\xi\xi_{\psi}\mathcal{H}^2\bigg)\tilde{h}_{_{R,L}}\simeq2\psi\mathcal{H}
\bigg(h'_{_{R,L}}-\mathcal{H} h_{_{R,L}}\pm k\xi h_{_{R,L}}\bigg),\nonumber\\
\end{eqnarray}
in which we have parity odd terms that have different signs for the right- and left-handed polarizations. Using the slow-roll relation \eqref{xi-eq} in the above and recalling that $h_{_{R,L}}\propto a$, we realize that the RHS of \eqref{v-eq} vanishes in the long wavelength limit.
In sec. \ref{tensor-section}, we solve the field equations of $\{h_{_{R,L}},\tilde{h}_{_{R,L}}\}$ and study tensor fluctuations during the slow-roll inflation.
\section{Scalar perturbations}\label{Scalar perturbations}
In the scalar sector, we have six independent fields and six equations.
Upon using variable redefinition \eqref{T-def}, it is straightforward to see that all of our equations can be written in terms of $\tilde{\tau}$ and $\tilde{\mathcal{H}}$. For instance, we can write the field equation of $\delta\varphi$ ( Eq. \eqref{deltachi-eq}) as
\begin{eqnarray}\label{delta-varphi-}
&&(a\delta\varphi)_{\tilde{\tau}\x}+\bigg(1-(2-3\eta-\epsilon)\tilde{\mathcal{H}}^2\bigg)a\delta\varphi\simeq\frac{6\dot{\varphi}}{H}\tilde{\mathcal{H}}^2 a\Phi-\frac{3\lambda g\psi^3}{fH}\bigg(\tilde{\mathcal{H}}^2(\frac{2a\delta\psi}{\psi}-a\Phi)-\tilde{\mathcal{H}}\big(\frac{a\delta\psi}{\psi}\big)_{\tilde{\tau}}\nonumber\\
&-&\frac{1}{3\psi^2}(\frac{2aM}{\xi_{\psi}^2}+a\tilde M)\bigg).
\end{eqnarray}
Assuming slow-roll inflation, all of the coefficients in our equations are slow varying with time and approximately constant up to the dominant order in slow-roll. Thus, all of our six fields are functions of $\tilde{\tau}$ with a coefficient of $k$ which is given by the initial value. Setting the initial value of the canonically normalized fields by the standard Bunch-Davis, solutions has the following formal forms
$$X_{I}(\tau,k)=\frac{1}{\sqrt{k}}f_I(\tilde{\tau})\quad \textmd{and}\quad Y_{J}=\frac{1}{\sqrt{k^3}}\tilde{f}_J(\tilde{\tau}) \quad \textmd{where}\quad \tilde{\tau}\equiv -k\tau,$$
where $X_I$ are canonically normalized fields and $Y_I$ are non-dynamical fields which are governed by the constraint equations. Using constraints to eliminate non-dynamical quantities, and solving the equations, we can decompose the dynamical fields as
\begin{equation}
X_I(\tau,k)=X_I^{G}(\tau,k)+X_I^S(\tau,k),
\end{equation}
where $X_I^{G}(\tau,k)$ is the solution of the homogeneous equation and $X_I^{S}(\tau,k)$ is the particular part which is sourced by the other dynamical fields. Formally, we have
\begin{equation}
X_I^S(\tau,k)=\frac{1}{\sqrt{k}}\int_0^{\tilde{\tau}}G_{I}(\tilde{\tau},\tilde{\tau}')S_{I}(\tilde{\tau}')d\tilde{\tau}',
\end{equation}
where $G_I(\tilde{\tau},\tilde{\tau}')$ and $S_{I}(\tilde{\tau}')$ are the Green's function and source term of equation $I$ respectively. As we may expect, using the \textit{Mukhanov-Sasaki variable}
\begin{equation}\label{MS-var}
a\delta\varphi_{\Psi}\equiv a(\delta\varphi+\frac{\dot{\varphi}}{H}\Psi),
\end{equation}
and using the constraint equations in \eqref{delta-varphi-}, we obtain the field equation of the homogeneous part of $a\delta\varphi_{\Psi}$ as
\begin{equation}
(a\delta\varphi_{\Psi}^G)_{\tilde{\tau}\x}+\bigg(1-(2+5\epsilon-3\eta)\tilde{\mathcal{H}}^2\bigg)a\delta\varphi_{\Psi}^G=0,
\end{equation}
which is the standard equation of a single scalar field model. Imposing the standard Banch-Davis initial value for $a\delta\varphi_{\Psi}^G$, we can solve the above equation in terms of Hankel functions as
\begin{equation}\label{free-axion}
a\delta\varphi_{\Psi}^G(k,\tau)=\frac{\sqrt{\pi\tilde{\tau}}}{2\sqrt{k}}H^{(1)}_{\nu_{G}}(\tilde{\tau}), \quad \textmd{where} \quad \nu_{G}\simeq\frac32+3\epsilon-\eta.
\end{equation}
In order to study the contribution of the gauge field to the perturbations and determine the dynamics of the system, we will write the equations in two asymptotic limits of deep inside horizon ($\tilde{\tau}\gg1$) and super-horizon ($\tilde{\tau}\ll1$). The former gives us the canonically normalized fields, $\{X_I\}$s, as well as the non-dynamical fields, $\{Y_I\}$s, while the latter determines the spectral tilt and super-horizon behavior of the solutions. The validity of our super-horizon limit analysis is crucially dependent on the stability of the scalar perturbations in the intermediate regime. That issue should be established by means of numerical study and we will address that matter in the last subsection.
\vskip 0.5 cm
\subsection{Canonically normalized fields}
At this point, after using the constraints to eliminate the non-dynamical fields in the second order action, we determine the canonically normalized fields. Setting the Banch-Davis vacuum for them, we then obtain the initial value of the rest of the variables. In the deep inside horizon limit in which $\tilde{\tau}\gg1$, the constraint equation \eqref{dq} is \begin{equation}\label{constsub}
\tilde{\tau}\partial_{\tilde{\tau}}(\Psi-M)+\psi\delta\psi+\frac12\frac{\dot{\varphi}}{H}\delta\varphi=0,
\end{equation}
while the combination of \eqref{drho} and \eqref{A02} can be written as below
\begin{subequations}\label{const--}
\begin{align}
&\frac{\dot{\varphi}}{H}\partial_{\tilde{\tau}}\delta\varphi-\tilde{\tau}(\Phi+\Psi)=0,\\\label{2nd-com}
&\partial_{\tilde{\tau}}(\psi\delta\psi+\frac12\frac{\dot{\varphi}}{H}\delta\varphi)-\tilde{\tau}(\Psi-M)\simeq0.
\end{align}
\end{subequations}
From the combination of constraints \eqref{constsub} and \eqref{2nd-com}, up to the dominant order, we obtain
\begin{equation}\label{dyn1}
\partial^2_{\tilde{\tau}}\big(\psi\delta\psi+\frac12\frac{\dot{\varphi}}{H}\delta\varphi\big)+\big(\psi\delta\psi+\frac12\frac{\dot{\varphi}}{H}\delta\varphi\big)\simeq0.
\end{equation}
Inserting \eqref{constsub} and \eqref{dyn1} into \eqref{dP} leads to $\partial_{\tilde{\tau}}^2\Psi+\Psi=0$, which combining with \eqref{constsub} gives
\begin{equation}
\partial_{\tilde{\tau}}^2M+M=0.
\end{equation}
Moreover, the field equation of $\delta\varphi$ \eqref{deltachi-eq} at the deep inside horizon reads as
\begin{equation}\label{dc}
\partial_{\tilde{\tau}}^2\delta\varphi+\delta\varphi=0.
\end{equation}
The second order action up to the leading orders in $\tilde\tau$ is given by
\begin{eqnarray}
\delta\!_{_{2}}S&=&\int a^3d^3kdt\bigg(\frac12\delta\dot\varphi^2_{\Psi}-\frac12\frac{k^2}{a^2}\delta\varphi^2_{\Psi}+\frac{3}{2}\delta\dot \psi^2-\frac{k^2}{a^2}\delta\psi^2+\frac{k^2}{a^2}\frac{(\dot M^2-\frac{k^2}{a^2}M^2)}{g^2\psi^4}+\frac{1}{2}\frac{k^4}{a^4}\frac{\tilde M^2}{H^2\psi^2}\nonumber\\
&+&\frac{k^2}{a^2}(-\frac{\tilde M}{H\psi}+\frac{2\lambda\varphi}{f}\frac{M}{g\psi^2})\delta\dot \psi+\frac{2k^2}{a^2}\frac{\lambda\varphi}{f}\frac{\dot M}{g\psi^2}\delta\psi
\bigg).
\end{eqnarray}
Using constraint \eqref{const--}, we can simply that to the following quadratic action
\begin{eqnarray}
\delta\!_{_{2}}S\!\simeq\!\frac{1}{2}\int\!k^2d^3kd\tau\bigg[(a\varphi_{\Psi})_{\tilde{\tau}}^2-(a\varphi_{\Psi})^2+2\bigg((a\delta\psi)_{\tilde{\tau}}^2-(a\delta\psi)^2\bigg)
+\frac{2}{(\xi_{\psi}\psi)^2}\bigg((\tilde{\tau} aM)_{\tilde{\tau}}^2-(\tilde{\tau} aM)^2\bigg)\bigg].\nonumber
\end{eqnarray}
The quadratic action above specifies our 3 canonically normalized (dynamical) fields as $$X_{I}=\{a\varphi_{\Psi},\sqrt{2}a\delta\psi,-i\sqrt{2}/(\psi\xi_{\psi})\tilde{\tau} aM\}.$$
As a result, the non-dynamical fields are
$$Y_{J}=\{a\Psi,a\Phi,a\tilde M\}.$$
Finally, imposing the standard Banch-Davis vacuum condition specifies our initial conditions as follows\footnote{It is noteworthy to mention that the above initial conditions leads to a non-vanishing scalar anisotropy
\begin{equation}
a^2\pi^S=\frac{iH\psi(1+\sqrt{\xi_{\psi}})}{\sqrt{k^{3}}}e^{i\tilde{\tau}},
\end{equation}
which is of the order of $\Psi$ and $\Phi$ themselves.}
\begin{equation}\label{BD}
a\delta\varphi_{\Psi}=\frac{e^{i\tilde{\tau}}}{\sqrt{2k}}, \quad a\delta\psi=\frac{e^{i\tilde{\tau}}}{2\sqrt{k}} \quad \textmd{and} \quad aM=\frac{i\psi\xi_{\psi}}{\tilde{\tau}}\frac{e^{i\tilde{\tau}}}{2\sqrt{k}}.
\end{equation}
\vskip 0.5cm
\subsection{Long wavelength Limit and scalar spectrum}
We now turn to study the long wavelength behavior of the scalar fluctuations. The validity of our analytical calculations depends on the stability of scalar perturbations which should be established by means of numerical study. We tackle that issue in the next subsection.
At the super-horizon limit, the constraint equation \eqref{drho} has the following form
\begin{equation}\label{const-sup}
V_{\varphi}\delta\varphi+6H(\dot{\Psi}+H\Phi)-\dot\varphi^2\Phi+3\big(\frac{\dot\phi^2}{a^2}
+2\frac{g^2\phi^4}{a^4}\big)\frac{\delta\psi}{\psi}=0,
\end{equation}
and the constraint equation \eqref{A02} is
\begin{eqnarray}\label{AAA}
\bigg(H\dot\varphi+\frac{\lambda g\phi^2\dot{\phi}}{fa^3}\bigg)\delta\varphi+\frac{\dot{\phi}^2}{a^2}\frac{\delta\psi}{\psi}
-2H(\dot\Psi+H\Phi)=0.~~~~~
\end{eqnarray}
From them, we then have
\begin{subequations}\label{relation-sh}
\begin{align}
&2\bar{\rho}_{_{\rm YM}}\frac{\delta\psi}{\psi}+\dot{\varphi}^2\Phi+\ddot{\varphi}\delta\varphi=0,\\\label{super-dq}
&V_{\varphi}\delta\varphi+6(\dot{\Psi}+H\Phi)\simeq0,
\end{align}
\end{subequations}
where the former is the combination of \eqref{const-sup} and \eqref{AAA}, while the latter is simply equation \eqref{const-sup} up to dominate orders in slow-roll.
From \eqref{dq}, \eqref{MS-var} and \eqref{super-dq}, we therefore have comoving curvature perturbation $(\mathcal{R}=\Psi-\frac{H}{\rho+P}\delta q_g)$ as
\begin{equation}\label{super-R}
\mathcal{R}\simeq\frac{H}{\dot{\varphi}}\delta\varphi_{\Psi},
\end{equation}
in terms of the Mukhanov-Sasaki variable.
In \eqref{free-axion}, we have the homogeneous part of $\delta\varphi_{\Psi}$, $\delta\varphi^{G}_{\Psi}$, which in super-horizon is
\begin{equation}
\delta\varphi^{G}_{\Psi}(\tau,k)\simeq\frac{H}{\sqrt{2}k^{\frac32}}\tilde{\tau}^{-(3\epsilon-\eta)}.
\end{equation}
Moreover, the long wavelength value of the special part\footnote{It is noteworthy to mention that in our \textit{non-Abelian gauge theory}, $\delta\varphi^{S}_{\Psi}$ is coming from the contribution of linearized $F^a\tilde F_a$ to the field equation of $\delta\varphi$. In case of \textit{U(1) gauge field}, however, the linearized $F\tilde F$ vanishes and the contribution of the Abelian gauge field starts from $\delta_{2}(F\tilde F)$. In that setup, the U(1) gauge field sources the axion via inverse decay, which is now a very well studied mechanism \cite{Barnaby:2010vf, Barnaby:2011vw,Barnaby:2011qe}.
}, $\delta\varphi^{S}_{\Psi}$ can be parametrized as
\begin{equation}\label{sourced-R}
\delta\varphi^{S}_{\Psi}(\tau,k)=\alpha(\xi_{\psi},\tilde{\tau})\frac{H}{\sqrt{2}k^{\frac32}}\tilde{\tau}^{-(3\epsilon-\eta)},
\end{equation}
in terms of $\alpha(\xi_{\psi},\tilde{\tau})$ which is a function of $\tilde{\tau}$ and the parameter $\xi_{\psi}$. We emphasis that \eqref{sourced-R} is only a relation between wave functions, while their operators are uncorrelated. In case of stable solutions, $\alpha(\xi_{\psi},\tilde{\tau})$ would be a slow-varying function\footnote{Note assuming slow-roll inflation, we neglect the time variation of background parameters during the first few e-folds in which CMB fluctuations have been generated.} of $\tilde{\tau}$, \textit{i.e.} $\alpha(\xi_{\psi},\tilde{\tau})\propto \tilde{\tau}^{\mathcal{O}(\xi_{\psi},\epsilon)}$.
In order to determine $\alpha(\xi_{\psi},\tilde{\tau})$ and its contribution to the spectral tilt $\frac{d\ln\alpha}{d\ln k}$, we need to do numerical analysis. In the next subsection, we present the details of our numerical study of a system with $\frac{\rho_{\rm YM}}{\rho}=\epsilon^2$ and here we only summarize the final results. The homogeneous part of the comoving curvature is given as $\mathcal{R}^G=\frac{H}{\dot{\varphi}}\delta\varphi^G_{\Psi}$ which is an adiabatic mode and hence constant after horizon crossing. However, from the combination of \eqref{super-R} and \eqref{sourced-R}, we can present the special part as $\mathcal{R}^S=\alpha(\xi_{\psi},\tilde{\tau})\mathcal{R}^G$ (which is a functional parametrization, while the operators are uncorrelated.). Due to the prefactor $\alpha$, $\mathcal{R}^S$ can have some deviations from adiabaticity.
Our scalar perturbations are stable and almost adiabatic for $\xi_{\psi}\gtrsim\sqrt{2}$ while otherwise deviates from the adiabatic solution. In particular for the parameter regime $\xi_{\psi}\gtrsim\sqrt{2}$, $\alpha(\xi_{\psi},\tilde{\tau})$ is almost a numerical factor of the order one ($\frac{d\ln\alpha}{d\ln k}\lesssim 10^{-3}$).
Therefore, in the parameter regime $\xi_{\psi}\gtrsim\sqrt{2}$, we have the formal form of super-horizon power spectrum $\mathcal{R}$ as
\begin{equation}\label{power-R}
P_{\mathcal{R}}=\frac{4\pi k^3}{(2\pi)^3}(\mid\!\mathcal{R}^G\!\mid^2+\mid\!\mathcal{R}^S\!\mid^2)\simeq\frac{(1+\alpha^2(\xi_{\psi}))}{2\epsilon}\bigg(\frac{H}{2\pi}\bigg)^2,
\end{equation}
and up to the leading order in the slow-roll parameters, the spectral tilt is
\begin{equation}
n_{\mathcal{R}}-1\simeq-2(3\epsilon-\eta).
\end{equation}
As a result, the total comoving curvature is almost adiabatic. For smaller values of $\xi_{\psi}$, the prefactor $\alpha$ can not be considered as a numerical factor as $\frac{d\ln\alpha}{d\ln k}\gtrsim 10^{-3}$. For instance, in $\xi_{\psi}=1.2$, we have $\frac{d\ln\alpha}{d\ln k}= 10^{-2}$ and it increases rapidly as we approach smaller $\xi_{\psi}$s (see figure \ref{figScalar}).
\subsection{Stability analysis of scalar perturbations}
In the previous subsections, we analytically studied the system in two limits of sub- and super-horizon regimes. An important question that may arise and the validity of our long wavelength study tightly depends on it is the stability of scalar fluctuations in the intermediate regime. In this part, we address this important question and find the inhomogeneous solution of axion fluctuation $\delta\varphi^S$ in the presence of the gauge field. Here, we neglect the time variation of the slow-roll parameters and the metric perturbations. These slow-roll suppressed corrections may be relevant in super-horizon scales and add some small corrections to the spectral index of $\varphi^S_{\Psi}(\tilde{\tau})$ which we leave for future work.
The special part of the axion field, $\delta\varphi^S_{\Psi}(\tilde{\tau})$, is sourced by the gauge field through the Chern-Simons interaction. The source term is proportional to $\frac{\lambda\psi}{f\tilde{\tau}}$ which since $\frac{\lambda\psi}{f}\sim\sqrt{\epsilon}$, it is mostly relevant after horizon crossing. Our numerical studies show that in small scales, $\delta\varphi^S_{\Psi}$ is negligible comparing to $\delta\varphi^G_{\Psi}$, while it gradually increases as the mode approaches the horizon. After horizon crossing, for modes with $\xi_{\psi}\gtrsim\sqrt{2}$, we have $\frac{d\ln\alpha}{d\ln k}\lesssim 10^{-3}$ and therefore $\mathcal{R}^s$ is almost adiabatic (Figure \ref{figScalar}). For smaller values of $\xi_{\psi}$, on the other hand, $\delta\varphi^S_{\Psi}$ deviates from adiabatic solution. In particular, for the parameter values $\xi_{\psi}=1.2$ and $\xi_{\psi}=1$, we have $\frac{d\ln\alpha}{d\ln k}\gtrsim 10^{-2}$ and $\frac{d\ln\alpha}{d\ln k}\gtrsim 10^{-1}$ respectively. Thus, the super-horizon scalar perturbations are not adiabatic at $\xi_{\psi}\lesssim1.2$ and not even stable at $\xi_{\psi}\lesssim1$. We can also see the $\xi_{\psi}\lesssim1$ instability in the amplitude of $\alpha(\xi_{\psi},\tilde{\tau})$ as well. In the left panel of figure \ref{figScalar}, we present $\alpha^2+1$ vs. $\xi_{\psi}$. This quantity is almost equal to one for $\xi_{\psi}>3$, while it is around and larger than one for $\sqrt{2}<\xi_{\psi}<3$. As a result, our scalar perturbations are stable and almost adiabatic for $\xi_{\psi}\gtrsim\sqrt{2}$. In smaller values of $\xi_{\psi}$, however, it deviates from adiabatic solution and eventually becomes unstable at long wavelengths.
\begin{figure}[h!]
\includegraphics[width=0.5\textwidth]{scalar-n-gravity.pdf}\includegraphics[width=0.53\textwidth]{dlnalpha-dlnk.pdf}
\caption{The amplitude and the spectral index of $\delta\varphi_{\Psi}^S/\delta\varphi_{\Psi}^G$ after horizon crossing. Left panel shows $1+\alpha^2(\xi_{\psi},\tilde{\tau})$ and $\frac{d\ln\alpha}{d\ln k}$ at $\tilde{\tau}=10^{-3}$ with respect to $\xi_{\psi}$. In the right panel, we present $\frac{d\ln\alpha}{d\ln k}$ vs. $\tilde{\tau}$ for different values of $\xi_{\psi}$. The small box in the left panel shows that for $\xi_{\psi}\geqslant\sqrt{2}$, the values of $\frac{d\ln\alpha}{d\ln k}$ is less than $10^{-3}$ and therefore we can approximately consider $\alpha(\tilde{\tau},\xi_{\psi})$ as a numerical prefactor. However, as we go to smaller values of $\xi_{\psi}$, both of $\alpha$ and $\frac{d\ln\alpha}{d\ln k}$ increases quickly. }\label{figScalar}
\end{figure}
\section{Tensor perturbations}\label{tensor-section}
Working out the field equations of tensor fluctuations in section \ref{perturbation}, here, we turn to study the evolution of gravitational waves. The spin-2 fluctuation of the SU(2) gauge field contributes to the anisotropic stress and acts as a source term for gravitational waves.
The field equation of $h_{_{R,L}}(\tau,\textbf{k})$ in Eq. \eqref{T-gf} can be read as
\begin{equation}\label{eq-h--}
\partial^2_{\tilde{\tau}}h_{_{R,L}}+\bigg(1-\big(2-\epsilon+2\psi^2\big)\tilde{\mathcal{H}}^2\bigg)h_{_{R,L}}\simeq S^T_{_{R,L}}(\tilde{h}_{_{R,L}}),
\end{equation}
where $S^T_{_{R,L}}(\tilde{h}_{_{R,L}})$ is given by the linear source term given in \eqref{T-gf}
\begin{equation}
S^T_{_{R,L}}(\tilde{h}_{_{R,L}})\simeq2\psi\tilde\mathcal{H}\bigg(\partial_{\tilde{\tau}}\tilde{h}_{_{R,L}}+(\xi_{\psi}^2\tilde{\mathcal{H}}\mp\xi_{\psi})\tilde{h}_{_{R,L}}\bigg).
\end{equation}
The solution of equation \eqref{eq-h--} can be written as
\begin{equation}\label{hR}
h_{_{R,L}}(\textbf{k},\tilde{\tau})=h^{G}_{_{R,L}}(\textbf{k},\tilde{\tau})+h^{S}_{_{R,L}}(\textbf{k},\tilde{\tau}),
\end{equation}
where $h^{G}$ is the homogeneous part, coming from vacuum fluctuations while $h^{S}$ is the particular part coming from the gauge field spin-2 fluctuation. We can expand $h^{G}_{R}(\textbf{k},\tilde{\tau})$ and $\tilde{h}_{R}(\textbf{k},\tilde{\tau})$ as below in terms of the creation and annihilation operators\footnote{In \eqref{v-eq}, one can negligent the RHS of the equation. Therefore, gravitational waves has negligible effect on evolution of the tensor wave $\tilde{h}_{R,L}$ (see equation \eqref{v-eq-app}). }
\begin{subequations}\label{hG-S}
\begin{align}
h^G_{R}(\tau,\textbf{k})&=\frac{1}{\sqrt{k}}\bigg(\hat{a}^{\dag}_{\textbf{k}}h(\tilde{\tau})+\hat{a}_{-\textbf{k}}h^{*}(-\tilde{\tau})\bigg),\\\label{s-h-norm}
\tilde{h}_{R}(\tau,\textbf{k})&=\frac{1}{\sqrt{k}}\bigg(\hat{b}^{\dag}_{R,\textbf{k}}\tilde{h}_{R}(\tilde{\tau})+\hat{b}_{L,-\textbf{k}}\tilde{h}_{L}^{*}(-\tilde{\tau})\bigg),
\end{align}
\end{subequations}
where the creation and annihilation operators satisfy the standard commutation relations
\begin{equation}
[b_{\sigma,\textbf{k}},b_{\sigma,\textbf{k}'}^{\dag}]=\delta_{\sigma,\sigma'}\delta^{(3)}(\textbf{k}-\textbf{k}'), \quad [b_{\sigma,\textbf{k}},b_{\sigma,\textbf{k}'}]=[b^{\dag}_{\sigma,\textbf{k}},b^{\dag}_{\sigma,\textbf{k}'}]=0.
\end{equation}
By definition, the left-handed polarization is given as $h_{L}(\tau,\textbf{k})=h^{*}_{R}(\tau,-\textbf{k})$.
Note that the mode functions $\frac{1}{\sqrt{k}}\tilde{h}_{_{R,L}}(\tilde{\tau})$ and $\frac{1}{\sqrt{k}}h(\tilde{\tau})$ satisfy the Banch-Davis normalization, \textit{i.e.} $\frac{1}{k}\big(h(\tilde{\tau})h^{*'}(\tilde{\tau})-h^{'}(\tilde{\tau})h^{*}(\tilde{\tau})\big)=i$.
As a result, the particular part of the gravitational wave can be expanded in terms of $b_{\sigma}$ and $b^{\dag}_{\sigma}$ as
\begin{equation}\label{sourced-GW}
h^S_{R}(\tau,\textbf{k})=\frac{1}{\sqrt{k}}\bigg(\hat{b}^{\dag}_{R,\textbf{k}}h^{\!^{s}}_{R}(\tilde{\tau})+\hat{b}_{L,-\textbf{k}}h_{L}^{\!^{s}*}(-\tilde{\tau})\bigg),
\end{equation}
Note that the general solution of the tensor modes are unpolarized and is specified by one function $h(\tilde{\tau})$.
After imposing the Banch-Davis inertial condition to \eqref{hG-S}, we have $h$ as
\begin{equation}\label{h-hankel}
h(\tilde{\tau})\simeq-\sqrt{\frac{\pi\tilde{\tau}}{2}}H^{^{(1)}}_{\nu_T}(\tilde{\tau}) \quad \textmd{for} \quad \nu_T\simeq\frac32+\epsilon.
\end{equation}
In order to solve the particular part of gravitational wave $h^{\!^{s}}_{_{R,L}}(\tilde{\tau})$, we need to determine $\tilde{h}_{_{R,L}}(\tilde{\tau})$ in the following.
\subsection{Particular gravitational waves}
During the slow-roll, we can neglect RHS of \eqref{v-eq}, and the field equation of $\tilde h_{_{R,L}}$ is
\begin{equation}\label{v-eq-app}
\partial^2_{\tilde{\tau}}\tilde{h}_{_{R,L}}(\textbf{k},\tau)+\bigg(1\mp\frac{2(\xi+\xi_{\psi})}{\tilde{\tau}}+\frac{2\xi\xi_{\psi}}{\tilde{\tau}^2}\bigg)\tilde{h}_{_{R,L}}(\textbf{k},\tau)\simeq0,
\end{equation}
in which we used the slow-roll relations \eqref{T-def}. Upon re-definitions below
\begin{equation}\label{re-def}
z=-2i\tilde{\tau},\quad \kappa_{_{R,L}}=\mp i\big(\xi+\xi_{\psi}\big) \quad \textmd{and} \quad \mu^2=\frac14-2\xi\xi_{\psi},
\end{equation}
we can rewrite \eqref{v-eq-app} in form of the Whittaker equation
\begin{equation}
\partial^2_{z}W_{\kappa,\mu}(z)+(-\frac14+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2})W_{\kappa,\mu}(z)=0.
\end{equation}
The most general solutions of the above equation are Whittaker functions $W_{\kappa,\mu}(z)$ and $M_{\kappa,\mu}(z)$
\begin{equation}
\tilde{h}_{\sigma}(\tilde{\tau})=c_1 W_{\kappa\!_{\sigma},\mu}(-2i\tilde{\tau})+c_2 M_{\kappa\!_{\sigma},\mu}(-2i\tilde{\tau}).
\end{equation}
Imposing the usual Minkowski vacuum state for the gauge field's canonically normalized field $\tilde{h}_{_{R,L}}$ in the asymptotic past\footnote{The $W_{\kappa,\mu}(z)$ has the following asymptotic from at the limit $\mid z\mid\rightarrow\infty$
\begin{equation}
W_{\kappa,\mu}(z)\rightarrow z^{\kappa}e^{-z/2}, \quad M_{\kappa,\mu}(z)\rightarrow \Gamma(2\mu+1)\bigg(\frac{i(-1)^{\mu-\kappa}z^{\kappa}e^{-z/2}}{\Gamma({-\kappa+\mu+\frac12})}+\frac{z^{-\kappa}e^{z/2}}{\Gamma({-\kappa+\mu+\frac12})}\bigg) \quad \textmd{for} \quad \mid \arg z\mid<\frac32\pi.
\end{equation}
Thus, the function $W_{\kappa,\mu}(-2i\tilde{\tau})$ represents the positive frequency solutions.}, we obtain $\tilde{h}_{_{R,L}}(\tilde{\tau})$
\begin{equation}\label{tilde-h}
\tilde{h}_{\sigma}(\tilde{\tau})=e^{i\kappa_{\sigma}\pi/2} W_{\kappa\!_{\sigma},\mu}(-2i\tilde{\tau}),
\end{equation}
up to a phase factor. Moreover, the particular part of the solution is given as below
\begin{equation}\label{h-sigma}
h^{\!^{s}}_{_{R,L}}(\tilde{\tau})=\int_{\tilde{\tau}}^{\infty}G(\tilde{\tau},\tilde{\tau}')S^T_{_{R,L}}(\tilde{\tau}')d\tilde{\tau}',
\end{equation}
in which $G(\tilde{\tau},\tilde{\tau}')$ is the retarded Green's function of Eqn. \eqref{eq-h--}
\begin{eqnarray}
\label{Green}
G(\tilde{\tau},\tilde{\tau}')\simeq\bigg(\frac{\tilde{\tau}'-\tilde{\tau}}{\tilde{\tau}'\tilde{\tau}}\cos(\tilde{\tau}'-\tilde{\tau})-(1+\frac{1}{\tilde{\tau}\x'})\sin(\tilde{\tau}'-\tilde{\tau})\bigg)\Theta(\tilde{\tau}'-\tilde{\tau}),
\end{eqnarray}
where $\Theta(\tilde{\tau}-\tilde{\tau}')$ is the Heveside's delta function.
It is useful to parametrize $h^{\!^{s}}_{_{R,L}}(\tilde{\tau})$ in \eqref{h-sigma} as below
\begin{eqnarray}\label{sourced-h--}
h^{\!^{s}}_{_{R,L}}(\tilde{\tau})= \bigg(\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\bigg)^{\!\frac12}\mathcal{G}_{_{R,L}}(\kappa,\mu,\tilde{\tau})h_{_{\rm deS}}(\tilde{\tau}),
\end{eqnarray}
where $h_{_{\!\rm{deS}}}(\tilde{\tau})$ is the homogeneous solution of \eqref{h-hankel} in de Sitter space
\begin{equation}
\frac{1}{\sqrt{k}}h_{_{\!\rm{deS}}}(\tilde{\tau})=\frac{1}{\sqrt{2k}}(1+\frac{i}{\tilde{\tau}})e^{i\tilde{\tau}},
\end{equation}
and $\mathcal{G}_{_{R,L}}(\kappa,\mu,\tilde{\tau})$ is defined as
\begin{equation}\label{math-G}
\mathcal{G}_{_{R,L}}(\kappa,\mu,\tilde{\tau})=\frac{e^{i\kappa_{_{R,L}}\!\pi/2}}{\sqrt{(1+\xi_{\psi}^2)/32}}\int^{\infty}_{\tilde{\tau}}\frac{G(\tilde{\tau},\tilde{\tau}')}{h_{_{\!\rm{deS}}}(\tilde{\tau})\tilde{\tau}'}\!\biggl(\!\partial_{\tilde{\tau}'}+(\frac{\xi_{\psi}^2}{\tilde{\tau}'}\mp\xi_{\psi})\biggl)\!W_{\kappa_{_{R,L}},\mu}(\!-2i\tilde{\tau}'\!)d\tilde{\tau}'.
\end{equation}
\begin{figure}[h!]
\includegraphics[width=0.54\textwidth]{GW-1.pdf}\includegraphics[width=0.5\textwidth]{GW-2.pdf}
\caption{The right panel shows $\tilde{\gamma}{_{R,L}}$ with respect to $\tilde{\tau}$ where the solid (red) line shows the right-handed and dashed (black) one presents the left-handed polarization. In the left panel, we plotted the particular part of gravitational waves $\gamma^s_{R}$ vs. $\tilde{\tau}$. In this system, we choose $\rho_{_{\rm YM}}=\epsilon^2H^2$ and $\xi_{\psi}=\sqrt{3}$ and since $\psi>0$, the right-handed circular polarization is enhanced by evolution. }\label{figGW}
\end{figure}
Before analytically computing the integral \eqref{math-G} and working out the explicit form of $h^{\!^{s}}_{_{R,L}}(\tilde{\tau})$, here we summarize the qualitative properties of the solutions.
As indicated by \eqref{v-eq-app}, the frequency of $\tilde{h}$ gets negative for one of the polarizations for a short period before horizon crossing.
Thus, that particular polarization of $\tilde{h}_{\sigma}$ experiences a short phase of tachyonic growth which eventually leads to its sharp decay after horizon crossing. The polarization with the tachyonic phase acts as an impulse function for its corresponding polarization of $h^s_{\sigma}$. That then enhances the amplitude of one of the polarizations while keeps the other polarization unchanged. In fig. \ref{figGW}, we presented the result of the numerical study of tensor fluctuations. In the following, we determine the analytic form of the particular solution of gravitational waves \eqref{h-sigma}, in the long wave length limit of the power spectrum.
\subsubsection*{$\rhd$~ super-horizon behavior of $h^{s}_{_{R,L}}$ }
In order to study the super-horizon behavior of gravitational waves, one needs to do the Green's integral \eqref{math-G} in the limit that $\tilde{\tau}\ll1$. We presented details of calculations in Appendix \ref{Green-int} and in the following we only report the final result. The particular solution of gravitational wave function in \eqref{sourced-h--} has the following super-horizon form
\begin{equation}\label{super-h}
h^{\!^{s}}_{_{R,L}}(\tilde{\tau})\simeq \bigg(\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\bigg)^{\!\frac12}\mathcal{G}_{_{R,L}}(\xi_{\psi})h_{_{\!\rm{deS}}}(\tilde{\tau}),
\end{equation}
where the explicit form of $\mathcal{G}_{_{R,L}}$ is presented in \eqref{Int-IV}. Depending on the sign of $\psi$, the prefactor $\mathcal{G}_{_{\sigma}}$ is subleading for one of the polarization states in which $i\kappa_{\sigma}$ is negative, while it can be significant for the other one in which $i\kappa_{\sigma}>0$. We call the former integral $\mathcal{G}_{_{-}}$ and the latter one $\mathcal{G}_{_{+}}$ and have
\begin{subequations}\label{super-h-}
\begin{align}
h^{\!^{s}}_{_{R,L}}(\tilde{\tau})&\simeq \bigg(\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\bigg)^{\!\frac12}\mathcal{G}_{_{\pm}}(\xi_{\psi})h_{_{\!\rm{deS}}}(\tilde{\tau}) \quad \textmd{where} \quad \psi>0,\\
h^{\!^{s}}_{_{R,L}}(\tilde{\tau})&\simeq \bigg(\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\bigg)^{\!\frac12}\mathcal{G}_{_{\mp}}(\xi_{\psi})h_{_{\!\rm{deS}}}(\tilde{\tau}) \quad \textmd{where} \quad \psi<0.
\end{align}
\end{subequations}
In the left panel of figure \ref{IR-Gamma}, we present $\mathcal{G}_{_{\pm}}$ with respect to $|\xi_{\psi}|$. Here, we rescaled $\mathcal{G}_{_{\pm}}$ to make a more straightforward connection between the amplitude of $h^{\!^{s}}$ and $h_{_{\!\rm{deS}}}$ (in our model $\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\lesssim\epsilon^2$).
\begin{figure}[h!]
\includegraphics[width=0.54\textwidth]{IR-IL-Gamma.pdf}\includegraphics[width=0.54\textwidth]{ns_T.pdf}
\caption{ The left panel shows the pre-factor $\mathcal{G}_{_{\pm}}(|\xi_{\psi}|)$ with respect to $|\xi_{\psi}|$. Since $h^s_{_{\pm}}/h_{\rm deS}=\big(\frac{\rho_{\rm YM}}{\rho}\big)^{\frac12}\mathcal{G}_{_{\pm}}$ where $\big(\frac{\rho_{\rm YM}}{\rho}\big)^{\frac12}\lesssim10^{-2}$ in our model, here we presented the rescaled $\mathcal{G}_{_{\pm}}$. In the right panel, the spectral tilt of the enhanced particular mode $n_{\gamma_{+}^s}$ is illustrated with respect to $\tilde{\tau}$ which damps like $a^{-\frac32}$.}\label{IR-Gamma}
\end{figure}
As we see, $\mathcal{G}_{_{-}}$ is always subleading and we can ignore it.
However, $\mathcal{G}_{_{+}}$ has a significant value (except around $|\xi_{\psi}|=\frac32$) and its explicit form is
\begin{eqnarray}\label{Int-expres}
\mathcal{G}_{_{+}}(\xi_{\psi})\!&\simeq&\!e^{\frac{i\pi}{2}\kappa_{_{\!+}}}\frac{2\sqrt{(1+\xi^2_{\psi})}}{\xi^2_{\psi}} \bigg(
\frac{(i\xi_{\psi}+1)\Gamma(-\kappa_{_{\!+}})}{\Gamma(\frac12-\kappa_{_{\!+}}-\mu)\Gamma(\frac12-\kappa_{_{\!+}}+\mu)}+\frac{(i\xi_{\psi}-1)}{\Gamma(1-\kappa_{_{\!+}})}\bigg)\Gamma(\frac12-\mu)\Gamma(\frac12+\mu),\nonumber
\end{eqnarray}
where $i\kappa_{+}=\frac{1+2\xi_\psi^2}{|\xi_{\psi}|}$.
As a result, the particular solution of gravitational waves are circularly polarized. In fact, depending on the sign of $\psi$, one of its polarizations gets sizeable around and after horizon crossing, while the other polarization is very small and negligible. Recalling that $\gamma^s_{\sigma}(\tau,k)=\frac{\sqrt2h^s_{\sigma}(\tilde{\tau})}{a}$, we have the super-horizon form for the gravitational waves ($k\tau\ll1$)
\begin{equation}\label{super-hs}
\gamma^{\!^{s}}_{_{+}}(\tau,k)\simeq \bigg(\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\bigg)^{\!\frac12}\mathcal{G}_{_{+}}(\xi_{\psi})\frac{H}{k^{\frac32}} \quad \textmd{and} \quad \gamma^{\!^{s}}_{_{-}}(\tau,k)\simeq 0.
\end{equation}
The power spectrum of the particular solution of gravitational waves is given as
\begin{equation}\label{plz-power}
P_{\gamma^{s}_{+}}=\frac{8\pi k^3}{(2\pi)^3}|\gamma^s_{+}|^2\simeq \bigg(\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\bigg) \mathcal{G}^2_{_{+}}(\xi_{\psi})\bigg(\frac{H}{M_{\rm pl}\pi}\bigg)^2 \quad\quad \textmd{and} \quad P_{\gamma^{s}_{-}}(\tau,k)\simeq 0,
\end{equation}
which is circularly polarized, unlike the unpolarized vacuum fluctuation.
Due to its prefactor $(\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}})^\frac12 \mathcal{G}_{_{+}}(\xi_{\psi})$ in \eqref{super-hs}, $\gamma^s_{+}$ does not \textit{exactly} freeze out after horizon crossing, but it evolves slowly as
\begin{equation}
\frac{d\ln\gamma^s_{+}(\tau,k)}{d\ln\tau}=-\vartheta-(\epsilon+\vartheta)\frac{d\ln(\sqrt{(1+\xi^2_{\psi})}\mathcal{G}_{_{+}})}{d\ln\xi_{\psi}}.
\end{equation}
and therefore is slightly deviates from the adiabatic solution, $\frac{d\ln\gamma^s_{+}(\tau,k)}{d\ln\tau}=\mathcal{O}(\epsilon)$. The spectral tilt of $\gamma^{\!^{s}}_{_{+}}$ has a rather complicated behavior
which is presented in the right panel of figure \ref{IR-Gamma}. It has damped oscillations which decays as $a^{-\frac32}$ at large scales and fades away.
\subsection{Modified Lyth bound and tensor spectrum}
Given the fact that $h^{G}$ and $h^{S}$ are uncorrelated and working out \eqref{h-hankel} and \eqref{super-h-}, we obtain the power spectrum of gravitational waves as
\begin{equation}\label{power-T}
P_{_{T}}\simeq \bigg(2+\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\mathcal{G}^2_{_{+}}(\xi_{\psi})\bigg)\big(\frac{H}{\piM_{\rm pl}}\big)^2 .
\end{equation}
In fact, the gauge field's tensor fluctuations modified the gravitational waves power spectrum proportional to $\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}$ and a function of $\xi_{\psi}$. However, the tensor spectral tilt of vacuum fluctuations is the same as the standard one
\begin{equation}
n_T=-2\epsilon,
\end{equation}
One of the polarization states of $\gamma_{ij}$ has the power spectrum of graviton vacuum fluctuations, $P_{_{vac}}(\tilde{\tau})\simeq\big(\frac{H}{\piM_{\rm pl}}\big)^2$, while the other is enhanced by the gauge field ( see equation \eqref{plz-power}). We can parametrize the chirality of CMB power spectrum by the dimensionless parameter
\begin{equation}
\chi\equiv\frac{P_{_{R}}-P_{_{L}}}{P_{_{vac}}}=s\mathcal{G}^2_{_{+}}(\xi_{\psi})\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}, \quad \textmd{where} \quad
s=\textmd{sign}(\psi).
\end{equation}
In the left panel of figure \ref{r}, we present $\chi$ with respect to $\xi_{\psi}$. As we see, it is negligible if $\mid\!\xi_{\psi}\!\mid\lesssim\frac32$, however it increases monotonously for $\mid\!\xi_{\psi}\!\mid>\frac32$.
The other important observational quantity is tensor to scaler ratio $r$ and using \eqref{power-R} and \eqref{power-T}, the prediction of our models is
\begin{equation}
r=16\epsilon\beta \quad \textmd{where} \quad \beta\equiv\bigg(\frac{1+\frac{\bar{\rho}_{_{\rm YM}}}{2\bar{\rho}}\mathcal{G}^2_{_{+}}(\xi_{\psi})}{1+\alpha^2(\xi_{\psi})}\bigg).
\end{equation}
The right panel of figure \ref{r}, shows $\beta$ for $\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\sim\epsilon^2$ with respect to $\xi_{\psi}$.
As we see here, $\beta$ increases by $\mid\!\xi_{\psi}\!\mid$ for $\mid\!\xi_{\psi}\!\mid<\frac32$ and $\mid\!\xi_{\psi}\!\mid>2.5$. $\beta$ is less than one for $\mid\!\xi_{\psi}\!\mid<2.5$, while is more than one and increases sharply by $\mid\!\xi_{\psi}\!\mid$ otherwise.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.55\textwidth]{chi-xi.pdf}\includegraphics[width=0.5\textwidth]{tensor-to-scalar-ratio.pdf}
\caption{The chirality parameter $\chi$ and $\beta$ with respect to $\tilde{\tau}$ for a system with $\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}=\epsilon^2$.}\label{r}
\end{center}
\end{figure}
Lyth (1997) noted that for standard single scalar slow-roll inflation, we can relate the change in the inflaton during inflation, $\Delta\varphi$, to the tensor to scalar ratio and the number of e-folds $N$, as $\Delta\varphi\simM_{\rm pl} N\sqrt{\frac{r}{8}}$ \cite{Lyth:1996im}. In our setup, slow-roll inflation is driven by the axion potential. The SU(2) gauge field is negligible on the background level, however, it has a significant contribution on the scalar and tenor perturbations. Therefore, our model satisfies in the following modified version of Lyth bound
\begin{equation}
\Delta\varphi\simM_{\rm pl} N_{CMB}\sqrt{\frac{r}{8\beta}},
\end{equation}
which relates the axion excursion and $r$.
\subsection{Generic features of tensor fluctuations}
In this subsection, we summarize the generic features of the tensor perturbations in our model.
\begin{itemize}
\item{We have two tensor fluctuations $\gamma_{ij}$ and $\tilde{\gamma}_{ij}$ which are coupled to each other. The former is the \textit{gravitational wave} coming form the perturbed metric while the latter is the spin-2 fluctuations of the perturbed SU(2) gauge field, \textit{tensor waves}.}
\item{The sound speed of both $\gamma_{ij}$ and $\tilde{\gamma}_{ij}$ are equal to one.}
\item{Our system is diagonalized in terms of the circular polarizations. In particular, there are parity odd terms in the perturbed action which have different signs for the right- and left-handed polarization states.}
\item{Due to its parity odd interactions, one of the polarization states of $\tilde{\gamma}_{ij}$ experiences a short period of tachyonic growth before horizon crossing, around $\frac{k}{aH}=2(\xi_{\psi}+\xi)\sim\mathcal{O}(1)$. Shortly after that, however, it starts to decay and fade away.}
\item{The effective mass of $\tilde{\gamma}_{ij}$ is equal to $2(1+\xi_{\psi}^2)H^2$ which leads to decay of its \textit{both} polarizations after horizon crossing.}
\item{$\tilde{\gamma}_{ij}$ contributes to the anisotropic stress $\pi^T_{ij}$ and acts as a source term for the gravitational waves. Thus we can decompose $\gamma_{ij}$ into its vacuum fluctuations, $\gamma^{G}_{ij}$, and the particular solution $\gamma^{S}_{ij}$ which is sourced by the SU(2) gauge field.}
\item{Our vacuum solutions $\gamma^{G}_{ij}$ is \textit{unpolarized} and has the same amplitude as the standard vacuum gravitational waves in the scalar inflationary models. }
\item{The particluar part of gravitational waves, $\gamma^{S}_{ij}$, is circularly polarized. Both of its polarization states are subdominate inside the horizon. However, one of its polarizations $\gamma^s_{+}$, is enhanced around horizon crossing while the other one, $\gamma^s_{-}$, is always negligible. }
\item{If $\psi$ is positive/negative, the right-/left-handed polarization of $\gamma^{S}_{\sigma}$ would get enhanced by its corresponding $\tilde{\gamma}_{\sigma}$ field around the horizon crossing. Therefore, the total tensor power spectrum is modified by a factor proportional to $\frac{\bar\rho_{\rm YM}}{\bar{\rho}}$. Since this modification is only on one polarization state, that generates a chirality equal to $\frac{P_{_{R}}-P_{_{L}}}{P_{_{vac}}}=\textmd{sign}(\psi)\mathcal{G}^2_{_{+}}(\xi_{\psi})\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}$.
As a result, our setup predicts non-vanishing parity odd CMB correlations, $\evdel{TB}$ and $\evdel{EB}$.}
\item{Because of the spin-2 fluctuations of the SU(2) gauge field, the total power spectrum is enhanced with respect to the vacuum fluctuations, \textit{i.e.} $P_{\rm T}=(1+\frac{\bar{\rho}_{\rm YM}}{2\bar{\rho}}\mathcal{G}_{+}^2)P_{\rm T}^{vac}$. That breaks the direct relation between the power spectrum of the gravitational waves and the scale of inflation.}
\item{The tensor to scalar ratio and the Lyth bound are also modified. In particular, the tensor to scalar ratio and the axion excursion are now given as $r=16\beta\epsilon$ and $\Delta\varphi\simM_{\rm pl} N\sqrt{\frac{r}{8\beta}}$ where $\beta$ is presented in figure \ref{r}.}
\end{itemize}
\section{Discussion}\label{conclusion}
In this paper, we have studied the very well-motivated axion inflation models in the presence of an SU(2) gauge field with a small (but non-vanishing) vev. We found that although the gauge field has a small energy density $\rho_{_{\rm YM}}\lesssim\epsilon^2H^2$, yet it leads to a rich phenomenology and new observables in the CMB anisotropy. The inflaton field is the axion $\varphi$ which for the sake of generality has an arbitrary potential. Thanks to the non-Abelian nature of the gauge field, it can have a homogeneous and isotropic solution and therefore a background energy density. Moreover, the Chern-Simons interaction ($\frac{\lambda\varphi}{4f}\tilde F^aF_a$) breaks the conformal invariance of the gauge field and prevents its decay during inflation. As the axion rolls down its potential, $\dot{\varphi}/H$ increases and part of the energy of the axion gradually injects to the gauge field, hence $\rho_{_{\rm YM}}$ slowly increases during inflation. After the end of inflation, on the other hand, $\dot{\varphi}$ starts oscillating around the minimum of the potential and the gauge field acts like a dark radiation, $\rho_{_{\rm YM}}\propto a^{-4}$. Therefore, in this scenario, inflation ends in a self-interacting dark radiation dominated Universe which may have interesting features for the (pre)reheating era. Moreover, the interaction $\varphi F^a\tilde F_a$ provides a natural decay channel for the inflaton during (pre)reheating which is beyond the scope of this paper. The slow-roll dynamics of the gauge field requires that $\frac{\lambda}{f}\sim\frac{\mathcal{O}(10)}{M_{\rm pl}}$. Since large coupling is hard to achieve in a controlled string compactification \cite{Baumann:2014nda}, here we are interested in small values of $\lambda$.
The SU(2) gauge field has a negligible contribution to the inflation dynamics, however, it leaves notable features on the cosmic perturbations. Its fluctuations can be decomposed into scalar, vector and tensor modes. The scalar perturbations are modified by the gauge field at large scales while the vector fluctuations are still damping and unimportant. The scalar perturbations are stable and almost adiabatic for $\xi_{\psi}\gtrsim\sqrt{2}$ while otherwise deviates from the adiabatic solution. Moreover, in the parameter regime $\xi_{\psi}\lesssim1$, the scalar perturbation is unstable. Tensor perturbations are also modified by the gauge field. In particular, the SU(2) gauge field has a spin-2 perturbation which is coupled to the primordial gravitational waves. This new tensor fluctuation explicitly breaks the parity between the left- and right-handed polarization states. Our gravitational waves are the standard vacuum fluctuations plus the particular solution coming from the spin-2 fluctuations of the gauge field. The former has the standard power spectrum $P_T^{\rm vac}=2\big(\frac{H}{\piM_{\rm pl}}\big)^2$ while the latter has a polarized power, proportional to the background energy density of the gauge field and a prefactor function of $\xi_{\psi}$, $P^{+}_T=\frac{\bar{\rho}_{_{\rm YM}}}{\bar{\rho}}\mathcal{G}^2_{_{+}}(\xi_{\psi})\big(\frac{H}{\piM_{\rm pl}}\big)^2$. $P^{+}_T$ is the circularly polarized part of the gravity waves power spectrum and quantifies the amounts of chirality in the super-horizon power spectrum. That results in parity odd CMB correlations between E and B-modes and T and B-models. In the parameter regime $\sqrt{2}<\xi_{\psi}<3$, the gauge field generates simultaneously a detectable chiral gravitational wave signal with negligible contribution to the scalar fluctuations, in agreement with the current CMB observations. Hence the axion excursion satisfies in a modified version of the Lyth bound and scale of inflation is not directly related to the tensor power spectrum.
We emphasise that the perturbed SU(2) gauge field is linearly coupled to the gravitational wave. This is in contrast to the case of U(1) gauge field in which the Abelian gauge field quanta is mixed to the gravitational waves at the nonlinear level through $\varphi F\tilde F$. In that construction of axion driven inflations, the U(1) gauge field quanta are also coupled to the curvature and generates large amounts of non-Gaussianity. Therefore, the resulting gravity wave signal is correlated to the large scale non-Gaussianity \cite{Barnaby:2011qe, Barnaby:2011vw}. In the non-Abelian case, however, the mixing between the gauge field and perturbations in the scalar and tensor sectors i) are coming from different fluctuations and ii) at the linear order. Hence, the enhancement of gravitational wave and the modification in the scalar perturbations are uncorrelated. Given the mixing between the inflaton field and the SU(2) gauge field, perhaps the most important question that is left to answer is the non-Gaussianity of this scenario, which we postpone for future work.
One of the interesting and robust features of this setup is the generation of intrinsic chiral gravity waves which makes it distinguishable from the unpolarized vacuum fluctuations. Interestingly, the spin-2 fluctuations of the SU(2) gauge field provide a source of CP violation during inflation. Inspiring by the gravitational leptogenesis scenario introduced in \cite{Alexander:2004us}, one may explore the possibility of the lepton production during inflation. In \cite{Maleknejad:2016dci}, using the gravitational anomaly in the standard model of particle physics, we studied that possibility.
We found that this setup can serve as a leptogenesis mechanism during inflation and explain the observed baryon asymmetry in the Universe.
\section*{\small Acknowledgment}
It is a pleasure to thank Peter Adshead, Lorenzo Bordin, Paolo Creminelli, Tomohiro Fujita and Marco Peloso for helpful discussion. I am grateful to the hospitality of Stanford University where this work has been initiated and the Galileo Galilei Institute for theoretical physics (GGI) and INFN during its completion. I acknowledge support from Allameh Tabatabaii grant of Boniad Melli Nokhbegan Iran.
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1604.03229
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\section{Introduction}
As a fascinating object, a black hole is increasingly popular in the classical and quantum gravity theories~\cite{XC,BS}. Its thermodynamics has been studied~\cite{JDB1,JMB,RW,SC}, especially in the AdS spacetime where the geometry plays a pivotal role in recent developments of theoretical physics~\cite{SH,RGC}. The AdS/CFT duality~\cite{JMM,EW} offers a powerful tool to tackle nonperturbative features of a variety
of physical systems, and the holographic principle~\cite{LSLS,RB} provides a way to study dynamics of a general gravitational system. The most typical example is the area law relating the entropy of a semi-classical black hole to its horizon area.
Moreover, the introduction of a negative cosmological constant injects new vitality into the research of thermodynamics of black holes, i.e. the variation of the cosmological constant enters the first law of thermodynamics~\cite{BPD,CEJM,KM,DSJT,BPD2}. When the cosmological constant is interpreted as the thermodynamic pressure $P$ in the equation of state,
\begin{equation}
P=-\frac{\Lambda}{8\pi}=\frac{(n-1)(n-2)}{16 \pi l^2},\label{pres}
\end{equation}
where $n$ stands for the dimension of spaetime and $l$ represents the curvature radius of the AdS spacetime, the mass of black holes is identified with the enthalpy rather than the internal energy. As is known in some more fundamental theory, physical constants, such as the Yukawa coupling, the gauge couplings, the Newton gravitational constant, and the cosmological constant, are not fixed but vary as dynamical variables, which arises from vacuum expectation values. As a result, variations of these so-called constants should be included in the first law of black hole thermodynamics, as shown in the Born-Infeld black hole~\cite{GKM,NB} and the Gauss-Bonnet black hole~\cite{RCLY,XXZ}.
Recently, the thermodynamics of a noncommutative black hole has received wide attention~\cite{PNSS,ESS,ANSS,SSN,AESS,TG,RBPN,LMWZ,NM,MY,PN}.
Noncommutative black holes, i.e. noncommutative geometry inspired black holes, originate from the possibility of implementing an effective minimal length responsible for delocalization of the point-like object in general relativity. The main idea is that the noncommutativity of spacetime would be an intrinsic rather than a super-imposed property of manifold. To realize such an idea, one can modify the energy-momentum tensor in terms of smeared matter distributions, such that this modified tensor describes a kind of anisotropic fluid rather than the perfect fluid, but keep the Einstein tensor unchanged in the field equations of gravity~\cite{RBPN}. As a result, the metric of noncommutative black holes is the solution of the modified Einstein's equations. There are two archetypical features of such noncommutative black holes. One is that no singularity exists at the origin which can be smeared out by a specific matter distribution. The other is that it naturally contains an extreme configuration of black holes with a minimal length which originates from the horizon radius of the extreme black hole. The features eliminate the unfavorable divergency of the Hawking radiation.
In this paper, based on a new extended phase space associated with a noncommutative parameter as an intensive variable, we investigate the thermodynamics of the noncommutative high-dimensional Schwarzschild-Tangherlini AdS black hole~\cite{ST} with the non-Gaussian smeared matter distribution~\cite{Park,POS,MX}. By making a dimensional analysis of the noncommutative parameter and an analogy of it with the cosmological constant~\cite{BPD,CEJM,KM,DSJT,BPD2}, the Born-Infeld parameter~\cite{GKM,NB}, and the Gauss-Bonnet coupling constant~\cite{RCLY,XXZ} that are all dealt with as a kind of thermodynamic pressure, we regard the noncommutative parameter as an independent thermodynamic variable called {\em the noncommutative pressure}. Under this assumption, such pressure will appear in the generalized Smarr relation and its variation will be included in the first law of thermodynamics.
In particular, we find that the noncommutative pressure and the thermodynamic pressure make opposite effects in the phase transition of the noncommutative black hole, where the former corresponds to the ultra-violet physics while the latter to the infra-red physics, respectively. Moreover, we show that the reverse isoperimetric inequality~\cite{CGKP,AKMS} remains valid for the noncommutative black hole, and also indicate that the noncommutative black hole with the Gaussian smeared matter distribution holds the maximum entropy for a given thermodynamic volume.
The organization of the present paper is as follows. In section \ref{sec2}, we regard the noncommutative parameter as a new pressure variable and discuss the thermodynamic behaviors of the noncommutative black hole in the extended phase space associated with this intensive variable and its conjugate one. In section \ref{sec3}, the maximum entropy problem of the noncommutative black hole is discussed in detail. Finally, we make our conclusions in section \ref{sec4}. Note that the geometric units, $\hbar=c=k_{_B}=G=1$, are adopted throughout this paper.
\section{Thermodynamics of noncommutative black holes in an extended phase space}\label{sec2}
Based on noncommutative black holes incorporating effects of quantum gravity in the high energy or short distance regime of gravitational fields,
the authors of ref.~\cite{ASES11} have indicated that a smeared matter density in a noncommutative manifold is governed by a Gaussian distribution, no longer by a Dirac delta function. From a physical point of view, the noncommutative geometry is described as a fluid diffused around the origin rather than squeezed at the origin. This observation gives the result that the modified energy-momentum tensor with smeared matter distributions corresponds to an anisotropic fluid rather than a conventional isotropic fluid. Moreover, the recent studies~\cite{Park,POS,MW} show that the Gaussian smeared matter distribution is not always required, as long as the distribution has a sharp peak at the origin like a Dirac delta function, and that the integration of the distribution function takes a finite value. Hence, a general smeared matter density was proposed~\cite{Park},\footnote{The general smeared matter density was applied in three dimensions. Here we extend it to the $n$-dimensional spacetime based on the achievement~\cite{TG} for the case of $k=0$.}
\begin{equation}
\rho(r)={\tilde A}\, r^k e^{-\left(\frac{r}{2\sqrt{\theta}}\right)^2},\label{fenbu}
\end{equation}
where $\sqrt{\theta}$ is the noncommutative parameter with the dimension of length which is associated with the effective minimal length, ${\tilde A}$ is a normalization constant\footnote{The normalization constant ${\tilde A}$ can be fixed by using the constraint: $\int_0^{\infty}\rho(r) \text{d}V_{n-1}=M$, where the parameter $M$ is the ADM mass of black holes, and $\text{d}V_{n-1}$ is an $(n-1)$-dimensional volume element. One thus has the complete form of the general smeared matter density, $\rho(r)=\frac{\Gamma\left(\frac{n-1}{2}\right)}{\Gamma\left(\frac{n+k-1}{2}\right)}\frac{Mr^k e^{-r^2/(4\theta)}}{\pi^{\frac{n-1}{2}}\left(2\sqrt{\theta}\right)^{n+k-1}}$.} and $k$ is a non-negative integer, $k=0, 1, 2, \cdots$. For instance, the Gaussian distribution corresponds to $k=0$, Rayleigh distribution to $k=1$, and Maxwell-Boltzmann distribution to $k=2$, etc.
For a static and spherically symmetric noncommutative black hole, the geometry structure of solutions has to satisfy the two conditions: (i) Radial matter distribution function $\rho=\rho(r)$, which is required by spherical symmetry, and (ii) Covariant energy conservation $\nabla_{\mu}T^{\mu\nu}=0$, which is required by the effective Einstein equations eq.~(\ref{einfield}). In addition, the third condition, (iii) Schwarzschild-like solution $g_{00}g_{rr}=-1$, is added here for discussing the simplest case. Without it, the noncommutative geometry-inspired dirty black holes can be obtained~\cite{MV,EPES12}. Thus, one can find the modified energy-momentum tensor~\cite{RBPN},
\begin{equation}
{T^{\mu}}_{\nu}=\text{Diag}(-\rho, p_r, p_{\perp}, p_{\perp}),\label{emten}
\end{equation}
which represents an anisotropic fluid with the density $\rho$, the radial pressure $p_r=-\rho$, and the tangential pressure $p_{\perp}=-\rho-(r/2)(\text{d}\rho/\text{d}r)$, rather than the perfect fluid. We must notice that the non-vanishing radial pressure would balance the inward gravitational pull and prevent the smeared matter collapsing into a point. This effect coincides accurately with the one caused by an effective minimal length in spacetime.
Therefore, we consider the modified Einstein equations with the energy-momentum tensor of an anisotropic fluid eq.~(\ref{emten}) in the AdS background,
\begin{equation}
R_{\mu\nu}-\frac12 R g_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi T_{\mu\nu},\label{einfield}
\end{equation}
and the metric with the spherical symmetry in $n$ dimensions~\cite{MX},
\begin{equation}
\text{d}s^2=-f(r)\text{d}t^2+\frac{\text{d}r^2}{f(r)}+r^2 \text{d}\,{\Omega}^{2}_{n-2}, \label{metrice}
\end{equation}
where $\text{d}\,\Omega_{n-2}^2$ is the square of line element on an $(n-2)$-dimensional unit sphere. By solving eqs.~(\ref{emten})-(\ref{metrice}) together with the smeared matter density eq.~(\ref{fenbu}), one can obtain the specific form of function $f(r)$,
\begin{equation}
f(r)=1-\frac{16\pi m(r)}{(n-2)\omega r^{n-3}}+\frac{r^2}{l^2}. \label{metric}
\end{equation}
Note that $\omega$ denotes the area of an $(n-1)$-dimensional unit sphere, $\omega=2\pi^{\frac{n-1}{2}}/\Gamma\left(\frac{n-1}{2}\right)$, and $m(r)$ is related to the mass distribution function of black holes\footnote{The mass distribution function is calculated in ref.~\cite{MX} by utilizing its definition $m(r) \equiv \int_0^{r}\rho(r^{\prime}) \text{d}V_{n-1}$ and the smeared matter distribution eq.~(\ref{fenbu}).}
\begin{equation}
m(r)=\frac{M}{\Gamma\left(\frac{n+k-1}{2}\right)}\gamma\left(\frac{n+k-1}{2},\frac{r^2}{4\theta}\right),\label{masfenbu}
\end{equation}
where $\Gamma(x)$ is the gamma function and $\gamma(a,x)$ is the lower incomplete gamma function.
For the non-Gaussian mass density distribution (eq.~(\ref{fenbu})), the matter mean radius reads
\begin{equation}\label{meanrad}
\bar{r}=\int_0^{\infty}r\, \frac{\rho(r)}{M} dV_{n-1}=2\sqrt{\theta}\, \frac{\Gamma(\frac{n+k}{2})}{\Gamma(\frac{n+k-1}{2})}.
\end{equation}
The larger the parameter $\theta$ is, the more diffuse the matter distribution is, while the smaller the parameter $\theta$ is, the more concentrated the matter distribution is. In fact, the noncommutativity of spacetime is a small effect superposed on the ordinary spacetime if it exists. In the limit $\theta\rightarrow 0$, the matter mean radius goes to zero and the matter mass eq.~(\ref{masfenbu}) goes to the total mass $M$, which implies that the matter distribution collapses into a point and the noncommutativity of spacetime disappears. Hence the noncommutative effect is mainly embodied in the region nearby the matter mean radius.
The location of black hole horizons is $r_h$, which is thought to be the largest real root of $f(r)=0$. The literature~\cite{PNSS,ESS,ANSS,SSN,AESS,TG,RBPN,LMWZ,NM,MY,PN,Park,POS,MX} has pointed out that the black holes with smeared matter distributions possess a significant feature, i.e. the existence of an extreme configuration of black holes, in short an {\em extreme black hole}. The temperature of such a black hole vanishes, indicating that it is in a frozen state. The thermodynamic behaviors of the noncommutative black holes have been discussed in the ordinary phase space, see our previous work~\cite{MX} for the details. Here we only mention the thermodynamic enthalpy $H=M$ for our later use. Utilizing eqs.~(\ref{metric}) and (\ref{masfenbu}), we can get the enthalpy in terms of the horizon radius of black holes $r_h$,
\begin{equation}
M=\frac{(n-2)\omega \Gamma\left(\frac{n+k-1}{2}\right)}{16\pi \gamma\left(\frac{n+k-1}{2},\frac{r_h^2}{4\theta}\right)}\left(r_h^{n-3}+\frac{r_h^{n-1}}{l^2}\right). \label{enth}
\end{equation}
We now analyze the thermodynamics of noncommutative black holes with a new perspective.
At first, let us have a closer look at the noncommutative parameter $\theta$. It is obvious that $\sqrt{\theta}$ gives the characteristic length of noncommutative spacetimes and its dimension is $\text{Length}$.
By analogy with the cosmological constant~\cite{BPD,CEJM,KM,DSJT,BPD2} and the Gauss-Bonnet coupling constant~\cite{RCLY,XXZ} that are all dealt with as a kind of thermodynamic pressure, we regard the noncommutative parameter as a new intensive thermodynamic variable $P_{NC}$ called the noncommutative pressure,
\begin{equation}
P_{NC}=\frac{1}{4\pi\theta}, \label{ncpress}
\end{equation}
which will appear in the Smarr relation. It is known that the thermodynamic pressure $P$ (eq.~(\ref{pres})) is associated with the action of the AdS background spacetime to the thermodynamic system of black holes. Similarly, the above noncommutative pressure $P_{NC}$ (eq.~(\ref{ncpress})) can be regarded as the action of the self-gravitating droplet of anisotropic fluid~\cite{PNSS,PN} to the thermodynamic system of black holes.
With the help of eq.~(\ref{enth}) and the relevant thermodynamic relation, we have its conjugate extensive variable, i.e. noncommutative volume $V_{NC}$,
\begin{equation}
V_{NC}=\left(\frac{\partial M}{\partial P_{NC}}\right)_{S,\,P}
=-\frac{(n-2)\omega \theta \Gamma\left(\frac{n+k-1}{2}\right)}{8 \gamma\left(\frac{n+k-1}{2},\frac{r_h^2}{4\theta}\right)}\left(r_h^{n-3}+\frac{r_h^{n-1}}{l^2}\right)G\left(n,k;\frac{r_h}{2\sqrt{\theta}}\right), \label{ncvol}
\end{equation}
where the function $G(n,k;x)$ is defined by
\begin{equation}
G(n,k;x):=\frac{2 x^{n+k-1} e^{-x^2}}{\gamma\left(\frac{n+k-1}{2},
x^2\right)}. \label{tezheng}
\end{equation}
Therefore, the pair of conjugate variables ($P_{NC}$, $V_{NC}$) is used in the physical system that consists of the black hole together with the self-gravitating droplet of anisotropic fluid as a background.
Secondly, we suggest that the entropy $S$ of the noncommutative black hole is still the standard Bekenstein-Hawking entropy which equals $1/4$ of the event horizon area $A$,
\begin{equation}
S =\frac{A}{4}=\frac{\omega r_h^{n-2}}{4}. \label{entro}
\end{equation}
According to eqs.~(\ref{enth}) and (\ref{entro}), we can obtain the temperature $T_h$,
\begin{eqnarray}
T_h&=&\left(\frac{\partial M}{\partial S}\right)_{P, \,P_{NC}}\nonumber \\
&=&\frac{\Gamma\left(\frac{n+k-1}{2}\right)}
{4\pi\gamma\left(\frac{n+k-1}{2},\frac{r_h^2}{4\theta}\right)}\left\{\frac{n-3-G\left(n,k;\frac{r_h}{2\sqrt{\theta}}\right)}{r_h}+\frac{r_h\left[n-1-G\left(n,k;\frac{r_h}{2\sqrt{\theta}}\right)\right]}{l^2}\right\}. \label{temp}
\end{eqnarray}
In light of eqs.~(\ref{entro}) and (\ref{temp}) we make some comments as follows.
\begin{itemize}
\item For the black holes with smeared matter distributions, the extreme configuration exists and its corresponding temperature vanishes. We must guarantee a vanishing entropy for the extreme black hole in accordance with the third law of thermodynamics. The entropy of the extreme black hole is a constant for a given spacetime structure and a fixed matter distribution. Such a constant can be absorbed into a redefinition of entropy. Hence, eq.~(\ref{entro}) is acceptable as the expression of the entropy of the noncommutative black hole.
\item As the temperature and entropy are conjugate thermodynamic variables to each other, it is reasonable to first define the entropy of the noncommutative black hole as that of the commutative black hole in form only. Then we can obtain the temperature according to eq.~(\ref{temp}). In fact, both the temperature and entropy are noncommutatively corrected because the horizon radius $r_h$ is implicitly $\theta$-dependent. We note that such a treatment coincides with that of the pair of conjugate variables ($P_{NC}$, $V_{NC}$), see, for instance, eqs.~(\ref{ncpress}) and~(\ref{ncvol}). This feature is understandable because the two pairs of conjugate variables appear in the forms of $T_h dS$ and $V_{NC}dP_{NC}$ in the first law of thermodynamics.
\item The temperature eq.~(\ref{temp}) turns back to that of the commutative black hole~\cite{BC} under the limit $\theta\rightarrow 0$, which shows the consistency of the noncommutative generalization.
\end{itemize}
At last, the thermodynamic volume $V$ corresponding to the pressure $P$ (eq.~(\ref{pres})) takes the form,
\begin{eqnarray}
V=\left(\frac{\partial M}{\partial P}\right)_{S,\,P_{NC}}
=\frac{\Gamma\left(\frac{n+k-1}{2}\right)}
{\gamma\left(\frac{n+k-1}{2},\frac{r_h^2}{4\theta}\right)}\frac{\omega}{n-1}r_h^{n-1}. \label{volu}
\end{eqnarray}
We can see that the volume $V$, like $V_{NC}$, is also noncommutatively corrected.
In accordance with the above scenario,
the first law of thermodynamics for the noncommutative black hole in the extended phase space that contains the variation of the noncommutative pressure can be written as
\begin{equation}
dM=T_h dS+VdP+V_{NC}dP_{NC}. \label{fl}
\end{equation}
From the dimensional scaling of the variables, $[M]=L^{n-3}$, $[T_h]=L^{-1}$, $[S]=L^{n-2}$, $[P]=L^{-2}$, $[V]=L^{n-1}$, $[P_{NC}]=L^{-2}$, and $[V_{NC}]=L^{n-1}$, we infer the generalized Smarr relation,
\begin{equation}
(n-3)M=(n-2)T_h S-2VP-2V_{NC}P_{NC}, \label{smarr}
\end{equation}
which is a simple consequence of the first law eq.~(\ref{fl}) in the $n$-dimensions spacetime. It can be verified easily when we
substitute eqs.~(\ref{pres}), (\ref{enth})-(\ref{ncvol}), and (\ref{entro})-(\ref{volu}) into eq.~(\ref{smarr}).
Now we have two pairs of similar conjugate thermodynamic quantities, i.e. the thermodynamic pressure and thermodynamic volume ($P$,$V$) and the noncommutative pressure and the noncommutative volume ($P_{NC}$, $V_{NC}$). Then let us make a comparison about these two pairs of conjugate variables.
\begin{itemize}
\item We notice that the thermodynamic volume eq.~(\ref{volu}) is positive, while the noncommutative volume eq.~(\ref{ncvol}) is negative. They appear in the work terms, $PV$ and $P_{NC}V_{NC}$, respectively, see eq.~(\ref{smarr}), and have contributions to the enthalpy (or thermodynamic characteristic function) which can be thought of as the total energy of the black hole thermodynamic system. For the negative noncommutative volume eq.~(\ref{ncvol}), it can be interpreted as the self-gravitating droplet of anisotropic fluid doing work to the thermodynamic system by pushing against the system and reducing its volume to create the noncommutative black hole via a thermodynamic process. The similar peculiar feature, i.e. the negative volume puzzle, has also appeared in the AdS-Taub-NUT case~\cite{CVJ}. For the thermodynamic volume eq.~(\ref{volu}), it means that a certain part of space is excised to `make a place' for forming a black hole at the cost of energy $P V$ as reported in ref.~\cite{CVJ}.
\item The thermodynamic volume eq.~(\ref{volu}) just depends on the parameters $\theta$, while the noncommutative volume eq.~(\ref{ncvol}) depends on the parameters $\theta$ and $l$. In the limit $l\rightarrow \infty$, the thermodynamic pressure goes to zero, which implies no AdS background spacetime, while the noncommutative volume becomes
\begin{equation*}
V_{NC}\rightarrow -\frac{(n-2)\omega \theta \Gamma\left(\frac{n+k-1}{2}\right) r_h^{n-3}}{8 \gamma\left(\frac{n+k-1}{2},\frac{r_h^2}{4\theta}\right)} \cdot G\left(n,k;\frac{r_h}{2\sqrt{\theta}}\right),
\end{equation*}
and plays a major role in a thermodynamic process. In the limit $\theta \rightarrow 0$, although the noncommutative pressure diverges, the work term $P_{NC} V_{NC}$ goes to zero, which implies no noncommutative effect, while the thermodynamic volume eq.~(\ref{volu}) turns back to the commutative one, $V_{\text{commutative}}=\omega r_h^{n-1}/(n-1)$. Hence such an asymptotic behavior is acceptable for this new pair of conjugate variables ($P_{NC}$, $V_{NC}$) in the first law of thermodynamics under the limit $\theta \rightarrow 0$. From the minus sign of the noncommutative volume, we predict that the two pressure variables ($P$ and $P_{NC}$) will play an opposite role in the thermodynamic process, which will be confirmed in the behaviors of the Gibbs free energy.
\item Due to the observation mentioned above that the noncommutative effect is mainly concentrated in the vicinity of the matter mean radius eq.~(\ref{meanrad}), we find that our model can be regarded as an analogue to the evaporation of liquid droplets in which the two work terms $\sigma \text{d}\mathscr{A}$ (surface tension $\sigma$ and surface area $\mathscr{A}$) and $\mathscr{P} \text{d}\mathscr{V}$ (vapor pressure $\mathscr{P}$ and volume $\mathscr{V}$) play an opposite role in order to keep the liquid droplet in equilibrium and both of them appear in the first law of thermodynamics. In our model, the corresponding two work terms are $P_{NC}\text{d}V_{NC}$ and $P \text{d}V$ that appear in the first law of thermodynamics for the noncommutative black hole, see eq.~(\ref{fl}).
\end{itemize}
Next we consider the Gibbs free energy that is defined as the Legendre transform of the enthalpy (eq.~(\ref{enth})),
\begin{equation}
G:=M(r_h)-T_h (r_h)S(r_h). \label{dgibb}
\end{equation}
By inserting eqs.~(\ref{enth}), (\ref{entro}), and (\ref{temp}) into the above definition, we can obtain the exact expression of the Gibbs free energy,
\begin{equation}
G=\frac{\omega \Gamma\left(\frac{n+k-1}{2}\right)}
{16\pi \gamma\left(\frac{n+k-1}{2},\frac{r_h^2}{4\theta}\right)}\left\{\left[G\left(n,k;\frac{r_h}{2\sqrt{\theta}}\right)+1\right]r_h^{n-3}+\left[G\left(n,k;\frac{r_h}{2\sqrt{\theta}}\right)-1\right]\frac{r_h^{n-1}}{l^2}\right\}. \label{gibb}
\end{equation}
Incidentally, the Gibbs free energy tends to the commutative formula~\cite{RD} under the limit $\theta\rightarrow 0$,
\begin{equation}
G \rightarrow \frac{\omega}{16 \pi}\left(r_h^{n-3}-\frac{r_h^{n-1}}{l^2}\right).
\end{equation}
The behaviors of the Gibbs free energy governed by eq.~(\ref{gibb}) are plotted in Figures \ref{tu1} and \ref{tu2}. In Figure \ref{tu1}, the phase transition is described for the constant thermodynamic pressure $P$ but the varying noncommutative pressure $P_{NC}$. The critical noncommutative pressure corresponds to $\theta_c=0.815$, see the fifth diagram of Figure \ref{tu1}. Once the noncommutative pressure $P_{NC}$ is lower than the critical noncommutative pressure, or the noncommutative parameter is larger than $\theta_c$, the characteristic swallowtail behavior disappears, which means that no first order phase transitions occur. In Figure \ref{tu2}, the phase transition is depicted for the constant noncommutative pressure $P_{NC}$ but the varying thermodynamic pressure $P$. The critical thermodynamic pressure corresponds to $l_c=7.855$, see the yellow curve in Figure \ref{tu2}. When the thermodynamic pressure $P$ is larger than the critical thermodynamic pressure, or the curvature radius of the AdS spacetime is smaller than $l_c=7.855$, no first order phase transitions occur. As a result, the thermodynamic pressure $P$ and noncommutative pressure $P_{NC}$ give the opposite contributions to the first order phase transition. Physically, $P$ dominates the infra-red regime (lower than its critical thermodynamic pressure) while $P_{NC}$ does the ultra-violet regime (larger than its critical noncommutative pressure). As our special examples, we also give the asymptotic behaviors of the Gibbs free energy under the limits $\theta\rightarrow 0$ and $l \rightarrow \infty$ in Figures \ref{tu1} and \ref{tu2}, respectively. The first limit leads to the case of the pure AdS spacetime without noncommutativity, and the second one to the noncommutative case without the AdS background as reported in ref.~\cite{AESS}.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=60mm]{1} &
\includegraphics[width=60mm]{2} \\
\includegraphics[width=60mm]{3} &
\includegraphics[width=60mm]{4} \\
\includegraphics[width=60mm]{5} &
\includegraphics[width=60mm]{6} \\
\includegraphics[width=60mm]{7} &
\includegraphics[width=60mm]{8}
\end{tabular}
\end{center}
\caption{For $n=5$ and $k=3$, plots of the relation of $G$ with respect to $T_h$ at different noncommutative pressure $P_{NC}$ but at the constant pressure $P$ that corresponds to $l=10$. The dashed curve in the last diagram is the commutative case.}
\label{tu1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=70mm]{9}
\caption{For $n=5$ and $k=3$, plots of the relation of $G$ with respect to $T_h$ at the constant noncommutative pressure $P_{NC}$ that corresponds to $\theta=0.5$ but at different thermodynamic pressure $P$ that corresponds to $l=7$ $(\text{{\color{black}Black}})$, 7.855 $(\text{{\color{yellow}Yellow}})$, 9 $(\text{{\color{red}Red}})$, 11 $(\text{\textcolor[rgb]{1.00,0.50,0.00}{Orange}})$, 13 $(\text{{\color{green}Green}})$, and 15 $(\text{\textcolor[rgb]{0.50,0.00,0.51}{Purple}})$, respectively. The \text{{\color{blue}blue}} dashed curve is associated with the vanishing thermodynamic pressure that corresponds to the asymptotic Minkowski spacetime with $l\rightarrow\infty$.}
\label{tu2}
\end{figure}
From eq.~(\ref{gibb}) we notice a special point where the Gibbs free energy vanishes. This special point dubbed by $r_g$ satisfies the following equation,
\begin{equation}
r_g=\left[{\frac{1+G\left(n,k;\frac{r_g}{2\sqrt{\theta}}\right)}{1-G\left(n,k;\frac{r_g}{2\sqrt{\theta}}\right)}}\right]^{1/2}l.
\end{equation}
Under the commutative limit, it comes back to the known result $r_g=l$ given in ref.~\cite{BPD3}. At this point, the first order Hawking-Page phase transition shows up between the thermal radiation and the large black hole. If $r_h < r_g$, the noncommutative AdS spacetime is more stable, which means that the black hole will evaporate. If $r_h > r_g$, the Gibbs free energy is negative, indicating that the black hole is in a more stable thermodynamical configuration.
\section{Isoperimetric inequality}\label{sec3}
In the Euclidean space, the isoperimetric inequality implies that the geometric volume $\mathcal{V}$ and area $\mathcal{A}$ satisfy the following inequality for a given $(n-1)$-dimensional connected domain,
\begin{equation}
\mathcal{R}:=\left[\frac{(n-1)\mathcal{V}}{\omega}\right]^{\frac{1}{n-1}}\cdot \left(\frac{\omega}{\mathcal{A}}\right)^{\frac{1}{n-2}}\leq 1,
\end{equation}
where the equality holds if and only if the domain is a standard round ball.
As a counterpart, the reverse isoperimetric inequality has been asserted~\cite{CGKP,AKMS} in any asymptotic AdS black holes,
\begin{equation}
\mathcal{R}:=\left[\frac{(n-1)V}{\omega}\right]^{\frac{1}{n-1}}\cdot \left(\frac{\omega}{A}\right)^{\frac{1}{n-2}}\geq 1, \label{reiso}
\end{equation}
where $V$ denotes the thermodynamic volume and $A$ the horizon area of black holes.
Note that the equality is attained for the ordinary Schwarzschild-AdS black hole.
Physically, the reverse isoperimetric inequality indicates that the entropy of black holes is maximized for the Schwarzschild-AdS spacetime at a given thermodynamic volume.
Up to now, the above statement has been verified for a variety of black holes with the horizon of spherical topology~\cite{CGKP} and black rings with the horizon of toroidal topology~\cite{AKMS}. The only counterexample, as we know, is the ultra-spinning black hole~\cite{hm}. However, the reverse isoperimetric inequality still remains valid for noncommutative black holes with smeared matter distributions. Let us check it.
In our case, with the help of the entropy (eq.~(\ref{entro})) and the thermodynamic volume (eq.~(\ref{volu})) of noncommutative black holes, we can obtain
\begin{equation}
\mathcal{R}=\left[\frac{\Gamma\left(\frac{n+k-1}{2}\right)}{\gamma\left(\frac{n+k-1}{2},\frac{r_h^2}{4\theta}\right)}\right]^{\frac{1}{(n-1)(n-2)}}.
\end{equation}
Considering the property of the lower incomplete gamma function, we yield $\mathcal{R}\geq 1$ and the equality is attained for taking the commutative limit $\theta\rightarrow 0$, i.e. for the ordinary Schwarzschild-AdS black hole. In other words, for a black hole with a kind of smeared matter distributions, its entropy is smaller than that of the black hole without any distributions at the same thermodynamic volume.
Next, we turn to the discussion about which kind of distributions corresponds to the maximum entropy for the black holes with smeared matter distributions. We compare the two kinds of distributions with the power index $k$ and $k+1$, respectively, see eq.~(\ref{fenbu}), in the $n$-dimensional spacetime. The ratios are denoted by $\mathcal{R}_{n,k}$ and $\mathcal{R}_{n,k+1}$, respectively, and the ratio of the two ratios can be calculated to be
\begin{equation}
\frac{\mathcal{R}_{n,k}}{\mathcal{R}_{n,k+1}}=\left[\frac{\Gamma\left(\frac{n+k-1}{2}\right)}{\Gamma\left(\frac{n+k}{2}\right)}\cdot
\frac{\gamma\left(\frac{n+k}{2},\frac{r_h^2}{4\theta}\right)}{\gamma\left(\frac{n+k-1}{2},\frac{r_h^2}{4\theta}\right)}
\right]^{\frac{1}{(n-1)(n-2)}}.
\end{equation}
We can prove $0< \mathcal{R}_{n,k}/\mathcal{R}_{n,k+1}< 1$, which means that the black hole with the Gaussian smeared matter distribution ($k=0$) holds the maximum entropy for a fixed spacetime dimension at a given thermodynamic volume, as expected. In other words, the more concentrated the matter distribution of black holes is, the greater the entropy of black holes is.
\section{Conclusion}\label{sec4}
Based on our recent work~\cite{MX} about the high-dimensional Schwarzschild-Tangherlini AdS black hole with the non-Gaussian smeared matter distribution, we deal with the noncommutative parameter as an independent thermodynamic variable called the noncommutative pressure. As a result, the noncommutative pressure together with its conjugate volume appears in the generalized Smarr relation and its variation comes out in the first law of thermodynamics. Through analyzing the Gibbs free energy, we point out that the noncommutative pressure and the thermodynamic pressure make the opposite effects in the phase transition of noncommutative black holes. Physically, the noncommutative pressure dominates the UV physics while the thermodynamic pressure does the IR physics. We also discuss the first order Hawking-Page phase transition. Furthermore, we calculate the reverse isoperimetric inequality for the noncommutative black holes and indicate that the noncommutative black hole with the Gaussian smeared matter distribution ($k=0$) holds the maximum entropy at a fixed thermodynamic volume.
\section*{Acknowledgments}
Z.-M. Xu would like to thank Y.-M. Wu and L. Zhao for their helpful discussions.
This work was supported in part by the National Natural
Science Foundation of China under grant No.11675081. At last, the authors would like to thank the anonymous referee for the helpful comment that indeed greatly improves this work.
|
1908.09977
|
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\newcommand\F{\mathbb{F}_1}
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\newcommand\MP{\mathcal{M}_{\mathbb{P}^2, H}}
\newcommand\MF{\mathcal{M}_{\mathbb{F}_1, E+F}}
\newcommand\MFF{\mathcal{M}_{\mathbb{F}_1, F}}
\renewcommand\L{\mathbb{L}}
\newcommand\Smu{S^{\mu} (\gamma_1, \cdots, \gamma_l ; F, E+F)}
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\fancyfoot[L]{
\footnotesize
2010 \textit{Mathematics Subject Classification.} Primary: 14D20, 14J60. Secondary: 14J26, 14F45.\\
\textit{Key words and phrases.} Moduli space of sheaves, Betti numbers, stable cohomology, rational surfaces.
}}
\begin{document}
\thispagestyle{abcd}
\begin{center}
\begin{Large}
On the stabilization of the Betti numbers of the moduli space of sheaves on $\P$\\
\end{Large}
$\qquad$\\
\begin{large}
Sayanta Mandal
\end{large}
\end{center}
\paragraph{\normalfont\textsc{Abstract.}}
{\linespread{0.1}\footnotesize
Let $r \geq 2$ be an integer, and let $a$ be an integer coprime to $r$. We show that if $c_2 \geq n + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right\rfloor$,
then the $2n$th Betti number of the moduli space $M_{\P,H}(r,aH,c_2)$ stabilizes, where $H = c_1(\mathcal{O}_{\P}(1))$.
}
\section{\textsc{Introduction}}
Let $X$ be a smooth projective surface over an algebraically closed field $\mathbb{K}$, and let $H$ be an ample divisor on $X$. We denote the Chern character of a torsion-free coherent sheaf on $X$ by $\gamma = (r,c,\Delta)$, where $r$ is the rank, $c$ is the first Chern class, and $\Delta = \frac{ch_1^2 - 2\cdot r\cdot ch_2}{2r^2}$ is the discriminant. We denote by $M_{X,H}(\gamma)$, the moduli-space parameterizing slope-$H$-semistable sheaves with Chern character $\gamma$. These spaces were constructed by Gieseker \cite{gie} and Maruyama \cite{mar}, and play a central role in many areas of mathematics including algebraic geometry, topology, representation theory, etc. For example, they are used to study linear systems on curves and in the Donaldson theory of $4$-manifolds.
A crucial step to understand the geometry of these moduli spaces is by scrutinizing the cohomology groups associated with them. Consequently, determining the Betti numbers of these spaces are of utmost importance. In this paper, we look at the special case when $X = \P$ and $H = c_1(\mathcal{O}_{\P}(1))$. We show that \\
{}\\
\textbf{Theorem} (Theorem \ref{theorem26})\textbf{.} \textit{Assume that the rank $r$ and the first Chern class $aH$ are coprime. If $c_2 \geq N + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right\rfloor $, then the $2N$th Betti numbers of the moduli space $M_{\P,H}(r,aH,c_2)$ stabilizes.}\\
The general philosophy of Donaldson, Gieseker and Li is that the geometry of the moduli space $M_{X,H}(\gamma)$ behaves better as $\Delta$ tends to infinity. O'Grady \cite{ogr} showed that $M_{X,H}(\gamma)$ is irreducible and generically smooth if $\Delta$ is sufficiently large. Li \cite{li} showed the stabilization of the first and the second Betti numbers of $M_{X,H}(\gamma)$ when the rank is two. When the rank is one, the moduli space $M_{X,H}(1,c,\Delta)$ is isomorphic to $Pic^c(X) \times X^{[\Delta]}$, where $X^{[n]}$ denotes the Hilbert scheme of $n$ points in $X$. The Betti numbers of $X^{[n]}$ were computed by G{\"{o}}ttsche \cite{got90}. Using the K{\"{u}}nneth formula, Coskun and Woolf \cite{cos}[Proposition 3.3] showed that the Betti numbers of $M_{X,H}(1,c,\Delta)$ stabilizes as $\Delta $ tends to infinity. In general, we don't know much about the Betti numbers of $M_{X,H}(\gamma)$.
Yoshioka \cite{yos95}, \cite{yos} and G{\"{o}}ttsche \cite{got96} computed the Betti and Hodge numbers of $M_{X,H}(\gamma)$ when $X$ is a ruled surface and the rank is two. Yoshioka \cite{yos95}, \cite{yos96a} observed the stabilization of the Betti numbers for rank two bundles on ruled surfaces. G{\"{o}}ttsche \cite{got99} extended his results to rank two bundles on rational surfaces with polarizations which are $K_X$-negative. The stabilization of the Betti numbers is known for smooth moduli space of sheaves on $K3$ surfaces. By works of Mukai \cite{muk}, Huybrechts \cite{huy03}, and Yoshioka \cite{yos99}, smooth moduli spaces of sheaves on a $K3$ surface $X$ are deformations of the Hilbert scheme of points on $X$ of the same dimension. In particular, they are diffeomorphic to the Hilbert scheme of points, and hence, their Betti numbers stabilizes. Yoshioka \cite{yos01} obtained similar results for moduli spaces of sheaves on abelian surfaces. A smooth moduli space of sheaves $M_{X,H}(\gamma)$ on an abelian surface $X$ is deformation equivalent to the product of the dual abelian surface of $X$ and a Hilbert scheme of points on $X$. Consequently, the Betti numbers stabilizes.
In the special case when $X = \P$ and $H = c_1(\mathcal{O}_{\P}(1))$, Yoshioka \cite{yos94},\cite{yos} computed the Betti numbers of $M_{\P,H}(2, -H,\Delta)$ and showed that they stabilizes as $\Delta$ tends to infinity. Manschot \cite{man11},\cite{man} computed the Betti numbers of $M_{\P,H}(3, -H, \Delta)$, and later, building on the work of Mozgovoy \cite{moz}, produced a formula to determine the Betti numbers for any rank and computed them in case of rank four. By looking at the tables present in the papers of Yoshioka and Manschot, one would expect the Betti numbers to stabilize as $\Delta$ tends to infinity, for any given rank and first Chern class. Coskun and Woolf \cite{cos} showed that this is indeed the case, and furthermore, they determined the generating function for the stable Betti numbers. Our goal in this paper is to produce lower bounds for the Betti numbers to become stable, and since we know the generating function for the stable Betti numbers, we can determine the Betti numbers for a large collection of moduli spaces.
\paragraph{Organization of the paper.} In section \ref{section2}, we set-up the notation and review some basic facts on slope-semistable sheaves and their moduli space. In section \ref{section3}, we look at the Betti numbers of the moduli space of rank one sheaves on $\P$. In section \ref{section4}, we determine lower bounds for vanishing of the coefficient of $\L^{-N}q^{\Delta}$ in the generating function $\tilde{G}_{r,\tilde{c}}(q)$ (see equation \ref{generatingfndefn}) defined over the ring $A^-$ (see equation \ref{ringdefn}). In section \ref{section5}, we determine lower bounds for vanishing of the coefficient of $\L^{-N}q^{\Delta}$ in the generating function $G_{r,c}(q)$ (see equation \ref{generatingfndefn}). In section \ref{section6}, we determine lower bounds for $c_2$ for the stabilization of the Betti numbers of the moduli space $M_{\P,H}(r,aH,c_2)$.
\paragraph{Acknowledgements.} I am extremely grateful to my advisor Prof. Izzet Coskun for invaluable mathematical discussions, correspondences, and several helpful suggestions.
\section{\textsc{Preliminaries}}\label{section2}
Let $X$ be a smooth projective surface over an algebraically closed field $\mathbb{K}$ of characteristic zero, and let $H$ be an ample divisor on $X$. Throughout this paper, we are going to assume that all sheaves are coherent and torsion free. Given a sheaf $\mathcal{F}$, we define the $H$\textit{-slope} of $\mathcal{F}$ as
\[ \mu_H(\mathcal{F}) = \frac{ch_1(\mathcal{F})\cdot H}{ch_0(\mathcal{F}) \cdot H^2} \]
Additionally, we define the \textit{Chern character} of $\mathcal{F}$ as $\gamma = (r,c,\Delta)$ where $r$ is the rank, $c$ is the first Chern class, and $\Delta$ is the discriminant defined as
\begin{equation}
\label{deltadefn}
\Delta(\mathcal{F}) = \frac{ch_1(\mathcal{F})^2 - 2 ch_0(\mathcal{F}) ch_2(\mathcal{F})}{2 ch_0(\mathcal{F})^2}
\end{equation}
We define a sheaf $\mathcal{F}$ to be $\mu_H$\textit{-semistable} if for every proper subsheaf $\mathcal{E}$, we have $\mu_H(\mathcal{E}) \leq \mu_H(\mathcal{F})$. Likewise, we define a sheaf $\mathcal{F}$ to be $\mu_H$\textit{-stable} if the inequality is strict. Every $\mu_H$-semistable sheaf has a Jordan H{\"{o}}lder filtration with the sub-quotients being $\mu_H$-stable \cite{huy}[Proposition 1.5.2]. We say two $\mu_H$-semistable sheaves are \textit{S-equivalent} if the corresponding direct sum of subquotients appearing in the Jordan H{\"{o}}lder filtration are isomorphic.
Given a Chern character $\gamma = (r,c,\Delta)$, we denote by $M_{X,H}(\gamma)$ the moduli space of S-equivalence classes of $\mu_H$-semistable sheaves with Chern character $\gamma$. We denote by $\mathcal{M}_{X,H}(\gamma)$ the moduli stack of $\mu_H$-semistable sheaves with Chern character $\gamma$. When $X$ is smooth projective surface and $H$ is ample divisor with $K_X \cdot H <0$, the moduli space $M_{X,H}(\gamma)$ is smooth at every stable sheaf $\mathcal{F}$ because $ext^2(\mathcal{F},\mathcal{F}) = hom(\mathcal{F},\mathcal{F} \otimes K_X) = 0$. Consequently, if all $\mu_H$-semistable sheaves with Chern character $\gamma$ are $\mu_H$-stable, then $M_{X,H}(\gamma)$ is a smooth projective variety of dimension $ext^1(\gamma,\gamma) = 1 - \chi(\gamma,\gamma)$.
Assume that $M_{X,H}(\gamma)$ is smooth, e.g. when $r \cdot H^2$ and $c \cdot H$ are coprime. To understand the Betti numbers of $M_{X,H}(\gamma)$, we look at the polynomial
\[ P_{M_{X,H}(\gamma)}(t) = \sum_{i=0}^{2(1 - \chi(\gamma,\gamma))} b_i(M_{X,H}(\gamma)) t^i \]
In general, consider a collection of polynomials $P_d(t) = \sum_{i=0}^{s_d} a_{i,d}t^i$ indexed by integers $d \geq N$, for some integer $N$. We look at the corresponding collection of shifted polynomials $\tilde{P}_d(t) = \sum_{j=-s_d}^0 b_{j,d}t^j$, where $b_{j,d} = a_{j+s_d,d}$.
\begin{defn}\label{definition1}
We say that the collection of polynomials $P_d(t)$ \textit{stabilize} if for each $j$ there exists an integer $d_0(j)$ such that for all $d \geq d_0(j)$ we have $b_{j,d} = b_{j,d+1}$. In this case, we define the \textit{stable limit} to be $\tilde{P}_\infty (t) = \sum_{j=-\infty}^0 \beta_j t^j$, where $\beta_j = b_{j,d}$ for any $d \geq d_0(j)$.
\end{defn}
In our case, we fix $r$ and $c$ and look at the collection of polynomials $P_{M_{X,H}(r,c,\Delta)}$ for $\Delta \geq 0$. If this collection of polynomials stabilize, we say that the Betti numbers of $M_{X,H}(r,c,\Delta)$ \textit{stabilize}.
Consider the generating function
\begin{equation}\label{defnFtilde}
\tilde{F}(q,t) = \sum_{d=N}^\infty \tilde{P}_d(t) q^d
\end{equation}
We have
\begin{propn}[\cite{cos}, Proposition 3.1]\label{proposition1}
The polynomials $P_d(t)$ stabilize iff the coefficient of $t^i$ in $(1-q)\tilde{F}(q,t)$ is a Laurent polynomial in $q$. Moreover, if the polynomials stabilize, the stable limit is obtained by evaluating $(1-q)\tilde{F}(q,t)$ at $q=1$.
\end{propn}
The proof of Proposition \ref{proposition1} due to Coskun and Woolf \cite{cos} essentially follows from the following Lemma.
\begin{lem}\label{lemma3}
For any $j\geq 0$, the coefficient of $t^{-j}q^d$ in $(1-q)\tilde{F}(q,t)$ is zero for $d \geq d_0(j)$ iff $b_{j,d} = b_{j,d+1}$ for all $d \geq d_0(j) -1$.
\end{lem}
\begin{proof}
Let us define $b_{j,d}=0$ for $j < -s_d$. It follows from equation \ref{defnFtilde} that \[ \tilde{F}(q,t) = \sum_{d \geq N, \, j \leq 0} b_{j,d}t^j q^d \] whence,
\[ (1-q)\tilde{F}(q,t) = \sum_{d \geq N, \, j \leq 0} (b_{j,d} - b_{j,d-1})t^jq^d\]
\end{proof}
Additionally, let
\[ F(q,t) = \sum_{d=N}^\infty P_d(t) q^d \]
and assume that the polynomials $P_d(t)$ satisfy Poincar{\'{e}} duality i.e. $t^{s_d}P_d(t^{-1}) = P_d(t)$ for $d \gg 0$, then we have
\begin{cor}[\cite{cos}, Corollary 3.2]
The polynomials $P_d(t)$ stabilize iff the coefficient of $t^i$ in $(1-q)F(q,t)$ is a Laurent polynomial in $q$, and in this case, we get the generating function for the stable coefficients by evaluating $(1-q)F(q,t)$ at $q=1$.
\end{cor}
Let $K_0(var_k)$ denote the Grothendieck ring of varieties over the field $k$ of characteristic zero. The Poincar{\'{e}} polynomials for smooth varieties induces \cite{joy07} the \textit{virtual Poincar{\'{e}} polynomial} map
\[ P(t) : K_0(var_k) \rarrow \mathbb{Z}[t] \]
Let $\L$ denote the class $[\mathbb{A}^1]$ in $K_0(var_k)$. Consider the ring $R = K_0(var_k)[\L^{-1}]$. We have a $\mathbb{Z}$-graded filtration $\mathfrak{F}$ on $R$, where for any given variety $Y$, we have
\[ [Y]\L^{a} \in \mathfrak{F}^i \quad\text{ iff } \quad dim(Y) + a \leq -i \]
We define the ring $A^{-}$ to be the inverse limit
\begin{equation}\label{ringdefn}
A^- := \varprojlim_{i \geq 0} R/(\mathfrak{F}^i \otimes_{\mathfrak{F}^0} R)
\end{equation}
Our notion of dimension extends from $K_0(var_k)$ to $A^-$. Similarly, the virtual Poincar{\'{e}} polynomial extends to $R$ and $A^-$ where it takes values in $\mathbb{Z}[t,t^{-1}]$ and $\mathbb{Z}((t^{-1}))$ respectively.
\begin{defn}
We say that a sequence of elements $a_i$ in $A^-$ for $i \geq 0$ \textit{stabilize} to $a$ iff the sequence $a_i \L^{-dim(a_i)}$ converges to $a$.
\end{defn}
Given any smooth projective variety $Y$ of dimension $d$, it follows from Poincar{\'{e}} duality that we have
\begin{equation}\label{poincaredualeqn}
P_{[Y]}(t) = t^{2d}P_{[Y]}(t^{-1}) = P_{[Y]\L^{-d}}(t^{-1})
\end{equation}
Therefore, we have
\begin{lem}\label{lemma6} Given a collection of smooth projective varieties $[X_i]$ of dimension $d_i$, if they stabilize in $A^-$ then their respective Poincar{\'{e}} polynomials also stabilize.
\end{lem}
Moreover, we know
\begin{propn}[\cite{cos}, Proposition 3.6]\label{proposition6}
A sequence of elements $a_i \in A^-$ for $i \geq 0$ converges to $a$ iff the generating function $(1-q) \sum_{i\geq 0} a_i q^i$ is convergent at $q=1$, and in this case, evaluating the generating function $(1-q)\sum_{i\geq 0} a_i q^i$ at $q=1$ yields $a$.
\end{propn}
In particular, we see that
\begin{rem}\label{remark8}
If for all $N \geq 0$, there exists $\Delta_0(N) > 0$ such that the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)\sum_{i\geq 0} [X_i]\L^{-d_i}q^i$ is zero, then for all $N \geq 0$ the coefficient of $\L^{-N}$ in $(1-q)\sum_{i\geq 0} [X_i]\L^{-d_i}q^i$ is a Laurent polynomial of $q$ of degree at most $\Delta_0(N)$. As a result, it follows from Proposition \ref{proposition6} that the generating function $(1-q)\sum_i [X_i]\L^{-d_i}q^i$ is convergent at $q=1$, whence Lemma \ref{lemma6} yields the Poincar{\'{e}} polynomials of $[X_i]$ also stabilize. Consequently, it follows from equation \ref{poincaredualeqn}, Lemma \ref{lemma3}, and definition \ref{definition1} that the $2N$th Betti number of $X_\Delta$ stabilize when $\Delta \geq \Delta_0(N)-1$.
\end{rem}
Let $\F \rarrow \P$ be blow-up of $\P$ at a point $p$. Let $E$ be the exceptional divisor and let $F$ be the fiber class. We are going to look at the moduli stacks $\MP(r,c,\Delta)$ and $\MF(r,\tilde{c}, \tilde{\Delta})$ where $\gamma = (r,c,\Delta)$ is Chern character on $\P$ and $\tilde{\gamma} = (r,\tilde{c}, \tilde{\Delta})$ is Chern character on $\F$. We define generating functions
\begin{equation}\label{generatingfndefn}
\begin{aligned}
& \qquad \qquad G_{r,c} (q) = \sum_{\Delta \geq 0} [\MP(r,c,\Delta)] \L^{r^2(1 - 2 \Delta)} q^{r\Delta} \\
& \text{and }\\
& \qquad \qquad \tilde{G}_{r,\tilde{c}}(q) = \sum_{\tilde{\Delta} \geq 0} [\MF(r,\tilde{c}, \tilde{\Delta})] \L^{r^2(1 - 2 \tilde{\Delta})} q^{r \tilde{\Delta}}
\end{aligned}
\end{equation}
Coskun and Woolf have shown that
\begin{thm}[\cite{cos}, Theorem 5.4, Corollary 5.5]
The generating function $(1-q)G_{r,c}(q)$ converges at $q=1$ to $\prod_{i=1}^\infty \frac{1}{(1 - \L^{-i})^3}$. Similarly, the generating function $(1-q)G_{r,\tilde{c}}(q)$ converges at $q=1$ to $\prod_{i=1}^\infty \frac{1}{(1 - \L^{-i})^4}$.
\end{thm}
Our goal is to determine lower bounds for the stabilization of the Betti numbers for the moduli space $M_{\P,H}(r,c,\Delta)$ in the special case when $r$ and $c \cdot H$ are coprime. The way we do this is by relating the stabilization of the Betti numbers with the convergence of the generating function $(1-q)G_{r,c}(q)$ at $q = 1$. A key ingredient in this method is to relate the classes of the moduli stack and the moduli space in $A$, which was shown by Coskun and Woolf, where $A$ is the quotient of $A^-$ by relations $[P] = [X][PGL_n]$ whenever $P \rarrow X$ is an {\'{e}}tale $PGL_n$-torsor.
\begin{propn}[\cite{cos}, Proposition 7.3]\label{proposition10}
The moduli stack and moduli space of $\mu_H$-stable sheaves on $X$, denoted $\mathcal{M}_{X,H}^s(\gamma)$ and $M_{X,H}^s(\gamma)$ respectively, are related in $A$ as follows:
\begin{equation} \label{stacktospace}
[M_{X,H}^s (\gamma)] = (\L -1) [\mathcal{M}_{X,H}^s (\gamma)]
\end{equation}
\end{propn}
By our assumption, $r$ and $c \cdot H$ are coprime, a posteriori, all $\mu_H$-semistable sheaves are $\mu_H$-stable. As a consequence, we can use Proposition \ref{proposition10} to relate the moduli stack and the moduli space.
\section{\textsc{Estimating the Generating Functions when the rank is one}}\label{section3}
In this section, our goal is to analyze the generating functions $G_{1,c}(q)$ and $\tilde{G}_{1,\tilde{c}}(q)$. More precisely, we are going to show that when $\Delta > 2N$ the coefficient of $\L^{-N}q^{\Delta}$ in the generating functions $(1-q)G_{1,c}(q)$ and $(1-q)\tilde{G}_{1,\tilde{c}}(q)$ is zero. As a consequence, we are going to show that the $2N$th Betti number of $M_{\mathbb{P}^2, H} (1,c,c_2)$ stabilize when $c_2 \geq 2N$.
Recall that given a smooth projective surface $X$ with an ample divisor $H$ on $X$, the moduli space $M_{X,H}(1,c,c_2)$ is isomorphic to $Pic^c(X) \times X^{[c_2]}$, where $Pic^c(X)$ is the abelian variety of line bundles on $X$ with first Chern class $c$, and $X^{[n]}$ is the Hilbert scheme of $n$ points on $X$. The Betti numbers of $X^{[n]}$ were computed by G{\"{o}}ttsche \cite{got90}. Using the K{\"{u}}nneth formula, Coskun and Woolf \cite{cos}[Proposition 3.3] showed that the Betti numbers of $M_{X,H}(1,c,c_2)$ stabilize as $c_2 $ tends to infinity. In the special case when $X = \mathbb{P}^2$, the moduli space $M_{\mathbb{P}^2, H}(1,c,c_2)$ is isomorphic to ${\mathbb{P}^2}^{[c_2]}$. Ellingsrud and Stromme \cite{ell}[Theorem 1.1, Corollary 1.3] computed the Betti numbers of ${\mathbb{P}^2}^{[c_2]}$ and showed that the $2N$th Betti number stabilize when $c_2 \geq 2N$. In this section, our goal is to re-derive this result in a flavor similar to the higher rank case.
We infer from equation \ref{generatingfndefn} that
\[ G_{1,c}(q) = \sum_{\Delta \geq 0} [\MP (r,c, \Delta)] \L^{(1-2\Delta)}q^\Delta \]
and
\[ \tilde{G}_{1,\tilde{c}}(q) = \sum_{\tilde{\Delta} \geq 0} [\MF (r,\tilde{c},\tilde{\Delta})] \L^{(1-2\tilde{\Delta})} q^{\tilde{\Delta}} \]
We have
\begin{propn}\label{clm1}
For $\Delta > 2N$, the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)G_{1,c}(q)$ is zero. Same for $(1-q)\tilde{G}_{r, \tilde{c}}(q)$.
\end{propn}
\begin{proof} We have the following equality of generating functions due to G$\ddot{o}$ttsche \cite{got01}[Example 4.9.1]
\[ \sum_{\Delta = 0}^\infty [(\P)^{[\Delta]}] q^{\Delta} = \prod_{m=1}^\infty \frac{1}{(1 - \L^{m-1}q^m)(1 - \L^m q^m)(1 - \L^{m+1}q^m)} \]
Replacing $q$ with $\L^{-2}q$ in above equation, we get
\[ \sum_{\Delta = 0}^\infty [(\P)^{[\Delta]}] \L^{-2 \Delta} q^{\Delta} = \prod_{m=1}^\infty \frac{1}{(1 - \L^{-(m-1)}q^{m})(1 - \L^{-m}q^m) (1 - \L^{-(m+1)}q^m)} \]
Note that we have \[ [\MP (1, c , \Delta)] = (\L -1)^{-1} [ (\P)^{[\Delta]}] \]
Thus, we get
\begin{align*}
(1-q)G_{1,c}(q) &= \frac{(1-q)\L}{(\L-1)} \sum_{\Delta = 0}^\infty [(\P)^{[\Delta]}] \L^{-2\Delta} q^{\Delta} \\
&= \frac{(1-q)}{(1 - \L^{-1})} \prod_{m=1}^\infty \frac{1}{(1 - \L^{-(m-1)}q^{m})(1 - \L^{-m}q^m) (1 - \L^{-(m+1)}q^m)} \\
&= \prod_{m_1 = 2}^\infty \frac{1}{(1 - \L^{-(m_1 -1)} q^{m_1})} \prod_{m_2 = 1}^\infty \frac{1}{(1 - \L^{-2 m_2}q^{m_2})} \prod_{m_3 = 0}^\infty \frac{1}{(1 - \L^{-(m_3 +1)}q^{m_3})} \\
&= \prod_{m_1 = 2}^\infty \left( \sum_{\alpha_1 = 0}^\infty \L^{-(m_1 -1)\alpha_1}q^{m_1 \alpha_1} \right) \times \prod_{m_2 = 1}^\infty \left( \sum_{\alpha_2 = 0}^\infty \L^{-m_2 \alpha_2}q^{m_2 \alpha_2} \right) \times \\
&\qquad \qquad \qquad \qquad \prod_{m_3 = 0}^\infty \left( \sum_{\alpha_3 = 0}^\infty \L^{-(m_3 + 1) \alpha_3}q^{m_3 \alpha_3} \right)
\end{align*}
Each non-zero term contributing to the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)G_{1,c}(q)$ arises from a pair of equations
\begin{align*}
\Delta &= \sum_{j=1}^{\delta_1} m_1^{(j)}\alpha_1^{(j)} + \sum_{j=1}^{\delta_2} m_2^{(j)}\alpha_2^{(j)} + \sum_{j=1}^{\delta_3} m_3^{(j)} \alpha_3^{(j)} \\
-N &= \sum_{j=1}^{\delta_1} -(m_1^{(j)}-1) \alpha_1^{(j)} + \sum_{j=1}^{\delta_2} -m_2^{(j)} \alpha_2^{(j)} + \sum_{j=1}^{\delta_3} - (m_3^{(j)} + 1) \alpha_3^{(j)}
\end{align*}
where $\alpha_1^{(j)}, \alpha_2^{(j)}, \alpha_3^{(j)} \geq 0$ for all $j\geq 1$, and $m_1^{(j)} \geq 2$, $m_2^{(j)} \geq 1$, $m_3^{(j)} \geq 0$ for all $j \geq 1$. Therefore, we see that
\[ \Delta - N = \sum_{j=1}^{\delta_1} \alpha_1^{(j)} - \sum_{j=1}^{\delta_3} \alpha_3^{(j)} \leq \sum_{j=1}^{\delta_1} (m_1^{(j)} -1) \alpha_1^{(j)} \leq N \]
Hence, for $\Delta > 2N $ the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)G_{1,c}(q)$ must be zero.
In a similar fashion as above, we use the following equality of generating functions due to G$\ddot{o}$ttsche \cite{got01}[Example 4.9.3]
\[ \sum_{ \tilde{\Delta} =0}^\infty [\F^{[\tilde{\Delta}]}] q^{\tilde{\Delta}} = \prod_{m=1}^\infty \frac{1}{(1 - \L^{m-1}q^m)(1 - \L^m q^m)^2 (1 - \L^{m+1}q^m)} \]
Replacing $q$ with $\L^{-2} q$ and using the fact $[\MF(1,\tilde{c},\tilde{\Delta})] = (\L -1)^{-1} [ \F^{[\tilde{\Delta}]} ]$, we obtain the following equation
\begin{align*}
(1-q) \tilde{G}_{1, \tilde{c}}(q)
&= \prod_{m_1 = 2}^\infty \left( \sum_{\alpha_1 = 0}^\infty \L^{-(m_1 -1)\alpha_1}q^{m_1 \alpha_1} \right) \times \prod_{m_2 = 1}^\infty \left( \sum_{\alpha_2 = 0}^\infty \L^{-m_2 \alpha_2}q^{m_2 \alpha_2} \right)^2 \times \\
&\qquad \qquad \qquad \qquad \prod_{m_3 = 0}^\infty \left( \sum_{\alpha_3 = 0}^\infty \L^{-(m_3 + 1) \alpha_3}q^{m_3 \alpha_3} \right)
\end{align*}
Each non-zero term contributing to the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)G_{1, \tilde{c}}(q)$ arises from a pair of equations
\begin{align*}
\Delta &= \sum_{j=1}^{\delta_1} m_1^{(j)}\alpha_1^{(j)} + \sum_{j=1}^{\delta_{2,1}} m_2^{(j,1)}\alpha_2^{(j,1)} + \sum_{j=1}^{\delta_{2,2}} m_2^{(j,2)}\alpha_2^{(j,2)} + \sum_{j=1}^{\delta_3} m_3^{(j)} \alpha_3^{(j)} \\
-N &= \sum_{j=1}^{\delta_1} -(m_1^{(j)}-1) \alpha_1^{(j)} + \sum_{j=1}^{\delta_{2,1}} - m_2^{(j,1)}\alpha_2^{(j,1)} + \sum_{j=1}^{\delta_{2,2}} - m_2^{(j,2)}\alpha_2^{(j,2)} + \sum_{j=1}^{\delta_3} - (m_3^{(j)} + 1) \alpha_3^{(j)}
\end{align*}
where $\alpha_1^{(j)}, \alpha_2^{(j,1)}, \alpha_2^{(j,2)}, \alpha_3^{(j)} \geq 0$ for all $j\geq 1$, and $m_1^{(j)} \geq 2$, $m_2^{(j,1)}, m_2^{(j,2)} \geq 1$, $m_3^{(j)} \geq 0$ for all $j \geq 1$. Therefore, we see that
\[ \Delta - N = \sum_{j=1}^{\delta_1} \alpha_1^{(j)} - \sum_{j=1}^{\delta_3} \alpha_3^{(j)} \leq \sum_{j=1}^{\delta_1} (m_1^{(j)} -1) \alpha_1^{(j)} \leq N \]
Hence, for $\Delta > 2N$ the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)\tilde{G}_{1,\tilde{c}}(q)$ must be zero.
\end{proof}
As a consequence of above Proposition \ref{clm1}, we have the following:
\begin{propn}
When $c_2 \geq 2N$, the $2N$-th Betti number of $M_{\mathbb{P}^2, H}(1,c,c_2)$ stabilize.
\end{propn}
\begin{proof}
Note that all $\mu_H$-semistable sheaves of rank one on $\mathbb{P}^2$ are $\mu_H$-stable, because the rank is coprime to the first Chern class. As a consequence, we can use Proposition
\ref{proposition10} due to Coskun and Woolf and the fact that $c_2 = r \Delta + \frac{r-1}{2r}c_1^2$ to get the following equality of generating functions
\[ (1-q)\sum_{c_2 \geq 0} [M_{\mathbb{P}^2,H}(\gamma)]\L^{-ext^1(\gamma,\gamma)} q^{c_2} = (1-\L^{-1})(1-q)G_{1,c}(q) \]
where $\gamma$ denotes the Chern character $(r, c, \Delta)$.
Each term contributing to the coefficient of $\L^{-N}q^d$ in $(1-\L^{-1})(1-q)G_{1,c}(q)$ comes from a pair of equations
\begin{align*}
d = \Delta
-N = \varepsilon - N'
\end{align*}
where $\varepsilon \in \{ -1,0 \}$ accounts for the contribution of the coefficient coming from $(1 - \L^{-1})$, and $(\Delta, N')$ accounts for the contribution coming from the terms in coefficient of $\L^{-N'}q^{\Delta}$ in $(1-q)G_{1,c}(q)$. It follows from Proposition \ref{clm1} that for the coefficient of $\L^{-N'}q^\Delta$ to be nonzero, we must have $\Delta \leq 2N'$. Consequently, we must have $d \leq 2N$. Hence, for $d>2N$, the coefficient of $\L^{-N}q^d$ in $(1-\L^{-1})(1-q)G_{1,c}(q)$ must be zero. Therefore, using Remark \ref{remark8}, we conclude that the $2N$th Betti number of $M_{\mathbb{P}^2, H}(1,c,c_2)$ stabilize for $c_2 \geq 2N$.
\end{proof}
\section{\textsc{Estimating the generating function $\tilde{G}_{r, \tilde{c}}(q)$ when rank is at least two}}\label{section4}
In this section, our goal is to show that there is a constant $C_0$ depending only on $r$ and $\tilde{c}$ such that when $\Delta > N + C_0$, the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)\tilde{G}_{r,\tilde{c}}(q)$ is zero. We are going to show this in a couple of steps. First, we are going to use Mozgovoy's theorem \cite{moz}[Theorem 1.1] and estimate a generating function in $A^-$ expressed in terms of the classes of the moduli stack $\MFF (\gamma)$. Then, we are going to use Joyce's theorem \cite{joy}[Theorem 6.21] to relate the classes of the moduli stacks $\MF (\gamma)$ and $\MFF (\gamma)$ in $A^-$. Lastly, we are going to use key ideas of Coskun and Woolf \cite{cos} and Manschot \cite{man11}, \cite{man} to derive our estimate (see Proposition \ref{boundMF}).
Throughout this section, we are going to assume that $r$ is at least two. We recall two theorems due to Mozgovoy \cite{moz} and Joyce \cite{joy} respectively.
Let $\MFF (\gamma)$ denote the moduli stack of torsion free $\mu_F$ semistable sheaves on $\F$ with Chern character $\gamma = (r, c , \Delta)$. We define generating function
\begin{equation}\label{eqnhrc}
H_{r,c}(q) = \sum_{\Delta \geq 0} [\MFF(r,c,\Delta)] q^{r\Delta} \end{equation}
Let $Z_{\mathbb{P}^1}(q) = \frac{1}{(1-q)(1-\L q)}$ be the motivic Zeta function for $\mathbb{P}^1$. Then, we have
\begin{thm}[\cite{moz}[Theorem 1.1]\label{hrc} If $r \nmid c \cdot F$, then $\MFF (\gamma)$ is empty, and hence $H_{r,c}(q) = 0$. Otherwise, we have
\[ H_{r,c}(q) = \frac{1}{(\L -1)} \prod_{i=1}^{r-1} Z_{\mathbb{P}^1}(\L^i) \prod_{k=1}^\infty \prod_{i=-r}^{r-1} Z_{\mathbb{P}^1}(\L^{rk+i}q^{k}) \]
\end{thm}
Before proceeding to Joyce's theorem, in a similar vein as in Proposition \ref{clm1}, we would like to show that for $\Delta \gg N$, the coefficient of $\L^{-N}q^{\Delta}$ in the generating function
$(1 - q) \sum_{ \Delta \geq 0} [\MFF(r,c,\Delta)] \L^{r^2( 1 - 2 \Delta)} q^{r \Delta}$ vanishes.
\begin{propn}\label{hrc2}
If $\Delta > N $, the coefficient of $\L^{-N}q^{\Delta}$ in the generating function
\[ (1 - q) \sum_{ \Delta \geq 0} [\MFF(r,c,\Delta)] \L^{r^2( 1 - 2 \Delta)} q^{r \Delta} \] is zero.
\end{propn}
\begin{proof} Clearly we can assume that $r \mid c \cdot F$, because otherwise by Mozgovoy's theorem (Theorem \autoref{hrc}) we have $[\MFF(r,c,\Delta)] = 0$.
Observe that
\begin{equation}\label{eqnhrc2}
(1 - q) \sum_{ \Delta \geq 0} [\MFF(r,c,\Delta)] \L^{r^2( 1 - 2 \Delta)} q^{r \Delta} = (1-q) \L^{r^2} H_{r,c} (\L^{-2r}q)
\end{equation}
Moreover, we have the following equations
\begin{align*}
\frac{1}{(\L -1)} \prod_{i=1}^{r-1} \frac{1}{(1 - \L^i)(1 - \L^{i+1})} &= \frac{L^{-r^2}}{(1 - \L^{-r})} \prod_{i=1}^{r-1} \frac{1}{(1 - \L^{-i})^2} \\
\prod_{k=1}^{\infty} \prod_{i = -r}^{r-1} \frac{1}{(1 - \L^{-rk +i}q^k)(1 - \L^{-rk + i + 1}q^k)} &= \prod_{k_1 = 1}^\infty \frac{1}{(1 - \L^{-(rk_1 +r)}q^{k_1})} \times \\
& \prod_{k_2 = 1}^\infty \prod_{i=-r+1}^{r-1} \frac{1}{(1 - \L^{-(rk_2-i)}q^{k_2})^2} \times \\
& \prod_{k_3 =1}^\infty \frac{1}{(1 - \L^{-(rk_3 -r)}q^{k_3})}
\end{align*}
Therefore, we have
\begin{align*}
(1-q) \L^{r^2} H_{r,c} (\L^{-2r}q) &= \left( \sum_{\alpha_1 = 0}^\infty \L^{-r \alpha_1} \right) \prod_{i=1}^{r-1} \left( \sum_{\alpha_2 = 0}^\infty \L^{-i \alpha_2} \right)^2 \prod_{k_1 = 1}^\infty \left( \sum_{\alpha_3 = 0}^\infty \L^{-(rk_1 + r) \alpha_3}q^{k_1 \alpha_3} \right) \times \\
& \prod_{k_2 = 1}^\infty \prod_{j = -r + 1}^{r-1} \left( \sum_{\alpha_4 = 0}^\infty \L^{-(rk_2 - j) \alpha_4} q^{k_2 \alpha_4} \right)^2 \prod_{k_3 = 2}^\infty \left( \sum_{\alpha_5 = 0}^\infty \L^{-(rk_3 - r)\alpha_5} q^{k_3 \alpha_5} \right)
\end{align*}
Each non-zero term contributing to the coefficient of $\L^{-N}q^{\Delta}$ in $(1 - q)\L^{r^2}H_{r,c}(\L^{-2r}q)$ corresponds to a pair of equations
\begin{align*}
\Delta &= \sum_{j_1 = 1}^{\delta_1} k_1^{(j_1)} \alpha_3^{(j_1)} + \sum_{j_2 = 1}^{\delta_2} \sum_{j = -r + 1}^{r-1} k_2^{(j_2, j)} (\alpha_4^{(j_2,j,1)} + \alpha_4^{(j_2, j, 2)}) + \sum_{j_3 = 1}^{\delta_3} k_3^{(j_3)} \alpha_5^{(j_3)} \\
-N &= -r \alpha_1 + \sum_{i = 1}^{r-1} -i (\alpha_2^{(i,1)} + \alpha_2^{(i,2)}) + \sum_{j_1 = 1}^{\delta_1} -(rk_1^{(j_1)} +r) \alpha_3^{(j_1)} + \\
& \qquad \qquad \sum_{j_2 = 1}^{\delta_2} \sum_{j = -r + 1}^{r-1} -(rk_2^{(j_2, j)} -j) (\alpha_4^{(j_2,j,1)} + \alpha_4^{(j_2, j, 2)}) + \sum_{j_3 = 1}^{\delta_3} -(rk_3^{(j_3)} -r) \alpha_5^{(j_3)}
\end{align*}
where all the $\alpha$'s are non-negative integers and all the $\delta$'s and $k$'s are positive integers except $k_3^{(j_3)}$ which is at least $2$, for all $1 \leq j_3 \leq \delta_3$. We see that
\[ r\Delta -N \leq \sum_{j_2 =1}^{\delta_2} \sum_{j = -r+1}^{r-1} j(\alpha_4^{(j_2,j,1)} + \alpha_4^{(j_2, j, 2)}) + \sum_{j_3 = 1}^{\delta_3} r \alpha_5^{(j_3)} \]
Since $j \leq r-1$ and $k_3^{(j_3)} \geq 2$, we see that $ (rk_2^{(j_2,j)} -j) \geq 1$ and $ (rk_3^{(j_3)}-r) \geq r$. Hence, we have
{\small
\begin{align*}
\sum_{j_2 =1}^{\delta_2} \sum_{j = -r+1}^{r-1} j(\alpha_4^{(j_2,j,1)} + \alpha_4^{(j_2, j, 2)}) + \sum_{j_3 = 1}^{\delta_3} r \alpha_5^{(j_3)} &\leq (r-1) \sum_{j_2 = 1}^{\delta_2} \sum_{j = -r + 1}^{r-1} (rk_2^{(j_2, j)} -j) (\alpha_4^{(j_2,j,1)} + \alpha_4^{(j_2, j, 2)}) \\
& \,\, + \sum_{j_3 = 1}^{\delta_3} (rk_3^{(j_3)} -r) \alpha_5^{(j_3)} \leq (r-1)N
\end{align*}}
Hence for $\Delta > N $, the coefficient of $\L^{-N}q^{\Delta}$ in
$(1 - q) \sum_{ \Delta \geq 0} [\MFF(r,c,\Delta)] \L^{r^2( 1 - 2 \Delta)} q^{r \Delta}$ is zero.
\end{proof}
We now proceed to state Joyce's theorem. Let $X$ be a surface with two ample line-bundles $H_1$ and $H_2$. Let $\mathcal{M}_{X,H_1}(\gamma)$ (respectively $\mathcal{M}_{X,H_2}(\gamma)$) denote the moduli stack of torsion free $\mu_{H_1}$ (respectively $\mu_{H_2}$) semistable sheaves on $X$ with Chern character $\gamma = (r,c, \Delta)$. Let $\gamma_1, \cdots, \gamma_l$ be Chern characters such that $\sum_{i=1}^l \gamma_i = \gamma$. Assume that $l \geq 2$, and consider the following conditions for all $1 \leq i \leq l-1$
\begin{equation}\label{defnSmu}
\begin{aligned}
& \qquad \text{A) } \mu_{H_1}(\gamma_i) > \mu_{H_1}(\gamma_{i+1}) \text{ and } \mu_{H_2}(\sum_{j=1}^i \gamma_j) \leq \mu_{H_2} ( \sum_{j=i+1}^l \gamma_j)\\
& \qquad \text{B) } \mu_{H_1}(\gamma_i) \leq \mu_{H_1}(\gamma_{i+1}) \text{ and } \mu_{H_2}(\sum_{j=1}^i \gamma_j) > \mu_{H_2} ( \sum_{j=i+1}^l \gamma_j)
\end{aligned}
\end{equation}
Let $u$ be the number of times that Case B occurs. We define
\begin{equation}\label{eqnSmu}
\begin{aligned}
S^\mu( \gamma_1, \cdots, \gamma_l ;H_1,H_2) = \begin{cases} 1, &\text{ if } l=1 \\
(-1)^u, &\text{ if } l\geq 2, \text{ and Case A or B occurs for all }1 \leq i \leq l-1\\
0, \qquad &\text{ otherwise } \end{cases}
\end{aligned}
\end{equation}
\begin{thm}[\cite{joy},{Theorem 6.21}]\label{joyce} If $H_1$ and $H_2$ are ample line-bundles on $X$ satisfying $K_X \cdot H_1 <0$ and $K_X \cdot H_2 < 0$, then we have the following equation
\[ [ \mathcal{M}_{X,H_2}(\gamma)] = \sum_{\sum_{i=1}^l \gamma_i = \gamma} S^\mu(\gamma_1, \cdots, \gamma_l; H_1, H_2) \L^{- \sum_{1 \leq i < j \leq l} \chi(\gamma_j, \gamma_i)} \prod_{i=1}^l [\mathcal{M}_{X,H_1} (\gamma_i)] \]
\end{thm}
In our case, we would like to take $X = \F$, $H_1 = F$ and $H_2 = E + F$. Clearly, since $K_{\F} = -2 E - 3 F$, we have $K_{\F} \cdot H_1 < 0$ and $K_{\F} \cdot H_2 < 0$. However, $H_1$ is not ample and so we cannot use Joyce's theorem (Theorem \autoref{joyce}) as stated. Luckily the following observation due to Coskun and Woolf \cite{cos}[Corollary 4.4] saves the day.
\begin{rem}
Joyce's theorem (Theorem \autoref{joyce}) holds if $H_1$ and $H_2$ are nef, as long as the sum on the right side of equation is convergent.
\end{rem}
Moreover, Coskun and Woolf shows \cite{cos}[Corollary 5.3] that we can use Joyce's equation in our case. Hence, we have
\begin{equation}\label{eqn}
\begin{aligned}
\sum_{\Delta \geq 0} \MF(\gamma) q^{r \Delta} &= \sum_{\Delta \geq 0} \;\; \sum_{\sum_{i=1}^l \gamma_i = \gamma} \Smu \; \L^{-\sum_{1 \leq i< j \leq l} \chi(\gamma_j, \gamma_i)} \,\times \\
& \qquad \qquad \qquad \qquad \left( \prod_{i=1}^l [\MFF (\gamma_i)] \right) q^{r\Delta}
\end{aligned}
\end{equation}
Let $\gamma_i = (r_i, c_i, \Delta_i)$ for all $1 \leq i \leq l$. Further, we define $\mu_i = \frac{c_i}{r_i}$ for all $1 \leq i \leq l$. We would like to manipulate equation \ref{eqn} so that the left hand side term of equation \ref{eqn} becomes $\tilde{G}_{r,c}(q)$ and get rid of $\Delta$ from the right hand side term of equation \ref{eqn}.
It is easy to see that
\[ - \sum_{1 \leq i < j \leq l} \chi(\gamma_j, \gamma_i) = -\frac{1}{2} \left( \sum_{i<j} \chi(\gamma_j, \gamma_i) + \chi(\gamma_i, \gamma_j) \right) -\frac{1}{2} \left( \sum_{i<j} \chi(\gamma_j, \gamma_i) - \chi(\gamma_i, \gamma_j) \right) \]
We now list down some equations expressing the various Euler characteristics
\begin{enumerate}[$\qquad\bullet$]
\item $ \chi(\gamma_j, \gamma_i) - \chi(\gamma_i, \gamma_j) = r_i r_j (\mu_j - \mu_i) \cdot K_{\F} $
\item $\chi(\gamma, \gamma) = r^2( 1 - 2\Delta)$, and $\chi(\gamma_i, \gamma_i) = r_i^2 ( 1 - 2\Delta_i)$ for all $1 \leq i \leq l$.
\item $\sum_{i<j} \chi(\gamma_j, \gamma_i) + \chi(\gamma_i, \gamma_j) = \chi(\gamma, \gamma) - \sum_{i=1}^l \chi(\gamma_i, \gamma_i) $
\end{enumerate}
Using the above equations we get
\begin{equation}\label{eqn2}
- \sum_{i<j} \chi(\gamma_j, \gamma_i) = -\frac{1}{2}r^2(1 - 2\Delta) + \frac{1}{2}\sum_{i=1}^l r_i^2 (1 - 2 \Delta_i) - \frac{1}{2} \sum_{i<j} r_i r_j (\mu_j - \mu_i)\cdot K_{\F}
\end{equation}
We now replace $q$ by $\L^{-2r}q$ in both sides of equation \ref{eqn}, multiply both sides of \ref{eqn} by $\L^{r^2}$, and use equation \ref{eqn2}. We get
\begin{equation}\label{eqn3}
\begin{aligned}
\sum_{\Delta \geq 0} [\MF (\gamma) ] \L^{r^2(1 - 2 \Delta)} q^{r\Delta} &= \sum_{\Delta \geq 0 } \;\; \sum_{\sum_{i=1}^l \gamma_i = \gamma} \Smu \;\times \\
& \L^{\frac{1}{2}r^2(1 - 2 \Delta) + \frac{1}{2}\sum_{i=1}^l r_i^2( 1 - 2 \Delta_i)}
\; \L^{- \frac{1}{2}\sum_{i<j}r_i r_j(\mu_j - \mu_i) \cdot K_{\F}} \;\times \\
& \left( \prod_{i=1}^l [\MFF (\gamma_i) ] \right) q^{r\Delta}
\end{aligned}\end{equation}
Note that we are yet to get rid of $\Delta$ from right hand side term in equation \ref{eqn3}. To do that, we need to use Yoshioka's relation for discriminants \cite{yos}[Equation 2.1]
\begin{equation}\label{eqnyos}
r\Delta = \sum_{i=1}^l r_i \Delta_i - \sum_{i=2}^l \; \frac{1}{2r_i \left( \sum_{j=1}^i r_j \right) \left( \sum_{j=1}^{i-1}r_j \right)} \left( \sum_{j=1}^{i-1} r_i c_j - r_j c_i \right)^2
\end{equation}
It follows from Yoshioka's relation that the difference $r \Delta - \sum_{i=1}^l r_i \Delta_i$ depends only on $(r,c)$ and $(r_i,c_i)$ for $1 \leq i \leq l$. So we rewrite the first exponent of $\L$ in equation \ref{eqn3}
\begin{equation}\label{eqn5}
\frac{1}{2}r^2(1 - 2 \Delta) + \frac{1}{2}\sum_{i=1}^l r_i^2 (1 - 2 \Delta_i) = \frac{1}{2}(r^2 + \sum_{i=1}^l r_i^2) - r(r \Delta - \sum_{i=1}^l r_i \Delta_i) - \sum_{i=1}^l r_i (r + r_i) \Delta_i
\end{equation}
{}\\
Using equation \ref{eqn5} back in equation \ref{eqn3} yields
\begin{equation}\label{eqn6}
\begin{aligned}
\tilde{G}_{r,c}(q) &= \sum_{\Delta \geq 0} \;\; \sum_{\sum_{i=1}^l \gamma_i = \gamma } \Smu
\L^{\frac{1}{2}\left( r^2 + \sum_{i=1}^l r_i^2 \right)} \;
\L^{-\frac{1}{2}\sum_{i<j}r_i r_j(\mu_j - \mu_i) \cdot K_{\F}} \;\times \\
& \qquad \qquad \qquad \qquad
\left( \L^{-r} q \right)^{r\Delta - \sum_{i=1}^l r_i \Delta_i}
\left( \prod_{i=1}^l [\MFF (\gamma_i)] \left( \L^{-(r + r_i)} q \right)^{r_i \Delta_i} \right)
\end{aligned}
\end{equation}
Observe that all the terms except the last one involving products on right hand side of equality in equation \ref{eqn6} depends only on $(r,c)$ and $(r_i, c_i)$ for $1 \leq i \leq l$, and the last term depends only on the $\Delta_i$'s for $1 \leq i \leq l$. Therefore, we have
\begin{equation}\label{eqn7}
\begin{aligned}
\tilde{G}_{r,c}(q) &= \sum_{\sum_{i=1}^l \gamma_i = \gamma} \Smu
\L^{\frac{1}{2}\left( r^2 + \sum_{i=1}^l r_i^2 \right)} \;
\L^{-\frac{1}{2}\sum_{i<j}r_i r_j(\mu_j - \mu_i) \cdot K_{\F}} \;\times \\
& \qquad \qquad \qquad
\left( \L^{-r} q \right)^{r\Delta - \sum_{i=1}^l r_i \Delta_i}
\sum_{\Delta_1, \cdots, \Delta_l}
\left( \prod_{i=1}^l [\MFF (\gamma_i)] \left( \L^{-(r + r_i)} q \right)^{r_i \Delta_i} \right)
\end{aligned}
\end{equation}
Recall that we previously defined in equation \ref{eqnhrc} the generating function
\[ H_{r,c}(q) = \sum_{\Delta \geq 0} [\MFF(r,c,\Delta)] q^{r\Delta} \]
The second summation term in equation \ref{eqn7} can be expressed in terms of $H_{r,c}(q)$ as follows
\begin{equation}\label{eqn8}
\sum_{\Delta_1, \cdots, \Delta_l}
\left( \prod_{i=1}^l [\MFF (\gamma_i)] \left( \L^{-(r + r_i)} q \right)^{r_i \Delta_i} \right)
= \prod_{i=1}^l H_{r_i, c_i} (\L^{-(r+r_i)}q)
\end{equation}
Therefore, we have
\begin{equation}
\label{eqn9}
\begin{aligned}
\tilde{G}_{r,c}(q) &= \sum_{\sum_{i=1}^l (r_i,c_i) = (r,c)} \Smu
\L^{\frac{1}{2}\left( r^2 - \sum_{i=1}^l r_i^2 \right)} \;
\L^{-\frac{1}{2}\sum_{i<j}r_i r_j(\mu_j - \mu_i) \cdot K_{\F}} \;\times \\
& \qquad \qquad \qquad
\left( \L^{-r} q \right)^{r\Delta - \sum_{i=1}^l r_i \Delta_i}
\prod_{i=1}^l \L^{r_i^2} H_{r_i, c_i} (\L^{-(r+r_i)}q)
\end{aligned}
\end{equation}
It follows from the definition of $\Smu$ in equation \ref{eqnSmu} and from Mozgovoy's theorem (Theorem \autoref{hrc}) that all the terms on right hand side of equality of equation \ref{eqn9} depends only on $(r,c)$ and $(r_i,c_i)$ for $1 \leq i \leq l$. Our next goal is to analyze the exponents of each of these terms further and show that for $\Delta \gg N$ the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)\tilde{G}_{r,c}(q)$ vanishes.
\begin{propn}\label{boundMF}
There is a constant $C_0$ depending only on $r$ and $c$ such that if $\Delta > N + C_0$, then coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)\tilde{G}_{r,c}(q)$ is zero. Moreover, we can take $C_0$ to be $\frac{1}{2}(r^2 + 1)$.
\end{propn}
\begin{proof}
Our approach is to look at each summand of $(1-q)\tilde{G}_{r,c}(q)$ corresponding to a equation \[ (r,c) = \sum_{i=1}^l (r_i,c_i) \]
and find a lower bound for $\Delta$ corresponding to the term
\begin{equation}\label{rhs}
\begin{aligned}
& (1-q)\Smu
\L^{\frac{1}{2}\left( r^2 - \sum_{i=1}^l r_i^2 \right)} \;
\L^{-\frac{1}{2}\sum_{i<j}r_i r_j(\mu_j - \mu_i) \cdot K_{\F}} \;\times \\
& \qquad \qquad \qquad
\left( \L^{-r} q \right)^{r\Delta - \sum_{i=1}^l r_i \Delta_i}
\prod_{i=1}^l \L^{r_i^2} H_{r_i, c_i} (\L^{-(r+r_i)}q)
\end{aligned}
\end{equation}
If $l=1$, then equation \ref{rhs} becomes
\begin{equation}
(1-q) S^\mu(\gamma; F, E+F)\L^{r^2} H_{r,c}(\L^{-2r}q)
\end{equation}
It follows from Proposition \ref{hrc2} and equation \ref{eqnhrc2} that for $\Delta > N $, the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)\L^{r^2} H_{r,c}(\L^{-2r}q)$ is zero.
Assume $l \geq 2$. We would like to estimate a lower bound for $\Delta'_i$ such that the coefficient of $\L^{-N'_i}q^{\Delta'_i}$ in $\L^{r_i^2}H_{r_i,c_i}(\L^{-(r+r_i)}q)$ is zero, and then use that to figure out a lower bound for $\Delta$ in equation \ref{rhs}. It follows from Mozgovoy's theorem (Theorem \ref{hrc}) that
\begin{align*}
\L^{r_i^2} H_{r_i,c_i}(\L^{-(r+r_i)}q) &= \L^{r_i^2} \frac{1}{(\L -1)} \prod_{j=1}^{r_i -1} Z_{\mathbb{P}^1}(\L^j) \prod_{k=1}^\infty \prod_{j=-r_i}^{r_i -1} Z_{\mathbb{P}^1} (\L^{-(rk -j)}q^k) \\
&= \frac{1}{(1-\L^{-r_i})} \left( \prod_{j=1}^{r_i -1} \frac{1}{(1 - \L^{-j})^2} \right)
\prod_{k=1}^\infty \Bigg\lbrace \frac{1}{(1 - \L^{-(rk + r_i)}q^k)} \;\times \\
& \qquad \qquad \qquad \left( \prod_{j= -r_i +1}^{r_i -1} \frac{1}{(1 - \L^{-(rk - j)}q^k)^2} \right) \frac{1}{(1 - \L^{-(rk-r_i)}q^k)} \Bigg\rbrace
\end{align*}
Thus, we get
\begin{align*}
\L^{r_i^2} H_{r_i,c_i}(\L^{-(r +r_i)}q) &= \left( \sum_{\alpha_1 =0}^\infty \L^{-r_i \alpha_1} \right)
\left( \prod_{j_1 =1}^{r_i -1} \left( \sum_{\alpha_2 =0}^\infty \L^{-j_1 \alpha_2} \right)^2 \right) \: \times \\
&
\prod_{k=1}^\infty \Bigg\lbrace
\left( \sum_{\alpha_3 =0}^\infty \L^{-(rk + r_i) \alpha_3}q^{k \alpha_3} \right)
\left( \prod_{j_2 = -r_i +1}^{r_i -1} \left( \sum_{\alpha_4 =0}^\infty \L^{-(rk - j_2) \alpha_4} q^{k \alpha_4} \right)^2 \right) \\
& \qquad \qquad
\left( \sum_{\alpha_5 =0}^\infty \L^{-(rk - r_i) \alpha_5} q^{k \alpha_5} \right) \Bigg\rbrace
\end{align*}
Each nonzero term contributing to the coefficient of $\L^{-N'_i}q^{\Delta'_i}$ in $\L^{r_i^2}H_{r_i,c_i}(\L^{-(r+r_i)}q)$ arises from a pair of equations
\begin{align*}
\Delta'_i &= \sum_{j=1}^\delta \left\lbrace k^{(j)}\alpha_3^{(j)} + \sum_{j_2 = -r_i +1}^{r_i -1} k^{(j)}( \alpha_4^{(j,j_2,1)} + \alpha_4^{(j,j_2,2)}) + k^{(j)}\alpha_5^{(j)} \right\rbrace \\
-N'_i &= -r_i \alpha_1 + \sum_{j_1=1}^{r_i -1} -j_1 (\alpha_2^{(j_1,1)} + \alpha_2^{(j_1,2)}) +
\sum_{j=1}^\delta \Bigg\lbrace -(rk^{(j)} +r_i) \alpha_3^{(j)} \; + \\
& \qquad
\left( \sum_{j_2 = -r_i +1}^{r_i -1} -(rk^{(j)} -j_2) (\alpha_4^{(j,j_2,1)} + \alpha_4^{(j,j_2,2)}) \right)
-(rk^{(j)} -r_i)\alpha_5^{(j)} \Bigg\rbrace
\end{align*}
where all the $\alpha$'s are non-negative integers, $\delta$ and the $k$'s are positive integers. Hence, we get
\[ r \Delta'_i - N'_i \leq \sum_{j=1}^\delta \left( \sum_{j_2 = -r_i +1}^{r_i -1} j_2 (\alpha_4^{(j,j_2,1)} + \alpha_4^{(j,j_2,2)} ) \right) + r_i \alpha_5^{(j)} \]
Since $j_2 \leq r_i -1$ and $k^{(j)} \geq 1$, we see that
$ j_2 \leq r_i (rk^{(j)} - j_2) $. Moreover, because $l \geq 2$ we have $r_i \leq (r-1)$, and so
$ r_i \leq r_i (rk^{(j)} -r_i) $. These two inequalities yield
\[ \sum_{j=1}^\delta \left( \sum_{j_2 = -r_i +1}^{r_i -1} j_2 (\alpha_4^{(j,j_2,1)} + \alpha_4^{(j,j_2,2)} ) \right) + r_i \alpha_5^{(j)} \leq r_i N'_i \]
In summary, we get $r \Delta'_i - N'_i \leq r_i N'_i \leq (r-1) N'_i$, a posteriori, $\Delta'_i \leq N'_i$.
Going back to equation \ref{rhs}, we see that each non-zero term contributing to the coefficient of $\L^{-N'}q^{\Delta'}$ in equation \ref{rhs} arises from a pair of equations
\begin{align*}
\Delta' &= \varepsilon + \left( r\Delta - \sum_{i=1}^l r_i \Delta_i \right) + \sum_{i=1}^l \Delta'_i \\
-N' &= \frac{1}{2}\left( r^2 - \sum_{i=1}^l r_i^2 \right) - \frac{1}{2} \left( \sum_{i<j} r_i r_j (\mu_j - \mu_i) \cdot K_{\F} \right) - r \left( r\Delta - \sum_{i=1}^l r_i \Delta_i \right) + \sum_{i=1}^l - N'_i
\end{align*}
where $\varepsilon \in \{ 0,1 \}$ which accounts for contribution to the coefficient coming from $(1-q)$, and $(\Delta'_i, N'_i)$ accounts for the contribution of terms to the coefficient of $\L^{-N'}q^{\Delta'}$ coming from terms of coefficient of $\L^{-N'_i}q^{\Delta'_i}$ appearing in $\L^{r_i^2}H_{r_i,c_i}(\L^{-(r+r_i)}q)$. Since $\Delta'_i \leq N'_i$ for all $1 \leq i \leq l$ and $\varepsilon \leq 1$, we see that
\begin{equation}\label{eqn15}
\Delta' \leq N' + 1 + \frac{1}{2}\left( r^2 - \sum_{i=1}^l r_i^2 \right) - \frac{1}{2} \left\lbrace \left( \sum_{i<j} r_i r_j (\mu_j - \mu_i) \cdot K_{\F} \right) + 2(r-1)\left( r\Delta - \sum_{i=1}^l r_i \Delta_i \right) \right\rbrace
\end{equation}
Clearly, to bound $\Delta'$, we need to bound the last term in above equation \ref{eqn15}. We are going to show later (in Lemma \autoref{lowerbound}) that
\[ 2(r-1) \left( r\Delta - \sum_{i=1}^l r_i \Delta_i \right) + \left( \sum_{i<j} r_i r_j (\mu_j - \mu_i) \cdot K_{\F} \right) \]
is bounded below by a constant $\kappa$ which depends only on $(r,c)$ and $r_i$ for all $ 1 \leq i \leq l$, except when $l=2$ and $\mu_F(\gamma_2) - \mu_F(\gamma_1) = -1$. Thus, we have
\[ \Delta' \leq N' + 1 + \frac{1}{2}\left(r^2 - \sum_{i=1}^l r_i^2 \right) - \frac{1}{2}\kappa \]
We would like to scrutinize the special case when $l=2$ and $\mu_F (\gamma_2) - \mu_F(\gamma_1) = -1$. Note that it follows from Mozgovoy's theorem (Theorem \ref{hrc}) that $H_{r,c}$ only depends on whether or not $r \mid c \cdot F$. Let $r = r_1 + r_2$, $c = aE + bF$, $c_1 = r_1a_1 E + b_1 F$ and $c_2 = r_2 a_2 E + b_2 F$. We will denote $H_{r_i,c_i}$ by $H_{r_i}$ for $i=1,2$ because we are assuming that $r_i \mid c_i \cdot F$ for $i=1,2$. It follows from equation \ref{eqnSmu} that for $S^\mu(\gamma_1,\gamma_2;F,E+F)$ to be nonzero, we must have $\mu_{E+F}(\gamma_1) \leq \mu_{E+F}(\gamma_2)$, or equivalently, we have $b_2 \geq \frac{br_2}{r}$. Furthermore, we see that
\[ -\frac{1}{2} r_1 r_2(\mu_2 - \mu_1)\cdot K_{\F} = r_1 r_2 (a_2 - a_1) + rb_2 - r_2 b \]
and
\[ r\Delta - r_1 \Delta_1 - r_2 \Delta_2 = \frac{r_1 r_2 }{2r}(a_2 - a_1)^2 - (a_2 - a_1)b_2 + b\frac{r_2(a_2 - a_1)}{r}\]
Using these equations together with the fact that $a_2 - a_1 = -1$, we see that equation \ref{rhs} transforms to
\[ (1-q)\L^{\frac{1}{2}(r^2 - r_1^2 - r_2^2)}\L^{-r_1 r_2}q^{\frac{r_1 r_2}{2r} - \frac{b r_2}{r}}q^{b_2} \prod_{i=1}^2 \L^{r_i^2}H_{r_i}(\L^{-(r+r_i)}q)
\]
whenever $b_2 \geq \frac{br_2}{r}$ and is zero otherwise. Adding all these terms for $b_2 \geq \frac{br_2}{r}$ yields
\begin{equation}\label{spcase}
\L^{\frac{1}{2}(r^2 - r_1^2 - r_1^2) - r_1 r_2} q^{\frac{r_1 r_2}{2r} - \frac{b r_2}{r}} q^{\left\lceil \frac{b r_2}{r}\right\rceil} \prod_{i=1}^2 \L^{r_i^2}H_{r_i}(\L^{-(r+r_i)}q)
\end{equation}
Each nonzero term appearing in the coefficient of $\L^{-N'}q^{\Delta'}$ in equation \ref{spcase} arises from a pair of equations
\begin{align*}
\Delta' &= \frac{r_1 r_2}{2r} - \frac{br_2}{r} + \left\lceil \frac{br_2}{r} \right\rceil + \Delta'_1 + \Delta'_2 \\
-N' &= \frac{1}{2}(r^2 - r_1^2 - r_2^2) - r_1 r_2 - N'_1 - N'_2 = -N'_1 - N'_2
\end{align*}
where $(\Delta'_i,N'_i) $ accounts for contribution coming from terms of coefficient of $\L^{-N'_i}q^{\Delta'_i}$ in $\L^{r_i^2}$ $ H_{r_i}(\L^{-(r+r_i)}q)$. We have shown before that we must have $\Delta'_i \leq N'_i$ for $i = 1,2$. Hence, we must have
\[ \Delta' \leq N' + \frac{r_1 r_2}{2r} + \left( \left\lceil \frac{br_2}{r}\right\rceil - \frac{br_2}{r}\right) \]
In conclusion, we have
\[ \Delta' \leq N' + C_0 \]
where $C_0$ is the supremum of $0$, the terms $1 + \frac{1}{2}\left( r^2 - \sum_{i=1}^l r_i^2 \right) - \frac{1}{2}\kappa $ corresponding to $l \geq 2$ and $r_1 + \cdots + r_l = r$, and the terms $\frac{r_1 r_2}{2r} + \left( \left\lceil \frac{br_2}{r}\right\rceil - \frac{br_2}{r}\right)$ corresponding to $l=2$, $r_1 + r_2 = r$, and $\mu_F(\gamma_2) - \mu_F(\gamma_1) = -1$.
It follows from equation \ref{kapval} that $\kappa$ is bounded below by $-(r-1)$. Clearly, $\left( r^2 - \sum_{i=1}^l r_i^2 \right)$ is bounded above by $r^2 - r$. Hence, we see that
\[ 1 + \frac{1}{2} \left( r^2 - \sum_{i=1}^l r_i^2 \right) - \frac{1}{2}\kappa \;\leq\; \frac{1}{2}\left( r^2 + 1 \right)\]
Clearly $\left( \left\lceil \frac{br_2}{r} \right\rceil - \frac{br_2}{r}\right) \leq 1$ and $\frac{r_1(r-r_1)}{2r}$ is bounded above by $\frac{r}{8}$, whence the terms corresponding to $r=r_1 + r_2$ and $\mu_F(\gamma_2 ) - \mu_F(\gamma_1) = -1$ are bounded above by $\frac{r}{8}+1$.
In summary, we can take $C_0$ to be $\frac{1}{2}(r^2 + 1)$. Hence, for $\Delta' > N' + \frac{1}{2}(r^2 + 1) $, the coefficient of $\L^{-N'}q^{\Delta'}$ in $(1-q)\tilde{G}_{r,c}(q)$ is zero.
\end{proof}
\begin{lem}\label{lowerbound}
The following expression
\begin{equation}\label{lbterm}
2(r-1) \left( r\Delta - \sum_{i=1}^l r_i \Delta_i \right) + \left( \sum_{i<j} r_i r_j (\mu_j - \mu_i) \cdot K_{\F} \right)
\end{equation}
is bounded below by some constant $\kappa$ which depends only on $(r,c)$ and $r_i$ for $1 \leq i \leq l$, except when $l=2$ and $\mu_F(\gamma_2 ) - \mu_F(\gamma_1) = -1$.
\end{lem}
\begin{proof}
We can assume that $r_i \mid c_i \cdot F$ for each $1 \leq i \leq l$, otherwise the entire summand (equation \ref{rhs}) vanishes due to Mozgovoy's theorem (Theorem \ref{hrc}). Let $c = aE + bF$ and for each $1 \leq i \leq l$, let $c_i = r_i a_i E + b_i F$. Note that every term in the generating function $\tilde{G}_{r,c}(q)$ is invariant under the action of tensoring by line bundles, whence, we can assume that $0 \leq a,b \leq (r-1)$. Furthermore, we define $s_i = \sum_{j=i}^l b_j$ for all $1 \leq i \leq l$.
Following Manschot \cite{man}[Proof of Proposition 4.1] we see that
\begin{equation*}
\begin{aligned}
r\Delta - \sum_{i=1}^l r_i \Delta_i &= \sum_{i=2}^l \frac{r_i}{2 \left( \sum_{j=1}^i r_j \right) \left( \sum_{j=1}^{i-1} r_j \right)}\left( \sum_{j=1}^{i-1} r_j(a_i - a_j) \right)^2 \; - \sum_{i=2}^l (a_i - a_{i-1})s_i \; \\
& \qquad \qquad \qquad \qquad + \, b \sum_{i=2}^l \frac{\sum_{j=1}^{i-1} r_i r_j(a_i - a_j)}{\left( \sum_{j=1}^i r_j \right) \left( \sum_{j=1}^{i-1} r_j \right)}
\end{aligned}
\end{equation*}
Similarly, following Manschot \cite{man}[Proof of Proposition 4.1] we see that
\begin{align*}
\sum_{i<j} r_i r_j (\mu_j - \mu_i) \cdot K_{\F} = \sum_{i<j} r_i r_j (a_i - a_j) -2 \sum_{i=2}^l (r_i + r_{i-1}) s_i + 2(r - r_1)b
\end{align*}
Using these two equations we get
\begin{equation}\label{eqn16}
\begin{aligned}
& 2(r-1) \left( r\Delta - \sum_{i=1}^l r_i \Delta_i \right) + \left( \sum_{i<j} r_i r_j (\mu_j - \mu_i) \cdot K_{\F} \right) = \\
& \left\lbrace 2(r-1) \sum_{i=2}^l \frac{r_i}{2 \left( \sum_{j=1}^i r_j \right) \left( \sum_{j=1}^{i-1} r_j \right)}\left( \sum_{j=1}^{i-1} r_j(a_i - a_j) \right)^2 +\sum_{i<j} r_i r_j (a_i - a_j) \right\rbrace \\
& \qquad \qquad \qquad + \left\lbrace 2(r-1)b \sum_{i=2}^l \frac{\sum_{j=1}^{i-1} r_i r_j(a_i - a_j)}{\left( \sum_{j=1}^i r_j \right) \left( \sum_{j=1}^{i-1} r_j \right)} \right. \; + \\
& \left. -2(r-1) \sum_{i=2}^l (a_i - a_{i-1}) s_i - 2 \sum_{i=2}^l (r_i + r_{i-1}) s_i + 2 (r-r_1) b \right\rbrace
\end{aligned}
\end{equation}
We would like to show that both the first and second summand of right hand side of equation \ref{eqn16} are bounded below. Let us call the first summand $S_1$ and the second summand $S_2$.
We now proceed to scrutinize $S_1$ to determine its lower bound. We are going to use the following identity of Manschot \cite{man}[Proof of Proposition 4.1]
\begin{equation}\label{iden1}
\sum_{i=2}^l \frac{r_i}{2 \left( \sum_{j=1}^i r_j \right) \left( \sum_{j=1}^{i-1} r_j \right)}\left( \sum_{j=1}^{i-1} r_j(a_i - a_j) \right)^2 = \frac{1}{2r}\left( \sum_{i=1}^l r_i (r-r_i)a_i^2 - 2 \sum_{1 \leq i<j \leq l} r_i r_j a_i a_j \right)
\end{equation}
Since $a = \sum_{i=1}^l r_i a_i$, it follows from equation \ref{iden1} that
\[ S_1 = (r-1)\sum_{i=1}^l r_i a_i^2 - \frac{r-1}{r} a^2 + \sum_{i=1}^l a_i r_i \left(\sum_{j=i+1}^l r_j - \sum_{j=1}^{i-1} r_j \right) \]
Consider the smooth polynomial function
\[ f(x_1, \cdots, x_l) = \sum_{i=1}^l r_i x_i^2 - \frac{1}{r}a^2 + \sum_{i=1}^l x_i \frac{r_i}{r-1}\left( \sum_{j=i+1}^l r_j - \sum_{j=1}^{i-1}r_j\right) \]
Clearly, the Hessian of $f$, given by $\left( \frac{\partial^2 f}{\partial x_j \partial x_i} \right)$ is positive definite. We define
\[ g(x_1, \cdots, x_l) = \sum_{i=1}^l r_i x_i - a \]
Our goal is to minimize $f$ along the locus of $g=0$ for integer values of the $x_i$'s. Using the Lagrange's multiplier method, we see that $f$ assumes minima at
\[ a_i = \frac{a}{r} - \frac{1}{2(r-1)}\left(\sum_{j=i+1}^l r_j - \sum_{j=1}^{i-1} r_j\right) , \qquad \text{ for } i=1, \cdots, l \]
Clearly $\left\vert \sum_{j=i+1}^l r_j - \sum_{j=1}^{i-1}r_j \right\vert \leq (r-1)$, and hence we get $\frac{a}{r}-\frac{1}{2} \leq a_i \leq \frac{a}{r} + \frac{1}{2}$ for all $1 \leq i \leq l$. Thus, to find a lower bound for $S_1$ we need to find the minimum value of $f$ when $x_i \in \left\lbrace -1,0,1,2 \right\rbrace$ for all $1 \leq i \leq l$. We have the following partition
\[ \left\lbrace 1, \cdots, l \right\rbrace = \left\lbrace i_\alpha \right\rbrace_{1\leq \alpha \leq p} \cup \left\lbrace j_\beta \right\rbrace_{1\leq \beta \leq q} \cup \left\lbrace k_\gamma \right\rbrace_{1\leq \gamma \leq s} \cup \left\lbrace m_\delta \right\rbrace_{1 \leq \delta \leq t} \]
where $x_{i_\alpha} = -1$, $x_{j_\beta} = 1$, $x_{k_\gamma} = 2$, and $x_{m_\delta} = 0$. We see that
\begin{align}\label{s1}
\begin{aligned}
r(r-1)f &= (12 r - 9) \left( \sum_{i_\alpha > k_\gamma} r_{i_\alpha} r_{k_\gamma} \right) + (6r-4)\left(\sum_{i_\alpha > j_\beta} r_{i_\alpha} r_{j_\beta} + \sum_{k_\gamma < m_\delta} r_{k_\gamma} r_{m_\delta}\right) \\
& + (2r-1) \left( \sum_{i_\alpha >m_\delta} r_{i_\alpha}r_{m_\delta} + \sum_{j_\beta > k_\gamma} r_{j_\beta} r_{k_\gamma} + \sum_{j_\beta < m_\delta} r_{j_\beta}r_{m_\delta} \right)\\
&+ (6r-9) \left( \sum_{i_\alpha < k_\gamma} r_{i_\alpha}r_{k_\gamma} \right) + (2r-4)\left(\sum_{i_\alpha < j_\beta} r_{i_\alpha} r_{j_\beta} + \sum_{k_\gamma >m_\delta} r_{k_\gamma} r_{m_\delta} \right) \\
&+ (-1)\left( \sum_{i_\alpha < m_\delta}r_{i_\alpha}r_{m_\delta} + \sum_{j_\beta < k_\gamma}r_{j_\beta}r_{k_\gamma} + \sum_{j_\beta > m_\delta}r_{j_\beta}r_{m_\delta}\right)
\end{aligned}
\end{align}
Note that since $r\geq 2$ all the summands in equation \ref{s1} except the last one have non-negative coefficient. By further examining the summands with non-negative coefficient, we see that together they must be bounded below by $(2r-4)$ because all the inequalities in the summations cannot be simultaneously compatible. Moreover, the negative summand is bounded below by $-(r^2 - r)$. Hence, $S_1$ is bounded below by $-r+3 -\frac{4}{r}$.
Our next goal is to determine a lower bound for $S_2$. We are going to use the following identities of Manschot \cite{man}[Proof of Proposition 4.1]
\begin{equation}\label{id2}
\sum_{i=2}^l \frac{r_i}{\left( \sum_{j=1}^i r_j \right) \left( \sum_{j=1}^{i-1}r_j \right)} \left( \sum_{j=1}^{i-1} r_j(a_i - a_j) \right) = \frac{1}{r}\left( \sum_{i=2}^l (a_i - a_{i-1})\left(\sum_{j=i}^l r_j \right)\right)
\end{equation}
and
\begin{equation}\label{id3}
\sum_{i=2}^l (r_i + r_{i-1})\left(\sum_{j=i}^l r_j\right) = (r-r_1)r
\end{equation}
The identities in equations \ref{id2} and \ref{id3} yields
\[ S_2 = 2\sum_{i=2}^l \left( (r-1)(a_i - a_{i-1}) + (r_i + r_{i-1})\right) \left( \frac{b}{r}\left( \sum_{j=i}^l r_j \right) - s_i \right) \]
Following Coskun and Woolf \cite{cos}[Proof of Theorem 5.4], we interpret the definition of $S(\{\gamma_\bullet \};F,$ $ E+F)$ (equation \ref{defnSmu}) in our current situation, we obtain for all $2 \leq i \leq l$
\begin{align}\label{defnSmuIn}
\begin{aligned}
& \qquad \text{A) } (a_i - a_{i-1}) <0 \text{ and } s_i \geq \frac{b}{r}\left( \sum_{j=i}^l r_j \right) \\
& \qquad \text{B) } (a_i - a_{i-1}) \geq 0 \text{ and } s_i < \frac{b}{r} \left( \sum_{j =i}^l r_j \right)
\end{aligned}
\end{align}
In Case A, we see that $(r-1)(a_i - a_{i-1}) + r_i + r_{i-1} \leq 0$ except when $l=2$ and $a_2 - a_1 = -1$, which is not possible by our assumption. Hence, the term
\begin{equation}\label{s2term}
\left( (r-1)(a_i - a_{i-1}) + (r_i + r_{i-1})\right) \left( \frac{b}{r}\left( \sum_{j=i}^l r_j \right) - s_i \right)
\end{equation}
is non-negative.
Similarly, in Case B, we see that $(r-1)(a_i - a_{i-1}) + r_i + r_{i-1} \geq (r_i + r_{i-1})$, hence the term in equation \ref{s2term} is non-negative. Additionally, by using the fact that $s_i$ are integers, it follows from equation \ref{defnSmuIn} that we have a slightly better bound of equation \ref{s2term}
\[ \left\vert (r-1) (a_i - a_{i-1}) + r_i + r_{i-1} \right\vert \left( 1 - sgn\left(a_i - a_{i-1} + \frac{1}{2}\right) \left( 1 - 2 \left\lbrace - \frac{b}{r}\sum_{j=i}^l r_j \right\rbrace\right)\right) \]
where $sgn$ is the sign function and $\lbrace \bullet \rbrace$ is the fractional part of any real number.
In conclusion, we can take $\kappa$ to be
\begin{align}\label{kapval}
\begin{aligned}
& -r + 3 - \frac{4}{r}\\
&+ \sum_{i=2}^l \left\vert (r-1) (a_i - a_{i-1}) + r_i + r_{i-1} \right\vert \left( 1 - sgn\left(a_i - a_{i-1} + \frac{1}{2}\right) \left( 1 - 2 \left\lbrace - \frac{b}{r}\sum_{j=i}^l r_j \right\rbrace\right)\right)
\end{aligned}
\end{align}
which is our lower bound for equation \ref{lbterm}.
\end{proof}
Now that we have shown that for $\tilde{\Delta} \gg \tilde{N}$, the coefficient of $\L^{-\tilde{N}}q^{\tilde{\Delta}}$ in $(1-q)\tilde{G}_{r,\tilde{c}}(q)$ vanishes (see Proposition \ref{boundMF}), our goal is to relate $G_{r,c}(q)$ with $\tilde{G}_{r,\tilde{c}}$ using the blow-up formula, and conclude a similar result for $G_{r,c}(q)$.
\section{\textsc{Estimating the generating function $G_{r,c}(q)$ when rank is at least two}}\label{section5}
In this section, our goal is to show that there is a constant $C$ depending only on $r$ and $c$ such that when $\Delta > N + C$, the coefficient of $\L^{-N}q^\Delta$ in $(1-q)G_{r,c}(q)$ is zero. To show this, we are going to look at the blow-up $\F \rarrow \P$ and use the blow-up formula due to Mozgovoy \cite{moz}[Proposition 7.3] to relate the generating functions $G_{r,c}(q)$ and $\tilde{G}_{r,\tilde{c}}(q)$ (see equation \ref{blowupeqn}) in $A^-$. We are going to scrutinize the terms appearing in this relation, and use Proposition \ref{boundMF} to derive our inequality (see Theorem \ref{clm14}).
Recall from section \ref{section2} that we have a blow-up $\F \rarrow \P$ at point $p \in \P$. Let $\gamma = (r,c,\Delta)$ be a Chern character on $\P$. Let $m$ be the multiplicity of $c$ at the point $p$. Let $\tilde{\gamma} = (r, c - mE, \tilde{\Delta})$ be a Chern character on $\F$. The blow-up formula due to Mozgovoy \cite{moz}[Proposition 7.3] is the following equation
\begin{equation}
\label{blowup}
\sum_{ch_2} [\MF (r,c-mE,ch_2)] q^{-ch_2} = F_m(q) \sum_{ch_2} [\MP (r, c , ch_2)] q^{- ch_2}
\end{equation}
where
\begin{equation}
\label{Fm}
F_m(q) = \left( \prod_{k=1}^\infty \frac{1}{(1 - \L^{rk}q^k)^r} \right) \left( \sum_{\substack{ \sum_{i=1}^r a_i = 0, \\ a_i \in \mathbb{Z} + \frac{m}{r}}} \L^{\sum_{i<j} \binom{a_j - a_i}{2}} q^{- \sum_{i<j} a_i a_j} \right)
\end{equation}
Note that on $\P$, we have $-ch_2(\gamma) = r \Delta - \frac{c^2}{2r}$, while on $\F$, we have $-ch_2(\tilde{\gamma}) = r \tilde{\Delta} - \frac{c^2}{2r} + \frac{m^2}{2r}$. Hence, we can rewrite the blow-up equation (equation \ref{blowup})
\begin{equation}\label{eqn19}
\sum_{\Delta \geq 0} [\MP (r, c, \Delta)] q^{r \Delta} = \frac{q^{\frac{m^2}{2r}}}{F_m(q)} \sum_{\tilde{\Delta} \geq 0} [\MF (r,c-mE,\tilde{\Delta})] q^{r\tilde{\Delta}}
\end{equation}
Replacing $q$ by $\L^{-2r}q$ and multiplying both sides by $\L^{r^2}$ in equation \ref{eqn19} yields
\begin{equation}\label{blowupeqn}
G_{r,c}(q) = \frac{(\L^{-2r}q)^{\frac{m^2}{2r}}}{F_m(\L^{-2r}q)} \tilde{G}_{r,c-mE}(q)
\end{equation}
It follows from equation \ref{blowupeqn} that in order to achieve our goal, we need to analyze $F_m(\L^{-2r}q)$ and find an estimate for $\Delta$ in this expression.
By examining the definition of $F_m$ in equation \ref{Fm}, we conclude that it depends only on the remainder of $m$ modulo $r$, which we shall denote by $\bar{m}$, which we will think of as an integer between $0$ and $r-1$.
We see that
\begin{equation}
\label{FmL}
F_{\bar{m}} (\L^{-2r}q) = \prod_{k=1}^\infty \frac{1}{(1 - \L^{-rk}q^k)^r} \,\sum_{\substack{ \sum_{i=1}^r a_i = 0, \\ a_i \in \mathbb{Z} + \frac{\bar{m}}{r}}}
\L^{\sum_{i<j} \binom{a_j - a_i}{2} + 2r \sum_{i<j}a_i a_j } q^{- \sum_{i<j}a_i a_j}
\end{equation}
Since $\sum_{i=1}^r a_i = 0$, we see that
\[ - \sum_{1 \leq i<j \leq r} a_i a_j = \frac{1}{2} \sum_{i=1}^r a_i^2 \]
and
\[ \sum_{1 \leq i<j \leq r} \binom{a_j - a_i}{2} + 2r \sum_{1 \leq i<j \leq r} a_i a_j = - \frac{r}{2} \left(\sum_{i=1}^r a_i^2 \right) - \left(\sum_{i=1}^r i a_i \right) \]
We now use the following substitutions
\begin{align*}
\qquad\qquad a_i &= b_i + \frac{\bar{m}}{r} \, , \text{ where } b_i \in \mathbb{Z}, \, \text{ for } 1 \leq i \leq r-1, \\
\qquad\qquad a_r &= - \sum_{i=1}^{r-1} \left( b_i + \frac{\mb}{r} \right)
\end{align*}
These substitutions yield the following equations
\begin{equation}\label{eqn23}
\begin{aligned}
- \frac{r}{2} \left(\sum_{i=1}^r a_i^2 \right) - \left(\sum_{i=1}^r i a_i \right) &= -r \left( - \frac{\mb^2}{2r} + \frac{\mb^2}{2} + \sum_{i=1}^{r-1} b_i^2 + \mb \sum_{i=1}^{r-1} b_i + \sum_{1 \leq i<j \leq (r-1)} b_i b_j \right) \\
& \qquad + \left( \frac{ (r-1) \mb}{2} + \sum_{i=1}^{r-1} (r-i)b_i \right) \\
\frac{1}{2} \sum_{i=1}^r a_i^2 &= \left( \frac{- \mb^2}{2r} + \frac{\mb^2}{2} + \sum_{i=1}^{r-1} b_i^2 + \mb \sum_{i=1}^{r-1} b_i + \sum_{1 \leq i<j \leq (r-1)} b_i b_j \right)
\end{aligned}
\end{equation}
Employing the above equations \ref{eqn23} leads to the following expression for $F_{\mb}(\L^{-2r}q)$
\begin{align*}
F_{\mb}(\L^{-2r}q) &= \left(\prod_{k=1}^\infty \frac{1}{(1 - \L^{-rk}q^k)^r} \right) \left( \L^{-r}q \right)^{- \frac{(r+1) \mb^2}{2r} } \L^{\frac{ (r-1) \mb}{2}} \; \times \\
& \sum_{b_1, \cdots, b_{r-1} \in \mathbb{Z}} \L^{\sum_{i=1}^{r-1} (r-i)b_i} \left( \L^{-r}q \right)^{ \mb^2 + \sum_{i=1}^{r-1}b_i^2 + \mb \sum_{i=1}^{r-1} b_i + \sum_{i<j} b_i b_j}
\end{align*}
For sake of convenience, we define
\begin{align}\label{Lambda}
\Lambda_d^{(\mb)} = \sum_{\substack{ b_1, \cdots, b_{r-1} \in \mathbb{Z}, \\ \mb^2 + \sum_{i=1}^{r-1} b_i^2 + \mb \sum_{i=1}^{r-1} b_i + \sum_{i<j} b_i b_j = d}} \L^{\sum_{j=1}^{r-1}(r-j)b_j}
\end{align}
Thus, we can think of the last summation term of $F_{\mb}(\L^{-2r}q)$ as a power series
\begin{align}\label{FmLL}
F_{\mb}(\L^{-2r}q) = \left( \prod_{k=1}^\infty \frac{1}{(1 - \L^{-rk}q^k)^r} \right) \left( \L^{-r}q \right)^{ - \frac{(r+1)\mb^2}{2r}} \L^{\frac{(r-1)\mb}{2}} \left( \sum_{d=0}^\infty \Lambda_d^{(\mb)} (\L) \left( \L^{-r}q \right)^d \right)
\end{align}
\begin{rem}\label{rmk8}
Recall that any power series of the form $f(x) = 1 + a_1 x + a_2 x^2 + \cdots$ is invertible, and its inverse is given by $1 + b_1 x + b_2 x^2 + \cdots$, where for any positive integer $n$, we have
\[ b_n = \sum_{\substack{n_1 + \cdots + n_l = n \\ n_i \in \mathbb{Z}_{>0}}} (-1)^l a_{n_1} \cdots a_{n_l} \]
\end{rem}
To analyze $G_{r,c}(q)$, we need to invert $F_{\mb}(\L^{-2r}q)$ (equation \ref{blowupeqn}), and a posteriori, we need to invert the power series $\sum_{d =0}^\infty \Lambda_d^{(\mb)}(\L) (\L^{-r}q)^d$. To do this, we need to figure out the least non-negative integer $d$ such that $\Lambda_d^{(\mb)}(\L)$ is nonzero.
\begin{lem}\label{clm9}
The smallest non-negative integer $d$ for which $\Lambda_d^{(\mb)}$ is nonzero, is $\frac{\mb^2 + \mb}{2}$. Additionally,
\[ \Lambda_{\frac{\mb^2 + \mb}{2}}^{(\mb)} (\L) = \L^{-r \mb} \; \sum_{\nu = \frac{\mb^2 + \mb}{2}}^{r \mb - \frac{\mb^2 - \mb}{2}} \rho_\nu \L^{\nu} \]
where $\rho_\nu$ is the cardinality of the set $\left\lbrace \left(j_1, \cdots, j_{\mb} \right) \,\vert\, 1 \leq j_1 < \cdots < j_{\mb} \leq r, \, j_1 + \cdots + j_{\mb} = \nu \right\rbrace$, when $\nu$ is a positive integer, and $\rho_0 = 1$.
\end{lem}
\begin{proof}
Note that
\[ \mb^2 + \sum_{i=1}^{r-1} b_i^2 + \mb \sum_{i=1}^{r-1} b_i + \sum_{i<j} b_i b_j = \frac{1}{2} \left( \mb^2 + \sum_{i=1}^{r-1}b_i^2 + \left( \mb + \sum_{i=1}^{r-1} b_i \right)^2 \right) \]
Consequently, we need to figure out the smallest value of $\mb^2 + \sum_{i=1}^{r-1} b_i^2 + \left( \mb + \sum_{i=1}^{r-1} b_i \right)^2$, where $b_i \in \mathbb{Z}$ for all $1 \leq i \leq r-1$.
If $\mb = 0$, we see that the equation $\sum_{i=1}^{r-1} b_i^2 + \left( \sum_{i=1}^{r-1} b_i \right)^2 = 0$ has only one solution, the trivial one. Thus, $\Lambda_0^{(0)}(\L) = 1$.
Assume $1 \leq \mb \leq r-1$. It follows from Lemma \ref{lowerbound2} (below), that the smallest value assumed by the expression $\sum_{i=1}^{r-1} b_i^2 + \left( \mb + \sum_{i=1}^{r-1} b_i \right)^2$ occurs at $b_1 = \cdots = b_{r-1} = - \frac{\mb}{r}$. As a result, we need to evaluate the expression when $b_i \in \left\lbrace -1,0 \right\rbrace$ for all $1 \leq i \leq r-1$, to figure out the minimum value of the expression for integer values. Suppose $k$ of the $b_i$'s are $(-1)$ and the remaining are zero, the expression becomes $k + \left( \mb -k \right)^2$. Clearly, the minimum value of $k + \left( \mb - k \right)^2$ for integer values of $k$ is $\mb$, which occurs when $k = \mb-1, \mb$.
In summary, when $1 \leq \mb \leq r-1$, the smallest value of the expression \[\frac{1}{2}\left( \mb^2 + \sum_{i=1}^{r-1} b_i^2 + \left( \mb + \sum_{i=1}^{r-1} b_i \right)^2 \right)\] for integer values of $b_i$ is $\frac{\mb^2 + \mb}{2}$, which occurs when $\mb-1$ or $\mb$ of the $b_i$'s are $(-1)$ and the remaining are zero. Hence, we have
\[ \Lambda_d^{(\mb)} (\L) = \sum_{1 \leq j_1 < \cdots < j_{\mb -1} \leq r-1} \L^{j_1 + \cdots + j_{\mb-1} - (\mb-1)r} + \sum_{1 \leq j_1 < \cdots < j_{\mb} \leq r-1} \L^{j_1 + \cdots + j_{\mb} - r \mb} \]
Factoring out $\L^{-r \mb}$ leads to
\[ \Lambda_d^{(\mb)} (\L) = \L^{-r\mb} \sum_{1 \leq j_1 < \cdots < j_{\mb} \leq r} \L^{j_1 + \cdots + j_{\mb}} \]
\end{proof}
Before proceeding further, we need to tie the loose ends of Lemma \ref{clm9} by analyzing the real valued polynomial function $y_1^2 + \cdots + y_n^2 + \left( A + y_1 + \cdots + y_n \right)^2$.
\begin{lem}\label{lowerbound2}
Consider the smooth real valued function
\[ f(y_1, \cdots, y_n) = y_1^2 + \cdots + y_n^2 + \left( A + y_1 + \cdots + y_n \right)^2 \]
where $A$ is any real number. The Hessian of $f$ is positive definite. Furthermore, the function $f$ has a global minima at $y_1 = \cdots = y_n = - \frac{A}{n+1}$, and the minimum value for $f$ is $ \frac{A^2}{n+1}$.
\end{lem}
\begin{proof}
Clearly, we see that for $1 \leq k \leq n$
\[ \frac{\partial f}{\partial y_k} = 2y_k + 2\left( A + y_1 + \cdots + y_n \right) \]
Subsequently, we see that for $1 \leq l \leq n$
\[ \frac{\partial^2 f}{\partial y_l \partial y_k} = \begin{cases} 2, \text{ if } k \neq l \\
4, \text{ if } k = l \end{cases} \]
Let $H$ be the $n \times n$ matrix with $H_{l,k} = \frac{\partial^2 f}{\partial y_l \partial y_k}$, then we see that
\[ \left( y_1 \; \cdots \; y_n \right) \cdot H \cdot \left( y_1 \; \cdots \; y_n \right)^{T} = 2 \left(\sum_{i=1}^n y_i^2 \right) + 2 \left( \sum_{i=1}^n y_i \right)^2 \]
Thus, $H$ is positive definite. As a consequence, $f$ has a global minimum when $\frac{\partial f}{\partial y_k} = 0$ for all $1 \leq k \leq n$. This system of linear equations has a unique solution $y_1 = \cdots = y_n = -\frac{A}{n+1} $. It follows that the minimum value for $f$ is $\frac{A^2}{n+1}$.
\end{proof}
Returning back to our track, we still need to analyze $F_{\mb}(\L^{-2r}q)$. Using equation \ref{FmLL} and Lemma \ref{clm9}, we see that
\begin{align*}
F_{\mb}(\L^{-2r}q) &= \left( \prod_{k=1}^\infty \frac{1}{(1 - \L^{-rk}q^k)^r} \right) \left(\L^{-r}q \right)^{- \frac{(r+1) \mb^2}{2r}} \L^{ \frac{(r-1) \mb}{2}} \\
& \qquad \qquad\Lambda_{\frac{\mb^2 + \mb}{2}}^{(\mb)} (\L) \left( \L^{-r}q \right)^{\frac{\mb^2 + \mb}{2}}
\sum_{d=0}^\infty \tilde{\Lambda}_{d}^{(\mb)} (\L) \left( \L^{-r}q \right)^d
\end{align*}
where $\tilde{\Lambda}_d^{(\mb)}(\L) = \left( \Lambda_{\frac{\mb^2 + \mb}{2}}^{(\mb)}(\L)\right)^{-1} \cdot \Lambda_{d + \frac{\mb^2 + \mb}{2}}^{(\mb)}(\L)$.
Finally, using remark \ref{rmk8}, we can invert $F_{\mb}(\L^{-2r}q)$.
\begin{equation}
\label{FmLinv}
\begin{aligned}
\left(F_{\mb} (\L^{-2r}q)\right)^{-1} &= \left( \prod_{k=1}^\infty (1 - \L^{-rk}q^k)^r \right) \left( \L^{-r}q \right)^{-\frac{r \mb - \mb^2}{2r}} \L^{-\frac{(r-1)\mb}{2}} \left( \Lambda_{\frac{\mb^2 + \mb}{2}}^{(\mb)} (\L) \right)^{-1} \\
& \qquad \qquad
\left( 1 + \sum_{d =1}^\infty \left( \sum_{\substack{d_1, \cdots, d_l \in \mathbb{Z}_{>0} \\ d_1 + \cdots + d_l = d}}(-1)^{l} \prod_{i=1}^l \tilde{\Lambda}_{d_i}^{(\mb)} \right) \left( \L^{-r}q\right)^{d} \right)
\end{aligned}
\end{equation}
Before tackling $G_{r,c}(q)$, we would like to analyze $F_{\mb}(\L^{-2r}q)^{-1}$ and produce bounds for $\Delta$ such that the coefficient of $\L^{-N}q^\Delta$ vanishes.
\begin{lem}\label{clm11}
If $\Delta > N - \frac{(r-\mb)\mb}{2r}$, then the coefficient of $\L^{-N}q^\Delta$ in $F_{\mb}(\L^{-2r}q)^{-1}$ is zero.
\end{lem}
\begin{proof}
We are going to produce an expression for $\left(\Lambda_{\frac{\mb^2 + \mb}{2}}^{(\mb)}(\L)\right)^{-1}$, and use it alongwith the expression for $F_{\mb}(\L^{-2r}q)^{-1}$ (see equation \ref{FmLinv}) to determine the bound for $\Delta$.
Using Lemma \ref{clm9} and factoring $\L^{r \mb - \frac{\mb^2 - \mb}{2}}$, we get
\[ \Lambda_{\frac{\mb^2 + \mb}{2}}^{(\mb)}(\L) = \L^{-\frac{\mb^2 - \mb}{2}} \,\sum_{\nu = 0}^{-(r\mb - \mb^2)} \rho_{\nu + r\mb - \frac{\mb^2 - \mb}{2}} \,\L^{\nu} \]
In a similar fashion as in remark \ref{rmk8}, it follows that
\[ \left(\Lambda_{\frac{\mb^2 + \mb}{2}}^{(\mb)}(\L)\right)^{-1} = \L^{\frac{\mb^2 - \mb}{2}}\left( 1+ \sum_{\nu = -1}^{-\infty} \left( \sum_{\substack{\nu_1 , \cdots, \nu_l \in \mathbb{Z}_{<0} \\ \nu_1 + \cdots + \nu_l = \nu}} (-1)^l \prod_{i=1}^l \rho_{\nu_i + r\mb - \frac{\mb^2 - \mb}{2}} \right) \L^{\nu} \right)\]
It follows from equation \ref{FmLinv} that
{\small
\begin{align*}
\left(F_{\mb}(\L^{-2r}q)\right)^{-1} &= \prod_{k=1}^\infty \left(\sum_{\alpha=0}^\infty (-1)^\alpha \binom{r}{\alpha} \L^{-rk\alpha}q^{k \alpha} \right) \times \left( \L^{-r}q \right)^{-\frac{r \mb - \mb^2}{2r}} \L^{-\frac{(r-1)\mb}{2}} \times
\left( \Big(\Lambda_{\frac{\mb^2 + \mb}{2}}^{(\mb)}(\L)\Big)^{-1} \right. \\
& \left.
+ \sum_{d = -1}^{-\infty} \left( \sum_{\substack{d_1, \cdots, d_l \in \mathbb{Z}_{<0} \\ d_1 + \cdots + d_l = d}} (-1)^l \left(\Lambda_{\frac{\mb^2 + \mb}{2}}^{(\mb)}(\L)\right)^{-(l+1)} \prod_{i=1}^l \Lambda_{d_i + \frac{\mb^2 + \mb}{2}}^{(\mb)}(\L)
\right) \left(\L^{-r}q\right)^d
\right)
\end{align*}}
Each nonzero term appearing in the co-efficient of $\L^{-N}q^{\Delta}$ in $F_{\mb}(\L^{-2r}q)^{-1}$ arises from a pair of equations
\begin{align*}
\Delta &= \left(\sum_{j=1}^\delta k^{(j)} \alpha^{(j)}\right) - \left(\frac{r\mb - \mb^2}{2r}\right) + d \\
-N &= \left(\sum_{j=1}^\delta -r k^{(j)} \alpha^{(j)}\right) + r \left( \frac{r\mb - \mb^2}{2r}\right) - \frac{(r-1)\mb}{2} + \\
& \;\qquad \qquad \left( \left( \frac{\mb^2 - \mb}{2} (l+1) + \sum_{i=1}^{l+1} \nu_i \right) + \sum_{i=1}^l \sum_{j=1}^{r-1} (r-j)b_j^{(i)} \right) - rd
\end{align*}
where the $\alpha$'s, the $\nu$'s, and $l$ are non-negative integers; the $k$'s are positive integers; and the $b_j^{(i)}$'s are integers satisfying
\[ \mb^2 + \sum_{j=1}^{r-1} \left( b_j^{(i)} \right)^2 + \left( \mb + \sum_{j=1}^{r-1} b_j^{(i)} \right)^2 = 2d_i + \mb^2 + \mb , \qquad \text{ for } 1 \leq i \leq l \]
Subsequently, we will show (in Lemma \ref{clm12}) that $\left(\sum_{j-1}^{r-1} (r-j) b_j^{(i)}\right) + \frac{\mb^2 - \mb}{2} \leq (r-1)d_i$, for all $1 \leq i \leq l$. Consequently, we have
\[ \left(\sum_{i=1}^l \sum_{j=1}^{r-1} (r-j) b_j^{(i)}\right) + \frac{\mb^2 -\mb}{2}l \leq (r-1) d \]
Therefore, we see that
\[ N + (r-1)\frac{r\mb - \mb^2}{2r} - \frac{(r-1)\mb}{2} + \frac{\mb^2-\mb}{2} \geq \left(\sum_{j-1}^\delta k^{(j)} \alpha^{(j)}\right) - \frac{r \mb - \mb^2}{2r} + d = \Delta \]
and hence,
\[ N - \frac{(r-\mb)\mb}{2r} \geq \Delta \]
\end{proof}
Before we continue, we need to wrap up the proof of Lemma \ref{clm11} by proving the following:
\begin{lem}\label{clm12}
Let $d$ be a non-negative integer, and $\mb $ be a non-negative integer less than $r$. Suppose $b_1, \cdots, b_{r-1}$ are integers satisfying
\begin{equation}\label{eqn27}
\mb^2 + \sum_{j=1}^{r-1} b_j^2 + \left( \mb + \sum_{j=1}^{r-1} b_j \right)^2 = 2d + \mb^2 +\mb
\end{equation}
Then, we have $\sum_{j-1}^{r-1} (r-j) b_j \leq (r-1)d$.
Furthermore, if $r \geq 3$ and $2 \leq \mb \leq (r-1)$, then we have
\[ \left(\sum_{j=1}^{r-1} (r-j)b_j\right) + \frac{\mb^2 - \mb}{2} \leq (r-1)d \]
\end{lem}
\begin{proof}
Before we begin the proof of Lemma \ref{clm12}, note that
\begin{rem}\label{ordering}
Let $r_1, \cdots, r_n$ be positive integers satisfying $r_1 > \cdots > r_n$, and let $b_1 , \cdots , b_n$ be integers satisfying $b_1 \geq \cdots \geq b_n$. Let $\sigma $ be any permutation of $\left\lbrace 1, \cdots, n \right\rbrace$. Then, we have
\[ r_1 b_{\sigma(1)} + \cdots + r_n b_{\sigma(n)} \leq r_1 b_1 + \cdots + r_n b_n \]
\end{rem}
Thus, if $b'_1, \cdots , b'_{r-1}$ be a rearrangement of $b_1, \cdots, b_{r-1}$ satisfying $b'_1 \geq \cdots \geq b'_{r-1}$, then we see that
\[ \sum_{j=1}^{r-1} (r-j) b_j \,\leq\, \sum_{j=1}^{r-1} (r-j) b'_j \]
Moreover, let $n_1, \, n_2,\, n_3$ be non-negative integers such that
\begin{enumerate}[$\qquad\qquad \bullet$]
\item $b'_{j_1} \geq \cdots \geq b'_{j_{n_1}} \geq 2$,
\item $b'_{j_{n_1 + 1}} = \cdots = b'_{j_{n_1 + n_2}} = 1 $,
\item $-1 \geq b'_{j_{n_1 + n_2 + 1}} \geq \cdots \geq b'_{j_{n_1 + n_2 +n_3}}$, and
\item $b'_j = 0$ for all $j \neq j_l, \, 1 \leq l \leq n_1 + n_2 + n_3$.
\end{enumerate}
Therefore, we have
\[ \sum_{j=1}^{r-1}(r-j) b'_j \,\leq\, \sum_{l=1}^{n_1} (r-j_{l}) b'_{j_{l}} + \sum_{l=n_1 +1}^{n_1 + n_2} (r - j_{l}) \leq \frac{(r-1)}{2} \left( 2 \left( \sum_{l=1}^{n_1} b'_{j_l} \right) + 2n_2 \right) \]
We observe that to complete our proof it is enough to show that
\[ \left(\sum_{l=1}^{n_1} 2b'_{j_l}\right) + 2n_2 \leq 2d \]
Since $\left(b'_{j_l}\right)^2 \geq 2 b'_{j_l}$ for $1 \leq l \leq n_1$ and $\left(b'_{j_l}\right)^2 = 1$ for $n_1 + 1 \leq l \leq n_1 + n_2$, it follows from equation \ref{eqn27} that it is enough to show that
\[ n_2 + \mb \leq \sum_{l=n_1 + n_2 +1}^{n_1 + n_2 + n_3} \left( b'_{j_l}\right)^2 + \left( \left( \mb + n_2 + \sum_{l=1}^{n_1} b'_{j_l}\right) + \sum_{l=n_1 + n_2 +1}^{n_1 + n_2 + n_3} b'_{j_l} \right)^2 \]
If $n_2 + \mb \leq n_3$, then we are done because $\left(b'_{j_l}\right)^2 \geq 1$ for all $n_1 + n_2 +1 \leq l \leq n_1 + n_2 + n_3$. Otherwise, it follows from Lemma \ref{lowerbound2} that
\[ \sum_{l=n_1 + n_2 +1}^{n_1 + n_2 + n_3} \left( b'_{j_l}\right)^2 + \left( \left( \mb + n_2 + \sum_{l=1}^{n_1} b'_{j_l}\right) + \sum_{l=n_1 + n_2 +1}^{n_1 + n_2 + n_3} b'_{j_l} \right)^2 \geq \frac{1}{n_3 +1} \left( \mb + n_2 + \sum_{l=1}^{n_1} b'_{j_l} \right)^2 \]
Since $b'_{j_l} \geq 2$ for $1 \leq l \leq n_1$ and $n_2 + \mb \geq n_3 +1$, we have
\[ \frac{1}{n_3 +1} \left( \mb + n_2 + \sum_{l=1}^{n_1} b'_{j_l} \right)^2 \geq n_2 + \mb \]
Now we are going to specialize to the case when $r \geq 3$ and $2 \leq \mb \leq r-1$. Clearly, since $\mb \geq 2$, we see that $\frac{\mb^2 - \mb}{2} = 1 + \cdots + (\mb-1)$. We define
\begin{align*}
b'_j = \begin{cases} b_j, &\text{ if }\; 1 \leq j \leq (r - \mb)\\ b_j +1 , &\text{ if }(r - \mb +1 )\leq j \leq (r-1) \end{cases}
\end{align*}
As a consequence, we see that
\[ \left(\sum_{j=1}^{r-1} (r-j) b_j\right) + \frac{\mb^2 - \mb}{2} = \sum_{j=1}^{r-1}(r-j)b'_j \]
Additionally, we can rewrite equation \ref{eqn27} in terms of $b'_j$'s as follows
\[ \sum_{j=1}^{r-1}\left(b'_j\right)^2 + \left( \sum_{j=1}^{r-1}b'_j \right)^2 + 2 \left( \sum_{j=1}^{r - \mb} b'_j \right) = 2d \]
As a result, to prove our claim, it is enough to show that
\[ \frac{(r-1)}{2}\left\lbrace \sum_{j=1}^{r-1}\left(b'_j\right)^2 + \left( \sum_{j=1}^{r-1}b'_j \right)^2 + 2 \left( \sum_{j=1}^{r - \mb} b'_j \right) \right\rbrace - \left( \sum_{j=1}^{r-1}(r-j)b'_j \right) \geq 0 \]
for integer values of $b'_j$, for all $1 \leq j \leq r-1$. Consider the smooth polynomial function
\[ f(x_1, \cdots, x_{r-1}) = \frac{(r-1)}{2}\left\lbrace \sum_{j=1}^{r-1}x_j^2 + \left( \sum_{j=1}^{r-1}x_j \right)^2 + 2\left(\sum_{j=1}^{r-\mb}x_j \right)\right\rbrace - \left( \sum_{j=1}^{r-1} (r-j)x_j \right) \]
We have
\begin{align*}
\frac{\partial f}{\partial x_k} = \begin{cases} \frac{(r-1)}{2}\left\lbrace 2x_k + 2\left( \sum_{j=1}^{r-1} x_j\right) + 2 \right\rbrace - (r-k), &\text{ if }\; 1 \leq k \leq (r-m) \\
\frac{(r-1)}{2}\left\lbrace 2x_k + 2 \left(\sum_{j=1}^{r-1} x_j\right)\right\rbrace - (r-k), &\text{ if }\; (r- \mb +1) \leq k \leq (r-1) \end{cases}
\end{align*}
and, the second partial derivatives are
\begin{align*}
\frac{\partial^2 f}{\partial x_l \partial x_k} = \begin{cases} 2\frac{(r-1)}{2}, &\text{ if }\; l \neq k\\
4\frac{(r-1)}{2}, &\text{ if }\; l=k \end{cases}
\end{align*}
Since $r \geq 3$ and the Hessian matrix for $f$ is $\frac{(r-1)}{2}$ times the Hessian matrix in Lemma \ref{lowerbound2}, we conclude that our Hessian matrix is positive definite. Thus, $f$ has a global minimum at the critical point
\begin{align*}
x_k = \begin{cases} -\frac{\mb}{r} -\frac{1}{2} + \frac{(r-k)}{(r-1)}, &\text{ if }\; 1 \leq k\leq (r-\mb)\\
-\frac{\mb}{r} + \frac{1}{2} + \frac{(r-k)}{(r-1)}, &\text{ if }\; (r-\mb+1) \leq k \leq (r-1)\end{cases}
\end{align*}
It follows from the bounds on $k$ that in either case, we have $-\frac{1}{2} \leq x_k \leq \frac{1}{2}$. Hence, to show that $f$ is non-negative for all integer values of $x_j$, for all $1 \leq j \leq (r-1)$, it is enough to show that $f$ is non-negative for every element of the set $\left\lbrace -1,0,1 \right\rbrace^{r-1}$. Let $(x_1, \cdots, x_{r-1})$ be an element of the set $\left\lbrace -1,0,1 \right\rbrace^{r-1}$. Furthermore, assume that for $1 \leq j \leq (r-\mb)$, $x$ of the $x_j$'s are $(+1)$ and $y$ of the $x_j$'s are $(-1)$. On a similar note, assume that for $(r-\mb +1) \leq j\leq (r-1)$, $z$ of the $x_j$'s are $(+1)$ and $w$ of the $x_j$'s are $(-1)$. It follows from Remark \ref{ordering} that
\begin{align*}
\sum_{j=1}^{r-1}(r-j)x_j &\leq (r-1) + \cdots + (r - x) - \left\lbrace \mb + (\mb+1) + \cdots + (\mb+y-1)\right\rbrace \\
&\qquad \qquad + (\mb-1) + \cdots + (\mb - z) - \left\lbrace 1 + \cdots + w \right\rbrace \\
&= rx - \mb y + \mb z - \frac{x^2 +x}{2} - \frac{y^2 -y}{2} - \frac{z^2 +z}{2} - \frac{w^2 +w}{2}
\end{align*}
Therefore, we have
\begin{align}\label{eqn29}
\begin{aligned}
f(x_1, \cdots, x_{r-1}) &\geq \frac{(r-1)}{2} \left\lbrace (x-y+z-w)^2 + 3x -y + z + w \right\rbrace\\
& - \left\lbrace rx - \mb y + \mb z - \frac{x^2 +x}{2} - \frac{y^2 -y}{2} - \frac{z^2 +z}{2} - \frac{w^2 +w}{2} \right\rbrace
\end{aligned}
\end{align}
For ease of notation, let's call the right hand side of inequality in equation \ref{eqn29} as $g(x,y,z,w)$. Upon further scrutinizing, we deduce that
\[ 2g(x,y,z,w) = (r-1)(x-y+z-w)^2 + (x^2 + y^2 + z^2 + w^2) + (r-2)x + (2\mb -r)y + (r-2\mb)z + rw \]
If $r = 2\mb$, then $2g(x,y,z,w) \geq 0$ because $x$ and $w$ are non-negative integers. If $r>2\mb$, then we see that
\[ 2g(x,y,z,w) \geq (r - 2\mb)\left\lbrace (x-y+z-w)^2 + (x-y+z-w) \right\rbrace \geq 0 \]
Similarly, if $r<2\mb$, then using the fact that $(r-1)> (2\mb -r)$, we get
\[ 2g(x,y,z,w) \geq (2\mb -r) \left\lbrace (-x+y-z+w)^2 + (-x+y-z+w)\right\rbrace \geq 0 \]
In conclusion, the function $f$ is non-negative for all integer values of $x_j$, for all $1 \leq j \leq r-1$.
\end{proof}
We are finally ready to analyze $(1-q)G_{r,c}(q)$.
\begin{thm}\label{clm14}
If $\Delta > N + \frac{(2 - 2r)\mb^2 - r\mb}{2r} + C_0$, where $C_0$ is the same constant as in Proposition \ref{boundMF}, then the coefficient of $\L^{-N}q^{\Delta}$ in $(1-q)G_{r,c}(q)$ is zero.
\end{thm}
\begin{proof}
Recall that if follows from the blow-up equation (equation \ref{blowupeqn}) that
\[ (1-q)G_{r,c}(q) = \left(\L^{-2r}q \right)^{\frac{m^2}{2r}} \times \left(F_m(\L^{-2r}q)\right)^{-1} \times (1-q)\tilde{G}_{r,c-mE}(q) \]
Each nonzero term appearing in the co-efficient of $\L^{-N}q^{\Delta}$ arises from a pair of equations
\begin{align*}
\Delta &= \frac{\mb^2}{2r} + \Delta_1 + \Delta_2 \\
-N &= -\mb^2 + \left(-N_1 \right) + \left( -N_2 \right)
\end{align*}
where $\left( \Delta_1, -N_1 \right)$ accounts for the contribution of terms from the co-efficient of $\L^{-N_1}q^{\Delta_1}$ in $\left(F_{\mb}(\L^{-2r}q)\right)^{-1}$, and $\left( \Delta_2, -N_2 \right)$ accounts for the contribution of terms from the co-efficient of $\L^{-N_2}q^{\Delta_2}$ in $(1-q)\tilde{G}_{r,c - m E}(q)$.
It follows from Lemma \ref{clm11} and Proposition \ref{boundMF} that
\begin{align*}
\Delta_1 \leq N_1 - \frac{(r-\mb)\mb}{2r}\,,\qquad \text{ and } \;\,
\Delta_2 \leq N_2 + C_0
\end{align*}
These inequalities yield
\[ \Delta \leq N + \frac{(2 - 2r)\mb^2 - r\mb}{2r} + C_0 \]
In conclusion, for $\Delta > N + \frac{(2 - 2r)\mb^2 - r\mb}{2r} + C_0 $, the co-efficient of $\L^{-N}q^\Delta$ in $(1-q) G_{r,c}(q)$ is zero.
\end{proof}
\section{\textsc{Bounds for stabilization of Betti numbers}}\label{section6}
In this section, our goal is to determine lower bounds such that the Betti numbers of the moduli space stabilize. More precisely, we look at $\P$ equipped with the ample divisor $H = c_1( \mathcal{O}_{\mathbb{P}^2}(1))$. We assume that $r$ and $a$ are coprime and consider the moduli space $M_{\mathbb{P}^2,H}(r,aH,c_2)$. Since $r$ and $a$ are coprime, all $\mu_H$-semistable sheaves are $\mu_H$-stable. Using Proposition \ref{proposition10} in conjunction with Theorem \ref{clm14}, we derive the lower bounds such that the Betti numbers of $M_{\P,H}(r,aH,c_2)$ stabilize. Lastly, we investigate some examples and show that we can improve this bound further.
\begin{thm}\label{theorem26}
Let $r$ be at least two. Assume that $r$ and $a$ be coprime. There is a constant $C$ depending only on $r$ and $a$ such that if $c_2 \geq N + C$, the $2N$th Betti number of the moduli space $M_{\P,H}(r,aH,c_2)$ stabilize. Moreover, we can take $C = \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right\rfloor$.
\end{thm}
\begin{proof}
Let $\gamma$ denote the Chern class $(r,aH, c_2)$. By our assumption, $r$ and $a$ are coprime, a posteriori, all $\mu_H$-semistable sheaves are $\mu_H$-stable. In this case, we know that $M_{\P,H}(\gamma)$ is a smooth projective variety of dimension $ext^1(\gamma,\gamma)$. We conclude using Remark \ref{remark8} that to show that the $2N$th Betti number stabilize for $c_2 \geq N + C$, it is enough to show that the coefficient of $\L^{-N}q^d$ in the generating function
\[ (1-q) \sum_{c_2 \geq 0}[M_{\P,H}(\gamma)] \L^{-ext^1(\gamma,\gamma)}q^{c_2} \]
is zero for $d > N + C$.
We note that $\chi(\gamma,\gamma) = 1 - ext^1(\gamma,\gamma)$ and $ c_2 = r\Delta + \frac{r-1}{2r} c_1^2$. Proposition \ref{proposition10} yields the following equality in $A$
\[ [M_{\P,H}(r,aH,c_2)] = (\L -1) [\MP (r,aH,c_2)] \]
Thus, we have the following equality of generating functions
\[ (1-q) \sum_{c_2 \geq 0 } [M_{\P,H}(\gamma)]\L^{-ext^1(\gamma,\gamma)} q^{c_2} = q^{\frac{r-1}{2r}a^2} (1 - \L^{-1})(1-q)G_{r,aH}(q) \]
Each term contributing to the coefficient of $\L^{-N}q^d$ in $q^{\frac{r-1}{2r}a^2} (1 - \L^{-1})(1-q)G_{r,aH}(q) $ arises from a pair of equations
\begin{align*}
d &= \frac{r-1}{2r}a^2 + \Delta' \\
-N &= \varepsilon - N'
\end{align*}
where $\varepsilon \in \{-1,0 \}$ accounts for the contribution to the coefficient of $\L^{-N}q^d$ coming from $(1-\L^{-1})$, and $(\Delta',N')$ accounts for the contribution coming from the coefficient of $\L^{-N'}q^{\Delta'}$ in $(1-q)G_{r,aH}(q)$. It follows from Theorem \ref{clm14} that for the coefficient of $\L^{-N'}q^{\Delta'}$ to be nonzero, we must have $\Delta' \leq N' + C_0$ (using $m = 0$). Moreover, it follows from Proposition \ref{boundMF} that we can take $C_0 = \frac{1}{2}(r^2 + 1)$. Consequently, for the coefficient of $\L^{-N}q^d$ in $q^{\frac{r-1}{2r}a^2} (1 - \L^{-1})(1-q)G_{r,aH}(q)$ to be nonzero, we must have
\[ d \leq N + \left\lfloor \frac{r-1}{2r}a^2 + C_0 \right\rfloor \]
\end{proof}
For the remainder of this section, we look at some examples. Yoshioka \cite{yos94}[Page 194] has computed the Betti numbers $b_{2N}(M_{\P,H} (2,-H,c_2))$, where $M_{\P,H}(2, -H, c_2)$ is the moduli space of $\mu_H$-stable sheaves with Chern classes $(2,-H, c_2)$, which we will denote by $\gamma$. We observe from the table in \cite{yos94}[Page 194] that the Betti numbers $b_{2N}(M_{\P,H} (\gamma))$ stabilize when $c_2 \geq N+1$. Since $r=2$ and $a = -1$, we get from Theorem \ref{theorem26} that the Betti numbers stabilize when $c_2 \geq N + 2$. Therefore, we need to improve our lower bound.
\begin{propn}\label{clm15}
If $c_2 \geq N+1$, the $2N$th Betti number of the moduli space $M_{\P,H}(2,-H, c_2)$ stabilize.
\end{propn}
\begin{proof}
Following the proof of Theorem \ref{theorem26}, it is enough to show that when $d > N+1$ ,the coefficient of $\L^{-N}q^d$ in $q^{\frac{1}{4}}(1 - \L^{-1})(1-q)G_{2,-H}(q)$ is zero.
Each term contributing to the coefficient of $\L^{-N}q^d$ in $q^{\frac{1}{4}}(1 - \L^{-1})(1-q)G_{2,-H}(q)$ arises from a pair of equations
\begin{align*}
d &= \frac{1}{4} + \Delta' \\
-N &= \varepsilon - N'
\end{align*}
where $\varepsilon \in \left\lbrace -1,0 \right\rbrace$ accounts for the contribution to the coefficient coming from $(1 - \L^{-1})$, and $(\Delta', N'')$ accounts for the contribution coming from terms in coefficient of $\L^{-N'}q^{\Delta'}$ in $(1-q)G_{2,-H}(q)$.
It follows from Theorem \ref{clm14} that for the co-efficient of $\L^{-N'}q^{\Delta'}$ to be nonzero, we must have $\Delta' \leq N' + C_0$. Consequently, we must have
\begin{equation}
d - \frac{1}{4} = \Delta' \leq N' + C_0 = N + \varepsilon + C_0 \leq N + C_0
\end{equation}
As a result, for $d > N + \left\lfloor \frac{1}{4} + C_0 \right\rfloor$, the coefficient of $\L^{-N}q^d$ in $q^{\frac{1}{4}}(1 - \L^{-1}) (1-q)G_{2,-H}(q)$ must be zero. Therefore, to complete the proof of our Claim, we need to figure out the value of $C_0$.
It follows from the proof of Proposition \ref{boundMF} that to compute $C_0$, we need to compute
\[ \frac{1}{2}\left(r^2 - \sum_{i=1}^l r_i^2 \right) - \frac{1}{2}\kappa \]
where $l=2$, $r=2$, $r_1 = r_2 = 1$, and $\kappa$ is a lower bound for
\[ 2\left( 2\Delta - \Delta_1 - \Delta_2 \right) + \left(c_2 - c_1 \right)\cdot K_{\F} \]
except for the case $l=2$ and $(c_2 - c_1)\cdot F = -1$.
Let $c_1 = a_1 E + b_1 F$ and $c_2 = a_2 E + b_2 F$. Since $c_1 + c_2 = -E-F$, we have $a_1 + a_2 = -1$ and $b_1 + b_2 = -1$. Moreover, we must have $a_2 - a_1 \neq -1$. Using Yoshioka's relation (equation \ref{eqnyos}) yields
\[ \left( 2 \Delta - \Delta_1 - \Delta_2 \right) = - \frac{1}{4}\left( c_1 - c_2 \right)^2
= \frac{1}{4}\left(2a_1 +1\right)^2 - \frac{1}{2}\left( 2a_1 +1\right) \left(2b_1 +1 \right) \]
Since $K_{\F} = -2E -3F$, we see that
\[ \left( c_2 - c_1 \right) \cdot K_{\F} = \left( 2a_1 +1 \right) + 2 \left( 2b_1 +1 \right) \]
Therefore, we have
\[ 2(2 \Delta - \Delta_1 - \Delta_2 ) + (c_2 - c_1)\cdot K_{\F} = 2 a_1^2 + 2 a_1 + 2b_1 - 4a_1 b_1 + \frac{5}{2} \]
Clearly $a_1^2 + a_1 \geq 0$ for all integer values of $a_1$. Thus, we need to find a lower bound for $2b_1(1-2a_1)$.
Recall that as per the definition of $S^{\mu}(\{1,c_1\},\{1,c_2\},F,E+F)$ (see equation \ref{defnSmu}, \ref{eqnSmu}) we have two cases
\begin{enumerate}[$\qquad \,$ A)]
\item $a_1 > -\frac{1}{2}$ and $b_1 \leq - \frac{1}{2}$
\item $a_1 \leq - \frac{1}{2}$ and $b_1 > - \frac{1}{2}$
\end{enumerate}
Since $a_1$ and $b_1$ are integers, in Case A, we see that $a_1 \geq 0$ and $-b_1 \geq 1$. When $a_1 =0$, we must have $a_2 = -1$, whence $a_2 - a_1 = -1$ which is not possible by our assumption. Hence, we must have $a_1 \geq 1$, which yields
\[ 2b_1(1 - 2a_1) = (2a_1 -1) (-2b_1) \geq \left( 2\left( 1\right) -1 \right) \left(2(1)\right) = 2 \]
Similarly, in Case B, we see that $-a_1 \geq 1$ and $b_1 \geq 0$, thereby yielding
\[ 2b_1(1 - 2a_1) \geq \left(2(0)\right) \left( 1 + 2(1)\right) = 0 \]
In either case we see that $2b_1(1 - 2a_1) \geq 0$, and hence we can take $\kappa = \frac{5}{2}$.
Clearly, in our case $r=2$ and $r_1 = r_2 = 1$, whence $\frac{1}{2}\left( r^2 - r_1^2 - r_2^2 \right) = 1$. Following the proof of Proposition \ref{boundMF}, we see that
\[ C_0 = \max \left\lbrace 0, 1 + 1 - \frac{1}{2}\kappa, 1 - \frac{3}{4} +\left(\left\lceil \frac{-1}{2}\right\rceil - \frac{-1}{2}\right) \right\rbrace = 2 - \frac{5}{4} \]
In summary, for the coefficient of $\L^{-N}q^d$ to be nonzero, we must have
\[d \leq N + \frac{1}{4} + 2 - \frac{5}{4} = N+1\]
In conclusion, when $d > N+1$, the coefficient of $\L^{-N}q^d$ in $q^{\frac{1}{4}}(1 - \L^{-1})(1-q)G_{2,-H}(q)$ is zero.
\end{proof}
Manschot \cite{man11}[Table 1], \cite{man}[Table 1] computed the Betti numbers of the moduli space $M_{\P,H}(3,-H,c_2)$ and the virtual Betti numbers of the moduli space $M_{\P,H}(4,2H, c_2)$. We observe from the tables in these papers that the Betti numbers of $M_{\P,H}(3,-H,c_2)$ stabilize when $c_2 \geq N + 2$ and the virtual Betti numbers of $M_{\P,H}(4,2H,c_2)$ stabilize when $c_2 \geq N+ 3$. In the first case, we have $r=3$ and $a = -1$, we get from Theorem \ref{theorem26} that the Betti numbers stabilize when $c_2 \geq N + 5$.
As our second example, we scrutinize the Betti numbers of the moduli space $M_{\P,H}(4,H,c_2)$. In this case, Theorem \ref{theorem26} yields the stabilization of the Betti numbers when $c_2 \geq N + 8$. We improve this bound in the following Proposition.
\begin{propn}\label{clm16}
If $c_2 \geq N + 5$, the $2N$-th Betti number of the moduli space $M_{\P,H}(4,H,c_2)$ stabilize.
\end{propn}
\begin{proof}
Following the proof of Theorem \ref{theorem26}, it is enough to show that when $d > N + 5$, the coefficient of $\L^{-N}q^d$ in $q^{\frac{3}{8}}(1 - \L^{-1})(1-q)G_{4,H}(q)$ is zero.
Each term contributing to the coefficient of $\L^{-N}q^{d}$ in $q^{\frac{3}{8}}(1 - \L^{-1})(1-q)G_{4,H}(q)$ arises from a pair of equations
\begin{align*}
d &= \frac{3}{8} + \Delta' \\
-N &= \varepsilon - N'
\end{align*}
where $\varepsilon \in \lbrace -1,0 \rbrace$ accounts for the contribution to the coefficient coming from $(1 - \L^{-1})$, and $(\Delta',N')$ accounts for the contribution coming from the terms in coefficient of $\L^{-N'}q^{\Delta'}$ in $(1-q)G_{4,H}(q)$.
It follows from Theorem \ref{clm14} that if the co-efficient of $\L^{-N'}q^{\Delta'}$ is non-zero, then we must have $\Delta' \leq N' + C_0$, whence, $d \leq N + \left\lfloor \frac{3}{8} + C_0 \right\rfloor$. Consequently, for $d > N + \left\lfloor \frac{3}{8} + C_0 \right\rfloor$, the coefficient of $\L^{-N}q^{d}$ in $q^{\frac{3}{8}}(1 - \L^{-1})(1-q)G_{4,H}(q)$ must be zero. Therefore, to complete our proof, we need to determine the value of $C_0$.
Adopting the notation used in proof of Proposition \ref{boundMF} and Lemma \ref{lowerbound} in our situation, we get $r = 4$, $a = b = 1$. Recall that $C_0$ is the maximum of the terms $1 + \frac{1}{2}\left( r^2 - \sum_{i=1}^l r_i^2\right) - \frac{1}{2}\kappa$ except the case when $l=2$ and $\mu_F(\gamma_2 ) - \mu_F(\gamma_1) = -1$ and the terms $\frac{r_1 r_2}{2r} + \left( \left\lceil \frac{br_2}{r}\right\rceil - \frac{br_2}{r}\right)$ for $r_1 + r_2 = r$, where $r = \sum_{i=1}^l r_i$, $a = \sum_{i=1}^l r_i a_i$, $s_i = \sum_{j=i}^l b_j$, $b=s_1$, and $ \kappa$ is lower bound for $S_1 + S_2$, where
\[ S_1 = (r-1)\sum_{i=1}^l r_i a_i^2 - \frac{r-1}{r}a^2 + \sum_{i=1}^l a_i r_i \left( \sum_{j=i+1}^l r_j - \sum_{j=1}^{i-1} r_j \right) \]
and
\[ S_1 = 2 \sum_{i=2}^l ( (r-1)(a_i - a_{i-1}) + r_i + r_{i-1}) \left( \frac{b}{r}\sum_{j=i}^l r_j - s_i \right) \]
When $l=2$ and $(r_1,r_2) = (3,1)$, we see that $S_1 \geq - \frac{3}{4}$ with equality occurring at $(a_1,a_2) = (0,1)$. At the point $(0,1)$ we get $S_2 \geq \frac{7}{2}$, and hence, $S_1 + S_2 \geq \frac{11}{4}$. Since there are no other points $(a_1,a_2)$ satisfying $3a_1 + a_2 = 1$ at which $S_1 < \frac{11}{4}$, we can take $\kappa = \frac{11}{4}$, and we get $1 + \frac{1}{2}\left( r^2 - r_1^2 - r_2^2 \right) - \frac{1}{2} \kappa = \frac{21}{8}$.
When $l=2$ and $(r_1,r_2) = (1,3)$, we see that $S_1 \geq \frac{21}{4}$ with equality occurring at $(a_1,a_2) = (1,0)$, and $S_2 \geq 1$. Thus, we can take $\kappa = \frac{25}{4}$, and we get $1 + \frac{1}{2}\left( r^2 - r_1^2 - r_2^2 \right) - \frac{1}{2} \kappa = \frac{7}{8}$.
When $l=2$ and $(r_1,r_2) = (2,2)$, there is no integer solution for $2 a_1 + 2a_2 = 1$. Thus, we ignore this case.
When $l=3$ and $(r_1,r_2,r_3) = (2,1,1)$, we see that $S_1 \geq - \frac{3}{4}$ with equality occurring at $(a_1,a_2,a_3) = (0,0,1)$. At this point we get $S_2 \geq 4$, whence $S_1 + S_2 \geq \frac{5}{2}$. The only other point $(a_1,a_2,a_3)$ with $S_1 \leq \frac{5}{2}$ is $(0,1,0)$ at which $S_1 = \frac{5}{4}$ and $S_2 \geq \frac{9}{2}$, and thus $S_1 + S_2 \geq \frac{23}{4}$. Therefore, we can take $\kappa = \frac{5}{2}$, and we get $1 + \frac{1}{2}\left( r^2 - r_1^2 - r_2^2 - r_3^2 \right) - \frac{1}{2} \kappa = \frac{19}{4}$.
When $l=3$ and $(r_1,r_2,r_3) = (1,2,1)$, we see that $S_1 \geq - \frac{3}{4}$ with equality occurring at $(0,0,1)$. At this point, we see that $S_2 \geq 6$, whence $S_1 + S_2 \geq \frac{21}{4}$. At every other point $(a_1,a_2,a_3)$ with $a_1 + 2a_2 + a_3 = 1$, we have $S_1 \geq \frac{21}{4}$. As a consequence, we can take $\kappa = \frac{21}{4}$, and we get $1 + \frac{1}{2}\left( r^2 - r_1^2 - r_2^2 - r_3^2 \right) - \frac{1}{2} \kappa = \frac{27}{8}$.
When $l =3$ and $(r_1,r_2,r_3) = (1,1,2)$, we see that $S_1 \geq \frac{5}{4}$ with equality occurring at $(a_1,a_2,a_3) = (-1,0,1)$. At this point, we see that $S_2 \geq 6$, and thus $S_1 + S_2 \geq \frac{29}{4}$. The other points $(a_1,a_2,a_3)$ satisfying $a_1 + a_2 + 2a_3 = 1$ at which $S_1 \leq \frac{29}{4}$ are $(0,1,0), \, (0,-1,1),\, (1,0,0)$. Analyzing $S_1$ and $S_2$ at these points, we see that $S_1 + S_2 $ may attain the least possible value $\frac{25}{4}$. Thus, we take $\kappa = \frac{25}{4}$, and we see that $1 + \frac{1}{2}\left( r^2 - r_1^2 - r_2^2 - r_3^2 \right) - \frac{1}{2} \kappa = \frac{23}{8}$.
When $l=4$ and $(r_1,r_2,r_3,r_4) = (1,1,1,1)$, we see that $S_1 \geq - \frac{3}{4}$ with equality occurring at $(a_1,a_2,a_3,a_4) = (0,0,0,1)$. At this point, we see that $S_2 \geq 6$, and thus $S_1 + S_2 \geq \frac{21}{4}$. The other points $(a_1,a_2,a_3,a_4)$ with $a_1 + a_2 + a_3 + a_4 = 1$ at which $S_1 \leq \frac{21}{4}$ are $(-1,0,0,2)$, $(0,-1,1,1)$, $(-1,1,1,0)$, $(-1,1,0,1)$, $(-1,0,1,1)$, $(1,0,0,0)$, $(0,1,0,0)$, and $(0,0,1,0)$. However, we see that at each of these points we have $S_1 + S_1 \geq \frac{21}{4}$. Hence, we can take $\kappa = \frac{21}{4}$, and we get $1 + \frac{1}{2}\left( r^2 - r_1^2 - r_2^2 - r_3^2 - r_4^2 \right) - \frac{1}{2} \kappa =\frac{35}{8}$.
Finally, since $b=1$, $r = 4$, and $1 \leq r_2 \leq 3$, we see that $\frac{(r-r_2)r_2}{2r} + 1 - \frac{r_2}{r}$ attains maximum value of $\frac{9}{8}$ at $r_2 = 1$.
In conclusion, we can take $C_0 = \frac{19}{4}$, and we get that when $d > N + 5$ the coefficient of $\L^{-N}q^d$ in $q^{\frac{3}{8}}(1 - \L^{-1})(1-q)G_{4,H}(q)$ is zero.
\end{proof}
In summary, as we see in our examples (Proposition \ref{clm15}, \ref{clm16}), the constant $C_0$ in Proposition \ref{boundMF} can be improved further, which will lead to better bounds for the stabilization of Betti numbers in Theorem \ref{theorem26}.
{\small
\renewcommand{\refname}{\textsc{References}}
|
2110.07205
|
\section{Introduction}
Starting with ELMo \cite{peters2018deep} and BERT \cite{devlin2018bert}, substantial work has shown that pre-trained models can bring significant improvements in various tasks, including natural language processing (NLP), image recognition, and speech processing \cite{radford2019language,lample2019cross,yang2019xlnet,dong2019unified,lewis2019bart,bao2021beit,baevski2020wav2vec}.
It is becoming a new principle to solve problems by pre-training a shared model with self-supervision tasks on a large amount of unlabeled data to learn universal representations.
Particularly, ``Text-To-Text Transfer Transformer'' (T5) \cite{raffel2019exploring} leverages a unified text-to-text framework and achieves state-of-the-art results on a wide variety of NLP tasks, including machine translation, question answering, sentiment classification, and so on. The basic idea of T5 is to treat every NLP problem as a ``text-to-text'' problem, and employ transfer learning to boost the performance of downstream tasks.
\begin{figure}[t]
\centering
\includegraphics[width=7.5cm]{speecht5.png}
\caption{An illustration of the proposed SpeechT5 framework, which treats all spoken language processing tasks as a speech/text to speech/text format, including voice conversion (VC), automatic speech recognition (ASR), text to speech (TTS), grapheme to phoneme (G2P), and so on.
}\label{speecht5}
\end{figure}
Within the same period, self-supervised speech representation learning has also been investigated and shown promising results, benefiting from richly learned representations \cite{chung2018speech2vec,chuang2019speechbert,song2019speech,baevski2020wav2vec,wang2021unispeech,hsu2021hubert,chung2021w2v}.
A prominent line of work has been proposed to improve acoustic encoder with speech pre-training, such as Wav2vec2 \cite{baevski2020wav2vec}, APC \cite{chung2020generative}, MPC \cite{jiang2021further}, and Hubert \cite{hsu2021hubert}. Another category of methods attempts to enhance a few spoken language understanding tasks by utilizing the speech-language pre-training \cite{chung2020splat,kim2021st,qian2021speech}.
However, most of these models rely on an encoder-only model similar to BERT, and have task-specific model architectures for different tasks.
How to design a unified encoder-decoder model which can take advantage of unlabeled speech and text data to improve various spoken language processing is not well explored.
Inspired by the T5 method, we attempt to convert each spoken language processing task into a speech/text to speech/text problem via an encoder-decoder framework, e.g., automatic speech recognition (ASR), text-to-speech (TTS), voice conversion (VC), and speaker identification (SID), as shown in Figure \ref{speecht5}, which enables us to use the same pre-trained model across diverse tasks.
To achieve this, we propose SpeechT5, a unified-modal encoder-decoder pre-training method for spoken language processing tasks.
The proposed SpeechT5 contains an encoder-decoder backbone network and modal-specific pre-post networks.
With the pre-nets, the input speech/text is embedded in a shared vector space, and the encoder-decoder backbone network models the sequence to sequence conversion, from which the model-specific post-nets generate the speech/text output.
SpeechT5 is pre-trained with a denoising sequence-to-sequence method by leveraging large-scale unlabeled text and speech corpus.
To align the textual and acoustic information into a unified semantic space, the proposed SpeechT5 model (1) maps text and speech representations into a shared vector quantization space, and (2) randomly mixes up the quantized latent representations and the contextual representations, which can explicitly guide the quantizer to learn the cross-modal information.
We fine-tune SpeechT5 on a wide variety of downstream spoken language processing tasks, including VC, ASR, TTS, and SID.
Extensive results show that the proposed SpeechT5 model achieves a significant improvement on these spoken language processing tasks when compared with strong baselines.
Specifically, the proposed SpeechT5 method performs better than the state-of-the-art voice transformer network (VTN) \cite{huang2021pretraining} on the VC task, and achieves the state-of-the-art result of 90.97\%. It also outperforms SpeechNet \cite{chen2021speechnet} and pre-trained models such as SUPERB \cite{yang2021superb} on the SID task.
Besides, SpeechT5 also obtains a gain of about 10.0\% and 6.5\% than the encoder-decoder based ASR model (i.e., Fairseq \cite{ott2019fairseq}) and some baseline model, respectively, on the ASR task, and obtains significant improvements over the strong Transformer TTS model \cite{li2019neural} by 13.4\% and 5.8\% in terms of the word error rate and mean option score on the TTS task.
The contributions of this paper are summarized as follows.
\begin{itemize}
\item To the best of our knowledge, this is the first work to investigate a unified encoder-decoder framework for various spoken language processing tasks.
\item We propose a cross-modal joint pre-training method, which learns potential alignment between acoustic and textual representation with large-scale unlabeled speech and text data.
\item Extensive experiments on spoken language processing tasks demonstrate the effectiveness and superiority of the proposed SpeechT5 method.
\end{itemize}
\begin{figure*}[!ht]
\centering
\includegraphics[width=13cm]{architecture.pdf}
\caption{Model architecture of SpeechT5, which contains an encoder-decoder module and six modal-specific pre/post networks. Most spoken processing tasks can be done by concatenating the encoder-decoder module and corresponding pre-net and post-net.
}\label{architecture}
\end{figure*}
\section{SpeechT5}
In this section, we introduce the proposed SpeechT5 method, a unified-modal framework for learning joint contextual representations of speech and text based on a shared encoder-decoder model.
It aims to derive generic representations
for spoken and natural language via pre-training on unlabeled speech and text.
In the following, We will first introduce the overall architecture of SpeechT5 and the details of individual components (i.e., Section \ref{sec_architecture}), and then present the pre-training method (i.e., Section \ref{sec_pretraining}) and the fine-tuning method (i.e., Section \ref{sec_finetuning}) for spoken language processing tasks.
\subsection{Model Architecture}
\label{sec_architecture}
All spoken language processing tasks take speech or text as the input or output.
Figure \ref{architecture} shows the model architecture of the proposed SpeechT5 model, which consists of an encoder-decoder module and six modal-specific pre/post networks.
The pre-nets convert the input speech/text to a unified space, and the shared encoder-decoder network models the sequence to sequence conversion. Then based on the output of the decoder, the post-nets generate the output in the speech/text modality.
\paragraph{Input/Output Representations}
To train a single model on a diverse set of spoken language processing tasks, we cast all the tasks we consider into a speech/text to speech/text format, where the input/output is a speech sequence or text sequence.
The 80-dimension log-Mel filterbank feature extracted from each frame with the librosa tool{\footnote[1]{https://librosa.org/}} is treated as a token.
If the output is in the speech modality, we employ the Vocoder \cite{kong2020hifi} to transform the log-Mel filterbank feature into a waveform.
For text, we split the text into a sequence of tokens by using a unigram language model \cite{kudo2018subword}.
\paragraph{Encoder-Decoder} This model is similar to the Transformer \cite{vaswani2017attention}. The encoder consists of a stack of blocks, each of which comprises two subcomponents: a self-attention layer, followed by a small feed-forward network.
Layer normalization \cite{ba2016layer} and residual connection \cite{he2016deep} are applied to the input of each subcomponent.
The decoder has a similar architecture to the encoder except that it includes a cross-attention mechanism after each self-attention layer that attends to the output of the encoder.
Besides, the self-attention mechanism in the decoder also uses a form of autoregressive or causal self-attention, which only allows the model to attend to past outputs.
We use simple relative position embedding \cite{shaw2018self} to enhance the model capabilities, in which we only add the relative position embedding to the dot-product weights of the self-attention.
\paragraph{Speech Pre/Post Net}
There are some differences between the speech-encoder pre-net and speech-decoder pre-net.
In the speech-encoder pre-net, we apply two convolutional layers (via convolution strides) to downsample them and process local relationships.
In the speech-decoder pre-net, the log-Mel filterbank is fed into a neural network composed of three fully connected layers with the ReLU activation.
To support multi-speaker TTS and VC, the speaker embedding, which is extracted by the public x-vector \cite{Snyder2018}, is concatenated with the output of the speech-decoder pre-net.
Then it is processed by a linear layer with the ReLU activation.
For the speech-decoder post network, we use two different linear projections to predict the log-Mel filterbank and the stop token, respectively, and use 5-layer 1-dimensional convolutional layers
to produce a residual to refine the reconstruction of the log-Mel filterbank.
\paragraph{Text Pre/Post Net}
We use shared embeddings as the text encoder pre-net and decoder pre/post networks.
The pre-net transforms the token index into an embedding vector, and the post-net transforms the hidden state into the probability of token distribution, which is normalized by the softmax function.
There is a shift in the input text of the decoder for the auto-regressive generation.
During the inference, the decoder uses its own past predictions to predict the next token.
\subsection{Pre-training}
\label{sec_pretraining}
With large scale collections of unlabeled speech and text corpus, we can pre-train the unified-modal model separately, and further align the textual and acoustic information into a unified semantic space by a joint pre-training method.
\paragraph{Speech Learning}
The goal of speech learning is to leverage unlabeled speech data to learn general speech representations for both speech understanding and generation tasks.
To this end, the SpeechT5 model is trained as a unified encoder-decoder model with two types of spoken language modeling tasks: bidirectional masked prediction and sequence-to-sequence generation.
Formally, the input to the speech module is a sequence of 80-dimensional log-Mel filterbank $X=(x_1,...,x_n)$.
The speech module, which consists of a speech-encoder pre-net and a Transformer encoder, produces hidden representations $S=(s_1,...,s_n)$.
Similar to the masked language modeling in BERT, we follow Hubert \cite{hsu2021hubert} to use the acoustic unit discovery model to provide frame-level targets{\footnote[2]{The target labels are generated by clustering the MFCC feature with the $k$-means clustering method.}} $Z=(z_1,...,z_n)$ for the output of the Transformer encoder.
Specifically, we use span mask strategies, where $p$\% of timesteps are randomly selected as start indices, and spans of $l$ steps are masked. The cross-entropy computed over masked timesteps is defined as
\begin{equation}
\begin{aligned}
\mathcal{L}_{mlm}^{s} = \sum_{t\in M} \log p_f (z_t|\hat{X}, t),
\end{aligned}
\end{equation}
where $\hat{X}$ denotes the corrupted version of $X$, and $M$ denotes the set of indices to be masked for the sequence $X$.
Furthermore, we propose to reconstruct the original speech through the speech-decoder pre-net, Transformer decoder, and speech-decoder post-net.
The decoder is autoregressivein that the output of the encoder $S=(s_1,...,s_n)$ and the previously generated features $y_{1:t-1}$ are considered when decoding current output $y_t$.
Inspired by the success of modern seq2seq TTS models \cite{li2019neural}, we enforce the corresponding output $y_t$ to be close to the original frame $x_t$ by minimizing their $L1$-distance as
\begin{equation}
\begin{aligned}
\mathcal{L}_1^{s} &= \sum_{i=1}^n ||y_i - x_i||_1.
\end{aligned}
\end{equation}
Besides, we also use a binary cross-entropy (BCE) loss $\mathcal{L}_{bce}^{s}$ for the stop token. To address the imbalance problem between stop tokens and normal tokens, we impose a positive weight on the tail positive stop token when calculating the BCE loss.
\paragraph{Text Learning}
The language module aims to offer contextual understanding and generation.
Overall, with unlabeled text data, SpeechT5 is trained by (1) corrupting text with an arbitrary noising function with a masked span, and (2) learning to produce a corresponding target that can reconstruct the original text or masked text.
For unsupervised objectives, we can use BART-style \cite{devlin2018bert} or T5-style \cite{raffel2019exploring} mask strategies and target sequences.
In the BART-style, the model aims to reconstruct the original text from the noisy source text.
However, in the T5-style, all selected fragments are removed from the text and concatenated as the target sequence, while the remaining parts are concatenated as the source sequence.
Formally, this model is trained to generate the target sequence $Y=(y_1,...,y_m)$ auto-regressively condition on the source sequence $X=(x_1,...,x_n)$ with the maximum likelihood estimation as
\begin{equation}
\mathcal{L}_{mle}^{t} = \sum_{t =1}^m \log p_f (y_t|y_{1:t-1},X),
\end{equation}
where the target sequence $Y$ can be the original text or all masked text fragments.
\begin{figure}[t]
\centering
\includegraphics[width=7.5cm]{codebook.png}
\caption{The joint pre-training approach. The shared quantization module over encoder representations produces cross-modality quantized latent speech/text units. Hidden states and latent units are mixed up and used as the inputs of the cross-attention module in the decoder. The model learns to share discrete tokens across modalities, creating bridges between speech and text.
}\label{codebook}
\end{figure}
\paragraph{Joint Pre-training}
The above pre-training methods can only leverage speech data or text data, individually, to model the acoustic information or language information.
However, some spoken language tasks need to build a cross-modality mapping between speech and text, such as ASR and TTS.
The alignment learning between the speech and text in pre-training will be beneficial to downstream tasks.
With this motivation, we propose a unified-modal pre-training method to learn representations that capture modality invariant information with discrete vector quantization.
Specifically, our goal is to utilize vector quantized embeddings as a bridge between speech and text as well as align speech representation and text representation through a shared codebook, as shown in Figure \ref{codebook}.
Inspired by VQ-VAE \cite{oord2017neural} and SemFace \cite{ren2021semface}, we first use the quantizer to turn these dense speech or text representations $s_i$ of the encoder output, into discrete representation $c_i$ from a fixed size codebook ${C}^K$ which contains $K$ learnable embeddings.
Formally, the nearest neighbor search is performed between the encoder output and the embedding of the latent code using the $L2$-distance metric as
\begin{equation}
c_i = \arg\min_{j\in[K]} ||s_i-c_j||_2,
\end{equation}
where $c_j$ is $j$-th quantized vector in the codebook.
Note that we do the same operation for the encoder output of speech and text with a shared codebook.
Then, we randomly replace a proportion of the contextual representations with quantized latent representations in the corresponding time steps, and calculate the cross-attention upon the mixed representations, which can explicitly guide the quantizer to utilize cross-modal information.
To encourage sharing more codebook, the diversity loss is used by maximizing the entropy of the averaged softmax distribution as
\begin{equation}
\mathcal{L}_d= \frac{1}{K} \sum_{k=1}^{K} p_k \log p_k,
\end{equation}
where $p_k$ is the averaged probability of choosing the $k$-th code in the codebook.
The final pre-training loss with unlabeled speech and text data can be formulated as
\begin{equation}
\mathcal{L} = \mathcal{L}_{mlm}^{s} + \mathcal{L}_{1}^{s} + \mathcal{L}_{bce}^{s} + \mathcal{L}_{mle}^{t} + \mathcal{L}_{d}.
\end{equation}
\subsection{Fine-tuning}
\label{sec_finetuning}
After pre-training, we fine-tune the encoder-decoder model with the corresponding loss of downstream tasks.
Our goal here is to measure general spoken learning abilities. As such, we study downstream performance on a diverse set of benchmarks, including ASR, TTS, VC, and SID.
Four speech processing tasks that we consider can be done by concatenating the encoder-decoder model and corresponding pre-net and post-net. For example, the speech-encoder pre-net, encoder-decoder, text-decoder pre-net, and text-decoder post-net can constitute the ASR model, and the training loss is the maximum cross-entropy loss.
\section{Experiments}
\subsection{Dataset and Evaluation Metrics}
For unsupervised speech pre-training, we use the full 960 hours of LibriSpeech audio \cite{panayotov2015librispeech}, which is derived from the LibriVox project that contains English recordings of copyright-free audiobooks by volunteers from the Internet.
For unsupervised text pre-training, we use the normalized language model training text of LibriSpeech as unlabeled data, which contains 400M sentences.{\footnote[3]{\url{https://www.openslr.org/11/}}}
In supervised fine-tuning, we use the commonly adopted dataset and evaluation metric for each task.
We train the ASR model with LibriSpeech training data,
and measure the performance of ASR by the word error rate (WER) on the standard Librispech
dev-other/clean and test-clean/other sets. A language model is trained by the same text data for pre-training, which is used for shallow fusion \cite{gulcehre2015using} during ASR inference.
For TTS, we finetune the pretrained model on the 460-hours LibriTTS clean sets \cite{zen2019libritts}, which is a multispeaker English corpus of read English speech from the audiobooks of the LibriVox project.
We trim the waveform as ESPnet recipe \cite{watanabe2018espnet}.
We evaluate the WER using a open-source ASR model wav2vec 2.0 CTC{\footnote[4]{\url{https://github.com/pytorch/fairseq/tree/main/examples/wav2vec}}} on all test set.
Moreover, we randomly select 200 fixed examples with various lengths (no overlap with training set) from our internal dataset as the evaluation set to evaluate the mean option score (MOS).
\begin{table*}[!htp]
\begin{center}
\begin{tabular}{l|cc|cc}
\toprule
\multirow{2}{*}{Model} & \multicolumn{2}{|c|}{WER} & \multicolumn{2}{|c}{MCD} \\
& BDL to SLT & CLB to SLT & BDL to SLT & CLB to SLT \\
\midrule
VTN w/ ASR \cite{huang2021pretraining} &11.1\% & 10.9\% & 6.50 & 6.11 \\
VTN w/ TTS \cite{huang2021pretraining} & \textbf{7.6\%} & 9.1\% & 6.33 & 6.02 \\
\midrule
Baseline & 21.5\% & 10.8\% & 6.26 & 6.16 \\
SpeechT5 & 10.1\% & \textbf{7.1\%} & \textbf{6.06} & \textbf{5.95} \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{exp_vc} Results of VC (speech to speech). BDL, CLB, and SLT mean three different speakers. VTN \cite{huang2021pretraining} is the state-of-the-art voice Transformer network model, which is fine-tuned from the pretrained ASR or TTS model.}
\end{table*}
For VC, we use CMU Arctic \cite{kominek2004cmu} corpus, which consists of speech recordings of four speakers, such as clb (female), bdl (male), slt (female), and rms (male), reading the same 1,132 phonetically balanced English utterances. We consider a many-to-many setting and use all speakers for training and evaluation. Thus, there are twelve different combinations of source and target speakers. For each speaker, the first 932, the last 100, and the rest 100 sentences of the 1,132 sentences are used for training, test, and validation, respectively.
The average of MCD (Mel-Cepstral Distortion) token along the DTW (dynamic time warping) path between the output and ground-truth mel-cepstra serves as the evaluation metric of VC. The smaller MCD indicates better performance.
Besides, we also use WER to evaluate the quality of generated voice with a public ASR model Hubert Large{\footnote[5]{\url{https://huggingface.co/espnet}}}, since the WER of the test set with this ASR model is comparable to that of VTN \cite{huang2021pretraining}.
For SID, VoxCeleb1 \cite{nagrani2017voxceleb} is adopted in our experiments, which contains over 100,000 speech records uttered by 1,251 celebrities extracted from videos uploaded to YouTube.
We use the official split of VoxCeleb1 for the speaker identification task, where the test set contains 8,251 utterances from these 1,251 celebrities.
The capability of identifying speakers is assessed by classifying an utterance into the ground-truth category.
The top-1 speaker classification accuracy is used as the evaluation metric of SID.
\subsection{Implementation Details}
\paragraph{Pre-training} All models in this paper are implemented in Fairseq{\footnote[6]{\url{https://github.com/pytorch/fairseq}}} \cite{ott2019fairseq}.
The encoder-decoder model contains 12 Transformer encoder blocks and 6 Transformer decoder blocks, where the model dimension is 768, inner dimension (FFN) is 3,072 and the number of attention heads is 12.
Speech-encoder pre-net is two 1-dimensional convolutional layers with strides [2,2], kernel size [5,5] and channel size [1024,1536], where each layer is followed by a gated linear unit \cite{yann2017glu}.
For speech-decoder pre-net and post-net, we use the same setting as the pre-net and post-net in \cite{shen2018tacotron2}, except the channel size of post-net is 256.
For text-encoder/decoder pre/post-net, a shared embedding table with dimension 768 is used.
For the quantization module, we use G=2 codebooks with V=100 entries for the shared codebook module, resulting in a theoretical maximum of 10k codewords.
The speech feature is a sequence of 80-dim log-Mel filterbank with 64 millisecond (ms) window, and 16 ms frame shift. It is normalized with utterance-level mean and variance when used as input data.
For the text data, we combine the text of LibriSpeech \cite{panayotov2015librispeech} and LibriTTS \cite{zen2019libritts} to get 10k unigram vocabulary \cite{kudo2018subword} and segment text by using SentencePiece{\footnote[7]{\url{https://github.com/google/sentencepiece}}}.
We optimize with Adam \cite{kingma2014adam}, warming up the learning rate for the first 10\% of updates.
We pretrain our model on 32 GPUs with 32 GB memory and set the update frequency to 4 for 40k steps.
\begin{table*}[!h]
\begin{center}
\begin{tabular}{l|l|l|cccc}
\toprule
Models & JD & LM & dev-clean & dev-other & test-clean & test-other \\
\midrule
Fairseq \cite{ott2019fairseq} & w/o & w/o & 3.00 & 7.50 & 3.20 & 7.50 \\
Espnet \cite{watanabe2018espnet} & w/o & w/o & 3.10 & 7.60 & 3.40 & 8.00 \\
\midrule
\multirow{3}{*}{Baseline} & w/o & w/o & 2.76 & 7.16 & 3.08 & 7.25 \\
& w/ & w/o & 2.60 & 7.15 & 2.88 & 7.06 \\
& w/ & w/ & 2.18 & 5.85 & 2.44 & 5.95 \\
\hdashline
\multirow{3}{*}{SpeechT5} & w/o & w/o & 2.74 & 6.94 & 2.88 & 7.07 \\
& w/ & w/o & 2.51 & 6.79 & 2.73 & 6.76 \\
& w/ & w/ & \textbf{2.15} & \textbf{5.63} & \textbf{2.32} & \textbf{5.73} \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{exp_asr} Results of ASR (speech to text). Fairseq \cite{ott2019fairseq} and Espnet \cite{watanabe2018espnet} are two open-source Transformer based encoder-decoder ASR models. JD and LM mean joint decoding and language model, respectively.}
\end{table*}
\paragraph{Fine-tuning and Inference} After pre-training, we fine-tune the learned representations on labeled data of downstream tasks.
The speech and text data are preprocessed in the same way as pre-training.
All fine-tuning experiments are conducted on 8 GPUs.
For ASR, we add an extra linear layer to calculate the CTC loss at the top of the encoder \cite{shinji2017hybrid}.
The loss weight is $0.3$ for CTC, and $0.7$ for cross-entropy.
We train our models for 200k steps with a batch size of up to 60000 tokens per GPU and a learning rate of 0.001.
We train a language model for ASR inference, which contains 12 blocks of transformer encoder and set the model dimension to 1024, inner dimension to 4096, and attention heads to 16.
During inference, the beam size is set to 5 for all experiments and we can also apply the joint CTC and decoder inference \cite{hori-etal-2017-joint} and language model (LM) to further improve the performance.
For TTS, we add an additional attention loss \cite{tachibana2018efficiently} to speed up model convergence besides $L1$ loss and BCE loss.
The model is updated for 300k steps with a learning rate of 0.0004, while each GPU processes up to 45000 tokens for a batch.
We utilize HiFi-GAN \cite{kong2020hifi} to produce the raw waveforms, which are capable of both efficient and high-fidelity TTS.
We train it in a speaker-independent manner using the training data of LibriTTS.
For VC, we apply the loss function as used in the fine-tuning of TTS. The model is trained by the Adam optimizer with a batch size of 20000 tokens per GPU. We assign the learning rate based on inverse square root with the maximum learning rate of $10^{-4}$ within 60k steps and apply 6k warm-up steps. For the waveform synthesis module, we use Parallel WaveGAN \cite{Yamamoto2020}, which is a non-autoregressive variant of the WaveNet vocoder. We train it in a speaker-dependent manner by conditioning on our acoustic features using the same training split.
For SID, we use cross-entropy loss and fine-tune all models by the Adam optimizer with a batch size of 256 segments per GPU and the inputs of 150 frames. We assign the learning rate based on one cycle of a triangular cyclical schedule between $10^{-8}$ and $5\times10^{-4}$ in 60k steps. During inference, the whole utterance inputs to the trained model to obtain the probability distribution among the known categories, where the maximum probability indicates the predicted speaker.
\subsection{Main Results}
\begin{table}[t]
\begin{center}
\begin{tabular}{l|cc}
\toprule
Model & WER & MOS \\
\midrule
Ground Truth & 2.17 & 3.86 $\pm$ 0.03 \\
Baseline & 1.72 & 3.45 $\pm$ 0.04 \\
SpeechT5 & \textbf{1.49} & \textbf{3.65} $\pm$ 0.04 \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{exp_tts} Results of TTS (text to speech). WER is evaluated on all test data, and MOS is evaluated on 200 randomly selected fixed sentences.}
\end{table}
\paragraph{VC}
We show the results of VC in Table \ref{exp_vc}.
In this table, we list the conversion from speaker BDL to SLT and speaker CLB to SLT, which are mainly reported in previous work.
The experimental results demonstrate that the proposed SpeechT5 achieves a significant improvement than baseline model.
Besides, our SpeechT5 outperforms the state-of-the-art VTN model in terms of MCD, where VTN \cite{huang2021pretraining} is a Transformer-based seq2seq VC model using pretrained ASR or TTS model parameters.
\begin{table*}[!ht]
\begin{center}
\begin{tabular}{l|l|c}
\toprule
Framework & Model & Top-1 ACC \\
\midrule
\multirow{3}{*}{SUPERB \cite{yang2021superb}}& wav2vec 2.0 Base \cite{baevski2020wav2vec} & 75.18\% \\
& HuBERT Base \cite{hsu2021hubert} & 81.42\% \\
& HuBERT Large \cite{hsu2021hubert} & 90.33\% \\
\midrule
\multirow{2}{*}{SpeechNet \cite{chen2021speechnet} } & Single Task & 86.00\% \\
& Multi-Task with TTS & 87.90\% \\
\midrule
\multirow{2}{*}{Ours} & Baseline & 87.72\% \\
& SpeechT5 & \textbf{90.97}\% \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{exp_sid} Results of SID (speech to class). SUPERB \cite{yang2021superb} is a leaderboard to benchmark the performance of a pre-trained model with minimal architecture changes and labeled data. SpeechNet \cite{chen2021speechnet} is a universal speech model with multi-task learning framework. }
\end{table*}
\paragraph{ASR} The performance of ASR are reported in Table \ref{exp_asr}. We also list the results of Transformer based encoder-decoder model from Fairseq \cite{ott2019fairseq} and Espnet \cite{watanabe2018espnet}.
As can be seen from the table, our baseline is much stronger than previous models.
Furthermore, the proposed SpeechT5 without LM achieves 3.5\%, 6.2\%, 5.2\%, and 4.2\% relative WER reduction with respect to baseline with same setting on dev-clean, dev-other, test-clean, and test-other, respectively, which demonstrates the effectiveness of our pre-training methods.
\paragraph{TTS}
Table \ref{exp_tts} shows the experimental results of TTS.
Our proposed SpeechT5 achieves the performance of 1.49\% WER and 3.65 MOS, getting a relative reduction of 13.37\% in WER and an gain of 0.2 in MOS with respect to the baseline model, respectively. It suggest the proposed pre-training technique achieves significant improvement.
\paragraph{SID}
The results of SID are shown in Table \ref{exp_sid}.
We also list the scores reported in SUPERB \cite{yang2021superb} and SpeechNet \cite{hsu2021hubert}.
In their leaderboard, wav2vec 2.0 \cite{baevski2020wav2vec} and Hubert \cite{hsu2021hubert} are two state-of-the-art pre-trained models.
Experimental results demonstrate that our Speech-T5 significantly outperforms strong baseline and previous work, and achieves the state-of-the-art performance of 90.97\% accuracy in SID task.
\section{Related Work}
Large-scale pre-training has drawn much attention in both the communities of NLP and speech, due to its strong capability of generalization and efficient usage of large-scale data.
Recent pre-trained models in NLP, such as BERT \cite{devlin2018bert}, RoBERTa \cite{liu2019roberta}, XLNet \cite{yang2019xlnet} and BART \cite{lewis2019bart}, have achieved the state-of-the-art performance on language understanding and generation tasks.
In spoken language processing, pre-trained speech models have also been applied to ASR \cite{hsu2021hubert,baevski2020wav2vec}, TTS \cite{hayashi2019pre}, speech translation \cite{li2020multilingual}, VC \cite{huang2021pretraining}, and so on.
However, the above-mentioned research effects gear towards single-modal learning, hence can only be used in either text or speech modeling.
Although some speech-language pre-training work \cite{chung2020splat,kim2021st,qian2021speech} attempts to improve spoken language understanding tasks, e.g., intent detection, dialog act classification, and spoken sentiment analysis, these methods can not be used for spoken generation tasks such as TTS or text generation.
We consider our work most related to T5 \cite{raffel2019exploring}. The core idea of the T5 model, a unified framework for a variety of text-based language problems, is to treat every text processing problem as a ``text-to-text" problem, i.e., taking the text as input and producing new text as output.
Unlike T5, SpeechT5 is a cross-modal encoder-decoder framework, whose input and output are speech or text through different pre/post networks.
Besides, we propose a new joint speech-text pre-training method to leverage
large-scale unlabeled text and speech dataset and align the textual and phonetic information.
SpeechT5 is also related to Speech Chain \cite{tjandra2020machine}, which leverages ASR model and TTS model to build a closed-loop machine speech chain and allows us to train model on the concatenation of both labeled and unlabeled data,
and SpeechNet \cite{chen2021speechnet}, which designs a universal modularized model to perform multiple speech processing tasks with multi-task learning.
SpeechNet shows that it can simultaneously learn several common and important speech processing tasks.
However, there are two big differences between SpeechNet and our SpeechT5. First, SpeechNet has different encoder and decoder for different modalities (e.g., speech and text), but SpeechT5 only uses one shared encoder-decoder model for all tasks. Second, SpeechNet aims to verify the multi-task learning in several speech tasks, but our SpeechT5 attempts to pre-train and improve the universal model with large-scale unlabeled text and speech data.
Another related work is SUPERB \cite{yang2021superb}, a benchmark to examine the capability of pre-trained models.
SUPERB collects various tasks with limited labeled data from speech communities to align with common research interests.
This paper focus on investigating a simple framework solving SUPERB tasks with a frozen, shared pretrained model, and lightweight prediction modules finetuned for each task.
In contrast, the goal of SpeechT5 is to achieve all speech tasks by fine-tuning a unified-modal encoder-decoder model which is pre-trained on unlabeled speech and text corpus.
\section{Conclusion and Future Work}
In this paper, we have presented SpeechT5 as a pre-trained encoder-decoder model for various spoken language tasks.
We convert all spoken language processing tasks into a speech/text to speech/text format, and propose a novel joint pre-training method to utilize cross-modal information by leveraging the unlabeled speech and text data.
Our unified model can support both spoken language understanding and generation tasks, such as speaker identification and voice conversion.
Experiments show that SpeechT5 significantly outperforms all baselines in several spoken language processing tasks.
For future work, we plan to investigate more efficient pre-training methods,
such as waveform learning representation via masked prediction like Hubert \cite{hsu2021hubert}, aligning text token and phoneme explicitly as unsupervised ASR \cite{baevski2021unsupervised}.
Besides, we will pre-train the SpeechT5 with a larger model and more unlabeled data, and fine-tune it on more spoken language processing tasks.
We are also interested in extending the proposed SpeechT5 framework to address the multilingual spoken language processing problem.
\bibliographystyle{acl_natbib}
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